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TECHNICAL
ALGEBRA
PART I.
HORACE WILMER MARSH
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MARSH'S CONSTRUCTIVE TEXT-BOOK
OF
PRACTICAL MATHEMATICS
For Use in Industrial, Manual Training, Tech-
nical Ilion Schools and Colleges, and
Apprentice and Evening Classes
THE PRACTICAL ESSENTIALS OF ARITHMBTIC, ALGEBRA, OBOH-
ETRY, TRIOONOMBTRir, ANALYTICS, AND CALCULUS; IN-
CLUDING THE EXTENSIVE USE OF LOGARITHMS
AND THE aLIDE-RULB« WITH THOUSANDS
OF EXAMPLES AND APPLIED PROBLEMS
BASED ON INDUSTRIAL DATA
Vul. I.
Vol. II.
Vol. III.
Vol. IV.
NOW READY
Industrial Mathematics, with tables... ne<
Technical Algebra, with tables, Part I. ,ndt
Technical Geometry net
Technical Trigonometry net
Interpolated Siz-place Tables nd
Mathematical Work-book for stu-
dents' use, removable blank sheets,
^ with instructions for use net
$2.00
$2.00
$1.25
$1.50
$1.25
.65
Each volume complete in itself. The entire course a unity. As
constructive, developing, and creative as shop work. Eklucates
tbroui!h self-activity, affords self-realization through self-expression,
and gives the knowledge and the use of the Mathematics of modem
Industries.
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CONSTRUCTIVE TEXT-BOOK
OF
PRACTICAL MATHEMATICS
BY
HORACE WILMER MARSH
Head of Department of Mathematiea, School of Science
and TecJinology, Pratt Institute
Volume II
TECHNICAL ALGEBRA
Part I
FIRST EDITION
NEW YORK
JOHN WILEY & SONS, Ino.
London: CHAPMAN & HALL, LiMrria)
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Copyright, 1913,
BY
HORACE WILMER MARSH
PRESS OF
BRAUNWORTH h. CO.
BOOKBINDERS AND PRINTERS
BROOKLYN, N. V.
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PREFACE
Professor John Dewey long ago emphasized the idea
that the school is truly educative only when it represents
actual, social conditions. In the trade and technical school
of secondary rank, such as the School of Science and Tech-
nology of Pratt Institute, we have been forced to recognize
and prove the truth of Professor Dewey's assertion. The
graduates of a school of this kind, although requiring an
acquaintance with books, must be able to do and to direct
others. Their mathematics must be of the sort which they
can use and apply in their daily work after graduation.
It must be presented in such form and manner during the
entire course that its use and application shall be an
acquired art the same as the use of any other tool.
This text in form and method is the result of an attempt
to solve the problem of the teaching of mathematics, not
from the point of view, therefore, of the mathematician
but from the necessities of the student and the demands
of the environment in which he is and into which he is to go.
The mathematician sees mathematics only; the student sees
the unknown and is estranged; the true teacher sees active
boys and girls or young men and women whose study must
be so directed that they shall become genuinely interested
in mathematics and shall have a feeling of confidence and
satisfaction in its use.
This direction is possible only when mathematics is
regarded and taught, not primarily as a means to mental
n33i5r;
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vi PREFACE
discipline, but as an instrument and a tool whose use can be
acquired only by continued practice on concrete pieces of
work. The student will so regard mathematics the moment
he feels its intimate relationship to his other studies and to
life. The things profoundly educative in the Uves of all
of us are the worthy things we love to do. What teacher
but would have boys and girls study mathematics for the
same reason that they play— because they love to? What
teacher but would have them experience something of the
same pleasure in preparing a lesson in algebra as in a whole-
some, lively game?
The mathematician has this pleasure as he follows the
unblazed trail which his genius illumines. This trail, still
seen dimly, I have endeavored to blaze in such manner in
this and in the other constructive and developing texts in
this course, through a study of the educational masters and
thirteen years' experiment and test of the mimeographed
text with upward of 2000 students, that the students who
use,^this book shall become genuinely enthusiastic and pro- /
ficient, for " He who sees without loving is only straining
his eyes in the dark."
The conventional, x-y-z mathematics — artificial in
subject-matter, notation, and method, the embodiment
and the outcome of the scholastic demand for mental dis-
cipline and of the educational ideal that the more difficult
the subject-matter the greater the discipline — has rarely
produced this result for the reason that it has few apparent
points of contact with the activities of the shop and labora-
tory or of industrial, scientific, engineering, and professional
pursuits.
To the beginner no study is significant whose use is not
apparent and which is not presented in a way to stimulate
creative effort. Algebra in its conventional form is a
striking illustration of this truth because it is the first branch
of mathematics to deal with generalizations. These can
have meaning only as they are the expression of individually
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PREFACE vii
investigated particulars which are intimately associated
with the student's interests. Students are interested tem-
peramentally in things which appeal to their activities;
hence their interest in machinery and in physics, chemistry,
or domestic science laboratories. Their interest in mathe-
matics is an unknown quantity, and he who likes it, is the
exception to whom a teacher and a special method are not
a necessity.
A serious disadvantage, therefore, confronts the teacher
of algebra. His success is conditioned by the interest of
his students, yet natural interest on their part is wanting.
In this emergency I know of but one interest to which the
teacher can appeal. This is the universal, human interest
in doing and in the creation of something worth while.
This text is therefore written in such form that from the
first, each student shall be busy on his own individual problem
with increasing interest and pleasure, because the con-
structive form of the text, the practical character of the
subject-matter, and its correlation with his other studies
give him an instinctive feeling that the problem, the form
in which the work is required, and the result to be found,
are worth the effort.
Even an unskilful and careless workman takes pride
in the results of his attempts to do a really good piece
of work. An indifferent student likewise becomes interested
and appreciative, the instant an enforced excellence of
form results in a piece of work which he is proud to own
and to have his friends see. If this be true, as I am
compelled to believe by repeated experiment, the advantage
and the necessity of a standard form for each student's
work is evident. The mathematics' work-book is an endeavor
to meet this requirement. This book consists of a note-
book cover with removable sheets, in which every student
writes in ink the work required in the text. It is desirable
that most of this work be done in the classroom and a pro-
vision for it by double periods for mathematics, is ideal.
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Tiu PREFACE
The educational purpose of the mathematics work-book
is two-fold. It demands a daily, finished product and the
written expression of the student's individual thought on
the topics assigned. Its incidental results are steady
improvement in spelling, penmanship, and lettering, the
use of correct English, a growing sense of method and order,
and an increasing ability to plan and work in a rational,
economic, and therefore businesslike manner.
One of the minor details of the method employed in this
text, which is of great value in explanation, is the require-
ment that in all work with equations every equation shall
be numbered. If not nmnbered or otherwise named, no
equation can be indicated in explanation at the blackboard,
except by reading. This not only lessens interest and wastes
time but imavoidably emphasizes symbols, instead of
operations and reasons and authorities for operations.
What the weak student must hear repeatedly and invariably,
is what kind of equation is given, by what operation each
subsequent equation is obtained, and by what mathematical
authorities. He can hear these, only when he hears nothing
else.
Students in industrial courses, many of whom have been
employed in technical industries, while fairly mature in
their thought, have usually a varied and sometimes inade-
quate school preparation. Others have taken advanced
courses in mathematics. In this text provision has been
made by examples and problems ranging from the simplest
to those of considerable difficulty, so that every student
whether beginner or advanced, with whatever accomplish-
ment or experience, shall have the work he is fitted to do,
and which shall be of most service both in his major, tech-
nical studies and subsequent employment.
In this connection attention is called to the absence of
the conventional, artificial problem whose very statement
necessitates a knowledge of the answer, whose data and
solution have absolutely nothing to do with science or
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PREFACE ix
mental development and whose principles have no applica-
tion to any industry in the universe.
Each chapter of the book contains specific instructions
for the order of procedure in all operations. Notable
instances are the chapters on complex fractions, factoring,
logarithms, and the slide-rule, for which there have com-
monly been no definite methods.
One of the most valuable chapters to the student is the
chapter on Transformation of Formulas, names imknown at
the time of preparation of this manuscripts Generations of
university graduates have had occasion to remark in their
first position that their first task was to forget all they
knew. The reason in so far as it concerns algebra, is
obvious. They studied books of one language, the conven-
tional. The mathematics of industry is written in an
apparently different language — the significant. The student
who knows this chapter needs not to learn a new mathe-
matical tongue in his employment or in his technical reading,
for he will find it " parent speech." In the conventional
treatment of algebra everything is labeled so that one
must never ask what is this, before attempting to solve.
In the chapter on transformation the formulas are arranged
so that the kind of equation must invariably be deter-
mined before operation is possible. By this means the
student gains the same power of analysis and classification
that is required in technical positions.
The author is confident that in other schools, as in his
own classes, this text will prove an integral part of a
student's industrial and technical courses, and an adequate
and unfaiUng interpreter of his technical studies. Since
one cannot write imtil he knows the alphabet, this text
will render greatest service if the principles of operation
required in any part of an applied study are presented
previous to their employment in the technical work.
Some of the most helpful suggestions for this text were
obtained from the shops, laboratories, and drafting rooms
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X PREFACE
t
of Pratt Institute. The men directing these kinds of work
do not " hear lessons "; their rooms are not " recitation "
rooms in which " classes " assemble to repeat what they have
learned from a book. The teachers of these useful crafts
do not assign for memorizing — definitions, directions, and
descriptions. The student who has once made a hexagonal
nut, or a milling cutter, or a propeller blade pattern, or
has worked out a hysteresis curve, knows them, not from
memorized definition or description which are vaguely or
entirely misunderstood, but from intimate acquaintance
and individual, creative work. He can define, make, and
direct the making of these things, not because he has memo-
rized another's thought, but because he has made them,
himself.
In the industrial or technical school it is inevitable that
the mathematics be no less constructive and creative than
shop-work and I am certain that in no school will the study
of mathematics be truly educative unless its conventional
study is deferred until its need is felt, and unless fundamental
propositions and principles are developed in a form to make
every student experience the joy of individual discovery
and creation.
This book aims to make the student proficient in the
fundamental, algebraic processes and in their application
to the numerous computations in technical industries.
That a student may have the best means of securing an
accurate result, the chapters on Logarithms and the Slide-
rule are the most complete yet written concerning every
use of these wonderful instruments with which no student
in this busy, industrial, twentieth century can afford to be
unfamiliar.
The ideals which have been the final authority in the
preparation of the Constructive Text-Book of which this
is the second volume, are perhaps best shown by contrast
with the old, as outlined *from a lecture by Professor F. M.
McMurry:
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PREFACE
XI
The Old Ideal:
Knowledge,
Stillness,
Machine,
Group,
Theory,
External Authority,
Restraint,
Self-consciousness,
Scholarship,
Motive:
Competition, marks, prizes.
Stimulus;
Compulsion,
Comparison with others.
The individual fact.
The Modern Ideal:
Development.
Motor Activity.
Germ.
Individual.
Executive Ability.
Inner control.
Freedom.
Unconscious effort.
Enlistment of interest.
Love for the subject.
The created thing in its
gradual creation.
Comparison of the student's
work with his own previous
work.
Unified knowledge, social ser-
vice and eflficiency.
In furtherance of these ideals which now dominate educa-
tion, the world over, mathematics work should be planned
with the same care and in much the same maimer as a man
must plan a piece of work which he is to do, or whose con-
struction he is to superintend in shop, factory, or other
enterprise. When so studied, mathematics is of the greatest
service in the development of those habits of thought which
qualify one for positions of responsibility. Thus consid-
ered, it is not a question whether one will use this or
that branch of mathematics in a certain position, but
whether one knows sufficient mathematics so that promo-
tion shall not be impossible because of the lack of it, and
whether it has been studied in such a way as to make one
increasingly valuable in any position.
Thoreau revealed a truth of tremendous import in educa-
tion, when he wrote:
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xii PREFACE
" The true poem is not what the public reads,
but is what the poet has become through his work."
So in the study of mathematics, the actual, supreme
result is not mathematical attainment, ability, or skill; it
is not the student's created product as represented in the
work-book, but is the student himself , as represented in what
he has become through the self-directed, systematic exer-
cise of neatness, accuracy, patience, seriousness, judgment,
native conmion sense, sincerity, and love cf work, which
are incidental and, I trust, inevitable in the method and
content of this text.
In conclusion I gratefully acknowledge my indebtedness
to my wife, Annie Griswold Fordycc Marsh, my constant
critic, co-worker, and counsellor without whose aid this
volume would not have been possible.
Horace Wilmer Marsh.
Brooklyn, New York,
August 19, 1913.
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CONTENTS
PAGE
The Work-book 3
CHAPTER I
Introduction
Section 1. Algebraic Notation. Exercise 1. Lettering. Exer-
cise 2. Ratio. Exercise 3. Compound Ratio. Exercise 4.
The Equation. Exercise 5. Coefficient and Exponent.
Section 2. Solution of a Simple Equation. Section 3. For-
mulation of Mathematical Laws. Section 4. Formulation
and Computation 11
CHAPTER II
Resolution and Composition op Forces
Section 1. Projection. Section 2. Graphical Resolution. Sec-
tion 3. Resolution by Computation. Section 4. Functions
of an Obtuse Angle, and Examples. Section 5. Resultant 60
CHAPTER III
Variation •» 77
CHAPTER IV
The Four Fundamental Operations
Section 1. Addition. Section 2. Subtraction. Section 3. Mul-
tiplication. Section 4. Division 101
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xiv CONTENTS
CHAPTER V
Laws of Numbers
PAoa
Section 1. The Square of the Sum of Two Numbers. Section 2.
The Square of the Difference of Two Numbers. Section 3.
The Product of the Sum and the Difference of Two Numbers.
Section 4. The Difference of Two Cubes. Section 6. The
Sum of Two Cubes. Section 6. The Square of any Polynomial
Section 7. The Exact Divisor of a Polynomial 120
CHAPTER VI
Factoring
Section 1. A Common Factor. Section 2. Grouping. Section 3.
The Difference of Two Squares. Section 4. The Difference of
Two Cubes. Section 6. The Sum of Two Cubes. Section
6. The Trinomial. Section 7. The Polynomial. Section 8.
Various Expressions 128
CHAPTER VII
Fractions
Section 1. Reduction. Section 2. Addition and Subtraction.
Section 3. Multiplication and Division. Section 4. The Com-
plex Fraction 144
CHAPTER VIII
The Quadratic Equation
Section 1. Introduction. Section 2. Solution by Factoring
Section 3. Solution by Completing the Square. Section 4.
Equations in Quadratic Form 155
CHAPTER IX
The Fractional Simple Equation
Section 1. Denominators Numerical. Section 2. Some De-
nominators Literal 166
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CONTENTS X?
CHAPTER X
SlBHTLTANEOUS SiMPLE EQUATIONS
PAQB
Section 1. Two Unknowns. Section 2. Three or More Un-
knowns 173
CHAPTER XI
Exponents
Section 1. A Zero Exponent. Section 2. A Negative Exponent.
Section 3. A Fractional Exponent 180
CHAPTER XII
The Binomial Theorem 191
CHAPTER XIII
Powers and Rooxs
Section 1. Powers. Section 2. Square Root. Section 3. Cube
Root. Section 4. Other Roots 200
CHAPTER XIV
Radicals f
Section 1. Reduction. Section 2. Addition and Subtraction.
Section 3. Multiplication and Division. Section 4. Ra-
tionalization. Section 5. Powers and Roots. Section 6. Rad-
ical Equations 207
CHAPTER XV
Logarithms
Section 1. Logarithm of a Number Greater than Unity. Section
2. Logarithm of a Number Less than Unity. Section 3.
Naperian or Hjrperbolic Logarithms. Section 4. Logarithm
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xvi CONTENTS
PAGS
of a Product. Section 5. Logarithm of a Quotient. Section
6. Logarithm of a Power. Section 7. Logarithm of a Root.
Section 8. Solution of an Exponential Equation. Section
9. Model Solutions. Section 10. Logarithmic Computa-
tion 221
CHAPTER XVI
The Slide Rule
Section 1. Introduction. Section 2. Sines. Section 3. Tangents. *
Section 4. Multiplication. Section 5. Division. Section 6.
Proportion. Section 7. Logarithms. Section 8. Powers
and Roots. Section 9. Gage-Points. Section 10. The Log
Log Rule 279
CHAPTER XVII
Tbanspormation op Formulas 350
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REFERENCE TABLES
PAGB
I. Measures of Length 391
II. Measures of Area 392
III. Measures of Volume 393
IV. Measures of Weight 394
V. Decimal Equivalents of fractions of an Inch 395
VI. U. S. and Metric Equivalents 396
VII. International Atomic Weights, 1913 397
VIII. Period Arrangement of the Elements, 1913 398
IX. Specific Gravities and Weights of Materials of Construction. 399
X. Wire Gage Sizes 401
XI. Four-place Logarithms of Numbers 402
XII. Four-place Trigonometric Functions 405
XIII. Three-place Trigonometric Functions 412
xvii
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TECHNICAL ALGEBRA
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THE WORK-BOOK
Note. The pages immediately following contain the detailed
instructions which govern the use of the mathematics' work-book
in the author's classes where one hour and a quarter of outside
preparation per class period is the maximum requirement. Even
in schools where written work does not seem feasible these pages
may perhaps suggest ways of making mathematics of the same
significance as shop and laboratory work in the student's prepara-
tion for life, and at the same time pleasurable and useful both to
student and teacher.
A student's work-book containing the first year's work.
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TECHNICAL ALGEBRA
THE WORK-BOOK
1. Description. The constructive form of this text
requires that each student shall prepare the work daily
in accordance with the suggestions, questions, and direc-
tions in numerous developing exercises, which force a student
not only to do his own thinking but to express his thought
in written form.
For this purpose the mathematics' work-book is used,
which both in size and form is the result of sixteen years
of experiment with over 2000 students. It consists of a
note-book cover, the daily record sheet described in a
subsequent paragraph, and 250 removable sheets of
16-pound, unruled, linen paper measuring 5^X8| inches,
with fasteners for attaching the sheets to the back cover
a,nd the student's written work to the front cover.
2. Instruments. In order to prepare the work in a
satisfactory manner and to secure the greatest educational
benefit each student will require the following equipment:
12-inch triangular scale with U. S. and metric gradu-
ations,
medium lead pencil,
ink and pencil erasers,
fountain pen,
ruling pen,
compasses,
3
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4 TECHNICAL ALGEBRA 8
protractor,
red, black, and India ink,
slide-rule.
3. Value of Careful Work. It is obvious that skill is
never acquired by careless, indifferent effort. Therefore,
as in manual training, the desideratum in every exercise,
example, and problem in this text, is perfection in the
finished work.
The instructions in the two subsequent paragraphs and
throughout the text have accordingly been written to
stimulate each student to a serviceable ideal of excellence
and efficiency and to give him an increasing ability and
enthusiasm for its realization.
4. Instructions for Work-book Entries. (1) What to
DO First. On the inside of the front cover of the work-
book write your full name, home and rooming address,
and name of school and course.
Attach all record sheets except one, to the back cover
under the blank sheets.
(2) Use of Ink. With the exception of the drawing
and the first exercise in lettering, all work is to be done
directly with pen and ink whether in the classroom or
outside.
(3) Date, and Page Number. Enter date on whict
work is prepared, in the upper right corner of the page
about one and one-half inches from the top.
Number each page in the lower right corner when
finished.
(4) Headings. Enter all work under the same heading
as in the text or as otherwise specified.
Begin paragraph numbers and headings about one-half
inch from the left margin.
(5) Lettering. The greater legibility of lettered head-
ings and the practical value of the ability to letter neatly
and rapidly, justify the requirement that title pages,
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4 THE WORK-BOOK 5
chapter, section, and paragraph headings, and problem
titles shall be lettered. The best style of letter for this
purpose, because the simplest and the most easily made, is
the Engineering News alphabet shown below and used
quite generally in drafting rooms throughout the country.
The only principle involved in learning to use this
alphabet is that each letter consists of straight lines, or
arcs of circles, or both, and that the width and the height
are the same.
ABCDEFGHIJKLMNOPQRSTUVWXYZ ft
1234567890 li 2131411 5i6& 7.039
ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234667690
- — iM — s — HK^Vilf-^
Be sure to observe that W is not an inverted il/, nor M
an inverted W. Observe also particularly how R and
G are made.
(6) Title Pages. Letter title pages in India ink
without punctuation, for the subject and for each chapter.
Insert these in the work-book preceded by a blank sheet.
(7) Spacing. Indicate a new topic both by heading
and by extra space. Keep all work in straight lines with
no irregular spacing between words or lines.
If straight lines are difficult without a guide, rule a
page of the work-book in India ink with lines from three-
sixteenths to one-fourth of an inch apart, and place it
under the page when writing.
Examples and problems are best separated by extra
space only, but if preferred the separation may be em-
phasized by a hair-line not over two inches long, lightly
drawn with straight-edge and ruling pea.
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6 TECHNICAL ALGEBRA 5
(8) Drawing. Drawings which are to be measured
should be drawn lightly and carefully with pencil and straight-
edge, and after measurement should be inked in with India
ink and a ruling pen. Other drawings may be done directly
in India ink.
As soon as possible learn to draw a lights smooth, drafts-
man's line.
(9) Symbols. Make parentheses, equality signs, and
other sjnnbols carefully: parentheses with regular curves
of the same height as the quantities inclosed; the lines
of the equality sign exactly the same length and about
one-eighth of an inch long.
Learn to make comparatively small, neat figures and
to draw free-hand, smooth, light, straight lines when
performing the four fundamental operations.
Work slowly, seriously, and steadily, and thereby
become expert by avoiding careless mistakes.
6. Instructions for the Record Sheet. Six daily record
sheets, a half year's supply, are furnished with the work-
book. These have columns for the instructor's stamp
and for the daily entry by the student, of date, paragraph
and problem numbers, and number of hours spent in
outside preparation of studies.
Submit Work for Inspection as Follows: On the
first day prepare a record sheet by filling in the blanks as
indicated. At the top of the time columns letter the
names of the studies in which outside preparation is
required, as Math., Phys., C. L. (chemistry laboratory), etc.
Observe that the record sheet provides for a complete
record of the mathematics work and is a time sheet for
all studies.
Whenever work is to be submitted attach it to the
record sheet on which fill in the entries denoted by the
column headings, making no entries in the remarks column.
When an additional record sheet is needed place it on
top of those already filled.
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9 THE WORK-BOOK 7
6. Excuse for Non-performance and Absence. If
unable to do assigned work, present to the instructor at
the beginning of the period a written excuse with date,
assignment, exact reason for failure, and signature. Enter
the date on the record sheet and write the word " Excuse "
in the remarks column.
In case of absence enter the date of each day's absence
and write " Absent '' in the remarks column.
7. Collection and Distribution of Work-books, (a)
Collection. On the stroke of the bell at the beginning
of the period each student will pass his work-book along
the row in reverse order from which the chairs or desks
are numbered, being sure to place it on top of the books
passed to him.
The student receiving the books of the last row will
collect each row's books and will place them in the file.
Work-books may be taken from the classroom only when
permission is noted on the record sheet by the instructor.
(6) Distribution. At the beginning of the mathe-
matics period the collector will place the books at the
end of the rows so that each student may remove his book
from the pile as it is passed.
Books of absentees will be reported ^ directly to the
instructor's desk by the collector. In the collector's absence
the next student in the row will attend to the books.
8. Inspection. The remarks column on the record
sheet is for the instructor's stamp. When the dater is
used instead of the " accepted " stamp it signifies that
the work is incomplete, or unsatisfactory, or incorrect.
Changes in such work unless obvious or indicated in the
book, must be arranged with the instructor before the
close of the period.
9. Corrected Work. Incorrect or rejected work is due in
correct form at the beginning of the next mathematics period.
Make corrections in red ink on the same page with the
incorrect work. If numerous mistakes have been made
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8 TECHNICAL ALGEBRA 10
prepare a new page and insert it following the incorrect
one on which write in red ink " Corrected on next page/'
with date of correction.
10. Solution of Equations. Number equations at the
left in a vertical column with Arabic numerals inclosed
in a parenthesis.
Separate equations from specified operations by a
horizontal rippled line not shorter than half an inch. It
must follow each equation except those taken from the
text or formed from the conditions specified in the problem.
The following model solution shows how to number,
how to use the rippled line, and how to abbreviate.
LCDS.
(1) 5x-24x=2— Mul (a) by 8
(2) -19x = 2 Col in (1)
(3) a:=-^— Div(2)by-19
As illustrated, equations from the text or from the
conditions of the problem are numbered with the first
letters of the alphabet.
Equations resulting from operation are numbered in
succession in Arabic numerals.
11. Indication of Results. Indicate final results in all
problems by double imderlining with parallel hair-lines
not over one-sixteenth of an inch apart. Do not write the
word " Answer.'*
On the same line with the result write a statement in
initial capitals specifying exactly what the result repre-
sents, whether niunber of revolutions per minute, horse-
power, speed, etc.
12. Index. At the end of the school year arrange all
the work-books of the year in order, make an alphabetic
index for your combined book, cut the work-book cover
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12 THE WORK-BOOK 9
in two through the back, and bind all together with one
pair of fasteners.
This shows a stuaent's work-book at the end of the second year.
On the front cover attach a label about 4"X5" with
a line border and lettered title enumerating the subjects
covered as suggested in the facsimile label below:
MATHEMATICS WORK-BOOK
FIRST YEAR
ALGEBRA
GEOMETRY
TRIGONOMETRY
WRITTEN BY
PRATT INSTITUTE
1911-1912 S.M.D.
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10 TECHNICAL ALGEBRA 13
13. The Axioms. An axiom is a self-evident truth
which does not depend on a simpler truth by means of
which it can be proved.
The axioms below are the basis of algebra and are
the sole authority for the operations necessary to the solu-
tion of an equation.
(1) Equality Axiom. ( = ity Ax.) Numbers equal to
the same number or to equal numbers, are equal to each
other.
(2) Addition Axiom. (Add Ax.) If equations are
added together, or if the same number is added to both
members of an equation, the result is an equation.
(3) Subtraction Axiom. (Sub Ax.) If one equation
is subtracted from another or if the same number is sub-
tracted from both members of an equation, the result is an
equation.
(4) Multiplication Axiom. (Mul Ax.) If equations
are multiplied together, or if both members of an equation
are multipUed by the same number, the result is an equation.
(5) Division Axiom. (Div Ax.) If one equation is
divided by another or if both members of an equation are
divided by the same number, the result is an equation.
(6) Power Axiom. (Power Ax.) If both members of
an equation are raised to the same power, the result is an
equation.
(7) Root Axiom. (Root Ax.) If the same root of both
members of an equation is indicated or extracted, the result
is an equation.
(8) Sum of Parts Axiom. (Sum Pts Ax.) The whole
of any quantity equals the sum of its parts and is therefore
greater than any of its parts when all the parts are positive.
Make no effort to memorize these axioms as they will
become familiar through repeated appUcation.
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CHAPTER I
INTRODUCTION
Section 1, Algebraic Notation. Exercise 1, Lettering.
Exercise 2, Ratio. Exercise 3, Compound Ratio. Exer-
cise 4, The Equation. Exercise 5, Coefficient and
Exponent. Section 2, Solution of a Simple Equation.
Section 3, Formulation of Mathematical Laws. Sec-
tion 4, Formulation and Computation.
§ 1. ALGEBRAIC NOTATION
Exercise 1. Lettering.
14. Lettering of LabeL* With pencil and straight-
edge rule ofif three spaces on the first page of the work-
book, the size of the gummed label to be attached to the
front cover, about IJ by 3^ inches.
Carefully letter with pencil each of the spaces in the
style of letter shown on page 5, in the following order:
Term
School Year
Subject
Student's name
Volume
Class Year of course and section
* See 4, page 4.
11
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12 TECHNICAL ALGEBRA 16
16. Lettering of Title Page. When the lettering has
been approved, lightly letter the label itself in pencil;
also letter the title page Technical Algebra.
Before the next class period in mathematics letter the
title page a second time with greater care than before
and attach the label to the front cover of the book. Also
write your name, home and rooming address on the
inside of the front cover.
Exercise 2. Ratio.
16.. Measurement. Carefully draw a straight line A
about two-thirds the width of the page. With the scale
measure it to the nearest sixty-fourth of an inch and to
the nearest millimeter.
Express the result as follows:
A = inches.
A= mm.
How was the length of A determined? *
A equals how many sixty-fourths of an inch?
When a line is measured in sixty-fourths of an inch,
what is the unit of measure?
A contains the unit of measure how many times?
When a line is measured in millimeters, what is the
miit of measure?
A contains this unit how many times?
Ratio is the quotient relation of one quantity to another,
expressed as a fraction.
Denoting the unit of measure by S, express the ratio
of Aio B.
Express the ratio of H to S; of £ to jB.
The ratio of two quantities of the same kind always
equals a fraction whose numerator is the iium15er~of times
* In answering all questions in the text include the question in the
answer. Thus: The length of A was determined, etc.
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le
INTRODUCTION
13
the first quantity contains the unit of measure, and whose
denominator is the mmiber of times the second contains
the unit of measure.
A
To what fraction is — equal? (Express with equality
B
sign.)
Draw a line x about the same length as A, but longer.
Measure it with the scale and write what it equals to
the nearest sixty-fourth of an inch and to the nearest
millimeter.
Draw a second line y considerably shorter than A.
Write what y equals to the nearest sixty-fourth of an
inch and millimeter.
Express the ratio oi x to y and indicate what it equals
X
in inches; also express what - equals in millimeters.
y
Fig. 1.
Carefully draw this figure in the work-book making
h and h' perpendicular to OD,
Determine and enter with an equality sign the follow-
ing measurements and ratios, both in U. S. and metric
units:
A, OD, W, 0D\ OV, OT,
Jl JL Jl JdL 9L 9IL
OD' OD" or OV" OD' OD''
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14 TECHNICAL ALGEBEA 17
Exercise 3. Compound Ratio.
17. Definition. A compound ratio is a product resulting
from the multiplication of two or more ratios.
A product is expressed in algebra by writing one quantity
following another; as, 5q/, lahx. Observe that the numer-
ical factor always precedes the literal, and that literal
factors are arranged in alphabetic order.
By the law of multipUcation of fractions, multiply
gby-; by^; by-^; by-j^; byy; by 3^; byj^.
Each of the results you have written is what kind of a
ratio? Why?
18. Factors. Factoring is the process of finding the
numbers whose product equals a given number. These
mmibers are called factors.
Thus,
18 = 2 times 9, or 3 times 6, or 3 times 2 times 3.
Factor the compound ratios,
llA 34ax ,m
12B' 50y ' .09r
Exercise 4. The Equation.
19. Definition. An equation is an expression of equality
between quantities.
Denoting by z the unit one sixty-fourth by which the
nr
line X was measured, to what fraction is -- equal?
oZ
y
To what fraction is — equal?
z
x = how many times 2?
2/ = how many times 2?
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21 INTEODUCnON 15
Why are your answers to the last four questions equa-
tions?
Number the equations in succession at the left, inclosing
the numbers in a small circle or parenthesis. (See para-
graph 10.)
20. Members. The members of an equation are the parts
separated by the equality sign. The first member is the
part to the left of the equality sign; the second member
is the part to the right of the equality sign.
An addition to an equation must always he made to both
memberSj otherwise the equality is destroyed.
An addition is indicated by a plies sign; a subtradionhy
a minus sign.
Add 5 to the first equation.
To how many members did you add it?
Add z to the second equation.
Subtract 13 from the third equation.
Both add 8 and subtract y from the fourth equation.
Number each of the resulting equations in succession
with the first four of paragraph 19.
21. Terms. The terms of an equation are the parts
separated by the signs, plus, minus, or equality, but no
other signs.
How many terms has the eighth equation?
Give reason for your answer.
22. Multiplication of an Equation. An equation is
multiplied by multiplying all of its terms. A numerical
multiplier is multiplied into the numerical factors of the
term; if a term has no numerical factor, the numerical
multiplier is written in front of the literal factors with
no sign between.
For example, 5 times 4fe = 206, the numerical multiplier
5 being multiplied into the numerical factor 4.
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16 TECHNICAL ALGEBRA 23
5 times a6 = 5afe, the numerical multiplier 5 being written
in front of the Uteral factors ab,
A literal multiplier is written in alphabetic order in the
term into which it is multiplied.
For example, d times 5acx = 5acdXy the literal multiplier
d being written in alphabetic order in the term 5acx.
Multiply the fifth equation by 7.
Multiply the sixth equation by z, the seventh by 3fe,
and the eighth by 5a.
Number the resulting four equations successively with
those previously written.
By what mathematical law do you know that when
an equation is multiplied, the result is an equation? (See
page 10.)
Exercise 6. Coefficient and Exponent.
23. Coefficient. A coefficient is a multiplier or a factor.
For example, in SaJ^c^y, 8 is the numerical coefficient
and is called the coefficient of the term.
8a2 is the coefficient of c^y. Why?
What is the coefficient of a^y? Of a^?
When no numerical coefficient is written, 1 is under-
stood.
In the third equation of paragraph 22, what is the
coefficient of the first term?
Of the second term?
24. Exponent. An exponent is a number, which, when
a positive integer, shows how many times a number affected
by it is used as a factor.
Exponents are written at the right of and slightly above
the numbers which they affect. When no exponent is
indicated exponent 1 is understood.
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25 INTRODUCTION 17
In a^, a is used how many times as a factor?
3 is what? What does it show?
In 5a^€?y, 5 is what? 2 is what?
In the same term, what is the coefficient of a^yt
What is the exponent of y1
a * times a times a = what?
Express a^ times a'* without exponents.
Does a^ times a^ = a^ or a®? Why?
Therefore when the same letters are multiplied together
should their exponents be multiplied, or added?
3x2 times y^ times x times y^ times 2/ = what?
5fe times fe times b^ times 4c6~^ = what?
r times r times r^ times 7r^ times 8r~^^=what?
7'-Himes y+Swhat?
26. Marks of Parenthesis. When more than one term
of an expression or when more than one of several factors
is affected by the same operation or exponent, the terms
and factors affected are inclosed by marks of parenthesis:
These are:
( ) the parenthesis,
{ } the brace,
[ ] the bracket,
the vinculum.
Q
— (5— 3i) indicates that the difference between 5 and
8
3j is to be multiplied by jq. This may be done in two
ways: Each term within the parenthesis may be mul-
g
tiplied by 75 ^^^ ^^® second product then subtracted from
the first, or the terms may first be subtracted and the
Q
remainder multiplied by — .
* In your answer use the multiplication symbol.
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18 TECHNICAL ALGEBRA ^6
In the work-book perform the operation in both ways.
Which way is preferable? Why?
Copy the following and perform the indicated opera-
tions:
1,
ll(^t-|).
2. ^(5J+2f)+i
4. 3a;(8+2x).
§ 2. SOLUTION OF A SIMPLE EQUATION
26. Definitions. A simple equation is an equation
having only the first power of the unknown quantity.
An unknown quantity is a quantity whose value maj
be determined by solution.
Unknown quantities are denoted conventionally by the
last letters of the alphabet.
In the application of mathematics any letter may
denote an unknown quantity, usually the initial letter of
the word by which the quantity is named, as V for velocity,
A for area, C for circumference, etc.
Copy the following equations in the work-book:
1. x+2x-15=0. 2. 7x-4x-6 = 12.
3. 6T^+x-S = i. 4. 5x+8-x=48.
Which of the four equations are simple? Why?
In the first equation what terms contain unknown quan-
tities?
What terms are unknown in the second?
What terms are unknown in the third?
What terms are unknown in the fourth?
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28 INTRODUCTION 19
27. How Solved. Simple equations like three of
those in paragraph 26 are solved as follows:
(1) Collect the unknown terms into the first member.
(2) Collect the known terms into the second member.
(3) Divide both members of the resulting equation by
the coefficient of the unknown quantity.
For example, in the first equation if the unknown terms
X and 2x are collected in the first member, we have 3x.
In collecting — 15 from the first member into the second,
the sign is * changed and we have 15 for the second member.
Therefore after collecting the unknown terms in the
first member and the known terms into the second member,
the equation becomes
3x = 15.
The value of x may now be determined by dividing
both members by the coefficient of x.
Therefore,
x = b.
28. Examples. Solve the following examples according
to the directions and form given on page 8, and as explained
in paragraph 27. If a result is not integral carry it to two
decimal places.
Those who have never studied algebra will work from
1 to 35 only and from 86 to 105. All others will begin
with example 30.
1. 5a:+8a;-7a:-a:=25. 2. 2x-x+^-^z-llx^U,
3. Sx+x- 12+9 -1=0. 4. 2a:+14-6a:+12a;=110.
5. 3a;-5+2a;-9a:=-6x+15. 6. 18+7a;-10=3^x+57-8.
•Whenever a quantity is trans poied from one member of an
equation to the other member, the sign of the quantity must be changed.
Actually there is no transposition. The word expresses what happens
when a quantity is eliminated from a member of an equation by
subtracting it from both members.
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20 TECBNICAL ALGEBRA ^$
7. |a:-132+8Jx=x-12. 8. 4(2x+7) -46=0.
o
9. 2x+9(2+3a:) =76. 10. 19+10x-5ix=3(28-2x). ^
11. 5(3a:+l)+2(2x+8)=42+4x.
12. 3j(4x-2)=3i(x+14).
13. ^x-3x+15(4-2x)=31Jx+80.
14. |(y^+64x) =41. 15. nh+3h+U^+lb\ -25=0.
16. 12r+|(^+^) =124. 17. 300+|c+lic+. 125c =425.
18. 2(18.5-30.2^+34.30=2^-16.6.
In the following examples divide both members by the coeffi-
cient of the unknown quantity, transposing when necessary.
A fractional coefficient must be inverted and used as a mul-
tiplier.
19. —V* -gs. Solve for v* and for v.
20. 2s -gV^, Solve for s and for g.
21. li =Q. Solve for i and for 7.
22. IK = y. Solve for 7 and for K,
23. Eff(ir+w;) =w;. Solve for EfT.
24. Ir—E- —p. Solve for 7 and for r.
26. — =7. Solve for E and 7^.
it
26. ^ = .0009477^. Solve for H.
27. 9C =5(/^ -32°). Solve for C and for F.
28. >S(n-l)=G. Solve for 5.
29. Ar =/?(Ai -il). Solve for R.
30. 7(r ns+/?) = J^ ns. Solve for 7.
31. RRi = {R+Ri)J. Solve for J.
32. Sn ^G+S. Solve for n and for G.
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28 INTRODUCTION 21
.7854
=4a6. Solve for A.
34. ^ = .5236. Solve for V and for d.
4
36. Fr* =r-x. Solve for V and for r.
o
36. 2|x =t{ ^+8 ) . Solve for x.
5. 2|x =1(1+8)
37. X =— ~2^. Solve for D.
Ji
38. — =-7-. Solve for E and for A,
e A
39. 2Kg = Aft;2. Solve for K and for t;.
40. 8 =— (r+0. Solve for p and for r.
41. FLD =9600000<2-'^ Solve for P and for D.
42. R^ = A«+52. Solve for 7^, for A, and for 5.
43. Q(a+6) =a(P+Q). Solve for Q and for 6.
44. 7 = — — — , Solve for t and for s.
46. A =^+6(5+n). Solve for 6 and for n.
46. A = jW^ -di^). Solve for rf.
'-ii(--f)
47. I =77^^3i22 -- ) . Solve for R and for 8,
48. i2 =\/t' Solve for 7 and for A.
49_R = ZiL__EL. Solve for d.
4
60. iV =L — -rrr: - 777. Solvc for a and for L.
100 144
61.* P =^^^- Solve for D and for iV. ,
*The equations for examples 51 to 70 are from formulas in
"Gearing," published by Brown & Sharpe Mfg. Co.
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22
TECHNICAL ALGEBRA
28
52. s =
N+2'
Solve for D and for N.
t^^P'. Solve for P'.
63.
64. s+f = .3683P'. Solve for s and for P\
66.
F =|+y. Solve for P and for A.
66. r' = — -2s. Solve for d' and for s.
67. r" =r'+D"+/. Solve for D" .
68. 0=^^'+^. Solve for P'.
69. Z =Z)"+2/+^. Solve for D" and for /.
o
C= — T- — s. Solve for D' and for s.
60.
61. 6 =7r(d -2s). Solve for d and for s
62.
iV
D =— +2s. Solve for s and for A/".
63.
64.
66.
66.
67.
P' =
iV+2*
Solve for N and for D.
937iV
D = ' „^ +4s. Solve for A'' and for s.
P' =
frC
70.
KA^a + iVi,)*
2)"+/ = 2s+--. Solve for « and for /.
1,2 =— 7^77— ^ Solve for W and for Gi.
2)" = f:i^ _/. Solve for £>" and
«+/=^(l+J). Solve for
28
INTRODUCTION
23
71. P =-^. Solve for S and for r.
r+l
72. jtD^p =TDtS. Solve for p and for D.
73. -=-^. Solve for /and fore.
C 04
/ 64
74. — = ■■ . Solve for / and for d.
c id
76. P =KN^^^^. Solve for AT.
76. J =— (d« -d,«). Solve for d.
S^d' Si,r(d,'-d,*)
~16~ iCdi ■ ^'veford.
78. B
\ V3.2n/
Solve for D,
79. .7CP=Sb
Solve for P and for b.
80. 'S = .. . ^ . Solve for F.
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24 TECHNICAL ALGEBRA 28
The following are examples in solution by substitution:
jD I r\ -Iff -Iff
^- ^="^57;;ir+T^+V- Compute the value of « when B=4",
^Uu lb o
2> = 16".
[87. In example 73 compute the value of — when Af = 19000,
/ = 15, ^4=9000.
88. W=VA+B+C+D-E, Compute the value of W when
A=2A, B = li, C=0, D~ E~
89. In example 76 compute the value of / when d =8", di =6".
90. S =-| (2t-l). Compute the value of t when S =530, a =32.2.
91. In example 71 compute the value of S when
r=5f, t~ P = 172.
W L
92. -Tr=77. Compute the value of F when T7 = 196, L=8,
t H
if=4i.
93. G(<-ii)x =(?(«! -<2). Compute the value of x when <=90,
fi=25, <2 = 18, G = 125.
94. In example 68 compute the value of / to four decimal places
when s = . 5570, P = 1.7952.
96. In example 69 compute the value of P' when (7=7.25,
iVa=28, iV6=72.
96. H.P.= ^^'~^^^ . Compute H.P. when !ri = 100, T^^^.ZTi,
7=700.
W 2R
97. 17 = ^ — . Compute the value of Pwhen 72=3.75, r=2.4,
T7 = 1250.
S8. A ^ /7r • Compute the vialue of h when g =32.2 iV =90.
47rW*
99. t* =7r2 — . Compute the value of t when L =39.1, ^ =32.2.
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2d INTRODUCTION 25
100. Kw = — . Compute the value of E when
ouOOO
G = .07, R = 1320, T7 =25000, Kw =65.
101. /S = y r-. Compute the value of S when
L=3400, P' = 1.2, /=5, C = .08, r=600, F = 1200.
._!H)
102. P' = — ^ — ^. Compute the value of P' when
a
P = 2i, d = 14.1, /=5i
103. W = .0357/P'(6.25n+.04n«). Compute the value of W when
/=3i, P'=2J, n=21.
' 104. Vi = — .^. , "L . Compute the value of Vi when
n{t +/)
F = 125, <'=95, n = 120, /=115, <=85.
2d
106. 2> =^ . Compute the value of D when
^tanA+1
Jtf
d =8, Af =6, 72/ =2, tan A = .5774.
§3. FORMULATION OF MATHEMATICAL LAWS
To THE Teacher. The aim in this section is to have the
student learn to formulate. Explanation of laws is therefore
unnecessary, will leave less time for formulation, and may con-
fuse rather than instruct.
29. Illustration. How would you compute the distance
traveled by a train which runs for a given number of hours
at a given rate per hour?
Distance = what times what?
If d = the distance in miles,
r=the rate in miles per hour,
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26 TECHNICAL ALGEBRA 80
and <=the time of running in hours,
then d=what in terms of rate and time?
(Express your answer entirely in the above symbols.)
Your answer to the last question, if correct, is a statement
in symbols, of the exact mathematical relation of distance to
time and rate. It is therefore a formula.
To be more definite it is a significant formula because the
symbols in which it is expressed are the first or initial letters
of the names of the quantities involved, as d for distance,
r for rate, and t for time.
The same law might be formulated conventionally by
denoting distance by x, rate by y, and time by z.
Then a: = what in terms of y and 2?
Of these two ways of writing a formula, do you prefer
the conventional or the significant? Why?
Write the definition of a formula.
30. Weight. In significant notation write a formula for
the law that Ws, the weight of any substance, equals its
volume F, times its specific gravity sp.gr., times the weight
TF of a unit volume of water.
Solve the weight formula for volume.
Solve it for specific gravity.
Solve the distance formula of paragraph 29, for rate.
Solve it for time.
31. Expression of Ratio. There are three ways of denot-
ing a ratio; one by the colon, one by the division symbol,
and the other by a fraction line.
Thus the ratio of 3 to 16 is expressed
3: 16, ;3-M6, ^, or 3/16.
The fraction form with a horizontal fraction line is always
to be preferred and should be used in all mathematical
calculations involving fractions.
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33 INTRODUCTION 27
Thus it is suggested that the ratio of a to 6 be indicated
by
T- and not by alb, or a 4- 6, or a/6.
The ratio of energy to momentum should be indicated by
enercv
^ — and not by energy : momentum,
momentmn '' ^'^ '
or energy -^ momentum,
or energy/momentum,
E
In symbols it may be indicated by — .
32. Specific Gravity. The specific gravity of a solid
substance equals the ratio of its weight to the weight of an
equal volume of W3,ter.
Express this law as an equation, omitting as many words
as possible but not using symbols.
Write a significant formula for specific gravity.
Write a conventional formula.
In both instances specify the notation employed.
Solve the significant formula for TF, and W.
33. Problems. Under proper heading and in significant
notation write the formulas for the following laws and solve
for each of the literal quantities in the second member.
When a figure is shown in the text, draw it in the work-
book.
1. Force. The force F in pounds, imparted by a moving
body, equals its mass M times its acceleration a.
2. Mass. The mass of a substance equals its weight W in
pounds, divided by 32.2.
3. Area of a Rectangle. The area of
a rectangle equals the length or base h
times the width or altitude h. ^ ^i
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28
TECHNICAL ALGEBRA
b
FiQ. 3.
• 4. Area of a Parallelogram.
The area of a parallelogram
equals the base b times the alti-
tude h.
6. Area of a Triangle. The area
of a triangle equals one-half the base b
times the altitude h.
7. Area of a Trapezoid.
two sides parallel and two
non-parallel.
The area of the trape-
zoid equals one-half the
sum of the parallel sides,
times the perpendicular
distance between them.
8. Area of a Trapezium.
6. Square of the Hypotenuse.
The hypotenuse of a right tri-
angle is the side opposite the
right angle.
The square of the hypotenuse
equals the sum of the squares of
the other two sides.
A trapezoid is a quadrilateral having
Fig. 7.
6
Fig. 6.
A trapezium is a quadrilateral
having none of its sides
parallel.
The area of a trape-
zium equals one-half
times one of its diag-
onals times the sum of
the perpendiculars to
that diagonal from the
vertices of the opposite
angles.
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INTRODUCTION
29
9. Area of a Regular Polygon. A
regular polygon is one which has equal
sides and equal angles.
The apothem is the perpendicular
distance from the center to any side.
The area of a regular polygon equals
one-half the perimeter times the apothem.
>9 y V
Fig. 8.
10. Angle of a Regular Polygon. An angle
6f of a regular polygon having n sides, equals
180° multiplied by n-2 and divided by n.
11. Central Angle of a Regular
Polygon. A central angle of a regular
polygon of n sides, equals 360° divided
by n.
Fig. 10.
12. Ratio of Circumference to Diam-
eter. The ratio of the circumference of
a circle to its diameter, equals 3.1416
approximately.
In the formula substitute t for 3.1416.
Formulate also in terms of radius.
Fig. 11.
13. Area of a Circle. The area of a
circle equals t timies the square of the
radius.
Formulate area also in terms of diam-
eter.
Fig. 12.
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TECHNICAL ALGEBRA
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14. Area of a Ring. The area of a
ring included between the circumferences
of two concentric circles equals t times
the difference between the squares of the
two radii.
Fig. 13.
In the remaining problems in this paragraph, do not
solve for the literal quantities in the second member, but
do as specified in each problem.
16. Length of Arc of Sector of a
Circle. A sector of a circle is a portion of
it bounded by two radii and the inter-
cepted arc.
The length of the arc of a sector of
Q
6° equals -— times the circumference of
360
the circle.
In the formula substitute for circum-
ference in terms of radius, and simplify.
16. Area of a Sector of a Circle. The area of a sector of a
circle equals one-half its radius times its arc.
In the formula, substitute for arc from the second formula
of problem 15.
Fig. 14.
Fig. 15.
17. Area of an Ellipse. The area of
an ellipse equals t times the product of
the two semi-axes.
In the formula denote the semi-axes
by a and 6, a being half the long axis
and b half the short axis.
18. Perimeter of an Ellipse. The perimeter of an ellipse
equals approximately 1.82 times the long axis, plus 1.315 times
the short axis
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INTRODUCTION
31
19. Total Area of a Cylinder. The total area
or surface of a cylinder, equals its circumference
times its length, plus twice the area of the base.
In the formula, substitute for circumference
and area of base in terms of radius of the base.
Formulate, also, the curved surface of a
cylinder.
Fig. 16.
20. Volume of a Cylinder. The volume of a cylinder equals
the area of the base times the length of the cylinder.
In the formula substitute for area of the base in terms of radius;
also in teems of diameter.
21. Total Area of a Regular Pjrramid.
A regular pyramid is one whose apex is
directly over the center of its base which
is a regular polygon.
The total area of a regular pyramid
equals the area of the base Ai,j plus the
area of the sides Ai^.
In the formula substitute for At from
problem 9, and for Ax, from problem 5.
22. Volume of a Regular Pyramid. The volume of a regular
pyramid equals one-third the area of the base times the altitude.
In the formula substitute from problem 9.
23. Total Area of a Cone. The total
surface or area of a cone equals the area of
the base, plus the lateral area.
In the formula substitute for area of the
base, in terms of radius of the base, and for
lateral area substitute one-half the circum-
ference of the base times the slant height.
In the formula thus obtained, substitute
for circumference in terms of radius.
Fig. 18.
24. Volume of a Cone. The volume of a cone equals one-
third the area of the base times the altitude.
In the formula substitute for area of base in terms of radius.
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TECHNICAL ALGEBRA
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26. Volumes of the Frustum of a Cone and a Pyramid. A
frustum of a cone and of a pyramid is the part which reniains when
the top is cut off parallel to the base.
Fig. 19.
The voliune of a frustum of a cone and of a pyramid equals
one-third the altitude times the following:
Area lower base Ab, plus area upper base Aj, plus the square
root of the product of the areas of the two bases.
26. Area of a Sphere. The area of a
sphere equals tt times the square of its
diameter.
Formulate also in terms of radius.
27. Volume of a Sphere. The volume
of a sphere equals two-thirds tt times the
square of the radius, times the diameter.
In the formula substitute for radius
in terms of diameter, and simplify; also
substitute for diameter in terms of radius, and simplify.
Fig. 20.
28. Volume of a Rectangu-
lar Solid. Write a law for the
volume of the solid here shown,
and formulate the law.
Fig. 21.
29. Volume of a Cylindrical Ring. The volume of a ring
with a circular cross-section, equals 2.4674 times the square
of the thickness, times the ^sum of the thickness and the inner
diameter.
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INTRODUCTION
33
30. Area of a Cylindrical Ring. The
surface or area of a cylindrical ring
equals 9.8696 times the thickness, times
the sum of the thickness and the inner
diameter.
By thickness is meant the diameter
of the materikl of which the ring is made.
Fig. 22.
Fig. 23.
31. Volume of a Spherical
Segment. The volume of a
spherical segment which is less
than a hemisphere equals .52367r
times the depth d, times the
following sum: Square of depth,
plus three times the square of
the radius of the base of the
segment.
32. Change in Velocity of a Falling Body. The change in
the velocity Vc of a falling body, equals its acceleration a, multiplied
by the time of motion.
33. Final Velocity of a Falling Body. The final velocity V/
equals the initial velocity F/, plus the change in velocity.
For the change in velocity substitute from the formula of prob-
lem 32, and for initial velocity substitute zero.
34. Average Velocity of a Falling Body. The average
velocity Va equals one-half the siun of the initial velocity and the
final velocity.
For initial velocity substitute zero and for final velocity sub-
stitute from the last formula of problem 33.
36. Space Traversed. The distance s equals the average
velocity multiplied by the time.
In this formula substitute for average velocity from the last
formula of problem 34, and simplify.
36. Velocity of Falling Body. The velocity 7 of a falling body
under the influence of gravity equals the force of gravity gr, mul-
tiplied by the time.
Solve the formula for time.
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34 TECHNICAL ALGEBRA S3
37. Velocity of a Falling Body. In the last formula of problem
35 substitute for time from problem 33 and solve the resulting
formula for velocity.
38. Law for Velocity. By reference to the last formula of
problem 37 write the law for velocity of a body falling freely from
rest.
39. Moment of Inertia. The moment of inertia of a particle
equals its mass times the square of the distance of the particle
from the center of rotation or the axis.
Denote moment of inertia by / and distance from center by r.
Solve for mass and r.
40. Centrifugal Force. The centrifugal force Fe of a rotating
body equals its mass times the square of its velocity in feet per
second, divided by the radius of revolution.
Solve for 7.
41. Centrifugal Force. In the first formula of problem 40
substitute for mass from the first formula of problem 2.
42. Work. The work done by a force which acts on a body
equals the force in pounds times the distance in feet through which
the force acts in the direction of displacement of the body; or
briefly, work equals force times distance.
43. Horse-Power. Horse-power, H.P., equals the force in
pounds times the distance in feet, divided by 33,000 * into the
time in minutes.
44. Horse-Power of a Steam Engine. This formula may
be written by substituting the following in the formula of problem 43 :
For force substitute PA, mean effective pressure of steam in
pounds per square inch times the area of the piston in square inches.
For distance substitute LN, the length of the stroke in feet
times the number of strokes per minute.
For time substitute 1 minute.
State the notation before writing the formula.
1000
45. A Kilowatt, Kw. One kilowatt equals ^— - horse-power.
1 kilowatt equals approximately how many thirds of a horse-power?
* In mathematics "into" is frequently used to denote times.
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INTRODUCTION
35
46. Elinetic Energy. The kinetic energ>' of a moving body
equals one-half its mass times the square of its velocity in feet per
second.
In the fonnula substitute for mass from problem 2.
47. Kinetic Energy. The kinetic energy of a rotating body
equals one-half its moment of inertia times the square of its angular
velocity a,
48. Momentum. The momentum m of a moving body equals
its weight in pounds times its velocity, divided by 32.2.
W
For -TT^ substitute from a preceding problem.
49. Coefficient of Friction. The force required to slide one
body on another equals the weight of the body times the coefficient
of friction /. Solve for /.
60. Law of Moments. In a
lever of whatever class, the force
P, times the force arm Pa^ equals
the weight times the weight arm
Solve this formula for each
quantity in both members. .
61. Ohm's Law. In an elec-
tric conductor through which a
current is flowing, the current /
in amperes equals the electro-
motive force E in volts, divided
by the resistance R in ohms.
Solve for E and /.
i<-W7r-
W
-Wa
-Ws
W
X:
w
FiQ. 24.
62. Current from a Series Battery.
Fig. 25.
In a battery arranged in
+ series, the current /
_ equals the number
of cells ns times the
electromotive force,
divided by the sum
of the internal and
external resistances,
nr and R.
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TECHNICAL ALGEBRA
33
63. Period of a Pendulum. The time T of oscil-
lation of a pendulum in seconds equals x times the
square root of the quotient of the length in feet and
the force of gravity g.
64. Current from a Multiple Battery. The cur-
rent from a battery arranged in parallel equals the
voltage (electromotive force) divided by the sum of
the external resistance and the ratio of the internal
resistance of one cell to the mmiber of cells np,
O 4 >
Fig. 26.
Fig. 27.
56. Current from Multiple-Series Battery. The current from
any multiple-series combination equals the voltage times the num-
ber of cells in series ns, divided by the external resistance plus the
ratio of the product of the number in series and the resistance of
each cell to the number in multiple arrangement np.
56. Current through a Shunted Galvanometer. The cur-
rent Is flowing through a galvanometer in a shunt circuit equals the
current / in the joint circuit, divided by 1 plus the ratio of the
galvanometer resistance Rgj to the shunt resistance Ra,
57. Electrical Efficiency. The electrical efficiency Eff of a
djmamo equals the number of kilowatts Kw which it delivers to
the circuit, divided by the number of kilowatts lost in the arma-
ture Kwa plus the number of kilowatts lost in the field coils Kwe
plus the number of kilowatts delivered to the circuit.
58. Determination of High Resistance. The resistance of a
circuit R equals the resistance of the voltmeter /2p, *into the
difference between the ratio of the two deflections 5i and 62, and 1.
* In mathematics and its applications the word "into" is often
used instead of times or multiplied by»
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34 INTRODUCTION 37
69. Power Required. The number of watts required for the
746
propulsion of a trolley car on a level road equals 5^7^ of the
weight in tons of the car as loaded, times the tractive force T
in pounds, times the speed of the car in feet per minute, divided
by the efficiency of the motor.
§ 4. FORMULATION AND COMPUTATION
To The Teacher. The sole purpose of this section is for-
mulation and computation. A student's work primarily is there-
fore the translation- of words into mathematical symbols, and the
more closely the attention is fixed on formulation to the exclusion
of externals, the better.
One of the means of securing this concentration is by the use
of laws regarding things of which the student has little or no knowl-
edge, .but with which he will become familiar in subsequent
mdustrial studies. At times the mathematics' instruction may
be made more significant by the use of a machine or by analjrsis of
mechanical principles in connection with students' work at the
blackboard.
In some instances it may be advisable to stimulate interest by
the use of the picture of the machine in a trade catalogue or tech-
nical periodical, when the machine itself is not available. In a
technical school arrangements may also be made from time to time
so that a class or section may visit the school shops or laboratories
and have a machine demonstrated by some advanced student or a
shop instructor.
None of these things is necessary, however, in teaching this
book, nor will they aid in formulation except indirectly. The
entire book may be well taught without them.
34. Definitions. A proportion is an equality of ratios.
Which of the equations written in your work-book, from exer-
cises 2 and 4, are proportions? Why?
Can a proportion be written which is not an equation? Give
reason.
J
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38 TECHNICAL ALGEBRA 35
A simple proportion is a proportion having two ratios.
A continued proportion is a proportion having more than
two ratios.
The proportions you have previously written are what
kind?
Why?
Write two more proportions of the same kmd.
The terms of a proportion are the quantities forming the
proportion.
How many terms has a simple proportion?
The extremes of a proportion are the first and last terms.
The means are the second and third terms.
What quantities are the extremes in your first proportion?
What quantities are the means in your last proportion?
36. Law of Proportion. In Chapter XXII, Part II, are
the various laws of proportion. Only one of these is neces-
sary in this part of algebra. You will recognize it as the
same law that was applied in the solution of problems in
proportion in your study of arithmetic.
This law is as follows:
The product of the means equals the product of the extremes.
Write the equations which are obtained by applying this
law to three proportions in your work-book, as written in
exercise 2, page 13.
36. A Simple-Geared Lathe. The drawing shows the
arrangement of gears on a simple-geared lathe. Through
the train of gears the revolution of the spindle to which the
work is attached, is transmitted to the lead-screw which
causes the carriage holding the cutting tool to move along
the lathe-bed.
When a lathe is simple-geared the gears required to cut a
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INTRODUCTION
39
given number of threads per inch may be determined from
the following law:
The pitch of the lead-screw is to the pitch of the screw
to be cut, as the number of teeth in the gear on the spindle
is to the number of teeth in the gear on the lead-screw.
Notation: L = pitch* of lead -screw,
5 = pitch of screw to be cut,
^5 = number of teeth in gear on spindle,
Tl = number of teeth in gear on lead-screw.
Spinflle gear
gear
Fig. 28.
In the work-book enter paragraph number and title,
and the notation.
Write the formula.
37. Problems. The first problems which follow illus-
trate the use of proportion in determining the gears which
are required on a screw-cutting lathe under various condi-
♦ Pitch as here used, means the number of threads to the inch.
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40 TECHNICAL ALGEBRA 87
tions. They are here introduced to show the simplicity
of solution by formulas and to help you to understand the
difference between arithmetic and algebra and the advantage
of both letters and figures in representing quantity.
In computation be satisfied with a correct result only
when obtained by the simplest and quickest method of
solution; in other words, by a method which minimizes
mistakes and saves time and energy.
In the solution of the following problems enter the work
under problem number and title with formula and solution
near the left margin of the page and data near the right.
1. Lead-screw Gear. A machinist desires to cut a 12-pitcii
screw with a 6-pitch lead-screw. If he uses a 24-gear on the spindle,
what gear must he use on the lead-screw?
D
miii
12-pitch screw. 24 spindle-gear.
Fig. 29.
Determine the lead-screw gear by substituting directly in the
formula of paragraph 36.
Cancel if possible.
Double underline the final result as specified in paragraph 11,
page 8.
Check. To test the correctness of a result by solution by another
method is called checking. Check your result as follows:
Before substituting the numerical values from the data, solve
the formula of paragraph 36 for the unknown quantity in two
ways:
(1) Multiply the formula by the * least common denominator
and divide the resulting equation by the coefficient of the imknown.
* The least common denominator is the smallest number in which
the given denominators are evenly contained.
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INTRODUCTION
41
(2) Apply the law of proportion regarding the product of the
means.
Substitute the numerical values, cancel if possible, and indicate
the result in the same form as before.
2. Spindle-Gear. A lathe is equipped as follows:
L=4,
5=10,
What gear must be used on the spindle?
3. Pitch of Screw. A lathe has a 28-gear on the spindle and
a lead-screw of 50-gear and 6-pitch.
It will cut a screw with what pitch, without changing the gears?
4. Standard Gears. In standard gears the number of teeth
ranges from 21 to 105 by intervals of 7, or from 24 to 120 by intervals
of 4.
Write down all the gears of both standards.
If the pitch of the lead-screw is 8, and 12 threads per inch are
to be cut, what must be the ratio of spindle and lead-screw gears?
If all the gears of both standards are in the shop equipment,
what three pairs of gears could be used under the ^ven conditions?
6. Cutting Speed. The speed S in feet per minute with which
a piece of work revolves under the cutting tool, equals ir times
the diameter of the work in inches, times the number of revolu-
tions per minute divided by 12.
A shaft 8J inches in diameter is turned with a cutting speed
of 18 feet per minute.
Compute the niunber of revolutions.
ji
^
<L
Fia. 30.
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TECHNICAL ALGEBRA
87
6. Time Formtila. The time in minutes required to turn
a revolving piece of work equals the length of the work in inches
divided by the product of the feed in inches per revolution and the
number of revolutions R per minute. Write the formula, sub-
stitute for R from the formula of problem 5, and simplify.
7. Time Required. With a feed of J of an inch and a cutting
speed of 21 inches per minute, a piece 10 feet long and 29 inches
in diameter is to be turned.
Find the time required in hours.
8.* Weight of a Pulley.
If R =the radius of a pulley in inches,
W =the width of the belt in inches,
and P =the approximate weight in pounds,
P = I7'[.163|+.015(|) +.00309(1)'].
then
Compute the approximate weight in pounds of a 29-inch pulley
carrying a 4-inch belt.
9. Diameter of a Pulley
Arm.
Z)=the approximate diameter in
inches of arm at rim,
N =the number of arms,
Tr=the width of the belt in
inches,
72= the radius of the pulley in
inches.
Fio. 31.
3 V 4 lOAT/
A draftsman is to design a pulley 18 inches in diameter and
having 4 arms, to carry a 3-inch belt.
Find the approximate diameter of arm at rim.
* The formulas in problems 8, 9, and 12 are from Cromwell's
"Belts and PuUeys."
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INTRODUCTION
43
10. Velocity of a Pulley. The circumferential velocity of a
pulley in feet per minute equals ir times the diameter of the pulley
in feet, times the number of revolutions per minute.
By the formula compute the number of revolutions per minute
of a 28-inch pulley whose velocity is 2100 feet per minute.
11. Circumferential Ve-
locity. A shaft carries two
fixed pulleys having diameters
of 9 J inches and 16| inches.
Compute the circumfer-
ential velocity of each pulley
when the shaft makes 125
revolutions per minute.
12. Diameter of a Shaft. The diameter of a pulley shaft in
inches equals the cube root of the product of the radius of the
pulley m inches and the number of pounds of force transmitted
by the pulley, provided the cube root is multiplied by the following
numbers for shafts of the material specified:
Fig. 32.
Material: Steel,
Cast-iron,
Wrought-iron,
Multiplier: .075
.108
Write the three formulas.
Compute the diameter of a shaft of each material to the nearest
64th of an inch for a pulley 12f inches in diameter and transmitting
a force of 900 pounds.
13. Horse-Power of
a Steam Engine. The
horse-power of an engine
equals the mean effective
pressure in pounds per
square inch times the
length of the stroke in
feet times the piston area
in square inches times
the number of strokes
per minute, divided by
Fig. 33. 33,000.
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TECHNICAL ALGEBRA
87
Notation: H.P.= horse-power,
N = number of strokes per minute =2 times num-
ber of revolutions,
A =area of piston in square inches,
P =mean effective pressure in pounds per square
inch,
L = length of stroke in feet.
An engine has a piston area of 212.9 square inches, the stroke
is 2J feet, and the mean effective pressure is 95 lbs. per square
inch. How many strokes per minute will be necessary in order
to develop 175 horse-power?
14. Radius of a Gear Wheel. The
radius of the pitch circle of a gear wheel
equals the number of teeth times the pitch *
divided by 2 tt.
Write the formula, using significant capital
letters.
A gear wheel of 3J inches pitch and
having 76 teeth, is to be drawn to a scale
of I inch = l foot.
What will be the length of the radius in
the drawing?
Fig. 34.
16. Taper of Keys.
Notation : T = taper per foot in inches,
L = length in inches,
a = large diameter or thickness in inches,
b = small diameter or thickness in inches.
Formula: ,^=»i
12, . f— ni
T=^(a-6).
Fig. 35.
Write the law.
A key 16 inches long, is 2} inches in diameter at one end and
and 3| inches at the other.
Required the taper per foot.
* Pitch is the distance between tooth centers, measured on the
pitch circle.
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S7
INTRODUCTION
45
.
1
w 1
1
1
i \
*1
(i
di
'
Fig. 36.
16. Center of Gravity. The distance of the common center
of gravity from the center of gravity of the larger of two bodies,
equals the distance be-
tween the centers of
gravity of the two bodies
times the weight of the
smaller body, divided
by the sum of the
weights of the two
bodies.
Illustrate by a dia-
gram with lettered dis-
tances denoting the quantities named, and write the formula.
Two bodies weighed 150 and 95 pounds respectively and their
centers of gravity were 46 inches apart.
