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###### MATHEMATICS FOR THE PRACTICAL MAN ######
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‹12‹EXPLAINING SIMPLY AND QUICKLY ¶ALL THE ELEMENTS OF››
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‹18‹ALGEBRA, GEOMETRY, TRIGONOMETRY, ¶LOGARITHMS, COÖRDINATE ¶GEOMETRY, CALCULUS››
‹19‹WITH ANSWERS TO PROBLEMS››
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‹13‹BY››
‹19‹GEORGE HOWE, M.E.››
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‹17‹ILLUSTRATED››
————————
‹17‹ELEVENTH THOUSAND››
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| Fig. 0.
‹8‹NEW YORK››
‹12‹D. VAN NOSTRAND COMPANY››
‹8‹25 Park Place››
‹8‹1918››
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=== Chapter Null - COPYRIGHT NOTICE - NONE ===
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‹13‹Copyright, 1911, by››
‹13‹D. VAN NOSTRAND COMPANY››
————————
‹13‹Copyright, 1915, by››
‹13‹D. VAN NOSTRAND COMPANY››
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‹8‹𝔖𝔱𝔞𝔫𝔥𝔬𝔭𝔢 𝔓𝔯𝔢𝔰𝔰››
‹8‹F. H. GILSON COMPANY››
‹8‹BOSTON. U.S.A.››
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=== Chapter Null - DEDICATION - NONE ===
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‹10‹Dedicated To››
‹18‹𝔅𝔯𝔬𝔴𝔫 𝔄𝔶𝔯𝔢𝔰, 𝔓𝔥.𝔇.››
‹10‹PRESIDENT OF THE UNIVERSITY OF TENNESSEE››
‹10‹“MY GOOD FRIEND AND GUIDE.”››
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=== Chapter Null - PREFACE - NONE ===
————————
In preparing this work the author has been prompted
by many reasons, the most important of which are:
The dearth of short but complete books covering the
fundamentals of mathematics.
The tendency of those elementary books which “begin
at the beginning” to treat the subject in a popular rather
than in a scientific manner.
Those who have had experience in lecturing to large
bodies of men in night classes know that they are composed
partly of practical engineers who have had considerable
experience in the operation of machinery, but no
scientific training whatsoever; partly of men who have devoted
some time to study through correspondence schools
and similar methods of instruction; partly of men who
have had a good education in some non-technical field of
work but, feeling a distinct calling to the engineering
profession, have sought special training from night lecture
courses; partly of commercial engineering salesmen, whose
preparation has been non-technical and who realize in this
fact a serious handicap whenever an important sale is to
be negotiated and they are brought into competition with
the skill of trained engineers; and finally, of young men
leaving high schools and academies anxious to become
engineers but who are unable to attend college for that
purpose. Therefore it is apparent that with this wide
difference in the degree of preparation of its students any
course of study must begin with studies which are quite
familiar to a large number but which have been forgotten
or perhaps never undertaken by a large number of others.
######################################################### p. iv ###
And here lies the best hope of this textbook. “It begins
at the beginning,” assumes no mathematical knowledge beyond
arithmetic on the part of the student, has endeavored
to gather together in a concise and simple yet accurate and
scientific form those fundamental notions of mathematics
without which any studies in engineering are impossible,
omitting the usual diffuseness of elementary works, and
making no pretense at elaborate demonstrations, believing
that where there is much chaff the seed is easily lost.
I have therefore made it the policy of this book that
no technical difficulties will be waived, no obstacles circumscribed
in the pursuit of any theory or any conception.
Straightforward discussion has been adopted; where
obstacles have been met, an attempt has been made to
strike at their very roots, and proceed no further until
they have been thoroughly unearthed.
With this introduction, I beg to submit this modest
attempt to the engineering world, being amply repaid if,
even in a small way, it may advance the general knowledge
of mathematics.
< GEORGE HOWE.
New York, September, 1910.
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=== 0 - TABLE OF CONTENTS - NONE ===
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Chapter Page
I. Fundamentals of Algebra. Addition and Subtraction 1
II. Fundamentals of Algebra. Multiplication and Division, I 7
III. Fundamentals of Algebra. Multiplication and Division, II 12
IV. Fundamentals of Algebra. Factoring 21
V. Fundamentals of Algebra. Involution and Evolution 25
VI. Fundamentals of Algebra. Simple Equations 29
VII. Fundamentals of Algebra. Simultaneous Equations 41
VIII. Fundamentals of Algebra. Quadratic Equations 48
IX. Fundamentals of Algebra. Variation 55
X. Some Elements of Geometry 61
XI. Elementary Principles of Trigonometry 75
XII. Logarithms 85
XIII. Elementary Principles of Coördinate Geometry 95
XIV. Elementary Principles of the Calculus 110
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=== CHAPTER I - Fundamentals of Algebra - Addition and Subtraction ===
As an introduction to this chapter on the fundamental
principles of algebra, I will say that it is absolutely
essential to an understanding of engineering that the
fundamental principles of algebra be thoroughly digested
and redigested,—in short, literally soaked into one’s
mind and method of thought.
Algebra is a very simple science—extremely simple
if looked at from a common-sense standpoint. If not
seen thus, it can be made most intricate and, in fact,
incomprehensible. It is arithmetic simplified,—a short
cut to arithmetic. In arithmetic we would say, if one
hat costs 5 cents, 10 hats cost 50 cents. In algebra
we would say, if one «𝒂» costs 5 cents, then 10 «𝒂» cost 50
cents, «𝒂» being used here to represent “hat.” «𝒂» is what
we term in algebra a symbol, and all quantities are
handled by means of such symbols. «𝒂» is presumed
to represent one thing; «𝒃», another symbol, is presumed
to represent another thing, «𝒄» another, «𝒅» another, and
so on for any number of objects. The usefulness
and simplicity, therefore, of using symbols to represent
objects is obvious. Suppose a merchant in the
furniture business to be taking stock. He would go
through his stock rooms and, seeing 10 chairs, he
would actually write down “10 chairs”; 5 tables, he
would actually write out “5 tables”; 4 beds, he would
actually write this out, and so on. Now, if he had at
the start agreed to represent chairs by the letter «𝒂»,
tables by the letter «𝒃», beds by the letter «𝒄», and so on,
he would have been saved the necessity of writing
down the names of these articles each time, and could
have written «10 𝒂», «5 𝒃», and «4 𝒄».
######################################################### p. 002 ###
Definition of a Symbol. — A symbol is some letter by
which it is agreed to represent some object or thing.
When a problem is to be worked in algebra, the first
thing necessary is to make a choice of symbols, namely,
to assign certain letters to each of the different objects
concerned with the problem,—in other words, to get
up a code. When this code is once established it must
be rigorously maintained; that is, if, in the solution of
any problem or set of problems, it is once stipulated
that «𝒂» shall represent a chair, then wherever a appears
it means a chair, and wherever the word chair would
be inserted an «𝒂» must be placed—the code must not
be changed.
######################################################### p. 003 ###
Positivity and Negativity. — Now, in algebraic thought,
not only do we use symbols to represent various objects
and things, but we use the signs plus (+) or minus (−)
before the symbols, to indicate what we call the positivity
or negativity of the object.
Addition and Subtraction. — Algebraically, if, in going
over his stock and accounts, a merchant finds that
he has 4 tables in stock, and on glancing over his
books finds that he owes 3 tables, he would represent
the 4 tables in stock by such a form as «+4𝒂», «𝒂»
representing table; the 3 tables which he owes he would
represent by «−3𝒂», the plus sign indicating that which
he has on hand and the minus sign that which he owes.
Grouping the quantities «+4𝒂» and «−3𝒂» together,
in other words, striking a balance, one would get «+𝒂»,
which represents the one table which he owns over and
above that which he owes. The plus sign, then, is taken
to indicate all things on hand, all quantities greater
than zero. The minus sign is taken to indicate all
those things which are owed, all things less than zero.
######################################################### p. 004 ###
Suppose the following to be the inventory of a certain
quantity of stock: «+8𝒂», «−2𝒂», «+6𝒃», «−3𝒄»,
«+4𝒂», «−2𝒃», «−2𝒄», «+5𝒄». Now, on grouping these
quantities together and striking a balance, it will be
seen that there are 8 of those things which are represented
by «𝒂» on hand; likewise 4 more, represented by
«4𝒂», on hand; 2 are owed, namely, «−2𝒂». Therefore,
on grouping «+8𝒂», «+4𝒂», and «−2𝒂» together, «+10𝒂»
will be the result. Now, collecting those terms representing
the objects which we have called «𝒃», we have
«+6𝒃» and «−2𝒃», giving as a result «+4𝒃». Grouping
«−3𝒄», «−2𝒄», and «+5𝒄» together will give 0, because
«+5𝒄» represents «5𝒄»’s on hand, and «−3𝒄» and «−2𝒄»
represent that «5𝒄»’s are owed; therefore, these quantities
neutralize and strike a balance. Therefore,
| «+ 8𝒂 − 2𝒂 + 6𝒃 − 3𝒄 + 4𝒂 − 2𝒃 − 2𝒄 + 5𝒄»
reduces to
| «+10𝒂 + 4𝒃».
This process of gathering together and simplifying a
collection of terms having different signs is what we
call in algebra addition and subtraction. Nothing is
more simple, and yet nothing should be more thoroughly
understood before proceeding further. It is obviously
impossible to add one table to one chair and thereby
get two chairs, or one book to one hat and get two
books; whereas it is perfectly possible to add one book
to another book and get two books, one chair to another
chair and thereby get two chairs.
Rule. — Like symbols can be added and subtracted, and
only like symbols.
«𝒂 + 𝒂» will give «2𝒂»; «3𝒂» + «5𝒂» will give «8𝒂»; «𝒂 + 𝒃»
will not give «2𝒂» or «2𝒃», but will simply give «𝒂 + 𝒃»,
this being the simplest form in which the addition of
these two terms can be expressed.
######################################################### p. 005 ###
Coefficients. — In any term such as «+8𝒂» the plus
sign indicates that the object is on hand or greater than
zero, the 8 indicates the number of them on hand, it
is the numerical part of the term and is called the
coefficient, and the «𝒂» indicates the nature of the object,
whether it is a chair or a book or a table that we
have represented by the symbol «𝒂». In the term «+6𝒂»,
the plus (+) sign indicates that the object is owned, or
greater than zero, the 6 indicates the number of objects
on hand, and the «𝒂» their nature. If a man has $20
in his pocket and he owes $50, it is evident that if he
paid up as far as he could, he would still owe $30. If
we had represented $1 by the letter «𝒂», then the $20 in
his pocket would be represented by «+20𝒂», the $50
that he owed by «−50𝒂». On grouping these terms
together, which is the same process as the settling of
accounts, the result would be «−30𝒂».
Algebraic Expressions. — An algebraic expression consists
of two or more terms; for instance, «+ 𝒂 + 𝒃» is
an algebraic expression; «+ 𝒂 + 2𝒃 + 𝒄» is an algebraic
expression; «+ 3𝒂 + 5𝒃 + 6𝒃 + 𝒄» is another algebraic
expression, but this last one can be written more
simply, for the «5𝒃» and «6𝒃» can be grouped together in
one term, making «11𝒃», and the expression now becomes
«+ 3𝒂 + 11𝒃 + 𝒄», which is as simple as it can be
written. It is always advisable to group together into
the smallest number of terms any algebraic expression
wherever it is met in a problem, and thus simplify the
manipulation or handling of it.
######################################################### p. 006 ###
When there is no sign before the first term of an
expression the plus (+) sign is intended.
To subtract one quantity from another, change the
sign and then group the quantities into one term, as just
explained. Thus: to subtract «4𝒂» from «+ 12𝒂» we
write «− 4𝒂 + 12𝒂», which simplifies into «+ 8𝒂». Again,
subtracting «2𝒂» from «+ 6𝒂» we would have «− 2𝒂 + 6𝒂»,
which equals «+4𝒂».
## PROBLEMS ##
Simplify the following expressions:
1. «10𝒂 + 5𝒃 + 6𝒄 − 8𝒂 − 3𝒅 + 𝒃».
2. «𝒂 − 𝒃 + 𝒄 − 10𝒂 − 7𝒄 + 2𝒃».
3. «10𝒅 + 3𝒛» «+ 8𝒃 − 4𝒅» «− 6𝒛 − 12𝒃» «+ 5𝒂 − 3𝒅»¶
«+ 8𝒛 − 10𝒂» «+ 8𝒃» « − 5𝒂 − 6𝒛» «+ 10𝒃».
4. «5𝒙 − 4𝒚» «+ 3𝒛 − 2𝒙» «+ 4𝒚» «+ 𝒙 + 𝒛» «+ 𝒂 − 7𝒙» «+ 6𝒚».
5. «3𝒃 − 2𝒂» «+ 5𝒄 + 7𝒂» «− 10𝒃 − 8𝒄» «+ 4𝒂 − 𝒃» «+ 𝒄».
6. «− 2𝒙 + 𝒂» «+ 𝒃 + 10𝒚» «− 6𝒙 − 𝒚» «− 7𝒂 + 3𝒃» «+ 2𝒚».
7. «4𝒙 − 𝒚» «+ 𝒛 + 𝒙» «+ 15𝒛 − 3𝒙» «+ 6𝒚 − 7𝒚» «+ 12𝒛».
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=== CHAPTER II - Fundamentals of Algebra - Multiplication and Division ===
We have seen how the use of algebra simplifies the
operations of addition and subtraction, but in multiplication
and division this simplification is far greater, and
the great weapon of thought which algebra is to become
to the student is now realized for the first time. If the
student of arithmetic is asked to multiply one foot by
one foot, his result is one square foot, the square foot
being very different from the foot. Now, ask him to
multiply one chair by one table. How can he express
the result? What word can he use to signify the result?
Is there any conception in his mind as to the appearance
of the object which would be obtained by multiplying
one chair by one table? In algebra all this is
simplified. If we represent a table by «𝒂», and a chair by
«𝒃», and we multiply «𝒂» by «𝒃», we obtain the expression «𝒂𝒃»,
which represents in its entirety the multiplication of a
chair by a table. We need no word, no name by which
to call it; we simply use the form «𝒂𝒃», and that carries
to our mind the notion of the thing which we call «𝒂»
multiplied by the thing which we call «𝒃». And thus the
form is carried without any further thought being given
to it.
######################################################### p. 008 ###
Exponents. — The multiplication of «𝒂» by «𝒂» may be
represented by «𝒂𝒂». But here we have a further short
cut, namely, «𝒂²». This 2, called an exponent, indicates that
two «𝒂»’s have been multiplied by each other; «𝒂 × 𝒂 × 𝒂»
would give us «𝒂³», the 3 indicating that three «𝒂»’s have
been multiplied by one another; and so on. The exponent
simply signifies the number of times the symbol has
been multiplied by itself.
Now, suppose «𝒂²» were multiplied by «𝒂²», the result would
be «𝒂⁵», since «𝒂²» signifies that 2 «𝒂»’s are multiplied together,
and «𝒂³» indicates that 3 «𝒂»’s are multiplied together; then
multiplying these two expressions by each other simply
indicates that 5 «𝒂»’s are multiplied together. «𝒂³ × 𝒂⁷»
would likewise give us «𝒂¹⁰», «𝒂⁴ × 𝒂⁴» would give us «𝒂⁸»,
«𝒂⁴ × 𝒂⁴ × 𝒂² × 𝒂²» would give us «𝒂¹²», and so on.
Rule. — The multiplication by each other of symbols
representing similar objects is accomplished by adding
their exponents.
######################################################### p. 009 ###
Identity of Symbols. — Now, in the foregoing it must
be clearly seen that the combined symbol «𝒂𝒃» is different
from either «𝒂» or «𝒃»; «𝒂𝒃» must be handled as differently
from «𝒂» or «𝒃» as «𝒄» would be handled; in other words, it is
an absolutely new symbol. Likewise «𝒂²» is as different
from «𝒂» as a square foot is from a linear foot, and «𝒂³» is
as different from «𝒂²» as one cubic foot is from one square
foot. «𝒂²» is a distinct symbol. «𝒂³» is a distinct symbol,
and can only be grouped together with other «𝒂³»’s. For
example, if an algebraic expression such as this were met:
| «𝒂² + 𝒂 + 𝒂𝒃 + 𝒂³ + 3𝒂² − 2𝒂 − 𝒂𝒃»,
to simplify it we could group together the «𝒂²» and the
«+ 3𝒂²», giving «+4𝒂²»; the «+𝒂» and the «−2𝒂» give «−𝒂»;
the «+𝒂𝒃» and the «−𝒂𝒃» neutralize each other; there is
only one term with the symbol «𝒂³». Therefore the
above expression simplified would be «4𝒂² − 𝒂 + 𝒂³».
This is as simple as it can be expressed. Above all
things the most important is never to group unlike
symbols together by addition and subtraction. Remember
fundamentally that «𝒂», «𝒃,» «𝒂𝒃», «𝒂²», «𝒂³», «𝒂⁴», are all
separate and distinct symbols, each representing a
separate and distinct thing.
Suppose we have «𝒂 × 𝒃 × 𝒄». It gives us the term
«𝒂𝒃𝒄». If we have «𝒂² × 𝒃» we get «𝒂²𝒃». If we have «𝒂𝒃 × 𝒂𝒃»,
we get «𝒂²𝒃²». If we have «2 𝒂𝒃 × 2 𝒂𝒃» we get «4𝒂𝒃»;
«6 𝒂²𝒃³ × 3𝒄», we get «18 𝒂²𝒃³𝒄»; and so on. Whenever two
terms are multiplied by each other, the coefficients are
multiplied together, and the similar parts of the symbols
are multiplied together.
######################################################### p. 010 ###
Division. — Just as when in arithmetic we write
down «⅔» to mean 2 divided by 3, in algebra we write «frac{𝒂÷𝒃}»
to mean «𝒂» divided by «𝒃». «𝒂» is called a numerator and
«𝒃» a denominator, and the expression «frac{𝒂÷𝒃}» is called a fraction.
If «𝒂³» is multiplied by «𝒂²», we have seen that the
result is «𝒂⁵», obtained by adding the exponents 3 and 2.
If «𝒂³» is divided by «𝒂²», the result is «𝒂», which is obtained
by subtracting 2 from 3. Therefore «frac{𝒂²𝒃÷𝒂𝒃}» would equal «𝒂»,
the «𝒂» in the denominator dividing into «𝒂²» in the numerator
«𝒂» times, and the «𝒃» in the denominator canceling
the «𝒃» in the numerator. Division is then simply the
inverse of multiplication, which is patent. On simplifying
such an expression as «frac{𝒂⁴𝒃²𝒄³÷𝒂²𝒃𝒄⁵}» we obtain «frac{𝒂²𝒃÷𝒄²}», and so on.
Negative Exponents. — But there is a more scientific
and logical way of explaining division as the inverse
of multiplication, and it is thus: Suppose we have the
fraction «frac{1÷𝒂²}». This may be written «𝒂⁻²», or the term «𝒃²»
may be written «frac{1÷𝒃⁻²}»; that is, any term may be changed
from the numerator of a fraction to the denominator by
simply changing the sign of its exponent. For example,
«frac{𝒂⁵÷𝒂²}» may be written «𝒂⁵ × 𝒂⁻²». Multiplying these two
terms together, which is accomplished by adding their
exponents, would give us «𝒂³», 3 being the result of
the addition of 5 and −2. It is scarcely necessary,
therefore, to make a separate law for division if one is
made for multiplication, when it is seen that division
simply changes the sign of the exponent. This should
be carefully considered and thought over by the pupil,
for it is of great importance. Take such an expression
as «frac{𝒂²𝒃⁻²𝒄²÷𝒂𝒃𝒄⁻¹}». Suppose all the symbols in the denominator
are placed in the numerator, then we have «𝒂²𝒃⁻²𝒄²𝒂⁻¹𝒃⁻¹𝒄»,
or, simplifying, «𝒂𝒃⁻³𝒄³», which may be further written
«frac{𝒂𝒄³÷𝒃³}». The negative exponent is very serviceable, and it
is well to become thoroughly familiar with it. The following
examples should be worked by the student.
######################################################### p. 011 ###
## PROBLEMS ##
Simplify the following:
1. «2𝒂 × 3𝒃 × 3𝒂𝒃».
2. «12𝒂²𝒃𝒄 × 4𝒄²𝒃».
3. «6𝒙 × 5𝒚 × 3𝒙𝒚».
4. «4𝒂²𝒃𝒄 × 3𝒂𝒃𝒄 × 𝒂⁵𝒃 × 6𝒃²».
5. «frac{𝒂²𝒃²𝒄³÷𝒂𝒃𝒄}».
6. «frac{𝒂⁴𝒃³𝒄²𝒅÷𝒂²𝒅²}».
7. «𝒂⁻² × 𝒃³ × 𝒂⁶𝒃²𝒄».
8. «𝒂𝒃𝒄² × 𝒃⁻²𝒂⁻¹𝒄⁵ × 𝒂³𝒃³».
9. «frac{𝒂⁴𝒃⁻⁶𝒄³𝒛÷𝒂²𝒃⁻²𝒄}».
10. «10𝒂²𝒃 × 5𝒂⁻¹𝒃𝒄⁻³ × frac{8𝒂𝒄⁻¹ ÷ 𝒃²𝒂⁻⁴} × 10⁻¹𝒂».
11. «frac{5𝒂²𝒃²𝒄²𝒅²÷45𝒂³ × 6𝒅³}».
[**NOTE: Small Empty Block]
######################################################### p. 012 ###
=== CHAPTER III - Fundamentals of Algebra - Multiplication and Division Continued ===
HAVING illustrated and explained the principles of
multiplication and division of algebraic terms, we will
in this lecture inquire into the nature of these processes
as they apply to algebraic expressions. Before doing
this, however, let us investigate a little further into the
principles of fractions.
Fractions. — We have said that the fraction «frac{𝒂÷𝒃}» indicated
that 𝒂 was divided by 𝒃, just as in arithmetic
«frac{1÷3}» indicates that 1 is divided by 3. Suppose we
multiply the fraction «frac{1÷3}» by 3, we obtain «frac{3÷3}», our procedure
being to multiply the numerator 1 by 3. Similarly,
if we had multiplied the fraction «frac{𝒂÷𝒃}» by 3, our result
would have been «frac{3𝒂÷𝒃}».
######################################################### p. 013 ###
Rule. — The multiplication of a fraction by any quantity
is accomplished by multiplying its numerator by
that quantity; thus, «frac{2𝒂²÷𝒃}» multiplied by 3𝒂 would give
«frac{6𝒂²÷𝒃}». Conversely, when we divide a fraction by a
quantity, we multiply its denominator by that quantity.
Thus, the fraction «frac{𝒂÷𝒃}» when divided by 2𝒃 gives «frac{𝒂÷2𝒃²}»
Finally, should we multiply the numerator and the
denominator by the same quantity, it is obvious that
we do not change the value of the fraction, for we have
multiplied and divided it by the same thing. From
this it must not be deduced that adding the same
quantity to both the numerator and the denominator of
a fraction will not change its value. The beginner is
likely to make this mistake, and he is here warned
against it. Suppose we add to both the numerator and
the denominator of the fraction «frac{1÷3}» the quantity 2. We will
obtain «frac{3÷5}», which is different in value from «frac{1÷3}», proving
that the addition or subtraction of the same quantity
from both numerator and denominator of any fraction
changes its value. The multiplication or division of
both the numerator and the denominator by the same
quantity does not alter the value of a fraction one whit.
Multiplying two fractions by each other is accomplished
by multiplying their numerators together and
multiplying their denominators together. Thus, «frac{𝒂÷𝒃} × frac{𝒅÷𝒄}»
would give us «frac{𝒂𝒅÷𝒃𝒄}».
######################################################### p. 014 ###
Suppose it is desired to add the fraction «frac{1÷2}» to the
fraction «frac{1÷3}». Arithmetic teaches us that it is first necessary
to reduce both fractions to a common denominator,
which in this case is 6, viz.: «frac{3÷6} + frac{2÷6} = frac{5÷6}», the numerators
being added if the denominators are of a common
value. Likewise, if it is desired to add «frac{𝒂÷𝒃}» to «frac{𝒄÷𝒅}», we must
reduce both of these fractions to a common denominator,
which in this case is «𝒃𝒅». (The common denominator of
several denominators is a quantity into which any one of
these denominators may be divided; thus 𝒃 will divide into
«𝒃𝒅», 𝒅 times, and 𝒅 will divide into «𝒃𝒅», 𝒃 times.) Our fractions
then become «frac{𝒂𝒅÷𝒃𝒅} + frac{𝒄𝒃÷𝒃𝒅}». The denominators now having a
common value, the fractions may be added by adding
the numerators, resulting in «frac{𝒂𝒅 + 𝒄𝒃÷𝒃𝒅}». Likewise, adding
the fractions «frac{𝒂÷3} + frac{𝒃÷2𝒂} + frac{𝒄÷3𝒂}», we find that the common
denominator in this case is 6𝒂. The first fraction
becomes «frac{2𝒂²÷6𝒂}» the second «frac{3𝒃÷6𝒂}» and the third «frac{2𝒄÷6𝒂}», the
result being the fraction «frac{2𝒂² + 3𝒃 + 2𝒄÷6𝒂}». This process
will be taken up and explained in more detail later, but
the student should make an attempt to apprehend the
principles here stated and solve the problems given at
the end of this lecture.
######################################################### p. 015 ###
Law of Signs. — Like signs multiplied or divided give
+ and unlike signs give −. Thus:
>> «+3𝒂 × +2𝒂» gives «+ 6𝒂²», ¶
also «−3𝒂 × −2𝒂» gives «+ 6𝒂²»,
>> while «+3𝒂 × −2𝒂» gives «−6𝒂²» ¶
or «−3𝒂 × + 2𝒂» gives «−6𝒂²»;
> furthermore «+8𝒂²» divided by «+2𝒂» gives «+4𝒂», ¶
and «−8𝒂²» divided by «−2𝒂» gives «+4𝒂» ¶
while «−8𝒂²» divided by «+2𝒂» gives «−4𝒂» ¶
or «+8𝒂²» divided by «−2𝒂» gives «−4𝒂».
Multiplication of an Algebraic Expression by a
Quantity. — As previously said, an algebraic expression
consists of two or more terms. «3𝒂», «5𝒃», are terms,
but «3𝒂 + 5𝒃» is an algebraic expression. If the stock
of a merchant consists of 10 tables and 5 chairs, and he
doubles his stock, it is evident that he must double the
number of tables and also the number of chairs, namely,
increase it to 20 tables and 10 chairs. Likewise, when
an algebraic expression which consists of «3𝒂 + 2𝒃» is
doubled, or, what is the same thing, multiplied by 2,
each term must be doubled or multiplied by 2, resulting
in the expression «6𝒂 + 4𝒃». Similarly, when an
algebraic expression consisting of several terms is
divided by any number, each term must be divided by
that number.
