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MATHEMATICS FOR THE PRACTICAL MAN
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EXPLAINING SIMPLY AND QUICKLY
ALL THE ELEMENTS OF
ALGEBRA, GEOMETRY, TRIGONOMETRY,
LOGARITHMS, COÖRDINATE
GEOMETRY, CALCULUS
WITH ANSWERS TO PROBLEMS
BY
GEORGE HOWE, M.E.
ILLUSTRATED
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~ELEVENTH THOUSAND~
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[IMAGE: Logo of D. Van Nostrand Company ]
NEW YORK
D. VAN NOSTRAND COMPANY
25 PARK PLACE
1918
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COPYRIGHT, 1911, BY
D. VAN NOSTRAND COMPANY
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COPYRIGHT, 1915, BY
D. VAN NOSTRAND COMPANY
𝔖𝔱𝔞𝔫𝔥𝔬𝔭𝔢 𝔓𝔯𝔢𝔰𝔰
F. H. GILSON COMPANY
BOSTON. U.S.A.
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DEDICATED TO
𝔅𝔯𝔬𝔴𝔫 𝔄𝔶𝔯𝔢𝔰, 𝔓𝔥.𝔇.
PRESIDENT OF THE UNIVERSITY OF TENNESSEE
“MY GOOD FRIEND AND GUIDE.”
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PREFACE
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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.
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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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TABLE OF CONTENTS
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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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MATHEMATICS
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.
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**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.
- - 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.
- - 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.
- - 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.
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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.
- - 007
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.
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**EXPONENTS. —** The multiplication of a by a may be represented
by aa. But here we have a further short cut, namely, a^2. This 2,
called an ~exponent~, indicates that two a’s have been multiplied
by each other; a × a × a would give us a^3, 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 a^2 were multiplied by a^2, the result would be a^5,
since a^2 signifies that 2 a’s are multiplied together, and a^3
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. a^3 × a^7 would likewise give us a^{17}, a^4 ×
a^4 would give us a^8, a^4 × a^4 × a^2 × a^2 would give us a^{12},
and so on.
**RULE. —** ~The multiplication by each other of symbols
representing similar objects is accomplished by adding their
exponents.~
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**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
a^2 is as different from a as a square foot is from a linear foot,
and a^3 is as different from a^2 as one cubic foot is from one
square foot. a^2 is a distinct symbol. a^3 is a distinct symbol,
and can only be grouped together with other a^3’s. For example, if
an algebraic expression such as this were met:
a^2 + a + ab + a^3 + 3a^2 − 2a −
ab,
to simplify it we could group together the a^2 and the + 3a^2,
giving +4a^2; the +a and the −2a give −a; the +ab and the −ab
neutralize each other; there is only one term with the symbol a^3.
Therefore the above expression simplified would be 4a^2 − a + a^3.
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, a^2, a^3, a^4,
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 a^2
× b we get a^2b. If we have ab × ab, we get a^2b^2. If we have 2 ab
× 2 ab we get 4ab; 6 a^2b^3 × 3c, we get 18 a^2b^3c; 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.
- - 010
**DIVISION. —** Just as when in arithmetic we write down ( 2 )⁄(
3 ) to mean 2 divided by 3, in algebra we write ( a )⁄( b ) to mean
a divided by b. a is called a numerator and b a denominator, and
the expression ( a )⁄( b ) is called a fraction. If a^3 is
multiplied by a^2, we have seen that the result is a^5, obtained by
adding the exponents 3 and 2. If a^3 is divided by a^2, the result
is a, which is obtained by subtracting 2 from 3. Therefore ( a^2b
)⁄( ab ) would equal a, the a in the denominator dividing into a^2
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 ( a^4b^2c^3 )⁄( a^2bc^5 ) we obtain ( a^2b )⁄( c^2 ), 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 ( 1
)⁄( a^2 ). This may be written a^{-2}, or the term b^2 may be
written ( 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, ( a^5 )⁄( a^2 ) may be written
a^5 × a^{-2}. Multiplying these two terms together, which is
accomplished by adding their exponents, would give us a^3, 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 ( a^2b^{-2}c^2 )⁄( abc^{-1 )}. Suppose all the
symbols in the denominator are placed in the numerator, then we
have a^2b^{-2}c^2a^{-1}b^{-1}c, or, simplifying, ab^{-3}c^3, which
may be further written ( ac^3 )⁄( b^3 ). 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.
- - 011
------------------------------ PROBLEMS ------------------------------
Simplify the following:
1. 2a × 3b × 3ab.
2. 12a^2bc × 4c^2b.
3. 6x × 5y × 3xy.
4. 4a^2bc × 3abc × a^5b × 6b^2.
5. ( a^2b^2c^3 )⁄( abc ).
6. ( a^4b^3c^2d )⁄( a^2d^2 ).
7. a^{-2} × b^3 × a^6b^2c.
8. abc^2 × b^{-2}a^{-1}c^5 × a^3b^3.
9. ( a^4b^{-6}c^3z )⁄( a^2b^{-2 )c}.
10. 10a^2b × 5a^{-1}bc^{-3} × ( 8ac^{-1} )⁄( b^2a^{-4 )} ×
10^{-1}a.
11. ( 5a^2b^2c^2d^2 )⁄( 45a^3 × 6d^3 ).
- - 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 ( a )⁄( b )
indicated that a was divided by b, just as in arithmetic ( 1 )⁄( 3
) indicates that 1 is divided by 3. Suppose we multiply the
fraction ( 1 )⁄( 3 ) by 3, we obtain ( 3 )⁄( 3 ), our procedure
being to multiply the numerator 1 by 3. Similarly, if we had
multiplied the fraction ( a )⁄( b ) by 3, our result would have
been ( 3a )⁄( b ).
- - 013
**RULE. —** ~The multiplication of a fraction by any quantity is
accomplished by multiplying its numerator by that quantity;~ thus,
( 2a^2 )⁄( b ) multiplied by 3a would give ( 6a^2 )⁄( b ).
Conversely, when we divide a fraction by a quantity, we multiply
its denominator by that quantity. Thus, the fraction ( a )⁄( b )
when divided by 2b gives ( a )⁄( 2b^2 ) 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 ( 1 )⁄( 3 ) the quantity 2. We will
obtain ( 3 )⁄( 5 ), which is different in value from ( 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, ( a )⁄( b ) × ( d )⁄( c ) would give
us ( ad )⁄( bc ).
- - 014
Suppose it is desired to add the fraction ( 1 )⁄( 2 ) to the
fraction ( 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.: ( 3 )⁄( 6 ) + ( 2 )⁄( 6 ) = ( 5 )⁄( 6 ),
the numerators being added if the denominators are of a common
value. Likewise, if it is desired to add ( a )⁄( b ) to ( 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 ( ad )⁄( bd ) +
( cb )⁄( bd ). The denominators now having a common value, the
fractions may be added by adding the numerators, resulting in ( ad
+ cb )⁄( bd ). Likewise, adding the fractions ( a )⁄( 3 ) + ( b )⁄(
2a ) + ( c )⁄( 3a ), we find that the common denominator in this
case is 6a. The first fraction becomes ( 2a^2 )⁄( 6a ) the second (
3b )⁄( 6a ) and the third ( 2c )⁄( 6a ), the result being the
fraction ( 2a^2 + 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.
- - 015
**LAW OF SIGNS. —** Like signs multiplied or divided give + and
unlike signs give −. Thus:
+3a × +2a gives + 6a^2,
also −3a × −2a gives + 6a^2,
while +3a × −2a gives −6a^2
or −3a × + 2a gives −6a^2;
furthermore +8a^2 divided by +2a gives +4a,
and −8a^2 divided by −2a gives +4a
while −8a^2 divided by +2a gives −4a
or +8a^2 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: 12x^2 + 9xy + 15z^2y, obtained
by multiplying each one of the separate terms by 3x successively.
- - 016
**DIVISION OF AN ALGEBRAIC EXPRESSION BY A QUANTITY. —** Dividing
the expression 6a^2 + 2a^2b + 4b^2 by 2ab would result in the
expression ( 3a^2 )⁄( b ) + a + ( 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 ( a + b )⁄(
a ) or ( 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 ( a + b )⁄( a ) is multiplied
by 3, it becomes ( 3a + 3b )⁄( a ). If the fraction ( a − b )⁄( a +
b ) is multiplied by ( 2 )⁄( 3 ) it becomes ( 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.
- - 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 /
Again, let us multiply
2a + b − 3c
a + 2b − c
____________________
2a² + ab − 3ac
4ab + 2b² − 6bc
− 2ac − bc + 3c²
_____________________________________
____________________________________________________________
\ Workup 3-2 /
and we have
2a² + 5ab − 5ac + 2b² − 7bc + 3c².
____________________________________________________________
\ Workup 3-3 /
- - 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 6a^2 + 17ab + 12b^2 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 /
3a will divide into 6a^2, 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, 6a^2 + 8ab, is placed beneath the dividend, as shown. A
line is then drawn and this quantity subtracted from the dividend,
leaving 9ab. The +12b^2 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 + 12b^2; 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.
- - 019
------------------------------ PROBLEMS ------------------------------
Solve the following problems:
1. Multiply the fraction ( 3a^2b^3c )⁄( 4x^2 )
by the quantity 3x.
2. Divide the fraction ( abc )⁄( 6d ) by the quantity 3a.
3. Multiply the fraction ( a^2b^2c^2 )⁄( xy^3 ) by
the fraction ( a^2b^2 )⁄( 6a ) by
the fraction ( x^2y )⁄( b ).
4. Multiply the expression 4x + 3y + 2z by the quantity 5x.
5. Divide the expression 8a^2b + 4a^3b^3 − 2ab^2 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 a^2 − 2ab + b^2 by a − b.
9. Divide the expression a^3 + 3a^2b + 3ab^2 + b^3 by a + b.
10. Multiply the fraction ( a + b )⁄( a − b ) by
( a − b )⁄( a − b ).
11. Multiply the fraction ( 3a )⁄( c + d ) by
( c − d )⁄( 2 ) by ( a + c )⁄( a − c ).
12. Multiply the fraction ( a^{-2}bc^3 )⁄( 4 ) by
( b )⁄( 3a^{-2 )} by ( a )⁄( b ).
- - 020
13. Add together the fractions ( 2a )⁄( b )
+ ( b )⁄( 4 ) + ( c )⁄( b ).
14. Add together the fractions ( 2 )⁄( 3a^2 )
− ( 4 )⁄( 2a ) + ( c )⁄( 6 ).
15. Add together the fractions ( 10a^2 )⁄( b )
+ ( b )⁄( 4b ) − ( x )⁄( 2c ) + ( d )⁄( 6 ).
16. Add together the fractions ( a + b )⁄( 2a )
+ ( b − c )⁄( 4b ).
17. Add together the fractions ( a )⁄( a + b )
− ( 2 )⁄( 5a ).
- - 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.
- - 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 6a^2b + 3a^3,
3a^3 is a common factor of both terms; therefore we may write the
expression, without changing its value, in the following manner:
3a^2(2b + a). The term 3a^2 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 +
4^3 + 6x^2z + 8xz, 2x is a common factor of each of the terms, and
the expression may be written 2x (6 y + 2x^2 + 3xz + 4z). It is
often desirable to factor in this simple manner.
Still further suppose we have a^2 + 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
6x^2 + 4xy − 3zx − 2zy
we have
2x(3x + 2y) − z(3x + 2y),
or,
(3x + 2y) × (2x − z).
- - 023
Now, suppose we have the expression a^2 − 2ab + b^2. 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,
9x^2 − 12xy + 4y^2 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 a^2 − b^2; 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, 2a^2 + 7ab + 3b^2 is readily detected to
be the product of 2a + b and a + 3b.
------------------------------ PROBLEMS ------------------------------
Factor the following:
1. 30 a^2b.
2. 48 a^4c.
3. 30 x^2y^4z^3.
4. 144 x^2a^2.
5. ( 12ab^2c^3 )⁄( 4a^2b^2 ).
6. ( 10xy^2 )⁄( 2x^2y ).
7. 2a^2 + ab − 2ac − bc.
- - 024
8. 3x^2 + xy + 3xc + cy.
9. 2x^2 + 5xy + 2xz + 5yz.
10. a^2 − 2ab + b^2.
11. 4x^2 − 12xy + 9y^2.
12. 81a^2 + 90ab + 25b^2.
13. 16c^2 − 48ca + 36a^2.
14. 4x^3y + 5xzy^2 − 10xzy.
15. 30ab + 15abc − 5bc.
16. 81x^2y^2 − 25a^2.
17. a^4 − 16b^4.
18. 144x^4y^2 − 64z^2.
19. 4a^2 − 8ac + 4c.
20. 16y^2 + 8xy + x^2.
21. 6y^2 − 5xy − 6x^2.
22. 4a^2 − 3ab − 10b^2.
23. 6y^2 − 13xy + 6x^2.
24. 2a^2 − 5ab − 3b^2.
25. 2a^2 + 9ab + 10b^2.
- - 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 a^2, a^4, a^6, and so on, will
be, respectively, ±a, ±a^2, and ±a^3, obtained by dividing the
exponents by 2. The plus-or-minus sign (±) shows that either +a or
−a when squared would give us ±a^2. 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 a^3, a^6, a^9, and so on, would be
respectively a, a^2 and a^3, obtained by dividing the exponents by
3. Similarly, the square root of 4a^4b^6 will be seen to be
±2a^2b^3, obtained by taking the square root of each factor of the
term. And likewise the cube root of −27a^9b^6 will be −3a^3b^2.
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.
- - 026
**SQUARE ROOT OF AN ALGEBRAIC EXPRESSION. —** Suppose we multiply
the expression a + b by itself. We obtain a^2 + 2ab + 6^2. 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 a^2 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 + b^2. 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 + b^2, 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 /
Likewise we see that the square root of 4a^2 + 12ab + 9b^2 is 2a +
3b, viz.:
4a² + 12ab + 9b² | 2a + 3b
4a² |________
____________________
4a + 3b | 2 ab + 9b²
| 2 ab + 9b²
|______________
____________________________________________________________
\ Workup 5-2 /
- - 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 a^3 + 3a^2b + 3ab^2 + b^2. 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 3a^2b + 3ab^2 + b^2. Now our trial divisor will consist
as follows: Square the part of the root already found and multiply
same by 3. This gives us 3a^2. Divide 3a^2 into the first term of
the remainder, namely, 3a^2b, 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, b^2. Our full divisor now becomes 3a^2 +
3ab + b^2. Now multiply this full divisor by b and subtract this
from the former remainder, namely, 3a^2b + 3ab^2 + b^2, 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 /
- - 028
Likewise the cube root of 27x^3 + 27x^2 + 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 /
------------------------------ PROBLEMS ------------------------------
Find the square root of the following expressions:
1. 16x^2 + 24xy + 9y^2.
2. 4a^2 + 4ab + b^2.
3. 36x^2 + 24xy + 4y^2.
4. 25a^2 − 20ab + 4b^2.
5. a^2 + 2ab + 2ac + 2bc + b^2 + c^2.
Find the cube root of the following expressions:
1. 8x^3 + 36x^2y + 54xy^2 + 27y^3.
2. x^3 + 6x^2y + 12xy^2 + 8y^3.
3. 27a^3 + 81a^2b + 81ab^2 + 27b^2.
- - 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
( a )⁄( c ) = ( 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.
- - 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.
- - 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 ( a )⁄( c ) = ( 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.
- - 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.
- - 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 = ( 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 = ( 40 − 2 )⁄( 4 + 5 ).
or,
x = ( 38 )⁄( 9 )
- - 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 = ( 40 − 4 )⁄( 8 + 5 ) = ( 36 )⁄(
13 ).
**CLEARING OF FRACTIONS. —** The above simple equations contained
no fractions. Suppose, however, that we are asked to solve the
equation
( x )⁄( 4 ) + ( 6 )⁄( 2 ) = ( 3x )⁄(
2 ) + ( 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~.
- - 035
As previously seen, in order to add together the fractions ( 1 )⁄(
2 ) and ( 1 )⁄( 3 ) we must reduce them to a common denominator, 6.
We then have ( 3 )⁄( 6 ) + ( 2 )⁄( 6 ) = ( 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 /
3 × 2 × 2 × 1 × 2 × 3 × 1 =
72.
- - 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, ( x )⁄( 4 ) + ( 6
)⁄( 2 ) = ( 3x )⁄( 2 ) + ( 5 )⁄( 6 ), we see that the common
denominator here is 12. Our equation then becomes ( 3x )⁄( 12 ) + (
36 )⁄( 12 ) = ( 18x )⁄( 12 ) + ( 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 = ( +26 )⁄( 15 ),
or
+ x = ( +26 )⁄( 15 ).
