A short introduction to graphical algebra

A

SHORT INTRODUCTION

TO

GRAPHICAL
ALGEBRA

H. 8, HALL

QA
219
H35
1903

A SHOET INTRODUCTION

TO

GRAPHICAL ALGEBRA

BY

H. S. HALL, M.A.

SECOND EDITION REVISED AND ENLARGED.

MACMILLAN AND CO., LIMITED

NEW YORK : THE MACMILLAN COMPANY

1903

All rights reserved

First Edition November, 1002.

Reprinted December, 1902.

Second Edition Revised and Enlarged January, 1903.
Reprinted March, 1903.

Oft

Ol.ASOOW. PRINTKD AT THK UNIVKK") I
BY ROBKRT MACLKHO8E AND CO.

PREFACE TO THE SECOND EDITION.

THE first edition of this little book was undertaken at very
short notice in order to meet a sudden demand. It was, in fact,
rather hastily compiled during a seaside holiday, and I had
neither time nor opportunity for adequately treating the prac-
tical side of graphical work. Consequently all questions dealing
with statistics and physical formulae were deliberately omitted
to enable me to present the analytical aspect of the subject in
sufficient detail within the limits of a few pages.

The present edition has been very considerably enlarged.
The additions are of two kinds : first, a further development of
the illustrations arising out of graphs of known functions ; and
secondly, the application to practical questions in which the
graph has to be obtained by plotting a series of values deter-
mined by observation or experiment.

The subject is practically inexhaustible ; but it is hoped that
a student who has worked intelligently through the following
pages will have added something useful and interesting to his
algebraical knowledge, and will find himself sufficiently equipped
to pursue the study further in the laboratory or workshop.

I am indebted to several friends for advice and suggestions.
In particular, I wish to express my thanks to Mr. D. Rintoul of
Clifton College, and to my former pupil Mr. E. A. Price
of Winchester.

H. S. HALL.

January, 1903.

CONTENTS.

ARTS. PAGE

1 — 6. AXES, COORDINATES. PLOTTING A POINT, 1

EXAMPLES I., - - 4

7 — 10. GRAPH OF A FUNCTION. STRAIGHT LINES, - 5

EXAMPLES II., ... 7

11 — 14. APPLICATION TO SIMULTANEOUS EQUATIONS, 9

15—18. GRAPHS OF QUADRATIC FUNCTIONS. ROOTS OF EQUA-
TIONS. MAXIMA AND MINIMA, - 11

EXAMPLES IV. - 15

19—26. INFINITE AND ZERO VALUES. ASYMPTOTES. GRAPHS

OF QUADRATIC AND HIGHER FUNCTIONS, 16

EXAMPLES V., 22

27 — 31. MEASUREMENT OF VARIABLES ON DIFFERENT SCALES.

ILLUSTRATIONS, 29

EXAMPLES VI., 30

32 — 35. PRACTICAL APPLICATIONS,- - - 32

EXAMPLES VII., 40

ANSWERS, - - - - . - - - 48

GEAPHICAL ALGEBRA.

[A considerable portion of this chapter may be taken at an early
stage. For example, Arts. 1-6 may be read as soon as
the student has had sufficient practice in substitutions in-
volving negative quantities. Arts. 7-14 may be read in
connection with Easy Simultaneous Equations. With the
exception of a few articles the rest of the chapter should be
postponed until the student is acquainted with quadratic
equations. References to Hall and Knight's Elementary
Algebra are given thus : " E. A., Art. 100."]

1. DEFINITION. Any expression which involves a variable
quantity x, and whose value is dependent on that of x, is called
a function of x.

Thus 3.^+8, 2#2 + 6#-7, ot-&xP + a?-9 are functions of x of
the first, second, and fourth degree respectively.

2. The symbol f(x) is often used to briefly denote a
function of x. If y=f(x\ by substituting a succession of
numerical values for x we can obtain a corresponding succession
of values for y which stands for the value of the function.
Hence in this connection it is sometimes convenient to call x
the independent variable, and y the dependent variable.