Rule. — An algebraic expression must be treated as a
unit. Whenever it is multiplied or divided by any quantity,
each term of the expression must be multiplied or
divided by that quantity. For example: Multiplying
the expression «4𝒙 + 3𝒚 + 5𝒙𝒚» by the quantity «3𝒙» will
give the following result: «12𝒙² + 9𝒙𝒚 + 15𝒛²𝒚», obtained
by multiplying each one of the separate terms by «3𝒙»
successively.
######################################################### p. 016 ###
Division of an Algebraic Expression by a Quantity. —
Dividing the expression «6𝒂² + 2𝒂²𝒃 + 4𝒃²» by «2𝒂𝒃»
would result in the expression «frac{3𝒂²÷𝒃} + 𝒂 + frac{2𝒃÷𝒂}», obtained
by dividing each term successively by «2𝒃». This rule
must be remembered, as its importance cannot be over-estimated.
The numerator or denominator of a fraction
consisting of one or two or more terms must be
handled as a unit, this being one of the most important
applications of this rule. For example, in the fraction
«frac{𝒂 + 𝒃 ÷ 𝒂}» or «frac{𝒂 ÷ 𝒂 + 𝒃}» it is impossible to cancel out the «𝒂» in
the numerator and denominator, for the reason that if
the numerator is divided by «𝒂», each term must be
divided by «𝒂», and the operation upon the one term «𝒂»
without the same operation upon the term «𝒃» would
be erroneous. If the fraction «frac{𝒂 + 𝒃 ÷ 𝒂}» is multiplied by 3,
it becomes «frac{3𝒂 + 3𝒃 ÷ 𝒂}». If the fraction «frac{𝒂 − 𝒃÷𝒂 + 𝒃}» is multiplied
by «frac{2÷3}» it becomes «frac{2𝒂 − 2𝒃÷3𝒂 + 3𝒃}»; and so on. Never
forget that the numerator (or denominator) of a fraction
consisting of two or more terms is an algebraic expression
and must be handled as a unit.
######################################################### p. 017 ###
Multiplication of One Algebraic Expression by Another. —
It is frequently desired to multiply an algebraic
expression not only by a single term but by another
algebraic expression consisting of two or more terms, in
which case the first expression is multiplied throughout
by each term of the second expression. The terms which
result from this operation are then collected together
by addition and subtraction and the result expressed in
the simplest manner possible. Suppose it were desired
to multiply «𝒂 + 𝒃» by «𝒄 + 𝒅». We would first multiply
«𝒂 + 𝒃» by 𝒄, which would give us «𝒂𝒄 + 𝒃𝒄». Then we
would multiply «𝒂 + 𝒃» by 𝒅, which would give us
«𝒂𝒅 + 𝒃𝒅». Now, collecting the result of these two multiplications
together, we obtain «𝒂𝒄 + 𝒃𝒄 + 𝒂d + 𝒃𝒅», viz.:
§ 𝒂 + 𝒃
𝒄 + 𝒅
_______
𝒂𝒄 + 𝒃𝒄
𝒂𝒅 + 𝒃𝒅
____________________
𝒂𝒄 + 𝒃𝒄 + 𝒂𝒅 + 𝒃𝒅
| [Workup 3-1 150w]
Again, let us multiply
§ 2𝒂 + 𝒃 − 3𝒄
𝒂 + 2𝒃 − 𝒄
____________________
2𝒂² + 𝒂𝒃 − 3𝒂𝒄
4𝒂𝒃 + 2𝒃² − 6𝒃𝒄
− 2𝒂𝒄 − 𝒃𝒄 + 3𝒄²
_____________________________________
| [Workup 3-2 300w]
and we have
§ «2𝒂² + 5𝒂𝒃 − 5𝒂𝒄 + 2𝒃² − 7𝒃𝒄 + 3𝒄²».
| [Workup 3-3 270w]
######################################################### p. 018 ###
The Division of one Algebraic Expression by Another. —
This is somewhat more difficult to explain and understand
than the foregoing. In general it may be said
that, suppose we are required to divide the expression
«6𝒂² + 17𝒂𝒃 + 12𝒃²» by «3𝒂 + 4𝒃», we would arrange
the expression in the following way:
§ 6𝒂² + 17𝒂𝒃 + 12𝒃² | 3𝒂 + 4𝒃
|_________
6𝒂² + 8𝒂𝒃 2𝒂 + 3𝒃
____________________
9𝒂𝒃 + 12𝒃²
9𝒂𝒃 + 12𝒃²
| [Workup 3-4 300w]
«3𝒂» will divide into «6𝒂²», «2𝒂» times, and this is placed
in the quotient as shown. This «2𝒂» is then multiplied
successively into each of the terms in the divisor, namely,
«3𝒂 + 4𝒃», and the result, namely, «6𝒂² + 8𝒂𝒃», is placed
beneath the dividend, as shown. A line is then drawn
and this quantity subtracted from the dividend, leaving
«9𝒂𝒃». The «+12𝒃²» in the dividend is now carried.
Again, we observe that «3𝒂» in the divisor will divide into
«9𝒂𝒃», «+3𝒃» times, and we place this term in the divisor.
Multiplying «3𝒃» by each of the terms of the divisor, as
before, will give us «9𝒂𝒃 + 12𝒃²»; and, subtracting this
as shown, nothing remains, the final result of the division
then being the expression «2𝒂 + 3𝒃».
This process should be studied and thoroughly understood
by the student.
######################################################### p. 019 ###
## PROBLEMS ##
Solve the following problems:
1. Multiply the fraction «frac{3𝒂²𝒃³𝒄 ÷ 4𝒙²}»¶
by the quantity «3𝒙».
2. Divide the fraction «frac{𝒂𝒃𝒄 ÷ 6𝒅}» by the quantity «3𝒂».
3. Multiply the fraction «frac{𝒂²𝒃²𝒄² ÷ 𝒙𝒚³}» by¶
the fraction «frac{𝒂²𝒃² ÷ 6𝒂}» by¶
the fraction «frac{𝒙²𝒚 ÷ 𝒃}».
4. Multiply the expression «4𝒙 + 3𝒚 + 2𝒛» by the quantity «5𝒙».
5. Divide the expression «8𝒂²𝒃 + 4𝒂³𝒃³ − 2𝒂𝒃²» by¶
the quantity «2𝒂𝒃».
6. Multiply the expression «𝒂 + 𝒃» by the expression «𝒂 − 𝒃».
7. Multiply the expression «2𝒂 + 𝒃 − 𝒄» by¶
the expression «3𝒂 − 2𝒃 + 4𝒄».
8. Divide the expression «𝒂² − 2𝒂𝒃 + 𝒃²» by «𝒂 − 𝒃».
9. Divide the expression «𝒂³ + 3𝒂²𝒃 + 3𝒂𝒃² + 𝒃³» by «𝒂 + 𝒃».
10. Multiply the fraction «frac{𝒂 + 𝒃 ÷ 𝒂 − 𝒃}» by¶
«frac{𝒂 − 𝒃÷𝒂 − 𝒃}».
11. Multiply the fraction «frac{3𝒂 ÷ 𝒄 + 𝒅}» by¶
«frac{𝒄 − 𝒅 ÷ 2}» by «frac{𝒂 + 𝒄 ÷ 𝒂 − 𝒄}».
12. Multiply the fraction «frac{𝒂⁻²𝒃𝒄³ ÷ 4}» by¶
«frac{𝒃 ÷ 3𝒂⁻²}» by «frac{𝒂 ÷ 𝒃}».
######################################################### p. 020 ###
13. Add together the fractions «frac{2𝒂 ÷ 𝒃}»¶
«+ frac{𝒃÷4} + frac{𝒄÷𝒃}».
14. Add together the fractions «frac{2÷3𝒂²}»¶
«− frac{4÷2𝒂} + frac{𝒄÷6}».
15. Add together the fractions «frac{10𝒂²÷𝒃}»¶
«+ frac{𝒃÷4𝒃} − frac{𝒙÷2𝒄} + frac{𝒅÷6}».
16. Add together the fractions «frac{𝒂 + 𝒃 ÷ 2𝒂}»¶
«+ frac{𝒃 − 𝒄÷4𝒃}».
17. Add together the fractions «frac{𝒂÷𝒂 + 𝒃}»¶
«− frac{2÷5𝒂}».
[**NOTE: Small Empty Block]
######################################################### p. 021 ###
=== CHAPTER IV - Fundamentals of Algebra - Factoring ===
Definition of a Factor. — A factor of a quantity is
one of the two or more parts which when multiplied
together give the quantity. A factor is an integral
part of a quantity, and the ability to divide and subdivide
a quantity, be it a single term or a whole expression,
into those factors whose multiplication has created
it, is very valuable.
Factoring. — Suppose we take the number 6. Its
factors are readily detected as 2 and 3. Likewise the
factors of 10 are 5 and 2. The factors of 18 are 9 and
2; or, better still, «3 × 3 × 2». The factors of 30 are
«3 × 2 × 5»; and so on. The factors of the algebraic expression
«𝒂𝒃» are readily detected as 𝒂 and 𝒃, because
their multiplication created the term «𝒂𝒃». The factors
of «6𝒂𝒃𝒄» are 3, 2, 𝒂, 𝒃 and 𝒄. The factors of «25𝒂𝒃» are 5,
5, 𝒂 and 𝒃, which are quite readily detected.
######################################################### p. 022 ###
The factors of an expression consisting of two or more
terms, however, are not so readily seen and sometimes
require considerable ingenuity for their detection. Suppose
we have an algebraic expression in which all of
the terms have one or more common factors,—that is,
that one or more like factors appear in the make-up of
each term. It is often desirable in this case to remove
the common factors from the several terms, and in
order to do this without changing the value of any of
the terms, the common factor or factors are placed
outside of a parenthesis and the terms from which they
have been removed placed within the parenthesis in
their simplified form. Thus, in the algebraic expression
«6𝒂²𝒃 + 3𝒂³», «3𝒂³» is a common factor of both terms;
therefore we may write the expression, without changing
its value, in the following manner: «3𝒂²(2𝒃 + 𝒂)».
The term «3𝒂²» written outside of the parenthesis indicates
that it must be multiplied into each of the separate
terms within the parenthesis. Likewise, in the
expression «12𝒙𝒚 + 4³ + 6𝒙²𝒛 + 8𝒙𝒛», «2𝒙» is a common
factor of each of the terms, and the expression may
be written «2𝒙 (6 𝒚 + 2𝒙² + 3𝒙𝒛 + 4𝒛)». It is often
desirable to factor in this simple manner.
Still further suppose we have «𝒂² + 𝒂𝒃 + 𝒂𝒄 + 𝒃𝒄»; we
can take 𝒂 out of the first two terms and 𝒄 out of the
last two, thus: «𝒂(𝒂 + 𝒃) + 𝒄(𝒂 + 𝒃)». Now we have
two separate terms and taking «(𝒂 + 𝒃)» out of each
we have «(𝒂 + 𝒃) × (𝒂 + 𝒄)». Likewise, in the expression
>> «6𝒙² + 4𝒙𝒚 − 3𝒛𝒙 − 2𝒛𝒚»
we have
>> «2𝒙(3𝒙 + 2𝒚) − 𝒛(3𝒙 + 2𝒚)»,
or,
>> «(3𝒙 + 2𝒚) × (2𝒙 − 𝒛)».
######################################################### p. 023 ###
Now, suppose we have the expression «𝒂² − 2𝒂𝒃 + 𝒃²».
We readily detect that this quantity is the result of
multiplying «𝒂 − 𝒃» by «𝒂 − 𝒃»; the first and last terms
are respectively the squares of 𝒂 and 𝒃, while the
middle term is equal to twice the product of 𝒂 and
𝒃. Any expression where this is the case is a perfect
square; thus, «9𝒙² − 12𝒙𝒚 + 4𝒚²» is the square of
«3𝒙 − 2𝒚», and may be written «(3𝒙 − 2𝒚)²». Remembering
these facts, a perfect square is readily detected.
The product of the sum and difference of two terms
such as «(𝒂 + 𝒃) × (𝒂 − 𝒃)» equals «𝒂² − 𝒃²»; or, briefly,
the product of the sum and difference of two numbers
is equal to the difference of their squares.
By trial it is often easy to discover the factors of
algebraic expressions; for example, «2𝒂² + 7𝒂𝒃 + 3𝒃²» is
readily detected to be the product of «2𝒂 + 𝒃» and
«𝒂 + 3𝒃».
## PROBLEMS ##
Factor the following:
1. «30 𝒂²𝒃».
2. «48 𝒂⁴𝒄».
3. «30 𝒙²𝒚⁴𝒛³».
4. «144 𝒙²𝒂²».
5. «frac{12𝒂𝒃²𝒄³ ÷ 4𝒂²𝒃²}».
6. «frac{10𝒙𝒚² ÷ 2𝒙²𝒚}».
7. «2𝒂² + 𝒂𝒃 − 2𝒂𝒄 − 𝒃𝒄».
######################################################### p. 024 ###
8. «3𝒙² + 𝒙𝒚 + 3𝒙𝒄 + 𝒄𝒚».
9. «2𝒙² + 5𝒙𝒚 + 2𝒙𝒛 + 5𝒚𝒛».
10. «𝒂² − 2𝒂𝒃 + 𝒃²».
11. «4𝒙² − 12𝒙𝒚 + 9𝒚²».
12. «81𝒂² + 90𝒂𝒃 + 25𝒃²».
13. «16𝒄² − 48𝒄𝒂 + 36𝒂²».
14. «4𝒙³𝒚 + 5𝒙𝒛𝒚² − 10𝒙𝒛𝒚».
15. «30𝒂𝒃 + 15𝒂𝒃𝒄 − 5𝒃𝒄».
16. «81𝒙²𝒚² − 25𝒂²».
17. «𝒂⁴ − 16𝒃⁴».
18. «144𝒙⁴𝒚² − 64𝒛²».
19. «4𝒂² − 8𝒂𝒄 + 4𝒄».
20. «16𝒚² + 8𝒙𝒚 + 𝒙²».
21. «6𝒚² − 5𝒙𝒚 − 6𝒙²».
22. «4𝒂² − 3𝒂𝒃 − 10𝒃²».
23. «6𝒚² − 13𝒙𝒚 + 6𝒙²».
24. «2𝒂² − 5𝒂𝒃 − 3𝒃²».
25. «2𝒂² + 9𝒂𝒃 + 10𝒃²».
[**NOTE: Small Empty Block]
######################################################### p. 025 ###
=== CHAPTER V - Fundamentals of Algebra - Involution and Evolution ===
We have in a previous chapter discussed the process
by which we can raise an algebraic term and even a
whole algebraic expression to any power desired, by
multiplying it by itself. Let us now investigate the
method of finding the square root and the cube root
of an algebraic expression, as we are frequently called
upon to do.
The square root of any term such as «𝒂²», «𝒂⁴», «𝒂⁶»,
and so on, will be, respectively, «±𝒂», «±𝒂²», and
«±𝒂³», obtained by dividing the exponents by 2.
The plus-or-minus sign («±») shows that either «+𝒂» or
«−𝒂» when squared would give us «±𝒂²». On taking the
square root, therefore, the plus-or-minus sign («±») is
always placed before the root. This is not the case in
the cube root, however. Likewise, the cube root of
such terms as «𝒂³», «𝒂⁶», «𝒂⁹», and so on, would be respectively
𝒂, «𝒂²» and «𝒂³», obtained by dividing the exponents
by 3. Similarly, the square root of «4𝒂⁴𝒃⁶» will be seen
to be «±2𝒂²𝒃³», obtained by taking the square root of
each factor of the term. And likewise the cube root
of «−27𝒂⁹𝒃⁶» will be «−3𝒂³𝒃²». These facts are so self-evident
that it is scarcely necessary to dwell upon them.
However, the detection of the square and the cube root
of an algebraic expression consisting of several terms is
by no means so simple.
######################################################### p. 026 ###
Square Root of an Algebraic Expression. — Suppose
we multiply the expression «𝒂 + 𝒃» by itself. We obtain
«𝒂² + 2𝒂𝒃 + 6²». This is evidently the square of «𝒂 + 𝒃».
Suppose then we are given this expression and asked to
determine its square root. We proceed in this manner:
Take the square root of the first term and isolate it,
calling it the trial root. The square root of «𝒂²» is
𝒂; therefore place 𝒂 in the trial root. Now square
𝒂 and subtract this from the original expression, and
we have the remainder «2𝒂𝒃 + 𝒃²». For our trial divisor
we proceed as follows: Double the part of the root
already found, namely, 𝒂. This gives us «2𝒂». «2𝒂» will
go into «2𝒂𝒃», the first term of the remainder, 𝒃 times.
Add 𝒃 to the trial root, and the same becomes «𝒂 + 𝒃».
Now multiply the trial divisor by 𝒃, it gives us «2𝒂𝒃 + 𝒃²»,
and subtracting this from our former remainder, we
have nothing left. The square root of our expression,
then, is seen to be «𝒂 + 𝒃», viz.:
§ 𝒂² + 2𝒂𝒃 + 𝒃² | 𝒂 + 𝒃
𝒂² |________
_______________
2𝒂 + 𝒃 | 2𝒂𝒃 + 𝒃²
| 2𝒂𝒃 + 𝒃²
|__________
| [Workup 5-1 260w]
Likewise we see that the square root of «4𝒂² + 12𝒂𝒃 + 9𝒃²»
is «2𝒂 + 3𝒃», viz.:
§ 4𝒂² + 12𝒂𝒃 + 9𝒃² | 2𝒂 + 3𝒃
4𝒂² |________
____________________
4𝒂 + 3𝒃 | 2 𝒂𝒃 + 9𝒃²
| 2 𝒂𝒃 + 9𝒃²
|______________
| [Workup 5-2 300w]
######################################################### p. 027 ###
The Cube Root of an Algebraic Expression. — If we
multiply «𝒂 + 𝒃» by itself three times, in other words,
cube the expression, we obtain «𝒂³ + 3𝒂²𝒃 + 3𝒂𝒃² + 𝒃²».
It is evident, therefore, that if we had been given this
latter expression and asked to find its cube root, our
result should be «𝒂 + 𝒃». In finding the cube root, «𝒂 + 𝒃»,
we proceed thus: We take the cube root of the first
term, namely, 𝒂, and place this in our trial root. Now
cube 𝒂, subtract the 𝒂 thus obtained from the original
expression, and we have as a remainder «3𝒂²𝒃 + 3𝒂𝒃² + 𝒃²».
Now our trial divisor will consist as follows: Square
the part of the root already found and multiply same
by 3. This gives us «3𝒂²». Divide «3𝒂²» into the first
term of the remainder, namely, «3𝒂²𝒃», and it will go
𝒃 times. 𝒃 then becomes the second term of the
root. Now add to the trial divisor three times the first
term of the root multiplied by the second term of the
root, which gives us «3𝒂𝒃». Then add the second term
of the root square, namely, «𝒃²». Our full divisor now
becomes «3𝒂² + 3𝒂𝒃 + 𝒃²». Now multiply this full divisor
by 𝒃 and subtract this from the former remainder, namely,
«3𝒂²𝒃 + 3𝒂𝒃² + 𝒃²», and, having nothing left, we see that
the cube root of our original expression is «𝒂 + 𝒃», viz.:
§ 𝒂³ + 3𝒂²𝒃 + 3𝒂𝒃² + 𝒃² | 𝒂 + 𝒃
𝒂³ |_______
____________________________
3𝒂² + 3𝒂𝒃 + 𝒃² | 3𝒂²𝒃 + 3𝒂𝒃² + 𝒃²
| 3𝒂²𝒃 + 3𝒂𝒃² + 𝒃²
|_____________________
| [Workup 5-3 350w]
######################################################### p. 028 ###
Likewise the cube root of «27𝒙³ + 27𝒙² + 9𝒙 + 1» is
seen to be «3𝒙 + 1», viz.:
§ 27𝒙³ + 27𝒙² + 9𝒙 + 1 | 3𝒙+ 1
27𝒙³ |_______
_______________________
27𝒙² + 9𝒙 + 1 | 27𝒙² + 9𝒙 + 1
| 27𝒙² + 9𝒙 + 1
|_________________
| [Workup 5-4 350w]
## PROBLEMS ##
Find the square root of the following expressions:
> 1. «16𝒙² + 24𝒙𝒚 + 9𝒚²».
> 2. «4𝒂² + 4𝒂𝒃 + 𝒃²».
> 3. «36𝒙² + 24𝒙𝒚 + 4𝒚²».
> 4. «25𝒂² − 20𝒂𝒃 + 4𝒃²».
> 5. «𝒂² + 2𝒂𝒃 + 2𝒂𝒄 + 2𝒃𝒄 + 𝒃² + 𝒄²».
Find the cube root of the following expressions:
> 1. «8𝒙³ + 36𝒙²𝒚 + 54𝒙𝒚² + 27𝒚³».
> 2. «𝒙³ + 6𝒙²𝒚 + 12𝒙𝒚² + 8𝒚³».
> 3. «27𝒂³ + 81𝒂²𝒃 + 81𝒂𝒃² + 27𝒃²».
[**NOTE: Small Empty Block]
######################################################### p. 029 ###
=== CHAPTER VI - Fundamentals of Algebra - Simple Equations ===
An equation is the expression of the equality of two
things; thus, «𝒂 = 𝒃» signifies that whatever we call 𝒂 is
equal to whatever we call 𝒃; for example, one pile of
money containing $100 in one shape or another is
equal to any other pile containing $100. It is evident
that if a quantity is added to or subtracted from one
side of an equation or equality, it must be added to or
subtracted from the other side of the equation or equality,
in order to retain the equality of the two sides;
thus, if «𝒂 = 𝒃», then «𝒂 + 𝒄 = 𝒃 + 𝒄» and «𝒂 − 𝒄 = 𝒃 − 𝒄».
Similarly, if one side of an equation is multiplied
or divided by any quantity, the other side must be
multiplied or divided by the same quantity; thus,
if
> «𝒂 = 𝒃»,
then
> «𝒂𝒄 = 𝒃𝒄»
and
> «frac{𝒂÷𝒄} = frac{𝒃÷𝒄}».
Similarly, if one side of an equation is squared, the
other side of the equation must be squared in order to
retain the equality. In general, whatever is done to
one side of an equation must also be done to the other
side in order to retain the equality of both sides. The
logic of this is self-evident.
######################################################### p. 030 ###
Transposition. — Suppose we have the equation
«𝒂 + 𝒃 = 𝒄». Subtract 𝒃 from both sides, and we have
«𝒂 + 𝒃 − 𝒃 = 𝒄 − 𝒃». On the left-hand side of the equation
the «+𝒃» and the «−𝒃» will cancel out, leaving 𝒂, and
we have the result «𝒂 = 𝒄 − 𝒃». Compare this with our
original equation, and we will see that they are exactly
alike except for the fact that in the one 𝒃 is on the
left-hand side of the equation, in the other 𝒃 is on
the right-hand side of the equation; in one case its sign
is plus, in the other case its sign is minus. This indicates
that in order to change a term from one side of
an equation to the other side it is simply necessary to
change its sign; thus,
>> «𝒂 − 𝒄 + 𝒃 = 𝒅»
may be transposed into the equation
>> «𝒂 = 𝒄 − 𝒃 + 𝒅»,
or into the form
>> «𝒂 − 𝒅 = 𝒄 − 𝒃»,
or into the form
>> «−𝒅 = 𝒄 − 𝒂 − 𝒃».
Any term may be transposed from one side of an equation
to the other simply by changing its sign.
######################################################### p. 031 ###
Adding or Subtracting Two Equations. — When two
equations are to be added to one another their corresponding
sides are added to one another; thus, «𝒂 + 𝒄 = 𝒃»
when added to «𝒂 = 𝒅 + 𝒆» will give «2𝒂 + 𝒄 = 𝒃 + 𝒅 + 𝒆».
Likewise «3𝒂 + 𝒃 = 2𝒄» when subtracted from
«10𝒂 + 2𝒃 = 6𝒄» will yield «7𝒂 + 𝒃 = 4𝒄».
Multiplying or Dividing Two Equations by one Another. — When
two equations are multiplied or divided
by one another their corresponding sides must be multiplied
or divided by one another; thus, «𝒂 = 𝒃» multiplied by
«𝒄 = 𝒅» will give «𝒂𝒄 = 𝒃𝒅», also «𝒂 = 𝒃» divided by «𝒄 = 𝒅»
will give «frac{𝒂÷𝒄} = frac{𝒃÷𝒅}».
Solution of an Equation. — Suppose we have such an
equation as «4𝒙 + 10 = 2𝒙 + 24», and it is desired that
this equation be solved for the value of 𝒙; that is, that
the value of the unknown quantity 𝒙 be found. In
order to do this, the first process must always be
to group the terms containing 𝒙 on one side of the
equation by themselves and all the other terms in the
equation on the other side of the equation. In this
case, grouping the terms containing the unknown
quantity 𝒙 on the left-hand side of the equation we
have
| «4𝒙 − 2𝒙 = 24 − 10».
Now, collecting the like terms, this becomes
| «2𝒙 = 14».
######################################################### p. 032 ###
The next step is to divide the equation through by the
coefficient of 𝒙, namely, 2. Dividing the left-hand
side by 2, we have 𝒙. Dividing the right-hand side
by 2, we have 7. Our equation, therefore, has resolved
itself into
| «𝒙 = 7».
We therefore have the value of 𝒙. Substituting this
value in the original equation, namely,
| «4𝒙 + 10 = 2𝒙 + 24»,
we see that the equation becomes
| «28 + 10 = 14 + 24»,
or
| «38 = 38»,
which proves the result.
The process above described is the general method of
solving for an unknown quantity in a simple equation.
Let us now take the equation
| «2𝒄𝒙 + 𝒄 = 40 − 5𝒙».
This equation contains two unknown quantities, namely,
𝒄 and 𝒙, either of which we may solve for. 𝒙 is usually,
however, chosen to represent the unknown quantity,
whose value we wish to find, in an algebraic expression;
in fact, 𝒙, 𝒚 and 𝒛 are generally chosen to represent
unknown quantities. Let us solve for 𝒙 in the above
equation. Again we group the two terms containing
𝒙 on one side of the equation by themselves and all
other terms on the other side, and we have
| «2𝒄𝒙 + 5𝒙 = 40 − 𝒄».
######################################################### p. 033 ###
On the left-hand side of the equation we have two terms
containing 𝒙 as a factor. Let us factor this expression
and we have
| «𝒙(2𝒄 + 5) = 40 − 𝒄».
