IN MEMORIAM
FLORIAN CAJORl
leath's Mathematical Monographs
Issued under the general editorship of
Webster Wells, S. B.
:i)fcssor of Mathematics in the Massachusetts Institute of Technology
GRAPHS
BY
ROBERT J. ALEY, A.M., Ph.D.
Professor or Mathematics in
Indiana University
. Heath & Co., Publishers
New York Chicago
Number 6
Price, Ten Cents
Heath's Mathematical Monographs
Number 6
GRAPHS
ALEY
Dr. Aley
has prepared a valuable
Chapter on Graphs
for
Wells's Essentials of Algebra
and
Wells's New Higher Algebra.
The editions containing this
chapter will be supplied
when specially ordered.
HEATH'S MATHEMATICAL MONOGRAPHS
Number 6
GRAPHS
BY
ROBERT J. ALEY, A.M., Ph.D.
■A
PROFESSOR OF MATHEMATICS IN
INDIANA UNIVERSITY
T
c-
BOSTON, U.S.A.
D. C. HEATH & CO., PUBLISHERS
1902
(an ' /
Copyright, 1902,
By D. C. Heath & Co.
CAJORI
INTRODUCTORY STATEMENT
At the present time the Graph is used so extensively
in many lines of work that it is necessary for the non-
technical reader to know something of it. When we
note that the fluctuations in the price of wheat, the
changes of temperature for a month, the age of conver-
sion in children, the advancement in learning a trade,
the strain on a girder under different loads, the death
rate at different ages, and the solutions of numerical and
algebraical problems have alike been subjected to graphi-
cal methods, we conclude that elementary mathematics
should take some note of the subject.
Graphical methods permeate so many subjects and
may be used so freely in the different parts of elemen-
tary mathematics that it seems that there may be use for
a brief treatment outside of the text-book. Such a treat-
ment in the hands of the teacher gives him the power to
use it whenever the occasion demands. Such a treat-
ment may also help the general reader to an understand-
ing of the graphical treatment now given to so many
subjects.
GRAPHS.
Definitions.
A Graph is a representation by means of lines,
straight or curved, of some set of measured or
numerically represented facts.
Axes. Two lines intersecting at right angles, as
in Fig. I, are called the Axes of Coordinates. OXy
the horizontal one, is the /-axis, or Axis of Abscis-
II
I
-l-ar,-f2/
—x.-y
III
+ a?,-2/
IV
Y'
Fig. I.
sas ; O Y, the vertical one, is the K-axis, or Axis of
Ordinates. O, the intersection of the axes, is the
origin.
Quadrants. The axes divide the plane into four
parts, called quadrants. These quadrants are num-
bered from I to IV, as in Fig. i.
I
2 Graphs.
Coordinates. A po,int is located when its distance
,ujid direcliori fr6m each of the axes is known. The
distance from OV is the ;r-distance, or abscissa.
The distance from OX is the ^-distance, or ordinate.
The two distances constitute the Coordinates of the
point. A point is denoted by the symbol (x, }^),
where ;r is the abscissa and j^ the ordinate.
Convention as to Signs. In the representation of
points, distances to the right of the F-axis are posi-
PJ-3.,3)
P3(-3,-l)
i
!
Pi(M)
KC 2.-3)
Fig. 2.
tive, to the left negative. Distances above the
JT-axis are positive, those below negative. An x in
the first or fourth quadrant is + , in the second or
third it is — . A _7 in the first or second quadrant
is +, in the third or fourth it is — . These are
indicated in Fig. i.
Plotting Points. To locate the point /*i(3, 4),
we measure 3 units to the right of (9Fand then
measure 4 units up from OX. (See Fig. 2.) The
point -PaC— 2, 3) is 2 units to the left oi OV and 3
Temperature Curve. 3
units above OX. The point Pjy— 3, — i)is 3 units
to the left of (9Fand i unit below OX. The point
P4(+ 2, — 3) is 2 units to the right oi OY and 3
units below OX.
Temperature Curve.
The temperature at noon on twenty successive
days was as follows: 60, 62, 64, 63, 61, 6^, 73, 75,
74, 72, 70, 68, 69, 70, 74, 70, 67, 65, 63, 64.
To show this graphically we take as our X-axis
(Fig. 3) a line which we let represent a temperature
of 60 degrees. Each unit along this line represents
one day, and each unit above the line one degree
of heat. The temperature on the first day, 60
Y
Fig. 3-
degrees, is at the origin, O. The temperature on
the second day is 62, and is shown at the point
(i, 2); that is, I to the right of o and 2 above the
X-axis. The temperature of the other 18 days is
shown in a similar way. If a smooth curve is
drawn through these 20 points, the result is the
temperature curve, or the graph of the noon tem-
perature for 20 days.