Again, let us now take the equation
( 2x )⁄( 5c ) + ( 10 )⁄( c^2 ) = ( x )⁄( 3 ).
- - 037
The ~least common denominator~ will at once be seen to be 15c^2.
Reducing all fractions to this common denominator we have
( 6cx )⁄( 15c^2 ) + ( 150 )⁄( 15c^2 ) = ( 5c^2x )⁄( 15c^2 ).
Canceling all denominators, we then have
6cx + 150 = 5c^2x.
Transposing, we have
6cx − 5c^2x = −150.
Taking x as a common factor out of both of the terms in which it
appears, we have
x(6c − 5c^2) = −150.
Dividing through by the parenthesis, we have
( −150 )⁄( 6c − 5c^2 )
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 = ( −150 )⁄( 12 − 20 ) = ( −150 )⁄( −8 ) = 18.75.
In this same manner all equations in which fractions appear are
solved.
- - 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. ( x )⁄( 5 ) + 3 = ( 8x )⁄( 2 ) + 4.
5. ( 4x )⁄( 3 ) + ( 3x )⁄( 5 ) + ( 7 )⁄( 2 ) =
( 11 )⁄( 3 ) + x.
6. ( x )⁄( c ) + ( 10 )⁄( 4c ) = ( x )⁄( 3 ) + ( x )⁄( 12c ).
Find the numerical value of x if c = 3.
- - 039
7. ( 10c )⁄( 3 ) − ( cx )⁄( c ) + ( 8 )⁄( 5c ) =
( 3cx )⁄( 10 ) + ( 15 )⁄( 2c ). Find the numerical
value of x if c = 6.
8. ( x )⁄( a + b ) − 2 + ( y )⁄( 3 ) = 1.
9. ( 2x )⁄( a ) + 3x + ( 2 )⁄( a − b ) = x − ( 3 )⁄( a^2 ).
10. ( x )⁄( a + b ) + ( x )⁄( a − b ) = 10.
11. Multiply ax + b = cx − b by 2a − x = c + 10.
12. Multiply ( a )⁄( 3 ) + b = ( c )⁄( d ) by x = y + 3.
13. Divide a^2 − b^2 = 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 15z^2 + x = 5 to 3x = −10y + 7.
18. Subtract 2a − d = 8 from 8a + d = 12.
19. Subtract 3x + 7 = 15x^2 + y from 6x + 5 = 18x^2.
20. Subtract ( 2x )⁄( 3a + b ) + c = 7 from
( 10x )⁄( 5y ) = 18.
21. Multiply ( x )⁄( 3a + b ) − ( x )⁄( 3 ) = c by
( x )⁄( c − d ) = ( 2a + b )⁄( c ).
22. Solve the equation ( 1 )⁄( x ) = −( 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?
- - 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?
- - 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:
- - 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.
- - 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.
- - 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.
- - 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.
- - 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.
- - 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. ( x )⁄( 3 ) + y = 10,
y + ( x )⁄( 5 ) = y − 3.
7. ( x )⁄( 4 ) + ( y )⁄( 3a ) = 100x + a if a = 8,
( 2x )⁄( 5 ) = y + 10.
8. 3x + y = 15,
x = 6 + 7y.
9. ( 9x )⁄( a + b ) = ( y )⁄( a − b ) − 7,
x + y = 5
if a = 6, b = 5.
10. 3x − y + 6x = 8,
y − 10 + 4y = x.
- - 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,
x^2 = 12 − 2x^2.
In solving this equation in the usual manner we obtain
3x^2 = 12,
x^2 = 4.
Taking the square root of both sides,
x = ± 2.
We first obtained the value of x^2 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:
4x^2 + 8x = 12.
- - 049
We see that none of the processes thus far discussed will do. We
must therefore find some way of grouping x^2 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 x^2 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 x^2. 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:
4x^2 + 8x = 12.
Dividing through by the coefficient of x^2, namely 4, we have
x^2 + 2x = 3.
Adding to both sides the square of one-half of the coefficient of
x, which is 2 in the term 2x,
x^2 + 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.
- - 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 x^2 and x on one side of the equation
alone, placing those in x^2 first.
(2) Divide through by the coefficient of x^2.
(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:
4x^2 = 56 − 20x,
4x^2 + 20x = 56,
x^2 + 5x = 14,
x^2 + 5x + ( 25 )⁄( 4 ) = 14 + ( 25 )⁄( 4 ),
x^2 + 5x + ( 25 )⁄( 4 ) = ( 81 )⁄( 4 ),
\bigl (x + ( 5 )⁄( 2 ) \bigr )^2 = ( 81 )⁄( 4 ).
- - 051
Taking the square root of both sides we have
x + ( 5 )⁄( 2 ) = ±( 9 )⁄( 2 ),
x = ±( 9 )⁄( 2 ) − ( 5 )⁄( 2 ),
x = 2 or −7,
~Example:~ Solve for x in the following equation:
2x^2 − 4x + 5 = x^2 + 2x − 10 − 3x^2 + 33,
2x^2 − x^2 + 3x^2 − 4x − 2x = 33 − 10 − 5,
4x^2 − 6x = 18,
x^2 −( 6x )⁄( 4 )6 = ( 18 )⁄( 4 ),
x^2 −( 3x )⁄( 2 ) = ( 18 )⁄( 4 ),
x^2 −( 3x )⁄( 2 ) + ( 9 )⁄( 16 ) = ( 18 )⁄( 4 ) + ( 9 )⁄( 16 ),
( x − ( 3 )⁄( 4 ) ) ^2 = ( 72 )⁄( 16 ) + ( 9 )⁄( 16 ),
( x − ( 3 )⁄( 4 ) ) ^2 = ( 81 )⁄( 16 ),
x − ( 3 )⁄( 4 ) = ± ( 9 )⁄( 4 ),
x = ± ( 9 )⁄( 4 ) + ( 3 )⁄( 4 ),
x = +3 or −1( 1 )⁄( 2 ).
- - 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 + 25a^2,
or,
2x = 25a^2 − 100a + 100 − 6,
x = ( (25a^2 − 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.
- - 053
Or the equation may be of the type
2a + 1 = ( 4 )⁄( √{a − x )}.
Squaring both sides we have
4a^2 + 4a + 1 = ( 16 )⁄( a − x )
Clearing fractions we have
−4a^2x − 4ax − x + a^2 + a = 16
−x(4a^2 + 4a + 1) = −4a^3 − 4a^2 − a + 16
x = ( 4a^3 + 4a^2 + a − 16 )⁄( 4a^2
+ 4a + 1 )
------------------------------ PROBLEMS ------------------------------
Solve the following equations for the value of x:
1. 5x^2 − 15x = −10.
2. 3x^2 + 4x + 20 = 44.
3. 2x^2 + 11 = x^2 + 4x + 7.
4. x^2 + 4x = 2x + 2x^2 − 8.
5. 7x + 15 − 2^2 = 3x + 18.
6. x^4 + 2x^2 = 24.
7. x^2 + ( 5x )⁄( a ) + 6x^2 = 10.
8. ( x^2 )⁄( a ) + ( x )⁄( b ) − 3 = 0.
9. 14 + 6x = ( 4x^2 )⁄( 2 ) + ( 2x )⁄( a ) − 7.
10. ( x^2 )⁄( a + b ) − 3x = 2.
11. 3x^2 + 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. ( x + a )⁄( x − a ) + ( x + b )⁄( x − b ) = 2.
- - 054
16. ( 3x + 7 )⁄( 2 ) − ( x + 2 )⁄( 6 ) = ( 12 )⁄( x + 1 ).
17. ( x^2 − 2 )⁄( 4x ) = ( x + 3 2x )⁄( 8 ).
18. ( x^2 − x − 1 )⁄( 4 ) = x^2 + 6.
19. 8 = ( 64 )⁄( √{x + 1 )}.
20. √{x + a} + 10a = 15.
21. ( x )⁄( a ) = √{x + 1}.
22. 3x + 5 = 2 + √{3x + 4}.
- - 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.
- - 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.
- - 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.
- - 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 ∝ ( 1 )⁄( p ),
then
v ∝ K × ( 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 ∝ b^2,
or
A = Kb^2.
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 ∝ ( 1 )⁄( d^2 ),
or
A = K( 1 )⁄( d^2 ),
- - 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 ∝ ( 1 )⁄( c ),
then
a ∝ ( 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.
- - 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 ∝ b^2, and the constant of the variation is 2205, what
is the value of b when a = 5?
3. a ∝ b; also a ∝ ( 1 )⁄( c ), or, a ∝ ( 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 × ( b )⁄( c ). If K = 15 and a = 6 and b = 2, what is
the value of c?
- - 061
CHAPTER X
_____________________ SOME ELEMENTS OF GEOMETRY ______________________
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.
[IMAGE: 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.]
- - 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.
[IMAGE: Two straight line segments (A-B and C-D) that do not
cross.]
IX. A definite portion of a surface or plane bounded by lines is
called a ~polygon~; thus, Fig. 3 shows a polygon.
[IMAGE: A polygon of 7 line segments, each ending at the
start of the next segment, meeting at various angles,
creating a closed figure.]
- - 063
X. A polygon bounded by three sides is called a ~triangle~ (Fig.
4).
[IMAGE: 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.]
XI. A ~polygon~ bounded by four sides is called a ~quadrangle~
(Fig. 5), and if the opposite sides are parallel, a ~parallelogram~
(Fig. 6).
[IMAGE: 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.]
[IMAGE: 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.]
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.
[IMAGE: 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.]
- - 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~.
[IMAGE: 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.]
- - 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).
[IMAGE: 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.]
[IMAGE: 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.]
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).
[IMAGE: A single line that maintains a distance from a single
point in the center of the figure, creating a circle.]
[IMAGE: 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.]
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.
- - 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~.
[IMAGE: 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.]
**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°
- - 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~.
[IMAGE: Two tringles of identical angles, but different
segment lengths.]
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.
[IMAGE: Two tringles of identical angles, and identical
segment lengths.]
- - 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.
[IMAGE: 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'.]
**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.
[IMAGE: A parallelogram of ABCD is shown with an additional
segment connecting A-C.]
- - 069
[IMAGE: A rectangle of ABCD.]
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.
[IMAGE: 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.]
- - 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.
[IMAGE: 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.]
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 = ( 1 )⁄( 2 ) BC × AH,
which means that the area of a triangle is equal to one-half of the
product of the base by the altitude.
- - 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
(\pi ). Therefore, the circumference of a circle is equal to \pi ×
the diameter.
circum. = \pi d
circum. = 2 \pi r
if r, the radius, is used instead of the diameter.
[IMAGE: A circle with a segment crossing the center, labeled
Diameter, 'd'.]
The area of a segment of a circle (Fig. 21), like the area of a
triangle, is equal to ( 1 )⁄( 2 ) of the product of the base by the
altitude, or ( 1 )⁄( 2 )a × 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 2\pi r, the area is
( 1 )⁄( 2 ) × 2\pi r × r = \pi r^2,
Area circle = \pi r^2.
[IMAGE: A segment of a circle, showing the radius as a dotted
line labeled 'r'.]
- - 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.
[IMAGE: Three squares (ABHK, BCRS, ACMN) with dotted lines
A-R, C-K, and B-N, B-M, B-F.]
To prove
ANMC = BCRS + ABHK,
or
length \overline{AC}^2 = length \overline{BC}^2 + length
\overline{AB}^2.
- - 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 = ( 1 )⁄( 2 )CR × BC
= ( 1 )⁄( 2 ) of the square BCRS,
triangle BCM = ( 1 )⁄( 2 ) CM × CO
= ( 1 )⁄( 2 ) of rectangle COFM.
Therefore
( 1 )⁄( 2 ) of square BCRS = ( 1 )⁄( 2 ) 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.
- - 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?
- - 075
CHAPTER XI
_______________ ELEMENTARY PRINCIPLES OF TRIGONOMETRY ________________
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.
[IMAGE: Figure 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:
- - 076
[IMAGE: Figure 24]
sine ∝ = ( a )⁄( c ),
cosine ∝ = ( b )⁄( c ),
tangent ∝ = ( 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.
- - 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 (
a )⁄( c ) being always the same wherever P may be chosen.
Likewise ( b )⁄( c ) and ( 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.
- - 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, ( a
)⁄( c ) and ( 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 ( a )⁄( b ) is less than 1; but
for angles between 45° and 90°, a is greater than b, and therefore
( 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.
- - 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 \alpha = ( a )⁄( c ) but if we take \beta , the
other acute angle of the right-angle triangle, we see that cosine
\beta = ( a )⁄( c ).
[IMAGE: Figure 25]
Remembering, always the fundamental definition of sine and cosine,
namely,
sine = ( Opposite side )⁄( Hypothenuse ),
cosine = ( Adjacent side )⁄( Hypothenuse ), we see that
the ~cosine \beta ~ is equal to the same thing as the ~sine
\alpha ~, therefore
sine \alpha = cosine \beta
.
- - 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 ∡\alpha + ∡\beta = 90°,
or 1 right angle. In other words ∡\alpha and ∡\beta 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 = ( b )⁄( a ),
Secant = ( c )⁄( b ),
Cosecant = ( c )⁄( a ).
- - 081
**OTHER RELATIONS OF SINE AND COSINE. —** We have seen that the
sine \alpha = ( a )⁄( c ) and the cosine \alpha = ( b )⁄( c ).
Also from geometry
a^2 + b^2 = c^2
(1)
Dividing equation (1) by c^2 we have
( a^2 )⁄( c^2 ) + ( b^2 )⁄( c^2 ) =
1
But this is nothing but the square of the sine plus the square of
the cosine of ∡\alpha , therefore
(sine \alpha )^2 + ( cosine \alpha
)^2 = 1.
Other relations whose proof is too intricate to enter into now are
sine 2 \alpha = 2\ \sin \alpha \ \cos\ \alpha ,
cos 2 \alpha = 1 − 2\ \sin^2 \alpha ,
or cos 2 \alpha = cos^2 \alpha − \sin^2 \alpha .
[IMAGE: Figure 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
\alpha . Now, as before seen, we know that
tangent\ \alpha = ( a )⁄( b
).
- - 082
Suppose b had been 1000 ft. and ∡\alpha was 40°, then
tangent\ 40° = ( a )⁄( 1000 ).
The tables show that the tangent of 40° is .8391;
then .8391 = ( a )⁄( 1000 ),
therefore a = 839.1 ft.
Thus we have found the distance across the river to be 839.1 ft.
[IMAGE: Figure 27]
Likewise in Fig. 27, suppose c = 300 and ∡\alpha = 36°, to find a
and b. We have
sine\ \alpha = ( a )⁄( c ),
or
sine\ 36° = ( a )⁄( 300 ).
- - 083
From the tables sine\ 36° = .5878.
.5878 = ( a )⁄( 300 )
a = .5878 × 300,
or
a = 176.34 ft.
Likewise
cosine\ \alpha = ( b )⁄( c ).
From table,
cosine\ 36' = .8090,
therefore
.8090 = ( b )⁄( 300 ),
or
b = 242.7 ft.
Now, if we had been told that a = 225 and b = 100, to find ∡\alpha
and c, we would have proceeded thus:
tangent\ \alpha = ( a )⁄( b ).
Therefore
tangent\ \alpha = ( 225 )⁄( 100 ),
tangent\ \alpha = 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 = ( a )⁄( c ),
sin\ 66° 4' = ( 255 )⁄( c
).
From tables, sine\ 66° 4' = .9140,
therefore
.9140 = ( 255 )⁄( c ),
or
c = ( 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.
- - 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 ∡\alpha = 30°, what are c and b?
8. If a = 500 ft. and b = 315 ft., what are ∡\alpha and c?
9. If c = 1250 ft. and ∡\alpha = 80°, what are b and a?
10. If b = 250 ft. and c = 530 ft., what are ∡\alpha and a?
- - 085
CHAPTER XII
_____________________________ LOGARITHMS _____________________________
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, 10^3) 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.
- - 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 10^2 is 100, and 10^3 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.
- - 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.
- - 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.
- - 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 ( 8 )⁄( 10 ) of the distance across the bridge between the two
given logarithms 3909 and 3927. The whole distance across is 16. (
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.
- - 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
a^2 × a^5 = a^{5+2}, or a^7,
and that
( a^8 )⁄( a^3 ) = a^{8-3}, or a^5.