Dividing through by the coefficient of 𝒙, which is the
parenthesis in this case, just as simple a coefficient to
handle as any other, and we have
| «𝒙 = frac{40 − 𝒄÷2𝒄 + 5}».
This final result is the complete solution of the equation
as to the value of 𝒙, for we have 𝒙 isolated on
one side of the equation by itself, and its value on the
other side. In any equation containing any number of
unknown quantities represented by symbols, the complete
solution for the value of any one of the unknowns is
accomplished when we have isolated this unknown on one
side of the equation by itself. This is, therefore, the whole
object of our solution.
It is true that the value of a above shown still contains
an unknown quantity, 𝒄. Suppose the numerical
value of 𝒄 were now given, we could immediately find
the corresponding numerical value of 𝒙; thus, suppose 𝒄
were equal to 2, we would have
| «𝒙 = frac{40 − 2÷4 + 5}».
or,
| «𝒙 = frac{38÷9}»
######################################################### p. 034 ###
This is the numerical value of 𝒙, corresponding to the
numerical value 2 of 𝒄. It 4 had been assigned as
the numerical value of 𝒄 we should have
| «𝒙 = frac{40 − 4÷8 + 5} = frac{36÷13}».
Clearing of Fractions. — The above simple equations
contained no fractions. Suppose, however, that we are
asked to solve the equation
| «frac{𝒙÷4} + frac{6÷2} = frac{3𝒙÷2} + frac{5÷6}».
Manifestly this equation cannot be treated at once in
the manner of the preceding example. The first step
in solving such an equation is the removal of all the
denominators of the fractions in the equation, this
step being called the Clearing of Fractions.
######################################################### p. 035 ###
As previously seen, in order to add together the
fractions «frac{1÷2}» and «frac{1÷3}» we must reduce them to a common
denominator, 6. We then have «frac{3÷6} + frac{2÷6} = frac{5÷6}». Likewise,
in equations, before we can group or operate upon any
one of the terms we must reduce them to a common
denominator. The common denominator of several
denominators is any number into which any one of the
various denominators will divide, and the least common
denominator is the smallest such number. The product
of all the denominators—that is, multiplying them all
together—will always give a common denominator, but
not always the least common denominator. The least
common denominator, being the smallest common denominator,
is always desirable in preference to a larger
number; but some ingenuity is needed frequently in
detecting it. The old rule of withdrawing all factors
common to at least two denominators and multiplying
them together, and then by what is left of the denominators,
is probably the easiest and simplest way to proceed.
Thus, suppose we have the denominators 6, 8,
9 and 4. 3 is common to both 6 and 9, leaving respectively
2 and 3. 2 is common to 2, 8 and 4, leaving
respectively 1, 4 and 2, and still further common to 4
and 2. Finally, we have removed the common factors
3, 2 and 2, and we have left in the denominators 1, 2,
3 and 1. Multiplying all of these together we have
72, which is the Least Common Denominator of these
numbers, viz.:
§ 3 | 6, 8, 9, 4
|____________
2 | 2, 8, 3, 4
|____________
2 | 1, 4, 3, 2
|____________
1, 2, 3, 1
| [Workup 6-1 160w]
| «3 × 2 × 2 × 1 × 2 × 3 × 1 = 72».
######################################################### p. 036 ###
Having determined the Least Common Denominator,
or any common denominator for that matter, the next
step is to multiply each denominator by such a quantity
as will change it into the Least Common Denominator.
If the denominator of a fraction is multiplied by any
quantity, as we have previously seen, the numerator
must be multiplied by that same quantity, or the value
of the fraction is changed. Therefore, in multiplying
the denominator of each fraction by a quantity, we
must also multiply the numerator. Returning to the
equation which we had at the outset, namely,
«frac{𝒙÷4} + frac{6÷2} = frac{3𝒙÷2} + frac{5÷6}», we see that the common denominator here is 12.
Our equation then becomes
«frac{3𝒙÷12} + frac{36÷12} = frac{18𝒙÷12} + frac{10÷12}».
We have previously seen that the multiplication or
division of both sides of an equation by the same
quantity does not alter the value of the equation.
Therefore we can at once multiply both sides of this
equation by 12. Doing so, all the denominators disappear.
This is equivalent to merely canceling all the
denominators, and the equation is now changed to the
simple form «3𝒙 + 36 = 18𝒙 + 10». On transposition
this becomes
>> «3𝒙 − 18𝒙 = 10 − 36»,
or
>> «−15𝒙 = −26»,
or
>> «−𝒙 = frac{+26÷15}»,
or
>> «+ 𝒙 = frac{+26÷15}».
Again, let us now take the equation
>> «frac{2𝒙÷5𝒄} + frac{10÷𝒄²} = frac{𝒙÷3}».
######################################################### p. 037 ###
The least common denominator will at once be seen to
be «15𝒄²». Reducing all fractions to this common denominator
we have
> «frac{6𝒄𝒙÷15𝒄²} + frac{150÷15𝒄²} = frac{5𝒄²𝒙÷15𝒄²}».
Canceling all denominators, we then have
> «6𝒄𝒙 + 150 = 5𝒄²𝒙».
Transposing, we have
> «6𝒄𝒙 − 5𝒄²𝒙 = −150».
Taking 𝒙 as a common factor out of both of the terms
in which it appears, we have
> «𝒙(6𝒄 − 5𝒄²) = −150».
Dividing through by the parenthesis, we have
> «frac{−150÷6𝒄 − 5𝒄²}»
This is the value of 𝒙. If some numerical value is
given to 𝒄, such as 2, for instance, we can then find
the corresponding numerical value of 𝒙 by substituting
the numerical value of 𝒄 in the above, and we have
> «𝒙 = frac{−150÷12 − 20} = frac{−150÷−8} = 18.75».
In this same manner all equations in which fractions
appear are solved.
######################################################### p. 038 ###
## PROBLEMS ##
Suppose we wish to make use of algebra in the solution
of a simple problem usually worked arithmetically,
taking, for example, such a problem as this: A man purchases
a hat and coat for $15.00, and the coat costs
twice as much as the hat. How much did the hat cost?
We would proceed as follows: Let 𝒙 equal the cost of
the hat. Since the coat cost twice as much as the hat,
then «2𝒙» equals the cost of the coat, and «𝒙 + 2𝒙 = 15»
is the equation representing the fact that the cost of
the coat plus the cost of the hat equals $15; therefore,
«3𝒙 = $15», from which «𝒙 = $5»; namely, the cost of the
hat was $5. «2𝒙» then equals $10, the cost of the coat.
Thus many problems may be attacked.
Solve the following equations:
> 1. «6𝒙 − 10 + 4𝒙 + 3 = 2𝒙 + 20 − 𝒙 + 15».
> 2. «𝒙 + 5 + 3𝒙 + 6 = − 10𝒙 + 25 + 8𝒙».
> 3. «𝒄𝒙 + 4 + 𝒙 = 𝒄𝒙 + 8». Find the numerical value¶
of 𝒙 if «𝒄 = 3».
> 4. «frac{𝒙÷5} + 3 = frac{8𝒙÷2} + 4».
> 5. «frac{4𝒙÷3} + frac{3𝒙÷5} + frac{7÷2}» =¶
«frac{11÷3} + 𝒙».
> 6. «frac{𝒙÷𝒄} + frac{10÷4𝒄} = frac{𝒙÷3} + frac{𝒙÷12𝒄}».¶
Find the numerical value of 𝒙 if «𝒄 = 3».
######################################################### p. 039 ###
> 7. «frac{10𝒄÷3} − frac{𝒄𝒙÷𝒄} + frac{8÷5𝒄}» =¶
«frac{3𝒄𝒙÷10} + frac{15÷2𝒄}». Find the numerical¶
value of 𝒙 if «𝒄 = 6».
> 8. «frac{𝒙÷𝒂 + 𝒃} − 2 + frac{𝒚÷3} = 1».
> 9. «frac{2𝒙÷𝒂} + 3𝒙 + frac{2÷𝒂 − 𝒃} = 𝒙 − frac{3÷𝒂²}».
> 10. «frac{𝒙÷𝒂 + 𝒃} + frac{𝒙÷𝒂 − 𝒃} = 10».
> 11. Multiply «𝒂𝒙 + 𝒃 = 𝒄𝒙 − 𝒃» by «2𝒂 − 𝒙 = 𝒄 + 10».
> 12. Multiply «frac{𝒂÷3} + 𝒃 = frac{𝒄÷𝒅}» by «𝒙 = 𝒚 + 3».
> 13. Divide «𝒂² − 𝒃² = 𝒄» by «𝒂 + 𝒃 = 𝒄 + 3».
> 14. Divide «2𝒂 = 10𝒚» by «𝒂 = 𝒚 + 2».
> 15. Add «2𝒂 + 10 = 𝒙 + 3 − 𝒅» to «3𝒂 − 7 = 2𝒅».
> 16. Add «4𝒂𝒙 + 2𝒚 = −10𝒙» to «2𝒂𝒙 − 7𝒚 = 5».
> 17. Add «15𝒛² + 𝒙 = 5» to «3𝒙 = −10𝒚 + 7».
> 18. Subtract «2𝒂 − 𝒅 = 8» from «8𝒂 + 𝒅 = 12».
> 19. Subtract «3𝒙 + 7 = 15𝒙² + 𝒚» from «6𝒙 + 5 = 18𝒙²».
> 20. Subtract «frac{2𝒙÷3𝒂 + 𝒃} + 𝒄 = 7» from¶
«frac{10𝒙÷5𝒚} = 18».
> 21. Multiply «frac{𝒙÷3𝒂 + 𝒃} − frac{𝒙÷3} = 𝒄» by¶
«frac{𝒙÷𝒄 − 𝒅} = frac{2𝒂 + 𝒃÷𝒄}».
> 22. Solve the equation «frac{1÷𝒙} = −frac{1÷𝒙 + 1}».
> 23. If a coat cost one-half as much as a gun and twice¶
as much as a hat, and all cost together $100, what¶
is the cost of each?
######################################################### p. 040 ###
> 24. The value of a horse is $15 more than twice the¶
value of a carriage, and the cost of both is¶
$1000; what is the cost of each?
> 25. One-third of Anne’s age is 5 years less than¶
one-half plus 2 years; what is her age?
> 26. A merchant has 10 more chairs than tables in¶
stock. He sells four of each and adding up stock¶
finds that he now has twice as many chairs as¶
tables. How many of each did he have at first?
[**NOTE: Small Empty Block]
######################################################### p. 041 ###
=== CHAPTER VII - Fundamentals of Algebra - Simultaneous Equations ===
As seen in the previous chapter, when we have a
simple equation in which only one unknown quantity
appears, such, for instance, as 𝒙, we can, by algebraic
processes, at once determine the numerical value
of this unknown quantity. Should another unknown
quantity, such as 𝒄, appear in this equation, in order
to determine the value of 𝒙 some definite value must
be assigned to 𝒄. However, this is not always possible.
An equation containing two unknown quantities
represents some manner of relation between these
quantities. If two separate and distinct equations representing
two separate and distinct relations which exist
between the two unknown quantities can be found,
then the numerical values of the unknown quantities
become fixed, and either one can be determined without
knowing the corresponding value of the other. The
two separate equations are called simultaneous equations,
since they represent simultaneous relations between
the unknown quantity. The following is an
example:
######################################################### p. 042 ###
>> «𝒙 + 𝒚 = 10». ¶
«𝒙 − 𝒚 = 4».
The first equation represents one relation between «𝒙»
and 𝒚. The second equation represents another relation
subsisting between «𝒙» and «𝒚». The solution for the numerical
value of «𝒙», or that of «𝒚», from these two equations,
consists in eliminating one of the unknowns, «𝒙» or «𝒚» as
the case may be, by adding or subtracting, dividing or
multiplying the equations by each other, as will be
seen in the following. Let us now find the value of
«𝒙» in the first equation, and we see that this is
>> «𝒙 = 10 − 𝒚».
Likewise in the second equation we have
>> «𝒙 = 4 + 𝒚».
These two values of «𝒙» may now be equated (things
equal to the same thing must be equal to each other),
and we have
>> «10 − 𝒚 = 4 + 𝒚»,
or,
>> «−2𝒚 = 4 − 10»,
>> «−2𝒚 = −6»,
>> «+2𝒚 = +6»,
>> «𝒚 = 3».
######################################################### p. 043 ###
Now, this is the value of «𝒚». In order to find the
value of «𝒙», we substitute this numerical value of
«𝒚» in one of the equations containing both «𝒙» and «𝒚»,
such as the first equation, «𝒙 + 𝒚 = 10». Substituting,
we have
>> «𝒙 + 3 = 10».
Transposing,
>> «𝒙 = 10 − 3»,
>> «𝒙 = 7».
Here, then, we have found the values of both «𝒙» and
«𝒚», the algebraic process having been made possible
by the fact that we had two equations connecting the
unknown quantities.
The simultaneous equations above given might have
been solved likewise by simply adding both equations
together, thus:
Adding
>> «𝒙 + 𝒚 = 10»
and
>> «𝒙 − 𝒚 = 4»,
we have
>> «𝒙 + 𝒚 + 𝒙 − 𝒚 = 14».
Here «+𝒚» and «−𝒚» will cancel out, leaving
>> «2𝒙 = 14», ¶
«𝒙 = 7».
Both of these processes are called elimination, the
principal object in solving simultaneous equations being
the elimination of unknown quantities until some equation
is obtained in which only one unknown quantity
appears.
######################################################### p. 044 ###
We have seen that by simply adding two equations
we have eliminated one of the unknowns. But suppose
the equations are of this type:
>> (1) «3𝒙 + 2𝒚 = 12», ¶
(2) «𝒙 + 𝒚 = 5».
Now we can proceed to solve these equations in one of
two ways: first, to find the value of «𝒙» in each equation
and then equate these values of «𝒙», thus obtaining an
equation where only «𝒚» appears as an unknown quantity.
But suppose we are trying to eliminate «𝒙» from
these equations by addition; it will be seen that adding
will not eliminate «𝒙», nor even will subtraction eliminate
it. If, however, we multiply equation (2) by 3, it becomes
>> «3𝒙 + 3𝒚 = 15».
Now, when this is subtracted from equation (1), thus:
>> «3𝒙 + 2𝒚 = 12» ¶
«3𝒙 + 3𝒚 = 15» ¶
________________ ¶
«−𝒚 = −3»
the terms in «𝒙», «+3𝒙» and «+3𝒙» respectively, will eliminate,
«3𝒚» minus «2𝒚» leaves «−𝒚», and 12 − 15 leaves −3,
or
>> «−𝒚 = −3»,
therefore
>> «+𝒚 = +3».
######################################################### p. 045 ###
Just as in order to find the value of two unknowns
two distinct and separate equations are necessary to
express relations between these unknowns, likewise to
find the value of the unknowns in equations containing
three unknown quantities, three distinct and separate
equations are necessary. Thus, we may have the
equations
>> (1) «𝒙 + 𝒚 + 𝒛 = 6», ¶
(2) «𝒙 − 𝒚 + 2𝒛 = 1», ¶
(3) «𝒙 + 3 − 8 = 4».
We now combine any two of these equations, for instance
the first and the second, with the idea of eliminating
one of the unknown quantities, as «𝒙». Subtracting
equation (2) from (1), we will have
>> (4) «2𝒚 − 𝒛 = 5».
Now taking any other two of the equations, such as the
second and the third, and subtracting one from the other,
with a view to eliminating «𝒙», and we have
>> (5) «−2𝒚 + 3𝒛 = −3».
We now have two equations containing two unknowns,
which we solve as before explained. For instance, adding
them, we have
>> «2𝒛 = 2», ¶
« 𝒛 = 1».
Substituting this value of 𝒛 in equation (4), we have
>> «2𝒚 − 1 = 5» ¶
«2𝒚 = 6», ¶
«𝒚 = 3».
######################################################### p. 046 ###
Substituting both of these values of 𝒛 and 𝒚 in equation
(1), we have
>> «𝒙 + 3 + 1 = 6», ¶
«𝒙 = 2».
Thus we see that with three unknowns three distinct
and separate equations connecting them are necessary in
order that their values may be found. Likewise with
four unknowns four distinct and separate equations
showing relations between them are necessary. In each
case where we have a larger number than two equations,
we combine the equations together two at a time, each
time eliminating one of the unknown quantities, and,
using the resultant equations, continue in the same
course until we have finally resolved into one final
equation containing only one unknown. To find the
value of the other unknowns we then work backward,
substituting the value of the one unknown found in an
equation containing two unknowns, and both of these in
an equation containing three unknowns, and so on.
The solution of simultaneous equations is very important
and the student should practice on this subject
until he is thoroughly familiar with every one of these
steps.
######################################################### p. 047 ###
## PROBLEMS ##
Solve the following problems:
> 1. «2𝒙 + 𝒚 = 8» ¶
«2𝒚 − 𝒙 = 6».
> 2. «𝒙 + 𝒚 = 7» ¶
«3𝒙 − 𝒚 = 13».
> 3. «4𝒙 = 𝒚 + 2» ¶
«𝒙 + 𝒚 = 3».
> 4. Find the value of «𝒙», 𝒚 and 𝒛 in the following equations:
> «𝒙 + 𝒚 + 𝒛 = 10», ¶
«2𝒙 + 𝒚 − 𝒛 = 9», ¶
«𝒙 + 2𝒚 + 𝒛 = 12».
> 5. Find the value of «𝒙», «𝒚» and «𝒛» in the following equations:
> «2𝒙 + 3𝒚 + 2𝒛 = 20», ¶
«𝒙 + 3𝒚 + 𝒛 = 13», ¶
«𝒙 + 𝒚 + 2𝒛 = 13».
> 6. «frac{𝒙÷3} + 𝒚 = 10», ¶
«𝒚 + frac{𝒙÷5} = 𝒚 − 3».
> 7. «frac{𝒙÷4} + frac{𝒚÷3𝒂} = 100𝒙 + 𝒂» if «𝒂 = 8», ¶
«frac{2𝒙÷5} = 𝒚 + 10».
> 8. «3𝒙 + 𝒚 = 15», ¶
«𝒙 = 6 + 7𝒚».
> 9. «frac{9𝒙÷𝒂 + 𝒃} = frac{𝒚÷𝒂 − 𝒃} − 7», ¶
«𝒙 + 𝒚 = 5» ¶
if «𝒂 = 6», «𝒃 = 5».
> 10. «3𝒙 − 𝒚 + 6𝒙 = 8», ¶
«𝒚 − 10 + 4𝒚 = 𝒙».
[**NOTE: Small Empty Block]
######################################################### p. 048 ###
=== CHAPTER VIII - Fundamentals of Algebra - Quadratic Equations ===
THUS far we have handled equations where the
unknown whose value we were solving for entered the
equation in the first power. Suppose, however, that
the unknown entered the equation in the second power;
for instance, the unknown «𝒙» enters the equation thus,
| «𝒙² = 12 − 2𝒙²».
In solving this equation in the usual manner we obtain
| «3𝒙² = 12»,
| «𝒙² = 4».
Taking the square root of both sides,
| «𝒙 = ± 2».
We first obtained the value of «𝒙²» and then took the
square root of this to find the value of «𝒙». The solution
of such an equation is seen to be just as simple in
every respect as a simple equation where the unknown
did not appear as a square. But suppose that we have
such an equation as this:
| «4𝒙² + 8𝒙 = 12».
######################################################### p. 049 ###
We see that none of the processes thus far discussed
will do. We must therefore find some way of grouping
«𝒙²» and «𝒙» together which will give us a single term in «𝒙»
when we take the square root of both sides; this device
is called “Completing the square in «𝒙».”
It consists as follows: Group together all terms in «𝒙²»
into a single term, likewise all terms containing «𝒙» into
another single term. Place these on the left-hand side
of the equation and everything else on the right-hand
side of the equation. Now divide through by the
coefficient of «𝒙²». In the above equation this is 4. Having
done this, add to the right-hand side of the equation
the square of one-half of the coefficient of «𝒙». If
this is added to one side of the equation it must likewise
be added to the other side of the equation. Thus:
| «4𝒙² + 8𝒙 = 12».
Dividing through by the coefficient of «𝒙²», namely 4, we
have
| «𝒙² + 2𝒙 = 3».
Adding to both sides the square of one-half of the
coefficient of «𝒙», which is 2 in the term «2𝒙»,
| «𝒙² + 2𝒙 + 1 = 3 + 1».
The left-hand side of this equation has now been made
into the perfect square of «𝒙 + 1», and therefore may be
expressed thus:
| «(𝒙 + 1)² = 4».
Now taking the square root of both sides we have
| «𝒙 + 1 = ± 2».
######################################################### p. 050 ###
Therefore, using the plus sign of 2, we have
| «𝒙 = 1».
Using the minus sign of 2 we have
| «𝒙 = −3».
The student will note that there must, in the nature of
the case, be two distinct and separate roots to a quadratic
equation, due to the plus and minus signs above
mentioned.
To recapitulate the preceding steps, we have:
(1) Group all the terms in «𝒙²» and «𝒙» on one side of the
equation alone, placing those in «𝒙²» first.
(2) Divide through by the coefficient of «𝒙²».
(3) Add to both sides of the equation the square of
one-half of the coefficient of the «𝒙» term.
(4) Take the square root of both sides (the left-hand
side being a perfect square). Then solve as for a simple
equation in «𝒙».
Example: Solve for «𝒙» in the following equation:
>> «4𝒙² = 56 − 20𝒙», ¶
«4𝒙² + 20𝒙 = 56», ¶
«𝒙² + 5𝒙 = 14», ¶
«𝒙² + 5𝒙 + frac{25÷4} = 14 + frac{25÷4}», ¶
«𝒙² + 5𝒙 + frac{25÷4} = frac{81÷4}», ¶
« \bigl (𝒙 + frac{5÷2} \bigr )² = frac{81÷4}».
######################################################### p. 051 ###
Taking the square root of both sides we have
>> «𝒙 + frac{5÷2} = ±frac{9÷2}», ¶
«𝒙 = ±frac{9÷2} − frac{5÷2}», ¶
«𝒙 = 2» or «−7»,
Example: Solve for «𝒙» in the following equation:
> «2𝒙² − 4𝒙 + 5 = 𝒙² + 2𝒙 − 10 − 3𝒙² + 33», ¶
«2𝒙² − 𝒙² + 3𝒙² − 4𝒙 − 2𝒙 = 33 − 10 − 5», ¶
«4𝒙² − 6𝒙 = 18», ¶
«𝒙² −frac{6𝒙÷4}6 = frac{18÷4}», ¶
«𝒙² −frac{3𝒙÷2} = frac{18÷4}», ¶
«𝒙² −frac{3𝒙÷2} + frac{9÷16} = frac{18÷4} + frac{9÷16}», ¶
« \bigl( 𝒙 − ¾ \bigr) ² = frac{72÷16} + frac{9÷16}», ¶
« \bigl( 𝒙 − ¾ \bigr) ² = frac{81÷16}», ¶
«𝒙 − ¾ = ± frac{9÷4}», ¶
«𝒙 = ± frac{9÷4} + ¾», ¶
«𝒙 = +3» or «−1frac{1÷2}.»
######################################################### p. 052 ###
Solving an Equation which Contains a Root. — Frequently
we meet with an equation which contains a
square or a cube root. In such cases it is necessary to
get rid of the square or cube root sign as quickly as
possible. To do this the root is usually placed on one
side of the equation by itself, and then both sides are
squared or cubed, as the case may be, thus:
Example: Solve the equation
> «√[2𝒙 + 6] + 5𝒂 = 10».
Solving for the root, we have
> «√[2𝒙 + 6] = 10 − 5𝒂».
Now squaring both sides we have
> «2𝒙 + 6 = 100 − 100𝒂 + 25𝒂²»,
or,
> «2𝒙 = 25𝒂² − 100𝒂 + 100 − 6»,
> «𝒙 = frac{(25𝒂² − 100𝒂 + 94)÷2}».
In any event, our prime object is first to get the square-root
sign on one side of the equation by itself if possible,
so that it may be removed by squaring.
######################################################### p. 053 ###
Or the equation may be of the type
> «2𝒂 + 1 = frac{4÷√[𝒂 − 𝒙]}».
Squaring both sides we have
> «4𝒂² + 4𝒂 + 1 = frac{16÷𝒂 − 𝒙}»
Clearing fractions we have
>> «−4𝒂²𝒙 − 4𝒂𝒙 − 𝒙 + 𝒂² + 𝒂 = 16»
>> «−𝒙(4𝒂² + 4𝒂 + 1) = −4𝒂³ − 4𝒂² − 𝒂 + 16»
| «𝒙 = frac{4𝒂³ + 4𝒂² + 𝒂 − 16÷4𝒂² + 4𝒂 + 1}»
## PROBLEMS ##
Solve the following equations for the value of «𝒙»:
> 1. «5𝒙² − 15𝒙 = −10».
> 2. «3𝒙² + 4𝒙 + 20 = 44».
> 3. «2𝒙² + 11 = 𝒙² + 4𝒙 + 7».
> 4. «𝒙² + 4𝒙 = 2𝒙 + 2𝒙² − 8».
> 5. «7𝒙 + 15 − 2² = 3𝒙 + 18».
> 6. «𝒙⁴ + 2𝒙² = 24».
> 7. «𝒙² + frac{5𝒙÷𝒂} + 6𝒙² = 10».
> 8. «frac{𝒙²÷𝒂} + frac{𝒙÷𝒃} − 3 = 0».
> 9. «14 + 6𝒙 = frac{4𝒙²÷2} + frac{2𝒙÷𝒂} − 7».
> 10. «frac{𝒙²÷𝒂 + 𝒃} − 3𝒙 = 2».
> 11. «3𝒙² + 5𝒙 − 15 = 0».
> 12. «(𝒙 + 2)² + 2(𝒙 + 2) = −1».
> 13. «(𝒙 − 3)² − 10𝒙 + 7 = 0».
> 14. «(𝒙 − 𝒂)² − (𝒙 + 𝒂)² = 3».
> 15. «frac{𝒙 + 𝒂÷𝒙 − 𝒂} + frac{𝒙 + 𝒃÷𝒙 − 𝒃} = 2».
######################################################### p. 054 ###
> 16. «frac{3𝒙 + 7÷2} − frac{𝒙 + 2÷6} = frac{12÷𝒙 + 1}».
> 17. «frac{𝒙² − 2÷4𝒙} = frac{𝒙 + 3 2𝒙÷8}».
> 18. «frac{𝒙² − 𝒙 − 1÷4} = 𝒙² + 6».
> 19. «8 = frac{64÷√[𝒙 + 1]}».
> 20. «√[𝒙 + 𝒂] + 10𝒂 = 15».
> 21. «frac{𝒙÷𝒂} = √[𝒙 + 1]».