What was the distance of their com-
mon center of gravity from the center
of gravity of the larger body?
17. Law for Theoretical Weight.
If there were no friction, the weight Wt
which could be lifted by a screw-jack, is
to the power, as the circumference (in
inches) of the circle in which the power
moves, is to the pitch of the screw.
Write the formula and solve for Wt .
18. Law for Actual Weight. The
approximate actual weight TTa, which
can be lifted by a screw-jack equals
the theoretical weight multiplied by the
fraction whose numerator is the pitch
of the screw and whose denominator is
the sum of the pitch and the diameter
(in inches).
Write the formula, substitute for Wt from problem 17, and
simplify.
19. A Screw-jack. A screw-jack has a screw 1| inches in
diameter with a pitch of \ of an inch.
What approximate actual weight can be raised by it with a
Fig. 37.
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4i5 TECHNICAL ALGEBRA 87
force of 125 pounds applied at the end of a lever 14^ inches long?
(Pitch as used in problems 17, 18, and 19 is the distance between
the threads.)
20. Joint Resistance. Formula:
K 7*1 7*2 • 7*8 7*4
Find the value of Ry when n =12, r2 =20, rs = 18, r^ = \,
21. Electrical Transmission. The size of copper wire in
circular mils required to transmit a given horse-power, equals
746 times 10.79 times the horse-power times the length of the
circuit, divided by the efficiency of the motor times the voltage
times the drop.
Compute the size of copper wire required for the transmission
of 260 horse-power through a circuit of 300 feet with a voltage of
190 volts, the drop being 9 volts and the efficiency of the motor
83 per cent.
22. Volume of a Hollow Column. The volume of the material
in a hollow column equals tt times the length times
the difference between the squares of the outer and the
inner radius.
The outside diameter of a hollow steel colunm is
12| inches, and the length Sf feet.
Compute the volume when the inside diameter is
\0\ inches.
23. Weight of a Hollow Column. The weight
of a hollow column in pounds equals 62.5 times the
specific gravity of the material times the volume in cubic feet.
Compute the weight of the colunm in problem 22, the specific
gravity of steel being 7.85.
24. Volume of a Sphere. The volume of a sphere equals
f times TT times the cube of the radius.
Compute the volume of a sphere whose radius is 2.5 centimeters.
Check by problem 27, paragraph 33.
25. Radius of a Sphere. By the formula of problem 24
compute the radius of a sphere whose volume is 5.64 cubic
centimeters.
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37 INTRODUCTION 47
26. Flux of an Electrical Field.
= flux or total number of lines of force in an electrical field,
Zf= field strength or number of lines of electrical force per
square centimeter,
A =area of field in square centimeters.
Write a formula for <t> in terms of A and H,
Compute when A = 125 and H =3000.
27. Law of Flux. By reference to the formula of problem 26
write a law for flux.
28. Ratio of Velocities. If two pulleys are connected by a
belt the number of revolutions of the first pulley is to the number
of revolutions of the second,
as the diameter of the second
is to the diameter of the
first.
A 13J-inch pulley is joined
by a belt to an 8A-inch Fig. 39.
pulley.
Compute the number of revolutions of the 8 A-inch pulley when
the.l3^-inch pulley is making 115 revolutions per minute.
29. Radius from Chord and Rise. The rise of an arc is the
perpendicular distance to the arc from
the center of its chord.
When the length of a chord of a circle
and the rise are known, the radius may
be determined by the following law:
The radius of the circle equals the
sum of the squares of the rise and the
half chord, divided by twice the rise.
By the formula determine the radius
Fig. 40. when the chord of a circle measures SJi
inches and the rise is 2^ inches.
30. Volume of a Spherical Segment. The volume of a segment
of a sphere equals .5236 times the square of the depth or height
of the segment times the difference between three times the diameter
of the sphere and twice the height of the segment.
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TECHNICxVL ALGEBRA
37
Compute the volume when the sphere is 10 inches in diameter
r.nd the height of the segment is ^ inches.
31. Elevation of Outside Rail. On a railway curve the eleva-
tion of the outside rail in feet
equals the distance between rail
centers in feet times the square of
the maximum velocity of the train
in feet per second divided by
32.2 times the radius of the curve
in feet.
Compute the elevation on a
curve having a radius of 1750 feet
and a distance of 4f feet between
rail centers, allowing a maximum velocity of 75 miles per hour.
32. Deflection of a Beam. The approximate deflection in
inches of a rectangular white oak beam supporting a central load
equals .00023 times the
Fig. 41.
S"
1
Fig. 42.
load in pounds times the
cube of the length in feety
divided by the width in
inches times the cube of
the depth in inches.
Determine the deflection of a white oak beam 36i feet long
whose width is 9 inches and depth 17 inches, when loaded at the
center with IJ tons.
33. Deflection of a Wrought-Iron Beam. The deflection in
inches of a rectangular wrought-iron beam, supported at the ends
and centrally loaded, approximately equals the load in pounds times
the cube of the length
in feet, divided by
H 104,000,000 times the
lH width in inches times
the cube of the depth
in inches.
Determine the de- ,
flection in a wrought- ,
iron beam (rectangular) 31 feet 8 inches long with a width of 4
inches and a depth of 16 J inches, under a central load of 2480 pounds.
D.
Fig. 43.
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37
INTRODUCTION
49
34. Safe Load on an I-Beam. The greatest safe, distributed
load on an I-beam which is supported at the ends, equals 3390
into the sectional area in square inches into the depth in inches,
divided by the length in feet between end supports.
TZL
HIE
OOOOOOOO
III
II I
]Z~L
I
TZI
n
Fig. 45.
Fig. 44.
Find the maximum safe load when the cross-sectional area is 19
square inches, depth 15 inches, the total length of the beam being
25 feet 5 inches, and the supports extending a distance of 5 inches
under the beam at each end.
36. Equilibrium on an Inclined Plane. A load of 1750
poimds rests on rollers on an
inclined plane 900 feet long,
the height of the plane being
198 feet.
Determine the force neces-
sary to prevent the load from
rolling down the plane, friction being disregarded.
Law: The force is to the load, as the height of the plane is
to the length of the plane.
36. Weight of Spur-Gear Blank. The weight of the blank
for a cut, cast-iron spur gear under 3J inches circular pitch, may
be computed approximately by the following formula from the
American Machinist Gear Book:
in which W = weight of gear blank in pounds,
pi = circular pitch = — ,
p= diametral pitch,
N =number of teeth,
/= width of face in inches.
Fig. 46.
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50 TECHNICAL ALGEBRA 37
Compute the weight of the blank for a gear of 50 teeth with a
4-iiich face and 2|-inch diametral pitch.
37. Resistance of a Copper Wire. The electrical resistance
of a copper wire 1 millimeter in diameter and 1 meter in length may
be determined by the following formula in which
R =the resistance of the wire in ohms,
<=the temperature of the wire in degrees Centigrade,
/e =.0203 (1 + . 00410.
Determine the resistance when the temperature is 23|° C.
38. Resistance of a Nickel Wire. The electrical resistance
of a nickel wire 1 millimeter in diameter and 1 meter in length is
expressed by the formula
/e = .1568(1 +.0062
Determine the resistance when the temperature is 18 J ° C.
39. Specific Heat of Mercury. The specific heat of mercury
in terms of temperature is expressed by the formula
H =0.033266 -0.0000092 t.
What is the specific heat of mercury when the temperature is
100° C?
40. Tension in a Rope. When a rope is suspended from
two supports the tension in pounds on the rope at the supports
approximately equals the weight of the rope per foot times the
square of the distance in feet between the supports, divided by
8 times the deflection of the rope in feet.
Fig. 47.
A rope weighing 2\ pounds per foot is suspended from two
supports 150 feet apart. The deflection is 12 J feet.
Determine the tension at the supports.
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87 INTRODUCTION 51
41. Diameter of a Piston-Rod. Th^ diameter in inches of a
long piston-rod equals the fourth root of the fraction whose nu-
merator is 640 times the square of the diameter of the cylinder in
inches times the square of the length of the rod in inches times
the maximum pressure on the pistion per square inch and whose
denominator is the coefficient of' elasticity times the square of t.
Determine the diameter of a steel piston-rod 42 inches long,
the coefficient of elasticity of steel being 30,000,000, the diameter
of the cylinder 18§ inches, and the maximum steam pressure on the
piston per square inch 140 pounds.
42. Safe Transmission by Hollow Shafts. The horse-power
which may be safely transmitted by a hollow shaft equals a con-
stant into the number of revolutions per minute into the fraction
whose numerator is the difference between
the fourth powers of the external and
internal diameters in inches, and whose
denominator is the external diameter in
inches.
Compute the horse-power which can Fig. 48.
be transmitted safely by a hollow steel
shaft majdng 50 revolutions per minute, whose diameters are 11
and 8§ inches.
For steel the constant is .028.
43. Volume of a Wedge. The volume of a wedge equals J
into the thickness into the length measured
perpendicular to the base, into the sum of
the length of the edge and twice the width of
the wedge.
The width of a wedge is 5 inches, the
thickness is 2f inches, the length of the edge
is 3 A inches, and the length of the wedge is 21 inches.
Compute the volume in cubic inches.
44. Horse-Power of a Shunt Motor. The horse-power of
a shimt motor may be determined from the formula
746H.P. = [/-^][/-H(/-f)J.
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52 TECHNICAL ALGEBRA 87
Compute the horse-power when
7=80, e = 100, r=30, R=—.
50
46. Resistance of a Shunt Winding. The resistance of
the shunt winding of a dynamo at a temperature of 15° was
45 ohms.
Compute the resistance when the temperature reached 58 J®
C, using the formula
R
^ / l + .0042r \
~^\1+. 0042^,7'
in which <o** =the lower temperature,
t° =the highdr temperature,
R =the resistance at t° C,
r=the resistance at to° C.
46.* Efficiency Formula of a Shunt Motor. The efficiency
of a shunt motor is expressed by the equation
. Power intake —losses
Power intake
in which Power intake = amperes volts,
and Losses =stray loss+shunt field loss+armature loss.
Copy the three equations and in the efficiency equation sub-
stitute for power intake and losses.
The formulas for the quantities in your last equation are as
follows:
Shunt field loss
Armature loss
Stray loss ==8,
Copy the formulas and by substitution write a complete formula
for efficiency.
♦Formulas in 46, 53, and 58 are from Franklin and Espy's "Direct
Currents."
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37 INTRODUCTION 53
47. Shunt Field Loss. Compute the shunt field loss when
iS? = 120, /e,=45.
48. Armature Loss. Compute the armature loss when • / =60,
^ = 120, /e,=45, /2a = .15
49. Efficiency of a Shunt Motor. Determine the efficiency
when the stray loss is 800 and the other losses are as computed
in problem 47 and 48.
60. Efficiency Formula of a Series Motor. In a series motor
the formula for efficiency is the same as for a shunt motor, as in
problem 46, except that shunt field loss becomes
Series field loss ^RJ^
and
Armature loss =RalK
Write the formula for efficiency.
61. Series Field and Armature Losses. Compute the series
field and armature losses when Re ^.10, / = 60, /2a = .15.
62. Efficiency of a Series Motor. Determine the efficiency
when the stray loss is the same as in problem 49 and the other
losses are as computed in problem 51, ^ being 120.
63. Efficiency Formtila of a Compound Motor (long-shunt).
This formula differs from that of problem 50, as follows:
Shunt field loss
Series field loss =i?«
Armature loss
Write the complete formula for efficiency.
64. Shunt Field Loss. Compute the shunt field loss when
& = 60, ^ = 120.
66. Series Field Loss. Compute the series field loss when
Rt and E have the same values as in problem 54, Re = .082, I = 60.
66. Armature Loss. Compute the armature loss with the
same data as in problems 54 and 55, and Ra=.l.
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54 TECHNICAL ALGEBRA 37
57. Efficiency Formula of a Compound Motor (long-shunt).
Determine the efficiency when the stray loss is the same as in the
preceding problems and the other losses as in problems 54, 55,
and 56.
58. Efficiency Formula of a Compound Motor (short-shunt).
This formula is the same as that of problem 50, with the following
exceptions:
Shunt field loss =rA — jr-^ j ,
Armature loss =/2a / - f — ^-^ j .
Write the complete formula for efficiency.
59. Armature Loss. Compute the armature loss with the
data of problems 54, 55, and 56.
60. Shunt Field Loss. Compute the shunt field loss with
the same data as in problems 54 and 55.
61. Efficiency of a Compound Motor (short-shunt). Com-
pute the efficiency when the stray loss is 800 and the other losses
arc as computed in problems 59 and 60.
62. A Compotmd-Geared Lathe. The law for the gears
required on a lathe when compound-geared, is as follows:
The ratio of the threads to be cut, to the threads on the lead-
screw, times the ratio of the number of teeth on the spindle-gear
to the number of teeth on the fixed stud-gear, equals the number
of teeth in the outer sleeve-gear times the number of teeth on the
leadscrew-gear, divided by the number of teeth on the change
stud-gear times the number of teeth on the inner sleeve-gear.
Notation: Ts = number of threads to be cut,
Tx,= number of threads on the lead-screw,
S = number of teeth on the spindle-gear,
F= number of teeth on the fixed stud-gear,
0= number of teeth on the outer sleeve-gear,
L= number of teeth on the leadscrew-gear,
C= number of teeth on the change stud-gear,
E =nmnber of teeth on the inner sleeve-gear.
Write the formula.
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87
INTRODUCTION
55
Fia. 50.
Leadscrew gear
FiQ. 51.
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66
TECHNICAL ALGEBRA
37
63. Ntunber of Threads per Inch.
Fig. 52. — Calorimeter.
Compute the number
of threads which may
be cut per inch when
rx,=6, F=48, iS =24,
L=36, C = 18, E^12,
0=36.
64. Weight of
Gaseous Steam. The
weight of gaseous steam
per cubic foot equals
2.7074 times the total
pressure in pounds per
square inch times the
specific density, di-
vided by 461 plus the
temperature Fahren-
heit.
Compute the weight
when the pressure is 580
pounds per square inch,
the specific density .622,
and the temperature
482^ F.
65. Per Cent of Moisture in Steam.
Notation: M =the per cent of moisture,
H = total heat of 1 pound of steam at boiler pressure,
Hi = the latent heat at boiler pressure,
/i =the total heat at reduced pressure,
Hs =the specific heat of saturated steam,
T= average calorimeter temperature,
t = temperature of steam at reduced pressure.
Formula;
_, ^H^h^H»iT-t)
M=ioa
Hl
Determine M when H = 1195.9, h = 1150.4, F, = .47, T =370.8° R,
Hi, = 852.7, ^=212° F.
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37
INTRODUCTION
57
66. Percentage Composition. The per cent of an element
in any compound equals 100 times the weight * of the element,
divided by the molecular weight of the compound.
The molecular weight of a compound equals the sum of the
weights of all the elements in the compound.
Formulate both laws, and in the first, substitute for moleculai
weight from the second.
67. Compute the per cent of copper, Cu, in crystallized copper
sulphate, CUSO4+5H2O, arranging the work as follows under
the problem heading and formula:
fk
Weight tCu =
04=
" . 5H2 =
50 =
Molecular weight =
68. Formulate, compute, and tabulate
in the work-book the percentage composi-
tion of such substances as may be
assigned in Table I., page 58.
Tabulate both problems and results.
69. Specific Gravity by Twaddell's
Hydrometer. The specific grayity of a
liquid heavier than water, equals .005
times the reading on Twaddell's hydrom-
eter Tw, plus 1.
Compute the specific gravity of a
liquid in which Twaddell's hydrometer
stands at 56.
70. A liquid is to be prepared having
Fig. 53. a specific gravity of 1.30. Fig. 54.
Twaddell What will be the Twaddell reading Baum6
Hydrometer, when this point is reached? Hydrometer.
* The weight of an element in a compound equals the weight of the
atomB of the element in the compound.
t Refer to the latest list of International Atomic Weights.
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68
TECHNICAL ALGEBRA
87
Table I.
PERCENTAGE COMPOSITION
Computed from International Atomic Weights for Year..
No.
Substance.
Name.
Per Cent of Element.
El. % El. % El,
MnOa
HgO
KCIO,
K,S04
KMnO*
MgS04
H2O
KOH
HCl
H,S04
ZnSO*
ZnCl,
HNO,
NH,
KNO,
CaCO,
C,H204
NaCl
KjCfjOt
AgNO,
Manganese peroxide . . . .
Mercuric oxide
Potassium chlorate
Potassium sulphate
Potassium permanganate
Magnesium sulphate . . . .
Water
Caustic potash
Hydrochloric acid
Sulphuric acid
Zinc sulphate
Zinc chloride
Nitric acid
Ammonia
Potassium nitrate ,
Calcium carbonate. ...
Oxahc acid
Sodium chloride
Potassium bichromate. .
Silver nitrate
O
Mn
Hg
K
K
K
M
H
H
H
H
Zn
Zn
H
H
K
Ca
H
Na
K-
Ag
CI
S
Mn
S
K
CI
s
s
CI
N
N
N
C
c
CI
Cr
N
71. A solution is to be evaporated until the reading is 55 Tw.
Compute the specific gravity when this point is reached.
72. Specific Gravity by Baum6's Hydrometer.
(a) The specific gravity of a liquid heavier than water at 15.55**
C. equals 145 divided by the difference between 145 and the Baum6
reading.
(6) The specific gravity of a liquid lighter than water at 15.55**
C. equals 140 divided by the sum of 130 and the Baum6 reading.
Write both formulas.
73. Compute the specific gravity of a solution heavier than
water, which reads 55 B6.
74. When sulphuric acid has a specific gravity of 1.842 what
will be the Baum6 reading?
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37 INTRODUCTION 59
75. Compute the Baum6 reading for a liquid whose specific
gravity is .95.
76. In a liquid lighter than water at 17.5** C,
the reading is 45 B6.
Compute the specific gravity of the liquid.
77. Specific Gravity by Pyknometer. The
specific gravity of a liquid by the pyknometer may
be computed from the following law:
Weight of bottle and liquid —weight of bottle
Weight of bottle and water —weight of bottle*
Simplify the second member of the equation, and j, --
formulate both the original equation and the simpli- ^ '
^ J ,. Pyknometer
fied equation. or Specific
78. Volume of Dry Air. Mendel6eff gives the Gravity
following formula for the volume of dry air at any Bottle,
temperature and pressure, when saturated with watery vapor:
^D= p ,
in which V^ = volume of dry air in cubic centimeters,
Vs = saturated volume of air in cubic centimeters,
P = barometric pressure of saturated air in millimeters.
Pa =. barometric pressure of the aqueous vapor in milli-
meters at the given temperature, Centigrade.
Copy the formula and write the law.
79. Compute the volume of dry air in a- saturated volume of
52.4 cc at 15.3° C under a pressure of 748.5 millimeters, the pres-
sure of aqueous vapor at this temperature being 12.9 millimeters.
80. Dry Volume at 760 mm. and 0**. The volume of dry
air in cubic centimeters under a barometric pressure of 760 milli-
meters at a temperature of 0° C, equals the moist voliune
times the given pressure in millimeters divided by 760, times 273
divided by the sum of 273 and the given temperature.
Formulate and compute the dry volume in problem 79 at 760
millimeters and O''.
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CHAPTER II
RESOLUTION AND COMPOSITION OF FORCES
Section 1, Projection. Section 2, Graphical Resolution.
Section 3, Resolution by Computation. Section 4, Func-
tions OP AN Obtuse Angle, and Examples. Section 5,
Resultant.
" Whenever an idea is constantly recurring the best which can be done for the
perfection of language and consequent advancement of knowledge is to shorten
as much as possible the sign which is- used to stand for that idea."
Db Morqan.
§ 1. PROJECTION
38. Definition. Under headings draw this figure to an
enlarged scale in the work-book.
Fia. 66.
From R draw a perpendicular to OA^ terminating in
OA at V. Then OV is the right-angled or orthogonal
projection of OR on OA.
Specify this under the figure.
Draw the same figure again in the work-book and from
A draw a perpendicular to OR, tenninating at C
What then is OC? (Write answer immediately under
the figure.)
60
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38 RESOLUTION AND COMPOSITION OF FORCES 61
Draw Fig. 57 to an enlarged scale in the work-book.
Fia. 57.
From R and L draw perpendiculars to AB.
The segment of AB lying between the feet of these
perpendiculars is the projection of LR on AB.
Specify this under the figure.
Observe that in orthogonal projection perpendiculars
are dravm to the line on which projection is to be made.
Draw the figure again in the work-book and project AB
on RL.
Specify the projection.
Draw Fig. 58 to an enlarged scale in the work-book and
draw perpendiculars to AB from and R, terminating in
AB at V and T.
Flq. 58.
Then VT is the projection of OR on AB.
Specify the projection.
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62
TECHNICAL ALGEBRA
39
Draw Fig. 59 to an enlarged scale in the work-book and from
R draw a perpendicular to OA
produced, terminating at V.
Then OV is the projection
oiORonOA.
Specify the projection.
Write the definition of the
orthogonal projection of one
line on another.
39. Examples. In the following examples draw each
figure to an enlarged scale and make both projections as
in the work required for Fig. 56, specifying the projection
under each.
\
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41 RESOLUTION AND COMPOSITION OF FORCES 63
§ 2. GRAPHICAL RESOLUTION OF FORCES
40. Components. A force is represented graphically
by a straight line whose length equals the magnitude of the
force and whose direction is the same as the direction of
the force.
In the figure / represents a force of 8 pounds acting on
W in the direction shown
by the arrow at the angle
of 30° with the hori-
zontal.
Copy the figure to
an enlarged scale in the
work-book, making / 8
units in length, and '
project / on a horizontal Fia. go.
line through the point
of application of the force on TF.
The projection is the horizontal component of / and the
perpendicular is the vertical component of /. In technical
language, / has been resolved into horizontal and vertical
components.
By the scale determine the magnitude of each component
in pounds.
What therefore is the vertical pull on Wi
What is the horizontal pull?
41. Axes of Reference. In the graphical determination
of components and of the resultant effect of forces acting at
the same point, it is customary to denote the horizontal
component by x and the vertical component by y. This
S3mabolisnx ori^nated in the consideration of a force as
acting at the point of intersection of a vertical and a horizontal
line, called axes of reference. The horizontal is denoted
conventionally by a capital X at the right extremity and
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G4
TECHNICAL ALGEBRA
42
+
Fig. 61.
the vertical by a capital Y at the upper extremity as shown
in the figure. The point of intersection is called the origin.
The projections of forces on the X axis are called X com-
ponents and the pro-
Y jections of forces on
the Y axis are called Y
components.
X components are
denoted by a;; Y com-
X ponents are denoted
by 2/.
As shown on the
figure, X components
are positive * to the
right of the Y axis.
Y components are
positive above theX axis.
X components are negative * to the left of the Y axis.
Y components are negative below the X axis.
42. Examples. As assigned, draw axes of reference, rep-
rezent graphically the forces in the following table, draw
vertical and horizontal components, measure with the scale,
determine and fill in the omitted entries in the table in
which
F = the force in pounds,
^l=rthe angle at which it acts with the horizontal,
y = the vertical component or projection on the Y axis,
a: = the horizontal component or projection on the X
axis.
♦ Positive numbers are indicated by a plus sign; negative, by a
minus sign.
t All angles are read counter-clockwise from the right segment of
the X axis.
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43 EESOLUTION AND COMPOSITION OF FOECES 65
Table II
RESOLUTION AND COMPOSITION OF FORCES
F
A
X
V
No.
F
X
V
No.
+
-
+
A
+
-
+
-
1
5
20
11
130
85
2
10
40
12
140
95
3
15
45
13
150
115
4
20
50
14
165
125
5
40
55
15
170
145
6
60
60
16
180
235
7
8
70
90
65
70
17
18
3
6
4
8
9
100
75
19
100
50
10
120
80
20
80
40
§ 3. RESOLUTION OF FORCES BY COMPUTATION
43. The Ratios of a Triangle. Suppose the Une OY to
revolve counter-clockwise about the point 0, in the plane
of the pdper, and let 07' be the original position of OY^
6 * being the angle of revolution in degrees.
If OV is projected as shown, a triangle is formed whose
sides are named as follows:
± perpendicular,
Proj projection,
RY radius vector.
♦ A letter of the Greek alphabet, pronounced theta.
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66
TECHNICAL ALGEBRA
44
The relations of these sides to each other are commonly
expressed in six ratios, of which only three will be considered
here.
These are named, sine dj cosine 9^ and tangent 6.
They are defined and abbreviated as follows:
(1) sin0 =
RV
(2) cos0=^
(3) tan^=
Proj
or
or
or
opp
byp'
adj
hyp'
opp^
adj*
Their value is always the same for a given angle, regard-
less of the length of RV.
44. Examples. Rule the following table in the work-
book. By reference to the table of natural trigonometric
functions determine and enter the sine, cosine, and tangent
of the angles to three decimal places:
Table III
NATURAL TRIGONOMETRIC FUNCTIONS
No.
d
sin
cos
tan
No.
e
Bin
COB
tan
1
10°
11
18°
2
20°
12
23°
3
30°
13
31°
4
40°
14
42°
5
50°
15
54°
6
60°
16
35°
7
70°
17
65°
8
75°
18
61°
9
45°
19
12°
10
25°
20
27°
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46 RESOLUTION AND COMPOSITION OF FORCES C7
46. Law of Components. Copy each equation in
paragraph 43 and clear it of fractions, observing the con-
vention that a multiplier of a sine, cosine, or tangent, is
written preceding and not following these functions.
The equations obtained by clearing (1) and (2) of fractions
may be used as formulas for computing the x and y com-
ponents of a force of unknown magnitude and direction.
If so used
The perpendicular represents what component?
The projection represents what component?
The radius vector represents what?
The Y component = the force times what?
The X component = the force times what?
Write a formula for the Y component; also a formula
for the X component, denoting the force by F and the compo-
nents by y and x.
46. Examples. By reference to the 4-place table of
natural trigonometric functions compute and fill in the
omitted entries in the following table ruled in the work-
book, in which
F denotes the force,
d the angle at which it acts to the horizontal (read
counter-clockwise) ,
y the vertical component,
X the horizontal component.
Enter sines and cosines to three decimal places, with
the third decimal figure increased by unity whenever the
fourth figure is 5 or greater.
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68
TECHNICAL ALGEBRA
47
Table IV
RESOLUTION BY COMPUTATION
V
X
No.
F
B
sin
cos
+
-
. +
-
1
22
32°
2
25
30°
3
75
35°
4
110
38°
5
115
41°
6
122
50°
7
214
54°
8
250
56°
9
300
90°
10
1050
45°
11
55.4
31.5°
12
GO. 5
36.4°
13
12.5
43.2°
14
11.75
51.1°
15
450
62°
16
67.2
29°
17
98.3
33°
IS
160
51°
19
78.4
63°
20
36.6
25°
I
§ 4. FUNCTIONS OF AN OBTUSE ANGLE
47. Signs. Suppose the line OV to have rotated from
its original position OT
until 6 the angle of rota-
tion is obtuse.
Draw the figure and
project OV on OT pro-
duced through the point
of revolution.
Fia. 63. Is the projection posi-
tive or negative? Why?
Is the perpendicular positive or negative?
(The radius vector is positive in any position.)
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48 RESOLUTION AND COMPOSITION OF FORCES G9
On the figure in the work-book prefix the proper signs
to the symbols for projection, perpendicular, and radius
vector.
Then sin d=what?
cos d=what?
tan d=what?
The law of signs for fractions is as follows:
When the numerator and denominator have unlike signs,
both may be written + provided the sign before the fraction
is changed.
Apply this law to two of the preceding equations.
Therefore is the sine of an obtuse angle, positive or
negative?
Is the cosine of an obtuse angle, positive or negative?
Is the tangent of an obtuse angle, positive or negative?
Explain fully each of your three answers.
Therefore when the sine of an obtuse angle is used in
computation, what sign is understood?
When the cosine of an obtuse angle is used from the
table, what sign should be prefixed?
When the tangent of an obtuse angle is used from the
table, what sign should be prefixed?
Will you remember this when using these functions?
48. How to Read Functions of an Obtuse Angle. The
3Upplement of an angle is the difference between ISO*' and
the angle.
Therefore the supplement of 110^ is 180^-110^=70^.
Draw a figure in your work-book like that in paragraph
47 and denote by S the angle which is the supplement of 0.
Under the figure state the relation of S to d.
Is S an acute or an obtuse angle?
sin S = what? sin ^= what?
cos S=what? *cos ^=what?
tan S = what? *tan = what?
* Don't forget what you promised to remember.
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TECHNICAL ALGEBRA
49
Therefore the functions of an obtuse angle equal nu-
merically the corresponding functions of an angle in what
relation to the obtuse angle?
49. Examples. Determine and record the omitted entries
in the following table:
Table V
NATURAL FUNCTIONS OF AN OBTUSE ANGLE
No.
A
sin
COB
tan
No.
A
sin
cos
tan
1
100**
21
96**
2
105**
22
107**
3
98**
23
118**
4
110**
24
123**
5
115**
25
180**
6
120**
26
111**
7
125**
27
126**
8
130**
28
136**
9
135**
29
124**
10
140**
30
144**
11
145 **
31
134**
12
148**
32
149**
13
150**
33
119**
14
152**
34
129**
15
155**
35
159**
16
160**
36
146**
17
162**
37
154**
18
165**
38
169**
19
168**
39
173**
20
94**
40
170**
i
50. Examples. Solve the following equations:
1. c =
118
tan 51°
2. sinL =
18
3. sin 50°=—.
6. cos50°=~.
X
1728
1893*
4. 63 sin 40° =
32.04'
6. cosL =
1728
1893'
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63 RESOLUTION AND COMPOSITION OF FORCES 71
7. 63 cos 40° =r:^. 8. tan 50° =— .
32.04 X
9. tan L =^. 10. 63 tan 40° = - ^
1893' * 32.04*
- .oo 14 ^^ 396 tan 51° ,
11. 638=-^ ', 12. -— =5c.
tan B 792
13. sin il tan il = Jc. il = 76°.
^^si^^ 27806 ^^^^„
tan P 5 cos F
16. 535.7c =|(cos2* /2+sin« R),
o
16. 98 cos S =.8476 sin S. S =30'.
17. «iE4 =2.724.
COS 5
18. 34.1 tan 7 = 129 times ^.
lb
19. ^^=429.8c. «=60«.
COS d
20. 17.46 sin 29° = 1.7966 tan 84°
^^ 200 tan 104° ^^^^ ^^^^
21. — . ,^,^ =5606 cos 104°.
sin 104°
22. -608sinl22°=-|-tanl22°.
o
^ sin 130°
23. rrTB = -3.14c.
cos 130
24. -5.29 tan 114i°=2.5(l+cos 130°)+c.
6 tnn 40°
26. 3 sin« 121°+3 cos« 121° = ^,^^ .
342
cos 34°+sin 56°
tan 20°
=6 tan 134°.
* Determine the value of this sum from a right triangle by writing
the value of sin* and cos*, and adding the two equations.
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72 TECHNICAL ALGEBRA 60
27. 3.21 sin 123° ^~ tan 34^
o
„„ -7 tan 141° „„,
^- sin 82° ''•^'-
29. 12.9cos34J° = -atanlllJ°.
^^ -34.6 cos 100° ,^^, . ^,«
30. jy^ =1726 sin 671°.
31. -81.2c cot Mer^^^'""^'
4 cos 84°'
32. -8(cos« ^+sin2 6) =64c tan 112J°.
„„ 7.25 tan 81° ,_
33. -— — = 190c tan 81°.
1.25 cos 52°
56 1
34. j-^cotl54°=-116i2.
QQ 1
36. -78.2(sin2 a+cos* «)c =^ tan 115°.
it
36. -450(cos« i2+sin2 R) J- f ^'"^ ^^°
8 2 cos 98°*
37. 41.8 = -129c cot 171° sin 81°.
38. 5 tan 24J° sin 42^° = 125c.
39. 12.6 cot 114° tan 28J° = -.63 h ^^^ ^f i°
2 cos 28J
^^ 6 tan 108°
^•l8^^U02-°=^^^«^^-
41. 200 cos 48J° = - 16f c tan 94i°.
12 cos 24° h
lo'
42. -122 cos 118° tan 118° =
43. -31.8 tan 123° =
4 tan 112°*
tan 12° sin 29°+c
cot 78°
♦ cot=— .
tan
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50 RESOLUTION AND COMPOSITION OF FORCES 73
18.4 tan 44° cos 81°
.135c
44. -7.9 cot 141° = -
^^ 8 cos 85° „^^ . ^„
46. T7- — —3 =8.26 sin 5°.
12 tan 45
46. -94.6a+12co8l8r= " ' >
cos IIOJ
47. y =7.22 cos 74° tan 118°+78.4 sin 85°.
^^ 155 sin 45*^ ,„ ^ , ^^o . ^^o
48. "^ --^=17.5 tan 75° cot 50°.
S cos 45
49. 18x =9.62 tan 134° cos 18°+ 104 tan 30°
18.75
60.
9.875(sin»^+cos«^)'"''*^''^^i°:
61. 146(cos« ^+sin2 ^)+.1896 -^^ =3468.8.
.u4o
5.867 sin 39° 14- 96.785
■ .6985 cos 39° 14' sin' a+cos« a"
7Qfi 5 S
64. 348.5 tan 81° 33'+sin 119° 38' = ' ,^^ .
9 tan 45°
^^ 8.283 cos 125° 41' 1267.005 tan 142° 5'
• .00466+68.21 ~ 15
12.785sin68°24'-1.005
^^- .0812 tan 15° 16' -63.08 = 185.0096.
^„ 34.56 sin 165° 19'+5 tan 83° 4' ,„^, / ^^
57. .fc^t:/ • ^ . o ^N = 1296.4 cos 6°.
.1575(sin2 <^+cos2 <t>)a
68. .054(17.5 sin 115° 29'+8Z) cos 43°) ^-^ f'^^^^lTo -
24.2 cos 75
,^ 39.604 tan 143° 31'+5.0954 cos 12° 19' , . ,,„,,
^^- 73.81+43.9 tan 128° 10' =^^^^^^ ^^
785.91 sin« 28° -51 cos 140° 25'
8A tan 71° 18' -1.8005 "l'-^^-
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74 TECHNICAL ALGEBRA 61
§ 6. RESULTANT
61. Square of the Hypotenuse. As accurately as pos-
sible draw a right triangle with the sides forming the right
angle equal to 3 and 4 inches respectively.
Measure the hypotenuse and determine and state as
an equation, the relation of the square of the hypotenuse
to the sum of the squares of the other two sides of the tri-
angle.
Draw another right triangle having one side extending
neariy the full width of the page.
Denote its sides on the figure by c, a, and 6, c being the
hypotenuse.
By measurement and computation, determine and formu-
late the relation of <? to a^+ft^
Therefore, what seems to be the law for the square of
the hypotenuse of a right triangle?
62. Law of Resultant. The resultant of two or more
forces is the amount of their combined effect in pounds.
When two forces act at the same point on a body at an
angle of 90® with each other, their resultant is the hypotenuse
of the right triangle whose other two sides are the two given
forces.
To what therefore, is the square of their resultant equal?
To what is their resultant equal?
Illustrate by a dimensioned diagram.
If two forces act on a body at any angle with each other,
their resultant may be represented by the diagonal of a
parallelogram constructed on the forces as adjacent sides,
as shown in figure 64.
Draw the figure in the work-book and by the law for
square of the hypotenuse write the value of R^ in terms
of h and {b+v).
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63 RESOLUTION AND COMPOSITION OF FORCES 75
But v = a times what?
In the formula for R^ substitute this value of v and
square the quantity within the parenthesis.
By the law for square of hypothenuse,
/i2 = what in terms of a and v?
In the formula for R^ substitute this value for h^. Collect
in the second member after substituting for ir^ in terms of a
and 6.
Fig. 64.
If you have made no mistake in your work the formula
will now read,
R2=a^+V^+2ab cos 6.
In this formula, R is the symbol for what?
a is the symbol for what?
b is the symbol for what?
d is the symbol for what?
By reference to the formula write the law for the square
of the resultant of two forces acting at any angle with each
other at the same point on a body.
63. Examples. Compute and record the omitted entries
in the following table in which the forces specified are sup-
posed to act as indicated in paragraph 52.
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78 TECHNICAL ALGEBRA 66
66. Direct Variation. Direct variation is a relation of
quantities in which the ratio of any two values of one quan-
tity equals the direct ratio of the two corresponding values
of the other quantity.
In other words it is a relation of quantities in which the
value of one of the quantities over a second value of the
same quantity equals the value of the second quantity
corresponding to the first value of the first, over the value
of the second corresponding to the second value of the first.
If BocP, by the definition of direct variation
Bi* Pi
B2 P2'
in which Bi and B2 are any two values of JB, and Pi and P2
are the two corresponding values of P.
Observe that no variation expression can be used for
purposes of computation until transformed by the definition
of variation into a proportion and therefore into an
equation.
Apply the definition to the following:
1. LocC. 3. Xocy.
2. QocS. 4. Ao^R^.
In expressing direct variation the word direct or directly
i^ omitted as here illustrated.
66. Indirect Variation. Indirect or inverse variation is a
relation of quantities in which the ratio of any two values
of one quantity equals the inverted ratio of the two cor-
responding values of the other quantity.
When two quantities vary inversely y therefore, one varies
as the reciprocal of the other,
* Read B one over B two.
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67 VARIATION 79
If V varies inversely as P, we write Foe—-, and by the
definition of inverse variation, obtain the equation
7l^P2
F2 Pi'
Observe that inverse variation is denoted by inverting
one of the ratios. The proportion may be stated in words
as follows:
The ratio of any two values of V equals the inverse
ratio of the two corresponding values of P.
Apply the definition of inverse variation to the following:
1. Ta^?- 2. /ioc-l. 3. Foci-.
W A H
4. loc-^. 5. Cocij. 6. Fcc^.
67. Arrangement of Work. The following illustrates the
arrangement of all work in variation:
The thickness of materials of the same rectangular
section area varies inversely as their width.
A piece of rectangular section is 2f inches wide and f
inch thick.
Compute the thickness of a piece 3| inches wide and of
the same rectangular section area.
(a) reel. T^^^^.
\^) ^2 Wi ^^ % 2, •
* The preceding equation was solved for Tj, because Tj is unknown,
as indicated by the data.
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80
TECHNICAL ALGEBRA
(3) T2=^. W2^3h"~
W 12 8^^^^^-4Q •
(5) T2 = A75" Thickness of Second Piece.
68
68. Problems, Solve the following problems in the
work-book under problem number and lettered title with
formulas and solution to the left and data near the right
margin as shown in the preceding model.
Reduce mixed numbers in the data to fractional form as
indicated.
In substituting in the formulas write fractional divisors
as such; then write them as inverted multipliers.
The order of work should be as follows:
At the left enter problem number and title, formula of
variation, and the proportion.
At the right, enter in a vertical column each quantity
of the proportion followed by an equality symbol and the
data.
1. Weight. The weight of a substance varies as its volume.
A steel bar containing 104 cubic inches weighs 466 ounces.
Compute the weight of a bar of the same material containing
500 cubic inches.
2. Circumference. A circle SJ inches in diameter has a circum-
ference of 26.7 inches.
Compute the circumference when the diameter is 14 J inches,
the law of variation being that the circumference varies as the
diameter.
3. Velocity of a Falling Body. The velocity of a falling body
varies as the time during which it is falling.
When a body falls from rest, its velocity at the end of 1 second
is approximately 32 feet per second.
Compute its velocity at the end of 12 J seconds.
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68
VARIATION
81
4. Space Traversed. The distance through which a body
falls from rest varies as the square of the time diuing which the
body is falling.
In one second a body falls from rest a distance of 16 feet.
In how many seconds will it fall 185 feet?
6. Time and Velocity. In the sunmier of 1908 a baseball
dropped from a window of the Washington monument was caught
at the base of the monument about 504 feet below.
In how many seconds did the ball strike the catcher's hands
4 feet above the ground?
, What was its velocity in feet per second when caught?
6. Relation of Time to Distance. When the speed is un-
changed, the time required for a train to cover any distance, varies
as the distance.
The Pennsylvania Special and the Twentieth Century Limited
for several years made the run between New York and Chicago
in 18 hours, the distance being 908.7 miles by the Pennsylvania
Road and 978 by the New York Central.
If this rate were maintained across the continent, in what time
would these trains make the run from New York to San Francisco
if both used the shortest rail route of 2275 miles from Chicago?
7. Velocity of a Pulley. The velocity of the ,rim of a pulley
varies as its diameter.
A 16-inch pulley has a rim velocity at a certain moment, of 155
feet per minute.
What is the rim velocity at the same moment, of a 9i inch
pulley which is keyed to the same shaft?
8. Shearing Stress and Resist-
ance of Rivets. The shearing stress
of a rivet varies as the square of its
diameter.
Solve the formula for di.
The resistance of a rivet to crush-
ing varies as its diameter.
Solve the formula for Ri and
for (fc. Fig. 66.
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82
TECHNICAL ALGEBRA
58
9. Pitch of Rivets. The number of rivets required for a
boiler seam varies inversely as the pitch (the distance between
rivet centers).
Fig. 67.
If 40 rivets are required when the pitch is 2| inches, determine
the pitch when 41 rivets are required for a boiler of the same size.
10. The Volume of a Gas. The volume of a gas varies in-
versely as the height of the mercury in the barometer.
The volume is 23^ cubic inches when the barometer registers
29.4 inches.
What is the volume when the barometer registers 30.7 inches?
11. Weight. The weight of a body varies inversely as the
square of its distance from the center of the earth.
If a substance weighs 40.8 pounds at sea level (3960 miles
from the center), compute its weight when on top of Mt. Everest,
29,000 feet* above sea level.
12. Generation of Hydrogen. In the generation of hydrogen
the amount of sulphuric acid (H2SO4) varies as the amount of
zinc (Zn).
* See National Geographic Magazine, June, 1909, p. 497.
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68
VARIATION
83
For complete action 65 grams of Zn require 98 grams of H2SO4.
How many grams of H2SO4 will be required for 100 grams of Zn?
Fig. 68.
13. Generation of Hydrogen. The amoimt of hydrogen
produced varies as the amount of zinc.
If 65.4 grams of zinc produce 2.016 grams of hydrogen, find the
amount of zinc required to liberate 75 grams of hydrogen.
14. Area of a Circle. The area of a circle varies as the square
of its diameter. ^^ni^ji^ '
When the diameter is 6f inches the area is (approximately) 32
square inches.
What is the radius when the area is 56 square inches?
15. Volume of a Sphere. The volume of a sphere varies as
the cube of its diameter. 1^
The volume is 1437.3 cubic inches when the radius is 7 inches.
Compute the volume when the radius is doubled.
16. The Resistance of a Wire. The resistance of a wire varies
inversely as the square of its diameter.
The resistance of a coil of copper wire A of an inch in diameter,
was 2.5 ohms.
What is the resistance of a copper wire of the same length, with
a diameter of f of an inch?
17. Generation of Oxygen. In the preparation of oxygen 13}
grams of mercuric oxide yield 1 gram of oxygen.
If the amount of oxygen produced varies as the amount of
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84
TECHNICAL AiXiEBRA
mercuric oxide, compute the number of grams of mercuric oxide
which will produce 150 grams of oxygen.
a w (^ o ^
Fia. 69.
18. The Hydratilic Press. In a hydraulic press the distances
through which the pistons move vary inversely as the areas of the
pistons.
Through what distance does the power piston 2J inches in
diameter, move, when the weight piston 2 feet in diameter moves
through a distance of 2.8 inches.
19. Power of a Hydraulic Press. In a hydrauUc press the
ratio of the weight to the power applied equals the ratio of the
piston areas.
Fig. 70.
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68
VARIATION
85
If the weight piston has an area of 214 square inches and the
power piston an area of 4A square inches, what power will be
required to lift a weight of 760 pounds?
20. Penetration of Armor. The depth of penetration of armor
by a projectile varies as the caliber of the gun.
If an 8-inch gun penetrates to a depth of 4} inches, compute
the penetration of a 14-inch
gun in the same armor at
the same distance.
21. Strength of the Cur-
rent. The current varies as
the tangent of the angle of
deflection (tan 5).
The current from a bat-
tery deflects the needle of
a tangent galvanometer 10°;
with a second battery, the
deflection is 25°.
Compute the ratio of the
strengths of the two cur-
rents.
22. Law of the Lever.
In a lever of whatever class
the power is to the weight,
as the weight's distance
from the fulcrum is to the
power's distance from the
fulcrum.
A lever 8^ feet long has the fulcrum 9 inches from one end.
What force in pounds applied at the long end will balance a
weight of 480 pounds, 2 inches from the short end?
23. Height in Communicating Tubes. The heights of liquids
at equilibrium in a U-tube vary inversely as their densities.
The height of the water in one of two communicating tubes is
14i inches.
Compute the height of the mercury in the other tube, mercury
being 13.6 times as heavy as water.
Fia. 71.
Tangent Galvanometer.
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86
TECHNICAL ALGEBRA
58
24. Weight of a Gun. The weight of a gun varies approxi-
inately as the cube of its caliber.
An 8-inch gun weighs 14.2 tons.
Determine the weight of a 12-inch gun of the same tjrpe.
25. Charles' Law. Under constant pres-
sure, the volume of a gas varies as the abso-
lute temperature.*
A gas with a volume of 90 cc. at 10° C.
is heated to 35° C.
Compute its volume at the higher tem-
perature, the pressure being the same at both
temperatures.
26. Osmotic Pressure. The osmotic
pressure of a solution varies as the molec-
ular mass M of the substance dissolved.
Solve the formula for Mi.
Fig. 72.
Osmosis.
27. A Falling Body. The distance
through which a body falls from rest varies
as the square of the time during which it is falling.
A body falls 175.6 meters in 6 seconds.
How far will it fall in J of a second ?
28. The Lever. The distances from the fulcrum of a lever, of
two weights which are in equilibrium, vary inversely as the weights;
that is, the distance varies inversely as the weight.
At the ends of a lever 10 feet long are two weights of 120 and
84 pounds respectively.
At what distance from the center must the fulcrum be placed in
order that the weights may balance?
See paragraph 37, problem 16.
29. Oscillation of a Pendulum. The number of oscillations
of a pendulum per time unit varies inversely as the square root of
its length.
♦Absolute temperature =273°-}- the recorded temperature Centi-
grade.
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68 VARIATION 87
The length of a pendulum which beats seconds at the sear
level, is 39.1 inches.
How long is a pendulum which oscillates 3 times a second?
30. Apparent Size. The apparent size of an object varies
inversely as its distance.
Solve the formula for Di.
31. Flow of Gas in Pipes. The volume of gas discharged from
a horizontal pipe under constant pressure and specific gravity,
varies inversely as the square root of the length of the pipe.
The volume in 3 J hours from a pipe 600 feet long was 1750
cubic feet?
Find the volume discharged per hour from a mile length of the
same pipe.
32. Vibration of Strings. The number of vibrations per time
unit, of a string under constant tension, varies inversely as the
length of the string.
^i^
^
Fig. 73.
When the length is 4 feet 8 inches, the string vibrates 256 times
per second.
. What must be the length of a string of the same material and
diameter, in order that the vibrations may be doubled?
33. Relation of Mass * to Velocity. The velocity imparted
to a body varies inversely as its mass.
A given force imparts a velocity of 75 feet per second to a
body whose weight is 644 pounds.
What velocity will the same force impart to a body whose mass
is 120?
34. Linear Velocity. Linear velocity varies directly as areal f
* » * weight in lbs.
* Mass = — .
32
t Areal velocity equals the rate at which an area increases.
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TECHNICAL ALGEBRA
68
velocity and inversely as the distance from the center of force to
the line of motion.
Solve the formula for D2.
35. Linear Velocity. The linear velocity of a particle per
time imit varies directly as the radius of revolution and inversely
as the time of one revolution.
Solve the formula for Ri.
36. Angular Velocity. The
angular velocity (co) of any
point in a revolving body
varies directly as the areal
velocity and inversely as the
square of the radius vector (R).
Solve for R2,
37. Centrifugal Force.
The centrifugal force at any
point in a revolving body
varies as the square of the
linear velocity.
Solve the formula for Fx,j.
38. Vibration of Light.
The number of vibrations of
light per second varies in-
versely as the wave length L.
Solve the formula for L2.
39. Intensity of Heat.
The intensity of heat H re-
ceived from a radiating body
varies inversely as the square
of the distance from the source
of heat.
Solve the formula for H2.
Fig. 74.
Boyle's Law Apparatus.
40. Boyle's Law. Under
constant temperature the vol-
ume of a gas varies inversely as the pressure.
If a gas have a volume of 106 cubic inches under a pres-
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5d VARIATION 89
sure of 14.7 pounds, compute the volume when the pressure
is 15.3 pounds more,
41. Bending of a Bar. The amount a bar will bend varies
inversely as the width when the thickness and the length
remain the same.
If a bar 3.5 inches wide bend through5° under theapphcationof a
given force, compute the amount of bending of a bar 5.1 inches wide,
of the same material, length, and thickness, imder the same force.
42. Centrifugal Force. The centrifugal force at any point
of a revolving body varies as the radius of revolution of the point.
The centrifugal force is 900 pounds when the radius is 24 inches.
Compute the force when the radius is 18 J inches.
43. Density and Volume. TJie density of a given weight of a
gas varies inversely as its volume.
If the density is .108 when the volume is 4.5 cubic feet, find the
density when the volume is reduced by pressure to 3.8 cubic feet.
44. Internal Resistance of a Bat-
tery. The total resistance varies ^s
the cotangent of the angle of deflection.
When the resistance is 9.8, the
angle of deflection is 45°.
Find the resistance when the angle
is 24°.
45. Area on a Globe. The area of
a country on a terrestrial globe varies as
the square of the diameter of the globe. ^^'
On a globe 28 inches in diameter, the area of a certain island
is 2.6' square inches.
Compute the area of the same island on a globe 36 inches in
diameter.
46. Period of a Pendulum. The time of vibration (period) of
a pendulum varies inversely as the square root of the force causing
it to vibrate.
Solve the formula for Fu
47. Force of a Moving Body. The force imparted by a moving
body varies as the mass of the body times its acceleration.
Solve the formula for Af 2.
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TECHNICAL ALGEBRA
68
48. Area of a Triangle. The area of a triangle varies jointly*
as its base and altitude.
The area of a triangle whose base is 19 feet and whose altitude
is 10 feet is 95 square feet.
Compute the altitude when the base is 24 feet and the area is
132 square feet.
Check by problem 5, paragraph 33.
49. Volume of a Cone. The volume of a cone varies jointly
as its altitude and the square of the diameter of its base.
The volume is 392.7 cubic inches when the altitude is 15 inches
and the diameter of the base is 10 inches.
Compute the diameter when the altitude is 10 feet and the
volume 28,160 cubic inches.
50. Weight of a Beam. The weight of a beam varies jointly as
its length, cross-sectional area, and material.
An iron bar 36 inches long and having a cross-sectional area
of 1 square inch weighs 10 pounds.
What is the weight of a wooden beam 18 feet long, whose material
is 1^5 as heavy, with a cross-sectional area of 64 square inches?
51. Wind Pressure on a Surface. The pressure of the wind
perpendicular to a
plane surface varies
jointly as the area of
the surface and the
square of the wind's
velocity.
Under a velocity
of 16 miles per hour,
the pressure on 1
square foot is 1 pound.
What is the veloc-
ity when the pres-
sure on 2 square
yards is 50 pounds?
52. Volume of a Pyramid. The volume of a pyramid varies
jointly as its height and the area of its ])ase.
* " Varies jointly as " is a conventional expression meaning
" as the product of."
Fig. 76.
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68
VARIATION
91
The height- of the great p3Tamid is 481 feet. Each side of its
base is approximately 756 feet.
What is its vokime if the volume of a pjo-amid whose height
is 20 feet and whose base is 9 feet square, is 540 cubic feet?
53. Weight of an Engine Pier. The weight of a concrete
pier varies jointly as its dimensions.
A pier 7' by 4^' by 3f ' weighs 15400 pounds.
What is the weight of a pier of the same concrete measuring
10rby5rby4r?
54. Friction. The friction between two surfaces varies j ointly as
the perpendicular pressure between them, the coefficient of friction,
and the area of the surface of contact.
The perpendicular pressure between
a journal arid its bearing is 12,000
pounds. The coefficient of friction is .3
and the area is 5 square inches.
If the area were twice as great howmany
times would the friction be increased?
Fig 77
55. Torque of a Magnetic Needle.
The torque (F) acting on a needle varies jointly as its magnetic
^'^t^:^'^^-vW'^j^
wmmmmM
Fig. 78.
moment (mZ), the horizontal intensity of the earth's magnetism
(H), and the sine of the angle of deflection (sin 5).
Express this law as an equation and solve for Fi.
56. Joint Variation, z varies jointly as x and y. When a: = 1
and2/=2, 0=4.
Compute the value of x when 2 =30 and y =3.
57. Inverse Variation. If x varies inversely as 2/^—2 and is
equal to 24 when y = 10, compute x when y=5.
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TECHNICAL ALGEBRA
68
58. Velocity of Sound in any Medium. The velocity of
sound in any medium varies directly as the square root of the elas-
ticity of the medium and inversely as the square root of its density.
Express this law as an equation and solve for E^
59. Transverse Vibration of Strings. The number of
vibrations (n) varies directly as the square root of the tension (T),
and inversely as the length (L) times the diameter {d) times the
square root of the density (D) .
Express this law as an equation and write the second member
in its simplest form.
60. Tensional Strength of
Shafting. The tensional strength
of a shaft varies as the cube of its
diameter Z).
Solve the formula for Di.
D i
1"
Fig. 79.
Fig. 80.— Tension Test.
61. Volume of Hydrogen. The volume of hydrogen varies
jointly as its weight and absolute temperature * and inversely as
the pressure.
Under 760 mm pressure at 0° C the volume of .08973 gram of
hydrogen is 1 liter.
What will be the volume of 18 grams under 1000 mm pressure
at 50°?
62. Illumination. The amount of illumination received by a
body varies directly as the intensity of the light and inversely
as the square of the distance from the light.
From a light of 16 candle-power the illumination is 6 at a dis-
tance of 5 feet.
Compute the illumination at a distance of 12 feet from a light
of 50 candle-power.
* Absolute temperature =273°+ recorded temperature, Centigrade.
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68
VARIATION
93
63. Volume of a Gas. The volume of a gas varies as the
absolute temperature and inversely as the pressure.
Under a pressure of 1 atmosphere and temperature of 260° C the
volume is 150 cubic inches.
Required the temperature when the pressure is 45 pounds per
square inch and the volume the same as in the first instance.
64. Pressure of a Gas. The pressure of a gas varies jointly
as the density and the absolute temperature.
At a recorded temperature of 25° C a given volume of a gas whose
density is .972, exerts a pressure of 2 pounds to the square inch.
Determine the pressure when the recorded temperature is 125°.
65. Offing at Sea. The distance of the offing at sea varies as
the square root of the height of the eye of the observer above the
sea level.
The offing is distant three miles when the eye r^\
is 6 feet above sea level.
Compute the distance of the offing when the
eye is 560 feet above sea level.
66. Elongation of a Spring. Within the elastic
limit the elongation of a spring varies as the load.
Under a load of 7 grams the elongation of a
spring was 23.15 millimeters.
Compute the load when the elongation of the
same spring is 31.53 millimeters.
67. Relation of Flow to Diameter. The amount of water that
will flow through a pipe varies as the square of the diameter of
the pipe.
How many gallons will flow through a 12-inch pipe if 40 gallons
flow through a 2-inch pipe
in the same time?
Fia. 81.
68. Resistance of a
Wire. The resistance of
a wire varies directly as
Fig. 82. ^^® length and inversely
as the cross-sectional area.
The resistance of 390 feet of i^^-inch copper wire is 1 ohm.
Compute the resistance of 2 miles of J-inch copper wire.
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94
TECHNICAL ALGEBRA
68
69. Intensity of a Magnetic Field. The intensity (H) of the
field varies directly as the number of vibrations and inversely
as the square of the distance of the magnet.
The intensity is .16 when the number of oscillations is 20 and
the distance .28 inch. Compute the intensity when the number
of vibrations is 100 and the distance is .0987 inch.
70. The Thompson Ammeter. The effect upon the needle
varies as the cube of the radius of the coil and inversely as the cube
of the mean distance between the coil and the needle.
Solve for Ri.
^mmvmvvvmm vvmmvmmvM^^^
Fia. 83. — ^Thompson Ammeter.
71. Visual Angle. The visual angle of a sphere varies directly
as the diameter and inversely as the distance of the sphere from
the observer.
Solve the formula for ft.
72. The Inclined Plane. The force acting parallel to the base
of an inclined plane varies directly as the height of the plane and
inversely as the length of the base.
Solve the formula for Hi,
73. Force of Gravity. The force of gravity at the surface of
any planetary body varies directly as the relative mass of the
body and inversely as the square of the radius of the body.
At the surface of the earth, whose radius is about 3960 miles
and whose relative mass is unity, the force of gravity is 32.
Compute the force of gravity at the surface of Jupiter whose
relative mass is estimated as 31L95 and whose radius is 43,000
miles.
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68 VARIATION 95
74. Intensity of lUiunination. The intensity of illumination
from a light varies directly as the sine of the angle at which the
rays fall (sin 6)y and inversely as the square of the distance of
the light.
Solve the formula for D%,
75. Law of Resistance. The resistance of a conductor varies
directly as its length into the
specific resistance of the material
and inversely as the square of its
diameter.
The resistance of a Ruhmkorff
coil wound with No. 12 copper
wire (Brown & Sharpe wire-
gage) was found to be 2.6 -pm. 84.
ohms.
The resistance of 2 meters length of the same wire was .006
ohm. Find the length of wire in the coil.
76. Thickness of a Hollow Cylinder. The thickness of a hollow
cylinder varies directly as the amount of material and inversely
as the length of the cylinder times the sum of the internal and
external radii.
Solve the formula for L2.
77. Elongation of a Wire. The elongation of a wire varies
jointly as the length and the force applied and inversely as the
square of its diameter.
The results of an experiment on the stretching of No. 12 iron
wire were as follows:
F L E
D
1st test
2 40 1.5
.26
2d test
2 30
.26
Find the value of E in the second test.
78. Torsion. The angle of torsion of a rod varies directly
as the twisting force in pounds times the length in inches times the
reciprocal of the fourth power of the diameter in inches.
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96
TECHNICAL ALGEBRA
68
A rod 36 inches long and an inch in diameter is twisted through
5** by a force of 10 pounds.
Find the force which would twist through 11** a rod of the
same material, 45 inches long and 2.3 inches in diameter.
Fia. 85.
79.^ Deflection of a Beam. The amount of deflection of a beam
when supported at both ends and loaded at the center varies directly
as the load times the cube of the length in feet, and inversely
as the width in inches times the cube of the depth in inches.
A beam 20' by 5" by 4" supported at the ends and loaded at the
center with 4 tons is bent downward 2i inches.
What must be the depth of a beam of the same material,
length, and width, that shall bend only J inch under the same load?
80. Stiffness of Shafting. The stiffness of a shaft varies directly
as the fourth power of its diameter and inversely as the load and
the cube of the length.
Solve the formula for dt,
81. Horse-Power of a Steamer. In steamers of the same type
the horse-power varies jointly as the cube root of the square of
the displacement in tons and the cube of the velocity.
If the horse-power is 1500 when the displacement is 1650 tons
and the velocity 14.4 knots per hour, determine the horse-power
developed in a vessel of the same type, whose displacement is
* 20,000 tons and whose speed is 23.2 knots per hour.
82. Heat Due to Current. The heat developed in a conductor
\aries jointly as the resistance of the conductor, the time the
current flows, and the square of the current.
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68
VABIATION
97
In 2i minutes a current of 3| amperes develc^d 1300 imits
of heat in a wire having 10^ ohms resistance.
Compute the resistance of a wire of the same size in which
22,000 units of heat were developed by a current of 7.11 amperes
in 3J minutes.
83. Weight of Cylinder. The weight of a cylinder varies
jointly as its length and the square of its diameter.
The weight of a cylinder 10.5 inches long and 3J inches in
diameter is 30.6 pounds.
Compute the diameter of a cylinder of the same material
and thickness, 12 inches long and weighing 82 pounds.
84. Rupture of a Beam. Under a central load the breaking
strength of beams of the same depth varies jointly as the width
in inches and the reciprocal of the length in feet.
Fig. 86.
A beam 8 feet long and 4 inches wide, supported at the ends,
breaks under a weight of 3000 pounds acting at the middle.
Under what weight will rupture occur in a beam of the same
material whose dimensions are 10' by 5"?
85. Resistance of Air. If a ball have a velocity greater
than 1100 feet per second the resistance of the air approximately
varies as the square of the diameter of the ball times the difference
between the velocity and 800. •
A ball .42 inch in diameter with a velocity of 2100 feet per
second meets an air resistance of 1.6 poimds.
Determine the resistance offered to the same ball when the
velocity is 1850 feet per second.
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98
TECHNICAL ALGEBRA
68
86. Horse-Power Transmitted. The horse-power transmitted
by a rope varies directly as the cube of the square root of the
tension and inversely as the square root of the weight of the rope
per foot.
Forty-five horse-power is transmitted when the tension is 1000
pounds and the weight per foot is .4 pound.
Compute the horse-power transmitted by the same rope when
the tension is reduced 200 poimds.
87. Variation of the Current. The
current varies inversely as the resist-
ance and the resistance varies directly
as the cotangent of the angle of deflec-
tion (cot 8),
Prove that the current varies in-
versely as the cotangent of the angle
of deflection.
88. Coulomb's Law. The force (/) exerted between two charges
of electricity varies as their product {qq') and inversely as the
square of the distance (r) between them.
Find the value of /2 when
HI 11^ — AWVV
Fig. 87.
/l= 1
n=15
g. = 12
g.'=9,'
<?»= 1
r,=2
89. Range of a Jet. The
range of a jet varies jointly
as the square root of the
head and the square root of
the vertical height of the orifice
above the level where the jet
strikes.
Solve for R2,
^^
90. Flow from an Orifice.
The amount of flow from an
orifice varies jointly as the area of the orifice and the velocity of
the flow.
Solve the formula for Fj.
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6d
VARIATION
99
91. An Inclined Plane. The force on an inclined plane varies
as the height of the plane, and the height of the plane varies as
the sine of the angle of inclination of the plane to the horizontal.
Prove that the force varies as the sine of the angle of inclination.
92. Lateral Surface of a Right Cylinder. In a
right cyUnder of constant altitude the lateral area
varies as the circumference of the base, and the cir-
cumference of the base varies as the radius.
Prove that the lateral surface varies as the radius.
Fia. 89.
93. Velocity of a Falling Body. The velocity of
a body falling freely under gravity varies as the
square root of the space traversed from rest.
When a body has fallen a distance of 16 feet its velocity is
approximately 32 feet per second.
Compute the velocity when the body has fallen 540 feet.
\
\
\
\
\
\
Fig. 90.
94. Diagonal of a Cube. The edge of a
cube varies as the diagonal of the face,
and the diagonal of the face varies as the
diagonal of the cube.
Prove that the diagonal of the cube varies
as the edge.
95. Diffusive Power of a Gas. The
rates of diffusion of gases are inversely proportional to the square
roots of their specific gravities.
Formulate as a proportion and compute the ratio of the rates
of diffusion of two gases whoso specific gravities are .97 and .0694.
96. Revolution of a Planet. The square of the time of the
revolution of a planet about the sun varies as the cube of its dis-
tance from the sun.
The earth at a mean distance of 92,000,000 miles from the
sun, makes a complete revolution about the sun in 365J days.
Compute the period of revolution of Neptune whose distance
from the sim is approximately 2,775,000,000 miles.
97. Period of a Pendulum. The time in which a pendulum
makes one vibration varies as the square root of the quotient
of the length and the force of gravity.
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100
TECHNICAL ALGEBRA
68
When the force of gravity is 32.14, a pendulum 39.1 inches
long vibrates in 1 second.
Compute the time of vibration of the same pendu-
lum on the moon where the force of gravity is .165.
98. Velocity of Sound. The velocity of sound in
gases varies directly as the square root of their elasticity
and inversely as the square root of their density.
Sound travels through air at the rate of 1080
feet per second.
The elasticity of air is 1.6X10* and its relative
density is 1. Fig. 91.
Compute the velocity of sound in hydrogen
whose relative density is .0694 and whose elasticity is 1.6X10*.
99. Indicated Horse-Power. The coal consumed in tons by
vessels of the same type varies jointly as the indicated horse-
power and the time of passage in days.
The displacement in tons varies as the coal consumed.
Show that the indicated horse-power varies as the displacement,
and inversely as the time of passage.
100. Initial Velocity of a Projectile. The square of the
initial velocity of a projectile in feet per second varies directly
as the charge of powder in pounds, and inversely as the weight
of the projectile in pounds.
If 21 pounds of powder will give a 40-pound projectile an
initial velocity of 2000 feet per second, compute the charge
required to hurl a 50-pound projectile with an initial velocity of
1800 feet per second.
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CHAPTER IV
THE FOUR FUNDAMENTAL OPERATIONS
Secton 1, Addition. Section 2, Subtraction. Section 3,
Multiplication. Section 4, Division.
69. Classification. The four fundamental operations are:
Addition,
Subtraction,
Multiplication,
Division.
They are called fundamental because they are basal in
mathematics and other seemingly different operations are
only these in various combinations. Their importance
and the necessity of speed and accuracy in their application
are therefore obvious.
60. Kinds of Terms. The terms of an algebraic expres-
sion are of two kinds, like and unlike.
Like terms are those having one or more letters the same
with the same exponents.
Unlike terms are those which do not have the same letters
with the same exponents.
What is meant by like terms can best be shown by an
illustration:
ox, to, rx, 5x, ex, and 2a;,
rioi
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102 TECHNICAL ALGEBRA 61
are like terms with respect to x, all having the same letter x,
affected by the same exponent 1.
5c2, 4a2c2, 10b3c2, and (u?,
are like terms with respect to c^.
Therefore to be like, terms must have one or more letters
the same with the same exponents,
ac and bc^, b(?, and 8c*,
although having the same letter c, are unlike because the
exponents of c are unlike.
§ 1. ADDITION
61. Definition. Addition is the process of finding the
sum of the terms of an expression.
To perform the addition it is convenient to write like
terms in the same vertical column, making as many columns
as there are unlike terms. The excess of plus or minus in
each column is then determined and the result is written
underneath.
62. Illustration. Suppose the following polynomials *
are to be added:
\27?-%x^+l^\ and6x4-llx-12x3__8^
Writing Uke terms in the same vertical column and
finding the excess of plus or minus in each column, we have
3x4+ 5a:3__6a.2^ ^^^ 1
-9x4_|_ a:3_|_53.2_i32.+ 1
l23?-%7?+ 16
6x4_i2a:3 -iix- 8
6x3-8x2-20x+ 8
* A polynomial is an algebraic expression of more than two terms.
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66 THE FOUR FUNDAMENTAL OPERATIONS 103
63. Terms with Literal Coefficients. Like terms witii
literal coefficients or with both literal and numerical are
added by inclosing the coefficients in a parenthesis followed
or preceded by the letter with respect to which the terms
were classified as like terms.