4 Graphs.
In the same way the graph may be used to show
the fluctuations in the price of wheat, the increase
in skill in learning a trade, or in fine anything that
may be represented by the combination of two
series of facts.
Solution of Problems by Graphs.
1. A travels 4 7niles an hour, B 6 miles an hour.
If A has 2 hours the start, when and where will B
overtake hint f
In this problem let each space along the JT-axis
(Fig. 4) represent i mile, and each space along the
F-axis represent one-half hour. At the end of
the first hour A is evidently at the point A. At the
end of the second hour he is at B, and so on. His
path in time and space is readily seen to be OP.
B does not start until 2 hours after A starts, so his
path begins at L, four spaces (2 hours) above O.
At the end of first hour B is at C, and his path is
LP. B overtakes A when his path crosses that of
A. This occurs at P. A perpendicular from P to
Solution of Problems by Graphs. 5
OX intersects it at M, 24 spaces to the right of O.
B, therefore, overtakes A 24 miles from starting-
place. The length of the perpendicular PM is 12
spaces. Hence B overtakes A 6 hours after A
starts, or 4 hours after he himself starts.
2. Two towns P and Q are 48 miles apart. A
walks from Y to (^ at the rate of 3 miles an Jiotir
a7td rides back at 12 miles a7i hour, B starts from,
Q two hours after A starts fvm P, a7td rides to P
at the rate of 8 miles an hour, and walks back at 4
miles a7i hour. When and where do A and B meet
the seco7id time ?
Figure 5 shows the solution to this. The paths
of A and B are marked and can be easily under-
Fig- 5.
stood from what has preceded. Their second
meeting place is M, which is seen to be 36 miles
from P and 12 miles from Q. The line ML is 17
6 Graphs.
spaces long, and so they meet 17 hours after A
starts, each vertical space here having been chosen
to represent one hour.
3. A, B, and C, travelling at 6, 8, and 12 miles
an hour^ start at the same time around aji island 48
miles in circumference. When and where are they
again all together ?
The graph in Fig. 6 shows the result at once.
The two perpendiculars OS and IR are 48 spaces
apart. Since the road is a circle, any point in IR
simply represents the completion of one circuit and
Fig. 6.
really represents a new starting-point in OS. A's
path is OPj PQ, QR, three complete circuits. B's
path is OL, LM, MN, NR, four complete circuits.
C's path is OD, DP, PM, MQ, QT, TS, six com-
Solution of Problems by Graphs. 7
plete circuits. At the end of these circuits they
are all together. The time IR is 24 hours.
4. A man walking from a town A to another B
at the rate of 4. miles aii hour, starts one hour before
a coach which goes 12 miles an hour and is picked
up by tJie coach. O71 arriving at B, he observes that
his coach journey lasted 2 hours. Find the distance
from A /^ B.
Let spaces (Fig. 7) to the right be miles and
spaces up be quarter hours. AP is the path of the
man while walking. The carriage path is CP. The
intersection P is 6 spaces to right and 6 spaces up.
The carriage picks the man up 6 miles from A
Fig. 7.
and i\ hours after he has started. The destina-
tion is reached 2 hours from P ; that is, the line CP
is continued until it cuts a time line 8 spaces above
P at B. AM equals 30 miles and is the distance
between the towns.
Graphs.
A Linear Equation in Two Variables.
3x-h4J^= 12.
We say in algebra that such an equation is inde-
terminate, for we can get an indefinite number of
values of x and j^ that will satisfy it. To get a
solution we need only to assign arbitrarily \.o x 3.
value and then solve for j. The following is a set
of solutions :
x= o y= 3
X— I
7=2i
X — 2
7=li
^=3
J=i
;ir=4
y^o
X
5
7 = -
The list could be extended indefinitely.
We can write these solutions as the points (o, 3),
Fig. 8.
A Linear Equation in Two Variables. 9
(I, 2i), (2, 1 1), (3, I), (4, o), and (5, - |)- These
points may be plotted as in Fig. 8. It will now be
noticed that these points are in a straight Hne.
The line is called the graph of the equation. If
the line be produced indefinitely and the x and y
of any point found by measurement from the
graph, the values thus found will satisfy the equa-
tion. For example, if we select the point Q, we
find that its x, OM, is - 4, and its y, MQ, is + 6.
These values satisfy the equation ^x + ^y— 12,
for 3(- 4) + 4(6)= 12.
An equation of the first degree in two variables
always has a straight line for its graph.
ax -{- by = c is a general linear equation in x
and y.