> 22. «3𝒙 + 5 = 2 + √[3𝒙 + 4]».
[**NOTE: Spacing Empty Block]
######################################################### p. 055 ###
=== CHAPTER IX - Fundamentals of Algebra - Variation ===
THIS is a subject of the utmost importance in the
mathematical education of the student of science. It
is one to which, unfortunately, too little attention is
paid in the average mathematical textbook. Indeed,
it is not infrequent to find a student with an excellent
mathematical training who has but vaguely grasped
the notions of variation, and still it is upon variation
that we depend for nearly every physical law.
Fundamentally, variation means nothing more than
finding the constants which connect two mutually
varying quantities. Let us, for instance, take wheat
and money. We know in a general way that the more
money we have the more wheat we can purchase.
This is a variation between wheat and money. But we
can go no further in determining exactly how many
bushels of wheat a certain amount of money will buy
before we establish some definite constant relation
between wheat and money, namely, the price per
bushel of wheat. This price is called the Constant of
the variation. Likewise, whenever two quantities are
varying together, the movement of one depending
absolutely upon the movement of the other, it is impossible
to find out exactly what value of one corresponds
with a given value of the other at any time,
unless we know exactly what constant relation subsists
between the two.
######################################################### p. 056 ###
Where one quantity, «𝒂», varies as another quantity,
namely, increases or decreases in value as another quantity,
𝒃, we represent the fact in this manner:
| «𝒂 ∝ 𝒃».
Now, wherever we have such a relation we can immediately
write
>> «𝒂» = some constant «× 𝒃», ¶
«𝒂 = 𝒌 × 𝒃».
If we observe closely two corresponding values of «𝒂» and
«𝒃», we can substitute them in this equation and find out
the value of this constant. This is the process which
the experimenter in a laboratory has resorted to in deducing
all the laws of science.
######################################################### p. 057 ###
Experimentation in a laboratory will enable us to
determine, not one, but a long series of corresponding
values of two varying quantities. This series of values
will give us an idea of the nature of their variation.
We may then write down the variation as above shown,
and solve for the constant. This constant establishes
the relation between «𝒂» and «𝒃» at all times, and is therefore
all-important. Thus, suppose the experimenter in
a laboratory observes that by suspending a weight of
100 pounds on a wire of a certain length and size it
stretched one-tenth of an inch. On suspending 200
pounds he observes that it stretches two-tenths of an
inch. On suspending 300 pounds he observes that it
stretches three-tenths of an inch, and so on. He at
once sees that there is a constant relation between the
elongation and the weight producing it. He then
writes:
>> Elongation «∝» weight. ¶
Elongation = some constant «×» weight. ¶
«E = K × W».
Now this is an equation. Suppose we substitute one of
the sets of values of elongation and weight, namely,
>> .3 of an inch and 300 lbs.
We have
>> .3 = «K × 300».
Therefore
>> «K = .001».
Now, this is an absolute constant for the stretch of that
wire, and if at any time we wish to know how much a
certain weight, say 500 lbs., will stretch that wire, we
simply have to write down the equation
| «E = K × W».
Substituting
>> elong. = «.001 × 500»,
and we have
>> elong. = «.5» of an inch.
Thus, in general, the student will remember that where
two quantities vary as each other we can change this
variation, which cannot be handled mathematically,
into an equation which can be handled with absolute
definiteness and precision by simply inserting a constant
into the variation.
######################################################### p. 058 ###
Inverse Variation. — Sometimes we have one quantity
increasing at the same rate that another decreases;
thus, the pressure on a certain amount of air increases
as its volume is decreased, and we write
>> «𝒗 ∝ frac{1÷𝒑}»,
then
>> «𝒗 ∝ 𝑲 × frac{1÷𝒑}»,
Wherever one quantity increases as another decreases,
we call this an inverse variation, and we express it in the
manner above shown. Frequently one quantity varies
as the square or the cube or the fourth power of the
other; for instance, the area of a square varies as the
square of its side, and we write
>> «𝑨 ∝ 𝒃²»,
or
>> «𝑨 = 𝑲𝒃²».
Again, one quantity may vary inversely as the square of
the other, as, for example, the intensity of light, which
varies inversely as the square of the distance from its
source, thus:
>> «𝑨 ∝ frac{1÷d²}»,
or
>> «𝑨 = 𝑲frac{1÷d²}»,
######################################################### p. 059 ###
Grouping of Variations. — Sometimes we have a
quantity varying as one quantity and also varying as
another quantity. In such cases we may group these
two variations into a single variation. Thus, we say
that
>> «𝒂 ∝ 𝒃»,
also
>> «𝒂 ∝ 𝒄»,
then
>> «𝒂 ∝ 𝒃 × 𝒄»
or,
>> «𝒂 = 𝑲 × 𝒃 × 𝒄».
This is obviously correct; for, suppose we say that the
weight which a beam will sustain in end-on compression
varies directly as its width, also directly as its depth,
we see at a glance that the weight will vary as the
cross-sectional area, which is the product of the width
by the depth.
Sometimes we have such variations as this:
>> «𝒂 ∝ 𝒃»,
also
>> «𝒂 ∝ frac{1÷𝒄}»,
then
>> «𝒂 ∝ frac{𝒃÷𝒄}».
This is practically the same as the previous case, with
the exception that instead of two direct variations we
have one direct and one inverse variation.
There is much interesting theory in variation, which,
however, is unimportant for our purposes and which
I will therefore omit. If the student thoroughly masters
the principles above mentioned he will find them
of inestimable value in comprehending the deduction
of scientific equations.
######################################################### p. 060 ###
## PROBLEMS ##
> 1. If «𝒂 ∝ 𝒃» and we have a set of values showing that
when «𝒂 = 500», «𝒃 = 10», what is the constant of this
variation?
> 2. If «𝒂 ∝ 𝒃²», and the constant of the variation is
2205, what is the value of «𝒃» when «𝒂» = 5?
> 3. «𝒂 ∝ 𝒃»; also «𝒂 ∝ frac{1÷𝒄}», or, «𝒂 ∝ frac{𝒃÷𝒄}». If we find that
when «𝒂 = 100», then «𝒃 = 5» and «𝒄 = 3», what is the constant
of this variation?
> 4. «𝒂 ∝ 𝒃». The constant of the variation equals 12.
What is the value of «𝒂» when «𝒃» = 2 and «𝒄» = 8?
> 5. «𝒂 = 𝑲 × frac{𝒃÷𝒄}». If 𝑲 = 15 and «𝒂» = 6 and «𝒃» = 2,
what is the value of «𝒄»?
[**NOTE: Small Empty Block]
######################################################### p. 061 ###
=== CHAPTER X - Some Elements of Geometry - NONE ===
In this chapter I will attempt to explain briefly some
elementary notions of geometry which will materially
aid the student to a thorough understanding of many
physical theories. At the start let us accept the following
axioms and definitions of terms which we will
employ.
Axioms and Definitions:
I. Geometry is the science of space.
II. There are only three fundamental directions or
dimensions in space, namely, length, breadth and depth.
III. A geometrical point has theoretically no dimensions.
IV. A geometrical line has theoretically only one
dimension,—length.
V. A geometrical surface or plane has theoretically
only two dimensions, namely, length and breadth.
VI. A geometrical body occupies space and has three
dimensions,—length, breadth and depth.
VI. An angle is the opening or divergence between
two straight lines which cut or intersect each other;
thus, in Fig. 1,
«∡𝒂» is an angle between the lines «AB» and «CD», and may
be expressed thus, «∡𝒂» or «∡BOD».
| [FIGURE - Two straight line segments (A-B and C-D) crossing over at point O. An angle marked 'a' is highlighted, defined by the divergence of segments O-B and O-D.]
| Fig. 1.
######################################################### p. 062 ###
VIII. When two lines lying in the same surface or
plane are so drawn that they never approach or retreat
from each other, no matter how long they are actually
extended, they are said to be parallel; thus, in Fig. 2,
the lines «AB» and «CD» are parallel.
| [FIGURE - Two straight line segments (A-B and C-D) that do not cross.]
| Fig. 2.
IX. A definite portion of a surface or plane bounded
by lines is called a polygon; thus, Fig. 3 shows a polygon.
| [FIGURE - A polygon of 7 line segments, each ending at the start of the next segment, meeting at various angles, creating a closed figure.]
| Fig. 3.
######################################################### p. 063 ###
X. A polygon bounded by three sides is called a
triangle (Fig. 4).
| [FIGURE - A polygon of 3 line segments, each ending at the start of the next segment, meeting at various angles, creating a closed figure of a triangle.]
| Fig. 4.
XI. A polygon bounded by four sides is called a
quadrangle (Fig. 5), and if the opposite sides are parallel,
a parallelogram (Fig. 6).
| [FIGURE - A polygon of 4 line segments, each ending at the start of the next segment, meeting at various angles, creating a closed figure termed a quadrangle.]
| Fig. 5.
| [FIGURE - A polygon of 4 line segments, each ending at the start of the next segment, and each segment is one of a pair with identical lengths and angles, creating a closed figure.]
| Fig. 6.
XII. When a line has revolved about a point until it
has swept through a complete circle, or 360°, it
comes back to its original
position. When it has revolved one quarter of a
circle, or 90°, away from its
original position, it is said
to be at right angles or
perpendicular to its original
position; thus, the angle «a» (Fig. 7) is a right angle
between the lines «AB» and «CD», which are perpendicular
to each other.
| [FIGURE - Two line segments (A-B and C-D) crossing at point O. An angle 'a' is shown, representing a 90-degree angle between O-B and O-D.]
| Fig. 7.
######################################################### p. 064 ###
XIII. An angle less than a right angle is called an
acute angle.
XIV. An angle greater than a right angle is called an
obtuse angle.
XV. The addition of two right angles makes a straight
line.
XVI. Two angles which when placed side by side or
added together make a right angle, or 90°, are said to
be complements of each other; thus, «∡30°» and «∡60°» are
complementary angles.
XVII. Two angles which when added together form
180°, or a straight line, are said to be supplements of
each other; thus, «∡130°» and «∡50°» are supplementary
angles.
XVIII. When one of the inside angles of a triangle is
a right angle, it is called a right-angle triangle (Fig. 8),
and the side AB opposite the right angle is called its
hypothenuse.
| [FIGURE - A polygon of 3 line segments (A-B, B-C, C-A), each ending at the start of the next segment, B-C and C-A meeting at 90 degrees, creating a closed figure of a right-angle triangle.]
| Fig. 8.
######################################################### p. 065 ###
XIX. A rectangle is a parallelogram whose angles
are all right angles (Fig. 9a), and a square is a rectangle
whose sides are all equal (Fig. 9).
| [FIGURE - A polygon of 4 line segments, each ending at the start of the next segment, and each segment is the same identical lengths and all segments meet at 90 degree angles, creating a square.]
| Fig. 9.
| [FIGURE - A polygon of 4 line segments, each ending at the start of the next segment, and each segment is one of two lengths and all segments meet at 90 degree angles, creating a rectangle.]
| Fig. 9a.
XX. A circle is a curved line, all points of which are
equally distant or equidistant from a fixed point called a
center (Fig. 10).
| [FIGURE - A single line that maintains a distance from a single point in the center of the figure, creating a circle.]
| Fig. 10.
| [FIGURE - Two straight line segments (B-M and C-N) that do not cross, and a third straight line (R-S) crossing over the previous segments at points A and O.]
| Fig. 11.
With these assumptions we may now proceed. Let us look at Fig. 11.
«BM» and «CN» are parallel lines cut by the common
transversal or intersecting line «RS». It is seen at a glance
that the «∡ROM» and «∡BOA», called vertical angles, are
equal; likewise «∡ROM» and
«∡RAN», called exterior interior angles, are equal;
likewise «∡BOA» and «∡RAN», called opposite interior
angles, are equal. These facts are actually
proved by placing one on the other, when they will
coincide exactly. The «∡ROM» and «∡BOR» are
supplementary, as their sum forms the straight line
«BM», or 180°. Likewise «∡ROM» and «∡MOS», or
«∡NAS», are supplementary.
######################################################### p. 066 ###
In general, we have this rule: When the corresponding
sides of any two angles are parallel to each other, the angles
are either equal or supplementary.
| [FIGURE - A triangle of (ABC) with angles of: 'a' (A-B and A-C), 'b' (A-B and B-C), 'c' (A-C and B-C). A dotted line (M-N), parallel to B-C is drawn through point A. Angle 'd' is formed from A-N and A-C. Angle 'e' is formed from the extension of A-B and A-N.]
| Fig. 12.
Triangles. — Let us now investigate some of the properties
of the triangle «ABC» (Fig. 12). Through «A»
draw a line, «MN», parallel to «BC». At a glance we see
that the sum of the angles «𝒂», «𝒅», and «𝒆» is equal to 180°,
or two right angles,—
«∡𝒂 + ∡d + ∡e = 180°»
But «∡𝒄» is equal to «∡𝒅», and «∡𝒃» is equal to «∡𝒆», as
previously seen; therefore we have
«∡𝒂 + ∡𝒄 + ∡𝒃 = 180°»
######################################################### p. 067 ###
This demonstration proves the fact that the sum of
all the inside or interior angles of any triangle is equal
to 180°, or, what is the same thing, two right angles.
Now, if the triangle is a right triangle and one of its
angles is itself a right angle, then the sum of the two
remaining angles must be equal to one right angle, or 90°.
This fact should be most carefully noted, as it is very
important.
When we have two triangles with all the angles of
the one equal to the corresponding angles of the other,
as in Fig. 13, they are called similar triangles.
| [FIGURE - Two tringles of identical angles, but different segment lengths.]
| Fig. 13.
When we have two triangles with all three sides of
the one equal to the corresponding sides of the other,
they are equal to each other (Fig. 14), for they may be
perfectly superposed on each other. In fact, the two
triangles are seen to be equal if two sides and the included
angle of the one are equal to two sides and the
included angle of the other; or, if one side and two
angles of the one are equal to one side and the corresponding
angles respectively of the other; or, if one side
and the angle opposite to it of the one are equal to one
side and the corresponding angle of the other.
| [FIGURE - Two tringles of identical angles, and identical segment lengths.]
| Fig. 14.
######################################################### p. 068 ###
Projections. — The projection of any given tract, such
as «AB» (Fig. I5), upon a line, such as «MN», is that space,
«CD», on the line «MN» bounded by two lines drawn
from «A» and «B» respectively perpendicular to «MN».
| [FIGURE - A segment of M-N is drawn from left to right, and a shorter, non-parallel segment of A-B is drawn above. A dotted line is drawn from point A to a point directly below on M-N, labeled 'C'. A similar dotted line is drawn from B to M-N and labeled 'D'.]
| Fig. 15.
Rectangles and Parallelograms. — A line drawn between
opposite corners of a parallelogram is called a
diagonal; thus, «AC» is a diagonal in Fig. 16. It is along
this diagonal that a body would move if pulled in the
direction of «AB» by one force, and in the direction «AD»
by another, the two forces having the same relative
magnitudes as the relative lengths of «AB» and «AD».
This fact is only mentioned here as illustrative of one
of the principles of mechanics.
| [FIGURE - A parallelogram of ABCD is shown with an additional segment connecting A-C.]
| Fig. 16.
######################################################### p. 069 ###
| [FIGURE - A rectangle of ABCD.]
| Fig. 17.
The area of a rectangle is equal to the product of the
length by the breadth; thus, in Fig. 17,
| Area of «ABDC = AB × AC».
This fact is so patent as not to need explanation.
Suppose we have a parallelogram (Fig. 18), however,
what is its area equal to?
The perpendicular distance «BF» between the sides «BC»
and «AD» of a parallelogram is called its altitude. Extend
the base «AD» and draw «CE» perpendicular to it.
| [FIGURE - A parallelogram of ABCD, with a vertical dotted line drawn from B down to AD, labeled point F, and a dotted line extending AFD. A third dotted line is drawn down from C to the extended line of AFD, creating point E.]
| Fig. 18.
######################################################### p. 070 ###
Now we have the rectangle «BCEF», whose area we know
to be equal to «BC × BF». But the triangles «ABF» and
«DCE» are equal (having 2 sides and 2 angles mutually
equal), and we observe that the rectangle is nothing else
than the parallelogram with the triangle «ABF» chipped
off and the triangle «DCE» added on, and since these are
equal, the rectangle is equal to the parallelogram, which
then has the same area as it; or,
Area of parallelogram «ABCD = BC × BF».
| [FIGURE - A triangle of ABC is shown. Dotted lines a drawn from points A and C to a point D which is situated opposite of A-B and C-B, but with equal lengths. A dotted line from A is drawn to point H on B-C.]
| Fig. 19.
If, now, we consider the area of the triangle «ABC»
(Fig. 19), we see that by drawing the lines «AD» and «CD»
parallel to «BC» and «AB» respectively, we have the parallelogram
«BADC», and we observe that the triangles «ABC»
and «ADC» are equal. Therefore triangle «ABC» equals
one-half of the parallelogram, and since the area of this
is equal to «BC × AH», then the
Area of the triangle «ABC = ½ BC × AH»,
which means that the area of a triangle is equal to
one-half of the product of the base by the altitude.
######################################################### p. 071 ###
Circles. — Comparison between the lengths of the
diameter and circumference of a circle (Fig. 20) made
with the utmost care shows that
the circumference is 3.1416 times as
long as the diameter. This constant,
3.1416, is usually expressed
by the Greek letter pi («π»). Therefore,
the circumference of a circle is
equal to «π ×» the diameter.
>> circum. = «π𝒅» ¶
circum. = «2 π𝒓»
if «𝒓», the radius, is used instead of the diameter.
| [FIGURE - A circle with a segment crossing the center, labeled Diameter, 'd'.]
| Fig. 20.
The area of a segment of a circle (Fig. 21), like the
area of a triangle, is equal to «½» of the product of the
base by the altitude, or «½𝒂 × 𝒓».
This comes from the fact that the
segment may be divided up into a
very large number of small segments
a whose bases, being very small, have
very little curvature, and may therefore
be considered as small triangles. Therefore, if we
consider the whole circle, where the length of the arc
is «2π𝒓», the area is
>> «½ × 2π𝒓 × 𝒓 = π𝒓²», ¶
Area circle« = π𝒓²».
| [FIGURE - A segment of a circle, showing the radius as a dotted line labeled 'r'.]
| Fig. 21.
######################################################### p. 072 ###
I will conclude this chapter by a discussion of one of
the most important properties of the right-angle triangle,
namely, that the square erected on its hypothenuse
is equal to the sum of the squares erected on
its other two sides; that is, that in the triangle «ABC»
(Fig. 22) «\overline{AC}² = \overline{AB}² + \overline{BC}²».
| [FIGURE - Three squares (ABHK, BCRS, ACMN) with dotted lines A-R, C-K, and B-N, B-M, B-F.]
| Fig. 22.
To prove
> «ANMC = BCRS + ABHK»,¶
or
> length «\overline{AC}²» = length «\overline{BC}²» + length «\overline{AB}²».
######################################################### p. 073 ###
This is a difficult problem and one of the most interesting
and historic ones that the whole realm of mathematics
can offer, therefore I will only suggest its solution
and leave a little reasoning for the student himself
to do.
> triangle «ARC» = triangle BMC, ¶
triangle «ARC» = «½CR × BC» ¶
= «½» of the square «BCRS», ¶
triangle «BCM» = «½ CM × CO» ¶
= «½ »of rectangle «COFM». ¶
Therefore
> «½» of square «BCRS» = «½» of rectangle «COFM»,¶
or
>> «BCRS = COFM».
Similarly for the other side
>> «ABHK = AOFN».
But
>> «COFM + AOFN» = whole square «ACMN».
Therefore
>> «ACMN = BCRS + ABHK». ¶
«(AC)² = (AB)² + (BC)²».
######################################################### p. 074 ###
## PROBLEMS ##
> 1. What is the area of a rectangle 8 ft. long by 12
ft. wide?
> 2. What is the area of a triangle whose base is 20 ft.
and whose altitude is 18 ft.?
> 3. What is the area of a circle whose radius is 9 ft.?
> 4. What is the length of the hypothenuse of a right-angle
triangle if the other two sides are respectively
6 ft. and 9 ft.?
> 5. What is the circumference of a circle whose diameter
is 20 ft.?
> 6. The hypothenuse of a right-angle triangle is 25 ft.
and one side is 18 ft.; what is the other side?
> 7. If the area of a circle is 600 sq. ft., what is its
diameter?
> 8. The circumference of the earth is 25,000 miles;
what is its diameter in miles?
> 9. The area of a triangle is 30 sq. ft. and its base is
8 ft.; what is its altitude?
> 10. The area of a parallelogram is 100 sq. feet and
its base is 25 ft.; what is its altitude?
[**NOTE: Small Empty Block]
######################################################### p. 075 ###
=== CHAPTER XI - Elementary Principles of Trigonometry - NONE ===
TRIGONOMETRY is the science of angles; its province is
to teach us how to measure and employ angles with the
same ease that we handle lengths and areas.
| [FIGURE - Two straight line segments (A-B and C-D) crossing over at point O. An angle marked 'a' is highlighted, defined by the divergence of segments O-B and O-D.]
| Fig. 23.
In a previous chapter we have defined an angle as the
opening or the divergence between two intersecting
lines, «AB» and «CD» (Fig. 23). The next question is, How
are we going to measure this angle? We have already
seen that we can do this in one way by employing
degrees, a complete circle being 360°. But there are
many instances which the student will meet later on
where the use of degrees would be meaningless. It
is then that certain constants connected with the angle,
called its functions, must be resorted to. Suppose we
have the angle «𝒂» shown in Fig. 24. Now let us choose
a point anywhere either on the line «AB» or «CD»; for
instance, the point «P». From «P» drop a line which will
be perpendicular to «CD». This gives us a right-angle
triangle whose sides we may call «𝒂», «𝒃» and «𝒄» respectively.
We may now define the following functions of the «∡𝒂»:
######################################################### p. 076 ###
| [FIGURE]
| Fig. 24.
>> sine «∝ = frac{𝒂÷𝒄}», ¶
cosine «∝ = frac{𝒃÷𝒄}», ¶
tangent «∝ = frac{𝒂÷𝒃}»,
which means that the sine of an angle is obtained by
dividing the side opposite to it by the hypothenuse; the
cosine, by dividing the side adjacent to it by the hypothenuse;
and the tangent, by dividing the side opposite
by the side adjacent.
These values, sine, cosine and tangent, are therefore
nothing but ratios,—pure numbers,—and under no circumstances
should be taken for anything else. This is
one of the greatest faults that I have to find with many
texts and handbooks in not insisting on this point.
######################################################### p. 077 ###
Looking at Fig. 24, it is evident that no matter where
I choose «P», the values of the sine, cosine and tangent
will be the same; for if I choose «P» farther out on the
line I will increase «𝒄», but at the same time «𝒂» will increase
in the same proportion, the quotient of «frac{𝒂÷𝒄}» being always
the same wherever «P» may be chosen.
Likewise «frac{𝒃÷𝒄}» and «frac{𝒂÷𝒃}» will always remain constant. The
sine, cosine, and tangent are therefore always fixed and
constant quantities for any given angle. I might have
remarked that if «P» had been chosen on the line «CD» and
the perpendicular drawn to «AB», as shown by the dotted
lines (Fig. 24), the hypothenuse and adjacent side simply
exchange places, but the value of the sine, cosine and
tangent would remain the same.
Since these functions, namely, sine, cosine and tangent,
of any angle remain the same at all times, they
become very convenient handles for employing the
angle. The sines, cosines and tangents of all angles of
every size may be actually measured and computed
with great care once and for all time, and then arranged
in tabulated form, so that by referring to this table one
can immediately find the sine, cosine or tangent of any
angle; or, on the other hand, if a certain value said to
be the sine, cosine or tangent of an unknown angle is
given, the angle that it corresponds to may be found
from the table. Such a table may be found at the end
of this book, giving the sines, cosines and tangents of
all angles taken 6 minutes apart. Some special compilations
of these tables give the values for all angles taken
only one minute apart, and some even closer, say 10 seconds
apart.
######################################################### p. 078 ###
On reference to the table, the sine of 10° is .1736, the
cosine of 10° is .9848, the sine of 24° 36' is .4163, the cosine
of 24° 36' is .9092. In the table of sines and cosines
the decimal point is understood to be before every value,
for, if we refer back to our definition of sine and cosine,
we will see that these values can never be greater than
1; in fact, they will always be less than 1, since the
hypothenuse «𝒄» is always the longest side of the right
angle and therefore «𝒂» and «𝒃» are always less than it.
Obviously, «frac{𝒂÷𝒄}» and «frac{𝒃÷𝒄}», the values respectively of sine and
cosine, being a smaller quantity divided by a larger,
can never be greater than 1. Not so with the tangent;
for angles between o° and 45°, «𝒂» is less than «𝒃»,
therefore «frac{𝒂÷𝒃}» is less than 1; but for angles between 45°
and 90°, «𝒂» is greater than «𝒃», and therefore «frac{𝒂÷𝒃}» is greater
than 1. Thus, on reference to the table the tangent
of 10° 24' is seen to be .1835, the tangent of 45° is 1,
the tangent of 60° 30' is 1.7675.
######################################################### p. 079 ###
Now let us work backwards. Suppose we are given .3437
as the sine of a certain angle, to find the angle.
On reference to the table we find that this is the sine
of 20° 6', therefore this is the angle sought. Again,
suppose we have .8878 as the cosine of an angle, to find
the angle. On reference to the table we find that this
is the angle 27° 24'. Likewise suppose we are given
3.5339 as the tangent of an angle, to find the angle.
The tables show that this is the angle 74° 12'.
When an angle or value which is sought cannot be
found in the tables, we must prorate between the next
higher and lower values. This process is called interpolation,
and is merely a question of proportion. It will
be explained in detail in the chapter on Logarithms.
Relation of Sine and Cosine. — On reference to Fig.
25 we see that the sine «α = frac{𝒂÷𝒄}»
but if we take «β», the other
acute angle of the right-angle
triangle, we see that cosine
«β = frac{𝒂÷𝒄}».
| [FIGURE - Right triangle of lines a, b, c, with the non-90 degree angles labeled with Greek characters 'alpha' and 'beta'.]
| Fig. 25.
Remembering, always the fundamental definition
of sine and cosine, namely,
>> sine = «frac{Opposite side÷Hypothenuse}»,
>> cosine = «frac{Adjacent side÷Hypothenuse}»,
we see that the cosine «β» is equal to the same thing as
the sine «α», therefore
| sine «α» = cosine «β».
######################################################### p. 080 ###
Now, if we refer back to our geometry, we will remember
that the sum of the three angles of a triangle
= 180°, or two right angles, and therefore in a right-angle
triangle «∡α + ∡β = 90°», or 1 right angle. In
other words «∡α» and «∡β» are complementary angles.