For example, if the smn of ^x—ax+cx—h^x is desired,
the result is written (5 — a+c — b^)^.
64. Law of Addition.
(1) Write like terms in the same vertical column.*
(2) Find the excess of plus or minus in each column.
65. Examples in Addition. Find the sum of the follow-
ing as specified by the law:
1. Zx'-6x^+2x*-Sx^y 3x'-6x*+2x2-x, Tx^+Oaj^-x^+lOa;,
- 12x2+ llx- 16x3.
2. y^+3y^-2y^+5y, 8y^-7y+ny^-my*, '-y^+2y*+l0y -Sy^,
7y^+15y^-y^+y.
3. 422-62+122^-202^+8, 62-1123-422+62^-9,
22*+323+9x2 -302+1, 823 -72+92* -322+6.
4. -34^+18^2 _7^4+8^3_9^ 7 _ 16^2+^ .8^3+9^4.
lU2_9^+14_7^3+3^^ ldt-ti+U*-l6+t\
6. ax+hx+cx+dx+ex.
6. 5x+3x-6x+cx+dx,
7. 6a-56+7a-d, 36 -a+d-c, d+2c+a-26, c-4d+9a-l,
l+8d+c-7a,
8. 5x2-4x2-6x2+3x+8x-x+cx.
9. a+5, 3a+8, -5a -7, -lla+16.
10. by+cy+dy+8y,
11. 4i/2+6i/2+ct/2.
12. 8x2+0x2+6x2-6x2-10x2+8.'
13. 3x2 -cx2+16x2-te2 -4x2+7.
* Usually unnecessary, since in algebraic work the addition of
like terms can be determined from the expression as written.
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104 TECHNICAL ALGEBRA 65
14. 5z^+dx^-kx^-3x^+7x-Zx+ax.
16. bx+cx-Sx+9x^+fx-cx+3x^+x.
16. a^x+9x^-b^-ax^+5x-dx.
17. sVx-^Vx+aVx-sVx.
18. sVbc+Vbc-ay/bc-sVbc.
19. 5x^-ax^+5bx^'-2x^+icx^.
20. ax^+3x^-x*+6x, 3x-hx^+x^+2x*.
21. (a+h)\/x-c\/x+dVx-5Vx.
22. 3(x+y) -3x^+i{x+y)+mx^-c{x+7/),
23. WbTc-Sy+6--2Vb+c+15y+hVb+c-5,
^b+c+ (r+s)y - uVb+c -dy+2^.
2
24. |aa;2-^aa;«-29+dr+3i/-^ax2,
^x+30 -Siy+Y^ax^ - 14+ (a+6)y.
25. 5x"-16x"+8Jx'+19aV+7a;"-10aV,
bx^'+cbf-la^* - 17a«2/» -6x^ -ax^
26. Safy" -8x^y^+xY+7x'y'+Six*y*,
-bafy'+12x^*-gxY+tx^*-y-K
27. 2462c»x2-30aa;2+12%'+1662c»x*,
-9by^+l5b^c^x^+33ax^+2lby^ -daxK
28. rx2-da:*+5x*-te«+3x*-5x2+cte2-x».
29. 5x^+Sx*-{a+b)x^+10x*-ax+5x+cx.
30. 0^x2 -ft'x* - Ax2+5x -Sx+kx+Sx^ -5x+18.
31. 24.8x2 -16.29x»+7.05x+24,
-16+3.17X -32.2x2+1.48x8 -18.36,
17.6x2+12.38x»+16.76-73.52x» -40.17.
32. a^+Zab*c-7a*bc, a^bc-Zab*c+a*y 15a^bc -Sab^c.
33. by--cy^+-ax\ ay+—cy*-—bx^, dy+—ay*-—dx\
34. r^x+s^—tr^x—ay+c^.
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67 THE FOUR FUNDAMENTAL OPERATIONS 105
36. 4.2a;»-3.3x«+1.5x+8.3, 2.62a;»-8.4a;«+3.17x-1.9,
2x*+ix^-7x+l, 5x»+1.33x2+21.6x2-1.91x, and
33.51x»+7.25x2 -6.5x+2.81.
36. 5c"'+.3a+86c, .4c"+7.3a"+6.q56c, c~+3.07a"~56c, and
3 13 1
37. 5ab'-2x+-rxy, .3x+.03ax-4.05a6, -ax+-xy—-ab,
4 o 4 6
7x+2.0ax+.2x, —xy+—ax ——xy,
5 4 o
38. .3(m-3x)»-|(m-3x)», 38(fn-3xr, -2.13(m-3x)».
39. x*-/-.4a^ .5x»-4/-.5a, .38x•+.5V-a^
40. j-ax«+|a«+-x»y+56», 4ax«+|x»y-7.5a«+|6»,
.8ax«+-x»y -;^a»+-6», — ax2+.25x»y --a« -— 6».
41. 8y-3(a+6), 5t/+8(a+6), 102/-.9(a+6), -.2y-3(a+6).
§2. SUBTRACTION
66. Definition. Subtraction is the process of taking one
expression from another.
Subtraction is indicated by the minus sign placed between
the quantities which are to be subtracted. This sign
indicates that the quantity following it is to be taken from
the quantity preceding it.
In the expression R—SXj the minus sign indicates that
Sx is to be subtracted from R, In bc—{x+2y), the minus
indicates that x+2y is to be subtracted from 6c.
67. The Subtrahend. The subtrahend is the quantity
which follows a minus sign.
In R—SXf what is the subtrahend? Why?
In bc-'{x+2y)j what is the subtrahend? Why?
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106 TECHNICAL ALGEBRA 68
What does the parenthesis indicate?
If written without the parenthesis what sign would be
necessary preceding 2y in order to show that it also is to be
taken from 6c?
Write the expression without the parenthesis so as to
indicate that both x and 2y are to be subtracted from be.
68. Laws of Subtraction.
(1) Write like terms in the same vertical column.
(2) Change signs of all terms in the subtrahend and add
it to the minuend.
69. Mental Change of Signs. The signs in the subtra-
hend should be marked changed only so long as may be
necessary for certainty of result. As soon as possible this
change should be effected mentally and the changed signs
should not be written.
70. Illustration. The following shows the application
of the laws of subtraction:
From ax^+c?y—4:by take 5by — 3ax^+2c^y
Applying (1) of the preceding law we have three vertical
columns of like terms.
By the law of subtrahends the signs of all the terms
in the subtrahend must be changed. The changed signs
are therefore entered under those in the example, as shown.
Observe that the changed signs are placed under the
original signs.
ax^+ c?y — 4by
zpSax^^2c^y^5by
4ax^— (?y — %y
71. Examples in Subtraction.
1. From 7a2-5624-8c, take4a2+462-9c.
2. From la^hc-%ah'^c+llahc^-c^
take 8c3+21a62c -Za''hc+I2abc^.
3. From ^x^-bxy-{-\%xHj''-2\xy^+^
take 6 -20xy^ -Sxy -6x^^ -9x\
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71 THE FOUR FUNDAMENTAL OPERATIONS 107
4. From Uacx-19'ad^+6a%^-r^+l
take - 19+42a262 - 17ad^ -40acx,
6. From 21z^+l3x*-x+l4: -bc+t
take 5t+x -60 -20x^+8x^ -be.
6. From 5d^x -Sgx^ -2lky^+lSx^y*
take 17^x2+20%3+5d2x -30.
7. From ISa^x-Ub'^+llc^^+Sld*
take Ud*-21a^x+%^y+l0c^K
8. From 314a: -722/2+18x^-1960
take 186c+ IQx*!/^ -300x+602/*.
9. From {a+b)x-(c+d)y+{e+f)z
take 2(a+6)x - (c+d)?/ -5(e+f)z.
10. From 3\/ a;-y +2\ /a;g-y 2-13aa;»+l
take e\/x-y-Vx^-y^+18+17ax\
11. From 16(r+s)«-20(6+c)»+21(d+c)*
take 32(d+e)*-19(r+s)2+52(6+c)3.
12. From ax2!/2+cx8+dx+92ar2
take 5ar^+3dx - 10ax22/24-4cx3.
13. From 23a^b^+30c^x*y -29b^xy^+U
take -51 +7263x2/' -22a^b^+55c^x%
14. From &r«+7x<-9x»-14ca:+15x2/-9
take -x2/+34x»+17x*+19c2;-100x2.
16. From inx^+.5ax^-.12xy^+73x
take 9.6aa;«+1.42x?/2 -15.4x22/ -1.8x.
16. From 5.9r2+3.4x22/2- 17.51/22+60
take -72+13.21/22 -5.9x22/2 -11.2r2.
17. From .61x2-9062x»-186x2+c*
take 1.61x2+8c*+10.562x'- 196x2.
3 11 3 11
18. From -5»--ar2--6x2 take jr6x2 -— s' -— ar«.
4 o o J lb o2
13 5
19. From — x*-— x»+j^-82/2+9cx«+6
3 3 3 11
take -ex* - 15+ j^ -^2 _ 2/+x».
20. From 24ar» -86c -42/2+ 16x2/+32
take 40 -xy+ 17ar» -96c+52/2.
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108 TECHNICAL ALGEBRA 71
21. From 12Ax^-31.5y^+ax+hx*+cy^
take - lQ,5y^+ax -6x« -cy« - 172.3x«.
22. From 32.34r«-17.25s»-12.84(«+1.9
take 3.42s^-7,9i^-d.79+15,Sr\
23. From 16.63m2~12.8n+42.54r«»2+7.4
take 1 1 .09m+ 12.7 -3.7m«+ l9.7r*vK
24. From l-L^-^V^'-^R^+iSW*
o 10 d4
take 15.9TF»+:^L«-7«+^i2«.
16 64
26. From 3.72x»-9.07x«+13.29x+41.6
take 5.18x2-4x5+12.8 -40.1a;.
26. From .7625-.9x*- 16.43+ 12.62y*+ 17.8
take -42.2+41.042/2+1.3x*-12.022».
27. From 10d*w;*-.llm»r+.78p5*-.9
take mm^'-.15dHD*+5pg*+5.09,
28. From 16.05x2 -.9^2+2x2/ -3x?/2+4.8
take -12.92xy+ 15.55x2 -3x2/.2+8.l2/2+7.92.
29. From 10.96c2-19.1acx2+32.76x»+6
take -9.26x3+1.846c2x-18.2acx2.
30. From -76.4^2x2 - 9.03A;x +4.752/2 -17^
take 54.82/2+l6.282/+15.5/i2a;2+ii.04ikx.
31. From |+^x»-|-x2»+4.1x22/«
take ——=r-x»+.25x2» -8.39x22/2.
lo 7J
32. From -12.18ar2-11.0862/2+.34x2/-9x*
take 9.17x*+5.92ar2 -.69X2/+1.9862/2.
33. From 1.08»2s2-4.19s22/2+8.84-7.296c«2_9.i
take -3.86 - .3956c<2 _ 297s^^+ Mv^sK
34. From dy/b^-c^+Wr^-s^-A^yz+y-M
take aV62-c2 - dVr^-s*+ .03 \^x+y - .96.
36. From 1 take 5y* -82/2+42/ and add the result to 8+5y Sy^+y*.
36. From l+3Vx+3x+Vx» take l-3Vx"+3x-Vxr
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73 THE FOUR FUNDAMENTAL OPERATIONS 109
37. From a+bVl-c* take a-6Vl-c«.
38. Subtract hy/x+y—ay/x-y from aVx+2/+6V x -y.
72. Meaning of a Minus Parenthesis. A minus pre-
ceding a parenthesis indicates that each term within the
parenthesis is to be subtracted from the quantity preceding
the parenthesis.
8a— (6a;+!/— 6a) is the mathematical way of stating
that 6x+2/— 6a is to be subtracted from 8a. In this expres-
sion, therefore, 8a is the minuend and 6a;+!/— 6a is the sub-
trahend.
By a minus parenthesis indicate that 3r— 5d+6 is to be
subtracted from 2r^,
Indicate the subtraction of 5ar+6y from 2x\ indicate the
subtraction of 7x2 _ le + 2y'^ from Sx^ _ g _ 42^2^
73. Examples in Removal of Parenthesis. The follow-
ing examples in removal of a negative parenthesis are merely
examples in subtraction and are therefore numbered con-
secutively with the preceding list.
39. 7aa;+46-c-8-(19aa;+46-c+2).
Copy this example in the work-book.
The minus sign preceding the parenthesis indicates that the
terms inclosed are what part of an example in subtraction?
Each of the inclosed terms is therefore subject to what law?
Therefore the parenthesis can be removed only by making
what change?
Rewrite the example with this change and add the terms.
40. 7aa;+46-c-8-(19ax+46-*c+2).
Copy this example. Write it with the parenthesis removed
but with the vinculum retained.
Write the result with the vinculum removed.
In the filial expression, which of the inclosed terms have signs
different from what they had in the example?
* The sign of c is -f- understood, the minus being the sign of the
culum only.
vinculum only,
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110
TECHNICAL ALGEBRA
73
Which of the inclosed terms have signs the same as they had in
the example?
The terms whose signs were changed, were effected by how many
minuses?
The terms whose signs were not changed were effected by how
many minuses?
Rewrite the example with the vinculum removed and the paren-
thesis retained.
Write the result with the parenthesis removed.
Is the final result the same as the result obtained when you
removed the parenthesis first and then the vinculum or is it
different?
41. IQxy -5x^+6r -5+t-{7x^ -z^ -ir).
Remove the marks of parenthesis by the two methods employed
in example 40, first removing from without inward, then from
within outward.
Are the results the same or different?
42. Rule the following table in India ink in the work-book
and make the entries indicated.
SOLUTION OF EXAMPLES 39 AND 40
Examples.
1
2
Number of
Minuses Affect-
ing Terms in
Column I.
Number of
Minuses Affect-
ing Terms in
Column 2.
Terms Having
Signs Changed.
Terms Having
Signs Unchanged.
39
40
In examples 39 and 40 were the signs of terms changed when
affected by an even number of minus signs or when affected by
an odd number?
All marks of parenthesis, therefore, may be removed in one
operation provided the signs are changed of all terms affected by
what number of minus signs?
Under a heading Law for Removal of Parenthesis, write a law
for the removal of negative parentheses.
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73 THE FOUR FUNDAMENTAL OPERATIONS 111
Remove the marks of parenthesis from the following examples
by one operation, applying the law regarding terms affected by
an odd or an even number of minuses, and then add the terms.
43. x+y-(2x+y) + {x-y)-(3x-2y+m).
44. 3y-(x2+32/)-2(3x-2/+8)-(x-^).
46. 6-(a+c)+a(c+6)-(a+c).
46. 5(a-6-c)-(5a+6+c)-3(x-y+2)-[3-(x+y-l)].
47. 2d-(3d-2e+/)-2d+5-(c-6d+3/).
48. 4a-{2a+c-l5a-(iS-b-c+d+30)]].
49. |--|a-(36+c)-|--(!;+s-?f^)J.
61. -4-{-5-[-2-(-6+5c+2a-8)l}+3-[6-(-7a)].
1
62. ix-— -Ih+Bc-iSb+s-Sa+c)]
8 4
53. ---^-b-j^-lb+c-(9c+ab-S)]
64. 3x-.2-(3a+2/-.2x)-|.08-^^-^j •.
66. 22/+5(x-32/+4)-[-2-(-x-22/-3)].
67. -3x-2(^-^+24)-(-3+2x).
58. -2-{ -a;-[3+x-(3x+8-2x+8)]}.
69. 5a+b{c-x+d-3+x+d)^[5a-{bc-bx+d)l
1
«.-i-
+5a:+l -[-5a:-3(a:-4x-3+22/)]
61. -(2x+y-3+j/-2a;)+5j/-{-2-I3-(6x+5)]}.
62. 6x»+3x-(8y+2V)-[32-5(x»-3)(5-x)]+82/+a;V.
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112 TECHNICAL ALGEBRA 73
63 - • ......2-6C 4+7c
66. 3x*-6(x8+5x«-8x+3) -(3a;*+9-6x8-2x«+7x).
66. - ( -2a+6 - -3a+5) -6x+7 -a(2 ~x -6+2).
3
67. — - 2a-[6-(2c-12+8a),-26] -4c
68. ^-j -2a:-[3x-(4a-c--x-2a)]-3a;-(2c+l)j.
^3 . 5x
^- -{7+-6-2-8a;)-{-3-[-2x-(6+5x-2+3x)]}.
70. 2a;-(3x-22/)-[-l+6x-(-2x-oj;4-5-d)+2x].
71. 5ax^-{-2ax^+c-[-Zax^-(2c+16-2ax^-7b-c-2)]},
72. -24^2- {3<-t3+7<*-[-2^+8-(3<8+6^*- -^24.^3)]}.
73. 3a;*-{ -3x2+x»^(-x« -x* -2x^+5) -8x- (9x2 -x)}.
74. -9y-{-y-l-y-(-y-y-y^-Sy-'-y+y^)]}.
76. 26^-(62+36-63-6-62)-{-6»-[-62-(62+6)]}.
77. -5(^-^-2+45^) -{-4j,-[-3y-(2v+9)]}.
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77 THE FOUR FUNDAMENTAL OPERATIONS 113
§3. MULTIPLICATION
74. Definition. Multiplication is the process of finding
the product of algebraic expressions.
Although not necessary it is a convenience to arrange
the terms of both multiplicand and multiplier in the order
of the ascending or descending powers of the same letter.
The first term of the multiplier is written under the first
term of the multiplicand and the others follow without
regard to like or unlike.
Each term of the multiplicand is multiplied in succession
by the terms of the multipUer, and the like terms of the
resulting products are written in the same vertical column
in order that they may be added for the total product.
76. Laws of Multiplication.
(1) Law of Signs: Like signs, plus; unlike signs, minus.
(2) Law of CoeflScients : Multiply numerical coefficients.
(3) Law of Letters: Bring down letters.
(4) Law of Exponents: Add exponents of the same
letters.
76. Illustration.
Multiply ir3+|r2+2r-i
by ir-2
|r*+ir3+|r2- uV (1st product)
- T^-JT^-^r+i (2d product)
ir4-|r3-fr2-4^V+i (total product).
77. ExplanatioYi. The first term of the first product is
|r*, obtained as follows:
The sign is plus; Like signs, plus.
The coefficient is J; Multiply numerical coefficients.
The letter is r; Bring down letter.
The exponent of r is 4; Add exponents of the same
letters.
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114
TECHNICAL ALGEBRA
78
Each of the terms may be explained in the same way.
Observe that before multiplying, the terms of both
expressions were arranged in descending order.
78. Examples in Multiplication. In the following exam-
ples enter the work as in the illustration given in paragraph
76 but do not name the products.
1. Multiply 2a+3b by 3a.
2. Multiply 3x^+x+Q by 2x+3.
3. Multiply 5x^-^^+x by x*+Sx.
4. Multiply ax^-2bx-dO by x*-xK
6. Multiply Qx*-5x+x*-Sx^ by 2x^-5x.
6. Multiply 2d2-3c»+8 by 6d-5c+l.
7. Multiply 2x^-3y^+x-2y^ by 2x2-4.
8. Multiply x+y by x+y,
9. Multiply x-y by x-y,
10. Multiply x+y by x-y.
11. Multiply 26c«+5e'+3a by 36c -8.
12. Multiply 3ax«-26x+24 by 5a*+10.
13. Multiply %«-16+3y by 6y^+2y+3.
14. Multiply 7x^-2x^+ixy^''Sy* by 2x-3y«.
16. Multiply 24x»+15x«-7x-8 by 5x-l.
16. Multiply 8x*-llx»-3x*+10x+6 by 7x-4.
17. Multiply 96*-156»+126«-26+6 by Gx^+x.
18. Multiply 7x*-5x+4x»+x2 by 5x«-2x.
19. Multiply 15bx-12cy+S by 2x-y-l.
20. Multiply 30d«--9c2-ll/« by 2d*-e*+Sf.
21. Multiply 2a+36+5c«+ll by 3a -6 -2c.
22. Multiply Zr^+2s-6v^ by 2r^+2s-6v*,
23. Multiply l2x+4y+9 by l2x+iy+9.
24. Multiply 8x»-52/+32/«-6y» by 2x«-j/«.
25. Multiply 3acx-2by+y* by 2ac-3j/-y.
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THE FOUR FUNDAMENTAL OPERATIONS 115
26. Multiply 2x»-9+8a;2-12x by Q-Sz+SzK
27. Multiply 2x'-3x*+4a;'; by Tx^-S.
28. Multiply 5ac2+126"-6c"-^ by 8a' -4.
29. Multiply 32z^-7z^-9z-6 by 2z^+4z-l.
30. Multiply x^-^+x^-^+x" by x"-^-5.
31. Multiply h'-'^-h^'+h by h'-2+h.
32. Multiply a^+a^-a^+a by a*.
33. Multiply x^-'+y*-3y-^+8y^-^' by x'-t/'.
34. Multiply 31x*-3ax2-5x~^-8 by 36x-7x«.
35. Multiply 21x*-20x'-^"'-15x"-H8x3 by 4x-*-2x-x-\
36. Multiply 5V^-aVy-hV7-9 by 2Vx-cVy+U,
37. Multiply 3y/x+y-5Vx-y-7Vx^-y* by Vx+y-Vx-y.
38. Multiply ax^+ay^'+l by ax^'-ay^'-l.
§4. DIVISION
79. Definition and Arrangement. Division is the process
of finding how many times one algebraic expression is con-
tained in another. It is the process of finding by what
multiplier one of two expressions must be multiplied to
give the other.
To perform division the terms of dividend and divisor
must be arranged in the order of the ascending or descend-
ing powers of the same letter. The divisor should be written
at the right of the dividend and the quotient should be
written under the divisor. This position of divisor and
quotient is an advantage both in writing the divisor and in
multiplying it by the successive terms of the quotient.
80. Laws of Division.
(1) Arrange terms of both dividend and divisor according
to the ascending or the descending powers of the same letter,
(2) Law of Signs: Like signs, plus; unlike signs, minus.
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116 TECHNICAL ALGEBRA 81
(3) Law of Coefficients: Divide numerical coefficients.
(4) Law of Letters: Bring down letters.
(5) Law of Exponents: Subtract the exponents of
letters in the divisor from the exponents of same letters in
the dividend.
81. Illustration. Divide 23x2-48+6x*-2x-31a:3 by
6+3x2«5a.
' 6x*-31a?+23ar^- 2x-48 (3rc2-5a:+6 (divisor)
-6x^=Fl0x^=i=12x2 2x2-7x-8 (quotient)
-21x3+lla:2« 2x
( -21x3+35r2-42a:
-24x2+40x-48
-24x2+40x-48
82. Explanation. 3x2 ^^^ gj^^ term of the divisor is
contained in 6ar* the first term of the dividend, 2x^ times,
obtained as follows:
The sign is plus; like signs, plus.
The coefficient is 2; divide numerical coefficients.
The letter is a:; bring down letters.
The exponent is 2; exponent of x in dividend minus
exponent of x in divisor.
Multiplying (all the terms of) the divisor by 2x^ we
obtain the first product.
Subtracting we obtain the first remainder.
3rc2 is contained in— 21x3, — 7a: times, obtained as follows:
The sign is minus; unlike signs minus.
The coefficient is 7; divide coefficients
The letter is x; bring down letters.
The exponent is 1 (understood); exponent of x in
dividend (first remainder) minus exponent of x in divisor.
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83 THE FOUR FUNDAMENTAL OPERATIONS 117
83. Examples in Division.
1. Divide 4x* -16x*+23z -20 by 2a; -5.
2. Divide 54x»+18a;«-6x-20 by 3a; -2,
3. Divide 3a;»+7ar«-2x+12 by 2a;+6.
4. Divide x*+3x*y+3xy^+y^ by x+y.
6. Divide 58a+7a» -21 -24a* by 7a -3.
6. Divide 21a;»-26a;»-12a;+32-51a;«+22x* by 3a;+4.
7. Divide 15a;*-32x»+50a;«-32a;+15 by 3a;«+5-4a;.
8. Divide x^—y* by x-y.
9. Divide x*+2xy+y* by x+y.
10. Divide x*-2xy+y* by x—y.
11. Divide x*+y* by x+y,
12. Divide h^—g* by h—g.
13. Divide 23a;«-48+6a;*-2a;-31a;» by 6+3a;«-5a;.
14. Divide a*— 6* by a— 6.
16. Divide a*— 6* by a+6.
16. Divide a«-&« by a+b.
17. Divide a«— &" by a— 6.
18. Divide a8-6« by a -6.
19. Divide a*— 6* by a+b.
20. Divide a^+fe* by a+6.
21. Divide a'^+b'^ by a+6.
22. Divide a^+¥ by a -6.
23. Divide a^+6^ by a -6.
24. Divide 625a* -816* by 5a+36.
26. Divide a+y by x^+yK
26. Divide x—y by x^—yK
27. Divide x8-816» by a;«-36.
28. Divide 2x*-xy*+6x^^'-Sx^+Qy* by x*-'2xy+3y\
29. Divide r7-3r«+l+2r»+7<-2r«+6r*+l-4r
by r«+l-r-r»+r«.
30. Divide 60a;+9x*+24-67a;« by a;-6+a;«.
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118 TECHNICAL ALGEBRA 84
31. Divide ^h^"" -h+21h!' -2bh^'' by 2A»-5.
32. Divide ^+2t*-7<*-l+4<-e*+4/» by t-\+t\
33. Divide a«+6« -6a«6+15a*6« -6a!>«+15a«6* --20o»6».
by a3-3a26+3a6»-6».
o^ T^ -^ 9 . 7 3 , ,4 , 16 , 3 , 8
34. Divide — x*—x^ — x'H — x-\ — bv —x*—x — .
16 4 4 3 9 -^ 2 3
or T^- ,1 « . 5 . 13 , 1 , 37 17 , 7 ^ ,.12
36. Divide a«+-a^--a»--a«+-a«-^+- by a^+^^-a.
36. Divide A:«-5ifc«-24A;»+15A;*+27A;»-13A;+5
by A;*-2A;+H-4A;2-2A;».
2 8 3 132
37. Divide -x^+-x^^+--xy^--—x^y^-7:^
o o 10 7o
2 14
by -x^+-x^--xHj.
38. Divide c'+^+c^A;+cA;^+Af +^ by d+l^.
39. Divide c2''--s2''+2sV-c2' by d'+s'-d.
40. Divide x«~l - x^-3a;2 by -2x^-x+x^-l.
41. Divide l+4«34.3^4 by {t-\-\)\
42. Divide r'^-l by r-"-r-'+r-5-l.
43. Divide a2*-62"+26V-c2'' by a'+U'-if.
44. Divide TF''+^+TF^F-Try^-r+^ by W-V,
46. Divide a^^+^-p^"^ by dF'^.
46. Divide dr—dir—diR by dii2.
47. Divide (a+6)3+l by (a+b) + h
84. Summary of Laws of the Fundamental Operations.
I. Addition.
(1) Write like terms in the same vertical column.
(2) Find excess of plus or minus in each column
II. Subtraction.
(1) Write like terms in the same vertical coliunn.
(2) Law of subtrahends:
(a) Change signs of all terms in the subtrahend.
(6) Add.
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84 THE FOUR FUNDAMENTAL OPERATIONS 119
Removal of Parenthesis.
Law: All marks of parenthesis may be removed from
any algebraic expression, provided:
(1) The signs are changed, of all terms affected by
an odd number of minus signs.
(2) The signs are not changed, of all terms aflfected
by an even number of minus signs.
(3) Other operations indicated by parenthesis, are
performed.
Abridged form of law: odd, change: even, don't.
in. Multiplication.
(1) Law of signs: like signs, plus; imlike, minus.
(2) Law of coefficients: multiply numerical coef-
ficients.
(3) Law of letters: bring down letters.
(4) Law of exponents: add exponents.
IV. Division.
(1) Arrange terms of both dividend and divisor
according to the ascending or the descending
powers of the same letter.
(2) Law of signs: like signs, plus; unlike, minus.
(3) Law of coefficients: divide numerical coefficients.
(4) Law of letters: bring down letters.
(5) Law of exponents: subtract exponents of letters
in the divisor from the exponents of the same
letters in the dividend.
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CHAPTER V
LAWS OF NUMBERS
Section 1, The Square op the Sum op Two Numbers. Section
2, The Square op the Difference of Two Numbers.
Section 3, The Product of the Sum and the Difference
of Two Numbers. Section 4, The Difference op Two
Cubes. Section 5, The Sum of Two Cubes. Section 6,
The Square of any Polynomial. Section 7, The Exact
Divisor of a Polynomial.
§ 1. THE SQUARE OF THE SUM OF TWO NUMBERS
85. Law. In (a+fe)^, what is denoted by the exponent
2?
Perform this operation.
Calling a the first and b the second, write the law for the
square of the sum of two numbers as determined by your
multiplication.
86. Examples for Sight Work. Copy the following
examples and by inspection write what each equals.
1. (r+a)\ 2. (3x+4y)\
3. (a6'+^')«. 4. (12+7s)K
5. (a^+9x-^)K 6. (5c^+Qb-^)\
7. (7r-'+9^)\ 8. (ll»«-i+12w;i-«)2.
d^'+r-O'
9. U"2/'+3-^"M • 10- (xV+13a-26-2)2.
11. (x-''+y-^-')\
12. Uxy+h^x'-^^y.
120
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88
T.AWS OF NUMBERS
13.
(c+sin x)'.
14. (6+4cosx)».
16.
(|B-+7iC-.)'.
16. (.8D-«+.05£?»-»)».
17.
(a;'-8+y»+»)«.
18. (0f-i+A'+2)..
19.
(a;+Ax*)«.
20. KL+U)+iP+Pi)]'
121
§ 2. THE SQUARE OF THE DIFFERENCE OF TWO
NUMBERS
87. Law. Perform the operation denoted by (a— 6)^.
How does the result differ from that obtained by squar-
ing (a+6)?
Write the law for the square of the difference of two
quantities.
88. Examples for Sight Work. Copy the following
examples and by inspection write what each equals.
1. (r-s)«. 2. (3x-42/)2.
3. (ab^-y^y. 4. (12-7s)«.
6. (a^-Qx-O*. 6. (8c*-116-2)«.
7. (Qr-^-Ms*)'. 8. {&tf''-12h^-^')K
9. (o^^-t^-m'- 10. (a363-20a-»6)2.
11. ((i-'-/-V-*)'.
12. (sab-j^^r+^y.
13. (3m-2sina;)«. 14. (46-5 cos a;)«.
16. (Ui-^-^isLnxj. 16. (.07a;-^~.62/«+2)2.
17. (x+2-.y-«-3)2. 18. (C^-i-6t+2)2.
19. (x-Ax)«. 20. [(L+Li)-(P+P01*.
* Read delta-x. A is the Greek capital D. Ax denotes one quantity
and its square is therefore Ax* and not A^x*.
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122 TECHNICAL ALGEBRA
§ 3. THE PRODUCT OF THE SUM AND THE DIF-
FERENCE OF TWO NUMBERS
89. Law. Perform the operation denoted by (a+b) (a — 6) .
Write the law.
90. Examples for Sight Work. Copy the following
examples and by inspection write what each equals.
1. {x+y){x-y). 2. (2r+3s)(2r-3s).
3. {ab+y)(ah-y). 4. (13+6r)(13-6r).
6. (8s-2+10r3)(8s-»-10r»). 6. (5aa;2+4y-»)(5aa:2-4y-»).
9. (x-«+32/»)(a;-*-32/»). 10. (5V'+4T^)i5V'-AT^).
11. (A;'*"Hp')(A;'*""^-p'). 12. (56+sin2a;)(56-sin2a:).
13. (26+ COS* x) (26 -cos8 x). 14. (46+tan x) (46 -tan x).
16. (.02x-»+.52/2)(.02x-»-.52/2). 16. (Aa;+.2A2/)(Ax-.2A2/).
17. (.8a2t;-3+.09i/)(.8a*z;-3_.o%).
18. (x^-*+15r-^y){x^y'^-15r-^).
19. (a;-"+3o2i/-i)(a:-"-3o2i/-i).
20. (^'"'+|s"«)(6--2_|,-sy
21. [{c^+d^)+(e'+PMc'+d') -(e'+/^)].
§ 4. THE DIFFERENCE OF TWO CUBES
91. Law. In the examples in division when you divided
the difference of two cubes by the difference of the roots,
what was the result?
Write the law.
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92
LAWS OF NUMBEES
123
92. Examples for Sight Work. Copy the following
examples in the work-book, followed by an equality sign,
and write the results by inspection.
x-y
3437»-8s«
4972+i4s7+4s2*
27<»~125
^- 3«-5 •
64^ 72r
4^"9^
9.
11.
13.
16.
17.
64d8n-27y-»
4dn*-32/-i *
g
27 ^
2
m— rTn^p*""*
1331g»-1728e»
12122+ 13262+ 1446»'
x-y
^^- ^.^
4.
6.
8.
10.
12.
14.
16.
18.
20.
64 -L»
4-L*
8-64L»
4+8L+16L2'
125ig»-2167»
25i2«+30i2F+36F«'
fe»r»-8t;--«
^2r«+2An;-«+4t>-«*
8a»6--»-512y»
2a6-i-82/ •
125<-»-216m;"
64a;»-125y-»
16a;2+20a;y-»+25y-«'
(3r+2s)8-(4p)»
(3r+2s)-4!; '
{x+yy-(a+hy
,(x+t/)-(a+6)-
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124
TECHNICAL ALGEBRA
93
§ 6. THE SUM OF TWO CUBES
93. Law. Refer to your work in division and write
the law.
94. Examples for Sight Work. Copy these examples in
the work-book, followed by an equality sign, and write
the results by inspection.
1.
6.
x+y'
216A»+343a;»
• 36A«-42/ix+49a;«*
27x^+125y*
9.
11.
13.
16.
17.
19.
3x+5y '
2162"+ 1
62^+1 •
216d-»+27e»
36(i-«-18d-»e+9e«'
64d»a;+8a;»y»
4dx^+2xy '
d»+17282»
d+12z '
8a;+64y
2x*+42/**
(5a;+2y)»+2»
(5x+22/)+2*
(x+yy+{v+zy
(x+y)+(v+z) •
2.
6.
8r»+278»
2r+3« •
512a«+7296»
8a»+96» •
64a-»+8fe«
4a-i+26«*
C+d
10.
12.
14.
16.
18.
20.
1728/ft+64r-«
144/g*-48i2«a;-«+16a;-**
g^-gh+h^'
512e-»+27g«
8e-i+32« *
r+{t+v)'
x+y
<^+</y'
(a+b)+{r+s)
\^a+h+\^r+s
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96 LAWS OF NUMBERS 125
§ 6. THE SQUARE OF ANY POLYNOMIAL
96. Law. Perform the operation indicated by
(2n+36+4a:3_ 52/2)2,
In the result do you have the sum of the squares of each
of the terms?
Do you also have plus twice the product of each term
into each which successively follow it?
Square any other polynomial and see if the result cor-
responds.
Write the law.
96. Examples for Sight Work. By application of the
law expand* the following:
1. {a+h'-2c+d)K 2. (e+Sf-Ag-h^.
3. 0*-^-5m-fi+4)«. 4. (3p-5g-2f«-9)«.
6. (V7+2y-4z«-l)«. 6. (3ax^+2y-5by+l)K
X -8-I6-2) •
9. (10-llx-12a;»-13a;»~14a;*)». 10. (4m+2r-3s-2t;~8)«.
11. (y-"+32/'*-*+92/'*-62/-'»-0». 12. (7a;*-6a;»-2a:«-5a;-9)«.
13. (\/d-V4f-3x»+7x~8Vx)*. 14. iax+hx+cx+ay+dy)\
16. (-3y*-62/-7y»-2y»-3)«. 16. {ax^+br^-2cx-d*-'9)K
17. (V3x-V^+3a-V46)« 18. (a;'»-a:»'»-2a;*'»-a;'*+i)«.
♦ Expand means to raise to the indicated power.
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126 TECHNICAL ALGEBRA 97
§ 7. THE EXACT DIVISOR OF A POLYNOMIAL
97. The Factor Law. What value of x will make
3(x— 7) equal to zero?
If any factor of a product is zero what does the product
equal?
What value of x will make a:— 4 equal to zero?
If a:— 4 is a factor of any expression will the expression
equal zero when x— 4 equals zero?
In (a:— 3)(aj — 6) name two values of x which will make
the expression equal to zero.
What value substituted for x will reduce to zero, every
expression in which a:— 2 is a factor?
An expression having a:— 9 as a factor will reduce to zero
when what is substituted for a:?
An expression having a: — 12 as a factor will reduce to
zero when what is substituted for. a:?
Therefore an expression having a:— a as a factor will
reduce to zero when what is substituted for a:?
Therefore if any rational integral expression containing
a:, does not reduce to zero when a is substituted for x, is it
divisible by a:— a?
Law. Any rational, integral expression in x, which reduces
to zero when x equals a, is exactly divisible by x -a.
98. Illustration. (1) To determine an exact divisor or
factor of 6x^ — 6a:^+l we substitute 1 for x in the ex-
pression.
Since it reduces to zero we know that it is exactly divisible
by a: — 1.
(2) To determine an exact divisor of x^+Gaj^+llar+G
we substitute for x, some factor of 6.
But since the signs are all positive it is evident that no
positive factor of 6 will reduce the expression to zero. We
therefore try —3 instead of +3 and obtain the following:
-27+54-33+6 = 0.
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99
LAWS OF NUMBERS
127
The expression is therefore exactly divisible by a:— (—3),
which by the law of subtrahends becomes x+S,
(3) To determine the exact divisor of x^— Tx^+lGx — 12
we substitute for z some factor of 12, say 3. When this is
substituted for x we have 27-63+48-12 = 0.
Therefore x— 3 is an exact divisor of the expression.
When the known term has several factors several trials
must sometimes be made before the correct factor is deter-
mined.
99. Examples. Determine an exact divisor or factor
of each of the following expressions:
1. x*+Qz+5.
3. a:»-43x+42.
6. 36a;«-61x+25.
7. x*+x^-20.
9. 22/»-%+39.
11. y»-13y-12.
13. x»-21x+20.
16. x»-10a;2+29a;-20.
17. 3A»-6A«-3A+6.
19. «<-252«+60«-36.
2. a;»-3x*+4.
4. 4c«-7x+3.
6. 16a:«+49x-60.
8. a:»-19a:+30.
10. 6a;8+7a;+13
12. 3y*-7y^-20.
14. a;»-14a;«+35a;-22.
16. r»+9r«+26r+24.
18. s*+s»-16s«-4s+48.
20. 4x*-19x»-37x«+44r-12.
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CHAPTER VI
FACTORING
Section 1, A Common Factor. Section 2, Grouping. Section
3, The Difference of Two Squares. Section 4, The
Difference of Two Cubes. Section 5, The Sum of Two
Cubes. Section 6, The Trinomial. Section 7, The
Polynomial. Section 8, Special Expressions.
100. Definitions. The factors of a number or expression
are the quantities whose product equals the number or
expression.
Factor means maker and factors are numbers which
make other numbers when multiplied together.
Factoring is the process of resolving an expression into
its factors. To resolve an expression into its factors is to
determine its factors.
101. Cases of Factoring. This subject will be presented
under the following cases:
Case 1. A common factor.
Case 2. Grouping.
Case 3. The difference of two squares.
Case 4. The difference of two cubes.
Case 5. The sum of two cubes.
Case 6. The trinomial.
Case 7. The polynomial.
Case 8. Special expressions.
128
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108 FACTORING 129
§1. A COMMON FACTOR
102. Illustration. In the expression
25x*+15x3«30x2-40x+100
what is the largest factor common to all the terms?
Write this factor before a parenthesis within which is
the result obtained by dividing the expression by this
common factor.
What is the largest factor which is evenly divisible in
Express the factors in the same form as in the first
illustration.
103. Examples. Factor the following expressions:
1. 3a«c-27a<c»+9a»c»-15a^^
[ 2. 5ax^—Sx*+cx\ In the result what is the coefficient of x'?
Why?
3.phy*-fh^^+ph%
4. Sz—hz+cx—tx. In the result what is the coefficient of x?
Why?
6. ^^^^^^^+Wu 6. ^+Wt.
7. Ci+Cin-2Ci. 8. ^x^-VIixy^—rx^yK
9. U^-ai^+U^--t^+tK 10. 8x«-ax«-7x»-(c+d)x«.
11. ay^-by^-Ay^+dy^-yK 12. 8a:«-16x«+24x-8.
13. acx*-hx^+dx^-x\ 14. -7 ^+^.
t Of 7t*
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130 TECHNICAL ALGEBRA 104
7s 2s» 6rs» ^ ^ az
17. C^-^+a-CK 18. ^-8.^..
7t;» 16r
125t;Ht;»
19. 5cH«' — - — — -. 20. 3rsma-6isma-12»sma.
8ar*
21. il tan ^+ila tan S+SA^ tan d.
22. 34oa;*-156a;*-rx«-x«.
23. 8a62cA~46«A+1662^»m-1246».
24. C0w«6a;-72m*62+126«m«.
26. 8tana+16tan«a-12tan»a.
26. a cos* ^ -3 cos 6+r cos^ 6.
27. 5V X —y —awx —y+h\/x — y.
28. 26V4a+«+3dV4a+« -7/V4a+i.
29. x^Vcos ^-1 -i/^Vcos ^-1 -Vcos d-1.
30. x«(x~t/)*--2xy(a;-2/)«+2/*(aJ-y)*.
31. 12x«-172&r»-156x+x*.
3a6x» _ 9a«&» ' 27a6»y
4t; 16 "^ 8x *
-_ Sfi 25fiX . ^^_
34r2^_51r5^_68r^*
' a6c 86c a6c '
9 5c
104. Removal of Common Factors. In the examples
in the subsequent cases and in all work involving factoring,
common factors if present should be removed before the attempt
is made to factor under other cases.
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106 PACTOKtNG 131
§2. GROUPING
106. Definition and Illustration. In conventional lan-
guage, grouping is the rearrangement of the terms of an
expression so that terms having common factors follow
each other, the terms which have a common factor being
inclosed (grouped) in parentheses. This makes errors
possible both in copying the expression and in changing
the signs of terms which are to be grouped in a negative
parenthesis. By the use of grouping linos as shown in
the illustration which follows, copying and inclosure and
therefore the mistakes mentioned, are avoided.
Factor ax — ay+cy — cx,
ax—ay+cy—cx =
a{x-y)-c{x-y) =
{a-c){x-y).
106. Explanation. In the preceding illustration the
first term and the second term have the common factor a;
therefore we group them.
The third term and the fourth term have a common
factor ~c; therefore we group them.
a is contained in the first term x times and in the second
term —y times. Therefore the factors of the first and the
second term are a and x—y.
The factor common to the third and to the fourth term
is — c; contained in the third term x times and in the fourth
term —y times. Therefore the factors of the third and the
fourth term are —c and x—y.
Adding the coefficients of x—y we obtain {a—c){x — y).
Observe that — c instead of c was used as a factor of the
second group because the factoring of the first group shows
us that x— 2/ is a required factor of the second group. — c
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132 TECHNICAL ALGEBRA 107
gives this by laws of division, while +c would give — x+y.
This means that in factoring by grouping, the signs of all
common factors after the first are determined by the first paren-
thesis and in no other way.
107. Another Solution. The expression in paragraph
105 may also be grouped and factored as follows:
ax—ay+cy--cx =
x{a'-c)—y{a—c) =
(a-c){x-y).
The first and last are grouped because they have x as
a common factor. The second and third are grouped be-
cause they have — y as a common factor.
X is contained in the first group, a—c times; — y is con-
tained in the second group, a—c times.
The factors are therefore a—c and x—y.
108. Examples. In the following indicate the grouping
by light grouping lines, and factor:
1. ac—as—rs+rc, 2. vx+yz—vy—xz.
3. v^x^-yz-v^+xH. 4. KL+KiLi-KLi-KiL,
5. gH^+wf-ft^-ghJO, 6. 2e^+^h^-'^eh-'2eh,
7. 4a5«+5a62-5a26-465«. «. ^0g''r^-2ay+\0ag'y-^rK
9. 1 Ic^r Vx -45a2c'p«r< - 15a«c2c'a;+33p*r<t? V.
10. * ax —csy+rx —ry+csx —ay,
11. 6ac-x2-3cx+2ax.. 12. 8x«-rc-2rx+4cx.
13. 2/'+2/*+2/+l. 14. cz'^-cz-cyz+cy+z-l.
15. ac-t-a-3c2-3c-4c-4. 16. pg-p-6g+6-<5+<.
17. 2a(i«-dV'+/+2d/-4ad-2a. 18. a{x+y)-h{x+y).
19. {s+c)k+{8+c)v. . 20. 7r-6r+6s-7s.
* Group by threes as follows : ax—csy-\-rx—ry-{-c8x—ay.
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Ill FACTORING 133
21. hx+l-h^x. 22. 12+182/ -72y«-48y«.
23. 2l2«+2+2+422«. 24. fx -bx -kx -kx^ -bx^+fxK
25. t{v-d)-(d-v). 26. 3(x-2/)«-9(2/-x).
27. x^+2xy+y^+cx-x-y+cy. 28. 6»+6«-36-3.
29. 1-^+i-^i. 30. 5-5t-t^-{-tK
§ 3. THE DIFFERENCE OF TWO SQUARES
109. Illustration. Perform the operation indicated by
{x+y){x-y).
By what law of numbers could you determine the prod-
uct without the necessity of multiplication?
Therefore the difference of two squares will always
resolve into what two factors?
Therefore
^2— c^ = (a-|-c)(a— c).
110. Explanation, a^—c^ is the difference of two squares.
It is therefore resolvable into two factors; one, the sum
of the roots, the other, the difference of the roots.
The first root is o, the second is c.
The sum is a+c; their difference a—c; therefore the
factors are a+c and a—c,
111. Examples. By inspection write the factors of
the following:
1. a»-6*. ^2. c»-d*. 3. r*-s»
4. C2-M 5. D^-EK 6. A*-/«.
7. V^-TK 8. V^-RK 9. W^-X^.
10. 64x8-81y-». 11. 2/*-2». 12. g^-bH^
13. 4x«-92/». 14. 25««-49r2s«. 15. x-y.
16. m^-n*. 17. s«-r«». 18. h^-k-K
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134 TECHNICAL ALGEBRA 112
19. z-^-y*. 20. 32/«-162»».
21. (x-y)«-(r+s)«. 22. (3t^+40»-(2c+6)«.
23. (x»-2/)«-(a+6-c)«. 24. 24(r-e«)«-24(A-3A;)«.
26. (3d+2e»)*-(7r*-5/)«. 26. (x+2/)«-s«.
27. 9x^'-3QaH*-2ixz+lQzK 28. r«-4s«+«*-2re.
29. l-x^+2xy-yK 30. a*+2a«6«+6«-a«6».
31. * D^+F^ -E^ -m+2DF-2EH.
32. *a2+2a6+62-(c«+2ce+e2).
33. *SF5+4-12i«J-52+9i«J«-16F«.
34. *2t;M;-t(;«+a;«+2/'-t^*+2xt/.
§ 4. THE DIFFERENCE OF TWO CUBES
112. Illustration. Divide t?—]^ by x—y. By what
law of numbers could you have written the quotient by
inspection?
Therefore the difference of two cubes may always be
resolved into what two factors?
Therefore 8C3-64r3 = (2C-4r)(4C2+8Cr+16T2).
113. Explanation. 803-647^ is the difference of two
cubes.
It is therefore resolvable into two factors: one, the
difference of the roots; the other, the sum of the squares
of the roots plus their product.
The first root is 2C; the second is 47.
Their difference is 2C— 47; the sum of their squares
plus their product is 4C2+8Cr+16T2.
' Therefore the factors are 2C-4r and ^C^+^CT+l&T^.
* Group the terms by threes in two parentheses so as to show the
difference of two squares.
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116 FACTORING 135
114. Examples. By inspection write the factors of the
following:
1. c*-dK 2. 8y»-27s».
3. x^-SzK 4. 64^»-r».
5. L»-125D«. 6. y3_2162-».
7. Sp-k-^ 8. 343a»-6- »*.
9. l-xK 10. 8y»-l.
11. l-1728r-»s». 12. 8a;»y"-272/*.
13. sin» a - cos» a. 14. 125 tan» 6 -343 sin» 6.
15. x'-l. 16. 27g-*-h\
17. l-64A;«-«. 18. A^'-B"
19. (p+g)*-(s-0*. 20. l-(l+2/)».
21. 64 cos» ^ -8 sin» e, 22. (3a+z)^ - 1.
23. (56«-6y)»-r«. 24. (x-y)» -(««-»«)».
26. C-"-(L2+8M«)«. 26. (7a6c+l)»-64d-».
27. (29ey-30)«-125^-«'. 28. x^-iSy^-lU^y^^
29. (k^+Ugy-Sr-^^. 30. 1 -(2x-32/+52)«.
31. l-(6a;«-a2-y)-«. 32. (116A-12B»+1)'-1.
33. 64 tan» 6 -343 cot» d. 34. (sin* d+ cos« ^)» -1.
35. 8(sin« <t>+ cos* <^)» -512 tan» <t>.
§ 5. THE SUM OF TWO CUBES
116. Illustration. Divide a^+l^ by a+b. By what
law of numbers could you have written the quotient by
inspection?
Therefore the sum of two cubes may always be resolved
into what two factors?
Therefore 27J53+125Z)3 = (3J5+5Z))(9J52-15J5Z)+25Z)2).
116. Explanation. 275^ + 1251)3 is the sum of two cubes.
It is therefore resolved into two factors: one, the sum
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136
TECHNICAL ALGEBRA
117
of the roots; the other, the sum of the squares of the roots,
minus their product.
The first root is SB; the second is 5D.
Their sum is 3J5+5D; the sum of their squares minus
their product is 9J52-15J5Z)+25Z)2.
Therefore the factors of 27B^+125I>^ are ZB+5D and
9B2-15J5Z)+25Z)2.
117. Examples. By
the following:
1. x^+y\
3. R*+S\
5. L»+F».
7. Sx'+l.
9. MA*+B*.
11. 125C-»+8i)-».
13. aY+^"d"".
15. x^+2lQz-».
17. l+2/».
19. 1728^»+729F-«.
21. (5 sin a— cos a) *+l.
23. (7a26»c+6d)»+(a;+y)«.
26. (V»-y)»+l.
inspection write the factors of
2. l + (l+2/)».
4. l+Mh^-^^\
6. (x+2/)»+(^+A)«.
8. (e-/)»+((7+/i)».
10. (6r+l)»+l.
12. l+(2ax+6-6«)».
14. (A;2-6^«)»+8A-»3^+«'.
16. (sm2^+cos*^)»+l.
18. 343+8(sin2 ^)».
20. {A^+B^)^+{E*^F*)K
22. l+(V2^+36)^
24. a-«6-9+(a-6+c2)3.
26. a;3+3a;22/+3a;i/2+2/3+l.
§ 6. THE TRINOMIAL
118. Illustration. Perform the operation denoted by
(a+3)(a-4).
The product has how many terms and therefore is
called what?
Each factor has how many terms and therefore is called
what?
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119 FACTORING 137
The first term of the product is the product of what
terms of the factors? (Answer first terms or last terms
as may be correct.)
The last term of the product is the product of what
terms of the factors?
The second term of the product is the product of the
inner terms plus the product of the outer terms as follows:
(a+3Xa-4).
The product of the inner terms is 3a
The product of the outer terms is— 4a
The sum of the products is — a which is what
term of the trinomial?
Therefore (a+3)(a-4) =what?
119. How a Trinomial is Factored. In the preceding
illustration a trinomial resulted from the multiplication of
two binomials.
The trinomials under this case of factoring are all re-
solvable into two binomial factors which may be determined
by the inverse process of that just given.
For example, L^ — 15L+56 is factored as follows:
This is a trinomial because it has three terms.
It is therefore resolvable into two binomial factors.
We therefore make two parentheses each large enough
to inclose a binomial.
This gives L2-15L+56 = ( )( ).
The first terms of these binomials are two numbers
whose product is L^, therefore L and L.
This gives L2-15L+56 = (L )(L ).
The last terms of these binomials are two numbers whose
product is 56 and the algebraic sum of whose products
when multiplied by the first terms of the binomials, is — 15L.
The last terms are therefore 7 and 8.
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138 TECHNICAL ALGEBRA 120
This gives L2-15L+56 = (L 8)(L 7).
The product of the inner terms (L 8)(L 7) is 8L.
The product of the outer terms (L 8)(L 7) is 7L.
But their sum must be — 15L. ^^*
Therefore both must be negative and the two minuses
will give +56 in the product.
Therefore the factors are L— 8 and L— 7.
This gives L2-15L+56 = (L-8)(L-7).
The order and method of operation here given will
factor any trinomial under this case.
120. Pairing of Terms. In a trinomial like 6+7Q-5Q2
care must be taken that the terms are properly paired and
the signs properly placed.
The factors of this trinomial are (3+5Q)(2— Q).
But in trying to determine them we might have written
(3-5Q)(2+Q), or (3+Q)(2-5Q), or different factors for
6 or different signs, none of which, however, would give
6+7Q-5Q2.
121. Special Instructions. In factoring a trinomial
observe first whether all terms have a common factor;
if they have, remove it and then factor as a trinomial.
Observe second whether the trinomial is a square. If
it is, it may be factored on sight by the special laws for the
square of the sum or the square of the difference of two num-
bers, more quickly than by the general method here given.
For instance ij^^—^rsx+a^ is the square of 2rs—x and
the factors {2rs^x){2rs—x) may be immediately written.
In a trinomial like L^— 15L+56, after the factors of
L2 are written, the problem is merely to determine two
numbers whose product is 56 and whose sum is —15.
This is always the case when the coefficient of the first
or the last term of the trinomial is unity.
122. Examples. Enter the following in the work-
'"ook with an equality sign and two parentheses, each large
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123
FACTORING
139
enough to inclose a binomial. If there is a common factor
place it before the first parenthesis.
1. z*+Sz+2.
3. 2S^+i2S+lSO.
5. A2+54A+729.
7. 9y'-2Wxy+mx.
9. 6r2+3r-45.
11. 4a;+4\/^+l.
13. 2p+l7f+35.
15. 12aA;«+69aA;+45a.
17. 9«2'-2-36r-V+36i
19. 7a2-14a5+76*.
21. 54A2-15A-56.
23. 3662-836d+35d«.
25. 21a^b*+20abd-9MK
27. 24a*x^-39aH'x-99i^
29. 15.T«+224x-15.
31. z*+it+V)z+tV.
^v
2. t?«-llt^+30.
4. c*+9c«f«+14<«.
6. c*-22c«d+105(i«.
8. 3W^+nW+6.
10. 12L2-68L+40.
12. 2y«+3a;2/-2a;«.
14. a:^+31x'-32.
16. 156«-776+10.
18. 9a;«+43a;-10.
20. 60a«62+300a6c+375c«.
22. xV -7x2/ -120.
24. 10R^+19RV -l^VK
26. 35x^+34x^2 -1442«.
28. «.+|a+^.
30. 44(7/i-20A«+15(7«.
32. (x+2/)«+2(x+2/)-15.
§ 7. THE POLYNOMIAL
123. Explanation. Since a poljrmonial by definition is
an algebraic expression of three or more terms and there-
fore includes the trinomial, it is necessary to state that the
poljTiomials which are considered under this case are of the
same type as those in paragraph 99. They are factored
by the factor law as given in paragraph 97. When one factor
has been determined by this law the second factor is deter-
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140 TECHNICAL ALGEBRA 124
mined by division. If this second factor is not prime
it is often factorable as a trinomial or by some other case
of factoring. Should it prove troublesome it may be read-
ily factored by the factor law.
124. Examples. For this list of examples use those
in paragraph 99. Copy each in the work-book and deter-
mine the factors.
§ 8. SPECIAL EXPRESSIONS
126. Sum or Difference of Same Power. The cases
presented in the previous sections include those that are
essential in the subsequent study of mathematics. Some-
times, however, one is asked to factor the sum or difference
of the same odd powers or the difference of the same even
powers when the odd powers are not cubes nor the even
powers squares. In such instance it is only necessary to
remember the following in which a and b denote any munbers
and n denotes an integer:
1. a**— 6** has the factor a—b whether n is odd or even.
2. a^—b^ has the factor a+b when n is even,
3. a^+b^ has the factor a+b when n is odd,
4. a^+b^ has not the factor a+6 or a— 6 when n is even.
In other words,
(1) The difference of the same even powers is divisible
by the difference of the roots and by the sum of the
roots.
(2) The difference of the same odd powers is divisible
by the difference of the roots and not by the sum.
(3) The sum of the same odd powers is divisible by the
sum of the roots and not by the difference.
(4) The sum of the same even powers is divisible by
neither the sum of the roots nor the difference of
the roots.
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126
FACTORING
141
Under the paragraph number and heading copy the fpl-
lowing and complete the entries, specifying whether the fac-
tor is a+b or a— b or both or neither, and give reason:
No.
Example.
Factor.
Reason.
No.
Example.
Factor.
Reason.
1
a^-b^
6
x^^y*
2
a^i-¥
7
x'^-y'^
3
a»-6»
8
x'^+y'
4
a»-h6«
9
R^-S^
5
x*-y*
10
/e«+/S«
126. Special Methods. Some expressions are made
easily factorable by the following methods:
(1) An expression may sometimes be factored by first
adding and subtracting the same quantity as in the case of
If the middle term were 2x^y^ the expression would be
factorable. We therefore add x^y^ and indicate the sub-
traction of sc^y^ as follows:
(a^+2a:2^+t/*)-xV.
The expression may now be factored as the difference
of two squares.
In the case of 4:a'^ — 21a^hf^+9b^ we add and denote the
subtraction of Qa^fr^ as follows:
(4a* - 12a262+9&4) -9a^1y^
when the expression is seen to be the difference of two squares.
Factor the following:
1. 4^:4-13x2+9,
2. Sly^-3^a^y^+y^.
3. 121R^-UR^S^+S\
4. x^+7^y^+y^.
5. c(^+^.
6. (^h^-2l(?h^+SQ.
7. 36eH49.
8. xio+642/2.
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142 TECHNICAL ALGEBRA 126
(2) The factoring of expressions like sfi—y^ is simplified
by observing that x^—y^ = {3i:!^)^ — {y^)^.
Write the factors.
(3) A polynomial which is a square may be factored
by the inverse of the law for the square of any polynomial.
For example
4x2+16x2/--402/z+162/2-20a;z+2522
is seen to be a square because three of its six terms are
squares and each of the other three terms is plus twice the
product obtained by multiplying each root into each of
those which follow.
Thus
4r2+162/2+2522+l6a:2/-2Oa:2-4O2/0 = (2a;+42/-52)2.
This inverse law * makes it possible to extract by inspec-
tion the square root of any algebraic expression which is
a square (usually called a perfect square by way of emphasis).
Determine the factors of the following:
1. 3ex^+z^+25y^+l2xz+myz+mxy.
2. (P+e^+^P+h^+2ed+Mf+2dh+^ef+2eh+^fh.
3. 9fc2+64m2+36n4-48fcm-36fcn2+96mn2.
4. 121p8+81gi«+49r6+16s4+198pV-56r3s2-126(/V
- 154p4r3+88pV+72g5^.
5. a^a^+10axz+2al^xy+25z^+l0b^yz+b^y^.
6. Mcdyz+Q^c^z^ - 2^d7?y+^7^ - A&cx^z+ IGcPi/^.
7. 100x2^i50a:2/3_|.35a65-4_96^^5-22^_^54^6__i20a36"2a;.
8. 25L2+169F2+150LS-4+130LF+225>S-8+390F>S-4.
9. 196a:2-252a:2/+8l2/2.
* See paragraph 95.
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126 FACTORING 143
10. 324^-756r-3^2+441r-^
11. 576x^-15S&a?y^+102^y^.
12. 9fc2+25j2+49m^+81g2+30fc;-42fcw2-54Jfcg-7Q/m2
-90ig+126m2g.
(4) A polynomial like 16y^-^0yz+25z^+32xy -4:0x2
+153? which at first sight might be thought to be a square,
may be factored as a trinomial by the foUowing^ arrange-
ment:
(42/-52)2+8x(4?/-52)+15a:2^ (Write the factors.)
(5) An expression like 32a^^-b^^ if written (2a3)5-(62)6
may be factored as the difference of the same odd i>owers
of two numbers.
Write the factors.
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CHAPTER VTI
FRACTIONS
Section 1, Reduction. Section 2, Addition and Subtraction.
Section 3, Multiplication and Division. Section 4,
The Complex Fraction.
§ 1. REDUCTION
127. Definition. A fraction is an indicated division of
one number or expression by another number or expression,
the first number being placed above the second with a hori-
zontal (fraction) line between.
A fraction may also be denoted by one number following
another with an oblique line between.
The terms of a fraction are the numerator and the
denominator.
The mathematical basis of most of the work of this
chapter is the
Fraction Axiom: If both terms of a fraction are
multiplied or divided by the same quantity the value of the
fraction is unchanged.
Reduction to lowest terms is the division of both nu-
merator and denominator by the same factor until they
can no longer be divided because prime to each other.
Numbers are prime to each other when not evenly
divisible by the same factor.
128. Illustration of Reduction. The fraction ^ » ^ _..,
is reduced to its lowest terms by factoring both numera-
144
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129
FRACTIONS
145
tor and denominator and by canceling the same factors
as follows:
(a;+5)(a;-6)
(x+5)(2x'-S)
129. Examples in Reduction. Reduce the following
fractions to their lowest terms. If cancellation marks
are shown slant them when possible, in the same direc-
tion. The factors may be placed above and below the
numerator and denominator as shown in the illustration,
or written following the given fraction.
Small, carefully made parentheses are desirable.
1.
3.
9.
11.
13.
16.
17.
h+k
4x«-19a;-5
2x*-7a;-15'
2x^+3x-U
l(te»+33x-r
2av+bt+2hv+(U
2cv-2dt+ct-4dv'
6ar~5A;s+15A;r-2fla
21H+s-3r-7st '
5a6+753«-135r-«
90a6-180s* '
Sx*-8y*
6x»+12xy+6y*'
15x»-6a;«-12x
6a«-246x
2.
6.
10.
12.
14.
16.
18.
mx^yY
x^+9x+U
' 3x«+21a;+30'
2x»-27x+13
3x«-37x-26*
6x«+7x~20
12x2+32x+5*
(c-2d)16?
400 '
17(^L»+17dLi»
85L+85Li •
96«+4o«-196r-«-12a6
4a-28r-*-66
15x»-llx«-10x+4
10x«-4x-8 '
2a;3-4y2-4xy+2x«y
ax*-'2ay+bx^+2by'
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146 TECHNICAL ALGEBRA
2a:»+7a;»-7a:-12
130
19<
20
21
22
24.
2x2+5a;-12 *
3ax+2r»-7x+6-6a
3x+eax-Q-12a '
20dr+25r^-30rx+id^ - 12dx+9x*
5r*-8rx+3x*+2dr-2dx
Mx+y)^+Sx+ S y--m
12a;+122/+60 *
2a«V^3-h tan ^-5V ^3+ t an e+h\^3+ tan g
7V^3+ tan ^
2fl(i2-rf2;-4flrf--f+2a+2d/
5/-10a
§ 2. ADDITION AND SUBTRACTION
130. Addition of Integer and Fraction. The algebraic
sum of an integer and a fraction is called a mixed expression.
The process of reduction to simple fractional form is the
same as the process of reducing a mixed number to an
improper fraction. Each is merely the addition of the
integer and the fraction.
5
Thus 3f is only a convenient expression for 3+^ and
o
5
the addition of the 3 to the x- gives the so-called improper
29
fraction -^. The process may be explained in two ways:
8
(1)
Therefore
1=8-
24/5^29
8'*"8 8'
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181 ■ FRACTIONS 147
(2) Every integer may be considered a fraction whose
denominator is 1.
Therefore 3f=3+|=|+|.
3
To reduce j to 8ths the denominator must be multiplied
by 8.
Therefore the numerator must also be multiplied by 8,
the principle involved being the
Fraction Axiom. If both numerator and denominator
are multiplied by the same quantity the value of the frac-
tion is unchanged.
The preceding explanation should make clear the reason
for the following rule of both arithmetic and algebra:
Rule. To reduce a mixed number or expression to
fractional form, that is, to add an integer to a fraction,
multiply the integral part by the denominator, add the
product to the numerator, and write the result over the
denominator.
Thus 3x4
x+5 x+5
. r o 7x-42/ 4a^lP-Uab+6-7x+4y
aft— 3 ao—3
131. Addition of Fractions. In order to add fractions
they must be reduced to the same denominator. That
the result may be as small as pospible, the fractions should
be reduced to the least common denominator which may be
defined as the smallest number in which each denominator
is evenly contained. The simplest way of determining
the least common denominator is to resolve each denomi-
nator into its factors and write each factor as many times
in the least common denominator as it is used in the given
denominator.
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148
Thus
TECHNICAL ALGEBRA
3a+5 a-1
id2
(a+x)(a+x) {a+x){a-ax+z^)
{3a+5)(a-ax+a^)-(a+z)(a-l)
{a+x)(a+z)(a—ax+a^)
As shown, a+x is used twice as a factor in the first
denominator; therefore it must be used twice as a factor in
the L.C.D.
Fractions, and fractions and integers, are therefore
added by determining the L.C.D., multiplying both terms
of each fraction by it, and collecting like terms in the
numerator of the resulting fraction.
132. Examples in Addition and Subtraction. Solve
the following examples:
'-•+¥-¥•
3. 2|/-3+
2y+5
y
5. 3-
7.
x+2y
X''2y
Bx-y
x-S
-3a;+2.
x+2
x'-x-e a:»-2a;-8 3a;
11.
a«-9
a+2
a«+3a o«+6a+9
+a+L
4 4 4 *
IB .J 3 _5_ 4_
^^' a«+l o«-l |l-o«^l+a
2. 3-+a-X-
4. ^J+2-6.
a—
2-t;*
8. x«-6a;+6+
5
3a;«-2
10. -- +'J^±^.
bi-si-sr+hr^ t+r
|/^4|/+7^16i/«-49'
* xy 22/ a:2 2*
16. — -;;
x*+3x 2a;«-2a;
+2.
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134 FRACTIONS 149
X 2a+26 a ^ 2^_1_
' a+h a«-a6+6« a»+6» (a+i;)« a«+r« 2*
^r. a 5X-46 ^ 7a;+26 ^ x 2a;+l . 3
19. ^^-—^2y^-—, 20. iIZ^-3i^+8.
oi o . . ^+y . 1 no 1 -49a;» . 1
21. 2-^+2,+2i^-2^,+— . 22. i^28x+49:c*+l+7i-
3a+6 _1_ ^^ 2^ 3 _21 _5_
^' a^-h^^ia^h)^^"^' '"^ 2^d-2^(d-2)«-
«^ c 1-c. 2c 1
25. 1 .
\+c c c* — 1 1—c
§ 3. MULTIPLICATION AND DIVISION
133. How Fractions are Multiplied. As in arithmetic,
fractions are multiplied by multiplying their numerators
together and multiplying their denominators together.