A set of solutions is as follows :
X = o y
x— I y
x= 2 y
^=3 y
b
_c — a
b
_c — 2a
~~~b
b
_ c — A,a
etc. etc.
lO
Graphs .
We see that a change of i in the value of x
makes a change
in y. If these points were
plotted, they would appear (Fig. 9) very much like
the side view of a uniform straight stairway, in
Fig. 9.
which the width of the steps is i, and the height
- • The points are readily seen to be in a straight
line.
A shorter way of getting the graph.
Since the linear equation always represents a
straight line, we can draw its graph if we know
two points upon it. In general, the two points
most easily determined are those where the graph
cuts the coordinate axes. The point on the Jf-axis
is found by putting 7 = and solving for x. The
point on the F-axis is found by putting x — o and
solving for^.
A Linear Equation in Two Variables, ii
Example. 2x— ^y— lo.
If J = o, .r = 5 ;
and if ;r = o, y — — 2.
Y
(6.0)
(0.-2)
Fig. 10.
The require(i graph cuts the X-axis at (5, o) and
the F-axis at (o, — 2). Plotting these two points,
the line is easily drawn as in Fig. 10.
Simultaneous linear equations.
(1) r5;r + 4;j/=22l
(2) \lx-\- 7= 9J
Draw the graphs of these two equations on the
same diagram as in Fig. 11. It is found that the
two lines intersect at a point P whose coordinates
are (2, 3). The x and y (2 and 3) of this point is
the solution of the equations.
Two simultaneous linear equations in x and y
have but one solution. Each equation represents
12
Graphs.
a straight Hne. The solution is the point common
to both hnes ; that is, the intersection of the hnes.
Fig. II.
Two straight Hnes can only intersect in one point,
so there is but one solution.
(1) r ^+ j=i
(2) L2.r + 2^= 7
If we undertake to solve the above equations, we
encounter a difficulty. We find that we cannot
Y
Fig. 12.
eliminate x without also eUminating y at the same
time.
The Quadratic Equation. 13
If we draw the graphs of these equations, we
find they are represented as in Fig. 12. The
graphs show at once where the difficulty is. The
Hues are parallel and so do not intersect at all. In
the language of mathematics, they intersect at in-
finity, which is just another way of saying that
they never intersect.
The Quadratic Equation.
ax^-\- bx + c =y is an equation which, whenj = o,
is the type form of the quadratic in a single variable.
If the quadratic in x is thought of in the above
form, it readily yields to graphical representation.
Graph of x'^ — 2 x — 2, = o.
We write x'^ — 2x — ■^= y. Solving this for x
in the usual way, we get
x= I ± V4 -\-y.
The following Hst of values for y and x are
readily found :
1. y = o x= I and — i
2. y = — I X = 2.y and — .7
Z. y — — 2 X— 2.4. and — 4
4. j/ = — 3 X = 2 and o
5. J/ = — 4 X = I and I
6. / = I ,r = 3.2 and — 1.2
H
Graphs.
7.
y =
2
X = 3.4 and
- 1-4
8.
y =
3
;r = 3.6 and
- 1.6
9.
y=.
4
;r= 3.8 and
- 1.8
10.
y^
5
;r = 4 and
— 2
11.
etc
12
X — ^ and
etc.
-3
Plotting these points carefully and connecting
them by a smooth curve, we get the result shown
in Fig. 13. It is seen that the graph in this case
is a curve, and that it cuts the axis of X in two
points. These points are at distances of 3 and
— I from the origin. 3 and — i are the two roots
of the quadratic x'^
2X — ^ — o.
A quadratic always represents a curve that can
be cut in two places by one straight line.
Write
X^ — 2X ■\- I =0.
2x •\- \ —y.
The Quadratic Equation.
15
Solving for x, we
J have
X = I
± Vj.
^ =
;r = I
7= I
X — 2 and
7 = 4
;r = 3 and
— I
y = 9
;r = 4 and
- 2
etc.
etc.
Plotting these points and drawing a smooth curve
through them, we have the curve shown in Fig. 14.
This curve does not cross the axis of X, but touches
it at the point (i, o). The first member of the
given equation being a perfect square, the equation
Fig. 14.
has two equal roots. The graph of a quadratic
having equal roots always touches the axis of X at
a distance from O equal to one of the equal roots.
If we consider the equation x'^—6x-\- 10 =j/
and treat it as the above, we get a graph shown in
1 6 Graphs.
Fig. 15. This curve does not touch the axis of X.
If in the equation x^ — 6x -\ 10 = j/ we put 7 =
Fig. 15.
and solve, we get imaginary roots for x. The graph
of a quadratic having imaginary roots does not
touch the axis of X.