We then have the following general law: “The sine of
an angle is equal to the cosine of its complement.” Thus,
if we have a table of sines or cosines from 0° to 90°, or
sines and cosines between 0° and 45°, we make use of
this principle. If we are asked to find the sine of 68°
we may look for the cosine of (90° − 68°), or 22°; or, if
we want the cosine of 68°, we may look for the sine of
(90° − 68°), or 22°.
Other Functions. — There are some other functions
of the angle which are seldom used, but which I will
mention here, namely,
>> Cotangent = «frac{𝒃÷𝒂}»,
>> Secant = «frac{𝒄÷𝒃}»,
>> Cosecant = «frac{𝒄÷𝒂}».
######################################################### p. 081 ###
Other Relations of Sine and Cosine. — We have seen
that the sine «α = frac{𝒂÷𝒄}» and the cosine «α = frac{𝒃÷𝒄}». Also from
geometry
| «𝒂² + 𝒃² = 𝒄²»
< (1)
Dividing equation (1) by «𝒄²» we have
| «frac{𝒂²÷𝒄²}» + «frac{𝒃²÷𝒄²}» = 1
But this is nothing but the square of the sine plus the
square of the cosine of «∡α», therefore
| «(»sine «α)² + (» cosine «α)² = 1».
Other relations whose proof is too intricate to enter
into now are
>> sine «2 α = 2\ \sin α\ \cos\ α», ¶
cos «2 α = 1 − 2\ \sin² α», ¶
or cos «2 α = cos² α − \sin² α».
| [FIGURE - Line drawing of a river with an overlayed right triangle of points A, B, C and lines a, b, and c.]
| Fig. 26.
Use of Trigonometry. — Trigonometry is invaluable
in triangulation of all kinds. When two sides or one
side and an acute angle of a right-angle triangle are
given, the other two sides can be easily found. Suppose
we wish to measure the distance «BC» across the
river in Fig. 26; we proceed as follows: First we lay off
and measure the distance «AB» along the shore; then by
means of a transit we sight perpendicularly across the
river and erect a flag at «C»; then we sight from «A» to «B»
and from «A» to «C» and determine the angle «α». Now, as
before seen, we know that
| «tangent\ α = frac{𝒂÷𝒃}».
######################################################### p. 082 ###
Suppose «𝒃» had been 1000 ft. and «∡α» was 40°, then
| «tangent\ 40° = frac{𝒂÷1000}».
The tables show that the tangent of 40° is .8391;
then «.8391 = frac{𝒂÷1000}»,
therefore «𝒂 = 839.1 ft».
Thus we have found the distance across the river to be
839.1 ft.
| [FIGURE - A right triangle of sides a, b, c, an angle 'alpha' noted at 36 degrees, and c labeled as 300 FT.]
| Fig. 27.
Likewise in Fig. 27, suppose «𝒄» = 300 and «∡α = 36°»,
to find «𝒂» and «𝒃». We have
| «sine\ α = frac{𝒂÷𝒄}»,
or
| «sine\ 36° = frac{𝒂÷300}».
######################################################### p. 083 ###
From the tables «sine\ 36° = .5878».
>> «.5878 = frac{𝒂÷300}»
>> «𝒂 = .5878 × 300»,
or
>> «𝒂 = 176.34 ft».
Likewise
>> «cosine\ α = frac{𝒃÷𝒄}».
From table,
>> «cosine\ 36' = .8090»,
therefore
>> «.8090 = frac{𝒃÷300}»,
or
>> «𝒃 = 242.7 ft».
Now, if we had been told that «𝒂 = 225» and «𝒃 = 100», to
find «∡α» and «𝒄», we would have proceeded thus:
>> «tangent\ α = frac{𝒂÷𝒃}».
Therefore
>> «tangent\ α = frac{225÷100}»,
>> «tangent\ α = 2.25» ft.
The tables show that this corresponds to the angle «66° 4'».
Therefore
> «𝒂 = 66° 4'».
Now to find «𝒄» we have
| «sin\ 𝒂 = frac{𝒂÷𝒄}»,
| «sin\ 66° 4' = frac{255÷𝒄}».
From tables, «sine\ 66° 4' = .9140»,
therefore
> «.9140 = frac{255÷𝒄}»,
or
> «𝒄 = frac{255÷.9140} = 248.5 ft».
And thus we may proceed, the use of a little judgment
being all that is necessary to the solution of the most
difficult problems of triangulation.
######################################################### p. 084 ###
## PROBLEMS ##
> 1. Find the sine, cosine and tangent of 32° 20'.
> 2. Find the sine, cosine and tangent of 81° 24'.
> 3. What angle is it whose sine is .4320?
> 4. What angle is it whose cosine is .1836?
> 5. What angle is it whose tangent is .753?
> 6. What angle is it whose cosine is .8755?
In a right-angle triangle—
> 7. If «𝒂» = 300 ft. and «∡α» = 30°, what are «𝒄» and «𝒃»?
> 8. If «𝒂» = 500 ft. and «𝒃» = 315 ft., what are «∡α» and «𝒄»?
> 9. If «𝒄» = 1250 ft. and «∡α» = 80°, what are «𝒃» and «𝒂»?
> 10. If «𝒃» = 250 ft. and «𝒄» = 530 ft., what are «∡α»
and «𝒂»?
[**NOTE: Small Empty Block]
######################################################### p. 085 ###
=== CHAPTER XII - Logarithms - NONE ===
I HAVE inserted this chapter on logarithms because
I consider a knowledge of them very essential to the
education of any engineer.
Definition. — A logarithm is the power to which we
must raise a given base to produce a given number.
Thus, suppose we choose 10 as our base, we will say
that 2 is the logarithm of 100, because we must raise 10 to
the second power—in other words, square it—in order
to produce 100. Likewise 3 is the logarithm of 1000,
for we have to raise 10 to the third power (thus, «10³») to
produce 1000. The logarithm of 10,000 would then be
4, and the logarithm of 100,000 would be 5, and so on.
The base of the universally used Common System of
logarithms is 10; of the Napierian or Natural System, «𝒆»
or 2.7. The latter is seldom used.
######################################################### p. 086 ###
We see that the logarithms of such numbers as 100,
1000, 10,000, etc., are easily detected; but suppose we
have a number such as 300, then the difficulty of finding
its logarithm is apparent. We have seen that «10²»
is 100, and «10³» equals 1000, therefore the number 300,
which lies between 100 and 1000, must have a logarithm
which lies between the logarithms of 100 and 1000,
namely 2 and 3 respectively. Reference to a table of
logarithms at the end of this book, which we will explain
later, shows that the logarithm of 300 is 2.4771,
which means that 10 raised to the 2.4771ths power will
give 300. The whole number in a logarithm, for example
the 2 in the above case, is called the characteristic;
the decimal part of the logarithm, namely, 4771, is
called the mantissa. It will be hard for the student to
understand at first what is meant by raising 10 to a
fractional part of a power, but he should not worry
about this at the present time; as he studies more
deeply into mathematics the notion will dawn on him
more clearly.
We now see that every number has a logarithm, no
matter how large or how small it may be; every number
can be produced by raising 10 to some power, and this
power is what we call the logarithm of the number.
Mathematicians have carefully worked out and tabulated
the logarithm of every number, and by reference
to these tables we can find the logarithm corresponding
to any number, or vice versa. A short table of logarithms
is shown at the end of this book.
Now take the number 351.1400; we find its logarithm
is 2.545,479. Like all numbers which lie between 100
and 1000 its characteristic is 2. The numbers which
lie between 1000 and 10,000 have 3 as a characteristic;
between 10 and 100, 1 as a characteristic. We therefore
have the rule that the characteristic is always one
less than the number of places to the left of the decimal
point. Thus, if we have the number 31875.12, we
immediately see that the characteristic of its logarithm
will be 4, because there are five places to the left of the
decimal point. Since it is so easy to detect the characteristic,
it is never put in logarithmic tables, the mantissa
or decimal part being the only part that the tables
need include.
######################################################### p. 087 ###
If one looked in a table for a logarithm of 125.60, he
would only find .09,899. This is only the mantissa of
the logarithm, and he would himself have to insert the
characteristic, which, being one less than the number of
places to the left of the decimal point, would in this
case be 2; therefore the logarithm of 125.6 is 2.09,899.
Furthermore, the mantissæ of the logarithms of
3.4546, 34.546, 345.46, 3454.6, etc., are all exactly the
same. The characteristic of the logarithm is the only
thing which the decimal point changes, thus:
>> log 3.4546 = 0.538,398, ¶
log 34.546 = 1.538,398, ¶
log 345.46 = 2.538,398, ¶
log 3454.6 = 3.538,398, ¶
etc.
Therefore, in looking for the logarithm of a number,
first put down the characteristic on the basis of the
above rules, then look for the mantissa in a table,
neglecting the position of the decimal point altogether.
Thus, if we are looking for the logarithm of .9840, we
first write down the characteristic, which in this case
would be −1 (there are no places to the left of the
decimal point in this case, therefore one less than none
is −1). Now look in a table of logarithms for the
mantissa which corresponds to .9840, and we find this
to be .993,083; therefore
| log .9840 = −1.993,083.
If the number had been 98.40 the logarithm would have
been +1.993,083.
######################################################### p. 088 ###
When we have such a number as .084, the characteristic
of its logarithm would be −2, there being one
less than no places at all to the left of its decimal point;
for, even if the decimal point were moved to the right
one place, you would still have no places to the left of
the decimal point; therefore
>> log .00,386 = −3.586,587, ¶
log 38.6 = 1.586,587, ¶
log 386 = 2.586,587, ¶
log 386,000 = 5.586,587.
Interpolation. — Suppose we are asked to find the
logarithm of 2468; immediately write down 3 as the
characteristic. Now, on reference to the logarithmic
table at the end of this book, we see that the logarithms
of 2460 and 2470 are given, but not 2468. Thus:
>> log 2460 = 3.3909, ¶
log 2468 = ? ¶
log 2470 = 3.3927.
######################################################### p. 089 ###
We find that the total difference between the two
given logarithms, namely 3909 and 3927, is 16, the
total difference between the numbers corresponding to
these logarithms is 10, the difference between 2460 and
2468 is 8; therefore the logarithm to be found lies «frac{8÷10}»
of the distance across the bridge between the two given
logarithms 3909 and 3927. The whole distance across
is 16. «frac{8÷10}» of 16 is 12.8. Adding this to 3909 we have
3921.8; therefore
| log of 2468 = 3.39,218.
Reference to column 8 in the interpolation columns to the
right of the table would have given this value at once.
Many elaborate tables of logarithms may be purchased
at small cost which make interpolation almost unnecessary
for practical purposes.
Now let us work backwards and find the number if
we know its logarithm. Suppose we have given the
logarithm 3.6201. Referring to our table, we see that
the mantissa .6201 corresponds to the number 417; the
characteristic 3 tells us that there must be four places
to the left of the decimal point; therefore
| 3.6201 is the log of 4170.0.
######################################################### p. 090 ###
Now, for interpolation we have the same principles
aforesaid. Let us find the number whose log is −3.7304.
In the table we find that
> log 7300 corresponds to the number 5370, ¶
log 7304 corresponds to the number ? ¶
log 7308 corresponds to the number 5380.
Therefore it is evident that
> 7304 corresponds to 5375.
Now the characteristic of our logarithm is −3; from
this we know that there must be two zeros to the left
of the decimal point; therefore
| −3.7304 is the log of the number .005375.
Likewise
> −2.7304 is the log of the number .05375, ¶
−7304 is the log of the number 5.375, ¶
4.7304 is the log of the number 53,750.
Use of the Logarithm. — Having thoroughly understood
the nature and meaning of a logarithm, let us
investigate its use mathematically. It changes multiplication
and division into addition and subtraction; involution
and evolution into multiplication and division.
We have seen in algebra that
>> «𝒂² × 𝒂⁵ = 𝒂⁵⁺²», or «𝒂⁷»,
and that
>> «frac{𝒂⁸÷𝒂³} = 𝒂⁸⁻³», or «𝒂⁵».
######################################################### p. 091 ###
In other words, multiplication or division of like symbols
was accomplished by adding or subtracting their exponents,
as the case may be. Again, we have seen that
| «(𝒂²)² = 𝒂⁴»,
or
| «∛[𝒂⁶] = 𝒂²».
In the first case «𝒂²» squared gives «𝒂⁴», and in the second
case the cube root of «𝒂⁶» is «𝒂²»; to raise a number to a
power you multiply its exponent by that power; to find
any root of it you divide its exponent by the exponent
of the root. Now, then, suppose we multiply 336 by
5380; we find that
| log of «336 = 10^{2.5263}», ¶
log of «5380 = 10^{3.7308}».
Then «336 × 5380» is the same thing as «10^{2.5263} × 10^{3.7308}»,
> But «10^{2.5263} × 10^{3.7308} = 10^{2.526310 + 3.7308} = 10^{6.2571}».
We have simply added the exponents, remembering that
these exponents are nothing but the logarithms of 336
and 5380 respectively.
Well, now, what number is «10^{6.2571}» equal to? Looking
in a table of logarithms we see that the mantissa .2571
corresponds to 1808; the characteristic 6 tells us
that there must be seven places to the left of the decimal;
therefore
> «10^{6.2571} =» 1,808,000.
######################################################### p. 092 ###
If the student notes carefully the foregoing he will see
that in order to multiply 336 by 5380 we simply find
their logarithms, add them together, getting another
logarithm, and then find the number corresponding
to this logarithm. Any numbers may be multiplied
together in this simple manner; thus, if we multiply
«217 × 4876 × 3.185 × .0438 × 890», we have
> log 217 = 2.3365 ¶
log 4876 = 3.6880 ¶
log 3.185 = .5031 ¶
log .0438 = −2.6415 [*] ¶
log 890 = 2.9494 ¶
------ ¶
Adding we get
>> 8.1185¶
[*] The −2 does not carry its negativity to the mantissa.
We must now find the number corresponding to the
logarithm 8.1185. Our tables show us that
| 8.1185 is the log of 131,380,000.
Therefore 131,380,000 is the result of the above multiplication.
To divide one number by another we subtract the
logarithm of the latter from the logarithm of the
former; thus, «3865 ÷ 735»:
>> log 3865 = 3.5872 ¶
log 735 = 2.8663 ¶
______ ¶
.7209
The tables show that .7209 is the logarithm of 5.259;
therefore
| «3865 ÷ 735 = 5.259».
######################################################### p. 093 ###
As explained above, if we wish to square a number, we
simply multiply its logarithm by 2 and then find what
number the result is the logarithm of. If we had
wished to raise it to the third, fourth or higher power,
we would simply have multiplied by 3, 4 or higher power,
as the case may be. Thus, suppose we wish to cube
9879; we have
> log 9897 = 3.9947 ¶
3 ¶
_____ ¶
11.9841 ¶
| 11.9841 is the log of 964,000,000,000;
| therefore 9879 cubed = 964,000,000,000.
Likewise, if we wish to find the square root, the cube
root, or fourth root or any root of a number, we simply
divide its logarithm by 2, 3, 4 or whatever the root may
be; thus, suppose we wish to find the square root of
36,850, we have
| log 36,850 = 4.5664. ¶
4.5664 ÷ 2 = 2.2832.
2.2832 is the log. of 191.98; therefore the square root of
36,850 is 191.98.
The student should go over this chapter very carefully,
so as to become thoroughly familiar with the
principles involved.
######################################################### p. 094 ###
## PROBLEMS ##
> 1. Find the logarithm of 3872.
> 2. Find the logarithm of 73.56.
> 3. Find the logarithm of .00988.
> 4. Find the logarithm of 41,267.
> 5. Find the number whose logarithm is 2.8236.
> 6. Find the number whose logarithm is 4.87175.
> 7. Find the number whose logarithm is −1.4385.
> 8. Find the number whose logarithm is −4.3821.
> 9. Find the number whose logarithm is 3.36175.
> 10. Multiply 2261 by 4335.
> 11. Multiply 6218 by 3998.
> 12. Multiply 231.9 by 478.8 by 7613 by .921.
> 13. Multiply .00983 by .0291.
> 14. Multiply .222 by .00054.
> 15. Divide 27,683 by 856.
> 16. Divide 4337 by 38.88.
> 17. Divide .9286 by 28.75.
> 18. Divide .0428 by 1.136.
> 19. Divide 3995 by .003,337.
> 20. Find the square of 4291.
> 21. Raise 22.91 to the fourth power.
> 22. Raise .0236 to the third power.
> 23. Find the square root of 302,060.
> 24. Find the cube root of 77.85.
> 25. Find the square root of .087,64.
> 26. Find the fifth root of 226,170,000.
[**NOTE: Small Empty Block]
######################################################### p. 095 ###
=== CHAPTER XIII - Elementary Principles of Coördinate Geometry - NONE ===
COÖRDINATE Geometry may be called graphic algebra,
or equation drawing, in that it depicts algebraic equations
not by means of symbols and terms but by means
of curves and lines. Nothing is more familiar to the
engineer, or in fact to any one, than to see the results of
machine tests or statistics and data of any kind shown
graphically by means of curves. The same analogy
exists between an algebraic equation and the curve
which graphically represents it as between the verbal
description of a landscape and its actual photograph;
the photograph tells at a glance more than could be
said in many thousands of words. Therefore the student
will realize how important it is that he master the
few fundamental principles of coördinate geometry which
we will discuss briefly in this chapter.
An Equation. — When discussing equations we remember
that where we have an equation which contains
two unknown quantities, if we assign some numerical
value to one of them we may immediately find the corresponding
numerical value of the other; for example, take
the equation
| «𝒙 = 𝒚 + 4».
######################################################### p. 096 ###
In this equation we have two unknown quantities,
namely, «𝒙» and «𝒚»; we cannot find the value of either
unless we know the value of the other. Let us say that
«𝒚 = 1»; then we see that we would get a corresponding
value, «𝒙 = 5»; for «𝒚 = 2», «𝒙 = 6»; thus:
>> If «𝒚 = 1», then «𝒙 = 5», ¶
«𝒚 = 2», «𝒙 = 6», ¶
«𝒚 = 3», «𝒙 = 7», ¶
«𝒚 = 4», «𝒙 = 8», ¶
«𝒚 = 5», «𝒙 = 9», etc.
The equation then represents the relation in value
existing between «𝒙» and «𝒚», and for any specific value of
«𝒙» we can find the corresponding specific value of «𝒚».
Instead of writing down, as above, a list of such corresponding
values, we may show them graphically thus:
Draw two lines perpendicular to each other; make one
of them the «𝒙» line and the other the «𝒚» line. These two
lines are called axes. Now draw parallel to these axes
equi-spaced lines forming cross-sections, as shown in
Fig. 28, and letter the intersections of these lines with
the axes 1, 2, 3, 4, 5, 6, etc., as shown.
######################################################### p. 097 ###
Now let us plot the corresponding values, «𝒚 = 1»,
«𝒙 = 5». This will be a point 1 space up on the «𝒚» axis
and 5 spaces out on the «𝒙» axis, and is denoted by letter
«A» in the figure. In plotting the corresponding values
«𝒚 = 2», «𝒙 = 6», we get the point «B»; the next set of values
gives us the point «C», the next «D», and so on. Suppose
we draw a line through these points; this line, called
the curve of the equation, tells everything in a graphical
way that the equation does algebraically. If this line
has been drawn accurately we can from it find out at
a glance what value of «𝒚» corresponds to any given value
of «𝒙», and vice versa. For example, suppose we wish to
see what value of «𝒚» corresponds to the value «𝒙 = 6½»;
we run our eyes along the 𝒙 axis until we come to «6½»,
then up until we strike the curve, then back upon the
𝒚 axis, where we note that «𝒚 = 2½».
| [FIGURE - A coordinate grid showing x and y axes. A line drawn through 3rd, 4th and 1st quadrants with points A thru J.]
| Fig. 28.
######################################################### p. 098 ###
Negative Values of 𝒙 and 𝒚. — When we started at o
and counted 1, 2, 3, 4, etc., to the right along the 𝒙
axis, we might just as well have counted to the left,
−1, −2, −3, −4, etc. (Fig. 28), and likewise we
might have counted downwards along the 𝒚 axis, −1,
−2, −3, −4, etc. The values, then, to the left of o
on the 𝒙 axis and below o on the 𝒚 axis are the negative
values of 𝒙 and 𝒚. Still using the equation 𝒙 = 𝒚 + 4,
let us give the following values to 𝒚 and note the corresponding
values of 𝒙 in the equation 𝒙 = 𝒚 + 4:
>> If «𝒚 = 0», then «𝒙 = 4», ¶
«𝒚 = −1», «𝒙 = 3», ¶
«𝒚 = −2», «𝒙 = 2», ¶
«𝒚 = −3», «𝒙 = 1», ¶
«𝒚 = −4», «𝒙 = 0», ¶
«𝒚 = −5», «𝒙 = −1», ¶
«𝒚 = −6», «𝒙 = −2», ¶
«𝒚 = −7», «𝒙 = −3».
The point «𝒚 = 0, 𝒙 = 4» is seen to be on the «𝒙» axis at
the point 4. The point «𝒚 = −1, 𝒙 = 3» is at point «E»,
that is, 1 below the «𝒙» axis and 3 to the right of the «𝒚»
axis. The points «𝒚 = −2, 𝒙 = 2» and «𝒚 = −3, 𝒙 = 1»
are seen to be respectively points «F» and «G». Point
«𝒚 = −4, 𝒙 = 0» is zero along the «𝒙» axis, and is therefore
at −4 on the «𝒚» axis. Point «𝒚 = −5, 𝒙 = −1»
is seen to be 5 below 0 on the «𝒚» axis and 1 to
the left of 0 along the «𝒙» axis (both «𝒙» and «𝒚» are now
negative), namely, at the point «H». Point
«𝒚 = −6, 𝒙 = −2» is at «J», and so on.
######################################################### p. 099 ###
The student will note that all points in the first
quadrant have positive values for both «𝒙» and «𝒚», all
points in the second quadrant have positive values for
«𝒚» (being all above 0 so far as the «𝒚» axis is concerned),
but negative values for «𝒙» (being to the left of 0), all
points in the third quadrant have negative values
for both «𝒙» and «𝒚», while all points in the fourth quadrant
have positive values of «𝒙» and negative values of «𝒚».
Coördinates. — The corresponding «𝒙» and «𝒚» values of
a point are called its coördinates, the vertical or «𝒚» value
is called its ordinate, while the horizontal or «𝒙» value is
called the abscissa; thus at point «A», «𝒙 = 5, 𝒚 = 1», here
5 is called the abscissa, while 1 is called the ordinate of
point «A». Likewise at point «G», where «𝒚 = −3, 𝒙 = 1»,
here −3 is the ordinate and 1 the abscissa of «G».
Straight Lines. — The student has no doubt observed
that all points plotted in the equation «𝒙 = 𝒚 + 4» have
fallen on a straight line, and this will always be the case
where both of the unknowns (in this case «𝒙» and «𝒚») enter
the equation only in the first power; but the line will
not be a straight one if either «𝒙» or «𝒚» or both of them
enter the equation as a square or as a higher power; thus,
«𝒙² = 𝒚 + 4» will not plot out a straight line because we
have «𝒙²» in the equation. Whenever both of the unknowns
in the equation which we happen to be plotting
(be they «𝒙» and «𝒚», «𝒂» and «𝒃», «𝒙» and «𝒂», etc.) enter the
equation in the first power, the equation is called a
linear equation, and it will always plot a straight line;
thus, «3𝒙 + 5𝒚 = 20» is a linear equation, and if plotted
will give a straight line.
######################################################### p. 100 ###
Conic Sections. — If either or both of the unknown
quantities enter into the equation in the second power,
and no higher power, the equation will always represent
one of the following curves: a circle or an ellipse, a
parabola or an hyperbola. These curves are called the
conic sections. A typical equation of a circle is «𝒙² + 𝒙²»
= «r²»; a typical equation of a parabola is «𝒚² = 4q𝒙»; a
typical equation of a hyperbola is «𝒙² − 𝒚² = r²», or,
also, «𝒙𝒚 = 𝒄²».
It is noted in every one of these equations that we
have the second power of «𝒙» or «𝒚», except in the equation
«𝒙𝒚 = 𝒄²», one of the equations of the hyperbola. In this
equation, however, although both «𝒙» and «𝒚» are in the
first power, they are multiplied by each other, which
practically makes a second power.
I have said that any equation containing «𝒙» or «𝒚» in
the second power, and in no higher power, represents
one of the curves of the conic sections whose type forms
we have just given. But sometimes the equations do
not correspond to these types exactly and require some
manipulation to bring them into the type form.
######################################################### p. 101 ###
Let us take the equation of a circle, namely,
«𝒙² + 𝒚² = 5²», and plot it as shown in Fig. 29.
| [FIGURE - A circle centered on point 'O' with the circle's formula noted as x-squared + y-squared equals five-squared.]
| Fig. 29.
We see that it is a circle with its center at the intersection
of the coördinate axes. Now take the equation
«(𝒙 − 2)² + (𝒚 − 3)² = 5²». Plotting this, Fig. 30,
we see that it is the same circle with its center at the
point whose coördinates are 2 and 3. This equation
and the first equation of the circle are identical in form,
but frequently it is difficult at a glance to discover this
identity, therefore much ingenuity is frequently required
in detecting same.
| [FIGURE - A circle centered on point x=2, y=3. with the circle's formula noted as (x minus 2)-squared + (y minus 3)-squared equals twenty-five.]
| Fig. 30.
######################################################### p. 102 ###
In plotting the equation of a hyperbola, «𝒙𝒚 = 25»
(Fig. 31), we recognize this as a curve which is met
with very frequently in engineering practice, and a
knowledge of its general laws is of great value.
Similarly, in plotting a parabola (Fig. 32), «𝒚² = 4𝒙»,
we see another familiar curve.
In this brief chapter we can only call attention to the
conic sections, as their study is of academic more than
of pure engineering interest. However, as the student
progresses in his knowledge of mathematics, I would
suggest that he take up the subject in detail as one
which will offer much fascination.
######################################################### p. 103 ###
| [FIGURE - Figure of a coordinate plane with axes x and y, and a line showing the values of x times y equals twenty five.]
| Fig. 31.
| [FIGURE - Figure of a coordinate plane with axes x and y, and a line showing the values of y-squared equals four times x.]
| Fig. 32.
######################################################### p. 104 ###
Other Curves. — All other equations containing unknown
quantities which enter in higher powers than the
second power, represent a large variety of curves called
cubic curves.