Before this is done, however, all numerators and denomi-
nators should be factored and all factors canceled as far
as possible.
When one fraction is to be divided by another, the divisor
should be inverted before factoring ,
134. Examples in Multiplication and Division of Frac-
tions. When possible factor both numerator and denomi-
nator of the following fractions; apply the law of divisors
if necessary, and simplify by cancelation:
2r«+3r 4r«-6r
4r» 12r+18'
18t?g -Zav -6a\ ^ !?» -2flt;+fl\ ^ 3a»+2a«
Zav-Za^ 92;«-4a« (2i;+a)(t;-a)*
. ^2-^S--20^/S*~/S-2 . S+l
««-25 ^/S2+25-8 S^+bS\
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150 TECHNICAL ALGEBRA 134
• x*-y* ^\ lh^x^-l¥y^ ldx-20y )'
L«~5L-14 . / 2L»-32L+126 L»-11L+18 \
(L-2)» " \c«L2-4c2L+4c*^ L*-4 /*
^ 3x» ^ / 2x»+2a:« 86x^ \
3+3a;** \ 2a; 46+46a;V*
fa+dy
3+3a;*
-d« 9a+9c
-c» ' 3a2+3ac+3c2'
a;g+a;?/+2/\ ^ 5x»+5y\ ^ 4a«
o. ~~; ; "X I r~X^
X*— xt/+y* x'— 2/' rx+ry
a;+2 ' x^^z^ \ x—y/'
a« -7fl+10 . 2a -4 8a»+64
a2-10o+25 a -5 o«-2a+4'
y.2_5s ' r«-9r+20 r2-10r4-2r
6(c-2d)» c^+5c^d^+4ri< ^ c^-4 c»d +4r2cg''
c(c2+4d')^ 3(i(c-4d2) • c34-8(i3
a»+2a6+&'-c\ ^ a-6+c
13. , 1 . X "
a+b+c a^-2ah-c^'
-- x^-y ^ (x-yy ^/ x^-'y^ x'^- xy+xjA
x*-2xy+y^ \x+y/ ' \x^-{-y^ x^+xy+ijy'
7a+21 o«-4 . 5a2
a+3 14a«+56a+56 3a2-6a-24
x+x^-e 2(x»+2x+4)(x^-x?/+2/^)
2x+2x*-12 x»+2/3
a«+5fl+6 a^-25
5a«+25a 2a2-8*
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136 FRACTIONS 151
^y^-i8y+90 2y'-S Qy^-'30y-{-24:'
xi-yi-zi-2yz ^ 2x^ -2y^+4xz+2z*
* x^+z^-2xz-y^ ' 4x2+82/2-422-42/*'
-(!-)(f-')(^.)-
§ 4. THE COMPLEX FRACTION
136. Definition. A complex fraction is a fraction
having fractions in either numerator or denominator, or
in both.
To simplify a complex fraction means to make it integral
or a simple fraction.
A simple fraction is a fraction whose numerator is
integral and whose denominator is integral; that is, a
fraction both of whose terms are integral.
In simplifying complex fractions application must be
made of the following law.
Law of Divisors:
(1) Invert every divisor.
(2) Use it as a multiplier.
136. Illustration. Given the complex fraction.
a ar
1-^.
This is a complex fraction because both terms contain
fractions.
To simplify it is to make it integral or one simple fraction,
i.e., a fraction whose numerator is integral and whose
denominator is integral.
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152 TECHNICAL ALGEBRA 136
Operations:
(1) Simplify the numerator.
(2) Simplify the denominator.
(3) Apply the law of divisors.
(1) Simplify the numerator.
To simplify the numerator is to make it integral or one
simple fraction.
The numerator is now a mixed expression because
it has both integral and fractional terms.
A mixed expression is simplified by adding its terms.
This is done by reducing them to a common denomi-
nator.
This is effected by multiplying the integral terms by
the least common denominator, adding the nimaerators,
and writing the result over the least common denominator.
The least common denominator is a^.
Multiplying 1 by a^ we have a^.
Adding to this the numerators we have a^— 2a6+&^.
Writing this over the denominator we have ^ •
The numerator is now simple because it is one simple
fraction.
(2) Simplify the denominiEitor.
To simplify the denominator is to make it integral or
one simple fraction.
The denominator is now a mixed expression.
Multiplying the integral part 1 by the denominator a^
and adding the numerator — 6^, we have a^—V^,
a^—}p
Writing it over the denominator we have — g — •
The denominator is now simple because it is one simple
fraction.
(3) Apply the law of divisors.
Invert every fractional divisor and use it as a multiplier
of that which it divided.
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137
The divisor is - o
FRACTIONS
153
Inverting it and factoring we
have
(a-6)(a-6)
a~&
(a+6)(d-6).
137. Examples. Simplify the following:
1.
7.
9.
c* c
24
2 ^h
36-9
6
11. 1+-
l+r+
2r«
1-r
L-1
+1
1-
L •
3.
5. d+
1-L
d
--ax
d+
1
4e
^+3
R-4+
8.
10.
R-
/g-f 1 . /g' -3/^+2
6 * i_^+5 •
72-1
1-
JB*-1
1
a;-?/
12.
1 . 1
x+-
1
y+
1 !/
13. -4^X?'-^'
6 a
a'+b'
(p«+f«-g»)«
nX ■
14.
4<»
{p+t)'-y'
(p-t+v)*
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154 TECHNICAL ALGEBRA 1S7
, 2zy S-2
15. ^I(i^. S-2-^'
17. '
^ h*-l ' ^„ 3d-2 3d+2
* T "• i '
20 I C^\^^CZA^(AC-1\
\A'C'-A*C^I\ ,24 /\AC+1/
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CHAPTER VIII
THE QUADRATIC EQUATION
Section 1, Introduction. Section 2, Solution by Factoring.
Section 3, Solution by Completing the Square. Sec-
tion 4, Equations in Quadratic Form.
§1. INTRODUCTION
138. Definition. A quadratic equation is an equation
having the second power of the unknown quantity but no
higher power when cleared of fractions and reduced to its
simplest form.
3x2+8a; = 10 is a quadratic because it contains the
second power of the unknown.
5x^—3i?+2x = S although having the second power of
the unknown, is not a quadratic because it has a higher
power also.
— r-^+x— 1 = 16 in its fractional form has only the
x-\-o
first power of the unknown and might seem not to be a
quadratic. When cleared of fractions, however, and
collected, it has a second power of the unknown and is
therefore a quadratic.
139. Classification. There are two classes of quadratics:
(1) Incomplete, or those having the second power
only.
(2) Complete, or those having both first and second
powers of the unknown quantity.
155
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156 TECHNICAL ALGEBRA 140
The names applied to these classes show that the second
power of the unknown constitutes a quadratic and that a
quadratic is regarded complete only when having the
first as well as the second power. When the first is want-
ing the quadratic is incomplete.
Quadratics of the first class are sometimes called pure;
those of the second class, adfected or affected.
140. Essentials to Solution. Two things are essential
to the solution of any equation:
(1) Its recognition as simple, simple fractional, or
quadratic.
(2) Knowledge of the operations by means of which
the value of unknown quantities can be found.
Observe that it is the highest power of the unknown
quantity which determines whether an equation shall be
called simple, quadratic, or cubic, etc., the powers of the
known quantities having nothing to do with it. One's
first thought, therefore, when an equation is to be solved,
should be what kind of an equation is it, and this is deter-
mined from the exponents of the unknown quantity.
141. Methods of Solution. (1) Incomplete Quadratics.
If a quadratic has only the second power of the unknown,
the value of that unknown is determined by the extraction
of the square root after solution for the second power.
In other words, an incomplete quadratic is solved by finding
the value of the square of the second power of the unknown
quantity, followed by extraction of the square root.
Thus 5x2-18 = 107
X=*i:5.
(2) Complete Quadratics. When an equation contains
both the first and the second power, solution may be effected
* Every square root must be written with a double sign.
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142 THE QUADRATIC EQUATION 157
either by transposing all terms to the first member and
factoring; or by transposing all known terms to the second
member and completing the square. The latter method
is employed when factoring is impossible or when the
factors are hard to determine and should not be used
under any other conditions.
There are therefore two methods of solving a complete
quadr&tic:
(1) Solution by factoring.
(2) Solution by completing the square.
§ 2. SOLUTION BY FACTORING
142. Illustration. To solve 5x2+6a:=32 by factoring
we transpose all terms to the first member, which gives
5x2+6a:-32=0
The first member is now a trinomial; therefore if fac-
torable it may be resolved into two binomial factors.
Therefore we have (5x+16)(x~2)=0.
Dividing both members of the equation by the first
factor 6a:+16 we have
a:-2=0,
.-. x=2.
Dividing both members by the second factor x— 2 we
have:
5x+16 =
16
x=2 or — =-.
5
Check: 6x2+6a:=32
Substituting 2 for x, 5 • 22+6 • 2 = 32
20+12=32.
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158 TECHNICAL ALGEBRA U3
1 n
Substituting — — for x,
o
i-WH-f)--
5
32 = 32.
Following is a complete model for the solution of a quad-
ratic by factoring:*
(a) 5x2+6x=32
(1) 53^+ex-d2=
—Trans 32.
(2) (5x+16)(a;-2)= 0—
Factor 1st member (1).
(3) x= 2—
-— Div (2) by 5a;+16 and trans.
(4) 5x+16=
— Div (2) by a; -2.
(5) x=-^-
In (4) trans 16 and Div by 5,
(6) .-. 1=2 or ■
-y— (3) and (5).
143. Short Method of Determination of Roots. If
the values of x in (6) are compared with the factors in (2)
it will be seen that x in (6) equals the second terms of the
binomials in (2) with the signs changed, divided by the
coefficient of x in each binomial.
For example, in (6) x = 2.
But —2 is the second term of the second binomial in (2).
Therefore x equals the second term -^2 with the sign
changed, divided by 1 which is the coefficient of x in that
binomial.
Therefore x = 2.
1 f\
X also equals — =-, 16 being the second term of the first
o
binomial in (2) and 6 the coefficient of x in that binomial.
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144 THE QUADRATIC EQUATION 159
— 1 A
With the sign of 16 changed this gives x=—=— which
o
1 /»
by the law of signs in division gives — =-.
5
Observe that when the numerator is negative it is written
as positive, and the minus sign is given to the fraction.
To solve a quadratic therefore by factoring, transpose
all terms to the first member.
If that member resolves into two binomial factors,
X (the unknown) will equal the second terms of these
binomials with the signs changed, divided by the coefiicients
of X,
It is therefore unnecessary to show the solution in the
long form of paragraph 142. Instead, use the form shown
in the following illustrations:
1. 3x2+121 =44a:
3x2-44x4-121=0
(3x-ll)(x-ll)=0
x = ll or—-.
3
2. 7r2-7cr = 84c2
7r2-7cr--84c2 =
r2-cr-12c2 =
(r-4c)(r+3c)=0
r = 4c or —3c.
3. 2v^-av-ab=-2bv
2v^'-av+2bv'-ab =
2v(v+b)-a(v+b)=0
{2v-a){v+b)=0
a
«;=— or —6.
2
144. Instructions for Solution by Factoring.
1. If the equation is fractional, clear it of fractions by
multiplying by the least common denominator.
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160
TECHNICAL ALGEBRA
146
2. Transpose all terms to the first member, arranged
in the order of the descending powers of the unknown quan-
tity, followed by the known term or terms.
3. If all terms have a common factor divide the equa-
tion by it.
4. Factor the first member.
6. Write the values of the unknown.
145. Examples. Solve the following by factoring.
Number all equations but do not specify operations.
1. a;«-12a:+30=3.
3. iS«- 169 =24/5 -144.
6. 5d«-6d-16=4d«-2d-4.
7. Gt^-6t+9=5t^+l.
9. |-|-42f=-20i.
11.
a.-^-5=16i.
4
13 2_5^^15
5 2i 4i«'
15. w^+2bw-2b =w (w unknown).
16. s*— cs+es— cc=0 (s unknown).
17. L2+r+(r+l)L=0 (L unknown).
18. k^ -2ek =k -2e (k unknown.)
19. P+-=a-\-- (P unknown).
a r
20. — |-T7= — hr (M unknown).
T M r
21. D«+eD+6D+6e=0 (D unknown).
2cF 2c^
22. F^-cF 1 =0 (/^unknown).
a a
ab
23. a+h=H+-zz (^unknown).
a
2. L«+16L-17=0.
4. 2F«-2F-9 = F«-1.
6. 4Q2+19Q-25=3Q«-4(?+25.
8. 25h^+^+?0h-Sl=9h*+ih.
10. |D«-|D-7f = -7.
12. y«-|t/+14i=16.
, . 5 3^+1 1 , 11
z z^ 4: 42*
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146 THE QUADRATIC EQUATION 161
24. Zb;+-J- =^ (Ax unknown). 26. 9'+^-^ =0.
26. /•+! +|-=0. 27. z«-(n+l)z= -n.
27 a*6*
30. 52/(2/-3)=j-122/. 31. -^ — a«=6«-x.
§ 3. SOLUTION BY COMPLETING THE SQUARE
146. How the Sqtiare is Completed. What must be
added to a^+2ab as a third term in order to make it a
trinomial square?
In a^+2ab what is the coefficient of a?
What is half the coefficient?
What is the square of half the coefficient?
Is this the term which must be added to make a^+2ab
a trinomial square?
What quantity must be added to a^—2ab to make it a
square?
Is this quantity the square of half the coefficient of a?
In the equation a;2_i2x=— 20 what is the coefficient
ofx?
What is half the coefficient of x?
What is the square of half the coefficient of x?
Indicate the addition of this quantity to the first member
as a third term.
Is the first member now a square? Why?
Was it a square in the original equation?
The square (of the first member) was completed by
adding what?
Completing the square therefore means to do what?
If a quantity is added to the first member of an equation
what must be added to the second member in order to
preserve equality?
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-Add Ax.
162 TECHNICAL ALGEBRA 147
Indicate the addition to the second member and collect
the terms in that member.
Extract the square root of the resulting equation.
Solve.
In the solution of a quadratic, do not fail to place the
double sign ± before the square root of the second member.
This will give two values of the unknown quantity; one
from the positive root, the other from the negative root.
Always show in full the operations by which the two roots
are obtained.
Thus: (a) x2-18x = 144
(1) x2-18x+81 = 144+81 = 225 Add Ax.
(2) a: - 9 = =t 15 ^Root Ax.
(3) a: = 9+15 = 24
(4) x=9-15=-6j
147. Instructions for Solving by Completing the Square.
1. Transpose:
(a) All unknowns to the first member, arranged in
the order of the descending powers of the
unknown.
(6) All knowns to the second member.
2. Collect the coefficients:
(a) Of the terms containing the second power of
the unknown number.
(6) Of the terms containing the first power of the
unknown number,
(c) Of the known terms when possible.
3. Divide the equation by the coefficient of the second
power of the unknown with its sign.
4. Indicate the addition to both members, of the square
of half the coefficient of the term containing the first power
of the unknown.
5. Combine* the terms in the second member.
* Very important in solution of conventional equations; usually
not done in solution of formulas.
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148 THE QUADRATIC EQUATION 163
6. Extract the square root of both members.
7. Solve.
8. Reduce the result to the simplest form.
148. Examples. Solve the following by completing the
square. When an equation in conventional symbols has
more than one letter, it is understood that the last letters
of the alphabet denote unknown quantities.
1. 5x«-69 = 10a;+7.
5 x^ X x^ 1 *
12 12' 6 6 4*
3. -^ = 10-3x.
. 20 4(x+5)
K "il ^ 1 9i
3x +24 12
2x«+ 2 "*2x"
*-5*-4x._2'2*-
'■'-1-^4
8. 5x^-l0ax-5^10a.
••f-'-f*
10. llfx-3Jx«+41i=0.
"■ f -f -■
12 -2^+64 -2^'
12. ig+t>4- ^g^.
13. x^-a+ax==z.
14. ax^-c=bx.
16. 2ax^b-'cx\
16. 36a«+4ax«=36aa;.
3a Qx(a-x)
4" 3a-2x'
18. a*-l=2a»x-o«x».
cr
19. (r+s)L2-cL=— -— (L unknown).
20. (e-h)R*-hR=h (/2 unknown).
„ . f« ht a\ab ^^ . .
21. 1 — =— :H — W unknown).
m* c m^ c
22. a;*-2ox = (6-c+a)(&-c-a).
* Solve also by factoring.
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164 TECHNICAL ALGEBRA 148
23. (F-l)«-(3F+8)«-(2F+5)«=0.
24. \/20+H-^«=2^-10.
Square both members to remove the radical.
26. -^=5-2\/F.
Clear this equation of fractions, transpose, arrange in the
order of the descending powers of D, and divide by the
coefficient of D. Complete the square by adding to both
members the square of half the coefficient of \/d,
26. aa:*+6x~c=0.
^ A;+l a+1 ,, , ^
27. —p= =—7=- (A; unknown).
y/k Va
Ao r..3a»x 6a»+a5-26« 6«x
28. a6x«H = —- .
c c^ c
^^ F-a F+a 5aF-3o-2 ,_ , ,
29-* f+^+rra'^^ZpT- (f unknown).
^^ /S«-2n5+2a5-n* 5+2n 1 ,« ,
^- ^^^:^i +^H^^+^«=S=-a (««^«--)-
«• 10 1 14
33.
6(r-2) 2 3(r-l)
8+C 8—C
36. aa;*+cx«=da;«H — — .
a+c
36. 56» = 156 - 1 1 (Determine result to nearest thousandth.)
* Before multiplying to clear of fractions change the signs in the
denominator of the second member. This will change the sign before
the fraction by the law of signs in division.
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148 THE QUADRATIC EQUATION 165
87./*+2a/+6«=26/+2af (/unknown).
38. Lz^^-^-Rx.
39. hite » -r .
g h
40. In example 39 compute the value of z when
W^j, flf=32.2, h^^, and ifc=0.02.
«. •+l_£+i.o.
43. \/6+i+V6^= ^^
5V6+X
44. V=-T-(ri«+rir2+ri*) (n unknown).
46. i2»=a«+6»+2a6cosa.
Solve for cos a.
Note. — ^If further practice in the solution of quadratics is
desired at this point, see Chapter XVII.
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CHAPTER IX
THE FRACTIONAL SIMPLE EQUATION
Section 1, Denominators Numerical. Section 2, Some
Denominators Literal.
§ 1. DENOMINATORS NUMERICAL
149. Definition. A fractional equation is an equation
which contains one or more fractions.
A fractional simple equation is a fractional equation
which has only the first power of the unknown quantity
when cleared of fractions and collected.
150. Solution. A fractional simple equation may be
solved by clearing it of fractions and by collecting and
dividing by the coefficient of the unknown.
It may be cleared of fractions by multiplication by
the least common denominator provided it contains only
simple fractions.
If any of its fractions are complex they should be
simplified before the least common denominator is
determined.
161. Instructions for the Solution of Fractional Equations.
1. Reduce all fractions to their lowest terms.
2. Simplify * all complex fractions.
3. Factor all literal denominators before attempting
to determine the least common denominator.
* It is sometimes better to multiply a complex fractional equa-
tion by the L.C.N.D. of the lower row of denominators.
166
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163 THE FRACTIONAL SIMPLE EQUATION 167
4. When a minus sign precedes a fraction, change the
signs of all terms resulting from the multiplication of the
fraction after canceling its denominator.
5. Never multiply any equation until all like terms
are collected.
6. When an eiquation has both literal and numerical
denominators,
(1) Factor literal denominators if factorable.
(2) Multiply every term of the equation by the
least common denominator of the numerical
denominators, L.C.N.D.
(3) Collect all integral terms.
(4) When possible, divide the equation by the
largest number which will evenly divide both
members.
(5) Multiply by the least conmaon denominator of
the literal denominators.
162. Change of Signs. In the subsequent work attention
is called to the following:
By the law of signs in division,
(1) All signs in either numerator or denominator
may be changed provided the sign before
the fraction is changed.
(2) All signs in both numerator and denominator
may be changed without changing the sign
of the fraction.
153. Examples. Solve the following equations:
z+2 14 3+5a; 5a; 5a; 9 S-x
*• 2 " 9 4 * 2 4 ~4 2 •
3. ^I^+2=a:-^. 4. 55^-?^=3.-14.
4 ' 2 * '2 3
r —
6
^ « 5a; -4 ^ l-2x ^ 2x+7 9a; -8 x-11
6. 2a;— r--=7— — — . 6. — z 7— = — ^r--
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168 TECHNICAL ALGEBRA 153
_ 7x+3 5a; -6 8 -5a; ^ 10a;+3 6a; -7 ,^, ,,
7. -6— ^=-12-- 8. _^— ^=10(x-l).
^ 8a;-15'llx-l 7a;+2 ^^ 5a; -3 9-x 5a; , 19, ,^
9. -3 ^ =-^. 10. — ^=-+-(x-4).
5x+3 3 -4x X ^31 9 -5a;
8 3 "*"2 ~ 2 6 •
7a;+9 3x+l _ 9a;-12 249-9a;
8 7-4 14 •
13 in? x_20_x-12 7 2x-?5±Z-i+i
^^' 2 +3"3 2+2- ^^-^"^ 11 "".2+^-
^, 6a;-4 ^ 18-4a; . ^^ 7a;+9 / 2x-l\ ^
n. '4^-(l-^) =7x. 18. ?^-?^-.4.-14i.
19.
4 \ 9 ./ '4
a; -3 2x -5 41 3a; -8 5a;+6
4 6 60"" 5 15 •
20. 7x+13f-|=|-8f+^.
2(x-8) 3(9 -x) 5(x-ll) ^ 3(x-17)
^^- 3 ~ 4 ~ 6 "^~ 8 •
22. ?5±^-??±Z+io4=0.
7 o o
23. ^(3x-4)+^(5x+3)=43-5x.
24. |(27-2x)=|~(7x-64).
«= = /7 2\ X 3x-(4
« 4x 7x
x-2_^-4^
.05 .0625
-(4-5x)
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158 THE FRACTIONAL SIMPLE EQUATION 169
lOo+ll 12a -13 7 -6a
6 3 ^" 4 •
4 5 4 3 ^
^ .e ..135a: -.225 .36 .09x-.18
30. .15.+-—^—=- —.
2(a;-18) 2(x+10)
33./-^.10|.
'^^ * I + 3 8 ~°-
5L-11 L-1 llL-1
85. ' ~'' '
5 12 '
A;-l 3fe-4 JL^ 6fc+7
o« 3 "^ 5 16 . 2
36.
8 4 ' 32 •
„.,+_L+,(|».j,j._V?_^.
m+3 3m -5
^^ 2 m-2 12 , 1
^- "1 — 4-=-r"+3-
3n-l 2n+l 2n-5 7n~l
^ ^ 4"^3 ^ 3'''8
39. 22 z =20-
«•¥-¥-+¥• "'Vn-SkllM.
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170
TECHNICAL ALGEBRA
153
§2. SOME DENOMINATORS LITERAL
4a;4-3 _ 2x-5 2x~l
' 10 "Sx-l' 5 '
14 "^6x4-2 " 7 '
^^ 9a;+20 4a; -12 x
36 5a: -4 ^4
,^ 17+4/ 7/+26 , f+13
^^ Qx+7 .7a: -13 2x+4
43. — h"
45.
47.
10-/
9 ' 6a:+3 3
3x+2 _ 2x-l X
6 ""3a:-7™2*
7x+16 x-\'S
21
4a:-ll~3*
49.
51.
53.
64.
55.
56.
57.
/+21 ' 7
2a: -4 2a: -1
21
6x+l ^
15 7x-16" 5 •
6a:+7 _ 2x-2 ^ 2x+l
15 Ix-Q" 6 •
10a:+17 12x+2 5a: -4
^^ 4a:+3 . 7x-29 8a:+19
60. — :; — \--
9 •5a:-12
18
6x+7 7a:-13 _ 2a:+4
9 "^ 6a:+3 3 *
18
llx-13
14
llc-13
14 28
4(a:+3) _ 8a:+37
9 " 18
x^+13 3x+5
9 *
13x+7
5a: -25
13a: -16
22a: -75 _
28 "2(3x+7)'
22c -75 _ 13C+7
"2(3c+7)'
_ 7x-29
5X-12"
2
5*
4a: -3
15
a: -3
59.
5 ■ 2a: -15 10
2x+H 2fa:-l x-^
5 ""50a: -10 - 2J '
-liV.
9x+5 8a:-7 _ 36x+15 lOj
^' 14 "^6^+2 56 "^14*
' See instruction 6, paragraph 151.
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153 THE FRACTIONAL SIMPLE EQUATION 171
2x+8^ _ 13a; -2 x Jx _ x-\-l^
9 17x-32"^3 "12 36 •
%-bx 7-2x^ _ l+3x lOx-ll 1
• 15 Uix-l)" 21 30 "^105'
18^-22 i + 16x 101 -64x
39 -6x ^ ^ 24 "^ 24
2d+8^ 13d -2 d _7d rf+16
• 9 17d-32'*"3 "12 36 '
65 6-5x 7-2a:« l+3x _ 2x-2i 1
' 15 14(a;-l)" 21 6 "^105*
66. -^^i:Ii+2x+y^=4A.i?i::55.
13-12x^ ^ 24 ^ 3
67. (a;+3)-?^=7x
.|3..fcM)'
0.25i~1.5 _2 1 , 2 _x+2
* 0.15C;-5) 1.5' 2"^a:+2 2x '
70. (x+l)«=x[6-(l-x)]-2. 71. ^ ^ ^
x-2 a;4-2 a:«-4'
„« x+1 x-1 2 „^ 1 1 x+1 ^
72. — ■ = . 73. — ' — =0.
x-1 x+1 x^-1 x-2 x+2 X2-4
3a; 2x _ 2x«-5 7 _ 6a;+l _3( l+2a;»)
2a;+3 2x-3"4a;2-9' a;-l"a;+l a;2-l '
^^ 3 x+1 x^ „.T 4 , 7 37
76. 7 —r^^z ;. 77. —77:+-
a;-l a;-l *1— x«' * a;+2 a;+3 x2-|-5a;+6*
„^ Sx+1 x-2 „^ x-7 2a;-15. 1
78. ■ — = . 79. =0.
3(x-2) a;-l a;+7 2a;-6^2a;+14
80. -^Z^^^Z^+l, 81. ,+^J^-2)(x+4)
4(x-l) 6(a;-l)^9 x-1 x+1
2(2x+3) 6 5x+l
2. 1-
9(7 -x) 7-x 4(7 -x)*
^ x«-(x+l) x«+x-l _ ^
X-l X— 1
* See paragraph 152.
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172 TECHNICAL ALGEBRA 153
Sx+1 x-1
86. A;+^ '
86.
87.
* 2x-3 z-2 3x+2"
3x-l 4a: -2 1
2:i;-l Sx-2 6*
1 1 x-1
2(x-3) 3(x-2) (x— 2)(x-3>*
• x+3 3x+9 ' ' x+7 2x-6 2(x+7)*
QA ^±i4.ii=3£±? 91 3a:-l 4x-2_l
Sx+S"^^ 2x+3' 2a:r-l 3x-2 6*
3+x_2+x_l+x^ _J 6_ J_
3-x 2-a; l-x 2x-3 3a;+2"^l-a;'
i
a;-8 2a;-16 24 3a;-24
X + 4:
r.= 1 ^'^ 1-X
5 x-1 3
g-3f 5-4g 3?-i(3-2g)
«, 2 7 1 , _, 2
97. ^ =3-9+7* .
4r 9 lf)r-81 6r-18 4r-9
Oft 5_2_ 6 _ 5 10_ JL
**• 3 6 , .4 9^2 ^4i*
t
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CHAPTER X
SIMULTANEOUS SIMPLE EQUATIONS
Section 1, Two Unknowns. Section 2, Three or More
. Unknowns.
164. Definition. .Simultaneous equations are equations
having the same unknown quantities with the same respect-
ive values.
Simultaneous simple equations, therefore, are simple
equations containing the same two or more unknown
quantities so related that the values of the unknown quan-
tities in one of the equations are the same respectively as
in the other equations.
155. Solution. Generally speaking, solution is possible
only when the number of given equations is the sam£ as the
number of unknown quantities.
For example, any given equation like 5x— Sy = 7 in which
there are two unknowns x and y, cannot be solved. To
make solution possible a second equation must be given in
which X and y have the same respective values as in the first
equation.
If the first equation contained three unknowns solution
would be possible only in case two more equations were
given, each having the same respective values of the
unknowns.
In other words, if there are two unknowns, two equa-
tions are necessary to solution; if there are three unknowns,
three equations are required, etc.
173
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174 TECHNICAL ALGEBRA 156
§ 1. TWO UNKNOWN QUANTITIES
156. Methods of Solution. In order to solve simultane-
ous equations having two unknown quantities, one of the
unknowns must be made to disappear. The process by
which this is accomplished is called elimination.
There are three methods of elimination:
(1) Elimination by addition or subtraction.
(2) Elimination by substitution.
(3) Elimination by comparison.
167. First Method. Elimination by addition or sub-
traction.
Under chapter and paragraph headings enter the fol-
owing example in the work-book:
2x+3y = 17
4x+5y=-31
Number the equations (a) and (6) respectively.
Multiply (a) by 2, giving (1).
Are the coefficients of x in (1) and (6) equal or unequal?
They were made so by what operation?
Are the signs of the x terms like or unlike?
In order to eliminate x must (1) and (6) be added, or
subtracted?
Perform the operation which will eliminate x. Solve
for y.
In either (a) or (6) substitute the value of y and solve
for X,
Why is this method called elimination by addition or
subtraction?
All of the examples which follow in paragraph 160 may be solved
by eliminating one of the unknowns by addition or subtraction.
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159 SIMULTANEOUS SIMPLE EQUATIONS 175
Usually both of the given equations must be multiplied in
order to make the coefficients of one of the unknowns equal, with-
out which elimination is impossible.
Sometimes less work is required by dividing one of the equations
by some number, or by adding or subtracting the given equations,
before multiplying.
Each example must be studied to determine the simplest and
quickest operation which will give equal coefficients of the same
unknown in two different equations.
When there are three or more unknowns solution is effected
by successive eliminations of the same unknown, care being taken
that each elimination involves a different pair of equations.
158. Second Method. Elimination by substitution.
Enter this heading.
Elimination by addition or subtraction is not always the
simplest means of solution. In some instances, especially in
work with formulas, it is better to solve the simpler equation
for one of its unknowns in terms of the other quantities and
to substitute the resulting value in the other given equation.
Solve (a) from paragraph 157 for x in terms of the other
quantities.
In (6) substitute the value of x and solve for y.
In either (a) or (6) substitute the value of y and solve
for X,
Why is this method called elimination by substitution?
169. Third Method. Elimination by comparison. Enter
this heading.
It is sometimes convenient to eliminate by solving
each given equation for the same unknown in terms of the
other quantities and by comparing the results by applying
the equality axiom.
Solve (a) for x or y. Solve (6) for the same quantity.
Apply the equality axiom and solve the resulting equation.
In (a) or (6) substitute the resulting numerical value
and solve.
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176
TECHNICAL ALGEBRA
160
160* Examples. Solve the first five by each of the three
methods; solve the others by any method. Time will be
saved in most examples by indicating the multiplication
which gives equal coeflScients, and not performing it.
1. 5x+y='21.
2x+3y=2i.
2. 3x+7y=27.
5i+2y=16.
8. l(te+9j/=-37;
9x-llj/=24.
4, 3z+4y = 10.
4x+y=9.
6. 7x-3y=67.
4x-10y=88.
6. 3x+y=3.
5x+2y=4,
7. I5x+8y = -3.
Qx -4j/ = -4.
8. 42-% =8.
6x-j/=28.
9. Sx-7y=8l.
10a;-24y=25.
10. 5x-5y=25.
7x+4j/=2.
11. 36x+63y=-Q9.
30x+5% = -60.
12. 7x+2y=31.
3x-4y=23.
13. 8a-c=34.
o+8c=53.
14. 9x+%=9.
2x-3y=24,
IB. 2e-5d=60.
7e-12d=342.
16. 6x-10j/=26.
5x-7j/ = 15.
17. 3h=7L.
12L-5A-1.
18. 2x-3y = -17.
6x+82/=66.
19. 8x-&y=2x+3Q.
llx+32/=80.
20. 12x-4y=84.
17x-19j/=79.
21. 9x-y=4x+Uy.
lZx-7y = U0.
22. 15x-13j/=20-5j/.
19x-8y=36.
23. 5x+7y=U.
3Jx-%=5.
24. 17x+7y = l(Xi.
llx-2y = 19.
26. 19ix-13y=65.
12x-14y=-2.
26. llA;-12y=-207.
-5A;-7y = -155.
27. 6m+3y=78.
12m -5y =2.
28. 17x-21j/=60.
24x-72/=130.
29. 5x-5y=70.
-7x+8y=-95.
30. 8x-9j/ = -31.
12x+2j/=62.
31. 9a: -3% = 111.
8x+13j/=118J.
32. 0+56=34,
7a -66 =33.
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160
SIMULTANEOUS SIMPLE EQUATIONS 177
33. 3c -86 =24.
8c-106=86.
35. 14€-62/=24.
13e+ll2/-188.
37. 39i~62/-36.
18i-9^=0.
39. 13§a;-152/=9.
-24a;+2l2/=-84.
41. 4§a;-3.5n=7.5.
ll|x-8n=32.
43. 23a: -5r =9.
17a;+4r=99.
46. 18.2«-5t;=-14.
29.6«-6t;=22.
47. -122+4x=-32.
-17z+8x = -15.
49. 3fA+4y=98.
12A-lli=-79. .
61. 1000.55 -1002) =49.026.
525+4002) =6.60.
63. |+|=3x-7y-37
x+3 S-y
5 " 4 •
55. 2r-.?±5=7+?i^^
4.^«^^=24i-?^.
10»+8+j|-^+«.
a
»2a6.
^+^-«+*-
34. 100d-18y=2.
33d -52^ = 11.
36. 120A-8y=0.
15^+61/ = 105.
38. 7x+92/ = 122.
101a;-102/=82.
40. 6x-52^ = 13.
lla;-8y=39.
42. 7.2a; -84.4p = 137.8.
5a;+26p = 138.
44. 3.375s+7a:=24.7.
7.75s -2.5a; =27.
46. 225.5m; -81 j2/= -111.
45.1m; -lOy =90.6.
48. 11.875L-9.56=-95.
3.375L+5.26 = 131.
60. 100a:-802=-15.
125:c+160 =39.25.
62. 2000a;+i2002/=46.4.
3000a: -9002/ =-16.8.
64. ?5+?l = l.
X y
56.
20(2+^) =7.
1 _ 11
L+P
L-P
=a.
7+a: 2a:— V
68. ^-^^+d=3y.
5a: -18+
4
4a:-3
5y-7
2©-r „^, 2r-59
60. r -:;::; — =20+-
23 -r
t;-- ^=30-
r-18
2
73 -3t;
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178 TECHNICAL ALGEBRA 161
61. ay -hx=0. 62. x+y=a,
h—y=a—x. hy=x,
63. X cos 70° -y cos 40° =0.* 64. R cos 20° -T cos 45° =2120.
X sin 70° -y sin 40° =2000. R sin 20°+^ sia45° =5520.
66. AR\ sin 25° -AB sin 67° =0.
AR cos 25° -AB cos 67° = 19.4.
§ 2. THREE OR MORE UNKNOWN QUANTITIES
161. How Solved. Simultaneous equations having any
number of unknown quantities, are solved by successive
eliminations of the same unknown by the methods already
described.
For example:
2x+42/+52=19. ^
-3ir+5i/+7z=8. ^
'8a;-3i/+5z = 23. ^
X may be eliminated in (a) and (6), then in (a) and (c),
or (6) and (c). By this means two equations having the
same two unknowns will be obtained.
The equations constituting an example are called a set or
* In equations of this kind, time is saved by indicating^ instead of
performing the multiplication which gives equal coefficients.
Thus to eliminate x the first equation is multiplied by sin 70° and
the second by cos 70°.
Therefore x cos 70° sin 70° -y cos 40° sin 70° =0.
X sin 70° cos 70° -y sin 40° cos 70° = 2000 cos 70°.
Subtracting, the x terms are eliminated because the coefficients
are equal and the signs ahke. It is therefore necessary to perform
the computations in the y terms only.
t AR and AB are unknown.
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162
SIMULTANEOUS SIMPLE EQUATIONS
179
162. Examples. Solve the following:
= -4.
66. 12a;+52/-42=29.
13a;-22/+52=58.
17a; -2/ -2 = 15.
68. a: -32/ -22 = 1.
2a;-32/+52=-19.
dx+2y-z^l2.
70. 3a:-5=2y.
32/ -7 =42.
42=2+5a;.
^^ a -6 c+b 1
g+fe c+fe
4 2
c— a_c— 6
~5~""~6~'
74. 1+^=5.
o; 2/
i-i-6.
2/ 2
2 a;
76. 3a; -22 =2.
42+32/ =41.
5w-72/ = ll.
2u+32=39.
77. i+f = 1.
22/ 4w
22/ 32*
4tA 32*
67. ay+bx=c.
oz+cx=b,
cy+bz =a.
69.A+3^6=443
4a; 2/ 2 24
1 =12a«.
6a; 2/ 2;
2 _^ 1_^85
X 32/2 "27*
72. a;— 03+0^2 =a2/.
x-M=^by-¥z.
c^z=cy—x.
73.^!^±^=.2-2.
2a-Aa;= — r — •
(o+A)i'+j/=oA(z+2+a;).
76.
2 1 3
c
c 3 a'
78.
a 6 2/
bay
y a b
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CHAPTER XI
EXPONENTS
Section 1, A Zero [Exponent. Section 2, A Negative
Exponent. Section 3, A Fractional Exponent.
163. Classification.' In paragraph 24 an exponent was
defined as a number which, when positive and integral,
shows how many times the number affected by it, is used
as a factor.
The appHcation of the latter part of this definition to a
negative or fractional number when written in the position
of an exponent, is meaningless. For example, by the expres-
sion T* we understand that x is used 5 times as a factor.
All attempts fail, however, to interpret expressions like
x^, x"^, or x*, in terms of factor, for the reason that a num-
ber used zero or —5 or i times as a factor is unthinkable.
The four expressions x^, x^, x~^, x*, represent the
Four Kinds of Exponents:
Positive integral)
Zero,
Negative,
Fractional.
The first is already understood; the significance of the
last three will now be determined.
180
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164 EXPONENTS ISl
§1. A ZERO EXPONENT
164. Meamng of A®. If a denotes any quantity whatever,
then aP denotes any quantity with an exponent zero.
The meaning of aP may be illustrated by the operations
of multiplication and division.
First Illustration. By the law of exponents in mul-
tiplication,
but oTxi =ar.
Therefore by the equality axiom,
arXaP=arxh^
Therefore d9=l.
Second Illustration. Under suitable heading write
this illustration in full in the work-book using the same
symbols as in the first illustration but dividing instead of
multiplying.
In working out the illustration number all equations
and specify all operations.
In both illustrations a represented what?
Therefore any quantity with a zero exponent equals
what?
Underline or otherwise emphasize your answer to this
question.
Therefore aP equals wljat?
999^ equals what?
.OOOl® equals what?
25x^y^ equals what?
(34^)2 equals what?
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182 TECHNICAL ALGEBRA 166
165. Zero Exponent Theorem. Any quantity with a zero
exponent equals unity.
Following is another proof of this theorem:
Given any quantity whose exponent is zero,
i.e., Given a^,
Prove aP = l.
(1)
a
Law of exponents in division.
(2)
but -=1 -™^
a
Law of division.
(3)
.-. 00=1-^
""- Equality axiom.
§2. A NEGATIVE EXPONENT
166. Meaning of A"*". If a denotes any quantity and r
denotes any quantity then a"'^ denotes any quantity with any
negative exponent.
The significance of the negative exponent will be dis-
covered by doing the work which follows.
Enter the work in standard form according to page 8,
paragraph 10.
Multiply a"*" by a*".
But aP equals what?
Equate and solve for a""*".
But a represents what?
And r represents what?
Therefore any quantity with any negative exponent is
equal to its reciprocal with the same exponent with what sign?
Letter, underline, or otherwise emphasize the answer.
167. Negative Exponent Theorem. Under this heading
write for the theorem your answer to the last question.
Under it write hypothesis and conclusion in the same
form as in paragraph 165.
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168 EXPONENTS 183
Write the demonstration in the same form also, as
follows:
a^X— equals what?
but (fi equals what?
Finish the demonstration.
168. Use of Negative Exponent Theorem. The fact that
a'
means that an algebraic expression having negative expo-
nents may be written with positive exponents by the appli-
cation of the negative exponent theorem.
' Thus (1) aH-^ = ^.
(2)
(3) 6-i+3a-2 = l+|.
(4)
%x^y-^ ^ 6x2
In like manner
by"
(5) i^ = 7a%-^ry^.
We therefore have the principle that any multiplier
may be written as a divisor and any divisor may be written
as a multiplier, provided the sign of its exponent is changed.
In other words any factor may be transferred from
numerator to denominator or from denominator to numerator,
provided the sign of its exponent is changed.
Thus .829_^829_
^•^"^ 10000 10* ^^^^" •
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184
TECHNICAL ALGEBRA
169
169. Examples.
and zero exponents:
1. a-K
3. tH"K
6. 7d-»/8.
7. 10-^«i>-».
9. ISz-^Vr-^
11. 3a-*c\
13. .00078960.
16. *10-*X75000000.
Express the following without negative
17,
19.
21.
25.
27.
29.
31.
2ar-»+5y-^
72a6"2c4 '
x-y
Acy^z " 2
Ic-Hj-^z^'
* Observe th'
denotes a shift of
of units in the c:
2.
x*y-K
4.
6ax-K
6.
34L«i>-».
8.
1
X-*'
10.
5bo.
12.
^oaH)-*.
14.
98299889799'.
16.
7a— d*/'.
18.
14r-V
20.
b-'+c'
3x-« •
22.
4x^-'
5x'yi+ab-i'
24.
7r»y-»
9r'y-'+4r-^''
26.
d'f-'
3e-»d-'-2a-»'
2r-2.'
171 EXPONENTS 186
33. -; — .^,, . . 34.
36.
1 -150/1-1 * 3am+6s-»»
18a6-\;-id« 8f-»c-7+3<»c-<
9o^»c-* ' 15«2c-i
§ 3. A FRACTIONAL EXPONENT
1^
170. Meaning of A**. Solve the following problems:
1. In the expression (a*)^ the exponent 3 denotes that
a* is used how many times as a factor, by the definition of
a positive integral exponent?
And by the law of exponents in multiplication,
a^Xa^Xa^ equals what?
Therefore a* is one of the three equal factors of a.
Therefore a* is what root of a?
Therefore the cube root of any number may be denoted
by what exponent?
2. Show that the square root may be denoted by expo-
nent ^.
3. Show that the fourth root may be indicated by ex-
ponent J.
4. How may the nth root be denoted by the use of a
fractional exponent?
171. Meaning of bpth Numerator and Denominator
of a Fractional Exponent.
1. In the expression (a*)^ the exponent 2 denotes that
a* is to be used how many times as a factor?
And a^Xa^ equals what?
But a number is squared by using it how many times
as a factor?
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186 TECHNICAL ALGEBRA 172
Therefore in a fractional exponent does the numerator
indicate a power, or a root?
What does the denominator of a fractional exponent
indicate?
2. In X* what does 3 denote? What does 4 denote?
3. In y* what does 1 in the fractional exponent denote?
What does 2 denote?
4. In 2""* what does the miniLS denote?
Write the expression without a negative exponent.
What does 2 denote?
What does 5 denote?
5. In 12 a®6*2/"^ what does the zero exponent denote?
What does the fractional exponent denote? What does
the negative exponent denote?
Write the expression as interpreted by the three laws.
6. What is the value of 64*?
By what law is its value determined?
7. Express x* with a radical. What law did you apply?
8. Express Sr^v^ with a radical.
Did you apply the same law as in problem 7? Why?
9. Express Qx~^y^z~^ without negative and without
fractional exponents.
How do you know that your result is correct?
8*a~^6~*c^ Simplify this fraction in two differ-
8 "^a -^6 ""^c ~^* ent ways :
(1) By applying the law of exponents in division.
(2) By applying the laws of negative and fractional
exponents.
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173 EXPONENTS 187
172. Fractional Exponent Theorem.
Given a".
Prove r denotes the power of a and n denotes the root of a.
W (a ) --a -^a | Law Exps in Mul.
i- n/-
(2) .-. an = Va (1) and Def nth Root.
/o\ XT -i r \\r [Def Pos Integ Exp and
(3) Now a''=(a'*y \ j ^ • ivyr i
^ ' [ Law Exps m Mul.
(4) /. In a ",r denotes
the power and,v.v,^,v I Def s Power and Root.
n denotes the
root. J
173. Examples. Express the following with radicals
and without negative exponents.
1. ax*. 2. {ax)K
3. 26*d*. 4. 5ris*.
6. 8(06)1. 6. ^j^^,
7. -^. . 8.
x-*y-
9. -7=. 10. 6o"*X2a-*.
11. 3a- «x*. 12. 7h-*xK
13. -^^. 14 "*'"*
16. (6c«a-»)^ 16. (6c«a-»)"^
17. 32*. 18. 32-».
19. 56-«c"*X86-i. 20. 21x-^y*Xix-^-K
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188 TECHNICAL ALGEBRA 174
174. Examples. Express the following with fractional
exponents in place of the radicals:
1. Vei^. 2. v^.
3. Vsix-*. 4. X^Sly^z-K
6. </xhi-\ 6. v^o^.
7. Vx^. 8. ^(a;+2/)*.
9. (r->5-»i*)*. 10. y/ia-hy.
13. \^XV^». 14. v^Tofcf+v^i.
17. V^64a;82/-". 18. V^32r-iop-«
19. ^7a*6-*. 20. V5a«6-»+2a-^
21. V^18a;-*+32/-». 22. V^6c*d-*.
23. V^(2a«-36-2)*. 24. V^(9x-*-52/)l
26. V^(15t;«-2«-»)*. 26. v^x-V+^a*.
27. 4V|a;»y-|x-*2/-*. 28. V^(x«-2/«)*+(a-6)V.
29. 3Vl46a;»-5aa;->. 30. v^(r» -«-»)» -2(a«+6»).
31. >J^(x«+2y»)*. 32. V7(4a6«-5a«6-«).
175. Root or Power of a Fraction. In raising a fraction
to any power and in extracting any root of a fraction be sure to
raise both numerator and denominator to the power and to
extract the root of both numerator and denominator.
rpu /3\2 9 , , 9 3
Thus I -r I =77: and not -r or
/3\2^^
U; 16
4 "* 16"
-r and not -tt- or -r-.
4 2 4
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176 EXPONENTS 189
\36 6-
F „ /T 3 • - 1
,-? = 3A/rv or -7= or 3-
15 \15 Vl5 Vl5"
A factor which is not a power of the indicated root is
left under the radical.
„. /i47 f~49 7 /3 .
Thus Vl8=V2X¥=3V2-
PRACTICAL METHOD
From the practical point of view the operation last shown
is ridiculous. If in any computation it were necessary to
/i47
determine \j^r^ no one would think of reducing it as indi-
cated but would directly determine it in one of six different
ways:
(1) Compute it on the slide rule.
(2) Find the quotient and extract the square root
arithmetically.
(3) Find the quotient and extract the square root by
logarithms.
(4) Find the quotient and read the square root from a
table.
(5) Read from a table the square roots of both 147
and 18 and then find the quotient.
49 1
(6) Reduce the fraction to its lowest terms -r-, take -
o o
of 49 by inspection, and extract the square root by one
of the several methods.
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190 TECHNICAL ALGEBRA 176
Since a square root is one of the two equal factors and a
cube root is one of the three equal factors of a number,
any ROOT of a monomial quantity is obtained by extracting
the required root of the numerical factors, and dividing the
exponents of all the literal factors by the index of the root
^
in which the exponents of the letters were divided by the
root index 2.
Similarly, any POWER of a monomial is obtained by rais-
ing the numerical factors to the required power, and multiply-
ing all the exponents of the literal factors by the exponent of
the power.
/ 3a;Vz-5 ^3_27^6yi2^-i5
^^""^ \ 2v^ ) ~' 8^6 •
176. Miscellaneous Examples. Express each of the
following in one or more different forms:
L a-^K 2. x-^K
3. sH\ 4. x^y-K
^ r - a-^c^
6. . 6. .
x-2 262
7. V36a26-*. 8. \^32g^^h-^.
9. V98a-»68. 10. {5a%-^)K
11. {2x-Y)^. 12. (3a-362c)l.
13. 4b-id^\/2x. U. (7-*a36J)«.
16. V^25o6-ic-*. 16. V2a^y^8r-*v^\
17. ^306 -id 18. V^729a-«6-«.
19. V 384x^2/ -10. 20. yj
32a-
15 -5c
1+X-*
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CHAPTER XII
THE BINOMIAL THEOREM
177. A New Symbol. In the study of this chapter we
are to derive and apply a formula whose expression is simpli-
fied by the use of a symbol not before used. In this formula
we shall wish to indicate 1 times 2, 1 times 2 times 3, 1
times 2 times 3 times 4, and so on. Whenever this is
necessary in mathematics, that is, when we wish to indicate
the yroduci of the integers in succession from unity ^ instead
of writing 1X2X3X4 or 1-2-3-4 and so on, we write 4!
or |4, each of which means 1X2X3X4 or 1-2-3-4, and is
called factorial 4.
Likewise 6 1 or |6^ is called factorial 5 and means
1X2X3X4X5.
In modem books the first or exclamation point form is
used almost exclusively to denote a factorial number
instead of the older or angle form shown in the two pre-
ceding illustrations.
As the exclamation form is easier to make and is in
common use it is preferable to the angle form.
Our new symbol, therefore, like other operation symbols,
is significant only when written in the proper position
with respect to the quantities affected by it. - Following a
word or sentence it denotes an exclamation; following a
symbol for quantity it denotes that the symbol is factorial
and therefore integral,
191
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192 TECHNICAL ALGEBRA 178
178. An Example in Addition. In the derivation of
the formula referred to in the preceding paragraph it will
be necessary to add terms involving a factorial denomi-
nator, like
n(n— l)(n— 2) ^ «,« , n(n— 1) ^ oi^
-^^ ^ ^a""2ft3.j — 1^ — ^a""-2b3.
Observe that with respect to a and b these are like
terms because a and b have the same respective exponents.
They may therefore be added by adding their coeflScients
n(n—l)(n— 2) , n(n— 1)
3! 2! *
Show that the sum is
n(n~l)(n~2) 3n(n-l)
3! "^ S\ '
Show that this reduces to -^^ by the two
following methods:
(1) By adding the terms.
(2) By removing the parenthesis and then adding.
In like manner, addland simplify
n(n-l)(n-2)(n-3) n(n~l)(n-2)
4! "^ 3!
179. Expansion. When an algebraic expression is oper-
ated upon as denoted by an exponent it is said to be
expanded and the operation is called expansion.
1. (a+6)2 can be expanded by what law of numbers?
Show the application of this law to the expression.
Expand the following, showing the full multiplication:
2. {a+b)\ 3. (a+6)*. 4. {a+b)\
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180 THE BINOMIAL THEOEEM 193
Each expansion has how many more terms than the
number of units in the exponents of the respective powers?
Therefore the expansion of (a+by^ would have how
many terms?
The expansion of (a+b)^ would have how many terms
provided n is a positive integer?
The first term of the binomials is in all terms of the
expansions, except what term?
The last term of the binomials is in all terms of the
expansions, except what term?
Which term of the binomials has increasing exponents
in the expansions?
Which has decreasing exponents?
What is the increase in each succeeding term?
What is the decrease?
In (a+b)^ what is the first term of the expansion?
What is the first term of the expansions of
(a+6)3, (a+6)^ {a+b)^ (a+by^l
How do the exponents of these first terms compare with
the respective exponents of the binomials?
Therefore what is the first term of the expansion of
(a+br?
How do the coefficients of the second term of the expan-
sions compare with the exponents of the respective powers?
The exponent of a is how much less in the second term
of the expansions than in the first term?
What is the exponent of b in the second term of all the
expansions?
Therefore what is the second term of the expansion of
{a+br?
180. Determination of the Coefficients of the Terms of
a Binomial Expansion by Inspection. What is the coefiicient
of the third term of the expansion of (a+b)^?
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194 TECHNICAL ALGEBRA 181
Show by expansion of (a +6)^ whether this coeflScient
equals the following:
Coef preceding termXEzp first letter in the term
Number of the term
Apply the same test to the coefficient of the third term
in the expansion of (a +6)^; also to the coeflScient of the
third term of (a+6)^.
Apply the same test to the coefficients of all the other
terms except the first in all of the expansions.
Law for the Coefficient. Write this law by answer-
ing the following question:
The coefficient of any term except the first in a binomial
expansion, equals what?
181. Summary of Facts Regarding the Expansion of a
Binomial.
In the work-book fill in the following outline:
In every binomial expansion.
(1) Number of terms =
(2) Coefiicient first term =
(3) Exponent first letter in first term =
(4) Exponent second letter in first term =
(5) In each succeeding term, decrease in exponent
first letter =
Increase in exponent second letter =
(6) Coefiicient any term except first =
182. The Binomial Theorem. This celebrated theorem
is a law for the expansion of a binomial. If written it
would state substantially what has been summarized in
paragraph 181, provided your summary is correct. The
theorem is true for all exponents whether integral or fractional,
positive or negative.
It will be best understood as summarized and therefore
need not be written in the work-book.
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IM THE BINOMIAL THEOREM 195
183. The Binomial Formula. This formula is best
expressed by the expansion of (a+t)", as indicated in
paragraph 181.
Thus (a+b)»=a"+na"-^b+^^^j^a"-V
_^ n(n-lKn-2) ^„_3^^3_^ . . .* to (n+l) terms.
Write the first six terms of this expansion.
184. Proof that the Formula Applies to any Ntimber
of Terms. By the expansion of (a+b)^ by multiplication
you have discovered a law for the expansion of any binomial
to the fifth power. This law was applied in the preceding
paragraph in obtaining six terms in the expansion of (a +6)**.
The work of this paragraph, if correctly done, will be a
proof that the law applies to all powers beyond the fifth.
Expand (a+b)^ to six terms and multiply both members
by a +6, writing like terms with respect to a and 6 under
each other.
Simplify by adding the like terms in the second member.
(If necessary refer to paragraph 178 in the work-book.)
You will now observe that
(1) Paragraph 183 gave the expansion of (a+b)** when
n = 5.
(2) This paragraph gives the expansion of (a +6)**"^^.
(3) The expansion of (a+6)'*"^\ follows exactly the
same law of development as (a +6)".
The law therefore applies to the expansion of
(a+6r+^ = (a+6)^+^ = (a+6)6.
But if n = 6, the law applies to the expansion of a power
1 greater and therefore to the seventh and eighth and ninth
and higher powers.
* Dots so written in an expression signify " and so on."
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196 TECHNICAL ALGEBRA 185
Therefore the statement of paragraph 182, has been
shown to be correct and it should be now clearly understood
why the binomial formula applies to all powers.
186. Change in the Formula when the Binomial is a
Difference Instead of a Simi. If instead of (a+6)'* the
binomial is of the form (a— 6)", will the terms of the expan-
sion containing even powers of b be positive or negative?
Why?
Will the terms of the expansion containing odd powers
of b be positive or negative? Why?
Write the first four terms of (a— 6)".
186. How to Apply the Formula. When the coefficients
and exponents of the given binomial are all unity the expan-
sion may be written directly without the use of the formula.
Thus (x+y)^=o^+Sa^y+Sxy^+y^.
and (x — 2/)^ = a:^ — Sx^y+dxr/^ — y^.
by the binomial theorem.
If however the binomial is of some other form, as
(2x2+4t/3)3^
the expansion may be obtained as shown, or as follows:
(1) Expand (a+b)^ to four terms.
(2) Substitute the given values, a = 2x^, 6=4i/3, n = 3,
and simpHfy the resulting equation.
Thus
/ I l.^» « . « ir . n(n— 1) ^n.o , n(n— l)(n— 2) ^ ,,,
{a+b)''=a''+na''^^b'\ — ^^ ^ a'^-^fe^H — ^^ ^ ^a^-^j^.
.-. (2x2+41/3)3 = (2^2)3+3(2x2)3-141/3+ . . . to 4 terms.
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187
THE BINOMIAL THEOREM
197
187. Examples. Solve the following examples either
by applying the binomial theorem or by substitution in the
binomial formula under the conditions stated in the pre-
ceding paragraph.
An expansion which will be too long for the width of
the page should be written lengthwise.
1. Copy and completely solve and simplify the last illustrative
example of paragraph 186.
2. ir+s):
3. (2r+8)».
4. (2r«+5s)«.
6. (2r»+5s«)».
6. (x»-y)«.
7. (»•-«;»)<
8. {bx+dy*)'.
9. (5x-'+j/«)«.
10. (t-d)'.
11. (g+h):
12. ixi-y^)'.
13. (3n-«-6A;«)'.
»■ H)'
»■ (f-)"-
.«. (t-^.y.
"• (^"-a-T-.
18. (2+2/)* to 3 terms.
19. (x-3)' to 4 terms.
20. (3v-2w*)-i to 4 terms.
21. (e«+2c»)-* to 3 terms
-(-.-)•■
23. lx-{ — j to 4 terms.
24. (2x-'+3j/«)».
- (-!?.)••
26. (7o»-26-»)»,
27. (9x»-7j/-«)».
28. (12»'-lla;«)-».
29. (4r»+5«»)> to 3 teiTOB,
30. (8c»+id-')».
31. (6x-»+3z')«.
32. (ll6-»+13d«)'.
«■ (-':?•
34. (3«.--) .
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198 TECHNICAL ALGEBRA 188
188. How to Determine any Term in a Binomial Expan-
sion. By reference to paragraph 183 write the fourth term
of (a+6)".
For n substitute 6 and simplify by performing the
indicated subtractions and denoting the multiplications
in the numerator of the coefficient.
Thus ^^a363.
1. Analyze the coefficient of this fourth term as follows:
(a) How does the number of factors in the numerator
compare with the number of the term?
(6) How does the first factor compare with the exponent
of the power?
(c) The last factor in the numerator is how many more
than the exponent of a?
(d) How does the number of units in the factorial num-
ber compare with the number of units in the exponent of 6?
(e) How docs the decrease in the exponent of a compare
with the number of units in the exponent of 6?
2. Write the third term of the same expansion and state
and illustrate whether these five facts are true in the third
term.
3. Write the fifth term of the same expansion of (a+b)^
and state whether the five facts recorded in 1 are true
for the fifth term.
4. Write the fourth term of (r+s)^ and determine
whether the analysis of 1 applies in this case.
5. By reference to the analysis of 1 write a law for any
term in a binomial expansion.
189. Examples. Solve the following by the preceding
law:
1. Write the 6th term of (2x -5y^yK
2. Write the 10th term of {5y^-Szy*.
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THE BINOMIAL THEOREM 199
3. Write the 7th term of {bx^-2d*y\
4. Write the 3rd term of (5a:»-82/)».
5. Write the 8th tenn
of (.+'-)%
6. Write the middle tenn of (2«»-t;»)".
7. Write the nth term of (a+h)\
8. Write the (n+l)th term of (x+y)\
9. Write the 8th term of (1+z^)-^^ and reduce it to its simplest
form.
10. Write the 5th term of (1 —a:*)-* and reduce it to its simplest
form.
11. Expand (c -ox) - ^ to four terms.
12. Expand (1 —nx)^ to three terms.
13. Compute the sum of the coefficients of the 4th and 7th terms
of (3a;+l)".
14. Compute the sum of the coefficients of the 8th and 12th terms
-H")'"
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CHAPTER XIII
POWERS AND ROOTS
Section 1, Powers. Section 2, Square Root. Section 3,
Cube Root. Section 4, Other Roots.
§ 1. POWERS
190. Power of a Monomial. In Chapter XI the rule was
given that a monomial may be raised to any power by multi-
pljring each exponent of its factors by the exponent of the power.
The reason for the rule will now be developed.
In (40^63)2 the exponent 3 denotes that ^a^l^ is to be
taken how many times as a factor?
Therefore 4 must be used how many times as a factor?
And a^ must be used how many times?
And 6^ how many times?
Therefore the cube of 4ia^lP equals what power of 4 times
what power of a times what power of 6?
Could these powers of 4, a, and 6, have been written
directly by multiplying the exponents of these three factors
by 3, the exponent of the power?
Therefore (4a263)3 equals what?
In order, therefore, to raise a monomial to any power
what operation should be performed
(1) on the numerical coefficient?
(2) on the exponents of its literal factors?
191. The Binomial Formula. As shown in the pre-
ceding chapter a binomial may be raised to any power
by substitution in the binomial formula. By the use of
200
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192
POWERS AND ROOTS
201
marks of parenthesis this formula may also be used for
detennining any power of a trinomial and an expression
having four terms.
Thus (a+b+c)^ = (^+c)^,
and (a+b+c+d)^ = {a+b+c+d^;
the vinculum being used in both illustrations to transform
the given expressions into binomials.
In this connection it should not be forgotten that the
square of any polynomial is most readily obtained by the
law given in Chapter V.
192. Examples. By inspection expand the following.
In each example state what law was applied.
1. i2x+3y-^y.
2. (2a-462+2c-5d')«.
3. (ix^-2a%)*.
4. (2a;-y+3c2-r«)*.
5. {5v-^+Qw^)-\
6. (ax'-cy«)'-
7. {2ab^+x^)^ to 4 terms.
8. (a;«-4a2y)-J to 4 terms,
9. {-2)K
10. (-2)-«.
11. (-2)3.
12. (-2)-'.
13. (-3a^x*zi)\
14. (-3oi^<3»)».
16. (-7a36'»d«-2)2.
16. (-6c-2diA"-3)3.
17. (5a26-4c)~.
18. (4a;i^-i)-3».
19. (^)^
xy
20. ^-«r
26*
21.W'.
2
22 ^'
22. 2 .
23. (^)"*.
4
26. (y-2):
»■ H);
«■ (!+-)•■
28. {y+-j to 4 terms.
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202 TECHNICAL ALGEBRA IM
29. (c+d-r-sy. 30. (x^-y^-v+w^)*,
31. iv+x-izy. 32. {a-3x^-iy-2c»y.
33. (^3r+26-2)-». 34. (46-i-2c-J)-2.
193. Any Root of a Monomial. A monomial may be
raised to any power by what operation on
(1) its numerical coefficient,
(2) the exponents of its literal factors?
The extraction of a root is the inverse of the determination
of a power. Therefore any root of a monomial may be
extracted by what operation on
(1) its numerical coefficient,
(2) the exponents of its literal factors?
Thus v^8xV equals what?
and v^lGa^fe-^c equals what?
Reduce each of the two results to a form having no
fractional and no negative exponents.
194. Sign of the Root. Perform the following indicated
operations:
(2x22/)2, (-2x22/)2, (-2x22/)4.
(2x22/)3, (-2x22/)3, (^2x^y)\
The results show that:
(1) Even powers have what sign?
(2) Odd powers have what sign?
(3) Even roots of positive quantities have what sign?
(4) Even roots of negative quantities are impossible.
Why?
(5) Odd roots have what sign?
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196 POWERS AND ROOTS 203
196. Examples. Extract the indicated roots of the
following and reduce the results to forms with no fractional
and no negative exponents:
1. V25a*6«c-». 2. (25a^«c-»)*.
3. \^8rH^K 4. (8r»<«z;«)*.
5. \^81x-^-K 6. (81x-^-»)*.
7. \^Slx-^-\ 8. (81x-^-»)"*.
9. V^(2r-3i2)io. 10. (2r-3/»)*.
11. <^-216a»6-». 12. (-216a»6-»)*.
^^ 5/ 243a:ioy-» .. /243xiV-*\*
^^- \"32i^- ^*- V^2^^/-
17. V-4a«6^ 18. (-4a«60*.
§2. SQUARE ROOT
196. Square Root of a Polynomial. The following
instructions should enable you to extract the square root
of any polynomial.
(1) Arrange the terms in descending or ascending order
with respect to the same letter.
(2) Enter at the right the square root of the first term
of the polynomial.
(3) Subtract its square from the first term and bring
down the next term of the polynomial.
(4) At the left set down a trial divisor determined
from the following law:
Trial divisor = 2 Xroot.
(5) Enter in the root the number of times the trial
divisor is contained in the first term. ^
(6) Add to the trial divisor to form the complete divisor,
last term of root.
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204 TECHNICAL ALGEBRA 197
(7) Multiply the complete divisor by the last term of
the root and subtract, bringing down more terms as may
be necessary.
(8) Repeat (4), (5), (6), and (7) until the root is
obtained.
Thus 9a^ - 24x3 « 14a;2+40x+25( 3x2-4x~5
9^*
6x2 -4a;) _ 24x3 - 14x2
24x3+16x2
6x2-8x-5) -30x2+40x4-25
-30x2+40x+25
197. Examples in Square Root. Extract the square
root of the following:
1. 166-24a+9a«+16+46«-12a6.
2. 25x«-120fox+1446«.
[3. v*-vy+jy\
L ^
'4. (x+2/)«-6(x+y)+9.
^^5. 12ac+9c*+6«+4a«+4a6+66c.
6. 9x-24x2/+16y.
7. 4x*+~-+|x+4x3+|x«.
r- 1
8. x+— to 3 terms.
[9. 5-2x to4 terms.
10. r*-6r»+^-2r+9?r2.
11. m^-SOtv' -2^t^v+m*v^+2^*.
12. a«+462-8cd+2ad+16c2-8ac+d«+4a6+4W-166c.
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199 POWERS AND ROOTS 205
§3. CUBE ROOT
198. Cube Root of a Polynomial. The following laws
should enable you to extract the cube root of any polynomial:
(1) Arrange the tenns in descending or ascending order
with respect to the same letter.
(2) Enter at the right the cube root of the first term
of the polynomial.
(3) Subtract its cube from the firslb term and bring
down the next two terms of the polynomial.
(4) At the left set down a trial divisor determined
from the following law:
Trial DivisoR=3Xroot2.
(5) Enter in the root the number of times the first term
of the trial divisor is contained in the first term of (3).
(6) Add to the trial divisor to form the complete divisor:
(a) 3 X last term root X all terms root except last,
(6) last2.
(7) Multiply the complete divisor by the last term
of the root, subtract, and bring down as many terms of
the polynomial as may be necessary.
(8) Repeat (4), (5), (6) and (7) until the root is obtained.
Thus 27a3-54a26+36a&2-863(3a--26^
27a3
27a2-18fl6+4b2 ) -54a26+36a&2-8b3
-54a2b+36ab2-8fe3
199. Examples in Cube Root. Extract the cube root
of the following:
1. x^+Sx^+3xy*+yK
2. 33x2-63x»+66a;*+l+8a;«-9a;-36x«.
3. -18a2+13a»+3a6+a«+8+12a+9a<.
4. 2/«-127/-1122/»-482/-f542/*-f 1082/2+8.
6. r«+3r5+2r»-3r7+3r«-6r*-3r+3r«+l.