Simultaneous Quadratics.
1. X +y = 2.
2. ;rj/ = — 15.
Square (i), subtract 4 times (2), and extract the
square root, and we have
{l)x-y= 8
{^) x-y = -^
In Fig. 16 the various Hues of the graph are
numbered to correspond with the numbers of the
equations.
Simultaneous Quadratics.
17
Equations (i) and (2) give a straight line and
the double-branched curve known as the hyper-
bola. These intersect at the points P, Q, whose
0\
i^
<
cs-r
^^s-r-
/
Fig. 16.
coordinates are (-3, 5) and (5, - 3). These are
the only solutions to the system of equations. The
auxiliary lines (3) and (4) intersect line (i) in P
and Q, and hyperbola (2) in R and 5.
\. xy = 12.
2. x^ -\- y^ = 40.
The auxihary equations appearing in the solution
are :
3. jr-f j/ = + 8.
4. X -^y = -^.
5. x—y = ^A,.
6. A'-j = -4.
i8
Graphs.
The graphs of all these equations are shown in
Fig. 17 by the corresponding numbers.
The solutions are at the points -P, Q^ R, and 5.
Y
Fig. 17.
The Complex Number.
In making general the treatment of quadratics,
the complex number becomes necessary. The
imaginary unit or z( = V— i) appears in the
extraction of the square root of a negative quan-
tity. E.g., V — 4 = V4 ( — I ) = V4 • V — I =
± 2 V— I =±22.
a + bi is the type of all complex numbers. The
imaginary and complex numbers are graphically
represented by means of Argand's diagram.
Two axes intersecting at right angles are used
just as in ordinary graphic work. The horizontal
one is the axis of reals, and the vertical the axis of
imaginaries.
The Complex Number.
19
In Fig. 18 CA and BD intersect at right angles
at O. 0A(— i) = - OA =0C. Hence we may
regard — i as an operator which reverses OA, or
which turns it about O through an angle of 180°.
We might then think of 2 = V— i as an operator
which turns OA through an angle of 90°, or OAz
= 0B.
Then OAz^ = OBi ^0C\
OAi^= OBi^=^ OCi ^0D\
OAi^ = OBP = Oa^= ODi = OA.
These results merely show that when we regard
i as an operator which turns a quantity through an
angle of 90°, we get results consistent with the
known algebraic set of facts :
•3 -^ _ V
I
\.i
I
i
— I
I.
20
Graphs.
To represent any complex number as ^ + iy, we
measure on OX a distance OM=x and a perpen-
dicular distance PM = y. The point P, or as is
iy
Fig. 19.
frequently more convenient, the line OP, is said
to represent x + iy. OP = V,t'2 + j/^ = r, radius
vector, and when taken with the positive sign is
called the modulus.
The angle MOP = ^ is called the amplitude.
X = OP cos 6 =^ r cos d ;
J = (9/^ sin ^ = r sin ^.
Hence ;r + 2^7 = r(cos ^ + ^ sin ^).
If OS represents x^ + /;/j,
and 6^7" represents x^ + /^/g*
then OP represents {x^ + iy^ + (;r2 4- ^3^2)'
OP being the diagonal of the parallelogram of
which OS and (97" are two adjacent sides.
The Complex Number.
21
The diagram shows at once that if OP repre-
"''"*" ^ + iy' (say),
then y = jTj + x^ and y = y\ + ^2'
and hence OP = OS + OT.
Fig. 20.
The diagram may be used to iUustrate nearly all
the principles of the complex number. Enough
has been given to show its adaptability.
FACTORING
AS PRESENTED IN
WELLS' ESSENTIALS <?/ALGEBRA
I. Advanced processes are not presented too early.
II. Large amount of practice work.
No other algebra has so complete and well-graded development of
this important subject, presenting the more difficult principles at those
stages of the student's progress when his past work has fully prepared
him for their perfect comprehension.
The Chapter on Factoring contains these simple processes :
CASE I. When the terms of the expression have a common monomial factor —
thirteen examples.
II. When the expression is the sum of two binomials which have a common
binomial factor — twenty examples.
III. When the expression is a perfect trinomial square — twenty-six examples.
IV. When the expression is the difference of two perfect squares — fifty-five
examples.
V. When the expression is a trinomial of the form x^-{-ax-i-l?
— sixty-six examples.
VI. When the expression is the sum or difference of two perfect cubes — twenty
examples.
VII. When the expression is the sum or difference of two equal odd powers of
two quantities — thirteen examples.
Ninety-three Miscellaneous and Review Examples.