The student may find the curve corresponding to
engineering laws whose equations he will hereafter
study. The main point of the whole discussion of this
chapter is to teach him the methods of plotting, and if
successful in this one point, this is as far as we shall go
at the present time.
Intersection of Curves and Straight Lines. — When
studying simultaneous equations we saw that if we had
two equations showing the relation between two unknown
quantities, such for instance as the equations
| «𝒙 + 𝒚 = 7», ¶
«𝒙 − 𝒚 = 3».
we could eliminate one of the unknown quantities in
these equations and obtain the values of «𝒙» and «𝒚» which
will satisfy both equations; thus, in the above equations,
eliminating «𝒚», we have
| «2𝒙 = 10», ¶
«𝒙 = 5».
######################################################### p. 105 ###
Substituting this value of 𝒙 in one of the equations, we
have
| «𝒚 = 2».
Now each one of the above equations represents a
straight line, and each line can be plotted as shown in
Fig. 33.
| [FIGURE - Figure of a coordinate plane with axes x and y, and one line showing the values of x minus y equals 3 and a second line showing x plus y equals 7.]
| Fig. 33.
Their point of intersection is obviously a point on
both lines. The coördinates of this point, then, «𝒙 = 5»
and «𝒚 = 2», should satisfy both equations, and we have
already seen this. Therefore, in general, where we
have two equations each showing a relation in value
between the two unknown quantities, 𝒙 and 𝒚, by combining
these equations, namely, eliminating one of the
unknown quantities and solving for the other, our
result will be the point or points of intersection of both
curves represented by the equations. Thus, if we add
the equations of two circles,
| «𝒙² + 𝒚² = 4²», ¶
«(𝒙 − 2)² + 𝒚² = 5²»,
and if the student plots these equations separately and
then combines them, eliminating one of the unknown
quantities and solving for the other, his results will be
the points of intersection of both curves.
######################################################### p. 106 ###
Plotting of Data. — When plotting mathematically
with absolute accuracy the curve of an equation, whatever
scale we use along one axis we must employ along
the other axis. But, for practical results in plotting
curves which show the relative values of several varying
quantities during a test or which show the operation
of machines under certain conditions, we depart
from mathematical accuracy in the curve for the sake
of convenience and choose such scales of value along
each axis as we may deem appropriate. Thus, suppose
we were plotting the characteristic curve of a shunt
dynamo which had given the following sets of values
from no load to full load operation:
######################################################### p. 107 ###
[**NOTE: START TABLE WITH TABS]
VOLTS AMPERES
122 0
120 5
118 10
116 15
114 19
111 22
107 25
[**NOTE: END TABLE WITH TABS]
| [FIGURE - A graph of the values from the table above.]
| Fig. 34.
We plot this curve for convenience in a manner as
shown in Fig. 34. Along the volts axis we choose a
scale which is compressed to within one-half of the
space that we choose for the amperes along the ampere
axis. However, we might have chosen this entirely at
our own discretion and the curve would have had the
same significance to an engineer.
######################################################### p. 108 ###
## PROBLEMS ##
Plot the curves and lines corresponding to the following
equations:
> 1. «𝒙 = 3𝒚 + 10».
> 2. «2𝒙 + 5𝒚 = 15».
> 3. «𝒙 − 2𝒚 = 4».
> 4. «10𝒚 + 3𝒙 = −8».
> 5. «𝒙² + 𝒚² = 36».
> 6. «𝒙² = 16𝒚».
> 7. «𝒙² − 𝒚² = 16».
> 8. «3𝒙² + (𝒚 − 2)² = 25».
Find the intersections of the following curves and
lines:
> 1. «3𝒙 + 𝒚 = 10», ¶
«4𝒙 − 𝒚 = 6».
> 2. «𝒙² + 𝒚² = 81», ¶
«𝒙 − 𝒚 = 10».
> 3. «𝒙𝒚 = 40», ¶
«3𝒙 + 𝒚 = 5».
######################################################### p. 109 ###
Plot the following volt-ampere curve:
[**NOTE: START TABLE WITH TABS]
VOLTS AMPERES
550 0
548 20
545 39
541 55
536 79
529 91
521 102
510 115
[**NOTE: END TABLE WITH TABS]
[**NOTE: Small Empty Block]
######################################################### p. 110 ###
=== CHAPTER XIV - Elementary Principles of the Calculus - NONE ===
It is not my aim in this short chapter to do more
than point out and explain a few of the fundamental
ideas of the calculus which may be of value to a
practical working knowledge of engineering. To the
advanced student no study can offer more intellectual
and to some extent practical interest than the advanced
theories of calculus, but it must be admitted
that very little beyond the fundamental principles ever
enter into the work of the practical engineer.
In a general sense the study of calculus covers an investigation
into the innermost properties of variable quantities,
that is quantities which have variable values as
against those which have absolutely constant, perpetual
and absolutely fixed values. (In previous chapters we
have seen what was meant by a constant quantity and
what was meant by a variable quantity in an equation.)
By the innermost properties of a variable quantity we mean
finding out in the minutest detail just how this quantity
originated; what infinitesimal (that is, exceedingly small)
parts go to make it up; how it increases or diminishes
with reference to other quantities; what its rate of
increasing or diminishing may be; what its greatest
and least values are; what is the smallest particle into
which it may be divided; and what is the result of
adding all of the smallest particles together. All of
the processes of the calculus therefore are either analysis
or synthesis, that is, either tearing up a quantity
into its smallest parts or building up and adding
together these smallest parts to make the quantity.
We call the analysis, or tearing apart, differentiation;
we call the synthesis, or building up, integration.
######################################################### p. 111 ###
#### DIFFERENTIATION ####
Suppose we take the straight line (Fig. 35) of length
«𝒙». If we divide it into a large number of parts, greater
than a million or a billion or any number of which we
have any conception, we say that each part is infinitesimally
small,—that is, it is small beyond conceivable
length. We represent such inconceivably small lengths
by an expression «Δ𝒙» or «δ𝒙». Likewise, if we have a
surface and divide it into infinitely small parts, and if
we call «𝒂» the area of the surface, the small infinitesimal
portion of that surface we represent by «Δ𝒂» or «δ𝒂».
These quantities, namely, «δ𝒙» and «δ𝒂», are called the
differential of «𝒙» and «𝒂» respectively.
| [FIGURE - A line segment showing minuscule divisions of the segment.]
| Fig. 35.
######################################################### p. 112 ###
| [FIGURE - A square with inset lines showing the minuscule segments of the sides.]
| Fig. 36.
We have seen that the differential of a line of the
length «𝒙» is «δ𝒙». Now suppose we have a square each of
whose sides is «𝒙», as shown in Fig. 36. The area of that
square is then «𝒙²». Suppose now we increase the length
of each side by an infinitesimally small amount, «δ𝒙»,
making the length of each side «𝒙 + δ𝒙». If we complete
a square with this new length as its side, the new square
will obviously be larger than the old square by a very
small amount. The actual area of the new square will
be equal to the area of the old square + the additions
to it. The area of the old square was equal to «𝒙²».
The addition consists of two fine strips each «𝒙» long by
«δ𝒙» wide and a small square having «δ𝒙» as the length of
its side. The area of the addition then is
| «(𝒙 × δ𝒙) + (𝒙 × δ𝒙) + (δ𝒙 × δ𝒙)» = additional area.
(The student should note this very carefully.) Therefore
the addition equals
| «2𝒙δ𝒙 + (δ𝒙)²» = additional area.
######################################################### p. 113 ###
Now the smaller «δ𝒙» becomes, the smaller in more rapid
proportion does «δ𝒙²», which is the area of the small
square, become. Likewise the smaller «δ𝒙» is, the thinner
do the strips whose areas are «𝒙δ𝒙» become; but the strips
do not diminish in value as fast as the small square
diminishes, and, in fact, the small square vanishes so
rapidly in comparison with the strips that even when
the strips are of appreciable size the area of the small
square is inappreciable, and we may say practically that
by increasing the length of the side «𝒙» of the square
shown in Fig. 36 by the length «δ𝒙» we increase its area
by the quantity «2𝒙δ𝒙».
Again, if we reduce the side «𝒙» of the square by the
length «δ𝒙», we reduce the area of the square by the
amount «2𝒙δ𝒙». This infinitesimal quantity, out of a
very large number of which the square consists or may
be considered as made up of, is equal to the differential of
the square, namely, the differential of «𝒙²». We thus see
that the differential of the quantity «𝒙²» is equal to «2𝒙δ𝒙».
Likewise, if we had considered the case of a cube instead
of a square, we would have found that the differential of
the cube «𝒙³» would have been «3𝒙²δ𝒙». Likewise, by more
elaborate investigations we find that the differential of
«𝒙⁴ = 4𝒙³δ𝒙». Summarizing, then, the foregoing results
we have
>> differential of «𝒙 = δ𝒙», ¶
differential of «𝒙² = 2𝒙δ𝒙», ¶
differential of «𝒙³ = 3𝒙²δ𝒙», ¶
differential of «𝒙⁴ = 4𝒙³δ𝒙».
######################################################### p. 114 ###
From these we see that there is a very simple and definite
law by which we can at once find the differential of
any power of «𝒙».
Law. — Reduce the power of «𝒙» by one, multiply by
«δ𝒙» and place before the whole a coefficient which is the
same number as the power of «𝒙» which we are differentiating;
thus, if we differentiate «𝒙⁵» we get «5𝒙⁴δ𝒙»; also,
if we differentiate «𝒙⁶» we get «6𝒙⁵δ𝒙».
I will repeat here that it is necessary for the student
to get a clear conception of what is meant by differentiation;
and I also repeat that in differentiating any
quantity our object is to find out and get the value of
the very small parts of which it is constructed (the rate
of growth). Thus we have seen that a line is constructed
of small lengths «δ𝒙» all placed together; that a
square grows or evolves by placing fine strips one next
the other; that a cube is built up of thin surfaces placed
one over the other; and so on.
Differentiation Similar to Acceleration. — We have just
said that finding the value of the differential, or one of
the smallest particles whose gradual addition to a quantity
makes the quantity, is the same as finding out the
rate of growth, and this is what we understood by the
ordinary term acceleration. Now we can begin to see
concretely just what we are aiming at in the term
differential. The student should stop right here, think
over all that has gone before and weigh each word of
what we are saying with extreme care, for if he understands
that the differentiation of a quantity gives us
the rate of growth or acceleration of that quantity he
has mastered the most important idea, in fact the keynote
idea of all the calculus; I repeat, the keynote idea.
Before going further let us stop for a little illustration.
######################################################### p. 115 ###
Example. — If a train is running at a constant speed
of ten miles an hour, the speed is constant, unvarying
and therefore has no rate of change, since it does not
change at all. If we call «𝒙» the speed of the train, therefore
«𝒙» would be a constant quantity, and if we put it in an
equation it would have a constant value and be called a
constant. In algebra we have seen that we do not
usually designate a constant or known quantity by the
symbol «𝒙», but rather by the symbols «𝒂», «k», etc.
Now on the other hand suppose the speed of the train
was changing; say in the first hour it made ten miles, in
the second hour eleven miles, in the third hour twelve
miles, in the fourth hour thirteen miles, etc. It is evident
that the speed is increasing one mile per hour each
hour. This increase of speed we have always called the
acceleration or rate of growth of the speed. Now if we
designated the speed of the train by the symbol «𝒙», we
see that «𝒙» would be a variable quantity and would have
a different value for every hour, every minute, every
second, every instant that the train was running. The
speed «𝒙» would constantly at every instant have added
to it a little more speed, namely «δ𝒙», and if we can find
the value of this small quantity «δ𝒙» for each instant of
time we would have the differential of speed «𝒙», or in
other words the acceleration of the speed «𝒙». Now let us
repeat, «𝒙» would have to be a variable quantity in order
to have any differential at all, and if it is a variable quantity
and has a differential, then that differential is the
rate of growth or acceleration with which the value of
that quantity «𝒙» is increasing or diminishing as the case
may be. We now see the significance of the term
differential.
######################################################### p. 116 ###
One more illustration. We all know that if a ball
is thrown straight up in the air it starts up with great
speed and gradually stops and begins to fall. Then as
it falls it continues to increase its speed of falling until
it strikes the earth with the same speed that it was
thrown up with. Now we know that the force of
gravity has been pulling on that ball from the time that
it left our hands and has accelerated its speed backwards
until it came to a stop in the air, and then
speeded it to the earth. This instantaneous change in
the speed of the ball we have called the acceleration of
gravity, and is the rate of change of the speed of the ball.
From careful observation we find this to be 32 ft. per
second per second. A little further on we will learn
how to express the concrete value of «δ𝒙» in simple
form.
Differentiation of Constants. — Now let us remember
that a constant quantity, since it has no rate of change,
cannot be differentiated; therefore its differential is zero.
If, however, a variable quantity such as «𝒙» is multiplied by
a constant quantity such as 6, making the quantity «6𝒙»,
of course this does not prevent you from differentiating
the variable part, namely «𝒙»; but of course the constant
quantity remains unchanged; thus the differential of
6 = 0.
######################################################### p. 117 ###
But the differential of «3𝒙 = 3δ𝒙»,
> the differential of «4𝒙² = 4» times «2𝒙δ𝒙 = 8𝒙δ𝒙»,
> the differential of «2𝒙³ = 2» times «3𝒙²δ𝒙 = 6𝒙²δ𝒙»,¶
and so on.
Differential of a Sum or Difference. — We have seen
how to find the differential of a single term. Let us now
take up an algebraic expression consisting of several
terms with positive or negative signs before them; for
example
| «𝒙² − 2𝒙 + 6 + 3𝒙⁴».
In differentiating such an expression it is obvious that
we must differentiate each term separately, for each
term is separate and distinct from the other terms, and
therefore its differential or rate of growth will be distinct
and separate from the differential of the other
terms; thus
######################################################### p. 118 ###
> The differential of «(𝒙² − 2𝒙 + 6 + 3𝒙⁴)»¶
>>> = «2xδ𝒙 − 2δ𝒙 + 12x³δ𝒙».
We need scarcely say that if we differentiate one side
of an algebraic equation we must also differentiate the
other side; for we have already seen that whatever operation
is performed to one side of an equation must be
performed to the other side in order to retain the
equality. Thus if we differentiate
| «𝒙² + 4 = 6𝒙 − 10»,
we get
| «2𝒙δ𝒙 + 0 = 6δ𝒙 − 0»,
or
| «2𝒙δ𝒙 = 6δ𝒙».
Differentiation of a Product. — In Fig. 37 we have
a rectangle whose sides are «𝒙» and «𝒚» and whose area is
therefore equal to the product «𝒙𝒚». Now increase its
sides by a small amount and we have the old area
added to by two thin strips and a small rectangle,
thus:
| New area = Old area + «𝒚δ𝒙 + δ𝒚δ𝒙 + 𝒙δ𝒚».
| [FIGURE - A rectangle showing minuscule segments of the sides.]
| Fig. 37.
######################################################### p. 119 ###
«δ𝒚 δ𝒙» is negligibly small; therefore we see that the
differential of the original area «𝒙𝒚 = 𝒙δ𝒚 + 𝒚δ𝒙». This
can be generalized for every case and we have the law
Law. — “The differential of the product of two variables
is equal to the first multiplied by the differential of
the second plus the second multiplied by the differential
of the first.” Thus,
| Differential «𝒙²𝒚 = 𝒙²δ𝒚 + 2𝒚𝒙δ𝒙».
This law holds for any number of variables.
| Differential «𝒙𝒚𝒛 = 𝒙𝒚δ𝒛 + 𝒙𝒛δ𝒚 + 𝒚𝒛δ𝒙».
Differential of a Fraction. — If we are asked to differentiate
the fraction «frac{𝒙÷𝒚}» we first write it in the form «𝒙𝒚⁻¹»,
using the negative exponent; now on differentiating we
have
> Differential «𝒙𝒚⁻¹ = −𝒙𝒚⁻²δ𝒚 + 𝒚⁻¹δ𝒙»
>> « = −frac{𝒙δ𝒚÷𝒚²} + frac{δ𝒙÷𝒚}»
Reducing to a common denominator we have
> Differential «𝒙𝒚⁻¹» or «frac{𝒙÷𝒚} = −frac{𝒙δ𝒚÷𝒚²} + frac{𝒚δ𝒙÷𝒚²}»
>> « = −frac{𝒚δ𝒙 − 𝒙δ𝒚÷𝒚²}»
######################################################### p. 120 ###
Law. — The differential of a fraction is then seen to
be equal to the differential of the numerator times the
denominator, minus the differential of the denominator
times the numerator, all divided by the square of the
denominator.
Differential of One Quantity with Respect to Another. —
Thus far we have considered the differential of a
variable with respect to itself, that is, we have considered
its rate of development in so far as it was itself alone
concerned. Suppose however we have two variable
quantities dependent on each other, that is, as one
changes the other changes, and we are asked to find the
rate of change of the one with respect to the other,
that is, to find the rate of change of one knowing the
rate of change of the other. At a glance we see that
this should be a very simple process, for if we know the
relation which subsists between two variable quantities,
this relation being expressed in the form of an equation
between the two quantities, we should readily be able
to tell the relation which will hold between similar
deductions from these quantities. Let us for instance
take the equation
| «𝒙 = 𝒚 + 2».
Here we have the two variables «𝒙» and «𝒚» tied together
by an equation which establishes a relation between
them. As we have previously seen, if we give any
definite value to «𝒚» we will find a corresponding value
for «𝒙». Referring to our chapter on coördinate geometry
we see that this is the equation of the line shown in
Fig. 38.
######################################################### p. 121 ###
| [FIGURE - A coordinate plane showing the line x equals y plus 2, with minuscule divisions between p and p1.]
| Fig. 38.
Let us take any point «P» on this line. Its coördinates
are «𝒚» and «𝒙» respectively. Now choose another
point «P₁» a short distance away from «P» on the same
line. The abscissa of this new point will be a little
longer than that of the old point, and will equal «𝒙 + δ𝒙»,
while the ordinate «𝒚» of the old point has been increased
by «δ𝒚», making the ordinate of the new point «𝒚 + δ𝒚».
From Fig. 38 we see that
| «tan α = frac{δ𝒚÷δ𝒙}».
Therefore, if we know the tangent «α» and know either «δ𝒚»
or «δ𝒙» we can find the other.
######################################################### p. 122 ###
In this example our equation represents a straight
line, but the same would be true for any curve represented
by any equation between «𝒙» and «𝒚» no matter how
complicated; thus Fig. 39 shows the relation between
«δ𝒙» and «δ𝒚» at one point of the curve (a circle) whose
equation is «𝒙² + 𝒚² = 25». For every other point of the
circle «tan α» or «frac{δ𝒚÷δ𝒙}» will have a different value. «δ𝒙» and
«δ𝒚» while shown quite large in the figure for demonstration’s
sake are inconceivably small in reality; therefore
the line «AB» in the figure is really a tangent of the
curve, and «∡α» is the angle which it makes with the
«𝒙» axis. For every point on the curve this angle will be
different.
| [FIGURE - a coordinate plane showing the tangent of a circle as A-B. Angle 'alpha' and segments X and differential x are labeled.]
| Fig. 39.
######################################################### p. 123 ###
Mediate Differentiation. — Summarizing the foregoing
we see that if we know any two of the three unknowns
in equation «tan α = frac{δ𝒚÷δ𝒙}» we can find the third.
Some textbooks represent «tan α» or «frac{δ𝒚÷δ𝒙}» by «𝒚ₓ» and, «frac{δ𝒙÷δ𝒚} by 𝒙_{𝒚}».
This is a convenient notation and we will use it here.
Therefore we have
>> «δ𝒙 tan α = δ𝒚»,
>> «frac{δ𝒚÷tan 𝒂} = δ𝒙»,
or
>> «δ𝒚 = δ𝒙 𝒚ₓ»,
>> «δ𝒙 = δ𝒚 𝒙_{𝒚}».
This shows us that if we differentiate the quantity
«3𝒙²» as to «𝒙» we obtain «6 𝒙δ𝒙», but if we had wished to
differentiate it with respect to «𝒚» we would first have to
differentiate it with respect to «𝒙» and then multiply by
«𝒙_{𝒚}», thus:
> Differentiation of «3𝒙²» as to «𝒚 = 6 𝒙 δ𝒚 𝒙_{𝒚}».
Likewise if we have «4𝒚³» and we wish to differentiate it
with respect to «𝒙» we have
> Differential of «4 𝒚³» as to «𝒙 = 12 𝒚² δ𝒙 𝒚ₓ».
This is called mediate differentiation and is resorted to
primarily because we can differentiate a power with
respect to itself readily, but not with respect to some
other variable.
######################################################### p. 124 ###
Law. — To differentiate any expression containing
«𝒙» as to «𝒚», first differentiate it as to «𝒙» and then multiply
by «𝒙_{𝒚}δ𝒚» or vice versa.
We need this principle if we find the differential of
several terms some containing «𝒙» and some «𝒚»; thus if we
differentiate the equation «2𝒙² = 𝒚² − 10» with respect
to «𝒙» we get
>> «4𝒙δ𝒙 = 3𝒚²𝒚ₓδ𝒙 + 0»,
or
>> «4𝒙 = 3𝒚²𝒚ₓ»,
or
>> «𝒚ₓ = frac{4𝒙÷3𝒚²}»,
Therefore
>> tan «α = frac{4𝒙÷3𝒚²}».
From this we see that by differentiating the original
equation of the curve we got finally an equation giving
the value «tan α» in terms of «𝒙» and «𝒚», and if we fill out the
exact numerical values of «𝒙» and «𝒚» for any particular point
of the curve we will immediately be able to determine
the slant of the tangent of the curve at this point, as we
will numerically have the value of tangent «α», and a is
the angle that the tangent makes with the «𝒙» axis.
######################################################### p. 125 ###
In just the same manner that we have proceeded
here we can proceed to find the direction of the tangent
of any curve whose equation we know. The differential
of «𝒚» as to «𝒙», namely «frac{δ𝒚÷δ𝒙}» or «𝒚ₓ», must be kept in
mind as the rate of change of «𝒚» with respect to «𝒙», and
nothing so vividly portrays this fact as the inclination
of the tangent to the curve which shows the bend of
the curve at every point.
Differentials of Other Functions. — By elaborate processes
which cannot be mentioned here we find that the
Differential of the sine «𝒙» as to «𝒙 =» cosine «𝒙 δ𝒙».
Differential of the cosine «𝒙» as to «𝒙 =» − sin «𝒙δ𝒙».
Differential of the log «𝒙» as to «𝒙 = frac{1÷𝒙}δ𝒙».
Differential of the sine «𝒚» as to «𝒙 =» cosine «𝒚𝒚ₓδ𝒙».
Differential of the cosine «𝒚» as to «𝒙 =» − sine «𝒚𝒚ₓδ𝒙».
Differential of the log «𝒚» as to «𝒙 = frac{1÷𝒚}𝒚ₓδ𝒙».
Maxima and Minima. — Referring back to the circle,
Fig. 39, once more, we see that
| «𝒙² + 𝒚² = 25».
Differentiating this equation with reference to «𝒙» we
have
> «2𝒙δ𝒙 + 2𝒚𝒚ₓδ𝒙 = 0»,
or
>> «2𝒙 + 2𝒚𝒚ₓ = 0»,
or
>>> «𝒚ₓ = −frac{𝒙÷𝒚}»,
Therefore
>> tan «α = −frac{𝒙÷𝒚}».
######################################################### p. 126 ###
Now when tan «𝒂 = 0» it is evident that the tangent to
the curve is parallel to the «𝒙» axis. At this point «𝒚» is
either a maximum or a minimum which can be readily
determined on reference to the curve.
| «0 = frac{𝒙÷𝒚}»,
| «𝒙 = 0».
Therefore «𝒙 = 0» when «𝒚» is maximum and in this particular
curve also minimum.
Law. — If we want to find the maximum or minimum
value of «𝒙» in any equation containing «𝒙» and «𝒚»,
we differentiate the equation with reference to «𝒚» and
solve for the value of «𝒙_{𝒚}»; this we make equal to 0 and
then we solve for the value of «𝒚» in the resulting
equation.
Example. — Find the maximum or minimum value
of «𝒙» in the equation
| «𝒚² = 14𝒙».
Differentiating with respect to «𝒚» we have
| «2𝒚δ𝒚 = 14𝒙_{𝒚}δ𝒚», ¶
«𝒙_{𝒚} = frac{2𝒚÷14}».
Equating this to 0 we have
| «frac{2𝒚÷14} = 0»,
or
| «𝒚 = 0».
In other words, we find that 𝒙 has its minimum value
when «𝒚 = 0». We can readily see that this is actually
the case in Fig. 40, which shows the curve (a parabola).
| [FIGURE - A coordinate plane showing the line for y-squared equals 14 times x.]
| Fig. 40.
######################################################### p. 127 ###
#### INTEGRATION ####
Integration is the exact opposite of differentiation.
In differentiation we divide a body into its constituent
parts, in integration we add these constituent parts
together to produce the body.
Integration is indicated by the sign «∫»; thus, if we
wished to integrate «δ𝒙» we would write
| «∫δ𝒙»
Since integration is the opposite of differentiation, if we
are given a quantity and asked to integrate it, our
answer would be that quantity which differentiated
will give us our original quantity. For example, we
detect «δ𝒙» as the derivative of «𝒙»; therefore the integral
«∫δ𝒙 = 𝒙». Likewise, we detect «4𝒙³δ𝒙» as the differential
of «𝒙⁴» therefore the integral «∫4𝒙³δ𝒙 = 𝒙⁴».
| [FIGURE - A coordinate plane showing area under a curve through the integration of minuscule segments under P-P1.]
| Fig. 41.
######################################################### p. 128 ###
If we consider the line «AB» (Fig. 35) to be made up of
small parts «δ𝒙», we could sum up these parts thus:
> «δ𝒙 + δ𝒙 + δ𝒙 + δ𝒙 + δ𝒙 + δ𝒙» . . . . . .
for millions of parts. But integration enables us to express
this more simply and «∫δ𝒙» means the summation
of every single part «δ𝒙» which goes to make up the line
«AB», no matter how many parts there may be or how
small each part. But «𝒙» is the whole length of the line
of indefinite length. To sum up any portion of the line
between the points or limits «𝒙 = 1» and «𝒙 = 4», we
would write
| «\int_{𝒙=1}ˣ⁼⁴ δ𝒙=(𝒙)_{𝒙=1}ˣ⁼⁴».
######################################################### p. 129 ###
Now these are definite integrals because they indicate
exactly between what limits or points we wish to find
the length of the line. This is true for all integrals.
Where no limits of integration are shown the integral
will yield only a general result, but when limits are
stated between which summation is to be made, then we
have a definite integral whose precise value we may
ascertain.