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206 TECHNICAL ALGEBRA 200
6. a»+3a+~+-.
a» o
7. 368+106«-126*+126^-106»-36+l+66«.
8. 42s^-9s-36s»+9s»-l+8s«-21s«.
9. 2/'-l to 3 terms.
10. 64r«-2to4tenns.
§4. OTHER ROOTS
200. What Roots Can be Extracted. If a root higher
than the cube root is required it can be determined by the
processes of this chapter only in case its index is factorable
into 2s and 3s.
Resolve the following root indexes into 2s and 3s as
far as possible:
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.
What root denoted by these indexes cannot be extracted
by applications of square root and cube root?
How can the 4th root be obtained?
201. Method of Signifying Successive Roots. If x
denotes any quantity the process of finding the 12th root
may be indicated as follows:
Indicate the process of finding the following roots:
1. v^ 2. v^ 3. \^x.
4. v^ 5. v^ 6. v^
In computation all roots are determined by logarithms and
checked on the slide-^ule.
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CHAPTER XIV
RADICALS
Section 1, Reduction. Section 2, Addition and Subtrac-
tion. Section 3, Multiplication and Division. Sec-
tion 4, Powers and Roots. Section 5, Rationalization.
Section 6, Radical Equations.
202. Operations with Radicals. In computation and
in the further study of mathematics little use is made of
radicals as conventionally presented.
In this chapter the following fundamental operations
are discussed and whenever the conventional and practical
methods of operation differ, both are shown.
(1) Reduction of radicals.
(2) Addition and subtraction of radicals.
(3) Multiplication and division of radicals.
(4) Powers and roots of radicals.
(5) Rationalization of radicals.
(6) Radical equations.
The study of this chapter will give excellent practice
in the simplest arithmetical processes, the use of expo-
nents, the interpretation of mathematical expression, and
an increased ability to transform the equation, all of which
is worth while even to the most practical.
203. Kinds of Radicals. A radical is an indicated root of
any quantity.
Thus V25, v^, 1728*, v1[%:
207
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208 TECHNICAL ALGEBRA 204
There are two kinds of radicals:
(1) Rational, or indicated roots which can be exactly
determined.
(2) Irrational or surd, or indicated roots which cannot
be exactly determined.
Thus V49, v^ are rational,
and VT?, V^156 are surd.
In two different ways denote
(1) four rational radicals,
(2) four surds.
State why the first four are rational and why the second
four are surds.
§ 1. REDUCTION
204. Processes of Reduction. A radical is said to be
reduced when changed to a different form having exactly the
same value.
Thus V98 = V2X49 = 7 V^.
The processes of reduction are as follows:
(1) Reduction to simplest form.
(2) Reduction to full radical form.
(3) Reduction to same degree.
205. Reduction to Simplest Form. A radical is in its
simplest form, only when the quantity under the radical is
(1) integial and
(2) as small as possible.
It is as small as possible only when
(a) it is not a power of the indicated root and
(Jb) has no factor which is a power of the indicated root.
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205 RADICALS 209
Thus 'vS has the first requirement of simplest form
because it is integral. It does not have the second require-
ment because it is not as small as possible. It is not as
small as possible because it is a power of the indicated root.
Thus v^8=2.
^^432 has the first requirement but not the second
because it has a factor which is a cube.
Thus v'l32 = v^2X216 = 6v^.
6v^ is in simplest form, being both integral and as
small as possible.
-W- is not in its simplest form because a fraction. It
can be reduced by multiplying both numerator and denomi-
nator by the smallest number which will make the denomi-
nator a power of the indicated root.
-^15 is in its simplest form because the quantity under
the radical is integral and as small as possible.
When the radical has a coefficienti the root of any factor
under the radical must be multiplied into the coefficient
Thus (o+6)V3(a2+2a6+62) = (a+b)^Vs,
and 8V32=8Vi6X2=8X4V2=32V2.
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210 TECHNICAL ALGEBRA 206
206. Examples in Simplest Form. Solve the following
by reduction to simplest form as indicated In the preceding
models:
.1. v^. 2. v^ 3. V72.
4. <^i, 6. Vm, 6. \^m,
[7. v^. 8. V96. 9. V20OO.
10. V392. 11. 3V^3456^ 12.T2V^12432.
13. {a+z)Vs{a+x)K 14. Vsx^-lGx.
16. V3z^+Qxy+Sy\ 16. a"'\/i^-
, 2(7^ ^^ 4/l6a«
19. 4a;'\/5aa;-*. 20. V{r^-s^)(r-s).
21. \^i728^. 22. xf^.
\x~y
\. ^I^\ 24. '/-^^^.
23
207. Simplest Form from the Practical Standpoint.
From the practical side the simplest form of a radical, as
of any other expression, is the form which is the simplest
for computation.
Thus V^192 whose conventional simplest form is
'>J^64X3 = 8v^, is in a simpler form before reduction than
afterwards for the reason that it is just as easy to determine
v^l92 as to determine v^
and in the latter form the root must be multiplied by 8.
Look over the examples worked in paragraph 206 and
if any would be simpler in computation in the form to
which they were reduced, specify which and^ state reason.
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209 RADICALS 211
208. Reduction to I*ull Radical Form. This process is
accomplished by introducing the coefficient of the radical,
under the radical sign. The operation required to do this
is determined by the root index.
If the index is 2 the coefficient must be squared; if the
index is 3 the coefficient must be cubed before being placed
under the radical sign. We therefore have the following
Rule. To introduce the coefficient of a radical, under the
radical sign, raise it to a power of the same degree as the
radical.
Thus 5 V2 = V25X2 = V50,
and 3 vY= v^27X7 = v^O",
and (3 +x) V3+2x = V(S+x)^(S+2x) .
From the practical standpoint this process is the height of
absurdity and would never be employed,
209. Examples. In the following introduce the coeffi-
cients under the radical; express the results in simplest
form.
1. 5V8. 2. 3\/i2.
3. (a+b)V^. 4. |v^.
2a -26
3
8 ' *" a;»'
9. |(2J»-2/^)*. 10. -XS-x)-K
11. (a+6)(a«+6«)-«. ^2. |^
8
1416*
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212 TECHNICAL ALGEBRA m
13. ZabxV{3abx)-K 14. ^V^.
o
15. 9^. 16. 5y-.^.
17. 3-iV27ai 18. 26*^346-*.
210. Reduction to the Same Degree. The degree or
order of a radical is denoted by the root index. With no
index the radical is of the second degree; with three as index
the radical is of the third degree, and so on.
Radicals are therefore reduced to the same degree by
transforming them to radicals of the same root index. This
may be easily done by the following process:
(1) Express the radical quantities with fractional
exponents.
(2) Reduce the fractional exponents to the least com-
mon denominator.
(3) Express the results with radical signs.
Thus V3, v^", and v^
are reduced as follows:
V3=3* = 3A = v^,
211. Examples in Reduction to Same Degree. By the
process of the preceding paragraph reduce the following to
equivalent radicals of the same degree. Express all final
results in simplest form.
1. V20, <^m, ^M. 2. ^, <^S, <^.
3. v^, v^, VK. 4. v^, Vxb, </y.
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li: RADICALS 213
6. Vsi, v^26, V^3y^ 6. v^2xS v^o+6.
7. Vsi, v^, v^. 8. ^|, \^{x-'S)'K
9. V^, V^2x-«. 10. \^x^+y*, y/2x^+ixy+2y*.
§ 2. ADDITION AND SUBTRACTION OF RADICALS
212. Similar Radicals. If radicals of the same degree
have exactly the same quantity under the radical sign, they
are similar.
Thus Sy/2ax, cV2ax are similar because both are of
the second degree and have the same quantity 2ax under
the radical sign.
Similar radicals are added by collecting their coefficients,
the same as in the addition of like terms.
Add the following radicals:
1. 4V5+2V5-5V5.
2. sVxy-aV^+bVxy-sVTy.
3. bVa-c+xVa-c+sVa+c-Va+c.
Radicals which are not similar may sometimes be made
similar by reduction to simplest form.
Thus V27+\/T8+V98 =
V9X3+V9X2+V49X2 =
3\/3+3V^+7V2=
3V3+IOV2.
In computation the original indicated roots would be
directly determined and the sum found wiih no attempt at
reduction to similar form.
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214 TECHNICAL ALGEBRA 213
213. Examples. Reduce the following to simplest form
and add:
1. V20+V45+ J|.
2. V24-Vi50+3V54.
3. V98+V2OOO-2V72.
4. 5V48-6V75-.-\/27.
6.^ 7V2(x+2/)2 - V 128+ Wm,
6. 2V^-oV^+6V^^.
7. rV^4(x-2/)3+t;V^108(r3+3r22/+3n/2+2/').
8. V5x2 - 10xy+5y^ - V20x2+20cc+l.
9. V^2a« - 6a26+ 6a62 _ 2b' + V^6466«.
11. V^+ V^TisS - \/2x^\
12. 4V^^-5V^8^+|^-.
§ 3. MULTIPLICATION AND DIVISION OF RADICALS
214. Multiplication. Radicals of the same degree are
multiplied by multiplying together
(1) their coefficients and
(2) the quantities under the radical.
Thus {a-b)V2xX{a+b)\/5xy =
(a2 - 62) Vl0x2// = x(a2 - 62) Vlb7.
When the radicals are of different degree the multipHca-
tion may be indicated; or the radicals may be reduced to
equivalent radicals of the same degree and then multiplied.
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216 RADICALS 215
Thus 3V5^X4v^ = 12 Vs^- \^,
or 3(5a)i = 3(5a)S = 3\/(5^
4(6a)» = 4(6a)» = 4v^(6^
It should be unnecessary to state that the second method
should not be used in the work which follows and that it wculd
not be tolerated in computation,
215. An Important Principle. The product obtained
when a radical is multiplied by itself is best shown by the
use of fractional exponents.
Thus •\/a+2xxVa+2x = (a+2x)*X(a+2x)» = a+2x.
If therefore, the square root of a quantity be multiplied
by itself the result is the quantity without the radical.
Thus V2x-32/XV2x-32/ = 2x-3?/,
and 2V29X3V29 = 6X29 = 174.
216. Examples. Perform the following indicated multi*
pUcations and express the results in the simplest form:
1. VT2XVJE. 2. V^xVa+6.
3. 8V3X2V5. 4. 2v^X7V^5l
6. v^XV^. 6. 4V^X3V^7!
7. VTsx^y/^. 8. aV^Xft^V^.
o
9. VxyX\^. 10. \^Sa^X\^isiu\
11. (V^+l)(vV-5). 12. (V2b-y){V2b+4)
13. (4V^-3)(5Vx+l). 14. (4V^+r)(4V^-r).
16. (VxT^+2)(V^-2). 16. (\/x2+3 -5){Vx^-\-'S -2).
17. (V^+^-\/^)2. 18. (2V^-4Vh-2)2.
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216
TECHNICAL ALGEBRA
217
217. Division. Since division is the inverse of multi-
plication it should be possible to solve the examples in the
following paragraph without special instructions.
One important principle may be noted:
a+x (a+xy f\ \. ^/ — I — '
Therefore if any quantity be divided by its square root
the quotient is the square root of the quantity.
Show this by two illustrations, one literal and the other
numerical. Number all equations and give the mathemati-
cal authorities for all operations.
218. Examples. Perform the following indicated divi-
sions and express the results in simplest form:
Observe that — ^ =
Vb'
1. V24^V6.
. V12T
2. V729^V8T.
V5OO
7.
10.
\/02.5'
6.
V24-A/i6+\/36
8.
V20*
V4
Wx-
■8
-2
</^ .
12. ,— ^ ,
14,
V28*
3.
6.
V58
V29*
<r^x
x-\-y
^x+y
11.
13.
15.
38
I9V3-57*
4V75
5V56'
7V24Sx*y-^
28V363x-2y**
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221 RADICALS 217
§4. POWERS AND ROOTS OF RADICALS
219. Any Power of a Monomial. Write the answers to
the following questions:
In order to square a monomial what operation is per-
formed
(1) on the numerical coefficient;
(2) on the exponents of the literal factors?
Illustrate by working out the square of bah''^(? with
all equations numbered and all mathematical authorities
for the operations.
In the same form work out the square of VGab.
Show the full work with authorities for (\/bx^y)^.
The square of the square root of a quantity therefore
equals what? .
The cube of the cube root of a quantity equals what?
A monomial may be raised to any power by what opera-
tion on its numerical coefficient?
By what operati on o n the exponents of its literal factors?
Illustrate by (VGox)®.
220. Any Power of a Binomial. By what law can a
binomial be raised to any power?
Illustrate by expanding {y/x+2VyY.
Can a trinomial be expanded by the same law?
Illustrate by (Va- Vb+Vc)^.
What is the law for the square of any polynomial?
Illustrate by (V2^-2V6+4V3c- Vd)2.
221. Examples in Powers. The following examples
may be worked in the form given or with the roots denoted
by fractional exponents, as preferred.
1. (aV2a6)«. 2. (-3aV^«)». 3. {-Za</wy.
4. {?x<^^y. . 6. (3-V6)«. 6. (3V2-4)«.
7. (x+2Vy)». 8. (2o+3V^)». 9. (3V^-4V6)«.
10. (V^-5V7)».
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218 TECHNICAL ALGEBRA 222
222. Roots of Monomials. Any root of a monomial
radical may be readily determined by the use of fractional
exponents in place of the radical signs. When this change
is made, in order to determine the root what operation
must be performed.
(1) on the numerical coefficient;
(2) on the exponents of the literal factors?
Illustrate by \^21a^x^
also by Va*»6*»+^.
§6. RATIONALIZATION
223. What Rationalization is. To rationalize any expres-
sion, is to free it from radical signs.
Thus Vx — 1=0 can be rationalized by squaring.
Why?
Show the complete solution for x.
In
\42/
the denominator can be rationalized by multiplication by a
quantity which will make it a power of the indicated root.
Show the process of rationalization of the denominator,
being sure to multiply the numerator by the same quantity
so that the value of the fraction shall not be changed.
224. Denominator a Binomial. When a radical denomi-
nator has two terms it is rationalized by multiplication by itself
with the sign changed of one of its terms, the numerator being
multiplied by the same rationalizing factor.
Thus
V3 - V5 ^ (V3 -VI) (\/2 - Vg) ^
\/2 + V6 (\/2 + Vg) (V2 - Vg)
(V3-V5)(V2-\/6) _ (\/3-V5)(\/2~\/6)
2-6 -4
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227
RADICALS
219
Show this illustration in full and reduce to the simplest
form.
State why the factor V2— Vg rationalized the denomi-
nator.
225. Examples. Rationalize the denominators of the
following:
6.
Va+Vb
-y
-Vx+y
^x—y+\^x+y
2.
6.
10.
3Vx"
_y36+2
V3b^4x^'
2-V3
V5+W2'
2-Va ^
3+Va^x
Vx^-x y +y^+2
Vx^+xy+y^*-2
§6. RADICAL EQUATIONS
226. Principle of Operation. A radical equation is one
in which the unknown quantity is under a radical sign. The
general method of solution is to transpose the radical to
one member (entirely by itself) and then to raise the
equation to a power of the same degree as the root index.
Illustrate by solving \^2x+l -3 = 0.
also by solving Vx+S =4.
227. Examples. Solve the following equations:
1. 2V^ = 14. 2. \V9+x^=3.
o
3. V6»-a;=7. 4. aVsb+x* =a».
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220 TECHNICAL ALGEBRA 227
6. 8-2VJ=6. 6. 7-4V3x=0.
7. 3vV+2 = vV+5. 8. 2V3x-14=8.
9. Va;»-7a;=a;-V5". 10. V5x«-10a;+20 = V5x».
When a radical equation is not cleared of radical signs
in one operation the operation is repeated after the remain-
ing radical is transposed to on e member entirely by itself.
Illustrate by solving Vx— 32-j- Vi = 16.
Solve the following:
11. V3y+10-V3y+25=-3.
12. ^4a;+12=6. 13. V2x-7-8= -V2x+9.
14. .L_pi-~=7. 16. 3VT-10=-V25+8A.
\ 5r+7
16. v^2y-8=2. 17. 3-Va+n = -Vo-3.
18. V2/+11-3 = VV^. 19. .^ =\/2a+9~V2a.
V2a+9
20. \/6"-V6^=-^.
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CHAPTER XV
LOGARITHMS
Section 1, Logarithm op a Number Greater than Unity.
Section 2, Logarithm op a Number Less than Unity.
Section 3, Naperian or Hyperbolic Logarithms. Sec-
tion 4, lyOGARITHM OP A PRODUCT. SECTION 5, LOGARITHM
OP A Quotient. Section 6, Logarithm op a Power.
Section 7, Logarithm op a Root. Section 8, Solu-
tion OP AN Exponential Equation. Section 9, Model
Solutions. Section 10, Logarithmic Computation.
228. Two Ways of Multiplying. 100 may be multi-
plied by 1000 as follows:
(1) 100X1000=100000,
(2) 102 X 103 = 10^ = 100000.
Observe that in (2) the product is obtained by addition
of the exponents of the powers of 10 which equal 100 and
1000. It is therefore possible to multiply together numbers
which are integral powers of 10, by addition of the exponents
of these powers.
In like manner 472 may be multiplied by 67.5 by addi-
tion of the exponents of the powers of 10 which equal 472
and 67.5. But here are two difficulties:
What powers of 10 equal 472 and 67.5, and what does
10 equal when raised to the sum of these two powers?
The answer* to this question and the removal of the
difficulties follow:
100=102
472 = 10^ +
1000 = 103
67.5 = 10^ +
10=101
221
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222 TECHNICAL ALGEBRA 229
472 is greater than the 2d power of 10 and less than the
3rd; 67.5 is greater than the 1st power of 10 and less than
the 2d.
Therefore 472 = 10^+" ^^^^™^
r*fj r 1 r\l +a decimal
A table of logarithms is an arrangement of numbers in
sequence with the decimal parts of the powers of 10 which
equal the numbers.
Taking the decimals from the table we have:
472=102-673942
67.5 = 10^-»2Q^Q^
Therefore 472X67.5 = 10^503246
By the table 104503246 = 31860.
Therefore ^472 X 67.5 = 3 1860.
In practice the labor of this method of multiplication
is reduced by setting down only the exponents of 10 as
follows:
472X67.5 = 31860 2.673942
1.829304
4.503246
In like manner by the use of a table of logarithms, one
number may be divided by another and any power or root
of a number determined.
229. What a Logarithm is. Logarithms are used as a
means of shortening and simplifying the mathematical
processes of multiplication, division, powers, and roots.
In every system of logarithms all numbers are regarded
as powers of another number which is called the hase of
the system.
Therefore the definition of a logarithm:
The logarithm of a number is the exponent of the power
to which the base of the system must be raised to equal the
ntmiber.
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230 LOGARITHMS 223
In the system of logarithms in common use, called the
common or Briggs' System, the base of the system is 10.
Hence in this system all numbers are regarded as powers o/ 10.
Consider any number, as 306. In the Briggs' system
the logarithm of 306 is the exponent of the power to which
the base 10 must be raised to equal 306.
Now 102 = 100 and 10^ = 1000. But 306 is greater
than 100 and less than 1000. Therefore in order to obtain
306 from 10, 10 must be raised to a power between the second
and the third.
Therefore 306 = 10^ +^ ^^"^"^^^
As shown 2+ a decimal is the exponent of the power
to which the base* 10 must be raised to equal 306.
But by definition the exponent of the power to which
the base must be raised to equal a given number is the
logarithm of that number.
Therefore log 306 = 2+a decimal.
This decimal is given in the table of the Logarithms of
Numbers.
§ 1. THE LOGAMTHM OF A NUMBER GREATER THAN
UNITY
230. A Number Having Three Figures. Direction I.
In the table of the Ix)garithms of Numbers find 306 in the
column headed N,
Place the index finger of the left hand directly under
306 and move the hand to the right in a horizontal line
until it is under the number in the column headed 0. This
number with the two figures prefixed (called leading figures)
which are immediately above the blank space to the left
of it in the same column, is the decimal part of the logarithm
306.
Therefore lo^ ^85721 ^ 3q^ . j^g ^^ ^ 2.485721.
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224 TECHNICAL ALGEBRA 231
The log of 306 therefore consists of two parts:
(1) Integral, called the characteristic,
(2) Decimal, called the mantissa.
The decimal part only, is given in the tables.
231. Accurate Use of the Tables. The above direction
for using the index finger of the left hand is given in order
to secure speed and accuracy in the use of the tables. It
makes possible the unobstructed use of the right hand for
writing the figures from the tables and the index finger can
be kept in its position on the page until the required number
has been written from the table and the written number compared
with the printed number.
Another excellent method is to move a straight-edge
or a blank sheet of paper up or down the page until the
required number can be read just above the upper edge.
232. Reason for the Characteristic. The integral part
of the logarithm of 306 is 2, but the number of figures in
306 is 3.
In this instance the integral part of the logarithm is
one less than the number of integral figures in 306, the
natural number.
This is also true regarding any other number, for example
4798.
This number is greater than 10^ and less than 10*.
Therefore as in the case of 306 the integral part of the
logarithm is one less than the number of integral figures
in the natural number.
233. Characteristic of the Logarithm of a Number
Greater than Unity. Rule 1. The characteristic (integral
part) of the logarithm of any number greater than 1 is one
less in unit value than the number of integral figures in
that number.
By this rule the characteristic of the logarithm of any
number greater than unity may be determined without
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286 LOGARITHMS 225
the necessity of locating it with respect to integral powers
of 10.
234. Position of Decimal Point. Since the division of
a number by 10 is made by moving the decimal point
one place to the left and division by 100 is made by moving
the decimal point two places to the left and so on, the position
of the dedrhal point affects the characteristic only.
For example, log 306 = 2.485721
log 30.6 = 1.485721
log 3.06 = .485721
log .306 =*I. 485721
log .0306= 2.485721
Each number, being one-tenth the preceding, represents
one less integral power of 10. Therefore, every shift of
decimal point to the left means one less integral power
of 10 and every shift to the right, one more integral power
of 10.
In both cases the mantissa is the same. In reading
mantissas, therefore, disregard decimal points in the natural
numbers.
236. A Negative Characteristic. The mantissas of the
logarithms of all numbers are positive but the character-
istics may be positive or negative. They are positive for
numbers greater than unity; they are negative for decimals.
It is therefore impossible to denote the logarithm of a
decimal by a minus sign written in the usual position since
that would indicate the entire logarithm as negative.
Accordingly negative characteristics are indicated by a minus
sign ABOVE them.
Thus log .401 = 1.603144.
*The minus sign above the characteristic is used to denote that
the characteriatic is negative but that the mantissa is positive.
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226
TECHNICAL ALGEBRA
237
236. Examples. About one inch and a half from the
top of the page rule the following table with the columns
long enough for twelve entries and sufficiently wide for the
niunbers in this paragraph and in paragraphs 238 and 243.
Table VII
LOGARITHMS
3 Figures.
4 Figures.
>4 Figures.*
Number.
Logarithm.
Number.
Levari thm.
Number.
Logarithm.
In the first column enter the following numbers: 375,
189, 208, 784, 999, 118, 510, 619, 200, 907, 666, 103.
In the second column enter:
(1) The characteristics (determined by rule I), and
(2) The mantissas (determined by direction I).
237. A Number Having Four Figures. Direction II.
To read the logarithm of a number consisting of four figures,
as for example, 5848, find the first three figures, 584, in
the N column.
Place the index finger of the left hand immediately
under 584 and move the hand in a horizontal line to the
right until it is under the number in the column headed 8.
This number with the leading figures prefixed from the
zero column is the mantissa of the log 5848.
The mantissa of the logarithm of any number containing
four figures is always to be found in the same horizontal
Hne as the first three figures, in the column headed by the
fourth figure of the number.
■ * This symbol means " greater than."
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2S9 LOGARITHMS 227
238. Examples. In Table VII enter the following
numbers and their logarithms: 2084, 3009, 6000, 1289,
9104, 1059, 7855, 8899, 5123, 7248, 6167, 1070.
239. A Number Containing any Number of Figures.
Consider the number 84678. This number is greater than
84670 and less than 84680.
log 84680=4.927781
and log 84670 = 4.927730
10^ 51
That is, a difference of 10 in these natural numbers
corresponds to a difference of 51 in their logarithms.
In other words a difference of 10 between 84670 and
84680 is expressed by a difference of 51 in their logarithms.
Since the difference of 10 in the natural numbers equals
a difference of 51 in their logarithms, a difference of 1 in
natural numbers equals a difference of ttt of 51, or 5.1 in
logarithms.
Therefore a difference of 8 in natural numbers equals
a difference of 8 times 5.1, or 40.8 in logarithms.
Therefore the logarithm of 84678 is 40.8 greater than
the logarithm of 84670.
Therefore log 84678 = 4.927771.
The same result would have been obtained by multi-
plying 51 (the difference between log 84670 and log
84680) by 8 the last figure of the given number 84678,
with a decimal point before that figure.
A continuance of this investigation with numbers of
more than four figures will give the following rule:
Rule. The logarithm of a number having any number
of figures equals the logarithm of the first four figures as
given in the tables, plus the product obtained by multiply-
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228 TECHNICAL ALGEBRA 240
ing the difference between the log of a number 1 greater
than the first four figures of the given number, by all the
remaining figures of the given number with a decimal
point before them.
Therefore, Direction III: To find the logarithm of a
number consisting of more than four figures, for example,
642147, find the logarithm of the first four figures, as in
Direction II.
Then multiply the difference between the mantissa
of the next higher number of four figures, 6422, and the
mantissa of 6421 (the first four figures), by the remaining
figures of the given number with a decimal point before
them.
Add this product to the logarithm of the first four figm-es,
placing it for performing the operation of addition so that
the right integral figure of it is under the right figure of the
mantissa of the logarithm to which it is to be added.
Thus
log 6422=
=807670
log 6421 =
=807603
difference
67
.47
469
268
31.49
.807603
log 642147 = 5.807634
240. Tabular Difference. In some tables the average
difference between successive mantissas, called tabular
difference, is given in a column headed D. In using such
tables it is customary not to ascertain the exact difference
by subtraction, but to multiply the tabular difference in the
same horizontal Une as the mantissa of the given number
(if there is no tabular difference in the same horizontal line,
use the tabular difference immediately above), by all the
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242 LOGARITHMS 229
figures of the natural number except the first four, with a
decimal point before them.
241, Proportional Parts. In modem tables of log-
arithms the last column is used for the proportional parts
which must be added to the mantissa when the number
has more than four figures.
The entries in the column are as follows:
The numbers at the top of each group are differences,
those at the left are the figures of the natural number,
and those in the several groups are the proportional parts
required for a fifth figure of a natural number, provided the
mantissa table gives readings for the first four significant
figures.
If the reading is for a sixth figure, one figure must be
pointed off in the number read; if the seventh figure, two
places must be pointed off, and so on. This is due to the
fact that the value of a figure in the sixth place is only
one-tenth of its value in the fifth place, and its value in
the seventh is only one-hundredth of its value in the fifth
place.
The column of proportional parts saves time and labor
and does away with the necessity of multiplying the dif-
ferences as required in the preceding paragraph.
In the Author's six-place tables, proportional parts
are given at the bottom of the page, arranged with differ-
ences at the top and figures of the natural number at the
left.
242. How to Use the fable of Proportional Parts. In
this paragraph the logarithm of 7145863 will be determined
by
(1) Direction III,
(2) The table of proportional parts.
(1) The logarithm of the first four figures 7145, is
6.85400i.
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230 TECHNICAL ALGEBRA 242
Multiplying the dijBference 61, from the D column by
the remaining figures with a decimal point before them
we have:
.863 6.854002
61 53
— 863 •'• log 7145863 = 6.854055
5178
52.643
(2) The readings from the table give:
6.854002
49
37
18
6.854055
In the reading,
49 was the reading for 8, the fifth figure of the given
number, and was therefore set down directly under the
mantissa;
37 was the reading for 6, the sixth figure of the given
number, and was therefore shifted one place to the right;
18 was the reading for 3, the seventh figure of the given
number, and was therefore shifted two places to the right
when set down for addition.
Observe that in the result only 6 places were retained
because the readings were taken from a 6-place table.
Always use exact difference. An inspection of any
page of the table of the logarithms of niunbers will show
that the entries in the D column are not exact but are the
average differences between successive mantissas for one
or more lines.
Therefore, whenever a reading is taken which requires
the use of the table of proportional parts, determine the dif-
ference between the last figure of the mantissa which is read
and the last figure of the mantissa in the next column, and
take the other figures of the difference from the D column.
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244 LOGARITHMS 231
When reading in the 9 column use the last figure of the
following zero column to determine the exact difference.
It is frequently the case that the table of proportional
parts gives no difference corresponding to the actual dif-
ference between the successive mantissas. No rule can
be given as to what difference to use in such cases, as some-
times the next larger difference will give a more nearly
correct reading, and sometimes the next smaller, but this
could not be determined without a table which gives the
next figure of the mantissas. To secure uniformity of
results it is suggested that the next smaller difference be
invariably used under such circumstances.
In other words, when the exact difference is not in the
proportional parts table, read proportional part under the
next smaller difference.
Therefore when using the table of proportional parts
(1) Note the actiuil difference between the last figure of
the mantissa read and the last figure of the mantissa in the
next column, and take the other figures from the D column.
(2) If this difference is not given in the proportional
parts table, take P.P. readings under the next smaller
difference.
243. Examples. In Table VII' enter the following num-
bers and their logarithms, taking all readings from the table.
Show all computations on a page following Table VII.
70829, 238495.06, 1007.61, 500.0089, 8799920.678,
4900.3240, 30005, 800062, 4300091, 699.714, 5.30792,
19.9049.
244. Miscellaneous Examples. On the same page with
Table VII rule Table VIII for the entry of the examples
below and those in paragraph 249. Make entries as in
Table VII and show all computation.
11007, 30000, 54431, 472, 7980, 3466145, 427214.96,
1C97, 20.0084, 7290.632, 3.008, 47.979.
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232 TECHNICxVL ALGEBRA 246
§ 2. THE LOGARITHM OF A NUMBER LESS THAN
UNITY
246. Sign of the Characteristic. Suppose that we wish
Co determine the characteristic of the logarithm of.l; in
other words suppose we wish to know what power of 10
will equal .1.
Now
100 = 1
and
101 = 10
but
1 = 10 times .1
and
10 = 100 times .1
When raised to the zero power, therefore, 10 becomes
10 times .1 and when raised to the first power it becomes
100 times .1.
As even a zero power gives a result ten times too large
it is evident that 10 can equal .1 only when raised to a
power less than 0, and therefore to some negative power.
This power may be determined as follows:
.1=^=10-.
01= — = — = 10-2
100 102 ^^ f
•^^ " 1000 " 103"^^"^'
Therefore .1 can be obtained by raising 10 to a power
indicated by the exponent —1, that is, .1 = 10"^.
But by definition, the exponent of the power to which
the base of the system must be raised to equal a given
number, is the logarithm of the number.
Therefore
(a) log .1=-1
and
(6) log .01 = -2
and
(c) log .001 = -3
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247 LOGARITHMS 233
In (a) it is seen that .1 is obtained from 10 by raising
10 to a power indicated by the exponent —1.
In (6) .01 is obtained by raising 10 to a power indicated
by the exponent —2.
In (c) .001 is obtained by raising 10 to a power indicated
by the exponent —3.
By continuing this analysis with any decimal it would
be foimd that the characteristic of the logarithm of a deci-
mal is always negative.
246. Unit Value of Characteristic. The unit value of
the characteristic may be determined from a consideration
of some decimal, for example .306.
It is evident that .306 is greater than .1 and less than 1.
But log .1= -1 and log 1 = 0.
Therefore .^06 can be obtained from 10 by raising 10
to a power indicated by an exponent greater than —1,
and less than 0, i.e., —1+ a positive decimal; in other
words the logarithm of .306 equals — 1+a decimal.
Therefore log .306 = 1.485721.
The minus sign is placed immediately above the character-
istic to show that the characteristic only, is negative, while the
mantissa is positive.
It must never be placed in front of but always above
the characteristic.
In the preceding work it has been shown that the char-
acteristic of the log .306 is —1. But the first figure in .306
is one place from the decimal point. In this instance
therefore, as in (a), (fe), and (c),the characteristic of the
logarithm of the decimal is negative and in unit value
equals the number of places of the first significant figure,
from the decimal point.
247. Another Illustration. The decimal .042 is less than
.1 which equals 10"^ and is greater than .01 which equals
10-2.
The power therefore to which 10 must be raised to give
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234 TECHNICAL ALGEBRA 248
.042 is less than the power indicated by the exponent — 1
and is greater than the power indicated by the exponent — 2.
But by definition, the exponent of the power to which 10
must be raised to equal a given number is thr logarithm
of the number.
Therefore the logarithm of .042 is less than —1 and
greater than —2.
Therefore .042 = 10"^+'*^*^^°^^
(See also paragraph 234.)
248. Characteristic of Logarithm of a Number Less than
Unity. It is evident from the preceding illustrations that
the characteristic of the logarithm of a decimal is negative
and in unit value equals the number of places of the first
significant figure of the decimal from the decimal point.
The first significant figure of .042 is two places from the
decimal point; the characteristic is —2. Therefore
Rule 11. The characteristic of the logarithm of a number
less than 1, in sign is negative and in unit value is equal to
the number of places of the first significant figure of the given
number from the decimal point.
249. Examples. Under the heading Decimals enter
the following with their logarithms in Table VIII and on a
subsequent page show all computations:
.428, .0343, .00072, .0700486, .00009915671, .000008409,
.17, .0056007, .95064, .70147, .314087, .0127078.
Show also the readings for the logarithms of the follow-
ing numbers:
1. 1.009678. 2. 581.639.
3. .00049637. 4. .072385.
5. 3891.739. G. .8276309. •
7. 18.46708. 0. 4.72381.
9. .008372008. 10. 542.39X10"*.
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261 LOGARITHMS 235
250. Summary. Following is a summary of the essen-
tial facts presented in the preceding pages of this
chapter.
1. Logarithms are exponents.
2. A logarithm consists of two parts:
(a) Integral, called the characteristic^ ascertained by
inspection;
(6) Decimal, called the mantissa, given in the tables.
3. The characteristic of the logarithm of a number
greater than 1, in sign is always positive and in unit value
is one less than the number of integral figures in the
number.
4. The characteristic of the logarithm of a number
less than 1, in sign is always negative and in unit value is
equal to the number of places of the first significant figure
of the number from the decimal point.
5. The mantissa of a number is not affected by the
position of the decimal point in that number.
6. The mantissa of the logarithm of a number having
less than four figures is in the column in the same hori-
zontal line with the given number.
7. The mantissa of the logarithm of a number having
four figures is in the horizontal line with the first three
figures of the number, in the column headed by the* fourth
figure of the given number.
8. The logarithm of a number having more than four
figures is obtained by adding to the logarithm of the first
four figures, either
(1) the product of the tabular difference by the remain-
ing figures preceded by a decimal point; or
(2) the readings from the table of proportional parts.
251. Antilogarithms. The natural number corresponding
to any given logarithm is called an antilogarithm. It can
be obtained by the inverse of the processes employed in
determining the logarithm of a natural number.
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236 TECHNICAL ALGEBRA 251
In the work-book enter the following:
Antilog 2.589703 =
Disregarding the characteristic 2, find the leading
figures 58 in the table of logarithms of numbers in the
column.
Place the index finger of the left hand under these figures
and move it down the column until it is under 9, the
third figure of the given mantissa .589703.
Move the finger to the right along the line or the line
above, until it is under the mantissa next smaller than
.589703.
Write this mantissa in the work-book under the given
mantissa.
After it write an equality sign followed by the natural
number to which it corresponds, the first three figures
of which are in the N column, in the same horizontal line,
and the fourth figure is at the top of the colmnn in which
the mantissa was read.
(1) Subtract the second mantissa from the given man-
tissa.
Determine the figure of the natural number to which
the remainder is equal, by the P.P. table as follows:
Subtract the mantissa read, from the mantissa in the
next column.
Under this difference in the table of proportional parts,
find the niunber equal to or smaller than the given remain-
der. Enter this niunber under the remainder obtained
in(l).
The figure at the left end of the line in which the number
was read is the fifth figure of the natural number.
Write the five figures now found, as the antilog*
2.589703 as follows: log-i* 2.589703 = .
* Antilog is symbolized log~^ the dash 1 signifying anti. Observe
that the S3anboI is not minus 1, but dash 1.
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263 LOGARITHMS 237
262. Position of Decimal Point. The number obtained
in paragraph 251 may be pointed off by reference to the
characteristic 2.
Does its positive sign indicate that the natural number of
whose logarithm it is the characteristic, is integral or decimal?
What is indicated by its unit value?
Point off the required number of integral figures and under
the work now entered write reason for the operation as follows:
Sign of the characteristic 2 is (state whether + or — ).
Therefore the natural number is (state whether greater
or less than 1).
The unit value of the characteristic is (state how many).
Therefore the natural number has (state how many
integral figures).
263. Examples. The following arrangement is suggested
for the work of this paragraph:
I.80G259
248 =*6401
11
7 =1
40
41=6
.640116
What to do when the leading figures change in the line
and when a difference gives no reading is shown below:
3.700364
58 =5016
6 =0
.00501607
60
60=7,
Determine and fill in the omitted entries in the table on
page 238, showing all readings.
* This symbol is a combination of a dash (from log-* meaning
antilog) and an equality sign. It means and should be read ** whose
antilog equals.''
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238
TECHNICAL ALGEBRA
254
Table IX
ANTILOGARITHMS
No.
Logarithm.
Number.
No.
Logarithm.
Number.
1
2
3
4
5
6
2.706184
2.706184
1.580462
3.418051
3.280096
4.121873
7
8
9
10
11
12
2.900173
1.607585
3.526319
i. 470008
1.342004
.087267
§ 3. NATURAL OR HYPERBOLIC LOGARITHMS
254. Systems of Logarithms. There are three kinds or
systems of logarithms:
1. Napierian logarithms, pubUshed 1614.
2. Common or Briggs' logarithms, pubUshed 1617.
3. Natural, hyperholic, or Speidelian logarithms, pubUshed
1619.
In elementary, practical computation by logarithms, com-
mon or Briggs' logarithms are used almost exclusively and are
the logarithms always meant when one speaks of logarithms.
The last two systems have the foUowing bases:
Briggs' or common, base 10.
Nattiral or hyperbolic, base e = 2.718284.
The logarithms of numbers in these systems are denoted
as follows:
Common, log 28.341.
Natural, log. 28,341 or hyp * log 23.341.
In the computations of higher mathematics where it
is understood that the logarithms employed in formulas
are to base e, a common logarithm is denoted by subscript.
Thus logi3 a.
255. Natural or HyperboUc Logarithms. Many en-
gineering formulas have been derived by higher mathe-
matics and therefore involve logarithms to base e.
* '* hyp " means hyperbolic.
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266
LOGARITHMS
239
When such logarithms are required they may be deter-
mined in two ways:
(1) From a table of hyperbolic logarithms.
(2) By multiplying the common logarithms by 2.3026,
generally used as 2.3.
From (2) it will be noted that the logarithm of any
number to base e equals 2.3 times the logarithm of the number
to base 10.
Thus loge a = 2.3 logio a.
Solve the equation for logio a.
Therefore to reduce Briggs' logarithms to Natural, mul-
tiply by what?
To reduce Natural to Briggs', divide by what?
266. Examples. Any number whose use as a multiplier
converts one quantity into another quantity is called a
coefficient, conversion factor, constant, or modulus.
The computation of the omitted entries in the following
table will give practice in the use of the conversion factor
2.3 in transforming common or Briggs' logarithms into
Natural or hyperboUc logarithms.
Rule the table in the work-book and compute and fill
in the omitted entries.
Table X
USE OF CONVERSION FACTOR
Ex.
Number.
^°«io
I^ogg
1
296.4
2
31.065
3
5.834
4
1796.3
5
85478
6
60.007
7
138.009
8
4.6342
9
71324.6
10
5009.61
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240
TECHNICAL ALGEBRA
267
257- How the Modulus is Obtained. Following is a
proof that the Natural ^o-ir'thm of a number equals
2.3 times its common logarithm.
Given n any positive nimiber,
a=logen,
and l>=logion.
Prove a = 2.30266,
or log«n = 2.3026 logj^n.
(1) a = log«n
and 6 = logjQn
(2) /. €f=n
and 10* =n
<=10*
6=10«
(3)
(4)
(5)
log,n« =r
Hyp
Def log
=ity Ax
-Root Ax
Def log
(6) /.
(7) But
(^) •'• "^logj^2.71828
(9) But log 2.71828 = .434294
(10) /.
a = i
logio^
6 = 2.71828
h
a = ;
a=-
.434294
(11) /. log* n = 2.3026 logj^n
Mul Ax and Div Ax
Notation
Subs Ax
Table
= 2.30266 Subs Ax
(1) and Subs Ax
The number in the denominator of (10) is known in
mathematics as the modulus of the common system of
logarithms becaitse by its use Natural logarithms are reduced
to common logarithms.
268. Hyperbolic Logarithm of a Decimal. A simple
way of determining the logarithm of a decimal to base e
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2G9 LOGARITHMS 241
by the use of the conversion factor 2.3026 or 2.3 is to
multiply the mantissa and characteristic separately as shown
below. Observe that the characteristic is written following
the mantissa.
Required the hjrp log .09685.
.986100-2
2.3 2.3
2958300
1972200
2.2680300-4.6
4.6
3.668030
hyp log .09685=3.6668030
259, Ways of Denoting Ciphers. When several ciphers
precede the first significant figure of a decimal they may be
denoted by:
(1) One cipher and a subscript whose unit value equals
the nimiber of ciphers.
(2) Indicated multiplication of the significant figures
by 10 with a negative exponent having the same
number of units as the number of decimal places.
(3) Indicated multiplication of the significant figures
as a decimal, by 10 with a negative exponent hav-
ing the same number of imits as the number of
ciphers immediately following the decimal point.
Thus .000498 and .0000007345 may be written
(1) .03498 and .067345, or
(2) 498X10-6 and 7345X10'!^ or
* (3) .498X10-3 and .7345X10-«.
If a number ends in several ciphers it may be denoted
by an indicated multiplication by 10 with an exponent
whose unit value equals the number of ciphers.
Thus 1832000000 is conveniently denoted by 1832 XIO^.
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242
TECHNICAL ALGEBRA
260
260. Examples. Compute the Natural logarithms of
the following numbers, arranging the work as in paragraph
256.
1. .0864.
4. .O3731.
7. .O48O95I.
10. .0,9812003.
13. .0,720041.
16. .00826054.
2. .48064.
6. .005614.
8. .0062149.
11. .817216.
14. .631142.
17. .049671.
3. .32196.
6. .073485.
9. .0(50009.
12. .0981467.
16. .05480072.
18. .0029346.
§ 4. LOGARITHM OF A PRODUCT
261. How Logarithms are Used. Logarithms are expo-
nents. Their use is therefore governed by the laws of
exponents in the following operations:
Multiplication,
Division,
Involution,
Evolution,
Solution of an Exponential.
. #
262. Multiplication by Logarithms. Since in the
Briggs' system all numbers are regarded as powers of 10,
the multiplication of two or more factors is only the multi-
plication of two or more 10s whose exponents are the
logarithms of the respective factors.
The exponents (logarithms) are therefore added.
Thus 43.4 X 2.91 = 10^ ^37490 x 10463893
101-637490
10 -463893
102.101383
Therefore 43.4X2.91 = 102ioi383
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262 LOGARITHMS 243
What this product is, may be determined by reading
the antilogarithm of 2.101383 from the table.
Thus 2.101383
059 sl262
324
310 =9
140
138=4
The characteristic 2 denotes three integral figures.
Therefore 43.4X2.91 = 126.294, which is exactly the
same as .would be obtained by arithmetical multiplication.
In practice, logarithms are never shown as powers of 10
bttt are set down directly.
Thus 43.4X2.91 = 126.294
1.637490
■463893
2.101383
059 =1262
324
310 =9
140
138=4
One or More Factors Decimal.
If one or more of the factors of a product are decimals
and their logarithms therefore have negative characteristics,
there are three ways of doing the work:
(1) Set down the logarithms directly with their char-
acteristics as determined.
(2) Both add and denote the subtraction of 10, before
adding the logarithms.
(3) Set down the logarithms of the decimal, increased
by 10.
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244 TECHNICAL ALGEBRA 263
us
2454.1 X. 00568
(1)
18
3.389875
3.754348
1.144241
(2) .
18
3.389875
7.754348-10
11.144241-10
(3)
18
3.389875
7.754348
*1 1.144241
3951 =15
291
281 s9
100
94s3
13.939*
Whenever logarithms having negative characteristics ore
to he added or subtracted it is suggested that they he set duun
as shown in (3).
263. Examples in Multiplication. Solve the following
as indicated in the preceding paragraph:
1. 10.09X687. 2. 381.56X16.9217.
3. 7.298X1.654. 4. .5341X13.908.
6. 181.96X31.148. 6. .00715X1.0083.
7. 5.037X236.84. 8. 29.53 X. 42159.
9. .0331287X12.64. 10. .0428961 X. 084507.
11. .81965X .037964X 15.823X .073854.
12. .067864X1384.19X.78113X144.58X.0834.
* Two integral figures are pointed off in the result because the
actual characteristic is 1, characteristic 11 being 10 too large.
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265 LOGARITHMS 24J
§6. LOGARITHM OF A QUOTIENT
264. Division by Logarithms^ or the Logarithm of a
Fraction. In the division of algebraic quantities the
exponents of the quantities in the divisor are subtracted
from the exponents of the same quantities in the dividend.
Therefore, in division by logarithms the logarithm of the
divisor is subtracted from the logarithm of the dividend.
A fraction is an indicated division of the numerator
by the denominator.
Therefore the value of a fraction may he determined by
subtracting the logarithm of the denominator from the logarithm
of the numerator and by reading the antilogarithm of the
remainder.
When the logarithm of the dividend is smaller than the
logarithm of the divisor it should be increased by one or
more 10s, as may be necessary.
Thus
(1) 3.728^145.73=^^558 (2) ^J=^^ffii£
10.571476 460
2.163549 89 7.754348 75
8.407927 3.389893 18
01_=2558 4.364455 ~
26 363 =2314
17^=1 92
90 75 =4
84=5 170
169=9
265. Examples in Division. Solve the following examples
by logarithms:
1. 12.803^1.728. 2. 39.1857 -1-7.264.
3. 3.7201-^14.96. 4. .082305 -^ 4.7812.
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246
TECHNICAL ALGEBRA
266
9.
11.
13.
15.
17.
19. ;^
54.<sro7
50.7002*
.O3I2O8
.047862*
.970008
1.20705'
42.984
.90358*
.30876
.00043581*
.083451
.0041486*
11.6309
.72184 '
45.8671
21.
23.
27.
29.
31.
33.
318.0075*
.014592
.0056847*
21.8429
178.0096'
1396.75
23815.62'
.063581
.0043217*
398.41 X 10 -»
.54189 X10»"
15.7847 X. 35685
.048129X47.842'
5173.96X10-'^
5300.48 X. 066814*
.084056 X. 0054734
6.
10.
12.
14.
16.
18.
20.
22.
21.
26.
28.
30.
32.
34.
1.9684
8.17201*
.00835
.74802'
.98305
14.0091'
1.72914
16.3429'
182.639
73.6425'
514.292
3084.43*
.3183
.074129'
.00096425
.0472638 '
.0690723
.O46I895'
.44571
.0059412"
408.039
3423.086'
.03465125
14962.9X10-*'
.076485X10-'
17.632 X. 340084*
.017358X5.63419
496.431 X. 0054866*
.914184X30.639X84.69
.071945X.18726X44.069*
35. 17.28X
.021968X1.70345
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267 LOGARITHMS 247
§ 6. LOGARITHM OF A POWER
266. Involution by Logarithms. Involution is the
process of finding any power of a number.
The process and the reason for it should be evident
from the following:
(3.968)3 = (10-598572)3 = 62.476
.598572
3
1.795716
672 =6247
44
41 =6
30
28=4
In practice the power of 10 is not shown.
Thus ( . 09867)4 =.0494785
2.994185
4
5.976740
17=9478
23
23=5
The characteristic of the product is 5 because 4X2=8: and
8+3 which was carried, gives 5.
267. Multiply Characteristic and Mantissa, Separately.
When the exponent of the power of a decimal has more than
one figure the computation is best made by separate multi-
plication of mantissa and characteristic.
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248 TECHNICAL ALGEBRA 268
Thus .006348 g^ = . 037300
.802637-3
.65.65
. 4013185
4815822
.52171405-1.95
1.95
5.571714
09^ =3730
50
^=4
In the two preceding illustrations the logarithm of the
given number was multiplied by the exponent of the power
* because to raise a number to a power is only ta raise 10
with some exponent to the given power. In Chapter XIII is it
shown that any power of a monomial is obtained by multi-
plying its exponent by the exponent of the power. (See
paragraph 190.) Therefore the following
Rule for Logarithm of a Power. To obtain any
power of a number, multiply its logarithm by the exponent of the
power and read the antilogarithm.
268. Examples in Powers. Determine the following
powers by logarithms as shown in the illustrations of the
two preceding paragraphs. The first four examples, as
will be seen, are different powers of x.
1. 3.1415926«.
2. 3.1415926».
3. 3.1415926*.
4. 3.1415926*.
6. .71834^
6. .05738^^
'■ a)*
'■ (^)'
9. 713.6042S.
10. .0,74209'.
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269
LOGARITHMS
249
11.
13.
/ i.i84 y
V,5631/ •
)■•
/7854
V231
^^ /.86211X42.19\»
^^- V 2.296 ) •
/ 1.2986X.124 Y
V .70201 / '
19. 1942.683*.
21. 37.6342X8.0097*.
23. 4317.92^X72834 J.
26. .63591 «^2.7314».
27. (.49518»X6.70158*)».
.5 4631 »X34.6 05»
696.432X. 0009248*
.82176 A »
31
/ 7.143^X.S
\ .00782
824* 7 •
33. 8.0053942«-6.7084«.
12. (^^'-)\
\. 012685/
14. (29.608X.60312)».
16
/ .08433XlO» y
\ 3.406 / •
^.' /27045X10-A*
^^ \ 2.298 ; •
20. 15761. 8*.
22. .08379«X. 92046*.
24. (2.7803»X. 84507*) «.
26. (1.7046»X3.9.5167)*.
17.964»X. 00952X1728
34.762«
/ 7rX9.345» y
• \ 4.00732 / '
21.436^.04573'
208.92* -^5.7814^•
19.783X.24685»-1.9632»
5609.41*
30.
32.
34.
§ 7. LOGARITHM OF A ROOT
269. Evolution by Logarithms. In Chapter XIII it is
shown that a root is only a power whose exponent is
fractional.
Write a rule for the determination of any root of a
number by logarithms.
How TO Extract the Root of a Decimal when
Root Index is Positive and Evenly Divisible. When the
root of a decimal is required and the root index is posi-
tive and evenly divisible in the negative characteristic, the
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250 TECHNICAL ALGEBRA 269
division may be performed with the negative characteristic
written in its usual place preceding the mantissa.
Thus v^.0072861 3) 5.862495 489
1.28749^ 6
V .0350187 2 )4.700591 531
2.350296 60
What to do when the Root Index is Negative, or
when not Evenly Divisible. When the root index is nega-
tive, or not evenly divisible in a negative characteristic,
there are several ways of determining the quotient as
shown in subsequent paragraphs.
The two best are the following:
(1) When the root index is positive and not evenly divi-
sible, both add to the logarithm of the given number, and
indicate the subtraction from it, of the smallest integral
multiple of the root index which will eliminate the negative
characteristic. Divide the result by the root index.
(2) When the root index is negative, even though it is
evenly divisible in the characteristic, find the excess of
the negative characteristic over the positive mantissa.
Divide the result by the root index.
(1) Positive Root Index, not evenly Divisible.
V.0007154 = .042896
4.854549
4.6
2.3 )1.454549 -4.6
.632413 -2
356=4289
57
61=6
In the preceding illustration, the smallest integral multiple
of 2.3 which will eliminate 4 is 2X2.3 =4.6.
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270
LOGARITHMS
251
4.6 was therefore added and its subtraction denoted, before
the division was performed.
The characteristic of the quotient, as shown, is 2.
(2) Negative Root Index.
"V.077584 = 236.247
8.879898
-3 ) -7.120102 *
2.373367
280 =2362
87
74 =4
130
129=7
270. Examples in Roots. Compute the following roots
by logarithms as shown in the illustrations of paragraph 269:
1. V.002948.
3. *V714629.
6. ^•V.004157.
2. V.052915.
4. V545.I7.
6. *V.061208.
7. V3.1415926.
9. "V.27806.
11. V.004248.
13. 'Vl294.63.
16. V.O46O92I8.
17. "V.012983.
IS. "V29.856.
21. V.017648.
23. "*V;0()63057.
8. V.7821.
10. V2.38007.
12. V.867502.
14. V.096317.
16. V12.OOO87.
18
20
22. Vl604.965
24. V.091872.
V.71405.
V.0678201.
* —7.120102 is the excess of minus over plu9 in 3.879398 »nd is
obtained by subtracting .879898 from 8.
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252
TECHNICAL ALGEBRA
871
26. 'V29.6294.
28. Vl78.4219.
26. *V341.738.
27. "V.013968.
29. "V.0085621.
81. V.83176.
33. V2.85009x3V.044216.
36. V.0017298 X V21.0634 x Vl42.833.
30. ^V4498.706.
32. *V37.642L
34. 175-2V.061287.
§ 8. SOLUTION OF AN EXPONENTIAL EQUATION
271. Definition and Illustration. An exponential equation
is one in which the unknown quantity is the exponent.
Thus 24.5*=?12.298
Equations of this kind are solved as follows:
(1) Base Greater than Unity.
log 24.5*
.'. X log 24.5
X-
X-
= log 12.298
= log 12.298
log 12.298
log 24.5
1.089835
1.389166
552
283
.037028
.142702
318
31
119
187
199
187
.037360
.142754 ,
.142754
1.894606
593=7845
13
lls2
x=. 78452
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271 LOGARITHMS 253
(2) Base Decimal. Since negative numbers have no
logarithms, the following illustrates what to do when both
of the known quantities are decimal and when therefore the
algebraic sum of characteristic and m/intissa is negative:
.0543''=. 2347
1.370513 "^
=
o=
2.734800
- .629487
-1.265200
.629487
1.265200
(3) One Term of Fraction, Negative. When only
one number is decimal, the operations are as follows:
1.342» = . 08763
2.942653
6=
6=
6=
.127753
-1.057347
.127753
1.057347
.127753
As shown b is negative. The value of the fraction is
therefore computed by logarithms and the antilog is written
with a minus sign before it.
In practice, work like that in the preceding illustrations
should be set down as follows:
(1) 24.5'= 12.298
1.089835
552
1.389166
283
and so on as shown in (1).
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254 TECHNICAL ALGEBRA 272
(2) .0543° = . 2347
1.370 513
2.734800
.629487
1.26520U
and so on as shown in (1).
272. When the Exponent is Negative. In case the
unknown exponent is negative , the operations are as follows:
1.342 -«»= .08763
_ 2^942653
.127753
1.057347
.127753
Observe that the last equation is obtained from the
second equation by multiplying by —1.
273. Examples. Solve the following equations:
1. 19.87' =345.2. 2. .2467*' = 12.675.
3. 3.7983^=449.31. 4. .00434' = .097504.
6. .01432^^ = .18335. 6. .645001'' =27.6404.
7. .5264-^ = 1.9234. 8. .01872 "'^ = .80441.
9. .0734-' = 1.11054. 10. .708-^ =22.007.
11. .00651-2' =3.7841. 12. 56.908' = .15925.
13. .4084' =7.0382. 14. .07329'' = .23194.
274. Summary of Laws with Formulas. There are jive
laws for logarithmic computationj which may be formulated
as follows:
I. Logarithm of a Product, ab.
log (ab) =log a+log b.
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276 LOGARITHMS 255
a
II. Logarithm of a Fraction, -- .
b
a
log r T^og a— log b.
b
III. Logarithm of]a Power, a^.
loga^=bloga.
IV. Logarithm of a Root, Va.
ft/- log a
log Va=— — .
b
V. Logarithm of an Exponential, a' =b.
X log a =log b.
If either b or a is decimal,
logb
log a'
If both 6 and o are decimal, or greater than 1,
logb
x=:
log a
§ 9. MODEL SOLUTIONS
275. Multiplication. (1) Factors Greater than
Unity.
346.84X93.72 = 32505.8
2.540079
50
1.971832
4.511961
883 =3250
78
67 =5
110
107=8
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256
TECHNICAL ALGEBRA
276
(2) Factors Decimal.
.034684 X .9372 = .032506
50
8.540079
9.971832
and so on as in (1).
276. Division.
346.84
2.511961
93.72
=3.7008
2.540129
1.971832
.034684
.9372
8.540129
9.971832
= .037008
.568297
2.568297
02=3700
02=3700
95
95
94s8
94=8
277. Powers. In the determination of a power by
logarithms, there are two cases:
1. Power of a ntimber greater than unity.
2. Power of a decimal.
Model solutions for each of these cases follow:
1. Power, Characteristic Positive or Zero.
3.46842.3 = 17^4702
.540129
2.3
1620387
1080258
1.2422967
3 =1747
47*
50=2
* Observe that figures in excess of the 6th are retained and used
in the differences.
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277 LOGARITHMS 257
2. Power, Characteristic Negative.
Following are illustrations of six different methods of
determining the power of a decimal:
(1) By logarithms.
(2) By separate multiplication of mantissa and char-
acteristic, and algebraic addition of the two products.
(3) By separate multiplication as in the second method,
but instead of algebraic addition of the products, sub-
traction of the product of the characteristic from the product
of the mantissa.
(4) Same as the third method, except that the mantissa
product is determined by logarithms instead of by arith-
metical multiplication.
(5) By multiplication after algebraic addition of char-
acteristic and mantissa.
(6) By multiplication of the logarithm increased by 10.
First Method. Entirely by Logarithms.
.900483-^1 = 0*.
log a* = 6 log a = fc X 1.954473
log a* =-*. 0455276.
logo* =-(.0455276).
log log a*= -(log .045527+log 3.51)
67
8.658202
.545307
9.203576
TL-
= 1598
logo'
= -.1598=
= 1.840200
169 s
56921
31
323
=5
.900488S1
= .69215
♦Obtained from T.954473 by finding the excess of the negative
characteristic over the positive mantissa.
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258 TECHNICAL ALGEBRA 277
As shown, — .1698 the logarithm of a*, is a negative decimal.
Therefore it is not in tabular form, is not a mantissa, and
its logarithm cannot be read in any table because all mantissas
are positive. It was therefore reduced to tabular form by sub-
traction from a number, one greater than its integral part, zero.
It was therefore subtracted from 1, which gave .840200
as follows:
1.
.1598
.840200
But this result is 1 too large; the subtraction of 1 is
therefore denoted by the negative characteristic I, giving
1.840200 as the tabular logarithm of a*.
Every negative logarithm may be reduced to tabular
form by the same process. Therefore the
Rule for Reducing a Negative Logarithm to
Tabular Form:
Subtract the negative logarithm from a number one greater
than its integral part and write for the negative characteristic
of the result, the number from which subtraction was made.
Thus - 3.285634 = 1.714366.
In the preceding solution, the mathematics of the entire
process has been shown.
In practice the following arrangement is preferable:
.90048351 = .69215
1.954473X3.51
-.045527X3.51
-(.045527X3.51) = -.1598 = 1840200
169=692.1
67 31
8.658202 32=5
.545307
9.203576
77 s 1598
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m LOGARITHMS 259
Second Method. Separate multiplication of mantissa
and characteristic, and the algebraic addition of the two
products.
.900483 51 = .69215
.954473 -1
3.51 3.51
954473
4772365
2863419
3.350200 -3.51
3.350200
• .159800
1.840200
169=6921
31
32s5
Third Method. Same as the second except that instead
of determining excess of negative over positive after multi-
plication, the product of the characteristic is subtracted directiy
from the product of the mantissa.
.900483" =.69215
.954473 -1
3.51 3.51
954473
4772365
2863419
3.350200 -3.51
3. 51
1.840200
169=6921
31
32=5
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260 TECHNICAL ALGEBRA 277
FouBTH Method. Same as the third except that the
product of the mantissa is determined by logarithms.
.900483 " = .69215
3.51 X. 954473 -3.51
9.979730
32
14
.545307
.525070
45=3350
25
26=2
3.350200-3.51
3.51
1.840200
169=6921
31
32s5
Fifth Method. Multiplication after algebraic addi-
tion of characteristic and mantissa.
.900483 51 = .69215
1.954473
-.045527
3.51
45527
227635
136581
-.15979977
' 1.840200
169s6921
31 '
32s5
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278 LOGAEITHMS 261
Sixth Method. Multiplication of the logarithm in-
creased by 10.
.900483" = .69215
9.954473
. 3.51
9954473
49772365
29863419
* 34.94020023
35.1
1.840200
169=6921
31
32=5
278. Roots. In the determination of a root by loga-
rithms there are two cases:
1. Root of a number greater than unity
2. Root of a decimaL
Following, are model solutions for each of these cases:
1 Root, Charactbbistic Positive or Zero.
''•V3;4684= 1.71727
By Division.
By Logarithms.
2.3).540129(.234839
46 770 =1717
.540129
9.732474
80
69
16
69
51 =2
73
111
180 ^ '
9.732497
92
177=7 J
.361728
192
1.370769
184
89
.234839 698 =2348
770 =1717 71
69
200
69
51=2
55=3
160
180
166=9
177=7
♦Log .90048 was increased by 10 and then multiplied by 3.51.
Therefore the starred number above, is 35.1 too large.
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262 TECHNIC.VL ALGEBRA 278*
2. Root, Characteristic Negative.
How to operate when the root of a decimal is wanted
depends on the root index which may be
(0) Evenly divisible in the negative characteristic,
(6) Not evenly divisible in the negative characteristic.
(a) Root Index Evenly Containable. A root index
evenly containable in the negative characteristic should be
directly divided into the logarithm without transformation
of any kind.
Thus v^.007286 and V.08723 and v/.00006504
3 )5.862498 ' 2 )2.940666 5) 5.813181
1.287499 1.470333 1.162637
(6) Root Index Not Evenly Containable. Following
are illustrations of seven different methods of determining
the root of a decimal when the root index is not .evenly divisible
in the negative characteristic.
(1) By algebraic addition of negative characteristic and
positive mantissa, with division by logarithms.
(2) Same as first except that the division is performed
arithmetically.
(3) By separate arithmetic division of mantissa and
characteristic, with algebraic addition of the two quotients.
(4) Same as the third except that the quotient of the
characteristic is subtracted directly from the quotient of
the mantissa.
(5) By separate logarithmic division of mantissa and
negative characteristic, with direct subtraction of the
characteristic quotient from the mantissa quotient.
(6) By adding to the logarithm of the given number,
and by denoting the subtraction from it, of the smallest
integral multiple of the root index which mil eliminate the
negative characteristic^ followed by arithmetical division,
and reduction as in the fifth method.
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278 LOGARITHMS 263
(7) Same as the sixth except that the division of the
positive number is performed by logarithms.
FiKST MsTHOD. Algebraic addition of the negative
characteristic and the positive mantissa, with division by log-
arithms.
, »/- log a 1.954473
1 .»/- - 045527
log Vo =
log Vo = ■
b
045527
b
log log v^a = -(log .045527 -log 3.51)
8.658269 202
.545307 67
8.112962
40=1297
22
33sl
log v^= - .012971 = 1.987029
6996 =9705
33
31=7
'•"V.90048 = . 970574 20
— 18=4
Tn the preceding, the mathematics of the entire process
has been shown in order that it may be clearly miderstood.
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264 TECHNICAL ALGEBRA 278
In practice, the following arrangement is preferable:
"V .90048= .970574
1.954473 -.045527
3.51
.045527
3.51
8.658269
.545307
8.112962
3.51
= -.012971 =
= 1.987029
6996 1
=9705
33
31 :
20
=7
= 19.07
18t
s4
22
33=1
Second Method. Same as the first except that the division
is performed arithmetically.
Show in full the application of this method to the example
employed in the first method.
Third Method. Separate arithmetical division of man-
tissa and characteristic.
3si\/.90048 = >?^a
1 »/- logo 1.954473
^-1 .954473
3.51"^ 3.51
^ .954473 1_
3.51 3.51
Show in full the application of this method to the example,
and reduce the result by algebraic addition of the two
quotients.
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278 LOGARITHMS 265
Fourth Method. Same as the third except that the
result is reduced by direct subtraction of the characteristic
quotient from the mantissa quotient.
Show in full the application of this method to the
example.
Fifth Method. Separate logarithmic division of mantissa
and characteristic, with direct subtraction of the quotient of
the characteristic from the quotient of the mantissa.
Show in full the application of this method to the
example.
Sixth Method. Addition to the logarithm of the given
number and by denoted subtraction from it, of the smallest
integral multiple of the root index which will eliminate the
negative characteristic, with arithmetical division by the
root index and direct subtraction of the quotient of the char-
acteristic from the quotient of the mantissa.
3-5V.90048 = ^
, 6A- log a 1.954473
1.954473
3.51
3.51 )3.464473-3.51
.987029-1
In the preceding/ 3.51 the root index was not evenly divisible
in the negative characteristic T.
Therefore, the negative characteristic was eliminated by add-
ing to it, 3.51.
This gave 3.464473.
But this was 3.51 too large.
Therefore, the subtraction of 3.51 was denoted as shown, and
the quotient obtained by arithmetic division.
Show in full the application of this method to the example.
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260 TECHNICAL ALGEBRA 279
Seventh Method. Same as the sixth except that the
division of the positive number, is performed by logarithms.
Show in full the application of this method to the example.
§10. LOGARITHMIC COMPUTATION
279. Miscellaneous Examples. This paragraph contains
complex and difficult work, but by patience and study of
the model solutions, even the student to whom mathematics
is particularly difficult can solve any example in the list and
will be able to make any computation from any formula
of whatever kind.
Many of the results will be found absurd from a practical
standpoint but are intentionally so, in order that it may
be definitely known that the position of the decimal point in
an antilogarithm is determined by the characteristic and
not by imagination of what the result ought to be.
However much you may use tables, never be satisfied
merely to copy a number. Always compare the copied
number with the number in the table and be sure it is
correct.
Do not dash ahead and take one or more readings the
moment an example is seen. Take no readings at all until
the example has been studied to determine exactly what
it is, and not only by what operations it may be solved
but in what order the operations are best performed and
what is the best method.
In each of the examples from 1 to 70 inclusive, the
first factor is a natural number and the second factor is a
logarithm. This means that the first seventy examples
in this paragraph are examples in powers. Why? Because
any power of a number is determined by multiplying its
logarithm by the exponent of the power, and reading the
antilogarithm of the product.
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279 LOGARITHMS 267
Thus in example 1 on page 268, .72X2.786340,
.72 is the exponent of the power,
2.786340 is the logarithm of the number whose .72 power is
wanted.