Further practice in the application of these principles is given in the two following
chapters — Highest Common Factor and Lowest Common Multiple.
In the discussion of Quadratic Equations, Solution of Equations by
Factoring is made a special feature.
Equations of the forms ;^*— 5.^ — 24 = 0, 2X^ — x = o,x^-\-4x^ — x — ^
— o, and ;^J— I =0 are discussed and illustrated by thirty examples.
The factoring of trinomials of the form ax^+lx-\-c and ax'^+I^x^-^c,
which involves so large a use of radicals, is reserved until Chapter
XXV, where it receives full and lucid treatment.
The treatment of factoring is but one of the many features of
superiority in Wells* Essendals of Algebra.
Ha/f Leather, 338 pages. Price $1.10.
D. C. HEATH & CO., Publishers
BOSTON NEWYORK CHICAGO
Wells's Essentials of Algebra is the
best book that can be placed in the hands of
a preparatory student. The one essential fea-
ture that I admire in the book is its simplicity.
So many authors now write algebras and geom-
etries to see how hard they can make them,
and the consequence is that only the brightest
students acquire any real knowledge of the
subject.
Prof. Wells leads the student step by step,
and his book is well named The Essentials,
as it contains all that is essential to a student
entering any of our great universities.
EDWIN W. RAND,
Principal Princeton University Academy.
The definitions ; the problems ; the chapter
on factoring ; the factoring of trinomials ; the
discussion of the principles of fractions ; the so-
lution of equations by factoring ; all combine to
make Wells's Essentials of Algebra a book of su-
perior worth. Such a book I can recommend.
E. MILLER,
Professor of Mathematics, University of Kansas.
Wells's
Essentials of Geometry
is far superior in
the arrangement of propositions,
the choice of proofs, and in
the method of presenting the subject.
The student is stimulated to help himself, to
make his own proofs, to gain logical power.
L. B. MULLEN, Ph.D.,
Dept. of Math., Central High School,
Cleveland, 0.
In the order of theorems,
the proof of corollaries,
the grading of the original exercises, and
the opportunity for original work,
Wells's Essentials of Geometry is notably
superior.
C. A. HAMILTON,
Dept. of Math., Boys' High School,
Brooklyn, N. T.
The Vital Difference
lies in this:
Wells's Essentials Plane and Solid
Geometry is characterized by the
inductive method of demonstration,
and always requires the student to do
for himself the maximum amount of
reasoning and thinking of which he is
capable, while most geometries require
of the pupil in his demonstrations little
personal power except that of memo-
rizing. They give the reasons for
statements ; while Wells, whenever the
student should be able to give the
reason for himself, always asks why
the statement is true. No other geom-
etry develops so strongly the power
of vigorous independent thought.
D. C. HEATH & CO., Publishers
BOSTON NEW YORK CHICAGO
Wells's Mathematical Series.
ALGEBRA.
Wells's Essentials of Algebra ..... $i.xo
A new Algebra for secondary schools. The method of presenting the fundamen-
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Wells's New Higher Algebra ..... 1.3a
The first part of this book is identical with the author's Essentials of Algebra.
To this there are added chapters upon advanced topics adequate in scope and
difficulty to meet the maximum requirement in elementary algebra.
Wells's Academic Algebra ..... 1.08
This popular Algebra contains an abundance of carefully selected problems.
Wells's Higher Algebra ...... 1.32
The first half of this book is identical with the corresponding pages of the Aca-
demic Algebra. The latter half treats more advanced topics.
Wells's College Algebra ...... 1.50
A modem text-book for colleges and scientific schools. The latter half of this
book, beginning with the discussion of Quadratic Equations, is also bound sep-
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Wells's University Algebra . . • • • 1.3a
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Wells's Essentials of Geometry — Plane, 75 cts.; Solid, 75 cts.;
Plane and Solid . . . . . . .1.25
This new text offers a practical combination of moxe desirable qualities than
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Wells's Stereoscopic Views of Solid Geometry Figures • .60
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Wells's Elements of Geometry — Revised 1894. — Plane, 75 cts.;
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Wells's Plane Trigonometry . . . . • '75
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THEIR HISTORY. By William W. Rupert, C.E.
i. The Greek Geometers, ii. The Pythagorean Proposition.
2. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert.
ii. The Pythagorean Proposition (concluded), iii. Squaring the Circle.
5. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert.
iv. Trisection of an Angle, v. The Area of a Triangle in Terms of
its Sides.
4. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert.
vi. The Duplication of the Cube. vii. Mathematical Inscription upon
the Tombstone of Ludolph van Ceulen.
6. ON TEACHING GEOMETRY. By Florence Milner.
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