3. Consider the function x(9-x2), and let its value be
represented by y.

Then, when #=0, y=0x9= 0,

ar=l, y = lx8= 8,

#=3, y = 3xO= 0,
a?=4, # = 4x(-7)=-28,
and so on.

2 GRAPHICAL ALGEBRA.

By proceeding in this way we can find as many values of the
function as we please. But we are often not so much concerned
with the actual values which a function assumes for different
values of the variable as with the way in which the value of the
function changes. These variations can be very conveniently
represented by a graphical method which we shall now explain.

4. Two straight lines XOX', TOY' are taken intersecting
at right angles in 0, thus dividing the plane of the paper into
four spaces XOY, YOX', X'OY', Y'OX, which are known as the
first, second, third, and fourth quadrants respectively.

Q

,1 ,

X'

0

X

F

*

s

Y'

Fig:, i.

The lines X'OX, TOY' are usually drawn horizontally and
vertically ; they are taken as lines of reference and are known
as the axis of X and y respectively. The point 0 is called the
origin. Values of x are measured from 0 along the axis of #,
according to some convenient scale of measurement, and are
called abscissae, positive values being drawn to the right of 0
along OX, and negative values to the left of 0 along OA'1.

Values of y are drawn (on the same scale) parallel to the axis
of y, from the ends of the corresponding abscissae, and are
called ordinates. These are positive when drawn above X'X>
negative when drawn below X'X.

5. The abscissa and ordinate of a point taken together
are known as its coordinates. A point whose coordinates are
x and y is briefly spoken of as " the point (#, ?/)."

The coordinates of a point completely determine its position
in the plane, Thus if we wish to mark the point (2, 3), we

PLOTTING A POINT. . 3

take #=2 units measured to the right of 0,y=3 units measured
perpendicular to the .r-axis and above it. The resulting point
P is in the first quadrant. The point ( - 3, 2) is found by taking
x=3 units to the left of 0, and y = 2 units above the #-axis. The
resulting point Q is in the second quadrant. Similarly the
points ( - 3, - 4), (5, - 5) are represented by R and S in Fig. 1,
in the third and fourth quadrants respectively.

This process of marking the position of a point in reference
to the coordinate axes is known as plotting the point.

6. In practice it is convenient to use squared paper ;
that is, paper ruled into small squares by two sets of equi-
distant parallel straight lines, the one set being horizontal and
the other vertical. After selecting two of the intersecting lines
as axes (and slightly thickening them to aid the eye) one or
more of the divisions may be chosen as our unit, and points
may be readily plotted when their coordinates are known.
Conversely, if the position of a point in any of the quadrants is
marked, its coordinates can be measured by the divisions on
the paper.

In the following pages we have used paper ruled to tenths of
an inch, but a larger scale will sometimes be more convenient.
See Art. 26.

Example. Plot the points (5, 2), (-3, 2), (-3, -4), (5, -4) on
squared paper. Find the area of the figure determined by these
points, assuming the divisions on the paper to be tenths of an inch.

Taking the points in the
order given, it is easily
seen that they are repre-
sented by P, Q, Ht S in
Fig. 2, and that they form
a rectangle which contains
48 squares. Each of these
is one-hundredth part of a
square inch. Thus the area
of the rectangle is '48 of a
square inch.

GRAPHICAL ALGEBRA.

EXAMPLES I.

[The following examples are intended to be done mainly by actual
measurement on squared paper ; where possible, they should
also be verified by calculation.]

Plot the following pairs of points and draw the line which joins
them :

1. (3,0), (0,6). 2. (-2,0), (0, -8).

3. (3, -8), (-2,6). 4. (5,5), (-2, -2).

5. (-2,6), (1, -3). 6. (4,5), (-1,5).

7. Plot the points (3, 3), ( - 3, 3), ( - 3, - 3), (3, - 3), and find
the number of squares contained by the figure determined by these
points.