Now log-^* 2.786340 =.061142. The operations, therefore,
are exactly the same as though the example were given in the
form
.061142'^^
By the first method of paragraph 277 we have the following
solution:
2.786340X.72 =
-1.213660X.72 =
-a.213660X.72) = -
.083861
215
. 215
9.857332
.873836 =
: 1.126164
31 = 1337
33 = 1
1.941430
12 =8738
18
15 =3
30=6
.13371
=.061142
.72
It is suggested that the results for the first seventy be
expressed as shown, that is, as an equation specifying to
what power of what number the final result is equal.
Confusion may be avoided by not attempting all six
methods of solution until at least twenty examples have
been worked by the first method and in every instance
checked by the second or third. Then at least ten should
*Log~^ denotes antilog. See page 236.
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268
TECHNICAL ALGEBRA
279
be worked by all methods, after which, use the method
which gives the least trouble and takes the least time.
1. .72X2.786340.
3. 2.19X3.717784.
5. 1.72X1.423380.
7. 3.17X2.463418.
9. 3.9X4.617388.
11. 4.08X5.980071.
13. 2.86X2.670531.
15. .83?X 3.405039.
17. 2.16X5.208764.
19. 3.076X1.181923.
21. 4.581X5.390067.
23. .735X1329671.
25. 3.17X1.007314.
27. 2.08X3.412953.
29. .389X1324615.
31. 1.78X2.814337.
33. 1.98X1.422891.
35. 4.9X2.909097.
37. 3.42X2.739451.
39. 2.76X5.238088
41. 9.07X2.734012.
43. 3.26X7.981264.
45. 1.56X2.448892.
47. 7^X2.352864.
Id
49. i^Xi.000465.
51. 3.742X7.007869.
2. 1.98XL301030.
4. 3.98X5.396318.
6. 5.32X6.223846.
. 8. 7.89X3.327864.
10. 3.7X1.385209.
12. 7.31X8.290004.
14. 9.87XL548119.
16. 3.009X2.084361.
18. 4.007X3.304812.
20. 2.98X6.423057.
22. 3.3X2.408568.
24. 6.219X3.700638.
26. 1.73X2.029975.
28. 3.07X1.730451.
30. 3.29X5.418927.
32. 7.91X6.317235.
34. 2.71X4.074698.
36. 3.99X7.089674.
38. 2.74X5.630425.
40. .889X5.329416.
42. 2.145X5.754112.
44. .69X1.726345.
46. ^X3.698113.
4
48. ltXS.147365.
50. 13.76X2.000091.
52. 1.69X2.396214.
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279
LOGAEITHMS
269
63. 4.097X^.720899.
55. 2.48X4.739542.
67. .3198 X. 009384.
59. 2.999X0.300179.
61. 5.008X2.940837.
63. .691X1.700008.
66. 2.471X7.900079.
67. .1892X2.147826.
69. .2567X0.842087.
71. '^V.oeiu.
73. ^•^V;00522i.
75. ^•^V.000001674.
77. ^'^ V.029068.
79. ^V .00041437.
81. ^•°V.000095515.
83. ^•^V.04683.
85. -^V .002541.
87. ^■^Vxi00013572.
89. ^•°^V.152028.
91. ^•^^V.000024551.
93. •^^V.21363r
95. ^•"V.00010169.
97. •^°V.00258794.
99. •^^V.211102.
101. ^•^V.065212.
103. ^^V .000264784.
106. ^•V.081113.
107. ^'^V.054885.
54. 1.782X2.047856.
56. .786X1.007694.
68. 2.^15X5.607284.
60. 1.583X2.078149.
62. 1.9004X4.931700.
64. .238X1.000001.
66. 2.128X5.300481.
68. 1.723X5.009308.
70. 1.2098X0.341723.
72. ^•^^v:2.
74. ^•®V.00002491.
76. ^•^V.25508.
78. ^-^V .0021747.
80. ^V .24272.
82. '•^V.000000019498.
84. ^^V.35328.
86. ^•^V.012144.
88. ^•°°V.0020175,
90. '^^V.00000264885.
92. ^•V.025619.
94. ^•^^V.00501925.
96. ^•^V.0107146.
98. ^"V .537590.
100. ^•^V.000262378.
102. ^^V.0000020764.
104. ^•^V.001187.
106. ^'®^V.000000122934.
108. ^•^V.0000426997.
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270
TECHNICAL ALGEBRA
279
109.
'•'"V.oooiTaoi.
110.
112.
114.
116.
US.
120.
122.
124.
126.
128.
130.
132.
134.
136.
138.
140.
142.
144.
146.
148.
160.
162.
164.
166.
168.
160.
■""V.00002135.
111.
'•"^.0542016.
""'V.00056171.
113.
''•*°V.0000009578.
■**V.53253.
116.
'•*"V.028112.
3
■'*V.0049901.
117.
wV.022535.
.408
'■V.001404.
13.76
119.
1W8V.000108107.
s«'V.0100021.
121.
'*"^V.0872632.
^"^V.000854476.
123.
V.501197.
V.100021.
126.
^•'"V.00000079448.
""'V.000019975.
127.
•'"'"V.014055.
' '^''V'.OOP0102166.
129.
•'^V6.95165.
i.^««V2.19646.
131.
'•"V.ouios.
•''"V398.726.
133.
■'"'V.3002.
•*'Vl.007.
136.
'•"^56.7859.
'"•'^V.087264.
1
137.
••"^Vl. 34009.
'V'.87744.
139.
•"™V.06349.
'■'"V.0093008.
141.
"''V5.30477.
•""V.9721.
143.
■""V.07263X1.3467«.
•*"V.02345.
146.
'•"V.000725.
'■''V.6345.
147.
'■"V.38961.
•''V.8397.
149.
161.
'•"Vl.2963.
.301
1.936^/ .0968.
'•'*Vl.307.
"V.425.
163.
1728 X "V4.965.
•*'*'V.098006.
166.
•■'"Vl.0007.
'^''V56.7859.
167.
•'*'V398.726.
■*^V.0017238.
169.
'•"^Vl. 34009.
'•''V.3417X. 04128.
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279
LOGARITHMS
271
161.
163.
166.
167.
169.
171.
173.
174.
175.
176.
177.
178.
179.
180.
181.
182.
183.
184.
185.
186.
^•*V.002873X 1.968.
•'''V.063492^
•^^V.00039253-28.
'®'''^V3.7346^»X.072.
2.760582
.52632-^* '
8" V92. 163 X. 009459.
(.008726X19.8005) 10-'-
3.008009
.00768* X.973i*
2 .723081
162.
1'
0004215
082428
164. ^•^V.0093008-®®^
166. 2:71 \/l.30843X .0009672.
168.^
170.
1. 923'' X. 8452
1
172. (32.0807 X.08267)-396.
9.400009
-4.761 X. 9983"
12.728406
®^V;3482l'
.573
1-208 V.00076008 + .8546 X 1.985.
i.600085
•^V.092807X.4062i'
5.726008
/ -. 07341 X log .401
\ 84.5
1.9672'*ixV9.126
.00765., ..56,+i^g4oi
396.72 X'^V3.127X.493X1.7842.
27.98 X "^^ V.0972 X .385^ X .0446^
^Vl348X "•^V.0726X.309-2.
.7204^X "•^^Vl.0732X.08293X6.19-*X10-8.
3.018724 X V7.218+.1728'.
1.073961 -329.8^-^^
^ V;0726
^•^ V.018292X .1928*X 10*.
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272 TECHNICAL ALGEBEA 279
187. 17.921^ " -34.96^ ""^xVs.lSOe.
3
188. 2V799l2xi.980072"<>»+.006834-2,
1.78
189. I7.634XV 1900.0082 X 10"*.
1
190. 7 76 V.084176J - 728.1009X 10 -5.
.007568
191.
193.
194.
.0075^ X 1000.967^"^^ X'V90.872»*
.000842007
..K)9615 \/72.389^
2.006346 +^-°V.07384»
1.19253X10-3
.08273 X. 09638 ^X. 01728-'
1705.61^^^
3Vl6.0892+*^V6.2401
19u.
197.
10-'^xV.01829
^7 8X10^+6.2172
1.7348-3X159.2008-^-ixV.0841
196. .00827 8X105+6.21731-729.
V.08293^-^^
3 2
198. 50.096"8 4-.5729"'^+V3.9638X10-2.
7
jgg 3.80193 X\ . 9731 5X3. 9204 «
V3.8172^+. 17292-7
>««• -Sx-«^2^<^^"-
201
202
4
4^
/.083622X^"V3.4'l69
72.684 -.08126^-^^ *
2.007345
V.028912X.7236*
203. log 83.21 93 X. 6873 '7 x^"V-.8o2P log .728.
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279 LOGARITHMS 273.
J3.1416«+.1892 8
3^
8
1.436 -*X^'V.T9307
204. \ y\_ 2.8 .- ==+log 7.9281 ^
206. .895-*Xl.586-3x7.64^*X --3^^.008226.
206. •*®^V54.43 log 64.15*.
207. .0l728-2-**8X.0361-2 39.
208. ^^V(.2481t log 39.41)+362.58 log 572.
00496»X 14.723*
209.
210.
.0896* X. 3516-**
4563^^^^
•^Vj3r9Xl8.5-i'
mV7431 X V.6245XV34.963
185.96 hyp lo£ .3461
212. .0329"*^^V.682""^^X'*V6.129X.000317-*.
213. I-X.98 log .31642 -.0892-*.
o
214. 66X*^V.0638X48.1*.
215. .00539 X 30.08 --^2^
216. 1.04^ loge 16.178X10-^ loge.0923-3-2^
^_ .4398-^^^X.045843-2«+42.97
128.1 hyp log .3973
218. " •^V.0008429X '"^^^VrndEQ X hyp log .4086 - «.
219. .0173"3"X25.09"'^^^X'^^V.00928^-'^V.09^.
220. .7583"*X.0046r^X7.69 hyp log •^V;217\
221. V.6018'iiX.0046l"^X7.69hyplog 'WaU.
222. '^^V3184.7-2X.897PX.09426-8-V;0142+Vj.
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274
823.
224.
226.
226.
227.
TECHNICAL ALGEBRA
1.41 r
0263 -«X
.08175
.075
L962X7^'.01298
396.34-^X4.0189 ? +.3118-*X7.914--^.
-3.18
v-i
.0268^X V-.74123 log .41722.
1.009 / f
V .00845 3 log 3.1 V -.4993xhyp log .0892-^^.
7.3219
3.17 log .07893
2.93
1-.
^ 6.4208 2.09 16
7196 .0823-2 018*
3.961 -•3»2_^.0458-»29 iQg Qg9i2
- 1.38
V.012963+ .01723 " -^^^ X .4779 " ^'
279
MODEL SOLUTIONS
174.
3.008009
- 2.991991
2.991991
.00768* X .973* " .00768* X .973* " .00768* X .973*
f^ log .00768 , log .973
3
= -34.454155
io,2.mm-(^^^+'^f-^)
.475816
131
13
20.475960
18.938719
1.537241
189 = 3445
52
50=4
20
13 = 1
70
63=5
70
3.885361
j4
2)1.885361 -4
.942681 -2
1.988113
3.
3 )2.9881 13-3
.996038-1
8.942681
9.996038
18.938719
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279
177.
1.600085
LOGARITHMS
.399915
-'V.092807X.4062* "^ V.092807 X .4062*
logc\
275
-16.48253
flog6+-
loga-l
.72
8.967548
33
8.967581
9.869580
18.837161
2.837161 1.162839
.72
.065206
299
112
336
.065520
9.857332
.208188
73 = 1615
15 =
150
134=5
160
161=6
.72
1.608740
J
3)2.608740-3
9.869580
= -1.615056=2.384944.
9.601951
11
55
9.601968
8.384944
1.217024
957 = 1648
67
53 = 2
140
132 = 5
80
79=3
r_ . 07341 X log .401 ^, j-aXi-h) . I ab
179. .00765=X^ -84.56^+l^r=N-?+(^=Nc-^
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276 TECHNICAL ALGEBRA 279
„n« 2.007345 L992655 oq raqoh
202. » , = - o / = -23.563211
V.0289PX.7236 V .02891 ^X. 7236 -
WlQQ9fi^^ 2 log .02891 +log .7236
.299289 ^^^ ^-^^^^^^ 3
131 2.461048
109 2
. 109 4.922096
_.299432 1.859499
2.927198 4.781595
1.372234 J
175=2356 3)2.781595-6
59 2.927198
55 = 3
40
37 = 2
30
18 = 1
. 20
206. •^^^V54.43 log 64.15^ = 1465.9
4)1.807197
.451799
3
1.355397
•^^''''V54.43X 1.355397
1.735838 .271377
.132064 46
96 209
288 10.271384
224 9.770859
1.868029 .500525
.59 n = 3166
14
3.166100
5838 =1465.
262
266 =9
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279
LOGARITHMS
'.2481* log 39.41+362.58 log 572
.394627-1
3
8)1.183881-1
1.595606
.147985
.202761
.375
163
9.772985
163
.202926
.202926
9.975911
891=9460
20
2.757396
18=4
.440437
20
47
140
999.777
94
.9460
.440499
1000.723
2.559308
96
3.000000
2.999903
304
870=9997
87
33
130
30=7
3.000314
30=7
129
.477121
43
14
.023258
58
_2=1055
477166
2.110590
2.366576
423 = 2325
153
149=8
= 1.055
^^V 1000.723 =
277
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278
TECHNICAL ALGEBRA
279
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CHAPTER XVI
THE SLIDE-RULE
Section 1, Introduction. Section 2, Sines. Section 3,
Tangents. Section 4, Multiplication. Section 5, Divi-
sion. Section 6, Proportion. Section 7, Logarithms.
Section 8, Powers and Roots. Section 9, Gage-Points.
Section 10, The Log Log Rule.
§ 1. INTRODUCTION
280. Historical. Within less than fifty years after
Napier's invention of logarithms in 1614, a slide-rule of
two fixed strips held together by brass plates at the ends
with a sliding strip between, the first duplex, had been
manufactured and sold in England.
Six years after Napier's invention John Gunter made
the first logarithmic scale, known ever since as Gunter's
scale. In addition to trigonometric scales this had a scale
of numbers from 1 to 10, the divisions of the scale being
proportional to the logarithms of these numbers exactly
as in the modern slide-rule. Instead, however, of having
some scales on a movable piece so that computations
could be performed by sliding one scale along another,
all the scales were on the same piece and computations
were made by means of a pair of compasses.
In the summer of 1630* while visiting his teacher WiUiam
Oughtred, an English clergyman and mathematician, a pupil
spoke to him regarding Gunter's scale. Oughtred replied
* See Cajori's " History of the Slide-rule."
279
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280 TECHNICAL ALGEBRA 281
that Gunter's scale was a " poor invention and cumber-
some in performance; '* and thereupon showed some mechani-
cal devices of his own, including a circular and a straight
slide-rule; the former so constructed that one disk rotated
within another with two pointers from the center, and the
latter having one scale on a piece which slid along another
scale. The pupil, Foster, in his account of this incident
says that Oughtred had invented these devices about
thirty years before, but as this would antedate the invention
of logarithms by fourteen years the statement is undoubtedly
incorrect. The probabilities are that William Oughtred
invented the slide-rule shortly after Gunter's scale first
appeared in 1620.
281. What the Slide-Rule is. A slide-rule is a mechani-
cal device for performing logarithmic computations, by
sliding (if straight), or rotating (if circular), the logarithm
of one number along the logarithm of another so that their
sum or difference may be determined.
It consists of a fixed part having logarithmic scales,
called the stock, a movable piece called the slide having
logarithmic scales, and a movable m^tal piece called the
runner having a hair-line for vertical readings of the scales.
282. Arrangement of Scales. The simplest arrange-
ment of scales, of which all others are modifications, is
that of the Mannheim rule which was designed about 1850
by Lieutenant Mannheim of the French army. The face
of the Mannheim rule has four logarithmic scales, two on
the stock and two on the slide. The two upper scales on
stock and slide, usually designated as the A and B scales,
are exactly alike and are used twice in the length of
the rule.
The lower scales on stock and slide, usually designated
as the C and D scales, are exactly alike, with the correspond-
ing divisions double the length of the divisions of the two
upper scales. The readings on the two upper scales are
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THE SLIDE-RULE
281
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282
TECHNICAL ALGEBRA
therefore the squares of the aligned readings on the two
lower scales when the indexes are in alignment.
On the reverse side of the slide are sine and tangent
scales and a scale of logarithms, which can be read by the
index marks on the end notches or transparent plate at the
back of the rule, or by inserting the slide turned over, and
using the runner.
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Fig. 93. — Section of Linear scale on beveled edge of Mannheim rule.
On the beveled edge of the Mannheim rule is an English
scale, and on the opposite edge a metric scale which, in some
rules, is continued along the channel of the rule.
1 •
5 € 7 , « , ?
'-.I'h'N-'imnMHimiilimiiiiih
Fig. 93A. — Section of Metric scale on lower edge of Mannheim rule.
Some of the modifications of this arrangement of scales
are as follows:
(1) An inverted scale in place of the direct B scale.
(2) The sine scale instead of being on the back of the
slide is placed in the middle of the face of the slide, between
scales B and C. In place of the sine scale on the back of
the slide » is an inverted B or C scale.
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288
THE SLIDE-RULE
283
(3) Instead of the metric scale on the edge of the rule
a scale of the cubes of readings on the D scale, the hair-line
being carried down on a tongue on the runner, as in the
POLYPHASE and the multiplex.
(4) The stock widened to cany the cube scale below
the D scale, A and C scales on the upper part of the stock,
and on the slide two C scales with inverted C or CI between,
as in the triplex.
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LnlHi^^
Fig. 94. — Section of an Inverted B scale.
(5) A two-faced rule with A, B, sine, C and D scales on
one face; and A, BI, tangent, C7, and D scales on the reverse,
with an L scale (scale of logarithms) on the lower edge or
lower stock, as in the duplex.
(6) Stock widened to carry in addition to the scales of
(5) on the reverse, a continuous log log scale mounted in
three sections on the upper stock, giving direct readings
for exponentials, logarithms to base e or any other base,
and any power or root, as in the log log duplex.
This rule has a C scale in place of the BI scale, and the
widened lower stock has the L scale on its face below the
D scale.
283. Graduation of the A and B Scales. Examine the
upper or A scale on your rule. Observe the graduation
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TECHNICAL ALGEBRA
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TECHNICAL ALGEBRA
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THE SLIDE-RULE
287
marked 1 at the beginning, center, and the end. These
marks are called indexes and are known as the left, center,
and right A indexes.
Fig. 101.— il-scale from 1 to 2.
The distance between the graduations nmnbered 1 and 2
is divided into how many large graduations?
Each of these large graduations is divided into how
many parts?
If the left index is taken as 1, the first large graduation
will denote what?
Each small graduation therefore denotes how many
hundredths?
Set down the reading for each graduation from 1 to 2
inclusive.
If the left index is taken as 10, each of the smallest
graduations therefore denotes how many tenths?
Set down the reading for each graduation from 10 to 11
inclusive.
If the left index is taken as 100, the first graduation
will be read what instead of 11? (If unable to answer,
notice what it must be in order that the graduation numbered
2 or 20 may be read 200.)
Write the reading for each graduation from 100 to 200
inclusive.
'i'
Fig. 102. — ^ A-scale. from 2 to 5.
Examine the A scale from 2 to 5.
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288 TEUiUNlUALi AAJJhllStiA 284
Is the number of the large graduations the same as from
lto2?
Are the small graduations the same?
Write the reading for each graduation from 2 to 3
inclusive; from 3 to 4; from 4 to 5; from 20 to 30 inclusive;
from 400 to 500 inclusive.
Miii|iiimiii|iiimiii|iiiiiii
10
Fig. 103. — ^A-scale from 5 to middle index.
Examine the A scale from 5 to the end.
Is the number of the large graduations the same as in
other parts of the scale?
Are the small graduations the same?
Write the reading for each graduation from 7 to 8
inclusive; from 70 to 80 inclusive; from 800 to 900 inclusive.
6 7 C 9 1 3
Fig. 104. — B scale showing left and middle indexes.
Examine the B scale. This is the second scale on the
rule, being the upper scale on the slide.
Is it the same as the A scale? How is it read?
284. Graduation of the C and D Scales. Examine the
D scale.
Fig. 105. — Section of D scale.
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286 THE SLIDE-RULE 289
This is the fourth scale on the rule, being the scale on
the lower stock.
By reference to the U. S. linear scale on the edge of
the rule state the relation of the distance from 1 to 2 on the
D scale, to the distance from 1 to 2 on the A scale.
What is the relation of the number of graduations on
the D scale from 1 to 2, to the number of graduations on
the A scale from 1 to 2?
Write the reading for each graduation on the D scale
between the following limits inclusive:
(1)
lto2,
10 to 20,
100 to 200.
(2)
3 to 4,
30 to 40,
200 to 300.
(3)
5 to 6,
50 to 60,
500 to 600.
(4)
7 to 8,
70 to 80,
600 to 700.
Examine the C scale. This is the third scale on the
rule, being the lower scale on the slide.
Fig. 106. — Section of C scale.
Is it exactly the same as the D scale? How is it read?
§ 2. SINES
286. Graduation of the S Scale. The S or sine scale
on most rules is the lower scale on the reverse of the slide
when the rule is held in position for reading the face scales.
When the rule is turned over without reversing ends the S
scale therefore is the upper scale, the lower scale being a
tangent scale, with a scale of logarithms between the two.
Draw the slide to the left and examine the left end of the
S scale. Looking back from the first numbered graduation
which is 40', observe that the scale begins at slightly less
than 35', or at 34' 18".
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290 TECHNICAL ALGEBRA 286
Write the readings for each graduation from 35' to 2°,
disregarding the irregularly placed graduations which
may immediately follow the 1° 10' and precede the 2**
graduations; these are gage-points as explained in para-
graph 290.
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Fig. 107. — Back of slide, showing the first 10® of the sine scale.
What is the graduation imit from 35' to 10®? From
10° to 20**? From 20° to 40°? From 40° to 70°? From
70° to 80°? From 80° to 90°?
Observ-e that the S scale is a scale of angles and not of sines.
286. How Sines are Read with Unreversed Slide. On
all rules sines are read on the B or the A scale. When the
back of the rule has a notch at the end, sines may be read
by moving the slide so as to bring the given angle on the S
scale directly under the index mark on the edge of the
notch in the right end of the rule.
The rule is then turned over, and the numerical value of
the sine of the angle is read on B, directly under the right
A index,
287. Rule for Pointing off a Sine Reading. (1) Place
.0 before all sines which are read on the first B scale.
(2) Place decimal point only, before all sines which are
read on the second B scale.
Thus sin 24° 15' is read as follows:
(1) Set 24° 15' under the right notch index.
(2) Turn the rule over.
(3) Move runner to right A index, and read .411 on B.
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288
THE SLIDE-RULE
291
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:inillnlilliiliilllii(JHiil!lliiiiM]il|jili|]!fi!ii!j|j||{|^ -'^ jj
Fig. 108. — Back of Mannheim rule, showing 24° 15' on S, under S
index on edge of notch.
A decimal point is placed before the reading because
it was read on the right or second B scale.
Verify this reading on your rule.
ii ill fmi
IjlTTTfMjMp—
B 9 _1
Fig. 109.— Reverse of Fig. 108, showing .411= sin 24° 15' under right
A index.
If your rule has an index mark on the edge of the notch
at the left end of the rule, set 24° 15' on S under the left
index mark and take the reading on B under the left A index.
Is the reading the same as before?
288. Examples in Sines. Enter the following examples
in the work-book with the readings for each.
1. sin 39° 10'. 2. sin 9° 12'. 3. sin 40° 25'.
4. sin 5° 20'. 5. sin 5° 55'. 6. sin 3° 15'.
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292
TECHNICAL ALGEBRA
289
7. sin 28° 30'.
10. sin 60° 35'.
13. sin 21° 24'.
16. sin 70° 31'.
19. sin 28° 40'.
8. sin 4° 25'.
11. sin 75°.
14. sin 8° 25'.
17. sin 45° 30'.
20. sin 64° 35'.
9. sin 19° 35'.
12. sin 4° 28'.
16. sin 32°.
18. sin 55° 25'.
21. sin 80° 45'.
i|i|^nrtiiui|un|iHijnii|ii^ii
Fig. 110.— Reading of sin 24** 15' on LL duplex.
24® 15' is under runner on 5; .411 is under runner on A and B.
289. Sines with Reversed Slide. Remove the slide and
insert it with the reverse side up and with the indexes of
all scales in alignment.
Move the runner so that the hair-line exactly covers
24° 15' on the S scale, and read the sine on A under the
hair-line.
Point ofif exactly the same as when sines are read on B,
Is the reading the same as in paragraph 288?
Read all the sines of the preceding paragraph with the
slide reversed.
Are the readings exactly the same as before? Why?
Which seems the simpler way to read sines, with the
slide unreversed or reversed?
290. Gage-Points for Sines and Tangents. If the S
scale is examined on some makes of rule theie will be found:
(1) A gage-point denoting seconds and therefore marked
with the seconds symbol, immediately following the 1° 10'
graduation.
(2) A gage-point denoting minutes and therefore marked
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891
THE SLIDE-RULE
293
with the minutes symbol, immediately preceding the 2°
graduation.
p|}rfpiifif!fi!|i}«ipiip!|r : '■\m
1
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liiite^^^ 1
Fig. 111. — Section of LL duplex showing seconds and minute gage-
points on S scale.
These gage-points on the S scale are used in reading
both sines and tangents of angles less than 34' 18".
291. Rules for Gage-Point Readings: (1) Angles less
THAN 1'.
Move the slide until the seconds gage-point is in align-
ment with the given number of seconds on the A scale.
With the runner read sine or tangent as required, on
A directly over the S index.
(2) Angles from 1' to 34' 17" inclusive. Move the
slide imtil the minutes gage-point is in alignment with the
given number of minutes on the A scale.
With the runner read sine or tangeiit as required on A
directly over the S index.
The following tabulation shows how many ciphers
precede readings for sines and tangents within the inclusive
limits:
Table XI
POINTING OF SINES AND TANGENTS
Angles from
Ciphers
Preceding
Angles from
Ciphers
Preceding
sin
tan
sin
tan
l"to 2"
3" to 20"
5
4
5
4
21" to 3' 26"
3' 27" to 34' 17"
3
2
3
2
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294 TECHNICAL ALGEBRA 292
292. Examples. Read the following functions on the
slide-rule by the use of the gage-points:
1.
sin 42".
2. tan 28".
3. smSO'.
4.
t^,n 6'.
5. sin 16'.
6. tan 41".
7.
sini^V.
8. tan 4' 15".
9. sin 34' 27".
10.
sin 45". ^
11. tan 18' 30".
§ 3. TANGENTS
12. tan 11' 12".
293. How Tangents are Read. How tangents are read
depends on whether the angle is
Less than 34.3',
Between 34.3' and 5° 42' 37" inclusive,
Between 5° 42' 38" and 45° inclusive,
Greater than 45° and less than 90°,
Greater than 90 and less than 180.
(1) Angles Less Than 34' 18". Use seconds or min-
utes gage-points on the S scale, as explained in the pre-
ceding paragraph.
(2) Angles Between 34.3' and 5° 42' 37" Inclusive.
Use S or sine scale, the readings being the same as those
for sines within the same limits, as may be verified by
reference to a table of natural functions.
(3) Angles Between 5° 42' 38" and 45° Inclusive.
Use T or Tangent scale as follows:
Mannheim, Slide Unreversed. Turn rule over and set
given angle on T under left notch index. On face of rule
read tangent on C over D index.
Mannheim, Slide Reversed; also Duplex. Align indexes.
Set runner to given angle on T; read tangent on D under
runner.
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293 THE SLIDE-RULE 295
Pointing Off. Place decimal point before all tangent
readings on D for angles from 5° 42' 38" to 45°.
Observe that tan 45° = L
^n^^TTTTTT^^
Mijto^^
Fig. 112. — Section of T scale on back of slide in Mannheim rule.
(4) Angles Greater than 45° and Less Than 90°.
The tangent scale necessarily reads only to 45° because
tan 45° = 1, and therefore the reading is the right D index.
By the definition of cotangent
tan 6 =
cot^ tan (90-^)*
Therefore if tangents above 45° and less than 90° are
required, use the reciprocal of the tangent of 90° minus
the given angle. This means that when the tangent of an
acute angle greater than 45° is required, the slide-rule
reading must be the reciprocal of the tangent of the com-
plement of the angle.
Mannheim. Invert * slide. Align indexes. Set runner
to 90°— given angle on T; read tangent on D under the
runner.
The tangent thus read is the tangent of the given angle because
read with one scale inverted, which always gives a reciprocal.
Duplex. Set runner to 90°— given angle on T; read
tangent on CI under the runner. The reading is:
1
= tan a.
tan (90 -a)
* Invert means remove, turn over, and insert the slide " the other
end to."
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296
TECHNICAL ALGEBRA
294
Observe that tan 90'' = oo*.
Pointiiig Oflf. Apply rule for a quotient.
(5) Angles Greater than 90° and Less than 180®.
The tangent of an obtuse angle equals minus the tangent of its
supplement. Therefore, if 6 is between 90 and 180°,
tan ^=-tan (180° -6^).
(180°-^) Less than 46°. Read exactly as in (3) or (2)
or (1).
(180° — 0) Greater than 46°. Read exactly as in (4).
Be sure to place minus before the readings.
See illustration in the next paragraph.
Observe that tan 180° =0.
294. Illustrations. (1) Required the tangent of 79°.
tan 79° =
1
1
cot 79° tan (90° -79°) tan 11
J = 5.145.
Fig. 113 shows this reading on a Mannheim rule with the
slide reversed but not inverted.
Fig. 113. — Tan 79° on Mannheim rule with slide reversed.
The runner is set at 11° on T. The reading 5.145 is on
D under the right T index.
The setting is diagrammed as follows:
T
11°
It
D
1
5.145
* oo denotes infinity.
1 1 denotes index.
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294
THE SLIDE-RULE
297
Fig. 114 shows the reading on a duplex rule. Observe
that the T scale is direct, the C scale inverted and therefore
marked CI at the left end; thus called CI scale and not C.
^^ ttiiiiiiiiiiiii
Fig. 114.— Tan 79** on duplex.
The runner is set at 11° on T. The reading 5.145 is on
CI under the runner.
This setting is diagrammed as follows:
T
*I
11°
CI
1
5.145
(2) Required the tangent of 134°.
tan 134°= -tan (180°-134°) = -tan 46°.
tan 46
o
1
1
cot 46° tan (90° -46°) tan 44
:= -1.0355.
Fig. 115. — Back of Mannheim rule showing 44° on T under right
notch index.
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298
TECHNICAL ALGEBRA
2M
The setting in Figs. 115 and 116 is diagrammed as
follows:
T
*44n°
C
1
D
-1.0355
Fig. 116.— Face of rule in Fig. 115 showing -1.0355= tan 134°, oa
D under sUde index.
Diagram of Same Reading on Duplex.
T
I
44°
CI
1
-1.0355
Observe that a reciprocal reading requires an inverted
scale. On a Mannheim rule which has no inverted scale,
this is obtained by inverting the slide and therefore the T
scale. On a duplex rule, which is a modification of the
plain Mannheim, the CI scale gives the reciprocal reading.
Readings from direct scales are called direct, and from
inverted scales are called inverse.
Fig. 117 shows a reading for
tan 134°=-:
tan 44°*
* n means notch index.
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296
THE SLIDE-RULE
299
on a plain Mannheim rule with the slide reversed but not
inverted.
r
' ^-^^-^■^'^^-•^
Fig. 117. — Tan 134° on Mannheim rule with slide reversed.
The figure at the right shows 44° on T in alignment with
the right D index.
The figure at the left shows the reading —1.0355 on D
under the left T index.
Diagrammatic setting.
T
I
440
D
-1.0355
1
—
295. Miscellaneous Examples in Ftmction Readings.
Write the readings for the following functions, diagram
each setting, and check by the table of functions.
1. tan 30° 45'
4. tan 18° 30'.
7. sin 14' 16".
10. tan 4° 29'.
13. tan 59°.
16. cos 63° 25'.
19. tan 22' 14".
22. tan 156° 20'.
25. tan 128° 40'.
28. tan 107° 45'.
2. sin 30° 45'.
6. tan 3° 12'.
8. tan 21'.
11. sin 4° 29'.
14. sin 75° 30'.
17. cos 31° 42".
20. sin 125°.
23. cos 130° 15'.
26. tan 131° 30'.
29. tan 98° 50' 30".
3. tan 30' 45".
6. tan 45".
9. tan 21° 18'.
12. sin 4' 29".
15. tan 72° 25'.
18. cot 27° 30'.
21. tan 165° 40'.
24. cos 172° 18'.
27. tan 112° 25'.
30. tan 155° 15' 45".
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300
TECHNICAL ALGEBRA
29e
§ 4. MULTIPLICATION
296. What Scales are Used in Multiplication. Numbers
are multiplied by the A and B scales, or by the C and D
scales. The advantage of using the C and D scales is that
they have a greater number of graduations than the A and B
scales, which makes it possible to read a result more closely
than on the A and B scales.
The disadvantage of using the C and D scales is that a
factor is frequently " off the rule,'* which makes it neces-
sary to change indexes. This disadvantage however is only
apparent and the C and D scales are recommended for all
multiplication except when one of the factors is a sine or a
cosine.
297. The Principle of Slide-Rule Multiplication. The
right and left ends of the logarithmic scales on the slide-
rule, marked 1, are called the indexes. The first graduation
is marked 1 because the graduations are proportional to the
logarithms of the numbers by which the graduations are
denoted, and the logarithm of 1 is zero.
The reason why the graduations continually decrease in
length as the scale advances, and the principles of slide«-rule
computation, will be evident from the following series of
numbers and their logarithms:
No.
1
2
3
4 5
6
7
8
9
10
log.
.301
.477
.602 .639
778
.845
.903
.954
1.0
Observe the decreasing interval between the logarithms.
Imagine the C and D scales divided into 1000 equal parts:
Then 1 is placed at the zero graduation,
2 at the 301st,
3 at the 477th,
4 atthe602d,
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1
299 THE SLIDE-RULE 301
5
at the 699th,
6
at the 778th,
7
at the 845th,
8
at the 903d,
9
at the 954th,
10
at the 1000th.
Therefore the product of 2X3 may be delennined by
placing 301 equal divisions of the D scale end to end with
477 equal divisions of the C scale, making a total of 778
which is the graduation marked 6 on the D scale.
298. Integral Figures. In computations on the slide-
rule, results may sometimes be pointed off by inspection.
In other cases, use is made of the integral figures of the
numbers involved.
For this purpose the integral figures of all numbers are
determined as follows:
(1) Number Greater Than 1. The number of integral
figures in a number greater than unity equals the number of
figures which precede the decimal point.
(2) Number less than 1. The number of integral figures
in a decimal equals minus the number of ciphers between the
decimal point and the first significant figure of the decimal.
Thus 113.92 has 3 integral figures,
.7896 has integral figures,
.00845 has —2 integral figures,
.O46I59 has —4 integral figures.
299. Rules for Pointing oflf a Product. All products
obtained on the slide-rule may be pointed off by the fol-
lowing rules:
(1) The number of integral figures in a product obtained
on C and D (or B and left A) with the slide to the left^ and on
B and right A with the slide to the right, equals the sum of
the integral figures in the factors.
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302 TECHNICAL ALGEBRA 300
(2) The number of integral figures in a product obtained
on C and D (or B and left A) with the slide to the right, or
on B and right A with the sUde to the leftj equals the sum
of the integral figures in the factors, minus 1.
Thus (a) L2X. 0005 = .0006.
SUde to the right; l-3-l = -3.
—3 means .000 precede the number read.
(6) 42.9 X. 854 = 36.6.
SUde to the left; 2+0 = 2.
2 means two figures precede the decimal
point.
300. Illustrations of Multiplication. The following exam-
ples illustrate the use of the slide-rule in the determination
of a product:
1. To multiply L25 by 17.3 on the sUde-rule.
Move the slide to the right until the index of the C scale
is exactly over (in alignment with) the graduation marked
L25 on the D scale. (Use the runner to align C index with
L25.)
Move the runner until the hair-line is in alignment with
the graduation marked 17.3 on C.
Under 17.3 on C read the product 216 on D,
Set the rule as specified and observe that log 17.3 is end
to end with log 1.25 and that graduation 216 is therefore
the product because it is the antilogarithm of the sum of the
logarithms.
The equation for pointing off, as determined by para-
graph 299, is as follows:
Slide to the right l-|-2-l = 2.
Therefore the product has 2 integral figures.
Fig. 118 shows the setting. The C index is over 1.25
on D; the runner is at 173 on C. The product 21.625 is
on D under the runner.
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300 THE SLIDE-RULE
The diagram is as follows:
c
1
17.3
D
1.25
216
*
>
1+2-1=2
303
Fig. 118.— 1.25X17.3 on LL duplex.
Determine the same product by a different setting,
write directions in the work-book for both settings, and
give the equation for pointing off.
2. To multiply 72.5 by .832, since .832 is off the rule
when the C index is set to 72.5 D, the slide is moved to the
MADE IN f^PPMAN
Fig. 119.— Section of Mannheim rule showing 72.5 on DX.832 on C.
The product 60.3 is on D under the runner.
left instead of to the right, until the right index of C is in
alignment with 72.5 on D.
Set the rule as specified and move the runner to bring
the hair-line into alignment with .832 on C.
* Arrow shows direction of slide.
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304
TECHNICAL ALGEBRA
801
Under .832 read the product 60.3 on D.
Fig. 119 shows the setting. Be sure to reproduce it on
your rule.
Diagram:
c
.832
1
D
60.3
72.5
2+0=2
The slide is to the left. Therefore the number of integral
figures in the product equals the sum of the integral figures
in the factors. See paragraph 299 (1).
Determine the same product by a different setting,
show diagram, arrow, and equation for pointing off.
3. To multiply 32.8 by 6.52 move the slide to the left
to bring the C index into alignment with 32.8 on.D.
Move the runner to bring the hair-line over 6.52 on C
and read the product 213.8 on Z>, under the hair-line.
Set the rule as specified.
Determine the product by another setting, and write
directions for, or diagram both settings.
c
1
6.52
D
32.8
213.8
2-Fl=3
301. Examples in Multiplication. Solve the following
examples on the slide-rule, showing all settings and equations
for pointing off. Check by logarithmic computation.
For pointing off see paragraph 299.
1. 18X34.2 2. .18X3.42. 3. .018X.342.
4. .018X.00342. 5. 18.8X3.8 6. 1.45X85.
7. 76.4X29. 8. 845X7.5. 9. 39.2X53.
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S02
THE SLIDE-RULE
305
10. .00059X.805.
13. 72X3.47.
16. .54 X. 0908.
19. .63X42.9.
22. 13.8X7.6.
26. 5.43X85.
28. 29.2X18.5.
11. 54.8X614.
14. 386X15.8.
17. 1.44X8.14.
20. .472X.129.
23. 31. 9 X. 0046.
26. .4309X3.7.
29. 74.4X.75.
12. .129X48.6.
15. 25.8X.0063.
18. 708X2.96.
21. .067X12.4.
24. .00657X15.
27. .0^512X392.
30. 391 X. 935.
302. How to Mtiltiply Sines and Tangents. Sines.
To multiply sines use A and S Scales with the slide reversed.
(Duplex requires no reverse.)
Thus to multiply 29.6 by sin 18 J ° take settings as shown
below:
29.6
18i°
Determine the product, give the equation for pointing
oflf, and explain why the setting gives the product.
Fig. 120 shows the setting as diagrammed.
Fig. 120. — Mannheim rule showing 29.6 sin 18 J ° with reversed slide.
Tangents. To multiply tangents use D and T scales with
the slide reversed.
Thus to multiply 19.5 by tan 21° set rule as follows:
Ij means left index.
T
Ii
21^
D
19.5
'
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am
TEGECNICAL ALGEBRA
Detefiniiie tbe product, gLve equation for poifoting off,
and explaitt why the setting gives the result.
Figs. 121 aDd 122 shows the setting.
Fig. 121. — Section of Mannheim rule with shde reversed showing
h T over 19.5 D.
Fig. 122. — Mffit end of mle in. Fig: 121 showing nimier at 21" on 7,
ami 7.485 oni L^ undie3r the runner.
303. fiitegrar Vtgfares in a Sine or Tangent. Tlie fol-
lowing summary of the number of integral figures in a sine
or tangent reading, will f iacilitate the solution of the examples
in the ne5it paragraph:
(1) The number of integral figures in a sine or a tangent
which is read from a guge^oint, may be determined from the
table in paragraph 291.
(2) A sine or a tangent reading on the first A scale is pre-
ceded by .0 and therefore has — I integral figures.
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SOS THE SLIDE-RULE 307
(3) A sine reading on the second A scale is preceded
directly by the decimal poiiit and therefore has zero integral
figures.
(4) A tangent reading on ^e D scale is preceded direcfly
by a declmfd point throughout the scale, except the reading for
tan 45° which is 1.
A tangent reading on the D scale therefore has zero integral
figures,
304. Examples. Solve the following examples on the
slide-rule, showing the diagrammatic settings and the equa-
tions for pointing oflf. Check by logarithmic or arithmetic
computation.
1. 34.19 sin 42°. 2. 19.21 sin 29° 20'.
3. 9.85 sin 51.6°. 4. .65 tan 14°.
5, 1,89 tan 34-5°. 6. 12.8 tan 42°.
7. 11.5 tan 45°. B. 7.26 sin 39.1°.
9. 427 tan 28.2°. 10. 57.2 sin 56° 18'.
11. 345 sin IS* tan 40°. 18. 18.7 fiin34i° tan 29i°.
13. 91.7 sin 50° tan 13.1°. 14. .639 sin 75° t.n 43°.
16. 450.2 sin 15". 16. 824.5 tan 34.5°.
17. 17.28 tan 20'. 16. .€71 sin 5° tan 50'.
200
16!'
200
19. 77^ sin 47.6°. ». 20.05X1.805 tan 118°.
305. Products of More than Two Factors. A product
having several factors may be read by the use of the scales
already specified.
On a rule having C, C/, and D scales on the same face,
the product may sometimes be obtained with fewer settings
than on rules not having this arrangement.
Following are general diagrammatic settings in which F
with subscripts denotes the factors in succession, R denotes
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308
TECHNICAL ALGEBRA
306
the rimner, and 1 denotes the index of C, either left or right
being used as may be necessary to give reading:
(1)
c
1
Rio Ft
1 to/2
RioFi
D
Fi
Product
(2)
c
F,
CI
Fi
D
F2
Product
Illustration. To determine 12.5X24X4.56, take set-
tings as follows:
(1)
c
1
ii;to24
ItoR
R to 4.56
D
12.5
1368
2+2-1+1=4
(2)
c
4.56
CI
12.5
D
24
1368
2+2-1+1=4
<
In the preceding, the product is read by the use of the
CI scale from one setting because the third factor is so located
on the slide that it does not project beyond the D scale.
Sometimes another setting must be made. This is always
the case when the third factor is located on the C scale so
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3i)6
THE SLIDE-RULE
309
that it projects beyond the D scale or is off the rule when
the setting is made for the first two factors.
Thus to determine 12.5X24X1.56, take settings as
follows:
(1)
c
1
Rto2A:
ItoR
Riol.m
CI
12.5
468
2+2-1+1-1=3
(2)
c
Rio I
1 toi2
R to 1,56
CI
12.5
D
24
468
2+2-1+1-1=3
< <
Take the preceding readings on your rule in one or both
of the two ways illustrated in each instance.
Take the readings also on the A and B scales, showing
the diagrammatic setting and the equation for pointing off.
306. One Factor a Sine or Cosine. If one of the factors
of a continued product is a sine or a cosine, the diagrammatic
setting is as follows:
A
Fi
Product
B
ItoR
RtoF2
S
1
Rtod*
* 6 denotes the angle.
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310
TECHNICAL ALGEBRA
307
Thus to determine 3.68X18 sin 45° take the following
settings:
A
3.68
46.8
B
lioR
18
S
1
R to 45°
1+0+2-1 = 2
307. One Factor a Tangent. If one of the factors of a
continued product is a tangent the diagrammatic setting
is as follows:
(1)
c
ItoR
RtoF2
T
1
RtoO
D
fi
Product
(2)
T
d
CI
Fi
D
F2
Product
Thus to determine 3.68X18 tan 40° take the following
settings:
(1)
c
ItoR
18
T
1
72 to 40
D
3.68
55.6
1+0+2-1=2
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THE SLIDE-RULE
311
(2)
T
40
CI
3.68
D
18
55.6
1+2-1+0 = 2
308. Caution about Setting the Runner. When several
settings are necessary, use special care so that accumulated
errors may not affect the final reading.
Keep the hair-line free from dust and take no reading
from the runner except when it is set so that the graduation
mark for the reading is exactly coincident with the hair-line.
When set properly no trace of the graduation can be seen.
309. Examples. Compute the following on the slide-
rule, showing the diagrammatic settings and equations for
pointing off.
Take two different settings for each example so that one
may be used as a check for the other. Check the slide-rule
computations by logarithmic solution from the tables.
1. 3.78X72.5 tan 24°.
3. 29.8X6.12 tan 18.4°.
6. 44.6X53.7 tan 29.3^
7. 842X9.65 tan 31.2°.
9. 55.6X.139 tan 22° 35'.
11. 17.28X16 sin 35°.
13. 472X12.8 sin 4° 12'.
16. 2630X.192sin2°34'.
17. 546X3.27 cos 18° 12'.
19. 8.27X58 cos 60° 20'.
21. 28.6X3.47X29.75.
23. 53.8X4.8X6.12.
25. 485X. 349X6315.
2. 17.4X51.8 tan 361°.
4. 34.7X.169tan9° 15'.
6. .789 Xtan 15.5°.
8. 51.8X1.64 tan 9° 20'.
10. 689X.185tan41° 10.5'.
12. 29.8X39 sin 56° 30'.
14. 584X3.125 sm 3° 8.5'.
16. 90.8X26.01 sin 1° 45'.
18. 3.86X.142cos40°31'.
20. 54.9X2.446 cos 72° 9'.
22. 72.6X5.408X12.9.
24. 8.126X73.5X17.
26. 6.07 X 50.9 X. 00845.
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312
TECHNICAL ALGEBRA
310
§ 6. DIVISION
310. How a Quotient is Determined on the Slide-Rule.
Since division is the inverse of multiplication, a quotient is
obtained by setting the rule so that the logarithm of the divisor
may be subtracted from the logarithm of the dividend. Division
may therefore be performed on the following scales:
A and B, and A and BI,
C and Z), and CI and D.
. BI means the B scale inverted, an inverted scale being graduated
from right to left instead of from left to right. If your rule has no
BI or CI scale,* it will have both if the slide is inserted with the
ends reversed. This is not reconunended, for the graduations
stamped on the slide are then upside down, the BI scale lies next
to the Z), and the CI next to the A. But, if the user of a plain
Mannheim rule must have an inverted scale, this is the only way
to get it without buying a more expensive rule.
Below are general diagrammatic settings for both C and
2>, and CI and D scales:
(1)
(2)
Illustration. To divide 76 by 4 take settings as
follows:
(1) . •
c
Divisor
1
D
Dividend
Quotient
CI
1
Divisor
D
Dividend
Qoutient
C
4
1
D
76
19
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310
THE SLIDE-RULE
313
CI
1
4
D
76
19
<2)
Enter all these diagrams in the work-book. Show also
corresponding diagrams for the A and B, and A and BI
scales, and take the readings on the rule for all settings.
Explain why these settings give the quotient.
Figs. 123, 124 and 125 show the setting on the multi-
plex, duplex, and log log duplex.
^- 7-^^^::r.--. 9 •■ 1 • n
16 17
■7-^
J
Tig. 123. — Multiplex rule, showing division of 76 by 4 on A and BI
scales.
|g{|^h!d!illihl| mjih
IB
mIp
76
Fig. 124. — Duplex rule showing — =19 by CI and D
4
tP
'■nm|iM!
Fig. 125. — LL duplex showing 76 divided by 4 on C and D.
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314
TECHNIOAL ALGEBRA
dU
311. Rule for Pointing Off a Quotient The rules for
pointing off a quotient obtained cm the slide-rule are the
inverse of the rules for pointing off a product:
C and D, or A and B scales.
(1) The number of integral figures in a quotient obtained
when the slide is moved to the left, equals the number of
integral figures in the dividend minus the number of integral
figures in the divisor.
(2) The number of integral figures in a quotient when
the slide is moved to the right, equals one more than the
difference between the number of integral figures in dividend
and divisor.
CI and D« or A and BI Scales.
(1) The number of integral figures in a quotient obtained
when the slide is moved to the left, equals one more than the
difference between the number of integral figures in the
dividend and divisor.
(2) The number of integral figures in a quotient obtained
when the dide is moved to the right, equals the difference
between the number of integral figures in dividend and
divisor.
These rules may be tabulated as follows:
Table XII
POINTING OFF RULES FOR DIVISION
Scales.
Direction of Slide.
Integral Figures in
Quotient.
C and Z> •
^ andB
Left
Difference
Right
DifFerence+1
C/andZ)
AI and B
Left
Difference +1
Right
Difference
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31S THE SUDE-RULE 315
312. Do not Memorize but Think. Be sure to observe
that when the divisor on C is set over the dividend on Z),
the quotient is always on D under the C index.
In other words, when the divisor is on (he slide the quotient
is on the stock.
Avoid thinking this must be remembered. Of all instru-
ments the slide-rule is the one which necessitates that
nothing be remembered, for everything is before you on the
rule. The one fact you must know for division is that in
division logarithms are subtracted, because logarithms are
exponents. For the rest, study the rule. The necessity
now is not to remember or to consult models or diagrams,
but to think. The models and diagrams are for reference
when you are tmable to think.
Do not be satisfied to solve the examples in this list only.
Have the ambition to masrier slide-rule division and other
operations. The work in shop, laboratory and classroom
will furnish the material. If not, try all sorts of numbers.
Work of this kind steadies the mind in idle moments,
makes one more intelligent, and will probably increase one's
earning capacity.
313. Examples in Division. Solve the following exam-
ples on the slide-rule, showing all diagrammatic settings
and equations for pointing off for one set of scales.
Check by logarithmic or arithmetic computation.
1. 18.94-3.1. 2. 144.5-^6.5. 3. 596-S-121
649 58.2 89.1
78.1* 3.21' 95 *
7.64 58.2 1728
2 28 • 2.61* 14.4*
./v 200 ., 4.39 ,^ .396
^^•16-f ^'•:212- ' ^^'WS'
5L6 m .m^
1.27 5.67 .349
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316 TECHNICAL ALGEBRA Sia
.^ ,608 ^„ 7.84 -^ 1626
16. . 17. . 18. .
L47 16.3 342
.A 12.96 ^ .2982 ^, 9.46
19. . 20. . 21. .
51.85 1.78 .575
200 im3 ^, .0484
ICf 18.5 •
25.
^«.
.6815'
27.
.483
sin 23" 30''
30.
743
sin 60.5°*
oo
743
420.5 .3917
8.075* .0562*
.275 31.4 sin 32**
• CSC 41.2°* tan 30° *
7.8 cos 18° . 1
sin 18° ' tan 18°' *^' sin 60.5°'
34. 41.5 tan 71°. 36. 2.28 tan 80.2°. 36. 5.25 taa 57° 40'.
«« 78.9 «. 29.6 ^^ .6871
37. . 38. . 39. .
3.18 .0875 12.9
^^ 5.62 ^, 4.76 ^„ .0328
40. . 41. . 42. .
.731 .0029 17.4
^^ .0096 ^^ 76.8 ^^ 2.46
43. . 44. . 46. .
.000751 125.2 .533
^^ 92.1 ^^ .805 ^^ 349
46. -— . 47. — — . 48.
.756' • .508* ; .1828'
,^ 5965 ^^ 27.15 ^/ 48.5
49. - — r—To' 60. : — -—I, 61.
tan 18.4°* ' tan 58^' ' cot 17° 15'
70.4 ^^ .704 ^^ .516
52. ; — TTx^v,. 63. 7— :^;^rTV7. 64.
tan 34° 17'* tan 20° 15'* tan 2° 18'"
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iiB
THE StIDE-RULE
3ir
§6. PROPORTION
314. How to Set the Rule. One of the simplest use»
of the slide-rule is for the solution of a simple proportion
in which two factors are divided by a third. The setting
is as follows:
(1)
(2)
When a proportion is given for solution by the slide-
rule there is no necessity of algebraic solution if one will
notice whether a mean or extreme is unknown. The
moment this is done the two factors and the divisor may be
immediately determined and the reading taken. .
. Thus if 12.5 : 8.2 :: 24.3 : R,
12^^24^
8.2" i2 '
c
Divisor
Ft
D
Fi
Result
CI
F,
Divisor
D
Fi
Result
or
the factors are 8.2 and 24.3 and the divisor is 12.5, since in
every numerical proportion the product of the means equals
the product of the extremes.
315. What to do when a Number is " Off the Rule."
If an attempt is made to determine R by the preceding
diagrammatic setting, the second factor 7^2 = 24.3 will be
found to be " off the rule," since when 12.5 on C is set over
8.2 on D, 24.3 on C is beyond the index of D. Whenever
this is the case the final reading can be taken only by a.
setting from the index which is off the rule.
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31S
TECHNICAL ALGEBRA
SM
This setting is always made as follows:
(a) Move the runner to the C index which is on the rule
denoted cm the diagram by /2 to 1.
This is necessary in order that the hair-line shall be set at the
number which is the product or quotient or other function of the
quantities involved in the previous setting. Do not read this
number, for it is not wanted.
(6) Keeping the runner stationary, bring the projecting
C index to the hair-line, denoted on the diagram by 1 to /2.
(c) Move the runner until the hair-line is over the
required reading.
Thus (1)
c
12.5
Riol
1 to/2
24.3
D
8.2
15.94
1-2+1+2 =
2
(2)
CI
24.3
i2tol
1 tofi
12.5
D
8.2
15.94
2+1-1=2
316. Explanation of the Preceding Process. The quick
and intelligent use of a slide-rule is possible only when a
mastery of the principles involved makes one independent
of prescribed models.
Therefore study diagrams (1) and (2) and the instructions
for the settings imtil the following is clearly understood:
In (1), 8.2 is divided by 12.5. Therefore the setting
is made so that from the logarithm of 8.2, is subtracted the
logarithm of 12.5, the difference being imder the left index
of C.
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317 THE SLIDE-RULE 319
The result is to be multiplied by 24.3.
If this were *^ on the rule " it would only be necessary
to move the runner to 24.3 on C because this would place
log 24.3 end to end with log 8.2 -log 12.5.
Since 24.3 is not ^* on the rule/'
(a) the runner is moved to log 8.2— log 12.5 under the
left C index,
(6) the slide is moved its whole length to bring the right
index under the hair-line,
(c) the logarithm of the reciprocal of 24.3 is subtracted
from the difference previously obtained,
(d) the antilogarithm of the final difference, which is R
in the proportion, is read on the D scale imder the left C
index.
In (2), 24.3 is multiplied by 8.2 on the CI scale. There-
fore, the setting is made so that from the logarithm of 8.2
on Z), the logarithm of the reciprocal of 24.3 on CI is sub-
tracted, the difference being under the left C index.
The result is to be divided by 12.5.
If 12.5 were on the rule it would only be necessary to
move the runner to 12.5 on Cly which would place the
logarithm of the reciprocal of 12.5 end to end with the dif-
ference of logs previously obtained. Since 12.5 on C is
off the rule, the final reading is taken only after a change of
indexes, as explained in (a), (6), and (c).
317. The Ratio Method. The usual method of solving
a proportion on the slide-rule is based on the principle that
whenever two numbers are aligned on C and D, or A and B,
all pairs of aligned numbers throughout the two scales are in
exactly the same ratio.
Thus, when the indexes are aligned,
12 3
the ratios are — ,— , -, etc., each being equal to 1, the ratio
1 ^ o
of the aligned indexes.
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320 TECHNICAL ALGEBRA 817
When 2 on C is aligned with 4 on Z),
1 4 3 15 , , . , 1 ^
the ratios are -, -, -, — , etc., each being equal to -, the
2 8 6- 30 2
ratio of 2 to 4.
Therefore the unknown term R in
12.5^24.3
8.2 R •
is determined by '* setting up *' the ratio
12.5
8.2
on A and B
or on C and D, The runner is then moved to 24.3 on the
wsame scale as 12.5, and the number aUgned with 24.3
under the runner is the unknown term.
Why? Because the ratio of 12.5 to 8.2 is exactly the
same as the ratio of 24.3 to the number in alignment with
it, or 15.94.
Fig. 126. — Sections of Mannheim rule illustrating ratio setting for
solution of a proportion.
The right figure shows the ratio 12.5 : 8.2. The left
figure shows the slide indexes interchanged, and the equal
ratio 24.3 : 15.94 under the runner.
It will be obvious that the numerators cf the two ratios
must be on one scale and the denominators on another.
This is undoubtedly the simplest way of solving a pro-
portion and is recommended.
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318 THE SLIDE-RULE 321
318. Examples. Solve the following proportions, with
diagrammatic settings and equations for pointing off:
1.
19 : 34::55 : x.
2.
75 : 29:: 11.8 : x.
3.
4.29 18.2
6.1 X '
4.
X 2.1
50.8 41.2*
6.
1.27 18.5
X 78.6*
6.
342 X
193 8.4*
7.
sin 47° 124.5
X 2.56 •
8.
9.1 50.9
tan 29° x '
9.
tan 32° x
18.24 "".61'
10.
398 tan 60°
X .536 '
11.
cos 48° 3' X
34.6 7.32
12.
8.69 7.06
tan 34° 16' " x '
13.
X tan 41.2°
93.4"" 4.29 *
14.
3.47 X
sin 18° 20' ".831'
16.
.723 .82
tan 29° 41' ~lx '
16.
13.45 914.9
4.21 " X *
17.
5.4 tan 75°=^.
X
18.
X 429
tan 27° 24' 61.7*
19.
306 X
cos 43° 8' "184'
20.
.0734 sin 54°
.00426" X '
21.
34 : 85::a; : 296.
22.
45.2 : 130.1:: 16 :
23.
534 29
296~>S'
24.
T 78.2
341 61.3 •
26.
182 608
71.3" T'
26.
556 19.2
V 4.75*
27.
70 760
y "740*
28.
125 760
112 mm
29.
.428 X
1.09 "3.76'
30.
♦22 X
7 "r
X.
♦Align 22 on A with 7 onB. The index 1 nearly aligns with what
gage-point? Why?
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322 TECHNICAL ALGEBRA 819
„/ 78.9 461 ^^ 308.5 12.84
* 594.5 «; 21.9 W '
„« .0893 X ^^ .00429 52.45
.624 192.5' * .00265 x '
319. Combined Multiplication and Division. In com-
putation it is sometimes necessary to determine a result
when several factors are divided by several other factors.
An analysis of the diagranmiatic settings for the examples in
the preceding paragraph will show how the result is obtained
under such conditions.
Thus, in the examples in proportion, the runner was set
to one of the factors on D and the slide was moved to bring
the divisor on C to the runner. The runner was then
moved to the second factor on C and the result was read on D,
Or, by the ratio method, the known ratio was set up.
The application of this process to the determination of
the value of
34X41X65X19
52X18X75X31.5*
is as follows:
(1) Set the riumer to a factor in the numerator, for
convenience the first factor on D. Bring a factor of the
denominator to the runner, for convenience the first factor
on C. This gives 52 C over 34 D,
(2) Bring runner to 41 C.
(3) Bring 18 C to the runner.
(4) Bring runner to 65 C.
(5) Bring 75 C to the runner.
(6) Bring runner to 19 C
(7) Bring 31.5 C to the runner.
(8) Read result on D under C index.
This means briefly:
Bring the runner to all multipliers.
Bring all divisors to the runner.
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320
THE SUDE-RULE
323
Whether multipliers and divisors are used alternately
as above, or whether all multipliers first and then divisors,
makes no difference in the result.
Following is the diagrammatic setting:
c
52
i2to41
18toi2
iJ to65
75 to R
i^to 19
31.5 to R
^
34
.778
2-2+2-2-hH-2-l -2+1+2-1 -2=0
< > <
Make these settings on the slide-rule with the equation
for pointing off.
Check by logarithmic computation from the table.
When the number of factors in the numerator is not the
same as the number of factors in the denominator, proceed
as follows:
(1) Bring the rimner to the C index for each missing
numerator factor.
(2) Bring the C index to the runner for each missing
denominator factor, in both cases using the C index that
is not " off the rule."
In applying (1) and (2) neither subtract nor add unity
to the number of integral figures, because a multiplier or
divisor of 1 does not change the result.
320. Examples. Solve the following examples on the
slide-rule and check by logarithmic computation from the
table.
1.
5.
29.2X77X12.3
3.45X53X8.6*
8.2X91.9X1 65
42X3.4X87 '
493X3.72X12.8X. 64
5.7X29.4X. 39X16.3*
50.5X31X18.2
48X91.2X3.75'
.78X14.5X22.6X83
15.2 X 14.9 X. 082 X. 634'
61.8X9.2X18.45 sin 35°
29.2 X. 806X5.36X25 cos 35**
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324
TECHNICAL ALGEBRA
321
7.
11.
13.
16.
17.
19.
492X18.3 tan 28° 20^
n.5X2.16sin29' '
217 X. 824
515 X. 00634X9. 12X11.6*
91.5X36X184.5X 75
33000
5.96 X 18.2 X. 349
.784X2.68X1.78'
.674X14.4 X8.12
3.39 X. 092X1721*
14.6X29.4X4.37
1296X15.9X6.72*
.581X84.2X13.9
64.1 X.128X. 049*
8.
10.
12.
14.
16. H^:
.0718X9.6X22.8 C08 35°
.0078X613
84.5X21.8 tan 63° 28^
4.19X.302X61.7 sec 29.4°
1875X21.4X63.2
746X 18.9k. 831*
.C09X 1345X78. 1
1.245X12.7X346.5*
.567 XI 8.9 X. 00436
18.
20.
7.47X9.02X. 00573
3.48 X .792X40.9
18.24X34.71X15.9'
72.8X6.55X1728
40.3X.64X3.09*
§ 7. LOGARITHMS
321. The L Scale of Logarithms. On the slide, usually
between the S and T scales, or on the face or edge of the
stock is an L scale or scale of logarithms.
' TilltliillliliillfilllllillililiiilllKlllilllillill
illllllllllllllllltllllllllillilllllllllllllllllllillllll^
Fig. 127. — Left end of L scale on edge of duplex rule.
When the zero of the L scale is in alignment with the index
of th3 D or C scales, the mantissa of the logarithm of any
numher on those scales may be read on the L scale: and
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321
THE SLIDE-RULE
325
inverselyj the antilogarithm of any number on the L scale may
be read on the D or C scales.
On some rules the D and L scales are both on the stock
with the L zero to the left of the D index. When this is
the case the D index is aligned with the L zero by a mark
on the tongue of the runnei .
Examine the L scale on the rule. (If the rule is a Mann-
heim, remove the shde.)
The entire length from zero to 10 has how many
graduations?
If the number of graduations were doubled, the scale
would then be divided into how many equal parts?
Therefore the 301st division would represent the loga-
rithm of what number? (See paragraph 297.)
[
1 1 2 3 * :> . 'i . . '
Fig. 128.
Insert the slide in its usual position so that the face
scales are A, B, C, and D. Move the runner to 2 on D and
bring the left slide index to the runner.
Turn the rule over and read the logarithm of 2 on the
L scale under the L index on the right notch or on the
celluloid plate. What is the reading? *
Fig. 129. — L notch (lower) in right end of Mannheim rule.
* Do not fail to notice in what direction the L scale is graduated.
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326
TECHNICAL ALGEBRA
If thie rule is a duplex, what is the reading on L when
the L and D indexes are aUgned and the runner is set at 2
onD?
How therefore is the logarithm of a number determined
by the slide-rule?
How is the antilogarithm determined?
What part of paragraph 297 exactly illustrates the read-
ing of logarithms on a sUde-rule?
322. Examples. By sUde-rule readings, fill in the omitted
entries in the following table, determining characteristics by
the same law as when a table of logarithms is used:
Table XIII
LOGARITHMS
No.
N
log.
No.
iV
log.
1
38
13
2.745
2
125
14
1.862
3
82.3
15
1.086
4
.546
16
.534
5
112
17
3.620
6
16
18
17
7
18
19
19
8
23
20
25
9
26
21
27
10
28
22
29
11
31
23
32
12
35
24
4.75
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323
THE SLIDE-BULE
327
§ 8. POWERS AND ROOTS
323. The Square of a Number. Move the runner to
each of the following numbers on D and enter in the table
the aUgned A readings:
D
A
1 D
A
D
A
2
1 3
11
4
5
13
6
7
15
8
9
; 16
The readings on A are what power of the readings on
D? Explain why.
Fig. 130.— 2« on duplex.
324. How a Square is Pointed Off. The number of
integral figures in a square when read oh the D and A scales
is as follows, when n denotes the number of integral figures
in the number to be squared :
(1) On the left A scale, 2n-l.
(2) On the right A scale, 2n.
Test this rule by reading the squares of the following
numbers, with the A and D scales: 13, 20, 60, 8, .1, .01, .31.
Is the law correct?
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328 TECHNICAL ALGEBRA 326
Read 18^ with the C and D scales and show the diagram-
matic setting with a statement of the principle involved.
If the principle is not clear, refer to paragraph 266.
325. The Square Root of a Number. As might be
inferred, the square root of a number on A may be read on
D. The process is as follows:
(1) When the number of integral figures is odd^ read
the given number on the left A scale.
(2) When the number of integral figures is even or zero,
read the given number on the right A scale.
326. How a Square Root is Pointed Off. The number
of integral figures in the square root of a number, read
(1) On the left A scale = ^.
Yl
(2) On the right A scale = ^ .
n denotes the number of integral figures in the number whose
square root is to be read.
327. Examples. When possible solve the following
both by the A and D scales and the C and D scales. Show
not less than one diagrammatic setting for left A, right Ay
and C and D readings, with equations for pointing off:
2. 282. 3. 522.
6. 28*. 6. V525^
8. Vs^ 9. 7.5*.
11. Vtan 18.5°. 12. Vcos 60** 14'.
14. 3082. 16. 842.
17. .00242. 18. .6252.
20. V.0046. 21. \/.0125.
23. Vtan 40° 20'. 24. (sin 52.3°)*.
1.
192.
4.
Vl9.
7.
31.8*.
10.
Vsin 35°.
13.
4.82.
16.
.0172.
19.
V.79.
22.
.00085*.
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THE SLIDE-EULE
329
328. How to Read a Cube. With a table of logarithms
the cube of a number is obtained by multiplying the logarithm
of the number by 3, and reading the antilogarithm. On
the slide'Tule the antilogarithm of three times the logarithm
is determined by the Z), A, and B scales as follows:
immmmmimiiiima
fi|i|rji|l(iti.|i|!K
" . : 3 7t
;ljllllllll(l|ll|llill!l|lll!l)lf
Fig. 131.— 19.68=2.7'.
/
(1) Set the runner to the number on D. Twice the loga-
rithm is then under the runner on A,
(2) If the given number is on the left half of D, move the
left index of the slide to the runner; if on the right half of
D, move the right index of the slide to the runner.