- - 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
(a^2)^2 = a^4,
or
√[3]{a^6} = a^2.
In the first case a^2 squared gives a^4, and in the second case the
cube root of a^6 is a^2; 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.
- - 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.
- - 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.
- - 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.
- - 095
CHAPTER XIII
____________ ELEMENTARY PRINCIPLES OF COÖRDINATE GEOMETRY ____________
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.
- - 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.
- - 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 = 6( 1 )⁄( 2 ); we
run our eyes along the x axis until we come to 6( 1 )⁄( 2 ), then
up until we strike the curve, then back upon the y axis, where we
note that y = 2( 1 )⁄( 2 ).
[IMAGE: Figure 28]
- - 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.
- - 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, x^2
= y + 4 will not plot out a straight line because we have x^2 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.
- - 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 x^2 + x^2 = r^2; a typical equation of a parabola is y^2
= 4qx; a typical equation of a hyperbola is x^2 − y^2 = r^2, or,
also, xy = c^2.
It is noted in every one of these equations that we have the second
power of x or y, except in the equation xy = c^2, 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.
- - 101
Let us take the equation of a circle, namely, x^2 + y^2 = 5^2, and
plot it as shown in Fig. 29.
[IMAGE: Figure 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 =
5^2. 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.
[IMAGE: Figure 30]
- - 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), y^2 = 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.
- - 103
[IMAGE: Figure 31]
[IMAGE: Figure 32]
- - 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.
- - 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.
[IMAGE: Figure 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,
x^2 + y^2 = 4^2,
(x − 2)^2 + y^2 =
5^2,
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.
- - 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:
- - 107
+-------------------------------+
|_____VOLTS_____|____AMPERES____|
| 122 | 0 |
| 120 | 5 |
| 118 | 10 |
| 116 | 15 |
| 114 | 19 |
| 111 | 22 |
| 107 | 25 |
+-------------------------------+
[IMAGE: Figure 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.
- - 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. x^2 + y^2 = 36.
6. x^2 = 16y.
7. x^2 − y^2 = 16.
8. 3x^2 + (y − 2)^2 = 25.
Find the intersections of the following curves and lines:
1. 3x + y = 10,
4x − y = 6.
2. x^2 + y^2 = 81,
x − y = 10.
3. xy = 40,
3x + y = 5.
- - 109
Plot the following volt-ampere curve:
+-------------------------------+
|_____VOLTS_____|____AMPERES____|
| 550 | 0 |
| 548 | 20 |
| 545 | 39 |
| 541 | 55 |
| 536 | 79 |
| 529 | 91 |
| 521 | 102 |
| 510 | 115 |
+-------------------------------+
- - 110
CHAPTER XIV
_______________ ELEMENTARY PRINCIPLES OF THE CALCULUS ________________
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~.
- - 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 \Delta x or Δ x. 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 \Delta a or Δ a. These quantities, namely,
Δ x and Δ a, are called the ~differential~ of x and a respectively.
[IMAGE: Figure 35]
- - 112
[IMAGE: Figure 36]
We have seen that the differential of a line of the length x is Δ
x. Now suppose we have a square each of whose sides is x, as shown
in Fig. 36. The area of that square is then x^2. Suppose now we
increase the length of each side by an infinitesimally small
amount, Δ x, making the length of each side x + Δ x. 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 x^2. The addition consists of two fine strips each x long by Δ x
wide and a small square having Δ x as the length of its side. The
area of the addition then is
(x × Δ x) + (x × Δ x) + (Δ x × Δ x)
= additional area.
(The student should note this very carefully.) Therefore the
addition equals
2xΔ x + (Δ x)^2 = additional
area.
- - 113
Now the smaller Δ x becomes, the smaller in more rapid proportion
does Δ x^2, which is the area of the small square, become. Likewise
the smaller Δ x is, the thinner do the strips whose areas are xΔ x
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 Δ x we increase
its area by the quantity 2xΔ x.
Again, if we reduce the side x of the square by the length Δ x, we
reduce the area of the square by the amount 2xΔ x. 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 x^2. We
thus see that the ~differential~ of the quantity x^2 is equal to
2xΔ x. Likewise, if we had considered the case of a cube instead of
a square, we would have found that the differential of the cube x^3
would have been 3x^2Δ x. Likewise, by more elaborate investigations
we find that the differential of x^4 = 4x^3Δ x. Summarizing, then,
the foregoing results we have
differential of x = Δ x,
differential of x^2 = 2xΔ x,
differential of x^3 = 3x^2Δ x,
differential of x^4 = 4x^3Δ x.
- - 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 Δ x 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 x^5 we get 5x^4Δ x; also, if we differentiate x^6 we
get 6x^5Δ x.
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 Δ x 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.
- - 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 Δ x, and if we can find the value
of this small quantity Δ x 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~.
- - 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 Δ x 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.
- - 117
But the differential of 3x = 3Δ x,
the differential of 4x^2 = 4 times 2xΔ x = 8xΔ x,
the differential of 2x^3 = 2 times 3x^2Δ x = 6x^2Δ x,
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
x^2 − 2x + 6 + 3x^4.
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
- - 118
The differential of (x^2 − 2x + 6 + 3x^4)
= 2xΔ x − 2Δ x + 12x^3Δ x.
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
x^2 + 4 = 6x − 10,
we get
2xΔ x + 0 = 6Δ x − 0,
or
2xΔ x = 6Δ x.
**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 + yΔ x + Δ yΔ x
+ xΔ y.
[IMAGE: Figure 37]
- - 119
Δ y Δ x is negligibly small; therefore we see that the differential
of the original area xy = xΔ y + yΔ x. 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 x^2y = x^2Δ y + 2yxΔ
x.
This law holds for any number of variables.
Differential xyz = xyΔ z + xzΔ y +
yzΔ x.
**DIFFERENTIAL OF A FRACTION. —** If we are asked to
differentiate the fraction ( x )⁄( y ) we first write it in the
form xy^{-1}, using the negative exponent; now on differentiating
we have
Differential xy^{-1} = −xy^{-2}Δ y + y^{-1}Δ x
= −( xΔ y )⁄( y^2 ) + ( Δ x )⁄( y )
Reducing to a common denominator we have
Differential xy^{-1} or ( x )⁄( y ) = −( xΔ y )⁄( y^2 ) + ( yΔ
x )⁄( y^2 )
= −( yΔ x − xΔ y )⁄( y^2 )
- - 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.
- - 121
[IMAGE: Figure 38]
Let us take any point P on this line. Its coördinates are y and x
respectively. Now choose another point P{_1} 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 + Δ x,
while the ~ordinate~ y of the old point has been increased by Δ y,
making the ~ordinate~ of the new point y + Δ y.
From Fig. 38 we see that
tan \alpha = ( Δ y )⁄( Δ x
).
Therefore, if we know the tangent \alpha and know either Δ y or Δ
x we can find the other.
- - 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 Δ x and Δ y at one point of the curve (a circle)
whose equation is x^2 + y^2 = 25. For every other point of the
circle tan \alpha or ( Δ y )⁄( Δ x ) will have a different value.
Δ x and Δ y 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
∡\alpha is the angle which it makes with the x axis. For every
point on the curve this angle will be different.
[IMAGE: Figure 39]
- - 123
**MEDIATE DIFFERENTIATION. —** Summarizing the foregoing we see
that if we know any two of the three unknowns in equation tan
\alpha = ( Δ y )⁄( Δ x ) we can find the third. Some textbooks
represent tan \alpha or ( Δ y )⁄( Δ x ) by yₓ and, ( Δ x )⁄( Δ y )
by x_{y}. This is a convenient notation and we will use it here.
Therefore we have
Δ x tan \alpha = Δ y,
( Δ y )⁄( tan a ) = Δ x,
or
Δ y = Δ x yₓ,
Δ x = Δ y x_{y}.
This shows us that if we differentiate the quantity 3x^2 as to x we
obtain 6 xΔ x, 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 3x^2 as to y = 6 x Δ y x_{y}.
Likewise if we have 4y^3 and we wish to differentiate it with
respect to x we have
Differential of 4 y^3 as to x = 12 y^2 Δ x 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.
- - 124
**LAW. —** To differentiate any expression containing x as to y,
first differentiate it as to x and then multiply by x_{y}Δ y 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
2x^2 = y^2 − 10 with respect to x we get
4xΔ x = 3y^2yₓΔ x + 0,
or
4x = 3y^2yₓ,
or
yₓ = ( 4x )⁄( 3y^2 ),
Therefore
tan \alpha = ( 4x )⁄( 3y^2 ).
From this we see that by differentiating the original equation of
the curve we got finally an equation giving the value tan \alpha
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 \alpha , and a is the angle that the tangent makes with the
x axis.
- - 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 ( Δ y )⁄( Δ x ) 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 Δ x.
Differential of the cosine x as to x = − sin xΔ x.
Differential of the log x as to x = ( 1 )⁄( x )Δ x.
Differential of the sine y as to x = cosine yyₓΔ x.
Differential of the cosine y as to x = − sine yyₓΔ x.
Differential of the log y as to x = ( 1 )⁄( y )yₓΔ x.
**MAXIMA AND MINIMA. —** Referring back to the circle, Fig. 39,
once more, we see that
x^2 + y^2 = 25.
Differentiating this equation with reference to x we have
2xΔ x + 2yyₓΔ x = 0,
or
2x + 2yyₓ = 0,
or
yₓ = −( x )⁄( y ),
Therefore
tan \alpha = −( x )⁄( y ).
- - 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 = ( 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
y^2 = 14x.
Differentiating with respect to y we have
2yΔ y = 14x_{y}Δ y,
x_{y} = ( 2y )⁄( 14 ).
Equating this to 0 we have
( 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).
[IMAGE: Figure 40]
- - 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 Δ x we would write
∫Δ x
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 Δ x as the derivative of x;
therefore the integral ∫Δ x = x. Likewise, we detect 4x^3Δ x as the
differential of x^4 therefore the integral ∫4x^3Δ x = x^4.
[IMAGE: Figure 41]
- - 128
If we consider the line AB (Fig. 35) to be made up of small parts Δ
x, we could sum up these parts thus:
Δ x + Δ x + Δ x + Δ x + Δ x + Δ x . . . . . .
for millions of parts. But integration enables us to express this
more simply and ∫Δ x means the summation of every single part Δ x
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
∫_{x=1}^{x=4} Δ
x=(x)_{x=1}^{x=4}.
- - 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
(x^2 − 1)_{x=2}^{x=3},
then
(x^2 − 1)_{x=2}^{x=3} = (3^2 − 1) − (2^2 − 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.
- - 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:
∫{x^2} Δ x=( x^3 )⁄( 3 ),
∫ 4 x^5 Δ x=( 4 x^6 )⁄( 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.
- - 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,
∫{x^2} Δ x + 2x Δ x + 3 Δ x
is the same thing as
∫ x^2 Δ x + ∫ 2 x Δ x + ∫ 3 Δ
x,
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
∫ cosine x Δ x = sine x,
∫ sine x Δ x = −cosine x.
(3) The integral of the reciprocal, which is
∫ ( 1 )⁄( x ) Δ x=\log{_e}x.[*]
[*] log{_e} 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 log{_e}.
**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.
- - 132
In Fig. 41 we have the curve PP{_1}, 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 Δ x
distance apart. Now we would have a large number of small strips
each Δ x wide and all having different heights, namely, y{_1},
y{_2}, y{_3}, y{_4}, etc.
The enumeration of all these areas would then be
y{_1} Δ x + y{_2} Δ x + y{_3} Δ x +
y{_4} Δ x, etc.
Now calculus enables us to say
Area wanted = ∫_{x=b}^{x=a} y Δ
x.
This integral ∫_{x=b}^{x=a} y Δ x cannot be readily solved. If it
were ∫ x Δ x we have seen that the result would be ( x^2 )⁄( 2 )
but this is not the case with ∫ x Δ x. 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
PP{_1} From this equation we find the value of y in terms of x; we
then substitute this value of y in the integral ∫ x Δ x, and then
having an integral of x as to itself we can readily solve it. Now,
if the equation of the curve PP{_1} 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 = ∫_{x=5 ft.}^{x=12
ft.} y Δ x
From the equation of this curve we have
xy = 10,
y = ( 10 )⁄( x ).
- - 133
[IMAGE: Figure 42]
Therefore, substituting we have
Shaded area = ∫_{x=5}^{x=12} ( 10 )⁄( x ) Δ x.
Area = 10 (\log{_e}x)_{x=5}^{x=12}
= 10 (\log{_e}12) − (\log{_e} 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.
- - 134
------------------------------ PROBLEMS ------------------------------
1. Differentiate 2x^3 as to x.
2. Differentiate 12x^2 as to x.
3. Differentiate 8x^5 as to x.
4. Differentiate 3x^2 + 4x + 10 = 5x^2 as to x.
5. Differentiate 4y^2 − 3x as to y.
6. Differentiate 14y^4x^3 as to y.
7. Differentiate ( x^2 )⁄( y ) as to x.
8. Differentiate 2y^2 − 4qx as to y.
Find yₓ, in the following equations:
9. x^2 + 2y^2 = 100.
10. x^3 + y = 5.
11. x^2 − y^2 = 25.
12. 5 xy = 12.
13. What angle does the tangent line to the circle x^2 + y^2 =
9 make with the x axis at the point where x = 2?
14. What is the minimum value of y in the equation x^2 = 15y?
15. Solve ∫2x^3 Δ x.
16. Solve ∫5x^2Δ x.
17. Solve ∫10axΔ x +5x^2Δ x + 3Δ x
18. Solve ∫ 3 sine x Δ x.
19. Solve ∫ 2 cosine x Δ x.
20. Solve ∫_{x=2}^{x=5} 3 x^2 Δ x.
21. Solve ∫_{x=2}^{x=18} y Δ x 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 ( y^2 )⁄( x^2 ) as to x.
- - 135
The following tables are reproduced from Ames and Bliss’s “Manual
of Experimental Physics” by permission of the American Book
Company.