Refer back to the expression «𝒙=(𝒙)_{𝒙=1}ˣ⁼⁴» in order to
solve this, substitute inside of the parenthesis the
value of «𝒙» for the upper limit of «𝒙», namely, 4, and substitute
and subtract the value of «𝒙» at the lower limit,
namely, 1; we then get
| «(𝒙)_{𝒙=1}ˣ⁼⁴ = (4 − 1) = 3».
Thus 3 is the length of the line between 1 and 4. Or,
to give another illustration, suppose the solution of
some integral had given us
>> «(𝒙² − 1)_{𝒙=2}ˣ⁼³»,
then
> «(𝒙² − 1)_{𝒙=2}ˣ⁼³ = (3² − 1) − (2² − 1) = 5».
Here we simply substituted for «𝒙» in the parenthesis its
upper limit, then subtracted from the quantity thus
obtained another quantity, which is had by substituting
the lower limit of «𝒙».
######################################################### p. 130 ###
By higher mathematics and the theories of series we
prove that the integral of any power of a variable as to
itself is obtained by increasing the exponent by one and
dividing by the new exponent, thus:
| «\int{𝒙²} δ𝒙=\frac{x³}{3}»,
| «\int 4 𝒙⁵ δ𝒙=\frac{4 𝒙⁶}{6}».
On close inspection this is seen to be the inverse of the
law of differentiation, which says to decrease the exponent
by one and multiply by the old exponent.
So many and so complex are the laws of nature and
so few and so limited the present conceptions of man
that only a few type forms of integrals may be actually
integrated. If the quantity under the integral sign by
some manipulation or device is brought into a form
where it is recognized as the differential of another
quantity, then integrating it will give that quantity.
######################################################### p. 131 ###
The Integral of an Expression. — The integral of an
algebraic expression consisting of several terms is equal
to the sum of the integrals of each of the separate terms;
thus,
| «\int{𝒙²} δ𝒙 + 2𝒙 δ𝒙 + 3 δ𝒙»
is the same thing as
| «\int 𝒙² δ𝒙 + \int 2 𝒙 δ𝒙 + \int 3 δ𝒙»,
The most common integrals to be met with practically
are:
(1) The integrals of some power of the variable
whose solution we have just explained
(2) The integrals of the sine and cosine, which are
>> «\int» cosine «𝒙 δ𝒙 =» sine «𝒙»,
>> «\int» sine «𝒙 δ𝒙 =» −cosine «𝒙».
(3) The integral of the reciprocal, which is
>> «\int \frac{1}{𝒙} δ𝒙=\logₑ𝒙».[*]
[*] «logₑ» means natural logarithm or logarithm to the Napierian base «𝒆»
which is equal to 2.718 as distinguished from ordinary logarithms to the
base 10. In fact wherever log appears in this chapter it means «logₑ».
Areas. — Up to the present we have considered only
the integration of a quantity with respect to itself.
Suppose now we integrate one quantity with respect to
another.
######################################################### p. 132 ###
In Fig. 41 we have the curve «PP₁», which is the graphical
representation of some equation containing «𝒙» and «𝒚».
If we wish to find the area which lies between the curve
and the «𝒙» axis and between the two vertical lines
drawn at distances «𝒙 = 𝒂» and «𝒙 = 𝒃» respectively, we
divide the space up by vertical lines drawn «δ𝒙» distance
apart. Now we would have a large number of small
strips each «δ𝒙» wide and all having different heights,
namely, «𝒚₁», «𝒚₂», «𝒚₃», «𝒚₄», etc.
The enumeration of all these areas would then be
| «𝒚₁ δ𝒙 + 𝒚₂ δ𝒙 + 𝒚₃ δ𝒙 + 𝒚₄ δ𝒙», etc.
Now calculus enables us to say
| Area wanted = «\int_{𝒙=𝒃}ˣ⁼ᵃ 𝒚 δ𝒙».
This integral «\int_{𝒙=𝒃}ˣ⁼ᵃ 𝒚 δ𝒙» cannot be readily solved. If
it were «\int 𝒙 δ𝒙» we have seen that the result would be
«frac{𝒙²÷2}» but this is not the case with «\int 𝒙 δ𝒙». We must then
find some way to replace «𝒚» in this integral by some expression
containing «𝒙». It is here then that we have to
resort to the equation of the curve «PP₁» From this
equation we find the value of «𝒚» in terms of «𝒙»; we then
substitute this value of «𝒚» in the integral «\int 𝒙 δ𝒙», and then
having an integral of «𝒙» as to itself we can readily solve
it. Now, if the equation of the curve «PP₁» is a complex
one this process becomes very difficult and sometimes
impossible.
A simple case of the above is the hyperbola «𝒙𝒚 = 10»
(Fig. 42). If we wish to get the value of the shaded
area we have
| Shaded area = «\int_{𝒙=5 ft.}^{𝒙=12 ft.} 𝒚 δ𝒙»
From the equation of this curve we have
| «𝒙𝒚 = 10»,
| «𝒚 = frac{10÷𝒙}».
######################################################### p. 133 ###
| [FIGURE a coordinate plane showing the area under the curve on x times y equals 10 between 5 and 12 feet.]
| Fig. 42.
Therefore, substituting we have
> Shaded area = «\int_{𝒙=5}ˣ⁼¹² \frac{10}{𝒙} δ𝒙».
>> Area = «10 (\logₑ𝒙)_{𝒙=5}ˣ⁼¹²»
>>> = «10 (\logₑ12) − (\logₑ 5)»
>>> = «10 (2.4817 − 1.6077)».
>> Area = 8.740 sq. ft.
Beyond this brief gist of the principles of calculus we
can go no further in this chapter. The student may not
understand the theories herein treated of at first—in
fact, it will take him, as it has taken every student,
many months before the true conceptions of calculus
dawn on him clearly. And, moreover, it is not essential
that he know calculus at all to follow the ordinary
engineering discussions. It is only where a student
wishes to obtain the deepest insight into the science that
he needs calculus, and to such a student I hope this
chapter will be of service as a brief preliminary to the
difficulties and complexities of that subject.
######################################################### p. 134 ###
## PROBLEMS ##
> 1. Differentiate «2𝒙³» as to 𝒙.
> 2. Differentiate «12𝒙²» as to 𝒙.
> 3. Differentiate «8𝒙⁵» as to 𝒙.
> 4. Differentiate «3𝒙² + 4𝒙 + 10 = 5𝒙²» as to 𝒙.
> 5. Differentiate «4𝒚² − 3𝒙» as to «𝒚».
> 6. Differentiate «14𝒚⁴𝒙³» as to «𝒚».
> 7. Differentiate «frac{𝒙²÷𝒚}» as to «𝒙».
> 8. Differentiate «2𝒚² − 4q𝒙» as to «𝒚».
Find «𝒚ₓ», in the following equations:
> 9. «𝒙² + 2𝒚² = 100».
> 10. «𝒙³ + 𝒚 = 5».
> 11. «𝒙² − 𝒚² = 25».
> 12. «5 𝒙𝒚 = 12».
> 13. What angle does the tangent line to the circle
«𝒙² + 𝒚² = 9» make with the «𝒙» axis at the point where
«𝒙 = 2»?
> 14. What is the minimum value of «𝒚» in the equation
«𝒙² = 15𝒚»?
> 15. Solve «∫2𝒙³ δ𝒙».
> 16. Solve «∫5𝒙²δ𝒙».
> 17. Solve «∫10𝒂𝒙δ𝒙 +5𝒙²δ𝒙 + 3δ𝒙»
> 18. Solve «∫ 3» sine «𝒙 δ𝒙».
> 19. Solve «∫ 2» cosine «𝒙 δ𝒙».
> 20. Solve «∫_{𝒙=2}ˣ⁼⁵ 3 𝒙² δ𝒙».
> 21. Solve «∫_{𝒙=2}ˣ⁼¹⁸ 𝒚 δ𝒙» if «𝒙𝒚 = 4».
> 22. Differentiate «10» sine «𝒙» as to «𝒙».
> 23. Differentiate cosine «𝒙» sine «𝒙» as to «𝒙».
> 24. Differentiate log «𝒙» as to «𝒙».
> 25. Differentiate «\frac{𝒚²}{𝒙²}» as to «𝒙».
[**NOTE: Spacing Empty Block]
######################################################### p. 135 ###
The following tables are reproduced from Ames and
Bliss’s “Manual of Experimental Physics” by permission
of the American Book Company.
[**NOTE: Small Empty Block]
######################################################### p. 136 ###
[**NOTE: Spacing Empty Block]
‹36‹REFERENCE MATERIAL››
‹24‹Tables of Logarithms and Trigonometry››
[**NOTE: Spacing Empty Block]
######################################################### p. 136 ###
#### LOGARITHMS 100 TO 1000 ####
[**NOTE: START TABLE WITH TABS - log]
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 Use preceding Table
11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 4 8 11 15 19 23 26 30 34
12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 7 10 14 17 21 24 28 31
13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 6 10 13 16 19 23 26 29
14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 3 6 9 12 15 18 21 24 27
15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 3 6 8 11 14 17 20 22 25
16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 3 5 8 11 13 16 18 21 24
17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 2 5 7 10 12 15 17 20 22
18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2 5 7 9 12 14 16 19 21
19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 2 4 7 9 11 13 16 18 20
20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 13 15 17 19
21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2 4 6 8 10 12 14 16 18
22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17
23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 2 4 6 7 9 11 13 15 17
24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16
25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 14 15
26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15
27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 2 3 5 6 8 9 11 13 14
28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2 3 5 6 8 9 11 12 14
29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13
30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 6 7 9 10 11 13
31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12
32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12
33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 5 6 8 9 10 12
34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11
35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 1 2 4 5 6 7 9 10 11
36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 1 2 4 5 6 7 8 10 11
37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10
38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10
39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 1 2 3 4 6 7 8 9 10
40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10
41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 1 2 3 4 5 6 7 8 9
42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 1 2 3 4 5 6 7 8 9
43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9
44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 1 2 3 4 5 6 7 8 9
45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9
46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 1 2 3 4 5 6 7 7 8
47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 1 2 3 4 5 5 6 7 8
48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8
49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 1 2 3 4 4 5 6 7 8
50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8
51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8
52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7
53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7
54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 137 ###
#### LOGARITHMS 100 TO 1000 ####
[**NOTE: START TABLE WITH TABS - log]
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7
56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7
57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7
58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7
59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7
60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6
61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6
62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6
63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6
64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6
65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6
66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6
67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6
68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6
69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 3 3 4 4 5 6
70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6
71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5
72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5
73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5
74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5
75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5
76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5
77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5
78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5
79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5
80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5
81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5
82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5
83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5
84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5
85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5
86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5
87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4
88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4
89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4
90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4
91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4
92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4
93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4
94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4
95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4
96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4
97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4
98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4
99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 0 1 1 2 2 3 3 3 4
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 138 ###
#### NATURAL SINES ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
0° 0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15
1° 0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15
2° 0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 3 6 9 12 15
3° 0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 3 6 9 12 15
4° 0698 0715 0732 0750 0767 0785 0802 0819 0837 0854 3 6 9 12 15
5° 0872 0889 0906 0924 0941 0958 0976 0993 1011 1028 3 6 9 12 14
6° 1045 1063 1080 1097 1115 1132 1149 1167 1184 1201 3 6 9 12 14
7° 1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 3 6 9 12 14
8° 1392 1409 1426 1444 1461 1478 1495 1513 1530 1547 3 6 9 12 14
9° 1564 1582 1599 1616 1633 1650 1668 1685 1702 1719 3 6 9 11 14
10° 1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 3 6 9 11 14
11° 1908 1925 1942 1959 1977 1994 2011 2028 2045 2062 3 6 9 11 14
12° 2079 2096 2113 2130 2147 2164 2181 2198 2215 2233 3 6 9 11 14
13° 2250 2267 2284 2300 2317 2334 2351 2368 2385 2402 3 6 9 11 14
14° 2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 3 6 8 11 14
15° 2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 3 6 8 11 14
16° 2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 3 6 8 11 14
17° 2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3 6 8 11 14
18° 3090 3107 3123 3140 3156 3173 3190 3206 3223 3239 3 6 8 11 14
19° 3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3 5 8 11 14
20° 3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3 5 8 11 14
21° 3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3 5 8 11 14
22° 3746 3762 3778 3795 3811 3827 3843 3859 3875 3891 3 5 8 11 13
23° 3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 3 5 8 11 13
24° 4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 3 5 8 11 13
25° 4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 3 5 8 11 13
26° 4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 3 5 8 10 13
27° 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 3 5 8 10 13
28° 4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 3 5 8 10 13
29° 4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 3 5 8 10 13
30° 5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 3 5 8 10 13
31° 5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 2 5 7 10 12
32° 5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 2 5 7 10 12
33° 5446 5461 5476 5490 5505 5519 5534 5548 5563 5577 2 5 7 10 12
34° 5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 2 5 7 10 12
35° 5736 5750 5764 5779 5793 5807 5821 5835 5850 5864 2 5 7 10 12
36° 5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 2 5 7 9 12
37° 6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 2 5 7 9 12
38° 6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 2 5 7 9 11
39° 6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 2 5 7 9 11
40° 6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 2 4 7 9 11
41° 6561 6574 6587 6600 6613 6626 6639 6652 6665 6678 2 4 7 9 11
42° 6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 2 4 6 9 11
43° 6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 2 4 6 9 11
44° 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 2 4 6 8 10
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 139 ###
#### NATURAL SINES ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
45° 7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 2 4 6 8 10
46° 7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 2 4 6 8 10
47° 7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 2 4 6 8 10
48° 7431 7443 7455 7466 7478 7490 7501 7513 7524 7536 2 4 6 8 10
49° 7547 7559 7570 7581 7593 7604 7615 7627 7638 7649 2 4 6 8 10
50° 7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 2 4 6 7 9
51° 7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 2 4 5 7 9
52° 7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 2 4 5 7 9
53° 7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 2 3 5 7 9
54° 8090 8100 8111 8121 8131 8141 8151 8161 8171 8181 2 3 5 7 9
55° 8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8
56° 8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 2 3 5 6 8
57° 8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 2 3 5 6 8
58° 8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 2 3 5 6 8
59° 8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 1 3 4 6 7
60° 8660 8669 8678 8686 8695 8704 8712 8721 8729 8738 1 3 4 6 7
61° 8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 1 3 4 6 7
62° 8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 1 3 4 5 7
63° 8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 1 3 4 5 7
64° 8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 1 3 4 5 6
65° 9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 1 2 4 5 6
66° 9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 1 2 4 5 6
67° 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 1 2 3 5 6
68° 9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 1 2 3 4 5
69° 9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 1 2 3 4 5
70° 9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 1 2 3 4 5
71° 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 1 2 3 4 5
72° 9511 9516 9521 9527 9532 9537 9542 9548 9553 9558 1 2 3 4 4
73° 9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 1 2 3 3 4
74° 9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 1 2 2 3 4
75° 9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 1 2 2 3 4
76° 9703 9707 9711 9715 9720 9724 9728 9732 9736 9740 1 1 2 3 4
77° 9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 1 1 2 3 3
78° 9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 1 1 2 2 3
79° 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 1 1 2 2 3
80° 9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 1 1 2 2 3
81° 9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 0 1 1 2 2
82° 9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 0 1 1 2 2
83° 9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 0 1 1 1 2
84° 9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 0 1 1 1 2
85° 9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 0 1 1 1 1
86° 9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 0 0 1 1 1
87° 9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 0 0 0 1 1
88° 9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 0 0 0 0 0
89° 9998 9999 9999 9999 9999 1,000¶nearly 1,000¶nearly 1,000¶nearly 1,000¶nearly 1,000¶nearly 0 0 0 0 0
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 140 ###
#### NATURAL COSINES ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
0° 1,000 1,000¶nearly 1,000¶nearly 1,000¶nearly 1,000¶nearly 0000 9999 9999 9999 9999 0 0 0 0 0
1° 9998 9998 9998 9997 9997 9997 9996 9996 9995 9995 0 0 0 0 0
2° 9994 9993 9993 9992 9991 9990 9990 9989 9988 9987 0 0 0 0 1
3° 9986 9985 9984 9983 9982 9981 9980 9979 9978 9977 0 0 0 1 1
4° 9976 9974 9973 9972 9971 9969 9968 9966 9965 9963 0 0 1 1 1
5° 9962 9960 9959 9957 9956 9954 9952 9951 9949 9947 0 1 1 1 1
6° 9945 9943 9942 9940 9938 9936 9934 9932 9930 9928 0 1 1 1 2
7° 9925 9923 9921 9919 9917 9914 9912 9910 9907 9905 0 1 1 1 2
8° 9903 9900 9898 9895 9893 9890 9888 9885 9882 9880 0 1 1 2 2
9° 9877 9874 9871 9869 9866 9863 9860 9857 9854 9851 0 1 1 2 2
10° 9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 1 1 2 2 3
11° 9816 9813 9810 9806 9803 9799 9796 9792 9789 9785 1 1 2 2 3
12° 9781 9778 9774 9770 9767 9763 9759 9755 9751 9748 1 1 2 2 3
13° 9744 9740 9736 9732 9728 9724 9720 9715 9711 9707 1 1 2 3 3
14° 9703 9699 9694 9690 9686 9681 9677 9673 9668 9664 1 1 2 3 4
15° 9659 9655 9650 9646 9641 9636 9632 9627 9622 9617 1 2 2 3 4
16° 9613 9608 9603 9598 9593 9588 9583 9578 9573 9568 1 2 2 3 4
17° 9563 9558 9553 9548 9542 9537 9532 9527 9521 9516 1 2 3 3 4
18° 9511 9505 9500 9494 9489 9483 9478 9472 9466 9461 1 2 3 4 5
19° 9455 9449 9444 9438 9432 9426 9421 9415 9409 9403 1 2 3 4 5
20° 9397 9391 9385 9379 9373 9367 9361 9354 9348 9342 1 2 3 4 5
21° 9336 9330 9323 9317 9311 9304 9298 9291 9285 9278 1 2 3 4 5
22° 9272 9265 9259 9252 9245 9239 9232 9225 9219 9212 1 2 3 4 5
23° 9205 9198 9191 9184 9178 9171 9164 9157 9150 9143 1 2 3 5 6
24° 9135 9128 9121 9114 9107 9100 9092 9085 9078 9070 1 2 4 5 6
25° 9063 9056 9048 9041 9033 9026 9018 9011 9003 8996 1 2 4 5 6
26° 8988 8980 8973 8965 8957 8949 8942 8934 8926 8918 1 3 4 5 6
27° 8910 8902 8894 8886 8878 8870 8862 8854 8846 8838 1 3 4 5 7
28° 8829 8821 8813 8805 8796 8788 8780 8771 8763 8755 1 3 4 5 7
29° 8746 8738 8729 8721 8712 8704 8695 8686 8678 8669 1 3 4 6 7
30° 8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 1 3 4 6 7
31° 8572 8563 8554 8545 8536 8526 8517 8508 8499 8490 2 3 5 6 8
32° 8480 8471 8462 8453 8443 8434 8425 8415 8406 8396 2 3 5 6 8
33° 8387 8377 8368 8358 8348 8339 8329 8320 8310 8300 2 3 5 6 8
34° 8290 8281 8271 8261 8251 8241 8231 8221 8211 8202 2 3 5 7 8
35° 8192 8181 8171 8161 8151 8141 8131 8121 8111 8100 2 3 5 7 8
36° 8090 8080 8070 8059 8049 8039 8028 8018 8007 7997 2 3 5 7 9
37° 7986 7976 7965 7955 7944 7934 7923 7912 7902 7891 2 4 5 7 9
38° 7880 7869 7859 7848 7837 7826 7815 7804 7793 7782 2 4 5 7 9
39° 7771 7760 7749 7738 7727 7716 7705 7694 7683 7672 2 4 5 7 9
40° 7660 7649 7638 7627 7615 7604 7593 7581 7570 7559 2 4 6 7 9
41° 7547 7536 7524 7513 7501 7490 7478 7466 7455 7443 2 4 6 8 10
42° 7431 7420 7408 7396 7385 7373 7361 7349 7337 7325 2 4 6 8 10
43° 7314 7302 7290 7278 7266 7254 7242 7230 7218 7206 2 4 6 8 10
44° 7193 7181 7169 7157 7145 7133 7120 7108 7096 7083 2 4 6 8 10
FOOTNOTE: N.B. - Numbers in difference column to be subtracted, not added.
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 141 ###
#### NATURAL COSINES ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
45° 7071 7059 7046 7034 7022 7009 6997 6984 6972 6959 2 4 6 8 10
46° 6947 6934 6921 6909 6896 6884 6871 6858 6845 6833 2 4 6 8 10
47° 6820 6807 6794 6782 6769 6756 6743 6730 6717 6704 2 4 6 9 11
48° 6691 6678 6665 6652 6639 6626 6613 6600 6587 6574 2 4 6 9 11
49° 6561 6547 6534 6521 6508 6494 6481 6468 6455 6441 2 4 7 9 11
50° 6428 6414 6401 6388 6374 6361 6347 6334 6320 6307 2 4 7 9 11
51° 6293 6280 6266 6252 6239 6225 6211 6198 6184 6170 2 5 7 9 11
52° 6157 6143 6129 6115 6101 6088 6074 6060 6046 6032 2 5 7 9 11
53° 6018 6004 5990 5976 5962 5948 5934 5920 5906 5892 2 5 7 9 12
54° 5878 5864 5850 5835 5821 5807 5793 5779 5764 5750 2 5 7 9 12
55° 5736 5721 5707 5693 5678 5664 5650 5635 5621 5606 2 5 7 10 12
56° 5592 5577 5563 5548 5534 5519 5505 5490 5476 5461 2 5 7 10 12
57° 5446 5432 5417 5402 5388 5373 5358 5344 5329 5314 2 5 7 10 12
58° 5299 5284 5270 5255 5240 5225 5210 5195 5180 5165 2 5 7 10 12
59° 5150 5135 5120 5105 5090 5075 5060 5045 5030 5015 2 5 7 10 12
60° 5000 4985 4970 4955 4939 4924 4909 4894 4879 4863 3 5 8 10 13
61° 4848 4833 4818 4802 4787 4772 4756 4741 4726 4710 3 5 8 10 13
62° 4695 4679 4664 4648 4633 4617 4602 4586 4571 4555 3 5 8 10 13
63° 4540 4524 4509 4493 4478 4462 4446 4431 4415 4399 3 5 8 10 13
64° 4384 4368 4352 4337 4321 4305 4289 4274 4258 4242 3 5 8 10 13
65° 4226 4210 4195 4179 4163 4147 4131 4115 4099 4083 3 5 8 11 13
66° 4067 4051 4035 4019 4003 3987 3971 3955 3939 3923 3 5 8 11 13
67° 3907 3891 3875 3859 3843 3827 3811 3795 3778 3762 3 5 8 11 13
68° 3746 3730 3714 3697 3681 3665 3649 3633 3616 3600 3 5 8 11 13
69° 3584 3567 3551 3535 3518 3502 3486 3469 3453 3437 3 5 8 11 14
70° 3420 3404 3387 3371 3355 3338 3322 3305 3289 3272 3 5 8 11 14
71° 3256 3239 3223 3206 3190 3173 3156 3140 3123 3107 3 6 8 11 14
72° 3090 3074 3057 3040 3024 3007 2990 2974 2957 2940 3 6 8 11 14
73° 2924 2907 2890 2874 2857 2840 2823 2807 2790 2773 3 6 8 11 14
74° 2756 2740 2723 2706 2689 2672 2656 2639 2622 2605 3 6 8 11 14
75° 2588 2571 2554 2538 2521 2504 2487 2470 2453 2436 3 6 8 11 14
76° 2419 2402 2385 2368 2351 2334 2317 2300 2284 2267 3 6 8 11 14
77° 2250 2233 2215 2198 2181 2164 2147 2130 2113 2096 3 6 9 11 14
78° 2079 2062 2045 2028 2011 1994 1977 1959 1942 1925 3 6 9 11 14
79° 1908 1891 1874 1857 1840 1822 1805 1788 1771 1754 3 6 9 11 14
80° 1736 1719 1702 1685 1668 1650 1633 1616 1599 1582 3 6 9 11 14
81° 1564 1547 1530 1513 1495 1478 1461 1444 1426 1409 3 6 9 11 14
82° 1392 1374 1357 1340 1323 1305 1288 1271 1253 1236 3 6 9 12 14
83° 1219 1201 1184 1167 1149 1132 1115 1097 1080 1063 3 6 9 12 14
84° 1045 1028 1011 0993 0976 0958 0941 0924 0906 0889 3 6 9 12 14
85° 0872 0854 0837 0819 0802 0785 0767 0750 0732 0715 3 6 9 12 14
86° 0698 0680 0663 0645 0628 0610 0593 0576 0558 0541 3 6 9 12 15
87° 0523 0506 0488 0471 0454 0436 0419 0401 0384 0366 3 6 9 12 15
88° 0349 0332 0314 0297 0279 0262 0244 0227 0209 0192 3 6 9 12 15
89° 0175 0157 0140 0122 0105 0087 0070 0052 0035 0017 3 6 9 12 15
FOOTNOTE: N.B. - Numbers in difference column to be subtracted, not added.