(3) Move the runner to the given number on that scale
of B whose index was set to the runner in (2).
The reading on A under the runner is the antilogarithm
of 3 times the logarithm of the given number and is there-
fore its cube.
Take the following setting on the rule and explain why
it gives the cube of 7:
A
343
B
1
7
D
7
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330
TECHNICAL ALGEBRA
329
Place the left B index over 27D and set runner as
shown in Fig. 131.
Why is the A reading under the runner, the cube of 2.7?
7 d B- 1
Fig. 132. — First third of a cube scale.
329. The Cube Scale. Some rules have a cube or K
scale on the stock, either on the vertical edge or on the face.
mi
mimmmiim
II
-tUi:
jnUltJiilllJJji/
Fig. 133. — 2.7» on Triplex rule. Observe 2.7 under runner on C;
the cube, 19.68 is on K (lower scale) under runner.
The K scale consists of three equal logarithmic scales placed
end to end, with the two end indexes in alignment with the
D indexes.
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830
THE SLIDE-RULE
331
The cube of any number on Z), therefore, is read on K
by the use of the runner on which the hair-line is carried
down to the K scale by an index mark on the side; and
inversely, the cube root of any number on K is read on D,
For convenience the three K scales may be denoted
from left to right by iCi, K2y and iCa.
330. How to Point Off the Cube of a Numb^. If
n denotes the number of integral figures in the number
whose cube is required, the number of integral figures in the
cube is as follows:
(1) When read on the K scale;
On Ki, 3n-2;
OniC2, 3n-l;
On Ksy 3n.
(2) When read on the A scale. (See Table XIV.)
Table XIV
POINTING OFF A CUBE
Cube read on.
Slide Pi ejecting
to the
Integral Figures
in Cube.
Left A
Right
3n*-2
Left
3n-l
Right A
Right
3n-l
Left
3n
The table applies only when readings on left A are the
cubes of numbers on the left half of D, and when readings
on right A are cubes of numbers on the right half of D,
(3) When the L scale is used. When a number is
cubed by multiplying its logarithm by 3, the mantissa being
*n denotes the number of integral figures in the given number.
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332
TECHNICAL ALGEBRA
331
read on the L scale, if c denotes the number of units in the
characteristic of the resulting logarithm, the number of
integral figures in the cube is as follows:
(a) When the characteristic is positive, c+1.
(6) When the characteristic is negative, — c.
331. Examples. By the use of the A^ B, and D scales,
determine and fill in the omitted entries in the following
table.
If your rule has a K scale, use it for checking the readings
on the A scale; if not, check by the L scale.
Table XV
SQUARES AND CUBES
No.
N
JV2
N*
No.
N
N*
isr«
1
13
6
3.4
2
9
7
53
3
14
8
.48
4
15
9
6.2
5
2.1
10
74
Read the squares and the cubes of the foUowng num-
bers and tabulate as above.
11. 4.
12. 9.1.
13. 5.4.
14. 7.
16. .21.
16. 3.7.
17. 11.
18. 1.3.
19. 6.5.
20. 1.5.
21. .6.
22. 2.4.
23. 12.
24. 8.
26. 7.6.
26. 6.1.
27. 4.3.
28. 5.
29. 3.2.
30. 2.9.
31. 2.5.
32. .4.
33. 7.2.
34. 5.3.
332. The Cube Root of a Number. By the principles
of logarithms the cube root of a number is obtained by
dividing its logarithm by 3, and by reading the antiloga-
rithm of the quotient.
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333 THE SUDE-EULE 333
The cube root of a number may therefore be computed
on the slide-rule, in four different ways:
(1) By ^, jB, and D scales.
(2) By the L and D scales.
(3) By the K and D scales.
(4) By the LL and CI or C scales.
Which of these ways should be used in determining a
cube root depends entirely upon the kind of slide-rule used.
If your rule has a K scale or a log log scale, the other ways
of computing the root will be of interest only as a means
to a more complete understanding of the principles of
operation.
333. Cube Root by the A, B, and D Scales. On all
plain Mannheim rules the A, B, and D scales are used for
the cube root. Before a reading can be taken, the number
whose cube root is required must be divided o£f as in arith-
metical cube root into periods of three figures each from the
decimal point.
Thus 3'917, .829'3, 1'.29, 14'.8, 42'674, .000763, .007'63,
.008'56, .091'5.
The part of the scale on which the readings are to be
taken is determined as follows:
When the first significant period has
(1) One significant figure, use left A and left C index;
(2) Two significant figures, use right A and left C index;
(3) Three significant figures, use right A and right C
index.
The first significant period is the first period beginning
at the left, which contains significant figures.
Pointing Off. The only rule required for pointing off
a cube root is the following:
Each period in the number whose root is required gives
ONE figure in the root.
Thus vT728=12, >J^.000^= .0928,
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334
TECHNICAL ALGEBRA
SU
334. Cube Root by the A, B, and D Scales. The prin-
ciples involved when the A, J5, and D scales are used, will
be evident from the setting for the cube of a number, which
always means the addition of once its logarithm to ttvice its
logarithm.
The cube of a number was therefore read on the A
scale by placing once the logarithm on jB, end to end with
twice the logarithm on Ay this latter being directly over
once the log on D,
Therefore when the cube was read on Aj the number on
B under the runner and the number on D under the sUde
index, were identical.
We therefore have the following
Rule for Cube Root on A, B, and D Scalm.
(1) Consider the given number to be separated into
periods of three figures each, in both directions from the
decimal point.
(2) Set the runner to the given number on A as specified
in the table below.
(3) Move the slide until the number on B under the runner
and the number on D under the C index are exactly the same.
This number is the cube root.
CUBE ROOT WITH A, B, AND D
Significant Figures in
First Period
Set Runner on
Read Root under
1
Left il
Left C index
2
Right A
3
Right C index
Owing to the difference in the graduation of the D and
B scales, the setting of the slide as required in (3) will give
some difficulty and requires considerable practice.
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836 THE SLIDE-RULE 335
In case the edge of the runner hides the reading of a
number under the slide index, the readings must be taken
without the runner.
335. Cube Root by the K and £) Scales. The cube root
of a number is read as follows on the D and K scales:
(1) Divide off the number, or consider it as divided off,
into periods of three figures each in both directions from
the decimal point.
(2) Set the runner to the number on K and read the
cube root on D under the hair-line, as follows:
When the first significant period has:
(a) One significant figure, set to Ki\
(h) Two significant figures, set to K2;
(c) Three significant figures, set to K3.
336. Examples in Cube Root. Extract the cube root
of the following numbers by the simplest method which
your rule permits.
Those who have a log log rule are referred to section 10,
page 342.
1. 2. 2. 3. 3. 4. 4. 3.9.
6. 78.5. 6. 100.5. 7. .00068. 8. .0068.
9. .068 10. 3.25. 11. 18.9. 12. 2024.
13. 15000. 14. 4.25. 16. 5.35. 16. 1.065.
17. 165. 18. 1800. 19. 78.5. 20. 341.6.
21. 3.456. 22. 5.09. 23. 61.4. 24. 49853.
26. 23.8. 26. 2095. 27. 1785. 28. 480.7.
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336 TECHNICAL ALGEBRA 887
§ 9. GAGE-POINTS
337. What Points are used. Computations involving t
are facilitated by the following gage-points:
IT =3. 1416.
- = 1.273.
^=.31.83.
miUL
kikikiiikitiiiijiiiii'iiiii-iiiifliii
Fig. 135. — r gage-point on A scale.
Almost any slide-rule has the w gage-point, but few have
the other two, both of which are an advantage in cylinder
computation.
which will be recognized as 100 X—, is used instead
TT IT
of the reciprocal of w for the following reasons:
When the w gage-point 3.1416 is marked on left A, —
or .3183 cannot be shown on the rule because the first
graduation on the rule, which is the left index, is then
unity.
If — or 3.183 were used, the gage-point would be so
w
close to 3.1416 that it would be confused with it.
100
Therefore — or 31.83 is used as the gage-point instead
TT
, 1 10
of — or — .
TT TT
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338
THE SLIDE-RULE
337
338. Reading from Gage-Point M. The gage-point
31.83 is located on right B and is denoted by M.
With a single setting it gives both the circumference
and the lateral area of a cylinder from the formulas
and
C=tD
A=tDL-
PL ^ lOODL ^ lOODL
1 100 31.83 •
25
30 M
35
40
25
30 M 35
Fig. 136. — M gage-point.
40
If your rule has no gage-point ilf , cut and letter it
carefully at the point specified, just under the B scale.
Observe that in the use of the slide-rule a factor of IQ
or an integral power of 10, affects only the position of the
decimal point in the result, and therefore is not included
in the settings.
Observe also, that a setting for
A =
lOODL
31.83
is also a setting for C = wD
because
1 100 31.83*
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S38
TECHNICAL ALGEBRA
339
Take the following setting on the rule, diagram setting
and readings, write the equation for pointing off, and explain
by the principle of logarithms why it gives the circum-
ference and the lateral area of a cylinder whose diameter
18 12" and whose length is 15".
A
12
Circumference
Lateral Area
B
M
100
15
339. Examples. In the following table,
D = diameter of cylinder in inches,
L = length in inches,
C = circumference ,
i4 =area curved surface.
Determine the circumference and lateral area by use of
the M gage-point, and fill in the omitted entries.
Table XVI
CYLINDERS. M GAGE-POINT
No.
D
L
C
A
No.
D
L
C
A
1
3i"
2i"
6
li"
r
2
sr
5r
7
2i"
¥'
3
11.4"
8.22"
8
7A"
3i"
4
23.8"
15.3"
9
9f"
41"
5
40.7"
20.2"
10
12.2"
30.5"
Diagram the settings and write the equations for point-
ing o£f.
340. Gage-Point C. This gage-point represents- = 1.273.
IT
When marked on the rule, it is located on the C scale
at a distance from the left C index corresponding to 1.273
on A, This is approximately at 1.1283 on the C scale. In
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S40 THE SLIDE-RULE 339
order that no readings involving this
point may be off the rule, a second
point Ci, is marked on the C scale at a
distance from C equal to the distance
from 1 to 10 on A, which is approximately
at 3.563 on the C scale. *
Take the following settings:
Align slide and stock indexes.
Move the runner to L273 on A. _ ^
Observe C scale reading under the ^^ ^ -^ ^
runner. _^ S
If your rule is accurately set, this -3- q| §
reading is where the gage-point C is
marked both on the C scale and on IS,
the D scale. " i m
Move the slide so that this reading ^ - ^ Mi A
is over left D index. ^ -^ i ^
Move runner to middle A index. «, " » I ^
Observe C scale reading under hair-
line.
If your rule is accurately set, this
reading is where gage-point Ci is marked
both on the C scale and on the D scale.
As will be seen from the next para-
graph, the proper location of C and Ci ?: ^- 2
is on the B scale. ^
In regard to gage-points in general,
every user of a slide-rule knows the diffi-
culty of an accurate reading from the
runner when the hair-line is between the
gage-point graduation and the regular
graduation. This difficulty would be
obviated by placing gage-points im- ziz
mediately above or below a scale instead
of across it.
:5^ 2
n --
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340
TECHNICAL ALGEBRA
341
Therefore if gage-points C and Ci are not graduated on
your rule, and you wish them there, carefully cut and ink
them just under the B scale.
341. Reading from C Gage-Point. The problem below
illustrates the use of the gage-point C
A cylinder is 22.5 inches in diameter and 38 inches long.
Formulate and compute the volume.
y ^^^^ 4 1.273-
A
.
Volume
B
38
C
gage-point C
D
22.5
Observe that when gage-point C is set over the diameter
22.5 on D by the use of the runner, 22.5^ is under the hair-
line on A and L273 is under the hair-line on B, Therefore
the distance on A from the left A index to the left B index,
represents
log 22.52 - log L273.
If the runner is now moved to the multiplier 38 on By
which is the length of the cyUnder, the volume may be read
on A under the runner, because to the difference above has
been added the logarithm of the length.
Attention is called to the fact that in this instance, gage-
point C was required on the B scale and nowhere else. If C
were so placed, as it should be, the diagrammatic setting
would be as in Table XVII.
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842
THE SLIDE-EULE
341
Table XVII
VOLUME OF A CYLINDER. C GAGE-POINT
A
Vol
A
Vol.
B
C
Length.
B
C
38
D
Diam.
D
22.5
342. Problems. Solve the following problems, in every
instance showing diagrammatic setting and equation for
pointing off.
1. Determine the volume of the cylinder in paragraph
341 and tabulate data and result.
2. Determine the volume of the following cyUnders both
in cubic inches, cubic feet, U. S. gallons, and liters.
Table XVIII
CYLINDERS. C GAGE-POINT
No.
L"
D"
Volume.
Cu. in.
Cu. ft.
Gal.
Liters.
1
18i
9i
2
12.8
6i
3
8.5
4f
4
13i
5i
-
5
26i
15
6
30
18.1
7
54.7
31.4
8
38.6
Hi
9
45
21.3
10
76
401
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342
TECHNICAL ALGEBRA
343
§10. THE LOG LOG RULE
343. A Log Log Scale. Any rule having the usual
scales is an effective instrument for the determination of
the square or the square root, and the cube of a number.
A cube root, as has been shown, may also be read, usually
with some difficulty and an unsatisfactory approximation.
Direct readings of any other powers or roots are impos-
sible. Indirectly, a power or root not the square or cube
may be determined by reading the logarithm of the number
on the L scale, transferring the reading to the D or the A
scale, and multiplying it on the rule by the exponent of the
power or dividing it by the root index.
The LL scale is a scale whose graduations are proportional
to the logarithms of the logarithms of numbers.
On the log log duplex rule, it is mounted in three sections,
marked LL 1, LL 2, and LL 3, making a continuous scale
from
to e
10
e denoting the Naperian base 2.71828.
Fig. 138. — Right index of back of LL duplex.
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843
THE SLIDE-RULE
343
The powers of e and the limiting numbers at the ends
of the three sections of the LL scale, are as follows:
LL 1; ei«o to e" or 1.01 to 1.105.
LL 2; eio to e, or 1.105 to 2.71828.
LL 3; e to e^", or 2.71828 to 22000.
By reference to Fig. 139 or the LL rule, write the value
of each graduation from 1.C8 to 1.09 and from 2000 to the
end of the LL scale.
1.08
iiiiiiii nil
2000
III
1.09
ililulil
lllllllllll
5000
1.10
II I I I IMIMlllll
I IIIIIIII
lllllllll ^
10000 20000
lllllll
Fig. 139. — Right end of log log scale.
Cube Root. The principles involved in the use of the
LL and CI scales, and LL and C, for cube root, will be
evident from the following illustration:
Required the cube root of 2C0. This may be represented
by the equation:
log a: =
log 260
/. log log a: = log log 260 — log 3.
What is wanted, therefore, is a rule so scaled that the
log 3 may be subtracted from log log 260. A rule with a
log log scale accomplishes this, and therefore makes the
determination of a power an addition, and the determination
of a root a subtraction.
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344
TECHNICAL ALGEBEA
844
If your rule has a log log scale, take the following settings
-and explain why they give the cube root of 260:
LL3
260
6.39
CI
1
3
LLS
260
6.39
C
3
1
Fig. 140. — Section cf LL rule showing setting for cube root of 260.
344. On What Section of LL, to Read the Root. When
the rule is set for the cube root of a number on any section
of the LL scale, if the reading on the same section in align-
ment with the C index or \vith 3 on C/, is greater than the
number whose root is required, the root must be read on
the preceding section; otherwise, read root on the same
section as the number.
Thus the cube root of 8 is read:
LLS
8
LL2
2
C
3
1
LLS 8
LL2
2
CI 1 1 1
3
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346
THE SLIDE-RULE
345
The root is read on LL 2 because the reading on LL 3
in alignment with 1 C and 3 CI, is greater than 8, the number
ivhose root is required.
The cube root 8000 is read:
LL3
8000
20
LLZ
8000
20
C
3
1
CI
1
3
In this instance, both the number and its root are read on
the same section LL 3, because the reading in alignment with
1 C and 3 CI is less than 8000, the number whose root is re-
quired.
Observe that the numbers on LL are absolute and must
be used without shift of decimal point.
The cube root or the square root of a number above or
below the limits of the LL scale may be read if the number
is separated into periods as in arithmetic root.
Thus v/.02942 = v^MF42.
Therefore read •v^2J.42.
345. Examples. Refer to paragraph 336 and solve all
the examples on LL and C, or LL and CI. Diagram, and
give equations for pointing off.
346. Any Power or Root. The equations in paragraph
343 indicate how any power or root may be read with the
log log scale.
Thus the determination of x^ is formulated as follows:
log x^ = 5 log X = log 5+log log X.
Therefore log 5 on C is placed end to end with log log x
on LL and the result is read on LL.
A log log scale, therefore, makes it as easy to obtain any
power or root as to perform multiplication and division, and
by the same process.
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346 TECHNICAL ALGEBRA 347
347. Examples. Determine the following powers and
roots by the use of the LL scale, and check by logarithmic
computation.
1. 4.7«.
2. 2.9*.
3. 4.7'.
4. 3.6- «,
6. 1.9«.
6. 11.5'.
7. V'l745.
8. V734.
11. -^9143.
9. V2448.
10. v' .07165.
12. V649.5.
13. V843.4,
14. V.764.
16. -^29.75.
16. 32.75*.
17. 464».
18. 5064'-
348. Logarithms to any Base. The common logarithm
of a number may be /"ead on a log log rule by the use
of the L and D scales, the same as on any other rule.
Since LL 3 begins at e the Naperian base, logarithms to
base e are read on C for any aligned number on LL,
Logarithms to any other base are read by aligning the
left C index with the required base on LL, If the reading
on C is off the slide, change indexes.
Thus, to read logio 40 set left C index to 10 on LLj
and under 40 on LL read 1.60 on C This is the common
log 40.
To read to base 5, set left C index to 5, etc.
348. How to Read a Naperian Logarithm.
1. Ahgn indexes. This sets left C index under e on LL.
2. Move runner to given number on LL,
3. Read complete Naperian logarithm on C under the
runner.
How to Point Off.
(1) Place .0 before all C readings from numbers on LL 1.
(2) Place decimal point only, before all C readings^
from numbers on LL 2.
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350
THE SLIDE-RULE
347
(3) Point off one integral figure in all C readings from
numbers on LL 3.
Observe that the C reading is the entire loge, and therefore
includes both characteristic and mantissa.
349. Illustration. The loge650 is diagrammed as
follows:
(1)
LLS
e
650
C
1
6.47
One integral figure is pointed off in the C reading
because 650 is on LL 3.
See preceding paragraph.
(2)
LL2
e
1.755
C
1
.562
Take this reading and explain the position of the decimal
point.
(3)
LLl
1.035
Complete this illustration and explain.
360. Napierian Logarithms of Numbers Less Than
1.01. The LL scale begins at 1.01 which is therefore the
lower limit of numbers whose logarithms can be read directly.
The logarithms of numbers below this Umit may be read
in two ways:
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348 TECHNICAL ALGEBRA ssa
L (1) Multiply the number by 10, or 100, or whatever
power of 10 is necessary to bring the number within the
range of the LL scale.
(2) Read log« of the number obtained by the multiplica-
tion.
(3) Subtract log* 10 or log* 100 (log« multiplier) from
the C reading.
The difference is the Naperian logarithm of the given
number.
Thus log. .585 = loge 5.85 - log* 10,
= 1.767-2.303.
Therefore log* .585 = T.464.
II. (1) Move runner to given number on C and read
its reciprocal on C/.
(2) Read log« reciprocal.
(3) Subtract the reading on C obtained in (2), from a
number one greater than its integral figure.
The difference thus obtained is the loge of the given
number.
Thus
.585 = -J-.
.585
Therefore
log. .585 = log«l log. ggg
= O-log.1.71.
= 0-.536
Therefore
log. .585= 1.464.
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361
THE SLIDE-RULE
34^
361. Examples in Naperian Logarithms. Read, and
diagram the reading for each example in the following table
and fill in the omitted entries.
Read examples 11 to 16 inclusive, both ways.
Table XIX
NAPERIAN LOGARITHMS
No.
Number.
Logarithme
No.
Number.
Logarithm^
1
1.0356
11
.5
2
1.062
12
.125
3
1.75
13
.456
4
2.17
14
.795
5
e
15
.028
6
4.58
16
.064
7
34.5
17
1.85
8
455
18
.185
9
2175
19
.0455
10
3 78
20
.3315
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CHAPTER XVII
TRANSFORMATION OF FORMULAS
352. Definition. A formula is a statement in significant
symbols, of the exact mathematical relation of two or more
quantities.
Transformation is the process by which formulas are
simplified, solved for any or all of their quantities, or other-
wise changed according to mathematical authority.
353. Method of Transformation. A point has now
been reached in the study of this book where it is necessary
to classify an equation before attempting to solve it. This
is due to the fact that while, in preceding chapters, equations
have been grouped under their various names so as to lessen
the confusion of learning the method of solution for each
kind, in this chapter hardly any two of the same kind are
grouped together and all are technically, instead of mathe-
matically, named.
Special methods of transformation adapted to various
conditions can be learned only by considerable practice.
There is a general method and order of operations, however,
which will reduce the work to a minimum.
Briefly, the method is as follows:
1. Observe where the unknown quantity is, in the
formula; whether in the first member or the second or both,
in numerator or denominator of a fraction, or under a radical,
and if so, whether as a factor or a term.
2. Classify the equation as simple, quadratic, etc.
3. Apply the method of solution which will give the
required transformation most directly and simply.
350
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364 TRANSFORMATION OF FORMULAS 351
Clearance of fractions is not always essential to solution,
and in many instances only increases the work.
The one essential to the transformation of any formula
is the determination of the coefficient of the unknown
quantity. If this can be determined without clearance,
as often happens, clearance should not be resorted to.
2cr
For example, 0G = — may be solved for r by multiplica-
tion by ^, the inverted coefficient of r.
76 5cb
625——=— may be solved for b by transposing, factor-
7 5c
ing, and dividing by ^+^r which is the coefficient of b.
As soon as possible all equations of this kind should be
solved mentally without the use of the pencil.
When elimination of one of the unknowns in one or
more simultaneous formulas is required, the method of
elimination by substitution is usually the simplest.
When solving for any quantity all others are regarded as
known.
Thus, when a^—ac = \/b + l is to be solved for a,
b and c are regarded as known quantities.
If solution is required for 6,
c and a are considered known.
It takes a long time to become skilful in transformation,
but it is well worth while for it is the key to the interpre-
tation of much of the technical material in books and
periodicals which help one to become intelligent and there-
fore of ^eater service.
354. Simplest Form of Result. A result is in its simplest
form, generally speaking, when so written that each quantity
is used as few times as possible. In other words, simplest
form is the form requiring fewest substitutions of known values
and the least work in computation.
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S52 TECHNICAL ALGEBRA 866
Illustration. Suppose a given formula has been solved
for Q with the result:
^ ri^+2rir+r^
If the values of ri and r are known, say ri = 5 and r = 8,
Q can be determined by six substitutions and seven
operations.
If however the equation is reduced to the form
Q
-m-
Q is determined by four substitutions and four operations.
Sometimes when a formula is reduced to a form requiring
fewest substitutions, more diflScult or longer operations are
involved after substitution is made.
The equation just solved may be reduced to a still,
simpler form as follows:
\ r\T ) \T\r r\r} \r r\)
Only two substitutions are now required to determine
Q, yet these might be offset by the increased time necessary
to perform the indicated operations. With a slide-rule, or
tables of reciprocals and squares, Q would be determined
with Uttle effort, whatever the values of r\ and r.
366. Problems. Enter each formula under the heading
given and solve as directed:
1. Total Heat of Vaporization.
^=108L94+.305«.
Solve for i,
2. Latent Heat of Vaporization.
Solve for q and r.
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355 TRANSFORMATION OF FORMULAS ' 353
3.. Net Pressure on the Piston.
CA-OD^DA.
Solve for OD.
Find the value of OD
when DA =45.3.
and CA =60.
4. Work m Foot-Pounds.
W=PV.
Solve for P and V.
5. Real and Apparent Cut-Off.
, h+i
i+t
Solve for i and ki.
6. The Pantograph.
AB^CD
L CE'
Solve for each term of both fractions.
7. Indicated Horse-Power.
jjj^JLAN
33000 •
Solve for LN.
8. Total Ratio of Expansion.
V
Solve for e, F, and v.
9. Stress.
Solve for P and A. ' ■ <
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354
TECHl
^ICA
L
10.
Strun.
8'
e
L
Solve for e and L.
11.
Coefficient of Elasticity.
E-
_S
s'
355
Substitute for S and s their values from 9 and 10 and
solve for E.
In the resulting formula, compute the value of E
when e =L and il = 1 sq. in.
12. Factor of Safety.
P-
Solve for /and -4.
13.
Pressure on a Pipe.
pd-
=2fcS.
Solve for d, t, p, and S.
14.
Cylinders.
P--
=j7rd»p
Solve for d.
16
4
Solve for d and p.
16.
St
Solve for r and t.
5
Compute the value of S when p = 163.8, r=7, i=r-.
o
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865 TRANSFORMATION OF FORMUX.AS 355
17. Pressure of a Mixture of Steam of Different Pressures.
VP=vp+ViPi.
Solve for P, ©, and pi.
18. Ratio of Cylinders.
V
^=2.72-.
V
Solve for v and V.
19. -=V^.
V
Solve for v and E,
20. The Mean Ordinate of Tangential Pressure on the Crank-
Pin.
2SV
Solve for SV.
21. Resistance.
AM^-^.
KL
Solve for d and L.
22. Measurement of Resistance.
dr=diR-\-dir,
Solve for i?, and cxpro&s the result in the simplest form.
23. Electrical Equivalent of Heat.
rr = .000^77 ECT,
Solve for C and T.
24. Magnetic Field within the Solenoid.
Solve for N and L.
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356 O'ECHNICAL ALGEBRA 356
26. Sum of the Moments of Horizontal Stresses.
Af--Ar«.
c
Solve for 8,C,A, r, and ArK
26. Moment of Inertia.
I^ArK
In 25, substitute the value of Ar* and solve for c.
27. Bending Moment.
7 ^'
Solve for — and/.
c
«.f.
Solve for L.
29. Deflection of Beams.
aWL*
Solve for a, W, E, and L.
30. Comparison of Strength and Stiffness of Beams.
bd*
D
Show that W«=Mi*.
31. Speed Cones.
Solve for d, Ui, and ni.
32. Locknuts.
a=l}d-l
Solve for d.
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866 TRANSFORMATION OF FORMULAS 357
S3. Proportion of Keys.
4
Find the value of t in terms of d, and solve the resulting
equation for d.
34. The Cylinder.
a = 1.21D+2e+1.22.
but e=.0003PD+.375.
In the first equation, substitute the value of e from the
second, and solve for D.
36.
The Piston.
Solve for D.
c=.18V2D-.
1875'
86.
The Connectmg Rod.
g-e'
. .32a;
*= h-
The equations are simultaneous. Solve for i, eliminating h.
37. The Law of the Lever.
w U
Solve for F, TT, L, and Li.
88. Quality of Steam.
W+R+w*
Solve for 5.
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358 TECHNICAL ALGEBRA 355
39. Factor of Evaporation.
W{H^t+32)
966.1
Solve for W and L
40. Thickness of a Fire-Box Plate.
Solve for t.
41. Strength of the Head of a Boiler Shell.
2r=p
Solve for p, r, and t.
42. Efficiency of a Riveted Joint.
h-d
2/=-
h
Solve for d a.nd h.
43. Double-Riveted Lap Joints.
•«)'■
Solve for d and t.
44. Rate of Combustion.
F=J2,25\/ll-l.
Solve for H,
46. Safety Valves.
Tf d+TTi a+W2C -pAa =0.
Solve for d and A.
46. Stress in Punch and Shear Frames.
Solve for Ct and P.
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3G5 TRANSFORMATION OF FORMULAS 359
47.
Diameter of a Pipe.
,^5^(/L+Jd)y.
Solve for h, d, and Q.
48.
Strength of a Stayed Surface.
-'!•
Solve for t and h.
49.
Pitch of a Boiler Stay.
a = .835d^.
Solve for T and d.
60.
Diameter of a Direct Boiler Stay.
Ap^^d'T.
Solve for d.
61.
Belting.
ws
^-900-
Solve for S and W.
62. Quantity of Air for Complete Combustion.
("-!)■
TF = 11.6C+34.8
Solve for H.
63. Thickness of Pipe.
. St
Solve for L
64. Pitch of a Screw in Terms of Outside Diameter.
V16D+IO -2.909
^■" 16.64
Solve for D,
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360 TECHNICAL ALGEBRA 365
66. The Area of a Segment of a Circle.
Solve for r, h, and C.
66. The Inclined Plane.
_ Wh
but 7 = what function of the angle a?
Therefore F= what?
67. Resistance Measured by a Wheatstone Bridge.
Ciri =Ctrt.
Cix ^CiR.
These equations are simultaneous.
Solve for — in terms of x and R,
68. Electromotive Force of a Battery.
E
R+r
E
Ri+r
These equations are simultaneous; E and r are unknown.
Solve for E, eliminating r.
69. ^=/(«+r).
Ei=lR.
Find the value of r in terms of everything except /.
60. Space Traversed in any Second.
Solve for t and g.
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366 TRANSFORMATION OF FORMULAS 361
61. The Screw.
Solve for TT, P, and r.
62. The Safety Valve.
Apb—Qa
Solve for W, p, and A.
63. The Compound Geared Lathe.
gchi
n = - ...
afj
Solve for t .
64. Temperature-Coefficient of Resistance.
Solve for n.
66. Location of a Fault in a Cable.
x+y
Solve for x.
66. Winding of Armatures.
y-l{i+'')-
Solve for C, 6, p, and a.
67. Unit Acceleration of a Pulley.
Solve for w.
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362 TECHNICAL ALGEBRA 366
68. Coefficient of Expansion of a Gas at Constant Pressure.
Vi = Vo(l+Bt,);
V.^Voil+BU).
These two equations are simultaneous.
B and Vo are unknown.
Solve for B, eliminating Fo.
69. Horizontal Intensity of the Earth's Magnetic Field.
(1) mhJ-^,
(2) but ^ = ^(i+?)'
M 1
(3) and -pz =^r* tan 8,
Divide (1) by (3).
In the resulting equation, substitute the value of K,
Solve for K,
70. Problem. By substitution in the preceding formula com-
pute the value of H to three decimal places when
r=30 W
= 17.22
5=3° 10' L
=49
«=3.65 a
= .09
71. Current in Series.
, Ens
ms+R'
Solve for n„ r, and R,
72. Current in Multiple-Series.
Up
Solve for n,, tip^ r, and R.
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855 TRANSFORMATION OF FORMULAS 363
73. Water Required by a Condenser.
W{ti-k)^H-U+32.
Solve for TT, Uy and H.
74. Potential Difference.
p n
(1)
(2)
(3)
Solve (1) and (3) as simultaneous equations, for P.D.,
eliminating /.
In the resulting equation, substitute the value of r from
(2), and simplify.
/=
' R '
/.
E-P.D.
r
1-
E
76.
Fly-Wheel Computation
yo=
2 '
E =
y. *
Solve for EV^K
76.
The Diameter of a
Wire
D =
w
.7854dL*
Solve for W and L.
77. Capacity of a Vessel Whose Form is the Frustum of a
Cone.
C=^(/2«+r«+/2r).
Solve for R and r.
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364 TECHNICAL ALGEBRA 866
78. Velocity of Sound in terms of Temperature.
F=333(l+.00370*.
Solve for t.
79. Measurement of Resistance of an Electrical Conductor.
R+r+g+X tan a
B'+r+g "tan o'*
C' =
B+r+g+X'
E
R'+r+g
C
Solve for —, in terms of tan a and tan o'.
80. Measurement of Electromotive Force of a Battery.
. _E E'
• R+r+g R''+r'+g'
E E'
R+R'+r+g R-+R^^'+r'+g
Prove
^- R"
81. Radius of Curvature of a Spherical Mirror.
a^b R-
Solve for R.
82.
"-6A+2-
Solve for h.
83. Thompson's Method of Comparison of Capacities of
Condensers.
Show that §=-'.
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366 TRANSFORMATION OF FORMULAS 365
84. Fall of Potential in a Series D3mamo.
E^e+I{Ra+Rf).
Solve for I and Ra.
85. Air-Gap in a Dynamo.
1,26^.
a ^^ XA,
In the first equation substitute the value of No from the
second, and solve for A.
86. Speed of a Continuous Current Motor. ,
2TTr:^ZNE-nZ*NK
Solve for n, and ZN.
87. Current in a Series Motor.
c-f-o.
Solve for C.
88. Focal Distance of a Concave Spherical Mirror.
— "T" — = — .
ajb r
Solve for /.
89. Radius of Curvature of a Convex Spherical Mirror.
a ^ jR
b 'c+iR'
Solve for R.
90. Principal Focal Distance of a Convex Lens.
Solve for r'.
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366 TECHNICAL ALGEBRA 355
91. Magnifying Power of a Lens. (Approximate formula.)
Solve for /.
92. The Focal Length of a Lens.
V Pi
Prove t / =
4L •
93. The Catenary.
Transpose so that the radical is the only term in one of
the members; then square to remove the radical, and
solve for m.
94. Stress Due to Longitudinal Tension.
R f-fi '
Solve for/i.
96. Wear of Wheels.
&
Solve for Ru
96. Stress in a Rectangular Plate.
Ri \Ri R2/
•' 2\L*-\-h*)t^
Solve for L.
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366 TRANSFORMATION OF FORMULAS 367
97. Effect of Temperature on Resistance.
K =
Solve for n.
98. Cable Testing.
riit^-tiY
X+y^r".
These equations are simultaneous.
Solve for X and y in terms of r, r', and r".
99. Thickness of Web of an I Beam.
A^td+2s{b-t) + ^^~^lQl
Solve for t and b.
100. Girder of two Spans with Uniformly Distributed Load.
^•-2r-4L(L+L.)J-
Solve for L and Li.
101. Radius of Gyration of a Ship.
Solve for TT, D, and L.
102. Diameter of Rivets in a Double-Riveted Joint.
Uird^ = lSt{P-d).
Solve for d and P.
103. Ratio of Expansion.
tt
Solve forr^""^ and r.
JXl.
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368 TECHNICAL ALGEBRA 355
104. Efficiency of Camot's Engine.
^^ . Work done
Efficiency =^~T n-
Heat expended
Work done =Heat expended -heat rejected.
Heat expended ^cUloggr.
Heat rejected ^cUhg^i,
But fj =r.
Derive the formula of efficiency.
For highest efficiency what must be the value of 1%^ the
temperature of the condenser?
105. The Rankine-Gordon Formula for Colimins.
P
S--7+S,,
Show that
Mc
M^Pf,
I=Ar*,
P S
A ,,!,«•
106. Internal Diameter of a Hollow Column.
P_ S
r'=^(i>'+d*).
In the first equation, substitute for A and r» and solve for d.
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S6S TRANSFORMATION OF FORMULAS 369
107. Stresses Due to Impact.
yQ
e P'
Solve these simultaneous equations for Q and y
In the formulas obt^ed find the values of Q and y,
when h=0;
also when h =4e and 12e.
108. Shrinkage of Hoops.
S
(D-d)
Substitute for 8 from the second equation and solve for E.
109. Elastic Resilience of Bars.
K^lPe.
PL
P=SAf and €=-7-=.
Alii
Show that ^=^^-
110. The Area of a Triangle.
^w (,-16— ;•
Solve for a, 6, and c.
111. The Area of a Trapezoid.
Solve for a, h, and h\
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370 TECHNICAL ALGEBRA 366
112. The Perimeter of an Ellipse. (Approximate Formula.)
p /P'+d' (D-d)'
^='\ 2 -"lis"-
Solve for D and d.
113. The Length of a Chord.
Solve for r and h.
114. The Helix.
Solve for L, d, and n.
116. The Surface of a Cone.
^=|[L(D+d)+i(D«+d«)].
Solve for D, //, and d.
116. The Volume of a Wedge.
V=lwh{a+b+c).
Solve for h and c.
117. The Volume of a Prismoid.
Solve for L and m.
118. The Volume of the Frustum of a Regular Pyramid.
V ^lh{A+a+VAa).
^ o
Solve for h and A.
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355 TRANSFORMATION OF FORMULAS
119. The Length of a Spiral.
Solve for t and R,
120. The Mean Velocity of Discharge of Water in Pipes.
371
4
7m =2.315
Solve for d and L.
121. The Diameter of a Shaft.
hd
fL+125d'
Solve for c and N.
-Kil
122. Relation of Temperature to Pressure.
U2\
to \po/
0-1
Compute the value of t when g = 1.408, to =520°,
p=30, po=15.
123.
Work Done by an Engine.
Work done =J(H-h),
Work done =what in terms of all except H?
124.
Pressure with Clearance.
--I
\^*-y]
[.+,T,J
Solve for v and V.
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. 372 TECHNICAL ALGEBRA 856
125. Point of Cut-Off.
vB „ vB
Initial pressure =-^+Bpz —^,
£jZ £i
In the first equation substitute the value of z from the
second, and prove
(2\/^ l\
F /•
126. Centers of the Valve Circles in Link Motions.
'4F''+C'ir)*'°«4
X--
Solve for k.
127. Exposed Surface of Cylinders.
Simplify this formula and solve for r and d.
128. Inductance of Transmission Lines.
L=^(9.211og..f+l).
Solve for Lu, and logio S.
129. Pressure of Saturated Steam.
4079 71
log p= 10.515354 - Z,' -0.00405096!r+0.ai39J9647'».
Compute p when !r = 575°.
130. The Specific Volume of Saturated Steam.
p7"=475.
Solve for log V,
Evaluate V when p = 129.
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356 TRANSFORMATION OF FORMULAS 373
131. Mean Efiiective Pressure.
_ l+hyplogjR
V =P ^ .
Compute the value of p when
p'=125,
and 12=3.
132. Thickness of a Boiler Tube or Flue.
^.18
p=960000(H-r.
La
in which L, d, and ^ are all in inches.
Compute the value of t when p = 170 lbs. per sq. in.
L =3i feet.
d =«4i inches.
133. Breadth of a Rectangular Column.
^= — ^ — •
Solve for L,
134. Effective Pull of a Belt.
T
hyplog^=/a,
a=27m,
P^Tx-Tt.
T
Compute ^ when / = .3 and n = .464,
and find P in terms of Ti.
135. The Area of a Parallelogram.
The diagonals of a parallelogram are 81 and 106, and
cross each other at an angle of 20° 18'.
Formulate and compute the area of the parallelogram.
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374
TECHNICAL ALGEBRA
366
136. Height of a Weighted Pendulum Governor.
BxXH =(B+Pr)r,
but Bx=XmMBrN\
Solve for H in terms of PT, JB, and N*.
137. The Equation of the Ellipse from the Trammel.
Fig. 141.
A trammel is a bar fitted with a pencil at D, and with pins
at F and G which slide in grooved pieces at right
angles to each other.
DF =6, the semi-minor axis of ellipse,
DG =a, the semi-major axis of ellipse.
The pencil Z>, in one complete revolution describes an
eUipse.
Prove
—+— = 1.
Write proof in full, with authorities.
138. Strength of a Riveted Joint.
{12 ^nd)tSi
P=~
Zird^Sz
12
Find d in terms of t, h, Si, and St,
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366 TRANSFORMATION OF FORMULAS 375
139. Deflection of a Magnetic Needle.
e^^iB-A)[c-\iA+B)].
Solve for B and m.
140. Potential of Needle in a Quadrant Electrometer.
Solve for C.
141. Charge of a Leyden Jar.
^-i^A
i+i+i
Ci C2 Ci
Solve for C2, and V,
142. Force between Parallel Electrified Plates.
Solve for 27rs*, the force per unit area.
143. Excess of Potential in Parallel Plates.
V=iirPh+^+iTP[d - {h+t)].
Simplify the second member.
144. Energy per Unit Area.
Using the equation in problem 142 derive a formula for
Js7, the energy per unit area of the plates.
145. Electric Capacity of Spheres.
Simplify the second member.
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376 TECHNICAL ALGEBRA 355
146. Difiference of Electric Energy in Leyden Jars.
Add the fractions, expressing the result in the simplest
form. ^
147. Refraction of Lines of Force.
Ri sin 01 =Ri sin 62.
KiRi cos 61 KjRi cos 6^
Derive the simplest formula for tan ^1.
148. Magnetic Force from a Small Magnet.
][f .
i2=— V4cos*^+sin«^.
ft
Substitute for sin 6 in terms of cos 6 and simplify.
149. Equilibrium of Couples.
HM' sm 6 = -^ .
Solve for tan 6,
150. Difference of Magnetic Potentials.
M
-Oi — ^(cos Bi -cos ^2),
Z = cos 01,
ft
Zi = — COS 6t.
ft
Solve for -Oi in terms of Z, Zi, and r.
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856 TRANSFORMATION OF FORMULAS 377
161. Work done by an Electric Field.
'^"oe^^^^-^Q)-
Simplify the second member.
162. Charges of two Concentric Spherical Shells.
a
Solve these simultaneous equations for Ei and Eu]
Solve these simultaneous equations for Ei and J^j.^
154. Total Charge on Two Concentric Spheres.
-^(-D-
V-
Solve for c' and c.
155. Relation of Temperature to Heat.
Qi-Q*
Ti
Determine the relation of Q to T and express the result
with the variation symbol.
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378 TECHNICAL ALGEBRA
166. Measurement of Coefficient of Self-Induction.
SJ6
Fig. 142.
Derive the formula for cos ft in terms of E^ Ety and Eu
157. Capacity of a Condenser.
C =
(
i^log.-
Solve for logio — .
168. Measurement of Power.
-H'-i^y-'-]
Solve for i^i.
159.
A Girder Stay.
phi* SM^
8 " 6/ •
Compute the value of d when
V = 18000 and b=\d.
f 4
160.
Effect of Variation of Resistance.
r ^
Vr^+L*w*
tantf=-.
In the first equation, substitute the value of Lw from the
second, and solve for tan 6,
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365 TRANSFORMATION OF FORMULAS
161. Coefficient of Self-induction.
/= current,
E = electromotive force,
I/W?= reactance,
/?= resistance,
A B C issi right triangle.
LW
379
The ratio of the electromotive force to the current equals
the impedance.
Write the formula for the current in terms of electro-
motive force, resistance, and reactance.
162. General Formula for Tangent Galvanometer.
ff tan 8
2wri^i 2'Kr<i^n2
Solve for tan 3.
163. Economic Coefficient in a Series Dynamo.
R
V =
R+ra+Tm
Solve for R.
164. Economic Coefficient in a Shunt Dynamo.
1
^ =
■^^^¥
Solve for r„ and ra.
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380 TECHNICAL ALGEBRA 866
166. Efficiency of a Motor.
Wt—We-jWi+Wf)
"-" Ws
Solve for W,.
166. Loss Through Heating of Conductors in a Shunt Dynamo.
(E\t E*
^-rJ ^+Ri
Solve for E and R/.
167. Theoretical Height of Chimney for Given Draft.
7.65 7.65'
Ta Tc
Solve for Tc.
168. Effective Area of a Chimney.
Vh
Solve for H and A.
169. Twisting Moment of a Shaft.
Ti^B+Vb^+TK
Solve for B.
170. Mutual Induction.
4^SM
L '
Determine the relation of L to S and use the variation
symbol in the result.
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366 TRANSFORMATION OF FORMULAS 381
"1. |=Z.
Substitute for B in formula 169, and solve the resulting
equation for Z and T.
172. Dimensions of a Journal.
Solve for P and p.
173. Bending Moment of a Wall Bracket.
Solve for h.
S^bh'.
2*
174. Solid
Flange Coupling.
(l)
(2)
2d=B-i
(3)
In the second equation, substitute the value of d from
the first and solve the resulting equation for B,
In the third, substitute for B and solve for d.
176. Outside Diameter of a Hollow Shaft.
di*-d2*
d^~
A
d.
Prove di=-d
31 1
l-m*
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3S2 TECHNICAL ALGEBRA 366
176. Strength of Gear Teeth.
, Sbt^
but t=-C,
L = .7(7.
In the first equation substitute these values and solve for C.
177. Power Transmitted by a Rope.
Solve for Z>.
178. Diameter of a Pi&ton-Rod.
IOttD^
*=.(|)'xf.
4
Solve for d,
179. Per Cent of Moisture in Steam.
Solve lot h, T, and Ti.
180. Strength of a Stayed Surface.
C(16^+l)*
p=-
s-6
Solve for t.
181. Intensity of Shearing.
(r-p) cos ^=5 sin B.
r sin ^=5 cos B.
Solve for 5* in terms of r and p.
Solve the q^ formula for r.
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355 TRANSFORMATION OF FORMULAS 383
182. Centrifugal Force of the Rim of a Fly-Wheel.
Wihw^
but h =
and w =
g '
Rn sin a
IT
V
R'
Therefore F= what when A and w are eliminated?
Solve for sin a.
183. Direct Tension on Each Arm of a Fly-Wheel.
frgR 2gr
Solve for TFi and W2.
184. Breaking Strength.
A
Solve for K.
185. Resistance of a Riveted Joint.
1 . A2L-dYiU
Solve for d.
186. Coefficient of Linear Expansion of a Solid.
Solve for L and K.
187. Stability of a Shaft.
Solve for L, T, E, and /.
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384 TECHNICAL ALGEBRA 865
188. A Boiler Plate.
Solve for D,
189. Length of a Journal.
a
Solve for P.
190. Strength of a Cotter.
Solve for 7), d, and L
191. Friction of a Pivot.
|d.A=^+l^iVd.
Solve for d.
192. Friction of Worm and Wheel.
l+/xtanfe
tan^— /i '
Solve for tan ^i.
193. Strength of a Crank.
Solve for A and m.
194. Radius of a Cast-Iron Spring-Ring of Unequal Thickness.
11 24pr«8in«g
Solve for t and p.
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366
TRANSFORMATION OF FORMULAS
385
195. Strength of Teeth.
Solve for p and P.
196. Equation of the Catenary.
Fig. 144.
^ 8 I ?
Solve for s.
197. Forces on the Crank Pin.
tmax=P^
Solve for L,
198. Sieman's Method of Comparison of Capacities of Con-
densers.
Solve for Ci.
•-'^"Vc+cy
199. Law of Deflection
.
Given
/ PO
H RO'
PO=AR,
f=2Mr:
Prove y
' 1
M=^fftan5.
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386 TECHNICAL ALGEBRA 355
200. Distribution of Magnetic Force on the Earth's Surface.
Approximate law:
Magnetic force oc V L+3 sin« L.
Express this law as an equation, denoting magnetic fore 3
hyH.
201. Induced Electromotive Force in a Closed Circuit.
E =
C
t
E
Solve these two simultaneous equations for C, eliminating E,
202. Electric Potential.
Fp-Fg=£-(n-r).
rri
Simplify the second member.
203. Force Exerted by a Charged Sphere.
Find the value of / in terms of tt, r, and p.
204. M. Love's Formula for Pressure.
p =5358150 -^+41900^^-1323 1.
Solve for t and d.
205. Capacity in Electrostatic Units of two Concentric
Cylinders.
Solve for logio r'. Y^°f^ r I
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356 TRANSFORMATION OF FORMULAS 387
206. Difference of Potentials in a Special Condenser.
Vn- ^ ,
K- ^
Solve for K in terms of r and r'.
207.
Capacity of two Condensers Joined in
Series.
Solve for — in terms of Ki and K%,
208. Path of a Projectile. (Equation of Trajectory.)
a:=dcos ao,
(1) Solve for y when t is eliminated.
But c=\/2i^.
(2) Find y when c is eliminated.
(3) Let J/ = 0, and solve for x,
209. The Connecting-Rod.
But n = .155D+.0623.
Solve for L eliminating n.
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388 TECHNICAL ALGEBRA 366
210. Stress on a Plate.
Solve for h.
211. Tension in a Bolt.
Q
2 1 ( p+nird \
3^'^2\Td-tJLpJ
R
d'
Solve for d and p.
212. Velocity of a Projectile.
VFTjTfr, ^ \/c*-2ctgBmao+gHK
Solve for t, c, and g,
213. The Attracted-Disk Electrometer.
Difference of potential = work,
But difference of potential = Fi — Vt,
And work = force times distance,
But force = attraction due to fixed plate
plus repulsion due to movable plate.
Attraction due to fixed plate =27rp.
Repulsion due to movable plate =27rp.
Distance =D.
Write the formula for Vi — Fj.
F=:2TpXSp,
In the formula, substitute the value of p from the last
equation and simplify.
214 Formula for Economizers.
x =
30 , /SW+GC\ •
-^+[-200-)'
Solve for y, C, and a.
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366 TRANSFORMATION OF FORMULAS 389
215. The Heat Balance Equation.
{W+w)T^W(L+ti)+wt.
Solve for W, w, U, and L.
216. Mean Pressure on the Piston.
Wt^PaLid+hyplogR).
Solve for p eliminating aLi, and W^
217. Adiabatic Expansion of Air.
pyl.iOS ^p^ylAOB
Write the formula for log P.
218. Work done by Air in Adiabatic Expansion.
pT^l.408 —py 1'408
Solve for L when Pi is eliminated.
219. Ratio of Absolute Temperatures of Air.
PVPrVi
T Ti'
Eliminate V and Vi through the use of the first formula
T
in 217 and solve for — .
ii
220. Relation of Temperatures in Adiabatic Expansion.
ftFs'^PiFA
Show that P2F2=PiFi
©'""•
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390 TECHNICAL ALGEBRA 356
221. Length of a Second's Pendulum.
'^'
in which r= period in seconds,
i = length of pendulum in feet,
g = acceleration due to gravity.
The period of a pendulum is the time required for one beat,
which is the time from the highest point on one side to the highest
point on the other. The period for any given length depends
on the latitude and the elevation above sea-level.
In the latitude of New York the force of gravity at the sea-
level is 32.14.
By substitution in the formula, determine the period of a
pendulum 39.1 inches long, in New York City.
Use 6-place table and check by the slide-rule.
222. Expected Mean Effective Pressure.
M.E.P.
F = — -. .
p,j,+(,+c)l0ge^^|-B.P.
(1) Solve for M.E.P.
(2) Compute the M.E.P.
when F = .8,
r = .118,
B.P.=5,
Pi =160,
c = .06.
(3) Formulate the law in problem 13, paragraph 37.
(4) In the formula obtained in (3), substitute the M.E.P. as
computed in (2), and compute the expected indicated horse-
power
when L = 36 inches ,
r.p.m. =85,
Z)* = 16 inches.
* D denotes the diameter of the piston.
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REFERENCE TABLES
391
TABLE I.— LENGTH
United States and English Linear Measure
Quantity.
Unit.
Symbol.
12 inches
3 ft.
5iyd.
16i ft.
40 rd.
8 fur. 1
320 rd.
5280 ft. J
Ifoot
1 yard
r Irod
Ipole
I 1 perch
1 furlong*
1 mile
ft.
yd.
rd.
P.
fur.
mi.
* Seldom used.
Surveyor's Long Measure
Quantity.
Unit.
Symbol.
7.92 inches
25 li.
4rd. \
66 ft./
80 ch.
Uink
1 rod
1 chain
1 mile
li.
rd.
ch.
mi.
Nautical Measure
Quantity.
Unit.
6 feet
1 fathom
120 fathoms
1 cable-length
7J cable-lengths \
880 fathoms /
1 statute mile
1 . 153 statute miles ]
1 minute of circum- \
1 geographic or nautical mile
ference of earth J
3 geographic miles
1 league
20 leagues |
60 geographic miles [
1 degree of earth's circumfer-
69.16 statute miles J
Iknot
1 nautical mile per hour.
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392
REFERENCE TABLES
TABLE L^Continued.
Metric Linear Measure
1 meter » 39.37 inches.
Quantity.
Unit.
Symbol.
10 mm.*
1 centimeter
cm.
10 cm.
1 decimeter
dm.
10 dm.
1 meter
m.
10 m.
1 dekameter
Dm.
10 Dm.
1 hectameter
Hm.
10 Hm.
1 kilometer
Km.
10 Km.
1 myriameter
Mm.
* mm. denotes millimeters.
TABLE XL— AREA ^
United States and English Square Measure
Quantity.
Unit.
144 square inches
9 sq. ft.
30J sq. yd. \
272J sq. ft. /
160 sq. rd. \
43560 sq. ft. /
1 square foot
1 square yard
1 square rod
1 acre
1
Surveyor's
Square Measure
Quantity.
Unit.
625 sq. li.
1 sq. rd.
16 sq. rd.
1 sq. ch.
10 sq. ch.
1 acre
640 A.
1 sq. mile
36 sq. mi.
1 township
(U. S. pubUc lands)
Metric Square Measure
Quantity.
Unit.
100 sq. mm.
100 sq. cm.
100 sq. dc.
100 sq. m.
100 sq. Dk.
100 sq. Hk.
100 sq. Km.
1 sq. centimeter
1 sq. decimeter
1 sq. meter
1 sq. decameter
1 sq. hektometer
1 sq. kilometer
1 sq. myriameter
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REFERENCE TABLES
'table III.— VOLUME
Cubic Measure
393
Quantity.
Unit.
1728 cubic inches
27 cu. ft.
24i cu. ft.*
1 CU. ft.
1 CU. yd.
1 perch* (masonry)
* In some states. Not a legal standard. Should always be specified.
Dry Measure
Quantity.
Unit.
2 pints
8qt.
4pk. \
2150.42 cu. in. /
2688 cu. in.
2218.19 cu. in.
1 quart
1 peck
^ g 1 bushel (struck)
i 1 bushel (heaped)
Eng. 1 bushel (struck)
Liquid Measure
Quantity.
Unit.
4 gills
2pt.
4qt. \
231 cu. in. /
277 J cu. in.
31i gal.
2 bbl. \
63 gal. J
42 gal.
1 pint
1 quart
1 gallon (U. S.)
1 Imperial gallon (Eng.)*
1 barrelt
1 hogshead
1 bbl. refined oil
♦ The exact imperial gallon has a capacity of 277.274 cubic inches,
t The sise of a barrel or cask varies to such an extent that the capacity in gallons
is sometimes stamped on the outside.
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394
REFERENCE TABLES
TABLE IV.— WEIGHT
Avoirdupois Weight
Quantity.
Unit.
Symbol.
16 drams
1 ounce
oz.
16 oz.
1 pound
lb.
100 lbs.
1 hundredweight
cwt.
20 cwt. \
2000 lbs. /
1 short ton (U. S.)
T.
112 lbs.
1 quarter (Eng.)
qr.
20 qrs. \
2240 lbs. /
1 long ton (Eng.)
T.
2204.6 lbs.
1 metric ton
T.
Troy Weight
Quantity.
Unit.
Symbol.
24 grains
20 pwt.
12 oz. \
6760 grains J
1 pennjrweight
1 ounce
1 pound
pwt.
oz.
lb.
Volume and Weight of Water
Unit.
Equivalent.
Approximate.
Exact.
1 gal. (U. S.)
1 cu.ft.
1 liter ]
1 kilogram
1 kilo J
/ .134cu.ft.
I 8i lbs.
/ 7i gal.
I 62.4 lb.
2.2 1b.
.13368 cu.ft.
8.3356* lb.
7.480517 gal.
62.35471* lb.
2.204622tlb.
* At 62° Fahrenheit, barometer 30".
t By Act of Congress. Distilled (pure) water at maximum density, barom-
eter 30".
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REFERENCE TABLES
395
TABLE v.— DECIMAL EQUIVALENTS OF PARTS OF AN
INCH
Fraction.
Decimal.
Fraction.
Decimal.
Fraction.
Decimal.
A
.01563
a
.32813
*f
.70313
ii
.03125
a
.34375
a
.71875
A
.04688
a
.35938
a
.73438
1-16
.0625
S-8
.375
3-4
.75
A
.07813
a
.39063
a
.76563
A
.09375
a
.40625
a
.78125
A
.10938
u
.42188
a
.79688
1-8
.125
7-16
.4375
13-16
.8125
A
.14063
a
.45313
a
.82813
A
.15625
a
.46875
a
.84375
il
.17188
H
.48438
a
.85938
3-16
.1875
1-2
.5
7-8
.875
«
.20313
ji'
.51563
a
.89063
A
.21875
a
.53125
a
.90625
U
.23438
a
.54688
a
.92188
1-4
.25
9-16
.5625
16-16
.9375
H
.26563
a
.57813
a
.95313
A
.28125
a
.59375
tt
'^. 96875
«
.29688
a
.60938
a
.98438
6-16
.3125
6-8
a
11-16
.625
.64063
.65625
.67188
.6875
1
1.00000
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396 i^' ^c^-" EEFERENCE TABLES
TABLE VL— U
. S. AND METRIC
EQUIVALENTS
Equivalent.
Unit.
Approximate.
Exact.
1 acre
. 40 hectares
.4047
1 bushel
35 i liters
35.24
1 cm.
.39 in.
.3937
Ice.
.Oo cu. in.
.0610
1 cu. ft.
.028 cum.
.0283
1 cu. in.
16.4 cc.
16.387
1 cum.
35.3 cu. ft.
35.31
1 cum.
1.3 cu. yd.
1.308
1 cu. yd. .
. 76 cum.
.7645
1ft.
30i cm.
30.48
1 gal. (U. S.)
3.8 Uters
3.785
1 grain
065 g.
.0648
1 gram
1 hectare
151 gr.
15. 4 i
2 . 5 acres
2.471
1 inch
2.5 cm.
2.54
Ikilo
2.2 lbs.
2.205
1km.
.62 mile
.6214
1 liter
.91 qt. (dry)
l.lqt. (liq.)
' .9081
1 liter
1,057
1 meter
3.3 ft.
3.281
1 mile
1.6 km.
1.6093
1 mm.
.039 in.
.03937
1 oz. (avoir.)
28i g.
28.35
1 oz. (troy)
31 g.
31.10
1 peck
8.8 liter
8.809
1 pint (liq.)
.47 1.
.4732
1 pound
.45 kg.
.4536
1 qt. (dry)
1.1 1.
1.101
1 qt. (liq.)
.95 1.
.9464
1 scm.
. 16 sq. in.
.1550
1 sq. ft.
.093 sm.
.0929
1 sq. in.
6 . 5 scm.
6.452
1 §q. mile
260 Ha.
259.
1 sm.
1.2 sq.yd.
1.196
1 sm.
11 sq. ft.
10.76
1 sq. rod
25.3 sm.
25.293
1 sq. yd.
.84 sm.
.8361
1 ton (U. S.)
.91 m. ton
.9072
1 ton (Eng.)
1 m. ton
1.017
1 ton (metric)
1.1 t. (U. S.)
1.102
1 ton (metric)
.98 t. (Eng.)
.9842
1yd.
.91 m.
.9144
See also, page 400.
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REFERENCE TABLES
397
rABLE VII.— INTERNATIONAL ATOMIC WEIGHTS, 1916
A Argon
... 39.88
Mo Molybdenum. . .
.. 96.0
Ag Silver
... 107.88
N Nitrogen
.. 14.01
Al Aluminum
As Arsenic
... 27.1
... 74.96
Na Sodium
.. 23.00
Nd Neodymium
.. 144.3
Au Gold
... 197.2
Ne Neon
.. 20.2
B Boron
... 11.0
Ni Nickel
.. 68.68
Ba Barium
... 137.37
Nt Niton
.. 222.4
Bi Bismuth
... 208.0
O Oxygen
.. 16.00
Br Bromine
... 79.92
Os Osmium
.. 190.9
C Carbon
... 12.005
P Phosphorus
.. 31.04
Ca Calcium
... 40.07
Pb Lead...
.. 207.20
Cb Colimibimn. . . .
... 93.5
Pd Palladium
.. 106.7
Cd Cadmium
... 112.40
Pr Praseodymium . .
.. 140.9
Ce Cerium
... 140.25
Pt Platinum
.. 195.2
CI Chlorine
... 35.46
Ra Radium
.. 226.
Co Cobalt
... 68.97
Rb Rubidium
.. 85.45
Cr Chromium
... 62.0
Rh Rhodium
.. 102.9
Cs Caesium
... 132.81
Ru Ruthenium
.. 101.7
Cu Copper
... 63.57
S Sulphur
.. 32.06
Dy Dysprosium. ..
... 162.5
Sa Samarium
.. 150.4
Er Erbium
... 167.7 '
Sb Antunony
.. 120.2
Eu Europium
... 152.0
Sc Scandium
.. 44.1
F Fluorine
... 19.0
Se Selenium
.. 79.2
Fe Iron
... 65.84
Si Silicon
.. 28.3
Ga Gallium
... 69.9
Sn Tin
.. 118.7
Gd Gadolinium
... 157.3
Sr Strontium
.. 87.63
Ge Germanium. . . .
... 72.6
Ta Tantalum
.. 181.1
Gl Glucinum
... 9.1
Tb Terbium
.. 159.2
H Hydrogen
... 1.008
Te Tellurium
.. 127.5
He Helium
... 4.00
Th Thorium
.. 232.4
He Mercurv
... 200.6
Ti Titanium
. . 48.1
Ho Holmium
... 163.6
Tl Thallium
.. 204.0
I Iodine
In Indium
... 126.92
... 114.8
Tm Thurium
.. 168.5
.. 238.2
U Uranium
Ir Iridium
... 193.1
V Vanadium
.. 61.0
K Potassium
... 39.10
W Tungsten
.. 184.0
Kr Krjrpton
... 82.92
Xe Xenon
.. 130.2
La Lanthanum. . . .
... 139.0
Yt Yttrium
.. 88.7
Li Lithium
... 6.94
Yb Ytterbium
.. 173.5
Lu Lutecium
... 175.0
Zn Zinc
.. 65.37
Mg Magnesium
... 24.32
Zr Zirconium
.. 90.6
Mn Manganese ....
. .. 64.93
Arranged Alphabetically According to Symbols
By W. A. Ballou
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398
BEFERENCE TABLES
(X4
§
O
H
a
o
Q
O
2
0. i
."
p .
Pi
o
op
P<0
•c
CO
M
«S5
?
2s
B
^8
S
O
"1
&
lil
sfi
-.^
G
Si
■"ft
Sg
IS
li
S2
I-
Il
2z
01
ir
lis
as
It
►^3
•SI
iS
ss
§o
|3
I!
1^
lis
§^ I
a. ^
Be
¥
8i
I?
S2
6!9
Zi
Is
«3
S"
a?"?
Sod
S I
a-
an
Is?
1"
^
S
SI
a^
1"
IS
Oi
1^
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REFERENCE TABLES ^
399
TABLE IX.— SPECIFIC GRAVITIES AND WEIGHTS OF
MATERIALS OF CONSTRUCTION
No.
Material.
Specific
Gravity.
Average Weight in Lbs.
Cu.in.
Cu.ft.
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
Aluminum, cast
'* hammered
* Ash, American white
" red
Asphalt
* Beech
* Birch
Brass, cast
'' rolled
Brick, best pressed
* * common, hard
" soft
Brickwork, pressed brick. . .
* ' medium brick . .
'* ordinary brick. .
Bronze, copper 8, tin 1 ... .
* Cedar, American
Cement, hydraulic, ground .
loose, Rosendale
Portland, loose
* Cherry
Concrete
Copper, cast
* * wire
♦Elm
Glass, common window. . . .
Granite
Gravel
* Hemlock
* Hickory
Iron, cast
' ' wrought
Lead, commercial
Limestone and marble
Mahogany, Spanish
' * Honduras
* Maple
1
2.7
1 to 1.8
7.8to8.4
8.4
8.5
8.6to8.8
8.8 to 9
2.52
2.56 to 2.9
.85
6.9to7.5
7.79
11.38
2.46 to 2.84
.85
.56
.075
.099
.29
.303
.306
.314
.321
.26
.278
125
170.6
38
40
87.3
43
45.5
504
524
150
125
100
140
125
112
529
34.5
56
42
130
542
555
35
157
170
117
25
53
450
480
709.
164.
53
35
49
* Seasoned.
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400
REFERENCE TABLES
TABLE IX.—Cantinued
No.
Material.
Specific
Gravity.
Average Weight in Lbs.
Cu.in.
Cu.ft.
37
38
39
40
41
42
43
44
45
46
47
48
48
50
51
52
53
54
Mortar, hardened
* Oak, live American . . .
* * red American. . . .
" white American.. .
* Pine, white
** yellow, Northern.
'* *' Southern.,
* Poplar
Sand, quartz
** wet...*
* ' well shaken
Sandstone, building . . .
Slate, American
* Spruce
Steel, structural
* Wabiut, black
Water
Zinc, cast
'' roUed
1.4tol.9
2.75
2.41
2.7to2.9
7.85
1
6.9
7.2
.28
103
59
40
50
25
34
45
39
90 to 106
120 to 140
99 to 117
151
175
25
490
38
t62.355
376
449
* Seasoned.
t At 62° Fahrenheit, barometer 30 inches.
Pressure and Parts Conversion Factors
Unit.
Equivalent.
Approximate.
Kxact
1 lb. per sq. in.
1 kg. per scm.
1 gram per gal.
. 07 kilograms per scm.
14.21b. persq. in.
26 parts per 100,000
.070308
14.223105
26.43737
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REFERENCE TABLES
401,
TABLE X.— WIRE GAGE SIZES
In Decimals of an Inch
Washburn
& Moen
British
Birming-
American
Manufac-
Trenton
Iron Co.
Wire
Gauge.
American
Imperial or
Number
ham or
or Brown &
turing Co.
Screw Co.
English
of
Stubs Iron
Sharpe
andJohnA.
Screw
Legal
Gauge.
Wire
Wire
Roebling's
Wire
Standard
Gauge.
Gauge.
Sons Co.
Gauge.
Wire.
Wire
Gauge.
Gauge.
0000000
.600
000000
.4600
.464
00000
.4300
"!456"
.432
0000
!454
'aqoooo
.3938
.400
.400
000
.425
.409642
.3625
.360
.0315
.372
00
.380
.364796
.3310
.330
.0447
.348
.340
.324861
.3065
.305
.0578
.324
1
.300
.289297
.2830
.285
.0710
.300
2
.284
.257627
.2625
.266
.0842
.276
3
.259
.229423
.2437
.245
.0973
.252
4
.238
.204307
.2253
.225
.1106
.232
5
.220
. 181940
.2070
.205
.1236
.212
6
.203
. 162023
.1920
.190
.1368
.192
7
.180
.144285
.1770
.175
.1600
.176
8
.165
.128490
.1620
.160
.1631
.160
.148
.114423
.1483
.145
.1763
.144
10
.134
. 101897
.1360
.130
.1894
.128
11
.120
.090742
.1205
.1175
.2026
.116
12
.109
.080808
.1055
.105
.2158
.104
13
.095
.071962
.0915
.0925
.2289
.092
14
.083
.064084
.0800
.0806
.2421
.080
16
.072
.057068
.0720
.070
.2662
.072
16
.065
.050821
.0625
.061
.2684
.064
17
.058
.045257
.0540
.0626
.2816
.056
18
.049
.040303
.0476
.045
.2947
.048
19
.042
.035890
.0410
.040
.3079
.040
20
.035
.031961
.0348
.035
.3210
.036
21
.032
.028462
.03175
.031
.3342
.032
22
.028
.025346
.0286
.028
.3474
.028
23
.025
.022572
.0268
.025
.3606
.024
24
.022
.020101
.0230
.0225
.3737
.022
25
.020
.017900
.0204
.020
.3868
.020
26
.018
.015941
.0181
.018
.4000
.018
27
.016
.014195
.0173
.017
.4132
.0164
28
.014
.012641
.0162
.016
.4263
.0148
29
.013
.011257
.0160
.015
.4395
.0136
30
.012
.010025
.0140
.014
.4526
.0124
31
.010
.008928
.0132
.013
.4658
.0116
32
.009
.007950
.0128
.012
.4790
.0108
33
.008
.007080
.0118
.011
.4921
.0100
34
.007
.006305
.0104
.010
.6053
.0092
35
.005
.005615
.0095
.0095
.5184
.0084
36
.004
.005000
.0090
.009
.6316
.0076
37
.004463
.0085
.0085
.6448
.0068
38
.003965
.0080
.008
.6579
.0060
39
.003631
.0075
.0075
.5711
.0062
40
.003144
.0070
.007
.6842
.0048
From the Cambria Handbook.