- - 136
REFERENCE MATERIAL
Tables of Logarithms and Trigonometry
- - 136
LOGARITHMS 100 TO 1000
+---------------------------------------+
|____|__0___|__1___|__2___|__3___|__4___|
| 10 | 0000 | 0043 | 0086 | 0128 | 0170 |
| 11 | 0414 | 0453 | 0492 | 0531 | 0569 |
| 12 | 0792 | 0828 | 0864 | 0899 | 0934 |
| 13 | 1139 | 1173 | 1206 | 1239 | 1271 |
| 14 | 1461 | 1492 | 1523 | 1553 | 1584 |
| 15 | 1761 | 1790 | 1818 | 1847 | 1875 |
| 16 | 2041 | 2068 | 2095 | 2122 | 2148 |
| 17 | 2304 | 2330 | 2355 | 2380 | 2405 |
| 18 | 2553 | 2577 | 2601 | 2625 | 2648 |
| 19 | 2788 | 2810 | 2833 | 2856 | 2878 |
| 20 | 3010 | 3032 | 3054 | 3075 | 3096 |
| 21 | 3222 | 3243 | 3263 | 3284 | 3304 |
| 22 | 3424 | 3444 | 3464 | 3483 | 3502 |
| 23 | 3617 | 3636 | 3655 | 3674 | 3692 |
| 24 | 3802 | 3820 | 3838 | 3856 | 3874 |
| 25 | 3979 | 3997 | 4014 | 4031 | 4048 |
| 26 | 4150 | 4166 | 4183 | 4200 | 4216 |
| 27 | 4314 | 4330 | 4346 | 4362 | 4378 |
| 28 | 4472 | 4487 | 4502 | 4518 | 4533 |
| 29 | 4624 | 4639 | 4654 | 4669 | 4683 |
| 30 | 4771 | 4786 | 4800 | 4814 | 4829 |
| 31 | 4914 | 4928 | 4942 | 4955 | 4969 |
| 32 | 5051 | 5065 | 5079 | 5092 | 5105 |
| 33 | 5185 | 5198 | 5211 | 5224 | 5237 |
| 34 | 5315 | 5328 | 5340 | 5353 | 5366 |
| 35 | 5441 | 5453 | 5465 | 5478 | 5490 |
| 36 | 5563 | 5575 | 5587 | 5599 | 5611 |
| 37 | 5682 | 5694 | 5705 | 5717 | 5729 |
| 38 | 5798 | 5809 | 5821 | 5832 | 5843 |
| 39 | 5911 | 5922 | 5933 | 5944 | 5955 |
| 40 | 6021 | 6031 | 6042 | 6053 | 6064 |
| 41 | 6128 | 6138 | 6149 | 6160 | 6170 |
| 42 | 6232 | 6243 | 6253 | 6263 | 6274 |
| 43 | 6335 | 6345 | 6355 | 6365 | 6375 |
| 44 | 6435 | 6444 | 6454 | 6464 | 6474 |
| 45 | 6532 | 6542 | 6551 | 6561 | 6571 |
| 46 | 6628 | 6637 | 6646 | 6656 | 6665 |
| 47 | 6721 | 6730 | 6739 | 6749 | 6758 |
| 48 | 6812 | 6821 | 6830 | 6839 | 6848 |
| 49 | 6902 | 6911 | 6920 | 6928 | 6937 |
| 50 | 6990 | 6998 | 7007 | 7016 | 7024 |
| 51 | 7076 | 7084 | 7093 | 7101 | 7110 |
| 52 | 7160 | 7168 | 7177 | 7185 | 7193 |
| 53 | 7243 | 7251 | 7259 | 7267 | 7275 |
| 54 | 7324 | 7332 | 7340 | 7348 | 7356 |
+---------------------------------------+
+---------------------------------------+
|____|__5___|__6___|__7___|__8___|__9___|
| 10 | 0212 | 0253 | 0294 | 0334 | 0374 |
| 11 | 0607 | 0645 | 0682 | 0719 | 0755 |
| 12 | 0969 | 1004 | 1038 | 1072 | 1106 |
| 13 | 1303 | 1335 | 1367 | 1399 | 1430 |
| 14 | 1614 | 1644 | 1673 | 1703 | 1732 |
| 15 | 1903 | 1931 | 1959 | 1987 | 2014 |
| 16 | 2175 | 2201 | 2227 | 2253 | 2279 |
| 17 | 2430 | 2455 | 2480 | 2504 | 2529 |
| 18 | 2672 | 2695 | 2718 | 2742 | 2765 |
| 19 | 2900 | 2923 | 2945 | 2967 | 2989 |
| 20 | 3118 | 3139 | 3160 | 3181 | 3201 |
| 21 | 3324 | 3345 | 3365 | 3385 | 3404 |
| 22 | 3522 | 3541 | 3560 | 3579 | 3598 |
| 23 | 3711 | 3729 | 3747 | 3766 | 3784 |
| 24 | 3892 | 3909 | 3927 | 3945 | 3962 |
| 25 | 4065 | 4082 | 4099 | 4116 | 4133 |
| 26 | 4232 | 4249 | 4265 | 4281 | 4298 |
| 27 | 4393 | 4409 | 4425 | 4440 | 4456 |
| 28 | 4548 | 4564 | 4579 | 4594 | 4609 |
| 29 | 4698 | 4713 | 4728 | 4742 | 4757 |
| 30 | 4843 | 4857 | 4871 | 4886 | 4900 |
| 31 | 4983 | 4997 | 5011 | 5024 | 5038 |
| 32 | 5119 | 5132 | 5145 | 5159 | 5172 |
| 33 | 5250 | 5263 | 5276 | 5289 | 5302 |
| 34 | 5378 | 5391 | 5403 | 5416 | 5428 |
| 35 | 5502 | 5514 | 5527 | 5539 | 5551 |
| 36 | 5623 | 5635 | 5647 | 5658 | 5670 |
| 37 | 5740 | 5752 | 5763 | 5775 | 5786 |
| 38 | 5855 | 5866 | 5877 | 5888 | 5899 |
| 39 | 5966 | 5977 | 5988 | 5999 | 6010 |
| 40 | 6075 | 6085 | 6096 | 6107 | 6117 |
| 41 | 6180 | 6191 | 6201 | 6212 | 6222 |
| 42 | 6284 | 6294 | 6304 | 6314 | 6325 |
| 43 | 6385 | 6395 | 6405 | 6415 | 6425 |
| 44 | 6484 | 6493 | 6503 | 6513 | 6522 |
| 45 | 6580 | 6590 | 6599 | 6609 | 6618 |
| 46 | 6675 | 6684 | 6693 | 6702 | 6712 |
| 47 | 6767 | 6776 | 6785 | 6794 | 6803 |
| 48 | 6857 | 6866 | 6875 | 6884 | 6893 |
| 49 | 6946 | 6955 | 6964 | 6972 | 6981 |
| 50 | 7033 | 7042 | 7050 | 7059 | 7067 |
| 51 | 7118 | 7126 | 7135 | 7143 | 7152 |
| 52 | 7202 | 7210 | 7218 | 7226 | 7235 |
| 53 | 7284 | 7292 | 7300 | 7308 | 7316 |
| 54 | 7364 | 7372 | 7380 | 7388 | 7396 |
+---------------------------------------+
+-------------------------------------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|__6___|__7___|__8___|__9___|
| 10 | __ Use preceding Table __ |
| 11 | 4 | 8 | 11 | 15 | 19 | 23 | 26 | 30 | 34 |
| 12 | 3 | 7 | 10 | 14 | 17 | 21 | 24 | 28 | 31 |
| 13 | 3 | 6 | 10 | 13 | 16 | 19 | 23 | 26 | 29 |
| 14 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
| 15 | 3 | 6 | 8 | 11 | 14 | 17 | 20 | 22 | 25 |
| 16 | 3 | 5 | 8 | 11 | 13 | 16 | 18 | 21 | 24 |
| 17 | 2 | 5 | 7 | 10 | 12 | 15 | 17 | 20 | 22 |
| 18 | 2 | 5 | 7 | 9 | 12 | 14 | 16 | 19 | 21 |
| 19 | 2 | 4 | 7 | 9 | 11 | 13 | 16 | 18 | 20 |
| 20 | 2 | 4 | 6 | 8 | 11 | 13 | 15 | 17 | 19 |
| 21 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
| 22 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 15 | 17 |
| 23 | 2 | 4 | 6 | 7 | 9 | 11 | 13 | 15 | 17 |
| 24 | 2 | 4 | 5 | 7 | 9 | 11 | 12 | 14 | 16 |
| 25 | 2 | 3 | 5 | 7 | 9 | 10 | 12 | 14 | 15 |
| 26 | 2 | 3 | 5 | 7 | 8 | 10 | 11 | 13 | 15 |
| 27 | 2 | 3 | 5 | 6 | 8 | 9 | 11 | 13 | 14 |
| 28 | 2 | 3 | 5 | 6 | 8 | 9 | 11 | 12 | 14 |
| 29 | 1 | 3 | 4 | 6 | 7 | 9 | 10 | 12 | 13 |
| 30 | 1 | 3 | 4 | 6 | 7 | 9 | 10 | 11 | 13 |
| 31 | 1 | 3 | 4 | 6 | 7 | 8 | 10 | 11 | 12 |
| 32 | 1 | 3 | 4 | 5 | 7 | 8 | 9 | 11 | 12 |
| 33 | 1 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 |
| 34 | 1 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 |
| 35 | 1 | 2 | 4 | 5 | 6 | 7 | 9 | 10 | 11 |
| 36 | 1 | 2 | 4 | 5 | 6 | 7 | 8 | 10 | 11 |
| 37 | 1 | 2 | 3 | 5 | 6 | 7 | 8 | 9 | 10 |
| 38 | 1 | 2 | 3 | 5 | 6 | 7 | 8 | 9 | 10 |
| 39 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 10 |
| 40 | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 |
| 41 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 42 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 43 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 44 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 45 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 46 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 8 |
| 47 | 1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 | 8 |
| 48 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 |
| 49 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 8 |
| 50 | 1 | 2 | 3 | 3 | 4 | 5 | 6 | 7 | 8 |
| 51 | 1 | 2 | 3 | 3 | 4 | 5 | 6 | 7 | 8 |
| 52 | 1 | 2 | 2 | 3 | 4 | 5 | 6 | 7 | 7 |
| 53 | 1 | 2 | 2 | 3 | 4 | 5 | 6 | 6 | 7 |
| 54 | 1 | 2 | 2 | 3 | 4 | 5 | 6 | 6 | 7 |
+-------------------------------------------------------------------+
- - 137
LOGARITHMS 100 TO 1000
+---------------------------------------+
|____|__0___|__1___|__2___|__3___|__4___|
| 55 | 7404 | 7412 | 7419 | 7427 | 7435 |
| 56 | 7482 | 7490 | 7497 | 7505 | 7513 |
| 57 | 7559 | 7566 | 7574 | 7582 | 7589 |
| 58 | 7634 | 7642 | 7649 | 7657 | 7664 |
| 59 | 7709 | 7716 | 7723 | 7731 | 7738 |
| 60 | 7782 | 7789 | 7796 | 7803 | 7810 |
| 61 | 7853 | 7860 | 7868 | 7875 | 7882 |
| 62 | 7924 | 7931 | 7938 | 7945 | 7952 |
| 63 | 7993 | 8000 | 8007 | 8014 | 8021 |
| 64 | 8062 | 8069 | 8075 | 8082 | 8089 |
| 65 | 8129 | 8136 | 8142 | 8149 | 8156 |
| 66 | 8195 | 8202 | 8209 | 8215 | 8222 |
| 67 | 8261 | 8267 | 8274 | 8280 | 8287 |
| 68 | 8325 | 8331 | 8338 | 8344 | 8351 |
| 69 | 8388 | 8395 | 8401 | 8407 | 8414 |
| 70 | 8451 | 8457 | 8463 | 8470 | 8476 |
| 71 | 8513 | 8519 | 8525 | 8531 | 8537 |
| 72 | 8573 | 8579 | 8585 | 8591 | 8597 |
| 73 | 8633 | 8639 | 8645 | 8651 | 8657 |
| 74 | 8692 | 8698 | 8704 | 8710 | 8716 |
| 75 | 8751 | 8756 | 8762 | 8768 | 8774 |
| 76 | 8808 | 8814 | 8820 | 8825 | 8831 |
| 77 | 8865 | 8871 | 8876 | 8882 | 8887 |
| 78 | 8921 | 8927 | 8932 | 8938 | 8943 |
| 79 | 8976 | 8982 | 8987 | 8993 | 8998 |
| 80 | 9031 | 9036 | 9042 | 9047 | 9053 |
| 81 | 9085 | 9090 | 9096 | 9101 | 9106 |
| 82 | 9138 | 9143 | 9149 | 9154 | 9159 |
| 83 | 9191 | 9196 | 9201 | 9206 | 9212 |
| 84 | 9243 | 9248 | 9253 | 9258 | 9263 |
| 85 | 9294 | 9299 | 9304 | 9309 | 9315 |
| 86 | 9345 | 9350 | 9355 | 9360 | 9365 |
| 87 | 9395 | 9400 | 9405 | 9410 | 9415 |
| 88 | 9445 | 9450 | 9455 | 9460 | 9465 |
| 89 | 9494 | 9499 | 9504 | 9509 | 9513 |
| 90 | 9542 | 9547 | 9552 | 9557 | 9562 |
| 91 | 9590 | 9595 | 9600 | 9605 | 9609 |
| 92 | 9638 | 9643 | 9647 | 9652 | 9657 |
| 93 | 9685 | 9689 | 9694 | 9699 | 9703 |
| 94 | 9731 | 9736 | 9741 | 9745 | 9750 |
| 95 | 9777 | 9782 | 9786 | 9791 | 9795 |
| 96 | 9823 | 9827 | 9832 | 9836 | 9841 |
| 97 | 9868 | 9872 | 9877 | 9881 | 9886 |
| 98 | 9912 | 9917 | 9921 | 9926 | 9930 |
| 99 | 9956 | 9961 | 9965 | 9969 | 9974 |
+---------------------------------------+
+---------------------------------------+
|____|__5___|__6___|__7___|__8___|__9___|
| 55 | 7443 | 7451 | 7459 | 7466 | 7474 |
| 56 | 7520 | 7528 | 7536 | 7543 | 7551 |
| 57 | 7597 | 7604 | 7612 | 7619 | 7627 |
| 58 | 7672 | 7679 | 7686 | 7694 | 7701 |
| 59 | 7745 | 7752 | 7760 | 7767 | 7774 |
| 60 | 7818 | 7825 | 7832 | 7839 | 7846 |
| 61 | 7889 | 7896 | 7903 | 7910 | 7917 |
| 62 | 7959 | 7966 | 7973 | 7980 | 7987 |
| 63 | 8028 | 8035 | 8041 | 8048 | 8055 |
| 64 | 8096 | 8102 | 8109 | 8116 | 8122 |
| 65 | 8162 | 8169 | 8176 | 8182 | 8189 |
| 66 | 8228 | 8235 | 8241 | 8248 | 8254 |
| 67 | 8293 | 8299 | 8306 | 8312 | 8319 |
| 68 | 8357 | 8363 | 8370 | 8376 | 8382 |
| 69 | 8420 | 8426 | 8432 | 8439 | 8445 |
| 70 | 8482 | 8488 | 8494 | 8500 | 8506 |
| 71 | 8543 | 8549 | 8555 | 8561 | 8567 |
| 72 | 8603 | 8609 | 8615 | 8621 | 8627 |
| 73 | 8663 | 8669 | 8675 | 8681 | 8686 |
| 74 | 8722 | 8727 | 8733 | 8739 | 8745 |
| 75 | 8779 | 8785 | 8791 | 8797 | 8802 |
| 76 | 8837 | 8842 | 8848 | 8854 | 8859 |
| 77 | 8893 | 8899 | 8904 | 8910 | 8915 |
| 78 | 8949 | 8954 | 8960 | 8965 | 8971 |
| 79 | 9004 | 9009 | 9015 | 9020 | 9025 |
| 80 | 9058 | 9063 | 9069 | 9074 | 9079 |
| 81 | 9112 | 9117 | 9122 | 9128 | 9133 |
| 82 | 9165 | 9170 | 9175 | 9180 | 9186 |
| 83 | 9217 | 9222 | 9227 | 9232 | 9238 |
| 84 | 9269 | 9274 | 9279 | 9284 | 9289 |
| 85 | 9320 | 9325 | 9330 | 9335 | 9340 |
| 86 | 9370 | 9375 | 9380 | 9385 | 9390 |
| 87 | 9420 | 9425 | 9430 | 9435 | 9440 |
| 88 | 9469 | 9474 | 9479 | 9484 | 9489 |
| 89 | 9518 | 9523 | 9528 | 9533 | 9538 |
| 90 | 9566 | 9571 | 9576 | 9581 | 9586 |
| 91 | 9614 | 9619 | 9624 | 9628 | 9633 |
| 92 | 9661 | 9666 | 9671 | 9675 | 9680 |
| 93 | 9708 | 9713 | 9717 | 9722 | 9727 |
| 94 | 9754 | 9759 | 9763 | 9768 | 9773 |
| 95 | 9800 | 9805 | 9809 | 9814 | 9818 |
| 96 | 9845 | 9850 | 9854 | 9859 | 9863 |
| 97 | 9890 | 9894 | 9899 | 9903 | 9908 |
| 98 | 9934 | 9939 | 9943 | 9948 | 9952 |
| 99 | 9978 | 9983 | 9987 | 9991 | 9996 |
+---------------------------------------+
+-------------------------------------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|__6___|__7___|__8___|__9___|
| 55 | 1 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 7 |
| 56 | 1 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 7 |
| 57 | 1 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 7 |
| 58 | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 |
| 59 | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 |
| 60 | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 6 |
| 61 | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 6 |
| 62 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 6 | 6 |
| 63 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 |
| 64 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 |
| 65 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 |
| 66 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 |