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 142 ###
#### NATURAL TANGENTS ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
0° .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15
1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15
2 .0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 3 6 9 12 15
3 .0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 3 6 9 12 15
4 .0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 3 6 9 12 15
5 .0875 0892 0910 0928 0945 0963 0981 0998 1016 1033 3 6 9 12 15
6 .1051 1069 1086 1104 1122 1139 1157 1175 1192 1210 3 6 9 12 15
7 .1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 3 6 9 12 15
8 .1405 1423 1441 1459 1477 1495 1512 1530 1548 1566 3 6 9 12 15
9 .1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 3 6 9 12 15
10 .1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 3 6 9 12 15
11 .1944 1962 1980 1998 2016 2035 2053 2071 2089 2107 3 6 9 12 15
12 .2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 3 6 9 12 15
13 .2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 3 6 9 12 15
14 .2493 2512 2530 2549 2568 2586 2605 2623 2642 2661 3 6 9 12 16
15 .2679 2698 2717 2736 2754 2773 2792 2811 2830 2849 3 6 9 13 16
16 .2867 2886 2905 2924 2943 2962 2981 3000 3019 3038 3 6 9 13 16
17 .3057 3076 3096 3115 3134 3153 3172 3191 3211 3230 3 6 10 13 16
18 .3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3 6 10 13 16
19 .3443 3463 3482 3502 3522 3541 3561 3581 3600 3620 3 7 10 13 16
20 .3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 13 17
21 .3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 3 7 10 13 17
22 .4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 3 7 10 14 17
23 .4245 4265 4286 4307 4327 4348 4369 4390 4411 4431 3 7 10 14 17
24 .4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 3 7 10 14 18
25 .4663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4 7 11 14 18
26 .4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 4 7 11 14 18
27 .5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 4 7 11 15 18
28 .5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 4 7 11 15 19
29 .5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 4 8 11 15 19
30 .5774 5797 5820 5844 5867 5890 5914 5938 5961 5985 4 8 12 16 20
31 .6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 4 8 12 16 20
32 .6249 6273 6297 6322 6346 6371 6395 6420 6445 6469 4 8 12 16 20
33 .6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 4 8 12 17 21
34 .6745 6771 6796 6822 6847 6873 6899 6924 6950 6976 4 8 13 17 21
35 .7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 4 9 13 17 22
36 .7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 4 9 13 18 22
37 .7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 5 9 14 18 23
38 .7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 5 9 14 19 24
39 .8098 8127 8156 8185 8214 8243 8273 8302 8332 8361 5 10 15 19 24
40 .8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 5 10 15 20 25
41 .8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 5 10 15 21 26
42 .9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 5 11 16 21 27
43 .9325 9358 9391 9424 9457 9490 9523 9556 9590 9623 5 11 16 22 28
44 .9657 9691 9725 9759 9793 9827 9861 9896 9930 9965 6 11 17 23 29
[**NOTE: END TABLE WITH TABS]
[**NOTE: Spacing Empty Block]
######################################################### p. 143 ###
#### NATURAL TANGENTS ####
[**NOTE: START TABLE WITH TABS]
0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5
45° .00000 0035 0070 0105 0141 0176 0212 0247 0283 0319 6 12 18 24 30
46 1.0355 0392 0428 0464 0501 0538 0575 0612 0649 0686 6 12 18 24 31
47 1.0724 0761 0799 0837 0875 0913 0951 0990 1028 1067 6 13 19 25 32
48 1.1106 1145 1184 1224 1263 1303 1343 1383 1423 1463 7 13 20 26 33
49 1.1504 1544 1585 1626 1667 1708 1750 1792 1833 1875 7 14 20 27 34
50 1.1918 1960 2002 2045 2088 2131 2174 2218 2261 2305 7 14 21 29 36
51 1.2349 2393 2437 2482 2527 2572 2617 2662 2708 2753 7 15 22 30 37
52 1.2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 8 15 23 31 39
53 1.3270 3319 3367 3416 3465 3514 3564 3613 3663 3713 8 16 24 33 41
54 1.3764 3814 3865 3916 3968 4019 4071 4124 4176 4229 8 17 26 34 43
55 1.4281 4335 4388 4442 4496 4550 4605 4659 4715 4770 9 18 27 36 45
56 1.4826 4882 4938 4994 5051 5108 5166 5224 5282 5340 9 19 28 38 48
57 1.5399 5458 5517 5577 5637 5697 5757 5818 5880 5941 10 20 30 40 50
58 1.6003 6066 6128 6191 6255 6319 6383 6447 6512 6577 10 21 32 42 53
59 1.6643 6709 6775 6842 6909 6977 7045 7113 7182 7251 11 22 33 45 56
60 1.7321 7391 7461 7532 7603 7675 7747 7820 7893 7966 12 23 35 48 60
61 1.8040 8115 8190 8265 8341 8418 8495 8572 8650 8728 12 25 38 51 64
62 1.8807 8887 8967 9047 9128 9210 9292 9375 9458 9542 13 27 40 54 68
63 1.9626 9711 9797 9883 9970 0̅057 0̅145 0̅233 0̅323 0̅413 14 29 43 58 73
64 2.0503 0594 0686 0778 0872 0965 1060 1155 1251 1348 15 31 46 62 78
65 2.1445 1543 1642 1742 1842 1943 2045 2148 2251 2355 16 33 50 67 84
66 2.2460 2566 2673 2781 2889 2998 3109 3220 3332 3445 18 36 54 72 91
67 2.3559 3673 3789 3906 4023 4142 4262 4383 4504 4627 19 39 58 78 99
68 2.4751 4876 5002 5129 5257 5386 5517 5649 5782 5916 21 42 64 85 108
69 2.6051 6187 6325 6464 6605 6746 6889 7034 7179 7326 23 46 70 94 118
70 2.7475 7625 7776 7929 8083 8239 8397 8556 8716 8878 25 50 76 103 130
71 2.9042 9208 9375 9544 9714 9887 0̅061 0̅237 0̅415 0̅595 28 56 84 114 144
72 3.0777 0961 1146 1334 1524 1716 1910 2106 2305 2506 31 62 94 127 160
73 3.2709 2914 3122 3332 3544 3759 3977 4197 4420 4646 34 69 105 142 179
74 3.4874 5105 5339 5576 5816 6059 6305 6554 6806 7062 39 78 118 160 202
75 3.7321 7583 7848 8118 8391 8667 8947 9232 9520 9812 44 89 135 182 230
76 4.0108 0408 0713 1022 1335 1653 1976 2303 2635 2972 50 102 154 209 265
77 4.3315 3662 4015 4373 4737 5107 5483 5864 6252 6646 58 118 179 243 308
78 4.7046 7453 7867 8288 8716 9152 9594 0̅045 0̅504 0̅970 68 138 210 285 363
79 5.1446 1929 2422 2924 3435 3955 4486 5026 5578 6140 81 164 251 341 434
80 5.6713 7297 7894 8502 9124 9758 0̅405 1̅066 1̅742 2̅432 *
81 6.3138 3859 4596 5350 6122 6912 7720 8548 9395 0̅264
82 7.1154 2066 3002 3962 4947 5958 6996 8062 9158 0̅285
83 8.1443 2636 3863 5126 6427 7769 9152 0̅579 2̅052 3̅572
84 9.5144 9.677 9.845 10.02 10.20 10.39 10.58 10.78 10.99 11.20
85 11.43 11.66 11.91 12.16 12.43 12.71 13.00 13.30 13.62 13.95
86 14.30 14.67 15.06 15.46 15.89 16.35 16.83 17.34 17.89 18.46
87 19.08 19.74 20.45 21.20 22.02 22.90 23.86 24.90 26.03 27.27
88 28.64 30.14 31.82 33.69 35.80 38.19 40.92 44.07 47.74 52.08
89 57.29 63.66 71.62 81.85 95.49 114.6 143.2 191.0 286.5 573.0
[**NOTE: END TABLE WITH TABS]
‹13‹[*] Difference columns cease to be useful, owing to the¶ rapidity with which the value of the tangent changes.››
[**NOTE: Spacing Empty Block]
[**TRANSCRIBER'S NOTE: In the second Natural Tangents table, overlines¶
are applied to values that exceed ten times the previous values in the¶
row. The first example is in cell for 63° 36', where the first 0 has an¶
overline. If the overlines are not present, please access the HTML¶
version of this eBook.]
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‹8‹[BLANK PAGE]››
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[**NOTE: Big Empty Block]
‹36‹ANSWERS TO PROBLEMS››
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‹8‹[BLANK PAGE]››
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##### ANSWERS TO PROBLEMS #####
#### CHAPTER I ####
1. «2𝒂 + 6𝒃 + 6𝒄 − 3d».
2. «−9𝒂 + 𝒃 − 6𝒄».
3. «3d − 𝒛 + 14𝒃 − 10𝒂».
4. «−3𝒙 + 6𝒚 + 4𝒛 + 𝒂».
5. «−8𝒃 + 9𝒂 − 2𝒄».
6. «−8𝒙 − 6𝒂 + 4𝒃 + 11𝒚».
7. «2𝒙 − 2𝒚 + 28𝒛».
#### CHAPTER II ####
1. «18𝒂²𝒃²».
2. «48𝒂²𝒃²𝒄³».
3. «90𝒙²𝒚²».
4 «144𝒂⁸𝒃⁵𝒄²».
5. «𝒂𝒃𝒄²».
6. «frac{𝒂²𝒃³𝒄²÷d}».
7. «𝒂⁴𝒃⁵𝒄».
8. «𝒂⁸𝒃²𝒄⁷».
9. «frac{𝒂²𝒄²𝒛÷𝒃⁴}».
10. «frac{40𝒂⁷÷𝒄⁴}».
11. «frac{𝒃²𝒄²÷54𝒂𝒅}».
#### CHAPTER III ####
1. «frac{9𝒂²𝒃³𝒄÷4𝒙}».
2. «frac{𝒃𝒄÷18d}».
3. «frac{𝒂⁴𝒃⁴𝒄²𝒙÷6𝒚²}».
4. «20𝒙² + 15𝒙𝒚 + 10𝒙𝒛».
5. «4𝒂 + 2𝒂²𝒃² − 𝒃».
6. «𝒂² − 𝒃²».
7. «6𝒂² − 𝒂𝒃 + 5𝒂𝒄 − 2𝒃² + 6𝒃𝒄 − 4𝒄²».
8. «𝒂 − 𝒃».
9. «𝒂² + 2𝒂𝒃 + 𝒃²».
10. «frac{𝒂 + 𝒃÷𝒂 − 𝒃}».
11. «frac{3𝒂²𝒄 − 2𝒂²d + 3𝒂𝒄² − 3𝒂𝒄d÷2𝒂𝒄 + 2𝒂d − 2𝒄² − 2𝒄d}».
12. «frac{𝒄³𝒃𝒂÷12}».
13. «frac{8𝒂 + 𝒃² + 4𝒄÷4𝒃}».
14. «frac{4 − 12𝒂 + 𝒂²𝒄÷6𝒂²}».
15. «frac{120𝒂²𝒄 + 3𝒃𝒄 − 6𝒃𝒙 + 2𝒃𝒄d÷12𝒃𝒄}».
16. «frac{3𝒂𝒃 − 𝒂𝒄 + 2𝒃²÷4𝒂𝒃}».
17. «frac{5𝒂² − 2𝒂 − 2𝒃÷5𝒂² + 5𝒂𝒃}».
######################################################### p. 148 ###
#### CHAPTER IV ####
1. «3, 2, 5, 𝒂, 𝒂, 𝒃».
2. «3, 2, 2, 2, 2, 𝒂, 𝒂, 𝒂, 𝒂, 𝒄».
3. «3, 2, 5, 𝒙, 𝒙, 𝒚, 𝒚, 𝒚, 𝒚, 𝒛, 𝒛, 𝒛».
4. «3, 3, 2, 2, 2, 2, 𝒙, 𝒙, 𝒂, 𝒂».
5. «3, 2, 2, frac{1÷2}, frac{1÷2}, 𝒂, frac{1÷𝒂}, frac{1÷𝒂}, 𝒃, 𝒃, frac{1÷𝒃}, frac{1÷𝒃}, 𝒄, 𝒄, 𝒄».
6. «2, 5, frac{1÷2}, 𝒙, frac{1÷𝒙}, frac{1÷𝒙}, 𝒚, 𝒚, frac{1÷𝒚}».
7. «(𝒂 − 𝒄)(2𝒂 + 𝒃)».
8. «(3𝒙 + 𝒚)(𝒙 + 𝒄)».
9. «(2𝒙 + 5𝒚)(𝒙 + 𝒛)».
10. «(𝒂 − 𝒃)(𝒂 − 𝒃)».
11. «(2𝒙 − 3𝒚)(2𝒙 − 3𝒚)».
12. «(9𝒂 + 5𝒃)(9𝒂 + 5𝒃)».
13. «(4𝒄 − 6𝒂)(4𝒄 − 6𝒂)».
14. «𝒙, 𝒚, (4𝒙² + 5𝒛𝒚 − 10𝒛)».
15. «5𝒃(6𝒂 + 3𝒂𝒄 − 𝒄)».
16. «(9𝒙𝒚 − 5𝒂) (9𝒙𝒚 + 5𝒂)».
17. «(𝒂² + 4𝒃²)(𝒂 + 2𝒃)(𝒂 − 2𝒃)».
18. «(12𝒙²𝒚 + 8𝒛)(12𝒙²𝒚 − 8𝒛)».
19. «(𝒂² − 2𝒂𝒄 + 𝒄) 2», «2».
20. «(4𝒚 + 𝒙)(4𝒚 + 𝒙)».
21. «(3𝒚 + 2𝒙)(2𝒚 − 3𝒙)».
22. «(40 + 56) (𝒂 − 26)».
23. «(3𝒚 − 2𝒙)(2𝒚 − 3𝒙)».
24. «(2𝒂 + 𝒃)(𝒂 − 3𝒃)».
25. «(2𝒂 + 5𝒃)(𝒂 + 2𝒃)».
#### CHAPTER V ####
### Square roots. ###
1. «4𝒙 + 3𝒚».
2. «2𝒂 + 6𝒃».
3. «6𝒙 + 2𝒚».
4. «5𝒂 − 2𝒃».
5. «𝒂 + 𝒃 + 𝒄».
### Cube roots. ###
1. «2𝒙 + 3𝒚».
2. «𝒙 + 2𝒚».
3. «3𝒂 + 3𝒃».
######################################################### p. 149 ###
#### CHAPTER VI ####
1. «𝒙 = 4frac{2÷3}».
2. «𝒙 = 2frac{1÷3}».
3. «𝒙 = 4».
4. «𝒙 = −frac{5÷19}».
5. «𝒙 = frac{5÷28}».
6. «𝒙 = 30».
7. «𝒙 = 6frac{133÷168}».
8. «𝒙 = frac{9𝒂 + 9𝒃 − 𝒂𝒚 − 𝒃𝒚÷3}».
9. «𝒙 = −frac{3(𝒂 − 𝒃) + 2𝒂²÷2𝒂(𝒂 − 𝒃)(𝒂 + 1)}».
10. «𝒙 = frac{10(𝒂² − 𝒃²)÷2𝒂}».
11. «2𝒂²𝒙 + 2𝒂𝒃 − 𝒂𝒙² − 𝒃𝒙 = 𝒄²𝒙 − 𝒃𝒄 + 10𝒄𝒙 − 10𝒃».
12. «frac{𝒂𝒙÷3} + 𝒃𝒙 = frac{𝒄𝒚÷d} + frac{3𝒄÷d}».
13. «𝒂 −𝒃 = frac{𝒄÷𝒄 + 3}».
14. «2 = frac{10𝒚÷𝒚 + 2}».
15. «5𝒂 + 3 = 𝒙 + d + 3».
16. «6𝒂𝒙 − 5𝒚 = 5 − 10𝒙».
17. «15𝒛² + 4𝒙 = 12 − 10𝒚».
18. «6𝒂 + 2d = 4».
19. «3𝒙 − 2 = 3𝒙² − 𝒚».
20. «8𝒙 − 10𝒄𝒚 = 20𝒚».
21. «frac{𝒙²÷(𝒄 − d)(3𝒂 + 𝒃)} − frac{𝒙²÷3(𝒄 − d)} = 2𝒂 + 𝒃».
22. «𝒙 = −frac{1÷2}».
23. Coat costs $28.57. ¶
Gun costs $57.14. ¶
Hat costs $14.29.
24. Horse costs $671.66. ¶
Carriage costs $328.33.
25. Anne’s age is 18 years.
26. 24 chairs and 14 tables.
#### CHAPTER VII ####
1. «𝒚 = 4, 𝒙 = 2».
2. «1 = 5, 𝒚 = 2».
3. «𝒙 = 1, 𝒚 = 2».
4. «𝒙 = 5, 𝒚 = 2, 𝒛 = 3».
5. «𝒙 = 3, 𝒚 = 2, 𝒛 = 4».
6. «𝒙 = −15, 𝒚 = 15».
7. «𝒙 = −.084, 𝒚 = -10.034».
8. «𝒙 = 5frac{1÷22}, 𝒚 = -frac{3÷22}».
9. «𝒙 = −1.1, 𝒚 = 6.1».
10. «𝒙 = 1frac{3÷22}, 𝒚 = 2frac{5÷22}».
######################################################### p. 150 ###
#### CHAPTER VIII ####
1. «𝒙 = 2» or «𝒙 = 1».
2. «𝒙 = frac{−2 ± 2√[19] ÷3}».
3. «𝒙 = 2».
4. «𝒙 = 4» or «−2».
5. «𝒙 = 3» or «1».
6. «𝒙 = ± 2» or «± √[−6] ».
7. «𝒙 = −frac{5 ± √[305] ÷ 14𝒂}».
8. «𝒙 = −frac{𝒂 ± √[12𝒂𝒃² + 𝒂²] ÷ 2𝒃}».
9. «𝒙 = −frac{1 − 3𝒂 ± √[51𝒂² − 6𝒂 + 1]÷2𝒂}».
10. «𝒙 = +frac{3(𝒂 + 𝒃) ± √[8 (𝒂 + 𝒃) + 9(𝒂 + 𝒃)²]÷2}».
11. «𝒙 = −frac{5 ± √[205]÷6}».
12. «𝒙 = −3».
13. «𝒙 = 4(2 ± √[3])».
14. «𝒙 = −frac{3÷4𝒂}».
15. «𝒙 = frac{2𝒂𝒃÷𝒂 + 𝒃}».
16. «𝒙 = −frac{27 ± √[2425]÷16}».
17. «𝒙 = −frac{3 ± √[−7]÷2}».
18. «𝒙 = −frac{1±√[−299]÷6}».
19. «𝒙 = 63».
20. «𝒙 = 100𝒂² − 301𝒂 + 225».
21. «𝒙 = frac{𝒂² ± 𝒂√[𝒂² + 4]÷2}».
22. «𝒙 = frac{−5 ± √[5]÷6}».
#### CHAPTER IX ####
1. «𝒌 = 50».
2. «𝒃 = √[frac{1÷441}]».
3. «𝒌 = 60».
4. «𝒂 = 192».
5. «𝒄 = 5».
#### CHAPTER X ####
1. 96 sq. ft.
2. 180 sq. ft.
3. 254.469 sq. ft.
4. Hypotenuse = «√[117]» ft. long.
5. 62.832 ft. long.
6. «√[301]» ft. long.
7. 27.6 ft. long.
8. 7957.7 miles.
9. Altitude = 7.5 ft.
10. Altitude = 4 ft.
######################################################### p. 151 ###
#### CHAPTER XI ####
1. sine = .5349; cosine = .8456; tangent = .6330.
2. sine = .9888; cosine = .1495; tangent = 6.6122.
3. «25° 36'».
4. «79° 25'».
5. «36° 59'».
6. «28° 54'».
7. «𝒄 = 600» ft.; «𝒃 = 519.57» ft.
8. «∡𝒂 = 57° 47'; 𝒄 = 591.01» ft.
9. «𝒂 = 1231» ft.; «𝒃 = 217» ft.
10. «∡𝒂 = 61° 51'; 𝒂 = 467.3» ft.
#### CHAPTER XII ####
1. 3.5879.
2. 1.8667.
3. −3.9948.
4. 4.6155.
5. 666.2.
6. 74430.
7. .2745.
8. .00024105.
9. 2302.5.
10. 9,802,000.
11. 24,860,000.
12. 778,500,000.
13. .000286.
14. .0001199.
15. 32.34.
16. 111.6.
17. .0323.
18. .03767.
19. 1,198,000.
20. 18,410,000.
21. 275,500.
22. .00001314.
23. 549.7.
24. 4.27.
25. .296.
26. 46.86.
#### CHAPTER XIII ####
Get cross-section paper and plot the following corresponding
values of 𝒙 and 𝒚 and the result will be the line or curve
as the case may be.
[BRACESTACKSTART]
[BRACESTART]
1. «𝒙 = 0; 𝒚 = −3frac{1÷3}». ¶
«𝒚 = 0; 𝒙 = 10». ¶
«𝒙 = 22; 𝒚 = 4». ¶
«𝒙 = −2; 𝒚 = −4».
[BRACE]
This is a straight line and only two¶
pairs of corresponding values of 𝒙 and¶
𝒚 are necessary to draw it.
[BRACEEND]
[BRACESTART]
2. «𝒙 = 0; 𝒚 = 3». ¶
«𝒚 = 0; 𝒙 = 7frac{1÷2}».
[BRACE]
This is also a straight line.
[BRACEEND]
[BRACESTACKEND]
######################################################### p. 152 ###
[BRACESTACKSTART]
[BRACESTART]
3. «𝒙 = 0; 𝒚 = −2». ¶
«𝒚 = 0; 𝒙 = 4».
[BRACE]
A straight line.
[BRACEEND]
[BRACESTART]
4. «𝒙 = 0; 𝒚 = −frac{8÷10}». ¶
«𝒚 = 0; 𝒙 = −2frac{2÷3}».
[BRACE]
A straight line.
[BRACEEND]
[BRACESTART]
5. «𝒙 = 0; 𝒚 = ±6». ¶
«𝒚 = 0; 𝒙 = ±6». ¶
«𝒙 = 1; 𝒚 = ±√[35]». ¶
«𝒙 = 2; 𝒚 = ±√[32]». ¶
«𝒙 = 3; 𝒚 = ±√[27]». ¶
«𝒙 = 4; 𝒚 = ±√[20]». ¶
«𝒙 = 5; 𝒚 = ±√[11]». ¶
[BRACE]
This is a circle with its center at the¶
intersection of the 𝒙 and 𝒚 axes and¶
with a radius of 6.
[BRACEEND]
[BRACESTART]
6. «𝒚 = 0; 𝒙 = 0». ¶
«𝒚 = 2; 𝒙 = ±√[32]». ¶
«𝒚 = 4; 𝒙 = ±8». ¶
«𝒚 = 6; 𝒙 = ±√[96]».
[BRACE]
This is a parabola and to plot it¶
correctly a great many corresponding¶
values of 𝒙 and 𝒚 are necessary.
[BRACEEND]
[BRACESTART]
7. «𝒚 = 0; 𝒙 = ±4». ¶
«𝒚 = ±1; 𝒙 = ±√[17]». ¶
«𝒚 = ±3; 𝒙 = ±5». ¶
«𝒚 = 5; 𝒙 = +√[41]». ¶
[BRACE]
This is an hyperbola and a great many¶
corresponding values of 𝒙 and 𝒚 are¶
necessary in order to plot the curve¶
correctly.
[BRACEEND]
[BRACESTART]
8. «𝒚 = 0; 𝒙 = ±√[7]». ¶
«𝒙 = 0; 𝒚 = +7» or «−3». ¶
«𝒙 = 1; 𝒚 = 2 ±√[22]». ¶
«𝒙 = 2; 𝒚 = 2 ±√[13]». ¶
[BRACE]
This is an ellipse with its center¶
at +2 on the 𝒚 axis. A great many¶
corresponding values of 𝒙 and 𝒚 are¶
necessary to plot it correctly.
[BRACEEND]
[BRACESTACKEND]
######################################################### p. 153 ###
###Intersections of Curves###
[BRACESTACKSTART]
[BRACESTART]
1. «𝒙 = 2frac{2÷7};» ¶
«𝒚 = 3frac{1÷7}».
[BRACE]
This is the intersection of 2¶
straight lines.
[BRACEEND]
[BRACESTART]
2. «𝒚 = −5 ± √[frac{31÷2}]»; ¶
«𝒙 = 5 ± √[frac{31÷2]}».
[BRACE]
This is the intersection of a¶
straight line and a circle.
[BRACEEND]
[BRACESTACKEND]
3. The roots are here imaginary showing that the two
curves do not touch at all, which can be easily shown by
plotting them.
#### CHAPTER XIV ####
1. «6𝒙² δ𝒙».
2. «24 × δ𝒙».
3. «40 × δ𝒙».
4. «6𝒙 δ𝒙 + 4 δ𝒙 = 15 𝒙² δ𝒙».
5. «8𝒚 δ𝒚 − 3𝒙_{𝒚} δ𝒚».
6. «42 𝒚⁴𝒙² δ𝒙 + 56 𝒙³𝒚³ δ𝒚».
7. «frac{2 𝒚𝒙 δ𝒙 − 𝒙² δ𝒚÷𝒚²}».
8. «4𝒚 δ𝒚 − 4q𝒙_{𝒚} δ𝒚».
9. «𝒚_{𝒙} = −frac{𝒙÷2𝒚}».
10. «𝒚_{𝒙} = −3𝒙²».
11. «𝒚_{𝒙} = frac{𝒙÷𝒚}».
12. «𝒚_{𝒙} = −frac{𝒚÷𝒙}».
13. «41° 48' 10''».
14. When «𝒙 = 0» at which time «𝒚» also «= 0».
15. «frac{𝒙⁴÷2}».
16. «frac{5𝒙³÷3}».
17. «5 𝒂𝒙² + frac{5÷3} 𝒙³ + 3𝒙».
18. «−3\ cos\ 𝒙».
19. «2\ sine\ 𝒙».
20. «117».
21. «8.7795».
22. «10\ cosine\ 𝒙 δ𝒙».
23. «cos²\ 𝒙 d𝒙 −\ sin²\ 𝒙 d𝒙».
24. «frac{1÷𝒙} d𝒙».
25. «frac{2𝒙²𝒚 δ𝒚 − 2𝒚²𝒙 δ𝒙÷𝒙⁴}».
[**NOTE: Small Empty Block]
######################################################### p. fin ###
=== Appendix A - Transcriber’s Notes - NONE ===
[**NOTE: START TRANSCRIBER NOTES]
New original cover art included with this eBook is granted to the public domain.¶
A few minor spelling errors and edits were made. (Page 8: “Indentity of Symbols.”; Page 85: “…the Naperian or…”; Page 117: “…the dffierential of…”)¶
Images and page breaks that originally broke paragraphs have been moved before or after the paragraph breaks as needed. The page numbers from the table of contents are still correctly associated.¶
The footnotes on pages 92, 131 and 143 have been placed directly following the elements that are referenced.¶
On pages 23, 28 and 46 a header for 'PROBLEMS' has been restored, corresponding with the other 13 chapters in the book. This will facilitate finding the these important sections.¶
Figures have been redrawn in order to improve the readability on both high-density screens and smaller physical sizes.¶
The overstroke numerals in the logarithm tables may not be visible in some reader clients and formats.¶
The typeface used in the logarithm and trigonometry tables has been set in a narrow typeface for readability.¶
Plain Text Note: the uppercase version of the Greek letter DELTA ('«Δ»') is used in place of the lowercase DELTA ('«δ»'), used to denote differential, in order to improve legibility.¶
Plain Text Note: tables for logarithms and trigonometry are very difficult to use, due to the limits of a 72 character screen width. In order to properly use these tables, please view any of the other versions of this eBook. If you must use the plain text version, they have been split into groups of columns in order to fit the text width.¶
Plain Text Note: Fractions are shown as ( _numerator_ )⁄( _denominator_ ), and square roots are shown as √{radican}, and roots with other indexes are shown as √[index]{radican}. (See page 91 for sole example.)
[**NOTE: END TRANSCRIBER NOTES]
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=== Appendix B - Figures - NONE ===
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