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402 REFERENCE TABLES
TABLE XL— FOUR-PLACE LOGARITHMS OF NUMBERS
N.
1
2
3
4
5
6
7
8
9
P.P.
10
11
12
13
0000
0043
0086
0128
0170
0212
0253 0294
0334
0374
22 1 21
0414
0792
1139
0453
0828
1173
0492
0864
1206
0531
0899
1239
0569
0934
1271
0607
0969
1303
0645
1004
1335
0682
1038
1367
0719
1072
1399
0755
1106
1430
1
2
3
4
5
2.2
4.4
6.G
8.8
11.0
2.1
4.2
6.3
8.4
10.5
14
15
16
1461
1761
2041
1492
1790
2068
1623
1818
2096
1653
1847
2122
1584
1875
2148
1614
1903
2175
1644
1931
2201
1673
1959
2227
1703
1987
2253
1732
2014
2279
6
7
8
9
13.2
15.4
17.6
19.8
12.0
14.7
16.8
18.9
17
18
19
20
21
22
23
2304
2553
2788
2330
2577
2810
2355
2601
2833
2380
2625
2856
2405
2648
2878
2430
2672
2900
2455
2695
2923
2480
2718
2945
2604
2742
2967
2529
2765
2989
1
2
3
4
5
6
7
8
9
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
18
19
1.9
3.8
5.7
7.0
o.r*
11.4
13.3
16.2
17.1
17
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
3222
3424
3617
3243
3444
3636
3263
3464
3655
3284
3483
3674
3304
3502
3692
3324
3522
3711
3345
3541
3729
3365
3560
3747
3385
3579
3766
3404
3598
3784
24
26
26
3802
3979
4150
3820
3997
4166
383S
4014
4183
3856
4031
4200
3874
4048
4216
3892
4065
4232
3909
4082
4249
3927
4099
4265
3945
4116
4281
3962
4133
4298
1
2
3
4
5
6
1.8
3.6
5.4
7.2
9.0
10.8
1.7
3.4
5.1
0.8
8.5
10.2
27
28
20
30
31
32
33
4314
4472
4624
4330
4487
4639
4346
4502
4654
4362
4518
4669
4378
4533
4683
4393
4548
4698
4409
4564
4713
4425
4579
4728
4440
4594
4742
4456
4609
4757
7
8
9
1
2
3
4
5
6
7
12.0
14.4
10.2
16
1.6
3.2
4.8
6.4
8.0
9.0
11 2
11.9
13.0
15.3
15
1.5
3.0
4.5
0.0
7.5
9.0
10.5
4771
4786
4800
4814
4829
4843
4857
4871
4886
4900
4914
5051
5185
4928
6065
6198
4942
5079
6211
4965
5092
6224
4969
5106
6237
4983
6119
6260
4997
6132
6263
6011
6145
6276
5024
5159
5289
5038
5172
5302
34
35
36
5315
6441
5663
5328
5453
6576
5340
5466
6587
5353
6478
6699
5366
6490
6611
6378
5502
6623
6391
6615
6636
6403
6527
5647
6416
6539
6658
5428
6551
6670
8
9
T
12.8
14.4
14
1.4
12.0
13.5
13
1.3
37
38
39
40
6682
5798
6911
5694
5809
5922
6706
6821
5933
6717
5832
6944
6729
6843
6956
6740
6855
6966
6762
6866
5977
6763
6877
6988
6775
6888
5999
6786
6900
6010
2
3
4
6
6
7
8
9
2.8
4.2
5.6
7.0
8.4
9.8
11.2
12.0
2.6
3.9
5.2
0.6
7.8
9.1
10.4
11.7
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
N.
1
2
3
4
6
6^
7
8
9
Note. — Proportional parts begin with N. 19. Determine P.P. from N. 10 to
N. 19 by multiplying the actual difference between the mantissa read and the man-
tissa immediately following, by all the figures of the natural number (except tho
first three) with a decimal point before them.
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REFERKNCE TABLES
TABLE XI.— Continued
403
N.
1
2
3
4
6
6
7
8
9
P.P.
40
41
42
43
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
6128
6232
6335
6138
6243
6345
6149
6253
6355
6160
6263
6365
6170
6274
6375
6180
6284
6385
6191
6294
6395
6201
6304
6405
6212
6314
6416
6222
6325
6425
44
45
46
6435
6532
6628
6444
6542
6637
6454
6551
6646
6464
6561
6656
6474
6571
6665
6484
6580
6675
6493
6590
6684
6503
6599
6693
6513
6609
6702
6522
6618
6712
47
48
49
50
51
52
53
6721
6812
6902
6730
6821
6911
6739
6830
6920
6749
6839
6928
6758
6848
6937
6767
6857
6946
6776
6866
6955
6785
6875
6964
6794
6884
6972
6803
6893
6981
1
2
3
4
5
6
7
8
9
18
1.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
11
1.1
22
3.3
4.4
5.5
6.6
7.7
8.8
9.9
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
707C
7160
7243
7084
7168
7251
7093
7177
7259
7101
7185
7267
7110
7193
7275
7118
7202
7284
7126
7210
7292
7135
7218
7300
7143
7226
7308
7152
7235
7316
54
55
50
7324
7404
7482
7332
7412
7490
7340
7419
7497
7348
7427
7505
7356
7435
7513
7364
7443
7520
7372
7451
7528
7380
7459
7536
7388
7466
7543
7396
7474
7551
57
58
59
GO
Gl
62
63
7559
7634
7709
7566
7642
7716
7674
7649
7723
7582
7657
7731
7589
7664
7738
7597
7672
7745
7604
7679
7752
7612
7686
7760
7619
7694
7767
7627
7701
7774
7782
7789
7796
7803
7810
7818
7825
7832
7839
7846
1
2
3
4
r»
0.9
1.8
2.7
3.6
4.5
8
0.8
1.6
2.4
3.2
4.0
7853
7924
7993
7860
7931
8000
7868
7938
8007
7875
7945
8014
7882
7952
8021
7889
7959
8028
7896
7966
8035
7903
7973
8041
7910
7980
8048
7917
7987
8055
64
65
66
8062
8129
8195
8069
8136
8202
8075
8142
8209
8082
8149
8215
8089
8156
8222
8096
8162
8228
8102
8169
8235
8109
8176
8241
8116
8182
8248
8122
8189
8254
6
7
8
5.4
6.3
7.2
8.1
4.8
5.6
6.4
7.2
67
68
69
70
8261
8325
8388
8267
8331
8395
8274
8338
8401
8280
8344
8407
8287
8351
8414
8293
8357
8420
8299
8363
8426
8306
8370
8432
8312
8376
8439
8319
8382
8445
8451
8457
8463
8470
8476
8482
8488J 8494
8500
8506
N.
1
2
3
4
6
6
7
8
9
Digitized by
Google
404
REFERENCE TABLES
TABLE XL— Continued
N.
3
6
8
P.P.
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
90
91
92
93
94
95
96
97
93
99
8451
8457
8463
8470
8476
8482
8488
8494
8500
8506
N,
8513
8573
8633
8692
8751
8808
8865
8921
8976
8519
8579
8639
8698
8756
8814
8871
8927
8982
8525
8585
8645
8704
8762
8820
8876
8932
8987
9031
9036 9042
9085
9138
9191
9243
9294
9345
9395
9445
9494
9090
9143
9196
9248
9299
9350
9400
9450
9499
9542
9547
9590
9638
9685
9731
9777
9823
9868
9912
9956
9595
9643
9689
9736
9782
9827
9872
9917
9961
9096
9149
9201
9253
9304
9355
9405
9455
9504
9552
9600
9647
9694
9741
9786
9832
9877
9921
9965
2
8531
8591
8651
8710
8768
8825
8882
8938
8993
8537
859Y
8657
8716
8774
8831
8887
8943
8998
8543
8603
8663
8722
8779
8837
8893
8949
9004
8549
8609
8669
8727
8785
8842
8899
8954
9009
8555
8615
8675
8733
8791
8848
8904
8960
9015
8561
8621
8681
8739
8797
8854
8910
8965
9020
8567
8627
8686
8745
8802
8859
8915
8971
9025
9047
9053
9058
9063
9069
9074
9079
9101
9154
9206
9258
9309
9360
9410
9460
9509
9106
9159
9212
9263
9315
9365
9415
9465
9513
9112
9165
9217
9269
9320
9370
9420
9469
9518
9117
9170
9222
9274
9325
9375
9425
9474
9523
9122
9175
9227
9279
9330
9380
9430
9479
9528
9128
9180
9232
9284
9335
9385
9435
9484
9533
9133
9186
9238
9289
9340
9390
9440
9489
9538
9557
9562
9566
9571
9576
9581
9586
9605
9652
9699
9745
9791
9836
9881
9926
9969
9609
9657
9703
9750
9795
9841
9930
9974
9614
9661
9708
9754
9800
9845
9890
9934
9978
9619
9666
9713
9759
9805
9850
9894
9939
9983
9624
9671
9717
9763
9809
9854
9943
9987
9628
9675
9722
9768
9814
9859
9903
9948
9991
9633
9680
9727
9773
9818
9863
9908
9952
9996
7
1
0.7
2
1.4
3
2.1
4
2.8
5
3.5
6
4.2
7
4.9
8
5.6
9
6.3
5
1
0.5
2
1.0
3
1.5
4
2.0
6
2.5
6
3.0
7
3.5
8
4.0
9
4.5
6
8
9
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
4
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
Digitized by
Google
REFERENCE TABLES
405
TABLE XII.— FOUR-PLACE TRIGONOMETRIC FUNCTIONS
Angle
Sine
Cosine
Tangent
Cotangent
Angle
o /
00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
o /
90 00
.0000
1.0000 0.0000
.0000
10
.0029 7.4637
1.0000 0000
.0029 7.4637
343.77 2.6363
50
20
.0058 7648
1.0000 0000
.0058 7648
171.89 2352
40
30
.0087 9408
1.0000 0000
.0087 9409
114.59 0591
30
40
.0116 8.0658
.9999 0000
.0116 8.0658
85.940 1.9342
20
50
.0145 1627
.9999 0000
.0145 1627
68.760 8373
10
1 00
.0175 8.2419
.9998 9.9999
.0175 8.2419
57.290 1.7581
89 00
10
.0204 3088
.9998 0999
.0204 3089
49.104 6911
50
20
.0233 3668
.9997 9999
.0233 3669
42.964 6331
40
30
.0262 4179
.9997 9999
.0262 4181
38.188 6819
30
40
.0291 4637
.9996 9998
.0291 4638
34.368 5362
20
60
.0320 6050
.9995 9998
.0320 5053
31.242 4947
10
2 00
.0349 8.5428
.9994 9.9997
.0349 8.5431
28.636 1.4569
88 00
10
.0378 6776
.9993 9997
.0378 5779
26.432 4221
60
20
.0407 6097
.9992 9996
.0407 6101
24.542 3899
40
30
.0436 6397
.9990 9996
.0437 6401
22.904 3599
30
40
.0465 6677
.9989 9995
.0466 6682
21.470 3318
20
60
.0494 6940
.9988 9995
.0495 6945
20.206 3055
10
3 00
.0523 8.7188
.9986 9.9994
.0524 8.7194
19.081 1.2806
87 00
10
.0552 7423
.9985 9993
.0553 7429
18.075 2571
50
20
.0581 7645
.9983 9993
.0582 7652
17.169 2348
40
30
.0610 7857
.9981 9992
.0612 7865
16.350 2135
30
40
.0640 8059
.9980 9991
.0641 8067
15.605 1933
20
60
.0669 8251
.9978 9990
.0670 8261
14.924 1739
10
4 00
.0698 8.8436
.9976 9.9989
.0699 8.8446
14.301 1.1554
86 00
10
.0727 8613
.9974 9989
.0729 8624
13.727 1376
50
20
.0756 8783
.9971 9988
.0758 8795
13.107 1205
40
30
.0785 8946
.9969 9987
.0787 8960
12.706 1040
30
40
.0814 9104
.9967 9986
.0816 9118
12.251 0882
20
60
.0843 9256
.9964 9985
.0846 9272
11.826 0728
10
5 00
.0872 8.9403
.9962 9.9983
.0875 8.9420
11.430 1.0580
85 00
10
.0901 9545
.9959 9982
.0904 9563
11.059 0437
50
20
.0929 9682
.9957 9981
.0934 9701
10.712 0299
40
30
.0958 9816
.9954 9980
.0963 9836
10.385 0164
30
40
.0987 9945
.9951 9979
.0992 9966
10.078 0034
20
60
.1016 9.0070
.9948 9977
.1022 9.0093
9.7882 0.9907
10
6 00
.1045 9.0192
.9945 9.9976
.1051 9.0216
9.5144 0.9784
84 00
10
.1074 0311
.9942 9975
.1080 0336
9.2553 9664
50
20
.1103 0426
.9939 9973
.1110 0453
9.0098 9547
40
30
.1132 0539
.9936 9972
.1139 0567
8.7769 9433
30
40
.1161 0648
.9932 9971
.1169 0678
8.5555 9322
20
60
.1190 0755
.9929 9969
.1198 0786
8.3450 9214
10
7 00
o /
.1219 9.0859
.9925 9.9968
.1228 9.0891
8.1443 0.9109
83 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
o /
Angle
Cosine
Sine
Cotangent
Tangent
Angle
Digitized by
Google
406
REFERENCE TABLES
TABLE XIL-— Continued
Anoud
Sine
Cosine
Tangent
Cotangent
Angle
e t
7 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
O f
83 00
.1219 0.0859
.9925 9.9968
.1228 9.0891
8.1443 0.9109
10
.1248 0961
.9922 9966
.1257 0995
7.9530 9005
50
20
. 1276 1060
.9918 9964
. 1287 1096
7.7704 8904
40
30
.1305 1157
.9914 9963
.1317 1194
7.5958 8806
30
40
. 1334 1252
.9911 9961
. 1346 1291
7.4287 8709
20
50
.1363 1345
.9907 9959
.1376 1385
7.2687 8615
10
8 00
.1392 9.1436
.9903 9.9958
.1405 9.1478
7.1154 0.8522
S2 00
10
. 1421 1525
.9899 9956
. 1435 1569
6.9682 8431
50
20
. 1449 1612
.9894 9954
.1465 1658
6.8269 8342
40
30
. 1478 1697
.9890 9952
. 1495 1745
6.6912 8255
30
40
. 1507 1781
.9886 9950
. 1524 1831
6.5606 8169
20
60
.1530 1863
.9881 9948
. 1554 1915
6.4348 8085
10
9 00
.1564 9.1943
.9877 9.9946
.1584 9.1997
6.3138 0.8003
81 00
10
. 1593 2022
.9872 9944
.1614 2078
6.1970 7922
50
20
.1622 2100
.9868 9942
. 1644 2158
6.0844 7842
40
30
. 1650 2176
.9863 9940
. 1673 2236
5.9758 7764
30
40
.1679 2251
.9858 9938
.1703 2313
5.8708 7687
20
50
. 1708 2324
.9853 9936
.1733 2389
5.7694 7611
10
10 00
.1736 9.2397
.9848 9.9934
.1763 0.2463
5.6713 0.7537
80 00
10
.1765 2468
.9843 9931
. 1793 2536
5.5764 7464
50
20
. 1794 2538
.9838 9929
. 1823 2600
5.4845 7391
40
30
.1822 2606
.9833 9927
.1853 2680
5.3955 7320
30
40
. 1851 2674
.9827 9924
. 1883 2750
5.3093 7250
20
50
.1880 2740
.9822 9922
. 1914 2819
5.2257 7181
10
11 00
.1908 9.2806
.9816 9.9919
.1944 9.2887
5.1446 0.7113
79 00
10
. 1937 2870
.9811 9917
. 1974 2953
5.0658 7047
50
20
.1965 2934
.9806 9914
.2004 3020
4.9894 6980
40
30
. 1994 2997
.9799 9912
.2035 3085
4.9152 6915
30
40
.2022 3058
.9793 9909
.2065 3149
4.8430 6851
20
60
.2051 3119
.9787 9907
.2095 3212
4.7729 6788
10
18 00
.2079 9.3179
.9781 9.9904
.2126 9.3275
4.7046 0.6725
78 00
10
.2108 3238
.9775 9901
.2156 3336
4.6382 6664
50
20
.2136 3296
.9769 9899
.2186 3397
4.5736 6603
40
30
.2164 3353
.9763 9896
.2217 3458
4.5107 6542
30
40
.2193 3410
.9757 9893
.2247 3517
4.4494 6483
20
50
.2221 3466
.9750 9890
.2278 3576
4.3897 6424
10
13 00
.2250 9.3521
.9744 9.9887
.2309 9.3634
4.3315 0.6366
77 00
10
.2278 3575
.9737 9884
.2339 3691
4.2747 6309
50
20
.2306 3629
.9730 9881
.2370 3748
4.2193 6252
40
30
.2334 3682
.9724 9878
. 2401 3804
4.1653 6196
30
40
.2363 3734
.9717 9875
.2432 3859
4.1126 6141
20
50
.2391 3786
.9710 9872
.2462 3914
4.0611 6086
10
14 00
o /
.2419 9.3837
.9703 9.9869
.2493 9.3968
4.0108 0.6032
76 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Anolb
Cosine
Sine
Cotangent
Tangent
Angle
Digitized by
Google
REFERENCE TABLES
407
I
TABLE Xll.^-Continued
AifOLB
SiNB
Cosine
Tangent
Cotangent
Angle
o /
14 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
o /
76 00
.2419 9.3837
.9703 9.9869
.2493 9.3968
4.0108 0.6032
10
.2447 3887
.9696 9866
.2524 4021
3.9617 5979
50
20
.2476 3937
.9689 9863
.2555 4074
3.9136 5926
40
30
.2504 3986
.9681 9859
.2586 4127
3.8667 5873
30
40
.2532 4035
.9674 9856
.2617 4178
3.8208 5822
20
60
.2560 4083
.9667 9853
.2648 4230
3.7760 5770
10
15 00
.2588 9.4130
.9659 9.9849
.2679 9.4281
3.7321 0.5719
75 00
10
.2616 4177
.9652 9846
.2711 4331
3.6891 5669
50
20
.2644 4223
.9644 9843
.2742 4381
3.6470 5619
40
30
.2672 4269
.9636 9839
.2773 4430
3.6059 5570
30
40
.2700 4314
.9628 9836
.2805 4479
3.5656 5521
20
50
.2728 4359
.9621 9832
.2836 4527
3.5261 5473
10
16 00
.2756 9.4403
.9613 9.9828
.2867 9.4575
3.4874 0.5325
74 00
10
.2784 4447
.9605 9825
.2899 4622
3.4495 5378
50
20
.2812 4491
.9596 9821
.2931 4669
3.4124 5331
40
30
.2840 4533
.9588 9817
.2962 4716
3.3759 5284
30
40
. 2868 4576
.9580 9814
.2994 4762
3.3402 5238
20
50
.2896 4618
.9572 9810
.3026 4808
3.3052 5192
10
17 00
.2924 9.4659
.9563 9.9806
.3057 9.4853
3.2709 0.5147
73 00
10
.2952 4700
.9555 9802
.3089 4898
3.2371 5102
50
20
.2979 4741
.9546 9798
.3121 4943
3.2041 5057
40
30
.3007 4781
.9537 9794
.3153 4987
3.1716 5013
30
40
.3035 4821
.9528 9790
.3185 5031
3.1397 4969
20
50
.3062 4861
.9520 9786
.3217 5075
3.1084 4925
10
18 00
.3090 9.4900
.9511 9.9782
.3249 9.5118
3.0777 0.4882
72 00
10
.3118 4939
.9502 9778
.3281 5161
3.0475 4839
50
20
.3145 4977
.9492 9774
.3314 5203
3.0178 4797
40
30
.3173 5015
.9483 9770
.3346 5245
2.9887 4755
30
40
.3201 5052
.9474 9765
.3378 5287
2.9600 4713
20
50
.3228 5090
.9465 9761
.3411 5329
2.9319 4671
10
19 00
.3256 9.5126
.9455 9.9757
.3443 9.5370
2.9042 0.4630
71 00
10
.3283 5163
.9446 9752
.3476 5411
2.8770 4589
50
20
.3311 5199
.9436 9748
.3508 5451
2.8502 4549
40
30
.3338 5235
.9426 9743
.3541 5491
2.8239 4509
30
40
.3365 5270
.9417 9739
.3574 5531
2.7980 4469
20
50
.3393 5306
.9407 9734.
.3607 5571
2.7725 4429
10
20 00
.3420 9.5341
.9397 9.9730
.3640 9.5611
2.7475 0.4389
70 00
10
.3448 5375
.9387 9725
.3673 5650
2.7228 4350
50
20
.3475 5409
.9377 . 9721
.3706 5689
2.6985 4311
40
30
.3502 5443
.9367 9716
.3739 5727
2.6746 4273
30
40
.3529 5477
.9356 9711
.3772 5766
2.6511 4234
20
50
.3557 5510
.9346 9706
.3805 5804
2.6279 4196
10
21 00
o »
.3584 9.5543
.9336 9.9702
.3830 9.5842
2.6051 0.4158
69 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Angle
Cosine
Sine
Cotangent
Tangent
Anglb^
Digitized by
Google
408^
REFERENCE TABLES
►
TABLE Xll.— Continued
Anolb
SiNB
Cosine
Tangent
Cotangent
Angle
o /
21 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
o /
69 00
.3584 9.5643
.9336 9.9702
.3839 9.5842
2.6051 0.4158
10
.3611 5576
.9325 9697
. 3872 6879
2.5826 4121
60
20
.3638 5609
.9315 9692
.3906 5917
2.6605 4083
40
30
.3665 5641
.9304 9687
.3939 5954
2.6386 4046
30
40
.3692 5673
.9293 9682
.3973 5991
2.5172 4009
20
50
.3719 5704
.9283 9677
.4006 6028
2.4960 3972
10
22 00
.3746 9.5736
.9272 9.9672
.4040 9.6064
2.4761 0.3936
68 00
10
.3773 5767
.9261 9667
.4074 6100
2.4545 3900
50
20
.3800 6798
.9250 9661
.4108 6136
2.4342 3864
40
30
.3827 5828
.9239 9656
.4142 6172
2.4142 3828
30
40
.3854 6859
.9228 9651
.4176 6208
2.3945 3792
20
60
.3881 6889
.9216 9646
.4210 6243
2.3760 3767
10
23 00
.3907 9.5919
.9205 9.9640
.4245 9.6279
2.3659 0.3721
67 00
10
.3934 5948
.9194 9635
.4279 6314
2.3369 3686
60
20
. 3961 5978
.9182 9629
.4314 6.348
2.3183 3652
40
30
.3987 6007
.9171 9624
.4348 6383
2.2998 3617
30
40
.4014 6036
.9159 9618
.4383 6417
2.2817 3683
20
60
.4041 6065
.9147 9613
.4417 6452
2.2637 3548
10
24 00
.4067 9.6093
.9135 9.9607
.4452 9.6486
2.2460 0.3514
66 00
10
.4094 6121
.9124 9602
.4487 6520
2.2286 3480
50
20
.4120 6149
.9112 9596
.4522 6553
2.2113 3447
40
30
.4147 6177
.9100 9590
.4667 6587
2.1943 3413
30
40
.4173 6205
.9088 9584
.4592 6620
2.1775 3380
20
50
.4200 6232
.9076 9579
.4628 6654
2.1609 3346
10
25 00
.4226 9.6259
.9063 9.9573
.4663 9.6687
2.1445 0.3313
65 00
10
.4253 6286
.9051 9567
.4699 6720
2.1283 3280
50
20
.4279 6313
.9038 9561
.4734 6752
2.1123 3248
40
30
.4305 6340
.9026 9555
.4770 6786
2.0965 3215
30
40
.4331 6366
.9013 9549
.4806 6817
2.0809 3183
20
60
.4358 6392
.9001 9543
.4841 6850
2.0655 3160
10
26 00
.4384 9.6418
.8988 9.9537
.4877 9.6882
2.0503 0.3118
64 00
10
.4410 6444
.8975 9530
.4913 6914
2.0353 3086
60
20
.4436 6470
.8962 9624
.4950 6946
2.0204 3064
40
30
.4462 6496
.8949 9518
.4986 6977
2.0057 3023
30
40
.4488 6521
.8936 9512
.6022 7009
1.9912 2991
20
60
.4514 6546
.8923 9505 .
.6059 7040
1.9768 2960
10
27 00
.4540 9.6570
.8910 9.9499
.5095 9.7072
1.9626 0.2928
63 00
10
.4566 6595
.8897 9492
.5132 7103
1.9486 2897
60
20
.4592 6620
.8884 9486
.6169 7134
1.9347 2866
40
30
.4617 6644
.8870 9479
.6206 7165
1.9210 2836
30
40
.4643 6668
.8857 9473
.6243 7196
1.9074 2804
20
60
.4669 6692
.8843 9466
.5280 7226
1.8940 2774
10
28 00
o /
.4695 9.6716
.8829 9.9459
.6317 9.7257
1.8807 0.2743
62 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Anolb
Cosine
SiNB
Cotangent
Tangent
Angle
Digitized by
Google
EEFERENCE TABLES
400
TABLE Xll— Continued
Angle
Sine
Cosine
Tangent
Cotangent
Angle
O f
28 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
62 00
.4695 9.6716
.8829 9.9469
.6317 9.7257
1.8807 0.2743
10
.4720 6740
.8816 9453 '
.5354 7287
1.8676 2713
50
20
.4746 6763
.8802 9446
.53D2 7317
1.8546 2683
40
30
.4772 6787
.8788 9439
.5430 7348
1.8418 2652
30
40
.4797 6810
.8774 9432
.5467 7378
1.8291 2622
20
60
.4823 6833
.8760 9425
.5505 7408
1.8165 2692
10
29 00
.4848 9.6856
.8746 9.9418
.6643 9.7438
1.8040 0.2562
61 00
10
.4874 6878
.8732 9411
.6581 7467
1.7917 25.33
50
20
.4899 C901
.8718 9404
.6619 7497
1.7796 2503
40
30
.4924 6923
.8704 9397
.5658 7626
1.7675 2474
30
40
.4950 6946
.8689 9390
.5696 7556
1.7556 2444
20
50
.4975 6968
.8675 9383
.5735 7585
1.7437 2415
10
30 00
.5000 9.6990
.8660 9.9376
.5774 9.7614
1.7321 0.2386
60 00
10
.5025 7012
.8646 9368
.5812 7644
1.7205 2356
50
20
.5050 7033
.8631 9361
.5851 7673
1.7090 2327
40
30
.5075 7055
.8016 9353
.5890 7701
1.6977 2299
30
40
.5100 7076
.8601 9346
.5930 7730
1.6864 2270
20
60
.5125 7097
.8587 9338
.6969 7759
1.6753 2241
10
31 00
.5150 9.7118
.8572 9.9331
.6009 9.7788
1.6643 0.2212
59 00
10
.5175 7139
.8557 9323
.6048 7816-
1.6534 2184
50
20
.5200 7160
.8542 9316
.6088 7845
1.6426 2155
40
30
.5225 7181
.8526 9308
.6128 7873
1.6319 2127
30
40
.5250 7201
.8511 9300
.6168 7902
1.6212 2098
20
50
.5275 7222
.8496 9292
.6208 7930
1.6107 2070
10
32 00
.5299 9.7242
.8480 9.9284
.6249 9.7958
1.6003 0.2042
58 00
10
.5324 7262
.8465 9276
.6289 7986
1.6900 2014
50
20
.5348 7282
.8450 9268
.6330 8014
1.5798 1986
40
30
.5373 7302
.8434 9260
.6371 8042
1.5697 1958
30
40
.5398 7322
.8418 9252
.6412 8070
1.5597 1930
20
50
.6422 7342
.8403 9244
.6453 8097
1.6497 1903
10
33 00
.5446 9.7361
.8387 9.9236
.6494 9.8126
1.5399 0.1875
57 00
10
.5471 7380
.8371 9228
.6536 8153
1.5301 1847
50
20
.5495 7400
.8355 9219
.6577 8180
1.5204 1820
40
30
.5519 7419
.8339 9211
.6619 8208
1.5108 1792
30
40
.5544 7438
.8323 9203
.6661 8235
1.6013 1765
20
50
.5668 7457
.8307 9194
.6703 8263
1.4919 1737
10
34 00
.5592 9.7476
.8290 9.9186
.6745 9.8290
1.4826 0.1710
56 00
10
.5616 7494
.8274 9177
.6787 8317
1.4733 1683
60
20
.5640 7513
.8258 9169
.6830 8344
1.4641 1656
40
30
.5664 7531
.8241 9160
.6873 8371
1.4550 1629
30
40
.5688 7550
.8225 9151
.6916 8398
1.4460 1602
20
60
.5712 7568
.8208 9142
.6959 8425
1.4370 1576
10
35 00
o /
.5736 9.7686
.8192 9.9134
.7002 9.8452
1.4281 0.1648
55 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Anqlb
Cosine
Sine
Cotangent
Tangent
Angle
Digitized by
Google
410
REFERENCE TABLES
TABLE XII.— Continued '
Anqud
SiNB
Cosine
Tanqevt
Cotangent
Angle
o t
35 00
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
o /
55 00
.5736 9.7586
.8192 9.9134
.7002 9.8462
1.4281 0.1648
10
.5760 7604
.8175 9125
.7046 8479
1.4193 1521
50
20
.5783 7622
.8168 9116
.7089 8606
1.4106 1494
40
30
.5807 7640
.8141 9107
.7133 8533
1.4019 1467
30
40
.5831 7667
.8124 9098
.7177 8569
1.3934 1441
20
60
.5864 7675
.8107 9089
.7221 8686
1.3848 1414
10
36 00
.5878 9.7692
.8090 9.9080
.7266 9.8613
1.3764 0.1387
54 00
10
.5901 7710
.8073 9070
.7310 8639
1.3680 1361
50
20
.5925 7727
.8066 9061
.7355 8666
1.3597 1334
^40
30
.5948 7744
.8039 9062
.7400 8692
1.3514 1308
30
40
.6972 7761
.8021 9042
.7445 8718
1.3432 1282
20
50
.6995 7778
.8004 9033
.7490 8746
1.3361 1256
10
37 00
.6018 9.7795
.7986 9.9023
.7536 9.8771
1.3270 0.1229
53 00
10
.6041 7811
.7969 9014
.7681 8797
1.3190 1203
50
20
.6065 7828
.7961 9004
.7627 8824
1.3111 1176
40
30
.6088 7844
.7934 8996
.7673 8860
1.3032 1160
30
40
.6111 7861
.7916 8985
.7720 8876
1.2954 1124
20
50
.6134 7877
.7898 8975
.7766 8902
1.2876 1098
10
38 00
.6157 9.7893
.7880 9.8965
.7813 9.8928
1.2799 0.1072
52 00
10
.6180 7910
.7862 8966
.7860 8954
1.2723 1046
50
20
.6202 7926
.7844 8946
.7907 8980
1.2647 1020
40
30
.6225 7941
.7826 8936
.7954 9006
1.2572 0994
30
40
.6248 7967
.7808 8926
.8002 9032
1.2497 0968
20
60
.6271 7973
.7790 8915
.8060 9058
1.2423 0942
10
39 00
.6293 9.7989
.7771 9.8905
.8098 9.9084
1.2349 0.0916
51 00
10
.6316 8004
.7753 8895
.8146 9110
1.2276 0890
50
20
.6338 8020
.7736 8884
.8196 9135
1.2203 0866
40
30
.6361 8036
.7716 8874
.8243 9161
1.2131 0839
30
40
.6383 8050
.7698 8864
.8292 9187
1.2069 0813
20
50
.6406 8066
.7679 8853
.8342 9212
1.1988 0788
10
40 00
.6428 9.8081
.7660 9.8843
.8391 9.9238
1.1918 0.0762
50 00
10
.6460 8096
.7642 8832
.8441 9264
1.1847 0736
50
20
.6472 8111
.7623 8821
.8491 9289
1.1778 0711
40
30
.6494 8125
.7604 8810
.8641 9316
1.1708 0685
30
40
.6517 8140
.7585 8800
.8591 9341
1 . 1040 0669
20
50
.6539 8156
.7566 8789
.8642 9366
1.1671 0634
10
41 00
.6561 9.8169
.7647 9.8778
.8693 9.9392
1.1504 0.0608
49 00
10
.6583 8184
.7628 8767
.8744 9417
1.1436 0683
50
20
.6604 S198
.7609 8756
.8796 9443
1.1369 0667
40
30
.6626 8213
.7490 8745
.8847 9468
1.1303 0532
30
40
.6648 8227
.7470 8733
.8899 9494
1.1237 0506
20
50
.667Q 8241
.7451 8722
.8952 9519
1.1171 0481
10
42 00
o /
.6691 9.8265
.7431 9.8711
.9004 9.9644
1.1106 0.0456
48 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Angle
Cosine
Sine
Cotangent
Tangent
Angle
Digitized by
Google
KEFERENCE TABLES
411
TABLE XIL— Con^inwed
Angle
Sine
Cosine
Tangent
Cotangent
Angle
o /
42 00
Nat. Log.
Nftt. Log.
Nat. Log.
Nat. Log.
o /
48 00
.6691 9.8255
.7431 9.8711
.9004 9.9544
1.1106 0.0456
10
.6713 8269
.7412 8699
.9057 9570
1.1041 0430
50
20
.6734 8283
.7392 8688
.9110 9695
1.0977 0405
40
30
.6756 8297
.7373 8676
.9163 9621
1.0913 0379
30
40
.6777 8311
.7353 8665
.9217 9646
1.0850 0354
20
60
.6799 8324
.7333 8653
.9271 9671
1.0786 0329
10
43 00
.6820 9.8338
.7314 9.8641
.9325 9.9697
1.0724 0.0303
47 00
10
.6841 8351
.7294 8629
.9380 9722
1.0661 0278
60
20
.6862 8365
.7274 8618
.9435 9747
1.0599 0253
40
30
.6884 8378
.7254 8606
.9490 9772
1.0538 0228
30
40
.6905 8301
.7234 8594
.9545 9798
1.0477 0202
20
50
.6926 8405
.7214 8582
.9601 9823
1.0416 0177
10
44 00
.6947 9.8418
.7193 9.8569
.9657 9.9848
1.0355 0.0152
46 00
10
.6967 8431
.7173 8557
.9713 9874
1.0295 0126
60
20
.6988 8444
.7153 8545
.9770 9899
1.0235 0101
40
30
.7009 8457
.7133 8532
.9827 9924
1.0176 0076
30
40
.7030 8469
.7112 8520
.9884 9949
1.0117 0061
20
50
.7050 8482
.7092 8507
.9942 9975
1.0058 0025
10
45 00
o r
.7071 9.8495
.7071 9.8495
1.0000 0.0000
1.0000 0.0000
45 00
o /
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
Angle
Cosine
Sine
Cotangent
Tangent
Angle
Digitized by
Google
412
KEFERENCE TABLES
TABLE Xin.— THREE-PLACE NATURAL TRIGONOMETRIC
FUNCTIONS
Deg.
sin
cos
tan
Deg.
sin
cos
tan
.000
1.000
.000
46
.719
.695
1.04
1
.017
.999
.017
47
.731
.682
1.07
2
.035
.999
.035
48
.743
.069
1.11
3*
.052
.999
.052
49
.755
.656
1.15
4
.070
.998
.070
50
.766
.643
1.19
5
.087
.996
.087
51
.777
.629
1.23
6
.105
.995
.105
52
.788
.616
1.28
7
.122
.993
.123
53
.799
.602
1.33
8
.139
.990
.141
54
.809
.588
1.38
9
.156
.988
.158
55
.819
.574
1.43
10
.174
.985
.176
56
.829
.559
1.48
11
.191
.982
.194
57
.839
.545
1.54
12
.208
.978
.213
58
.848
.530
1.60
13
.225
.974
.231
59
.857
.515
1.66
14
.242
.970
.249
60
.866
.500
1.73
15
.259
.966
.268
61
.875
.485
1.80
16
.276
.961
.287
62
.883
.469
1.88
17
.292
.956
.306
63
.891
.454
1.96
18
.309
.951
.325
64
.899
.438
2.05
19
.326
.946
.344
65
.906
.423
2.14
20
.342
.940
.364
66
.914
.407
2.25
21
.358
.934
.384
67
.921
.391
2.36
22
.375
.927
.404
68
.927
.375
2.48
23
.391
.921
.424
69
.934
.358
2.61
24
.407
.914
.445
70
.940
.342
2.75
25
.423
.906
.466
71
.946
.326
2.90
26
.438
.899
.488
72
.951
.309
3.08
27
.454
.891
.510
73
.956
.292
3.27
28
.469
.883
.532
74
.961
.276
3.49
29
.485
.875
.554
75
.966
.259
3.73
30
.500
.866
.577
76
.970
.242
4.01
31
.515
.857
.601
77
.974
.225
4.33
32
.530
.848
.625
78
.978
.208
4.70
33
.545
.839
.649
79
.982
.191
5.14
34
.559
.829
.675
80
.985
.174
5.67
35
.574
.819
.700
81
.988
.156
6.31
36
.588
.809
.727
82
.990
.139
7.12
37
.602
.799
.754
83
.993
.122
8.14
38
.616
.788
.781
84
.995
.105
9.51
39
.629
.777
.810
85
.996
.087
11.4
40
.643
.766
.839
86
.998
.070
14.3
41
.656
.755
.869
87
.999
.052
19.1
42
.669
.743
.900
88
.999
.035
28.6
43
.682
.731
.933
89
.999
.017
57.3
44
.695
.719
.966
90
1.000
,000
Inf.
45
.707
.707
1.000
Digitized by
Google
GREEK ALPHABET
Aa
Alpha
I L
Iota
Pp
Rho
B^
Beta
Kk
Kappa
Scr
Sigma
A^>
Gamma
AX
Lambda
Tr
Tau
AS
Delta
Mfi
Mu
Yv
Upsilon ^
E e
Epsilon
Nv
Nu
^<l>
Phi
Zf
Zeta
Bi
Xi
xx
Chi
Hr,
Eta
O o
Omicron
^,A
Psi
ee
Theta
Htt
Pi
Q 0)
Omega
In the applications of mathematics, angles are commonly
denoted by the small or lower case letters of the Greek alphabet.
Those most generally used are a, 6, <^, and 8.
413
Digitized by
Google
Digitized by
Google
INDEX
Absolute temperature, 86
Accurate use of tables, 224
Actual weight, 45
Addition, 102-105
Air, resistance, 97
dry volume, 59
Algebraic notation, 11-18
Alphabet, 5
Amperes, 35
Anemometer, 90
Angular velocity, 35, 88
Area, circle, 29
cone, 31
cylinder, 31
ellipse, 30
on a globe, 89
parallelogram, 28
pjrramid, regular, 31
regular polygon, 29
ring, 30
ring, cylindrical, 33
sector, 30
sphere, 32
trapezium, 28
trapezoid, 28
triangle, 27, 28
Areal velocity, 87
Armor penetration, 85
Atomic weights, 397
Avoirdupois weight, 394
Axes, ellipse, 30
of reference, 63-64
Axioms, 10
B
Barometer, 77
Battery, multiple-series, 36
parallel, 36
series, 35
Beam, deflection, 48, 96
rupture, 97
weight, 90
Bending of bar, 89
Binomial formula, 195
theorem, 191-199
Boyle's law, 88-89
Briggs' system, 223
Calorimeter, 66
Center of gravity, 45
Centrifugal force, 34, 89
Charles' law, 86
Check, 40
Circle, arc of sector, 30
area, 29
area sector, 30
ratio circumference to diameter,
29
Coefficient, 16
of friction, 35
Components, 63
Compound motor, 53-54
Cone, area, 31
frustum, 32
volume, 31
Continued proportion, 38
415
Digitized by
Google
416
INDEX
Convention of signs, 64
Copper wire, resistance, 50
Cosine, 66
Cotangent, definition, 72
Coulomb's law, 98
Cube, diagonal of, 99
Cube root, polynomial, 205-206
Cubic measure, 393
Current, series battery, 35
multiple battery, 36
multiple-series battery, 36
through shunted galvanometer,
36
variation, 98
Cutting speed, 41
Cylinder, area, 31
lateral surface, 99
slide-rule computation, 337-341
volume, 31
weight, 97
Cylindrical ring, area, 33
D
Definition, absolute temperature,
86,92
addition, 102
antilogarithm, 235
areal velocity, 87
checking, 40
coefficient, 16
complex fraction, 151
compound ratio, 14 ^
cotangent, 72 i
direct variation, 78
division, 115
equation, 14
expansion, 192
exponent, 16
exponential equation, 252
fraction, 144
fractional, simple equation, 166
factorial, 191
factoring, 14, 128
factors, 128
formula, 350
frustum, 32
Definition, functions of an angle, 66
grouping, 131
"into," 35
inverse variation, 78
involution, 247
"jointly," 90
known quantity, 18
LL scale, 342
least common denominator, 40
like terms, 101
logarithm, 222
members, 15
modulus, 239
molecular weight, 57
multiplication, 113
pitch of screw, 39
polynomial, 139
prime numbers, 144
proportion, 37
continued, 38
extremes, 38
means, 38
simple, 38
terms, 38
pjrramid, regular, 31
quadratic equation, 155
radical, 207
raciical equation, 219
ratio, 12, 26-27
rational quantity, 208
rationalization, 218
reduction, 208
regular polygon, 29
resultant, 74
rise of a chord, 47
sector, 30
significant formula, 26
similar radicals, 213
simple equation, 18
fraction, 151
simplest form of, radical, 208
result, 351
simultaneous equations, 173
subtraction, 105
subtrahend, 105
supplement of an angle, 69
surd quantity, 208
table of logarithms, 222
Digitized by
Google
INDEX
417
Definition, terms, 15
of a fraction, 144
transformation, 360
trapezium, 28
trapezoid, 28
unlike terms, 101
work, 34
Deflection of beam, 48
De Morgan, 60
Density of a gas, 89
Diameter, long piston-rod, 51
pulley arm, 43
shaft, 43
Diffusive power, 99
Division, 115-118
Dry air, volume, 59
Dry Measure, 393
Dynamo, resistance of shunt
winding, 52
E
Element, percentage composition,
57
Efficiency, shunt motor, 52
Electrical, efficiency, 36
transmission, 46
Elevation, outside rail, 48
Ellipse, area, 30
axes, 30
perimeter, 30, 370
Elongation, of a spring, 93
of a wire, 95
Equation, 14-16
quadratic, 155-165
simple, 18-25
simple fractional, 166-172
Equations, simultaneous simple,
173-179
trigonometric, 70-73
Equilibrium on inclined plane, 49
of liquids, 84
Equivalents, U. S. and metric, 396
Examples, addition, 103-105
binomial formula, 197-199
common factor, 129-130
difference of cubes, 123
Examples, division, 117-118
exponents, 184-185, 187-188,
190
factoring, 129-130, 132-134,
135-136, 139, 141-143
factor law, 127
fractions, 145-146, 148-151,
153-154
functions, acute angle, 66
obtuse angle, 70
grouping, 132-133
logarithmic computation, 266-
277
logarithms, 226, 227, 231, 234,
238, 239, 242, 244, 245-246,
248-249, 251-252, 254, 268-
274
multiplication, 114-115
powers, 201
product sum and difference, 122
projection, 62
quadratic, 160-161, 163-165
radicals, 210-217, 219-220
removal of parenthesis, 109-112
resolution, 64-65, 68
resultant, 76
roots, 203-206
simple equations, 19-25
simple fractional equation,
167-172
simultaneous simple equations,
176-179
shde-rule, 291-202, 294, 299,
304-305, oC7, 311, 315-316,
321-324, 326-328, 332, 335,
338,341,346,349
square 9f, any polynomial, 125
sum of two numbers, 120-
121
subtraction, 106-108
sum of cubes, 124
trigonometric equations, 70-73
Expansion, 192-197
Exponent, 16-17
Exponents, 180-190
fractional, 185-188
negative, 182-185
zero, 181-182
Digitized by
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418
INDEX
Factor, law, 126
with negative exponent, 183
Factorial symbol, 191
Factoring, 128-143
cases, 128
common factor, 129-130
difiFerence of two cubes, 134-135
difference two squares, 133-134
grouping, 131-133
polynomial, 139-140
special expressions, 140-143
methods, 141-143
sum of two cubes, 135-136
trinomial, 136-139
Factors, 14
Falling body, average velocity, 33
change in velocity, 33
final velocity, 33
space traversed, 33
Flow, from orifice, 98
of gas, 87
through a pipe, 87
Flux, 47
Force, 27
centrifugal, 34
Forces, 63-76
Formula, adiabatic expansion, 389
area, trapezoid, 369
triangle, 369
armatures, winding, 361
attracted disk electrometer, 388
battery, electromotive force of,
360
beams, 356
belt, effective pull, 373
belting, 359
bending moment, 356
bending moment of wall-brack-
et, 381
binomial, 195
boiler, pitch of, 359
plate, 384
stay, diameter, 359
tube, thickness of, 373
breaking strength, 383
cable, location of fault in, 361
cable testing, 367
Formula, capacity, concentric
cylinders, 386
condensers, 378
conical vessel, 363
Camot's engine, 368
catenary, 366, 385
chimney, effective area, 380
theoretical height, 380
chord, length of, 370
circular pitch, 49
closed circuit, 386
coefficient of, elasticity, 354
linear expansion, 383
self-induction, 378, 379
columns, breadth of rectangular,
373
Rankine-Gordon formula, 368
combustion, air required, 359
rate of, 358
condensers, 385, 387
water required, 363
cone, surface, 370
connecting rod, 367, 387
convex lens, focal length, 365,
366
magnifying power, 366
Cotter, 384
crank, strength of, 384
crank-pin, tangential pressure,
355
current from battery, 362
cut-off, point of, 372
real and apparent, 353
cylinders, 354
exposed surface, 372
ratio of, 355
ratio of expansion, 367
deflection, 385
difference of magnetic poten-
tial, 376
double-riveted joint, 367
dynamo, air-gap, 365
fall of potential in, 365
earth's magnetic force, 386
economic coefficient, series dy-
namo, 379
shunt motor, 379
economizers, 388
Digitized by
Google
INDEX
419
Formiila, effective pull of belt,
373
efficiency, Carnot's engine, 368
compound motor, 53-54
motor, 380
shunt motor, 52
elastic resilience of bars, 369
ellipee, from trammel, 374
perimeter, 370
equation of trajectory, 387
equilibrium of couples, 376
expansion, ratio of, 353
factor of, evaporation, 358
safety, 354
fire-box plate, 358
flange coupling, 381
force of a charged sphere, 386
forces on crank-pin, 385
fly-wheel, 363
centrifugal force, 383
tension in arms, 383
friction of, pivot, 384
worm and wheel, 384
gas, coefficient of expansion,
362
gear teeth, 382
girder, stay, 378
two-span, 367
head of boiler shell, 358
heat balance equation, 389
heat, electrical equivalent, 355
of vaporization, 352
helix, 370
hoops, shrinkage, 369
horsepower, indicated, 353, 390
of shunt motor, 51
I-beam, thickness of web, 367
inclined plane, 360
inductance of transmission
lines, 372
internal diameter, hollow col-
umn, 368
joint-resistance, 46
journal, dimensions, 381
length, 384
keys, proportion of, 357
lap-joints, double-riveted, 358
lathe, compound-geared, 361
Formula, lever, law of, 367
Leyden jar, charge of, 375
Leyden jars, difference of elec-
trical energy, 376
linkages, centers of valve circles,
372
locknuts, 356
loss, armature, 52
shunt field, 52
stray, 52
through heating, 380
magnetic field, earth's, 362
magnetic,- force, 376
needle, deflection of, 375
M.E.P., 373, 390
moisture in steam, 56
moment, bending, 356
of inertia, 356
moments of horizontal stresses,
356
mutual induction, 380
pantograph, 353
parallel electrified plates,
375
pipe, diameter, 359
discharge of water, 371
pressure on, 354
thickness, 359
piston, 367
mean pressure on, 389
net pressure on, 353
piston-rod, diameter, 382
pitch of screw, 359
potential, 386
difference, 363
of needle, 375
power, 378
pressure, 386
and clearance, 371
projectile, path of, 387
velocity, 388
pulley, acceleration, 351
punch and shear frames, stress
in, 358
radius of gyration of a ship,
367
Rankine-Gordon, for columns,
368
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420
INDEX
Formula, ratio of, absolute tem-
peratures, 389
ratio of expansion, 367
refraction of lines of force, 376
resistance, by Wheatstone
bridge, 360
copper wire, 50
effect of temperature, 367
joint, 46
measurement of, 355, 364
nickel wire, 50
shunt winding, 52
resultant, 75
riveted joint, efficiency, 358
resistance of, 383
strength of, 374
rope, power transmitted, 382
safety valve, 358, 361
screw, 361
segment of a circle, area of, 360
series motor, eflficiency, 53
losses, 53
shaft, diameter, 371
outside diameter, 381
stability of, 383
twisting moment, 380
shearing, 382
Siemens' method, 385
solenoid, magnetic field, 355
space traversed, 360
specific heat of mercury, 50
speed cones, 356
sphere electric capacity, 375
spherical mirror, focal dis-
tance, 365
radius of curvature, 364-365
spiral, length of, 371
spring-ring, 384
stayed surface, 382
strength of, 359
steam, cylinder, 357
per cent of moisture, 382
pressure of a mixture, 355
pressure of saturated, 372
quality of, 367
specific volume of saturated,
372
, strain, 353-354
Formula, stress, 353, 388
due to impact, 369
from longitudinal tension, 366
in rectangular plate. 366
tangent galvanometer, 379
teeth, strength of, 385
temperature, coefficient, 361
and heat, 377
and pressure, 371
tension in a bolt, 388
Thompson's method, 364
variation of resistance, 378
velocity of sound, 364
volume, dry air, 59
frustum of a pyramid, 370
prismoid, 370
wedge, 370
wear of wheels, 366
weight, spur-gear blank, 49
weighted pendulum governor,
height of, 374
Wheatstone bridge, resistance
by, 360
wire, diameter of, 363
work, 353
done by, an electric field, 377
an engine, 371
in adiabatic expansion, 389
Formulas, 20-25, 352-390
gearing, 21-22
laws of logarithms, 254-255
mensuration, 25-33
transformation of, 350-390
Formulation, 25-37
Formulation and computation,
37-59
mathematical laws, 25-59
Foster, 281
Four-place logarithms of numbers,
402-404
trigonometric functions, 405-
411
Fractional exponent, 185-188
Fractions, 144-154
addition and subtraction, 146-
149
complex, 151-154
law of signs, 69
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INDEX
421
Fractions, multiplication and divi-
sion, 149-151
reduction, 144-146
Friction, 91
Frustum volume, cone, 32
pyramid, 32
Functions, obtuse angle, 68-70
Fundamental operations, 101-119
G
Galvanometer, shunted, current
through, 36
Gear, leadscrew, 39
radius, 44
spindle, 39
weight formula, 49
Gears, standard, 41
Generation of oxygen, 83
Globe, area on, 89
Graphical resolution of forces,
63-65
Gravity, 94
Greek alphabet, 413
Gunter's scale, 279-281
H
Hollow, column, 46, 368
shaft, safe transmission, 51
Horizontal component, 63
Horsepower, 34
shunt motor, 51
steam engine, 43-44
steamer, 96
transmitted by rope, 98
Hydraulic press, 84
Hydrogen, generation of, 82-83
volume, 92
Hydrometer, Baum6, 57
Twaddell, 57
Hyperbolic logarithm, 238-242
Hypotenuse, 28
I-beam, 49
Inclined plane, 49
Illumination, 92
Inertia, moment of, 34
Instruments, 3
Intensity of heat, 88
"Into,** meaning of, 35
Introduction, 11-59
"Jointly," definition, 90
Joint resistance, 46
K
Keys, taper of, 44
Kilowatt, 34
Kinetic energy, 35
Knot, 391
Lathe, compound geared, 54-55
simple geared. 39
Law, actual weight, 45
addition, 103
angular velocity, 88
area, cone, 31
cylinder, 31
ellipse, 30
parallelogram, 28
regular pyramid, 31
ring, 30
ring, cylindrical, 33
sector, 30
sphere, 32
trapezium, 28
trapezoid, 28
triangle, 27, 28
battery, multiple, 36
multiple-series, 36
series, 35
bending of bar, 89
Boyle's, 88-89 •
centrifugal force, 34, 89
characteristic, 224, 234
Charles^ 86
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A22
INDEX
Law, circle, area, 29
area sector, 30
ratio circumference to diam-
eter, 29
coal consumption, 100
coefficient any term in binomial
expansion, 194
coefficient of friction, 35
common center of gravity, 45
components, 67
compound geared lathe, 54
cone, area, 31
volume, 31
Coulomb's, 98
cube root, 205
current, shunted galvanometer,
/ 36
xjutting speed, 41
cylinder, area, 30
volume, 31
deflection of beam, 48, 96
density of a gas, 89
diagonal of cube, 99
diameter, piston-rod, 50
shaft, 43
diffusion, 99
division, 115-116
dry volume, 59
efficiency of shunt motor, 52
electrical efficiency, 36
transmission, 46
elevation outside rail, 48
ellipse, area, 30
perimeter, 30, 370
elongation of, spring, 93
a wire, 95
equilibrium on inclined plane, 49
factor, 126
falling bodies, 33
flow from orifice, 98
flow of gas, 87
flux, 47
force, 27
fractional division, 151
friction, 91
frustum volume, 32
generation of, hydrogen, 82-83
oxygen, 83
Law, gravity, 94, 390
heating effect of current, 96
high resistance, 36
horsepower, 34, 43^4
indicated, 390
of steamer, 96
transmitted by rope, 98
hydraulic press, 84
illumination, 92
inclined plane, 99
intensity of heat, 88
joint resistance, 46
kilowatt, 34
kinetic energy, 35
lathe, compound geared, 39, 54
simple geared, 39
linear velocity, 88
Mass, 27
moment of inertia, 34
moments, 35
momentum, 35
multiple battery, 36
multiplication, 113
offing at sea, 93
Ohm's, 35
Osmotic pressure, 86
penetration of armor, 85
percentage composition, 57
perimeter ellipse, 30, 370
period of pendulum, 36, 89
pitch of rivets, 82
power, 37
pressure of a gas, 93
projectile, 100
proportion, 38
pulley, arm diametei , 43
velocities, 47
velocity, 42
pyramid, regular, area, 31
volume, 31
radius, from chord and rise, 47
gear wheel, 44
sphere, 46
range of jet, 98
reduction of a negative logarithm
to tabular form, 258
regular polygon, angle of, 29
area, 29
Digitized by
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INDEX
423
Law, regular '"polygon, central
angle, 29
resistance of air, 97
revolution of planet, 99
ring, cylindrical, area, 33
volume, 32
rupture of beam, 97
safe load on I-beam, 49
safe transmission by hollow
shafts, 51
shearing stress, 81
sigas for fractions, 69
specific gravity, 27
specific gravity by, Baum6, 58
pyknomet^r, 59
Twaddell, 57
sphere, area, 32
volume, 32
square hypotenuse, 28
root, 203-204
stiffness of shafting, 96
strength of current, 85
subtraction, 106
subtrahend, 106
summary fundamental opera-
tions, 118-119
taper of keys, 44
tension in rope, 50
tensional strength of shafting, 92
theoretical weight, 45
Thompson ammeter, 94
time, 42
torque of magnetic needle, 91
torsion, 95-96
variation of current, 98
velocity sound, 100
vibration of, hght, 88
strings. 87
visual angle, 94
volume cone, 31, 32
cylinder, 31
hollow column, 46
hydrogen, 92
pyramid, 31, 32
rectangular solid, 32
ring, 46
sphere, 32, 46
spherical segment, 33, 47
Law, wedge, 51
weight, 26
beam, 90
cylinder, 97
gaseous steam, 56
gun, 86
hollow column, 46
pulley, 42
work, 34
Laws of number, 120-127
difference of two cubes, 122
factor law, 126
product of sum and difference,
122
square of any polynomial, 125
square of difference, 121
square of sum, 120-121
sum of two cubes, 124
Leadscrew gear, 39, 55
I^ever, law of moments, 35
Linear measure, 391
velocity, 88
Liquid measure, 393
Ix)ad on I-beam, 49
Log-S 236
Logarithmic computation, 266-
277
Logarithms, 221-277
antilogarithms or log-i, 235-
238
base in higher mathematics, 238
base of common and Naperian
systems, 238
Briggs', 238
common, 238
conversion factor, 239-240
definition of a logarithm, 222
division, 245-246, 256
exponential equations, 252-254
formulas for operations, 254-255
how to reduce common to
Naperian, 239
how to take readings, 266
hyperhohc, 238-242
logarithm of a decimal, 232-235
mantissa, 235
meaning of hyp. 238
meaning of log- ^ , 236
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424
INDEX
Logarithms, methods of determin-
ing a, power, 257-261
root, 261-266
miscellaneous examples, 266-
277
model solutions, exponential
equation, 252-254
division, 245, 256
miscellaneous, 274-277
multiplication, 243-244, 255-
256,
powers, 247-248, 256-261,
267
roots, 250-251, 261-266
modulus of common system,
239-240
multiplication, 242-244, 255-
256
Naperian, 238-242
Natural, 238
negative characteristic, 234-235
negative logarithm, 258
number greater than unity,
223-231
number less than unity, 232-234
powers, 247-249, 256-261,
267-269
proportional partes, 229-231
reduction of a negative log-
arithm to tabular form, 258
roots, 249-252, 261-266,269-271
summary of laws, 254-255
systems of, 238-240
table- of proportional parts,
229-231
tabular difference, 228-229
Losses, series motor, 53
M
Mannheim, 281
Marks of parenthesis, 17
Mass, 27
Materials of construction, 399
Meaning of, log-i, 236
s,237
Measures, area, 392
length, 391
Measures, volume, 393
weight, 394
Mendel^eff, formula for volume of
dry air, 59
periodic table, 398
Mercury, specific heat, 50
Metric equivalents, 396
linear measure, 392
square measure, 392
Moisture in steam, 56
Molecular weight, 57
Moment of inertia, 34
Moments, law of, 35
Momentum, 35
Motor, compound, 53-54
series, 53
shunt, 52
Multiple battery, 36
Multiplication, 113-115
Multiplier, literal, 16
numerical, 15
N
Napier, 279
Nautical measure, 391
Negative exponent, 182-185
Nickel wire, resistance, 50
Notation, 11-18
O
Oak beam, 48
Obtuse angle functions of, 68-70
Offing at sea, 93
Ohm's law, 35
Orthogonal projection, 61
Oscillation of pendulum, 86-87
Osmotic pressure, 86
Oughtred, 279-281
Oxygen, generation of, 83
Parallelogram, area, 28
Parenthesis, 17
minus, 109-112
Pendulum, 36, 390 ^
Percentage composition, 57
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INDEX
425
Perimeter ellipse, 30, 370
Periodic table, 398
Piston-rod, diameter, 51
Pitch of rivets, 82
screw, 39
Polygon, regular, angle of, 29
central angle, 29
area, 29
Powers, 200-201
and roots, 200-206
Pressure and parts conversion, 400
Pressure of a gas, 93
Problems, formulation and com-
putation, 37-59
formulation, 25-37
area parallelogram, 373
mean effective pressure, 390
thickness of a boiler tube, 373
variation, 80-100
Product, 14
sum and difference, 122
Projection, GO-62
Proportion, 37
Pulley, arm diameter, 43
ratio velocities, 47
velocity, 42
weight of, 42
Pyknometer, 59
Pyramid, area, 31
frustum, 32
volume, 31
Q
Quadratics, 155-105
solution by, factoring, 157-161
completing the square, 161-165
R
Radicals, 207-220
addition and subtraction,
213-214
equations, 219-220
multipUcation and division,
214-216
powers and roots, 217-218
rationalization, 218-219
reduction, 208-213
Rail, elevation of, 48
Radius, gear, 44
sphere, 46
vector, 65
Range of a jet, 98
Ratio, 12-14
Rationalization, 218-219
Ratios of a triangle, 65-66
Reading of angles, 64
Record sheet, 6-7
Rectangular solid, volume, 32
Resistance, 36
air, 97
copper wire, 50
joint, 46
nickel wire, 50
shunt winding, 52
Resolution and composition, 60-76
by computation, 65
graphical, 63-65
Resultant, 74-76
formula, 75
Revolution, planet, 99
Ring, area, 30
cylindrical, area, 33
volume, 32
Rise of chord, 47
Roots, 202-206
Rope tension, 50
transmission, 98
Ruhmkorff coil, 95
S
Screw-jack, 45-46
Second's pendulum, 100, 370
Sector, area, 30
length of arc, 30
Segment, sphere, 33
Series battery, 35
motor, losses, 53
Shaft, diameter, 43
tensional strength, 92
stiffness, 90
Shunt dynamo, resistance of
winding, 52
motor, efficiency, 52
horsepower, 51
Digitized by
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426
INDEX
Signs of functions of obtuse
angle, 69
Simple proportion, 38
Simplest form, 351-352
sin* d-f cos* e, 71
Sine, 66
Slide-rule, 279-349
accumulated errors, 311
change of indexes, 317-318
combined multiplication and
division, 322-324
cube, 329-332
cube root, 332-335
cylinders, 337-341
description, 281
diagrammatic setting, 296-299,
303-305, 308-310, 312-313,
317-318, 322, 329, 338,
340-341, 344-345, 347
division, 312-316
duplex, 283
examples, 291-292, 294, 299,
304-305, 307, 311, 315-316,
321-324, 326-328, 332, 335,
338, 341, 346, 349
gage-points, 336-341
C and Ci, 338-341
M, 337-338
X, 336
second and minute, 292-293
graduation, A and B scales,
283-285
C and D scales, 288-289
5 scale, 289-290
historical, 279-281
hyperbolic logarithms, 346-349
integral figures, 301-302
sine or tangent, 306-307
K scale, 330-331
L scale, 324-325
logarithms, 324-326
to any base, 346
log log duplex, 283
rule, 342-349
hyperbolic logarithms,
346-349
logarithms to any base,
346
Slide-rule, log log rule, Naperian
logarithms, 346-349
powers, 345-346
roots, 343-346
Mannheim, 281
multiplex, 283
multiplication, 300-311
sines and tangents, 305-307
" off the rule," 317-319
pointing off, cube, 331-332
cube root, 333
division, 314
Naperian log, 346-^7
product, 301-302
seconds and minutes gage-
point readings, 293
sine, 290
square root, 328
tangent, 296
powers and roots, 327-335
principle of multiplication, 300-
301
product more than two factors,
307-311
proportion, 317-322
quotient, 312-314
ratio method for proportion,
319-320
scales, 281-283
used in multiplication, 300
sines, 289-294
with reversed slide, 292
with unreversed slide, 290-291
square, 327-328
square root, 328
tangent, reading, 294-299
of obtuse angle, 297-299
tangents, 294-299
triplex, 283
Solution of equations, quadratic,
155-165
simple, 19-25
simultaneous, 173-179
trigonometric, 70-73
Specific gravities, 399
Specific gravity, 27
by Baimi<^, 58
bottle, 5Q
Digitized by
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INDEX
427
Specific gravity, by pyknometer,
59
by Twaddell, 67
Specific heat of mercury, 50
Sphere area, 32
radius, 46
segment, volume, 33, 47
volume, 32, 46
Spindle gear, 39, 55
Square, ditterence two numbers,
121
measure, 392
root, polynomial, 203-204
Standard gears, 41
Steam cylinder, 43
engine, horse-power, 34
per cent of moisture, 56
weight of gaseous, 56
Subtraction, 105-112
Surveyor's long measure, 391
square measure, 392
Table, periodic,
avoirdupois weight, 394
cubic measure, 393
decimal equivalents, 395
dry measure, 393
mternational atomic weights,
397
linear measure, 391
liquid measure, 393
logarithms ot numbers, 402
metric linear measure, 392
metric square measure, 392
nautical measure, 391
Prerisure and parts conversion,
400
specific gravities and weights,
399
square measure, 392
surveyor's long measure, 391
surveyor's square measure, 392
trigonometric functions, 405-411
troy weight, 394
volume and weight of water,
394
Table, wire-gage, 401
Tangent, 66
galvanometer, 85
Taper of keys, 44
Tension in a rope, 50
Theoretical weight, 45
Thompson ammeter, 94
Time lomiula, 42
Torque of magnetic needle, 91
Torsion, 95-96
Trammel, 374
Transformation, definition, 350
methods, 350-351
of formulas, 350-390
Trapezium, area, 28
Trapezoid, 28
Triangle, area, 27, 28
functions of, 66
Trigonometric equations, 70-73
Troy weight, 394
Twaddell hydrometer, 57
Variation, 77-100
Velocity, falling body, 33
projectile, 100
pulley, 42
pulleys, 47
sound, 100
Vertical component, 63
Vibration of, light, 88
strings, 87
Visual angle, 94
Volts, 35
Volume, cone, 31
cone frustum, 32
cylinder, 31
dry air, 59
frustum cone, 32
frustum pyramid, 32
gas, 82
hollow column, 46
hydrogen, 92
pjrramid, 31
frustum, 32
rectangular solid, 32
ring, 32
Digitized by
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428
INDEX
Volume, segment of sphere, 33, 47
sphere, 32, 46
water, 394
wedge, 51
W
Wedge volume, 51
Weight, 26
actual, 45
atomic, 397
avoirdupois, 394
beam, 90
concrete pier, 91
cylinder, 97
gaseous steam, 56
gun, 86
hollow column, 46
materiab of construction, 390
molecular, 57
pulley, 42
Weight, spur-gear blank, 49
theoretical, 45
troy, 394
water, 394
Wind pressure, 90
Wire-gage table, 401
Work, 34
Work-book, 3-9
collection, 7
correction, 7
distribution, 7
inspection, 7
instructions for, entries, 4r6
record sheet, 6-7
indication of results, 8
index, 8-9
Wrought-iron beam, 48
Zero exponent, 181-182
FEB
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