| 67 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 |
| 68 | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 |
| 69 | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 |
| 70 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 6 |
| 71 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 |
| 72 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 |
| 73 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 |
| 74 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 |
| 75 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 5 | 5 |
| 76 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 5 | 5 |
| 77 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 78 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 79 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 80 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 81 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 82 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 83 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 84 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 85 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 86 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
| 87 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 88 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 89 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 90 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 91 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 92 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 93 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 94 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 95 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 96 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 97 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 98 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |
| 99 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4 |
+-------------------------------------------------------------------+
- - 138
NATURAL SINES
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
| 0° | 0000 | 0017 | 0035 | 0052 | 0070 |
| 1° | 0175 | 0192 | 0209 | 0227 | 0244 |
| 2° | 0349 | 0366 | 0384 | 0401 | 0419 |
| 3° | 0523 | 0541 | 0558 | 0576 | 0593 |
| 4° | 0698 | 0715 | 0732 | 0750 | 0767 |
| 5° | 0872 | 0889 | 0906 | 0924 | 0941 |
| 6° | 1045 | 1063 | 1080 | 1097 | 1115 |
| 7° | 1219 | 1236 | 1253 | 1271 | 1288 |
| 8° | 1392 | 1409 | 1426 | 1444 | 1461 |
| 9° | 1564 | 1582 | 1599 | 1616 | 1633 |
|10° | 1736 | 1754 | 1771 | 1788 | 1805 |
|11° | 1908 | 1925 | 1942 | 1959 | 1977 |
|12° | 2079 | 2096 | 2113 | 2130 | 2147 |
|13° | 2250 | 2267 | 2284 | 2300 | 2317 |
|14° | 2419 | 2436 | 2453 | 2470 | 2487 |
|15° | 2588 | 2605 | 2622 | 2639 | 2656 |
|16° | 2756 | 2773 | 2790 | 2807 | 2823 |
|17° | 2924 | 2940 | 2957 | 2974 | 2990 |
|18° | 3090 | 3107 | 3123 | 3140 | 3156 |
|19° | 3256 | 3272 | 3289 | 3305 | 3322 |
|20° | 3420 | 3437 | 3453 | 3469 | 3486 |
|21° | 3584 | 3600 | 3616 | 3633 | 3649 |
|22° | 3746 | 3762 | 3778 | 3795 | 3811 |
|23° | 3907 | 3923 | 3939 | 3955 | 3971 |
|24° | 4067 | 4083 | 4099 | 4115 | 4131 |
|25° | 4226 | 4242 | 4258 | 4274 | 4289 |
|26° | 4384 | 4399 | 4415 | 4431 | 4446 |
|27° | 4540 | 4555 | 4571 | 4586 | 4602 |
|28° | 4695 | 4710 | 4726 | 4741 | 4756 |
|29° | 4848 | 4863 | 4879 | 4894 | 4909 |
|30° | 5000 | 5015 | 5030 | 5045 | 5060 |
|31° | 5150 | 5165 | 5180 | 5195 | 5210 |
|32° | 5299 | 5314 | 5329 | 5344 | 5358 |
|33° | 5446 | 5461 | 5476 | 5490 | 5505 |
|34° | 5592 | 5606 | 5621 | 5635 | 5650 |
|35° | 5736 | 5750 | 5764 | 5779 | 5793 |
|36° | 5878 | 5892 | 5906 | 5920 | 5934 |
|37° | 6018 | 6032 | 6046 | 6060 | 6074 |
|38° | 6157 | 6170 | 6184 | 6198 | 6211 |
|39° | 6293 | 6307 | 6320 | 6334 | 6347 |
|40° | 6428 | 6441 | 6455 | 6468 | 6481 |
|41° | 6561 | 6574 | 6587 | 6600 | 6613 |
|42° | 6691 | 6704 | 6717 | 6730 | 6743 |
|43° | 6820 | 6833 | 6845 | 6858 | 6871 |
|44° | 6947 | 6959 | 6972 | 6984 | 6997 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
| 0° | 0087 | 0105 | 0122 | 0140 | 0157 |
| 1° | 0262 | 0279 | 0297 | 0314 | 0332 |
| 2° | 0436 | 0454 | 0471 | 0488 | 0506 |
| 3° | 0610 | 0628 | 0645 | 0663 | 0680 |
| 4° | 0785 | 0802 | 0819 | 0837 | 0854 |
| 5° | 0958 | 0976 | 0993 | 1011 | 1028 |
| 6° | 1132 | 1149 | 1167 | 1184 | 1201 |
| 7° | 1305 | 1323 | 1340 | 1357 | 1374 |
| 8° | 1478 | 1495 | 1513 | 1530 | 1547 |
| 9° | 1650 | 1668 | 1685 | 1702 | 1719 |
|10° | 1822 | 1840 | 1857 | 1874 | 1891 |
|11° | 1994 | 2011 | 2028 | 2045 | 2062 |
|12° | 2164 | 2181 | 2198 | 2215 | 2233 |
|13° | 2334 | 2351 | 2368 | 2385 | 2402 |
|14° | 2504 | 2521 | 2538 | 2554 | 2571 |
|15° | 2672 | 2689 | 2706 | 2723 | 2740 |
|16° | 2840 | 2857 | 2874 | 2890 | 2907 |
|17° | 3007 | 3024 | 3040 | 3057 | 3074 |
|18° | 3173 | 3190 | 3206 | 3223 | 3239 |
|19° | 3338 | 3355 | 3371 | 3387 | 3404 |
|20° | 3502 | 3518 | 3535 | 3551 | 3567 |
|21° | 3665 | 3681 | 3697 | 3714 | 3730 |
|22° | 3827 | 3843 | 3859 | 3875 | 3891 |
|23° | 3987 | 4003 | 4019 | 4035 | 4051 |
|24° | 4147 | 4163 | 4179 | 4195 | 4210 |
|25° | 4305 | 4321 | 4337 | 4352 | 4368 |
|26° | 4462 | 4478 | 4493 | 4509 | 4524 |
|27° | 4617 | 4633 | 4648 | 4664 | 4679 |
|28° | 4772 | 4787 | 4802 | 4818 | 4833 |
|29° | 4924 | 4939 | 4955 | 4970 | 4985 |
|30° | 5075 | 5090 | 5105 | 5120 | 5135 |
|31° | 5225 | 5240 | 5255 | 5270 | 5284 |
|32° | 5373 | 5388 | 5402 | 5417 | 5432 |
|33° | 5519 | 5534 | 5548 | 5563 | 5577 |
|34° | 5664 | 5678 | 5693 | 5707 | 5721 |
|35° | 5807 | 5821 | 5835 | 5850 | 5864 |
|36° | 5948 | 5962 | 5976 | 5990 | 6004 |
|37° | 6088 | 6101 | 6115 | 6129 | 6143 |
|38° | 6225 | 6239 | 6252 | 6266 | 6280 |
|39° | 6361 | 6374 | 6388 | 6401 | 6414 |
|40° | 6494 | 6508 | 6521 | 6534 | 6547 |
|41° | 6626 | 6639 | 6652 | 6665 | 6678 |
|42° | 6756 | 6769 | 6782 | 6794 | 6807 |
|43° | 6884 | 6896 | 6909 | 6921 | 6934 |
|44° | 7009 | 7022 | 7034 | 7046 | 7059 |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
| 0° | 3 | 6 | 9 | 12 | 15 |
| 1° | 3 | 6 | 9 | 12 | 15 |
| 2° | 3 | 6 | 9 | 12 | 15 |
| 3° | 3 | 6 | 9 | 12 | 15 |
| 4° | 3 | 6 | 9 | 12 | 15 |
| 5° | 3 | 6 | 9 | 12 | 14 |
| 6° | 3 | 6 | 9 | 12 | 14 |
| 7° | 3 | 6 | 9 | 12 | 14 |
| 8° | 3 | 6 | 9 | 12 | 14 |
| 9° | 3 | 6 | 9 | 11 | 14 |
|10° | 3 | 6 | 9 | 11 | 14 |
|11° | 3 | 6 | 9 | 11 | 14 |
|12° | 3 | 6 | 9 | 11 | 14 |
|13° | 3 | 6 | 9 | 11 | 14 |
|14° | 3 | 6 | 8 | 11 | 14 |
|15° | 3 | 6 | 8 | 11 | 14 |
|16° | 3 | 6 | 8 | 11 | 14 |
|17° | 3 | 6 | 8 | 11 | 14 |
|18° | 3 | 6 | 8 | 11 | 14 |
|19° | 3 | 5 | 8 | 11 | 14 |
|20° | 3 | 5 | 8 | 11 | 14 |
|21° | 3 | 5 | 8 | 11 | 14 |
|22° | 3 | 5 | 8 | 11 | 13 |
|23° | 3 | 5 | 8 | 11 | 13 |
|24° | 3 | 5 | 8 | 11 | 13 |
|25° | 3 | 5 | 8 | 11 | 13 |
|26° | 3 | 5 | 8 | 10 | 13 |
|27° | 3 | 5 | 8 | 10 | 13 |
|28° | 3 | 5 | 8 | 10 | 13 |
|29° | 3 | 5 | 8 | 10 | 13 |
|30° | 3 | 5 | 8 | 10 | 13 |
|31° | 2 | 5 | 7 | 10 | 12 |
|32° | 2 | 5 | 7 | 10 | 12 |
|33° | 2 | 5 | 7 | 10 | 12 |
|34° | 2 | 5 | 7 | 10 | 12 |
|35° | 2 | 5 | 7 | 10 | 12 |
|36° | 2 | 5 | 7 | 9 | 12 |
|37° | 2 | 5 | 7 | 9 | 12 |
|38° | 2 | 5 | 7 | 9 | 11 |
|39° | 2 | 5 | 7 | 9 | 11 |
|40° | 2 | 4 | 7 | 9 | 11 |
|41° | 2 | 4 | 7 | 9 | 11 |
|42° | 2 | 4 | 6 | 9 | 11 |
|43° | 2 | 4 | 6 | 9 | 11 |
|44° | 2 | 4 | 6 | 8 | 10 |
+---------------------------------------+
- - 139
NATURAL SINES
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
|45° | 7071 | 7083 | 7096 | 7108 | 7120 |
|46° | 7193 | 7206 | 7218 | 7230 | 7242 |
|47° | 7314 | 7325 | 7337 | 7349 | 7361 |
|48° | 7431 | 7443 | 7455 | 7466 | 7478 |
|49° | 7547 | 7559 | 7570 | 7581 | 7593 |
|50° | 7660 | 7672 | 7683 | 7694 | 7705 |
|51° | 7771 | 7782 | 7793 | 7804 | 7815 |
|52° | 7880 | 7891 | 7902 | 7912 | 7923 |
|53° | 7986 | 7997 | 8007 | 8018 | 8028 |
|54° | 8090 | 8100 | 8111 | 8121 | 8131 |
|55° | 8192 | 8202 | 8211 | 8221 | 8231 |
|56° | 8290 | 8300 | 8310 | 8320 | 8329 |
|57° | 8387 | 8396 | 8406 | 8415 | 8425 |
|58° | 8480 | 8490 | 8499 | 8508 | 8517 |
|59° | 8572 | 8581 | 8590 | 8599 | 8607 |
|60° | 8660 | 8669 | 8678 | 8686 | 8695 |
|61° | 8746 | 8755 | 8763 | 8771 | 8780 |
|62° | 8829 | 8838 | 8846 | 8854 | 8862 |
|63° | 8910 | 8918 | 8926 | 8934 | 8942 |
|64° | 8988 | 8996 | 9003 | 9011 | 9018 |
|65° | 9063 | 9070 | 9078 | 9085 | 9092 |
|66° | 9135 | 9143 | 9150 | 9157 | 9164 |
|67° | 9205 | 9212 | 9219 | 9225 | 9232 |
|68° | 9272 | 9278 | 9285 | 9291 | 9298 |
|69° | 9336 | 9342 | 9348 | 9354 | 9361 |
|70° | 9397 | 9403 | 9409 | 9415 | 9421 |
|71° | 9455 | 9461 | 9466 | 9472 | 9478 |
|72° | 9511 | 9516 | 9521 | 9527 | 9532 |
|73° | 9563 | 9568 | 9573 | 9578 | 9583 |
|74° | 9613 | 9617 | 9622 | 9627 | 9632 |
|75° | 9659 | 9664 | 9668 | 9673 | 9677 |
|76° | 9703 | 9707 | 9711 | 9715 | 9720 |
|77° | 9744 | 9748 | 9751 | 9755 | 9759 |
|78° | 9781 | 9785 | 9789 | 9792 | 9796 |
|79° | 9816 | 9820 | 9823 | 9826 | 9829 |
|80° | 9848 | 9851 | 9854 | 9857 | 9860 |
|81° | 9877 | 9880 | 9882 | 9885 | 9888 |
|82° | 9903 | 9905 | 9907 | 9910 | 9912 |
|83° | 9925 | 9928 | 9930 | 9932 | 9934 |
|84° | 9945 | 9947 | 9949 | 9951 | 9952 |
|85° | 9962 | 9963 | 9965 | 9966 | 9968 |
|86° | 9976 | 9977 | 9978 | 9979 | 9980 |
|87° | 9986 | 9987 | 9988 | 9989 | 9990 |
|88° | 9994 | 9995 | 9995 | 9996 | 9996 |
|89° | 9998 | 9999 | 9999 | 9999 | 9999 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
|45° | 7133 | 7145 | 7157 | 7169 | 7181 |
|46° | 7254 | 7266 | 7278 | 7290 | 7302 |
|47° | 7373 | 7385 | 7396 | 7408 | 7420 |
|48° | 7490 | 7501 | 7513 | 7524 | 7536 |
|49° | 7604 | 7615 | 7627 | 7638 | 7649 |
|50° | 7716 | 7727 | 7738 | 7749 | 7760 |
|51° | 7826 | 7837 | 7848 | 7859 | 7869 |
|52° | 7934 | 7944 | 7955 | 7965 | 7976 |
|53° | 8039 | 8049 | 8059 | 8070 | 8080 |
|54° | 8141 | 8151 | 8161 | 8171 | 8181 |
|55° | 8241 | 8251 | 8261 | 8271 | 8281 |
|56° | 8339 | 8348 | 8358 | 8368 | 8377 |
|57° | 8434 | 8443 | 8453 | 8462 | 8471 |
|58° | 8526 | 8536 | 8545 | 8554 | 8563 |
|59° | 8616 | 8625 | 8634 | 8643 | 8652 |
|60° | 8704 | 8712 | 8721 | 8729 | 8738 |
|61° | 8788 | 8796 | 8805 | 8813 | 8821 |
|62° | 8870 | 8878 | 8886 | 8894 | 8902 |
|63° | 8949 | 8957 | 8965 | 8973 | 8980 |
|64° | 9026 | 9033 | 9041 | 9048 | 9056 |
|65° | 9100 | 9107 | 9114 | 9121 | 9128 |
|66° | 9171 | 9178 | 9184 | 9191 | 9198 |
|67° | 9239 | 9245 | 9252 | 9259 | 9265 |
|68° | 9304 | 9311 | 9317 | 9323 | 9330 |
|69° | 9367 | 9373 | 9379 | 9385 | 9391 |
|70° | 9426 | 9432 | 9438 | 9444 | 9449 |
|71° | 9483 | 9489 | 9494 | 9500 | 9505 |
|72° | 9537 | 9542 | 9548 | 9553 | 9558 |
|73° | 9588 | 9593 | 9598 | 9603 | 9608 |
|74° | 9636 | 9641 | 9646 | 9650 | 9655 |
|75° | 9681 | 9686 | 9690 | 9694 | 9699 |
|76° | 9724 | 9728 | 9732 | 9736 | 9740 |
|77° | 9763 | 9767 | 9770 | 9774 | 9778 |
|78° | 9799 | 9803 | 9806 | 9810 | 9813 |
|79° | 9833 | 9836 | 9839 | 9842 | 9845 |
|80° | 9863 | 9866 | 9869 | 9871 | 9874 |
|81° | 9890 | 9893 | 9895 | 9898 | 9900 |
|82° | 9914 | 9917 | 9919 | 9921 | 9923 |
|83° | 9936 | 9938 | 9940 | 9942 | 9943 |
|84° | 9954 | 9956 | 9957 | 9959 | 9960 |
|85° | 9969 | 9971 | 9972 | 9973 | 9974 |
|86° | 9981 | 9982 | 9983 | 9984 | 9985 |
|87° | 9990 | 9991 | 9992 | 9993 | 9993 |
|88° | 9997 | 9997 | 9997 | 9998 | 9998 |
|89° | ~1k | ~1k | ~1k | ~1k | ~1k |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
|45° | 2 | 4 | 6 | 8 | 10 |
|46° | 2 | 4 | 6 | 8 | 10 |
|47° | 2 | 4 | 6 | 8 | 10 |
|48° | 2 | 4 | 6 | 8 | 10 |
|49° | 2 | 4 | 6 | 8 | 10 |
|50° | 2 | 4 | 6 | 7 | 9 |
|51° | 2 | 4 | 5 | 7 | 9 |
|52° | 2 | 4 | 5 | 7 | 9 |
|53° | 2 | 3 | 5 | 7 | 9 |
|54° | 2 | 3 | 5 | 7 | 9 |
|55° | 2 | 3 | 5 | 7 | 8 |
|56° | 2 | 3 | 5 | 6 | 8 |
|57° | 2 | 3 | 5 | 6 | 8 |
|58° | 2 | 3 | 5 | 6 | 8 |
|59° | 1 | 3 | 4 | 6 | 7 |
|60° | 1 | 3 | 4 | 6 | 7 |
|61° | 1 | 3 | 4 | 6 | 7 |
|62° | 1 | 3 | 4 | 5 | 7 |
|63° | 1 | 3 | 4 | 5 | 7 |
|64° | 1 | 3 | 4 | 5 | 6 |
|65° | 1 | 2 | 4 | 5 | 6 |
|66° | 1 | 2 | 4 | 5 | 6 |
|67° | 1 | 2 | 3 | 5 | 6 |
|68° | 1 | 2 | 3 | 4 | 5 |
|69° | 1 | 2 | 3 | 4 | 5 |
|70° | 1 | 2 | 3 | 4 | 5 |
|71° | 1 | 2 | 3 | 4 | 5 |
|72° | 1 | 2 | 3 | 4 | 4 |
|73° | 1 | 2 | 3 | 3 | 4 |
|74° | 1 | 2 | 2 | 3 | 4 |
|75° | 1 | 2 | 2 | 3 | 4 |
|76° | 1 | 1 | 2 | 3 | 4 |
|77° | 1 | 1 | 2 | 3 | 3 |
|78° | 1 | 1 | 2 | 2 | 3 |
|79° | 1 | 1 | 2 | 2 | 3 |
|80° | 1 | 1 | 2 | 2 | 3 |
|81° | 0 | 1 | 1 | 2 | 2 |
|82° | 0 | 1 | 1 | 2 | 2 |
|83° | 0 | 1 | 1 | 1 | 2 |
|84° | 0 | 1 | 1 | 1 | 2 |
|85° | 0 | 1 | 1 | 1 | 1 |
|86° | 0 | 0 | 1 | 1 | 1 |
|87° | 0 | 0 | 0 | 1 | 1 |
|88° | 0 | 0 | 0 | 0 | 0 |
|89° | 0 | 0 | 0 | 0 | 0 |
+---------------------------------------+
- - 140
NATURAL COSINES
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
| 0° |1,000 | ~1k | ~1k | ~1k | ~1k |
| 1° | 9998 | 9998 | 9998 | 9997 | 9997 |
| 2° | 9994 | 9993 | 9993 | 9992 | 9991 |
| 3° | 9986 | 9985 | 9984 | 9983 | 9982 |
| 4° | 9976 | 9974 | 9973 | 9972 | 9971 |
| 5° | 9962 | 9960 | 9959 | 9957 | 9956 |
| 6° | 9945 | 9943 | 9942 | 9940 | 9938 |
| 7° | 9925 | 9923 | 9921 | 9919 | 9917 |
| 8° | 9903 | 9900 | 9898 | 9895 | 9893 |
| 9° | 9877 | 9874 | 9871 | 9869 | 9866 |
|10° | 9848 | 9845 | 9842 | 9839 | 9836 |
|11° | 9816 | 9813 | 9810 | 9806 | 9803 |
|12° | 9781 | 9778 | 9774 | 9770 | 9767 |
|13° | 9744 | 9740 | 9736 | 9732 | 9728 |
|14° | 9703 | 9699 | 9694 | 9690 | 9686 |
|15° | 9659 | 9655 | 9650 | 9646 | 9641 |
|16° | 9613 | 9608 | 9603 | 9598 | 9593 |
|17° | 9563 | 9558 | 9553 | 9548 | 9542 |
|18° | 9511 | 9505 | 9500 | 9494 | 9489 |
|19° | 9455 | 9449 | 9444 | 9438 | 9432 |
|20° | 9397 | 9391 | 9385 | 9379 | 9373 |
|21° | 9336 | 9330 | 9323 | 9317 | 9311 |
|22° | 9272 | 9265 | 9259 | 9252 | 9245 |
|23° | 9205 | 9198 | 9191 | 9184 | 9178 |
|24° | 9135 | 9128 | 9121 | 9114 | 9107 |
|25° | 9063 | 9056 | 9048 | 9041 | 9033 |
|26° | 8988 | 8980 | 8973 | 8965 | 8957 |
|27° | 8910 | 8902 | 8894 | 8886 | 8878 |
|28° | 8829 | 8821 | 8813 | 8805 | 8796 |
|29° | 8746 | 8738 | 8729 | 8721 | 8712 |
|30° | 8660 | 8652 | 8643 | 8634 | 8625 |
|31° | 8572 | 8563 | 8554 | 8545 | 8536 |
|32° | 8480 | 8471 | 8462 | 8453 | 8443 |
|33° | 8387 | 8377 | 8368 | 8358 | 8348 |
|34° | 8290 | 8281 | 8271 | 8261 | 8251 |
|35° | 8192 | 8181 | 8171 | 8161 | 8151 |
|36° | 8090 | 8080 | 8070 | 8059 | 8049 |
|37° | 7986 | 7976 | 7965 | 7955 | 7944 |
|38° | 7880 | 7869 | 7859 | 7848 | 7837 |
|39° | 7771 | 7760 | 7749 | 7738 | 7727 |
|40° | 7660 | 7649 | 7638 | 7627 | 7615 |
|41° | 7547 | 7536 | 7524 | 7513 | 7501 |
|42° | 7431 | 7420 | 7408 | 7396 | 7385 |
|43° | 7314 | 7302 | 7290 | 7278 | 7266 |
|44° | 7193 | 7181 | 7169 | 7157 | 7145 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
| 0° | 0000 | 9999 | 9999 | 9999 | 9999 |
| 1° | 9997 | 9996 | 9996 | 9995 | 9995 |
| 2° | 9990 | 9990 | 9989 | 9988 | 9987 |
| 3° | 9981 | 9980 | 9979 | 9978 | 9977 |
| 4° | 9969 | 9968 | 9966 | 9965 | 9963 |
| 5° | 9954 | 9952 | 9951 | 9949 | 9947 |
| 6° | 9936 | 9934 | 9932 | 9930 | 9928 |
| 7° | 9914 | 9912 | 9910 | 9907 | 9905 |
| 8° | 9890 | 9888 | 9885 | 9882 | 9880 |
| 9° | 9863 | 9860 | 9857 | 9854 | 9851 |
|10° | 9833 | 9829 | 9826 | 9823 | 9820 |
|11° | 9799 | 9796 | 9792 | 9789 | 9785 |
|12° | 9763 | 9759 | 9755 | 9751 | 9748 |
|13° | 9724 | 9720 | 9715 | 9711 | 9707 |
|14° | 9681 | 9677 | 9673 | 9668 | 9664 |
|15° | 9636 | 9632 | 9627 | 9622 | 9617 |
|16° | 9588 | 9583 | 9578 | 9573 | 9568 |
|17° | 9537 | 9532 | 9527 | 9521 | 9516 |
|18° | 9483 | 9478 | 9472 | 9466 | 9461 |
|19° | 9426 | 9421 | 9415 | 9409 | 9403 |
|20° | 9367 | 9361 | 9354 | 9348 | 9342 |
|21° | 9304 | 9298 | 9291 | 9285 | 9278 |
|22° | 9239 | 9232 | 9225 | 9219 | 9212 |
|23° | 9171 | 9164 | 9157 | 9150 | 9143 |
|24° | 9100 | 9092 | 9085 | 9078 | 9070 |
|25° | 9026 | 9018 | 9011 | 9003 | 8996 |
|26° | 8949 | 8942 | 8934 | 8926 | 8918 |
|27° | 8870 | 8862 | 8854 | 8846 | 8838 |
|28° | 8788 | 8780 | 8771 | 8763 | 8755 |
|29° | 8704 | 8695 | 8686 | 8678 | 8669 |
|30° | 8616 | 8607 | 8599 | 8590 | 8581 |
|31° | 8526 | 8517 | 8508 | 8499 | 8490 |
|32° | 8434 | 8425 | 8415 | 8406 | 8396 |
|33° | 8339 | 8329 | 8320 | 8310 | 8300 |
|34° | 8241 | 8231 | 8221 | 8211 | 8202 |
|35° | 8141 | 8131 | 8121 | 8111 | 8100 |
|36° | 8039 | 8028 | 8018 | 8007 | 7997 |
|37° | 7934 | 7923 | 7912 | 7902 | 7891 |
|38° | 7826 | 7815 | 7804 | 7793 | 7782 |
|39° | 7716 | 7705 | 7694 | 7683 | 7672 |
|40° | 7604 | 7593 | 7581 | 7570 | 7559 |
|41° | 7490 | 7478 | 7466 | 7455 | 7443 |
|42° | 7373 | 7361 | 7349 | 7337 | 7325 |
|43° | 7254 | 7242 | 7230 | 7218 | 7206 |
|44° | 7133 | 7120 | 7108 | 7096 | 7083 |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
| 0° | 0 | 0 | 0 | 0 | 0 |
| 1° | 0 | 0 | 0 | 0 | 0 |
| 2° | 0 | 0 | 0 | 0 | 1 |
| 3° | 0 | 0 | 0 | 1 | 1 |
| 4° | 0 | 0 | 1 | 1 | 1 |
| 5° | 0 | 1 | 1 | 1 | 1 |
| 6° | 0 | 1 | 1 | 1 | 2 |
| 7° | 0 | 1 | 1 | 1 | 2 |
| 8° | 0 | 1 | 1 | 2 | 2 |
| 9° | 0 | 1 | 1 | 2 | 2 |
|10° | 1 | 1 | 2 | 2 | 3 |
|11° | 1 | 1 | 2 | 2 | 3 |
|12° | 1 | 1 | 2 | 2 | 3 |
|13° | 1 | 1 | 2 | 3 | 3 |
|14° | 1 | 1 | 2 | 3 | 4 |
|15° | 1 | 2 | 2 | 3 | 4 |
|16° | 1 | 2 | 2 | 3 | 4 |
|17° | 1 | 2 | 3 | 3 | 4 |
|18° | 1 | 2 | 3 | 4 | 5 |
|19° | 1 | 2 | 3 | 4 | 5 |
|20° | 1 | 2 | 3 | 4 | 5 |
|21° | 1 | 2 | 3 | 4 | 5 |
|22° | 1 | 2 | 3 | 4 | 5 |
|23° | 1 | 2 | 3 | 5 | 6 |
|24° | 1 | 2 | 4 | 5 | 6 |
|25° | 1 | 2 | 4 | 5 | 6 |
|26° | 1 | 3 | 4 | 5 | 6 |
|27° | 1 | 3 | 4 | 5 | 7 |
|28° | 1 | 3 | 4 | 5 | 7 |
|29° | 1 | 3 | 4 | 6 | 7 |
|30° | 1 | 3 | 4 | 6 | 7 |
|31° | 2 | 3 | 5 | 6 | 8 |
|32° | 2 | 3 | 5 | 6 | 8 |
|33° | 2 | 3 | 5 | 6 | 8 |
|34° | 2 | 3 | 5 | 7 | 8 |
|35° | 2 | 3 | 5 | 7 | 8 |
|36° | 2 | 3 | 5 | 7 | 9 |
|37° | 2 | 4 | 5 | 7 | 9 |
|38° | 2 | 4 | 5 | 7 | 9 |
|39° | 2 | 4 | 5 | 7 | 9 |
|40° | 2 | 4 | 6 | 7 | 9 |
|41° | 2 | 4 | 6 | 8 | 10 |
|42° | 2 | 4 | 6 | 8 | 10 |
|43° | 2 | 4 | 6 | 8 | 10 |
|44° | 2 | 4 | 6 | 8 | 10 |
+---------------------------------------+
N.B. - Numbers in difference column to be subtracted, not added.
- - 141
NATURAL COSINES
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
|45° | 7071 | 7059 | 7046 | 7034 | 7022 |
|46° | 6947 | 6934 | 6921 | 6909 | 6896 |
|47° | 6820 | 6807 | 6794 | 6782 | 6769 |
|48° | 6691 | 6678 | 6665 | 6652 | 6639 |
|49° | 6561 | 6547 | 6534 | 6521 | 6508 |
|50° | 6428 | 6414 | 6401 | 6388 | 6374 |
|51° | 6293 | 6280 | 6266 | 6252 | 6239 |
|52° | 6157 | 6143 | 6129 | 6115 | 6101 |
|53° | 6018 | 6004 | 5990 | 5976 | 5962 |
|54° | 5878 | 5864 | 5850 | 5835 | 5821 |
|55° | 5736 | 5721 | 5707 | 5693 | 5678 |
|56° | 5592 | 5577 | 5563 | 5548 | 5534 |
|57° | 5446 | 5432 | 5417 | 5402 | 5388 |
|58° | 5299 | 5284 | 5270 | 5255 | 5240 |
|59° | 5150 | 5135 | 5120 | 5105 | 5090 |
|60° | 5000 | 4985 | 4970 | 4955 | 4939 |
|61° | 4848 | 4833 | 4818 | 4802 | 4787 |
|62° | 4695 | 4679 | 4664 | 4648 | 4633 |
|63° | 4540 | 4524 | 4509 | 4493 | 4478 |
|64° | 4384 | 4368 | 4352 | 4337 | 4321 |
|65° | 4226 | 4210 | 4195 | 4179 | 4163 |
|66° | 4067 | 4051 | 4035 | 4019 | 4003 |
|67° | 3907 | 3891 | 3875 | 3859 | 3843 |
|68° | 3746 | 3730 | 3714 | 3697 | 3681 |
|69° | 3584 | 3567 | 3551 | 3535 | 3518 |
|70° | 3420 | 3404 | 3387 | 3371 | 3355 |
|71° | 3256 | 3239 | 3223 | 3206 | 3190 |
|72° | 3090 | 3074 | 3057 | 3040 | 3024 |
|73° | 2924 | 2907 | 2890 | 2874 | 2857 |
|74° | 2756 | 2740 | 2723 | 2706 | 2689 |
|75° | 2588 | 2571 | 2554 | 2538 | 2521 |
|76° | 2419 | 2402 | 2385 | 2368 | 2351 |
|77° | 2250 | 2233 | 2215 | 2198 | 2181 |
|78° | 2079 | 2062 | 2045 | 2028 | 2011 |
|79° | 1908 | 1891 | 1874 | 1857 | 1840 |
|80° | 1736 | 1719 | 1702 | 1685 | 1668 |
|81° | 1564 | 1547 | 1530 | 1513 | 1495 |
|82° | 1392 | 1374 | 1357 | 1340 | 1323 |
|83° | 1219 | 1201 | 1184 | 1167 | 1149 |
|84° | 1045 | 1028 | 1011 | 0993 | 0976 |
|85° | 0872 | 0854 | 0837 | 0819 | 0802 |
|86° | 0698 | 0680 | 0663 | 0645 | 0628 |
|87° | 0523 | 0506 | 0488 | 0471 | 0454 |
|88° | 0349 | 0332 | 0314 | 0297 | 0279 |
|89° | 0175 | 0157 | 0140 | 0122 | 0105 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
|45° | 7009 | 6997 | 6984 | 6972 | 6959 |
|46° | 6884 | 6871 | 6858 | 6845 | 6833 |
|47° | 6756 | 6743 | 6730 | 6717 | 6704 |
|48° | 6626 | 6613 | 6600 | 6587 | 6574 |
|49° | 6494 | 6481 | 6468 | 6455 | 6441 |
|50° | 6361 | 6347 | 6334 | 6320 | 6307 |
|51° | 6225 | 6211 | 6198 | 6184 | 6170 |
|52° | 6088 | 6074 | 6060 | 6046 | 6032 |
|53° | 5948 | 5934 | 5920 | 5906 | 5892 |
|54° | 5807 | 5793 | 5779 | 5764 | 5750 |
|55° | 5664 | 5650 | 5635 | 5621 | 5606 |
|56° | 5519 | 5505 | 5490 | 5476 | 5461 |
|57° | 5373 | 5358 | 5344 | 5329 | 5314 |
|58° | 5225 | 5210 | 5195 | 5180 | 5165 |
|59° | 5075 | 5060 | 5045 | 5030 | 5015 |
|60° | 4924 | 4909 | 4894 | 4879 | 4863 |
|61° | 4772 | 4756 | 4741 | 4726 | 4710 |
|62° | 4617 | 4602 | 4586 | 4571 | 4555 |
|63° | 4462 | 4446 | 4431 | 4415 | 4399 |
|64° | 4305 | 4289 | 4274 | 4258 | 4242 |
|65° | 4147 | 4131 | 4115 | 4099 | 4083 |
|66° | 3987 | 3971 | 3955 | 3939 | 3923 |
|67° | 3827 | 3811 | 3795 | 3778 | 3762 |
|68° | 3665 | 3649 | 3633 | 3616 | 3600 |
|69° | 3502 | 3486 | 3469 | 3453 | 3437 |
|70° | 3338 | 3322 | 3305 | 3289 | 3272 |
|71° | 3173 | 3156 | 3140 | 3123 | 3107 |
|72° | 3007 | 2990 | 2974 | 2957 | 2940 |
|73° | 2840 | 2823 | 2807 | 2790 | 2773 |
|74° | 2672 | 2656 | 2639 | 2622 | 2605 |
|75° | 2504 | 2487 | 2470 | 2453 | 2436 |
|76° | 2334 | 2317 | 2300 | 2284 | 2267 |
|77° | 2164 | 2147 | 2130 | 2113 | 2096 |
|78° | 1994 | 1977 | 1959 | 1942 | 1925 |
|79° | 1822 | 1805 | 1788 | 1771 | 1754 |
|80° | 1650 | 1633 | 1616 | 1599 | 1582 |
|81° | 1478 | 1461 | 1444 | 1426 | 1409 |
|82° | 1305 | 1288 | 1271 | 1253 | 1236 |
|83° | 1132 | 1115 | 1097 | 1080 | 1063 |
|84° | 0958 | 0941 | 0924 | 0906 | 0889 |
|85° | 0785 | 0767 | 0750 | 0732 | 0715 |
|86° | 0610 | 0593 | 0576 | 0558 | 0541 |
|87° | 0436 | 0419 | 0401 | 0384 | 0366 |
|88° | 0262 | 0244 | 0227 | 0209 | 0192 |
|89° | 0087 | 0070 | 0052 | 0035 | 0017 |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
|45° | 2 | 4 | 6 | 8 | 10 |
|46° | 2 | 4 | 6 | 8 | 10 |
|47° | 2 | 4 | 6 | 9 | 11 |
|48° | 2 | 4 | 6 | 9 | 11 |
|49° | 2 | 4 | 7 | 9 | 11 |
|50° | 2 | 4 | 7 | 9 | 11 |
|51° | 2 | 5 | 7 | 9 | 11 |
|52° | 2 | 5 | 7 | 9 | 11 |
|53° | 2 | 5 | 7 | 9 | 12 |
|54° | 2 | 5 | 7 | 9 | 12 |
|55° | 2 | 5 | 7 | 10 | 12 |
|56° | 2 | 5 | 7 | 10 | 12 |
|57° | 2 | 5 | 7 | 10 | 12 |
|58° | 2 | 5 | 7 | 10 | 12 |
|59° | 2 | 5 | 7 | 10 | 12 |
|60° | 3 | 5 | 8 | 10 | 13 |
|61° | 3 | 5 | 8 | 10 | 13 |
|62° | 3 | 5 | 8 | 10 | 13 |
|63° | 3 | 5 | 8 | 10 | 13 |
|64° | 3 | 5 | 8 | 10 | 13 |
|65° | 3 | 5 | 8 | 11 | 13 |
|66° | 3 | 5 | 8 | 11 | 13 |
|67° | 3 | 5 | 8 | 11 | 13 |
|68° | 3 | 5 | 8 | 11 | 13 |
|69° | 3 | 5 | 8 | 11 | 14 |
|70° | 3 | 5 | 8 | 11 | 14 |
|71° | 3 | 6 | 8 | 11 | 14 |
|72° | 3 | 6 | 8 | 11 | 14 |
|73° | 3 | 6 | 8 | 11 | 14 |
|74° | 3 | 6 | 8 | 11 | 14 |
|75° | 3 | 6 | 8 | 11 | 14 |
|76° | 3 | 6 | 8 | 11 | 14 |
|77° | 3 | 6 | 9 | 11 | 14 |
|78° | 3 | 6 | 9 | 11 | 14 |
|79° | 3 | 6 | 9 | 11 | 14 |
|80° | 3 | 6 | 9 | 11 | 14 |
|81° | 3 | 6 | 9 | 11 | 14 |
|82° | 3 | 6 | 9 | 12 | 14 |
|83° | 3 | 6 | 9 | 12 | 14 |
|84° | 3 | 6 | 9 | 12 | 14 |
|85° | 3 | 6 | 9 | 12 | 14 |
|86° | 3 | 6 | 9 | 12 | 15 |
|87° | 3 | 6 | 9 | 12 | 15 |
|88° | 3 | 6 | 9 | 12 | 15 |
|89° | 3 | 6 | 9 | 12 | 15 |
+---------------------------------------+
N.B. - Numbers in difference column to be subtracted, not added.
- - 142
NATURAL TANGENTS
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
| 0° |.0000 | 0017 | 0035 | 0052 | 0070 |
| 1 |.0175 | 0192 | 0209 | 0227 | 0244 |
| 2 |.0349 | 0367 | 0384 | 0402 | 0419 |
| 3 |.0524 | 0542 | 0559 | 0577 | 0594 |
| 4 |.0699 | 0717 | 0734 | 0752 | 0769 |
| 5 |.0875 | 0892 | 0910 | 0928 | 0945 |
| 6 |.1051 | 1069 | 1086 | 1104 | 1122 |
| 7 |.1228 | 1246 | 1263 | 1281 | 1299 |
| 8 |.1405 | 1423 | 1441 | 1459 | 1477 |
| 9 |.1584 | 1602 | 1620 | 1638 | 1655 |
| 10 |.1763 | 1781 | 1799 | 1817 | 1835 |
| 11 |.1944 | 1962 | 1980 | 1998 | 2016 |
| 12 |.2126 | 2144 | 2162 | 2180 | 2199 |
| 13 |.2309 | 2327 | 2345 | 2364 | 2382 |
| 14 |.2493 | 2512 | 2530 | 2549 | 2568 |
| 15 |.2679 | 2698 | 2717 | 2736 | 2754 |
| 16 |.2867 | 2886 | 2905 | 2924 | 2943 |
| 17 |.3057 | 3076 | 3096 | 3115 | 3134 |
| 18 |.3249 | 3269 | 3288 | 3307 | 3327 |
| 19 |.3443 | 3463 | 3482 | 3502 | 3522 |
| 20 |.3640 | 3659 | 3679 | 3699 | 3719 |
| 21 |.3839 | 3859 | 3879 | 3899 | 3919 |
| 22 |.4040 | 4061 | 4081 | 4101 | 4122 |
| 23 |.4245 | 4265 | 4286 | 4307 | 4327 |
| 24 |.4452 | 4473 | 4494 | 4515 | 4536 |
| 25 |.4663 | 4684 | 4706 | 4727 | 4748 |
| 26 |.4877 | 4899 | 4921 | 4942 | 4964 |
| 27 |.5095 | 5117 | 5139 | 5161 | 5184 |
| 28 |.5317 | 5340 | 5362 | 5384 | 5407 |
| 29 |.5543 | 5566 | 5589 | 5612 | 5635 |
| 30 |.5774 | 5797 | 5820 | 5844 | 5867 |
| 31 |.6009 | 6032 | 6056 | 6080 | 6104 |
| 32 |.6249 | 6273 | 6297 | 6322 | 6346 |
| 33 |.6494 | 6519 | 6544 | 6569 | 6594 |
| 34 |.6745 | 6771 | 6796 | 6822 | 6847 |
| 35 |.7002 | 7028 | 7054 | 7080 | 7107 |
| 36 |.7265 | 7292 | 7319 | 7346 | 7373 |
| 37 |.7536 | 7563 | 7590 | 7618 | 7646 |
| 38 |.7813 | 7841 | 7869 | 7898 | 7926 |
| 39 |.8098 | 8127 | 8156 | 8185 | 8214 |
| 40 |.8391 | 8421 | 8451 | 8481 | 8511 |
| 41 |.8693 | 8724 | 8754 | 8785 | 8816 |
| 42 |.9004 | 9036 | 9067 | 9099 | 9131 |
| 43 |.9325 | 9358 | 9391 | 9424 | 9457 |
| 44 |.9657 | 9691 | 9725 | 9759 | 9793 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
| 0° | 0087 | 0105 | 0122 | 0140 | 0157 |
| 1 | 0262 | 0279 | 0297 | 0314 | 0332 |
| 2 | 0437 | 0454 | 0472 | 0489 | 0507 |
| 3 | 0612 | 0629 | 0647 | 0664 | 0682 |
| 4 | 0787 | 0805 | 0822 | 0840 | 0857 |
| 5 | 0963 | 0981 | 0998 | 1016 | 1033 |
| 6 | 1139 | 1157 | 1175 | 1192 | 1210 |
| 7 | 1317 | 1334 | 1352 | 1370 | 1388 |
| 8 | 1495 | 1512 | 1530 | 1548 | 1566 |
| 9 | 1673 | 1691 | 1709 | 1727 | 1745 |
| 10 | 1853 | 1871 | 1890 | 1908 | 1926 |
| 11 | 2035 | 2053 | 2071 | 2089 | 2107 |
| 12 | 2217 | 2235 | 2254 | 2272 | 2290 |
| 13 | 2401 | 2419 | 2438 | 2456 | 2475 |
| 14 | 2586 | 2605 | 2623 | 2642 | 2661 |
| 15 | 2773 | 2792 | 2811 | 2830 | 2849 |
| 16 | 2962 | 2981 | 3000 | 3019 | 3038 |
| 17 | 3153 | 3172 | 3191 | 3211 | 3230 |
| 18 | 3346 | 3365 | 3385 | 3404 | 3424 |
| 19 | 3541 | 3561 | 3581 | 3600 | 3620 |
| 20 | 3739 | 3759 | 3779 | 3799 | 3819 |
| 21 | 3939 | 3959 | 3979 | 4000 | 4020 |
| 22 | 4142 | 4163 | 4183 | 4204 | 4224 |
| 23 | 4348 | 4369 | 4390 | 4411 | 4431 |
| 24 | 4557 | 4578 | 4599 | 4621 | 4642 |
| 25 | 4770 | 4791 | 4813 | 4834 | 4856 |
| 26 | 4986 | 5008 | 5029 | 5051 | 5073 |
| 27 | 5206 | 5228 | 5250 | 5272 | 5295 |
| 28 | 5430 | 5452 | 5475 | 5498 | 5520 |
| 29 | 5658 | 5681 | 5704 | 5727 | 5750 |
| 30 | 5890 | 5914 | 5938 | 5961 | 5985 |
| 31 | 6128 | 6152 | 6176 | 6200 | 6224 |
| 32 | 6371 | 6395 | 6420 | 6445 | 6469 |
| 33 | 6619 | 6644 | 6669 | 6694 | 6720 |
| 34 | 6873 | 6899 | 6924 | 6950 | 6976 |
| 35 | 7133 | 7159 | 7186 | 7212 | 7239 |
| 36 | 7400 | 7427 | 7454 | 7481 | 7508 |
| 37 | 7673 | 7701 | 7729 | 7757 | 7785 |
| 38 | 7954 | 7983 | 8012 | 8040 | 8069 |
| 39 | 8243 | 8273 | 8302 | 8332 | 8361 |
| 40 | 8541 | 8571 | 8601 | 8632 | 8662 |
| 41 | 8847 | 8878 | 8910 | 8941 | 8972 |
| 42 | 9163 | 9195 | 9228 | 9260 | 9293 |
| 43 | 9490 | 9523 | 9556 | 9590 | 9623 |
| 44 | 9827 | 9861 | 9896 | 9930 | 9965 |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
| 0° | 3 | 6 | 9 | 12 | 15 |
| 1 | 3 | 6 | 9 | 12 | 15 |
| 2 | 3 | 6 | 9 | 12 | 15 |
| 3 | 3 | 6 | 9 | 12 | 15 |
| 4 | 3 | 6 | 9 | 12 | 15 |
| 5 | 3 | 6 | 9 | 12 | 15 |
| 6 | 3 | 6 | 9 | 12 | 15 |
| 7 | 3 | 6 | 9 | 12 | 15 |
| 8 | 3 | 6 | 9 | 12 | 15 |
| 9 | 3 | 6 | 9 | 12 | 15 |
| 10 | 3 | 6 | 9 | 12 | 15 |
| 11 | 3 | 6 | 9 | 12 | 15 |
| 12 | 3 | 6 | 9 | 12 | 15 |
| 13 | 3 | 6 | 9 | 12 | 15 |
| 14 | 3 | 6 | 9 | 12 | 16 |
| 15 | 3 | 6 | 9 | 13 | 16 |
| 16 | 3 | 6 | 9 | 13 | 16 |
| 17 | 3 | 6 | 10 | 13 | 16 |
| 18 | 3 | 6 | 10 | 13 | 16 |
| 19 | 3 | 7 | 10 | 13 | 16 |
| 20 | 3 | 7 | 10 | 13 | 17 |
| 21 | 3 | 7 | 10 | 13 | 17 |
| 22 | 3 | 7 | 10 | 14 | 17 |
| 23 | 3 | 7 | 10 | 14 | 17 |
| 24 | 3 | 7 | 10 | 14 | 18 |
| 25 | 4 | 7 | 11 | 14 | 18 |
| 26 | 4 | 7 | 11 | 14 | 18 |
| 27 | 4 | 7 | 11 | 15 | 18 |
| 28 | 4 | 7 | 11 | 15 | 19 |
| 29 | 4 | 8 | 11 | 15 | 19 |
| 30 | 4 | 8 | 12 | 16 | 20 |
| 31 | 4 | 8 | 12 | 16 | 20 |
| 32 | 4 | 8 | 12 | 16 | 20 |
| 33 | 4 | 8 | 12 | 17 | 21 |
| 34 | 4 | 8 | 13 | 17 | 21 |
| 35 | 4 | 9 | 13 | 17 | 22 |
| 36 | 4 | 9 | 13 | 18 | 22 |
| 37 | 5 | 9 | 14 | 18 | 23 |
| 38 | 5 | 9 | 14 | 19 | 24 |
| 39 | 5 | 10 | 15 | 19 | 24 |
| 40 | 5 | 10 | 15 | 20 | 25 |
| 41 | 5 | 10 | 15 | 21 | 26 |
| 42 | 5 | 11 | 16 | 21 | 27 |
| 43 | 5 | 11 | 16 | 22 | 28 |
| 44 | 6 | 11 | 17 | 23 | 29 |
+---------------------------------------+
- - 143
NATURAL TANGENTS
+---------------------------------------+
|____|__0'__|__6'__|_12'__|_18'__|_24'__|
|45° |.00000| 0035 | 0070 | 0105 | 0141 |
| 46 |1.0355| 0392 | 0428 | 0464 | 0501 |
| 47 |1.0724| 0761 | 0799 | 0837 | 0875 |
| 48 |1.1106| 1145 | 1184 | 1224 | 1263 |
| 49 |1.1504| 1544 | 1585 | 1626 | 1667 |
| 50 |1.1918| 1960 | 2002 | 2045 | 2088 |
| 51 |1.2349| 2393 | 2437 | 2482 | 2527 |
| 52 |1.2799| 2846 | 2892 | 2938 | 2985 |
| 53 |1.3270| 3319 | 3367 | 3416 | 3465 |
| 54 |1.3764| 3814 | 3865 | 3916 | 3968 |
| 55 |1.4281| 4335 | 4388 | 4442 | 4496 |
| 56 |1.4826| 4882 | 4938 | 4994 | 5051 |
| 57 |1.5399| 5458 | 5517 | 5577 | 5637 |
| 58 |1.6003| 6066 | 6128 | 6191 | 6255 |
| 59 |1.6643| 6709 | 6775 | 6842 | 6909 |
| 60 |1.7321| 7391 | 7461 | 7532 | 7603 |
| 61 |1.8040| 8115 | 8190 | 8265 | 8341 |
| 62 |1.8807| 8887 | 8967 | 9047 | 9128 |
| 63 |1.9626| 9711 | 9797 | 9883 | 9970 |
| 64 |2.0503| 0594 | 0686 | 0778 | 0872 |
| 65 |2.1445| 1543 | 1642 | 1742 | 1842 |
| 66 |2.2460| 2566 | 2673 | 2781 | 2889 |
| 67 |2.3559| 3673 | 3789 | 3906 | 4023 |
| 68 |2.4751| 4876 | 5002 | 5129 | 5257 |
| 69 |2.6051| 6187 | 6325 | 6464 | 6605 |
| 70 |2.7475| 7625 | 7776 | 7929 | 8083 |
| 71 |2.9042| 9208 | 9375 | 9544 | 9714 |
| 72 |3.0777| 0961 | 1146 | 1334 | 1524 |
| 73 |3.2709| 2914 | 3122 | 3332 | 3544 |
| 74 |3.4874| 5105 | 5339 | 5576 | 5816 |
| 75 |3.7321| 7583 | 7848 | 8118 | 8391 |
| 76 |4.0108| 0408 | 0713 | 1022 | 1335 |
| 77 |4.3315| 3662 | 4015 | 4373 | 4737 |
| 78 |4.7046| 7453 | 7867 | 8288 | 8716 |
| 79 |5.1446| 1929 | 2422 | 2924 | 3435 |
| 80 |5.6713| 7297 | 7894 | 8502 | 9124 |
| 81 |6.3138| 3859 | 4596 | 5350 | 6122 |
| 82 |7.1154| 2066 | 3002 | 3962 | 4947 |
| 83 |8.1443| 2636 | 3863 | 5126 | 6427 |
| 84 |9.5144|9.677 |9.845 |10.02 |10.20 |
| 85 |11.43 |11.66 |11.91 |12.16 |12.43 |
| 86 |14.30 |14.67 |15.06 |15.46 |15.89 |
| 87 |19.08 |19.74 |20.45 |21.20 |22.02 |
| 88 |28.64 |30.14 |31.82 |33.69 |35.80 |
| 89 |57.29 |63.66 |71.62 |81.85 |95.49 |
+---------------------------------------+
+---------------------------------------+
|____|_30'__|_36'__|_42'__|_48'__|_54'__|
|45° | 0176 | 0212 | 0247 | 0283 | 0319 |
| 46 | 0538 | 0575 | 0612 | 0649 | 0686 |
| 47 | 0913 | 0951 | 0990 | 1028 | 1067 |
| 48 | 1303 | 1343 | 1383 | 1423 | 1463 |
| 49 | 1708 | 1750 | 1792 | 1833 | 1875 |
| 50 | 2131 | 2174 | 2218 | 2261 | 2305 |
| 51 | 2572 | 2617 | 2662 | 2708 | 2753 |
| 52 | 3032 | 3079 | 3127 | 3175 | 3222 |
| 53 | 3514 | 3564 | 3613 | 3663 | 3713 |
| 54 | 4019 | 4071 | 4124 | 4176 | 4229 |
| 55 | 4550 | 4605 | 4659 | 4715 | 4770 |
| 56 | 5108 | 5166 | 5224 | 5282 | 5340 |
| 57 | 5697 | 5757 | 5818 | 5880 | 5941 |
| 58 | 6319 | 6383 | 6447 | 6512 | 6577 |
| 59 | 6977 | 7045 | 7113 | 7182 | 7251 |
| 60 | 7675 | 7747 | 7820 | 7893 | 7966 |
| 61 | 8418 | 8495 | 8572 | 8650 | 8728 |
| 62 | 9210 | 9292 | 9375 | 9458 | 9542 |
| 63 | 0̅057 | 0̅145 | 0̅233 | 0̅323 | 0̅413 |
| 64 | 0965 | 1060 | 1155 | 1251 | 1348 |
| 65 | 1943 | 2045 | 2148 | 2251 | 2355 |
| 66 | 2998 | 3109 | 3220 | 3332 | 3445 |
| 67 | 4142 | 4262 | 4383 | 4504 | 4627 |
| 68 | 5386 | 5517 | 5649 | 5782 | 5916 |
| 69 | 6746 | 6889 | 7034 | 7179 | 7326 |
| 70 | 8239 | 8397 | 8556 | 8716 | 8878 |
| 71 | 9887 | 0̅061 | 0̅237 | 0̅415 | 0̅595 |
| 72 | 1716 | 1910 | 2106 | 2305 | 2506 |
| 73 | 3759 | 3977 | 4197 | 4420 | 4646 |
| 74 | 6059 | 6305 | 6554 | 6806 | 7062 |
| 75 | 8667 | 8947 | 9232 | 9520 | 9812 |
| 76 | 1653 | 1976 | 2303 | 2635 | 2972 |
| 77 | 5107 | 5483 | 5864 | 6252 | 6646 |
| 78 | 9152 | 9594 | 0̅045 | 0̅504 | 0̅970 |
| 79 | 3955 | 4486 | 5026 | 5578 | 6140 |
| 80 | 9758 | 0̅405 | 1̅066 | 1̅742 | 2̅432 |
| 81 | 6912 | 7720 | 8548 | 9395 | 0̅264 |
| 82 | 5958 | 6996 | 8062 | 9158 | 0̅285 |
| 83 | 7769 | 9152 | 0̅579 | 2̅052 | 3̅572 |
| 84 |10.39 |10.58 |10.78 |10.99 |11.20 |
| 85 |12.71 |13.00 |13.30 |13.62 |13.95 |
| 86 |16.35 |16.83 |17.34 |17.89 |18.46 |
| 87 |22.90 |23.86 |24.90 |26.03 |27.27 |
| 88 |38.19 |40.92 |44.07 |47.74 |52.08 |
| 89 |114.6 |143.2 |191.0 |286.5 |573.0 |
+---------------------------------------+
+---------------------------------------+
|____|__1___|__2___|__3___|__4___|__5___|
|45° | 6 | 12 | 18 | 24 | 30 |
| 46 | 6 | 12 | 18 | 24 | 31 |
| 47 | 6 | 13 | 19 | 25 | 32 |
| 48 | 7 | 13 | 20 | 26 | 33 |
| 49 | 7 | 14 | 20 | 27 | 34 |
| 50 | 7 | 14 | 21 | 29 | 36 |
| 51 | 7 | 15 | 22 | 30 | 37 |
| 52 | 8 | 15 | 23 | 31 | 39 |
| 53 | 8 | 16 | 24 | 33 | 41 |
| 54 | 8 | 17 | 26 | 34 | 43 |
| 55 | 9 | 18 | 27 | 36 | 45 |
| 56 | 9 | 19 | 28 | 38 | 48 |
| 57 | 10 | 20 | 30 | 40 | 50 |
| 58 | 10 | 21 | 32 | 42 | 53 |
| 59 | 11 | 22 | 33 | 45 | 56 |
| 60 | 12 | 23 | 35 | 48 | 60 |
| 61 | 12 | 25 | 38 | 51 | 64 |
| 62 | 13 | 27 | 40 | 54 | 68 |
| 63 | 14 | 29 | 43 | 58 | 73 |
| 64 | 15 | 31 | 46 | 62 | 78 |
| 65 | 16 | 33 | 50 | 67 | 84 |
| 66 | 18 | 36 | 54 | 72 | 91 |
| 67 | 19 | 39 | 58 | 78 | 99 |
| 68 | 21 | 42 | 64 | 85 | 108 |
| 69 | 23 | 46 | 70 | 94 | 118 |
| 70 | 25 | 50 | 76 | 103 | 130 |
| 71 | 28 | 56 | 84 | 114 | 144 |
| 72 | 31 | 62 | 94 | 127 | 160 |
| 73 | 34 | 69 | 105 | 142 | 179 |
| 74 | 39 | 78 | 118 | 160 | 202 |
| 75 | 44 | 89 | 135 | 182 | 230 |
| 76 | 50 | 102 | 154 | 209 | 265 |
| 77 | 58 | 118 | 179 | 243 | 308 |
| 78 | 68 | 138 | 210 | 285 | 363 |
| 79 | 81 | 164 | 251 | 341 | 434 |
| 80 | [*] See note below |
| 81 | |
| 82 | |
| 83 | |
| 84 | |
| 85 | |
| 86 | |
| 87 | |
| 88 | |
| 89 | |
+---------------------------------------+
[*] Difference columns cease to be useful, owing to the
rapidity with which the value of the tangent changes.
[**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.]
- - 144
[BLANK PAGE]
- - 145
ANSWERS TO PROBLEMS
- - 146
[BLANK PAGE]
- - 147
== 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. 18a^2b^2.
2. 48a^2b^2c^3.
3. 90x^2y^2.
4 144a^8b^5c^2.
5. abc^2.
6. ( a^2b^3c^2 )⁄( d ).
7. a^4b^5c.
8. a^8b^2c^7.
9. ( a^2c^2z )⁄( b^4 ).
10. ( 40a^7 )⁄( c^4 ).
11. ( b^2c^2 )⁄( 54ad ).
CHAPTER III
1. ( 9a^2b^3c )⁄( 4x ).
2. ( bc )⁄( 18d ).
3. ( a^4b^4c^2x )⁄( 6y^2 ).
4. 20x^2 + 15xy + 10xz.
5. 4a + 2a^2b^2 − b.
6. a^2 − b^2.
7. 6a^2 − ab + 5ac − 2b^2 + 6bc − 4c^2.
8. a − b.
9. a^2 + 2ab + b^2.
10. ( a + b )⁄( a − b ).
11. ( 3a^2c − 2a^2d + 3ac^2 − 3acd )⁄( 2ac + 2ad − 2c^2 − 2cd ).
12. ( c^3ba )⁄( 12 ).
13. ( 8a + b^2 + 4c )⁄( 4b ).
14. ( 4 − 12a + a^2c )⁄( 6a^2 ).
15. ( 120a^2c + 3bc − 6bx + 2bcd )⁄( 12bc ).
16. ( 3ab − ac + 2b^2 )⁄( 4ab ).
17. ( 5a^2 − 2a − 2b )⁄( 5a^2 + 5ab ).
- - 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, ( 1 )⁄( 2 ), ( 1 )⁄( 2 ), a, ( 1 )⁄( a ), ( 1 )⁄( a ),
b, b, ( 1 )⁄( b ), ( 1 )⁄( b ), c, c, c.
6. 2, 5, ( 1 )⁄( 2 ), x, ( 1 )⁄( x ), ( 1 )⁄( x ), y, y, ( 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, (4x^2 + 5zy − 10z).
15. 5b(6a + 3ac − c).
16. (9xy − 5a) (9xy + 5a).
17. (a^2 + 4b^2)(a + 2b)(a − 2b).
18. (12x^2y + 8z)(12x^2y − 8z).
19. (a^2 − 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.
- - 149
CHAPTER VI
1. x = 4( 2 )⁄( 3 ).
2. x = 2( 1 )⁄( 3 ).
3. x = 4.
4. x = −( 5 )⁄( 19 ).
5. x = ( 5 )⁄( 28 ).
6. x = 30.
7. x = 6( 133 )⁄( 168 ).
8. x = ( 9a + 9b − ay − by )⁄( 3 ).
9. x = −( 3(a − b) + 2a^2 )⁄( 2a(a − b)(a + 1) ).
10. x = ( 10(a^2 − b^2) )⁄( 2a ).
11. 2a^2x + 2ab − ax^2 − bx = c^2x − bc + 10cx − 10b.
12. ( ax )⁄( 3 ) + bx = ( cy )⁄( d ) + ( 3c )⁄( d ).
13. a −b = ( c )⁄( c + 3 ).
14. 2 = ( 10y )⁄( y + 2 ).
15. 5a + 3 = x + d + 3.
16. 6ax − 5y = 5 − 10x.
17. 15z^2 + 4x = 12 − 10y.
18. 6a + 2d = 4.
19. 3x − 2 = 3x^2 − y.
20. 8x − 10cy = 20y.
21. ( x^2 )⁄( (c − d)(3a + b) ) − ( x^2 )⁄( 3(c − d) ) = 2a + b.
22. x = −( 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 = 5( 1 )⁄( 22 ), y = -( 3 )⁄( 22 ).
9. x = −1.1, y = 6.1.
10. x = 1( 3 )⁄( 22 ), y = 2( 5 )⁄( 22 ).
- - 150
CHAPTER VIII
1. x = 2 or x = 1.
2. x = ( −2 ± 2√{19} )⁄( 3 ).
3. x = 2.
4. x = 4 or −2.
5. x = 3 or 1.
6. x = ± 2 or ± √{−6} .
7. x = −( 5 ± √{305} )⁄( 14a ).
8. x = −( a ± √{12ab^2 + a^2} )⁄( 2b ).
9. x = −( 1 − 3a ± √{51a^2 − 6a + 1} )⁄( 2a ).
10. x = +( 3(a + b) ± √{8 (a + b) + 9(a + b)^2} )⁄( 2 ).
11. x = −( 5 ± √{205} )⁄( 6 ).
12. x = −3.
13. x = 4(2 ± √{3}).
14. x = −( 3 )⁄( 4a ).
15. x = ( 2ab )⁄( a + b ).
16. x = −( 27 ± √{2425} )⁄( 16 ).
17. x = −( 3 ± √{−7} )⁄( 2 ).
18. x = −( 1±√{−299} )⁄( 6 ).
19. x = 63.
20. x = 100a^2 − 301a + 225.
21. x = ( a^2 ± a√{a^2 + 4} )⁄( 2 ).
22. x = ( −5 ± √{5} )⁄( 6 ).
CHAPTER IX
1. k = 50.
2. b = √{( 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.
- - 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.
_____________________________________________
/ \
1. x = 0; y = −3( 1 )⁄( 3 ).
y = 0; x = 10.
x = 22; y = 4.
x = −2; y = −4.
^^^^
This is a straight line and only two
pairs of corresponding values of x and
y are necessary to draw it.
\_____________________________________________/
_____________________________________________
/ \
2. x = 0; y = 3.
y = 0; x = 7( 1 )⁄( 2 ).
^^^^
This is also a straight line.
\_____________________________________________/
- - 152
_____________________________________________
/ \
3. x = 0; y = −2.
y = 0; x = 4.
^^^^
A straight line.
\_____________________________________________/
_____________________________________________
/ \
4. x = 0; y = −( 8 )⁄( 10 ).
y = 0; x = −2( 2 )⁄( 3 ).
^^^^
A straight line.
\_____________________________________________/
_____________________________________________
/ \
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}.
^^^^
This is a circle with its center at the
intersection of the x and y axes and
with a radius of 6.
\_____________________________________________/
_____________________________________________
/ \
6. y = 0; x = 0.
y = 2; x = ±√{32}.
y = 4; x = ±8.
y = 6; x = ±√{96}.
^^^^
This is a parabola and to plot it
correctly a great many corresponding
values of x and y are necessary.
\_____________________________________________/
_____________________________________________
/ \
7. y = 0; x = ±4.
y = ±1; x = ±√{17}.
y = ±3; x = ±5.
y = 5; x = +√{41}.
^^^^
This is an hyperbola and a great many
corresponding values of x and y are
necessary in order to plot the curve
correctly.
\_____________________________________________/
_____________________________________________
/ \
8. y = 0; x = ±√{7}.
x = 0; y = +7 or −3.
x = 1; y = 2 ±√{22}.
x = 2; y = 2 ±√{13}.
^^^^
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.
\_____________________________________________/
- - 153
== INTERSECTIONS OF CURVES ==
_____________________________________________
/ \
1. x = 2( 2 )⁄( 7 );
y = 3( 1 )⁄( 7 ).
^^^^
This is the intersection of 2
straight lines.
\_____________________________________________/
_____________________________________________
/ \
2. y = −5 ± √{( 31 )⁄( 2 )};
x = 5 ± √{( 31 )⁄( 2 )}.
^^^^
This is the intersection of a
straight line and a circle.
\_____________________________________________/
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. 6x^2 Δ x.
2. 24 × Δ x.
3. 40 × Δ x.
4. 6x Δ x + 4 Δ x = 15 x^2 Δ x.
5. 8y Δ y − 3x_{y} Δ y.
6. 42 y^4x^2 Δ x + 56 x^3y^3 Δ y.
7. ( 2 yx Δ x − x^2 Δ y )⁄( y^2 ).
8. 4y Δ y − 4qx_{y} Δ y.
9. y_{x} = −( x )⁄( 2y ).
10. y_{x} = −3x^2.
11. y_{x} = ( x )⁄( y ).
12. y_{x} = −( y )⁄( x ).
13. 41° 48' 10''.
14. When x = 0 at which time y also = 0.
15. ( x^4 )⁄( 2 ).
16. ( 5x^3 )⁄( 3 ).
17. 5 ax^2 + ( 5 )⁄( 3 ) x^3 + 3x.
18. −3\ cos\ x.
19. 2\ sine\ x.
20. 117.
21. 8.7795.
22. 10\ cosine\ x Δ x.
23. cos^2\ x dx −\ sin^2\ x dx.
24. ( 1 )⁄( x ) dx.
25. ( 2x^2y Δ y − 2y^2x Δ x )⁄( x^4 ).
- - fin
Appendix A
________________________ TRANSCRIBER’S 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 ('\Delta ') 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.)
- - fig
Appendix B
______________________________ FIGURES _______________________________
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