8. Plot the points (4, 0), (0, 4), (-4, 0), (0, -4), and find the
number of square units in the resulting figure.

9. Plot the points (0, 0), (0, 10), (5, 5), and find the number of
square units in the triangle.

10. Shew that the triangle whose vertices are (0, 0), (0, 6), (4, 3)
has an area of 12 square units. Shew also that the points (0, 0),
(0, 6), (4, 8) determine a triangle of the same area.

11. Plot the points (5, 6), (-5, 6), (5, -6), (-5, -6). If one
millimetre is taken as unit, find the area of the figure in square
centimetres.

12. Plot the points (1, 3), ( - 3, - 9), and shew that they lie on a
line passing through the origin. Name the coordinates of other
points on this line.

13. Plot the eight points (0, 5), (3, 4), (5, 0), (4, -3), ( - 5, 0),
(0, - 5), ( - 4, 3), ( - 4, - 3), and shew that they are all equidistant
from the origin.

14. Plot the two following series of points :

(i) (5,0), (5,2), (5,5), (5, -1), (5, -4);
(ii) (-4, 8), (-1,8), (0,8), (3,8), (6,8).

Shew that they lie on two lines respectively parallel to the axis of y,
and the axis of x. Find the coordinates of the point in which they
intersect.

GRAPH OF A FUNCTION. 5

15. Plot the points (13, 0), (0, - 13), (12, 5), ( - 12, 5), ( - 13, 0),
( - 5, - 12), (5, - 12). Find their locus, (i) by measurement, (ii) by
calculation.

16. Plot the points (2,2), (-3, -3), (4, 4), (-5, -5), shewing
that they all lie on a certain line through the origin. Conversely,
shew that for every point on this line the abscissa and ordinate are
equal.

Graph of a Function.

7. Let f(x) represent a function of x, and let its value be
denoted by y. If we give to x a series of numerical values we
get a corresponding series of values for y. If these are set off
as abscissae and ordinates respectively, we plot a succession of
points. If all such points were plotted we should arrive at a
line, straight or curved, which is known as the graph of the
function f(x\ or the graph of the equation y=f(x). The varia-
tion of the function for different values of the variable x is
exhibited by the variation of the ordinates as we pass from
point to point.

In practice a few points carefully plotted will usually enable
us to draw the graph with sufficient accuracy.

8. The student who has worked intelligently through the
preceding examples will have acquired for himself some useful
preliminary notions which will be of service in the examples on
simple graphs which we are about to give. In particular,
before proceeding further he should satisfy himself with regard
to the following statements :

(i) The coordinates of the origin are (0, 0).
(ii) The abscissa of every point on the axis of y is 0.
(iii) The ordiiiate of every point on the axis of x is 0.

(iv) The graph of all points which have the same abscissa is
a line parallel to the axis of y. (e.g. # = 2.)

(v) The graph of all points which have the same ordinate is
a line parallel to the axis of x. (e.g. y = 5.)

(vi) The distance of any point P(x, y) from the origin is
given by OP2=z2+y2.

GRAPHICAL ALGEBRA.

Example 1. Plot the graph of y = x.

When a; = 0, y = 0 ; thus the origin is one point on the graph.
Also, when x=l, 2, 3, ... -1, -2, -3, ...,
y=l, 2, 3, ... -1, -2, -3, ...

Thus the graph passes through O, and represents a series of points
each of which has its ordinate equal to its abscissa, and is clearly
represented by POP' in Fig. 3.

Example 2. Plot the graph of y=x+ 3.
Arrange the values of x and y as follows :

X

3

2

1

0

-1

-2

-3

y

6

5

4

3

2

1

0

...

Fit

By joining these points we
obtain a line M. N parallel to
that in Example 1.

The results printed in
larger and deeper type
should be specially noted
and compared with the
graph. They shew that the
distances ON, OM (usually
called the intercepts on the
axes) are obtained by separ-
ately putting x = Q, y — Q in
the equation of the graph.

Note. By observing that in Example 2 each ordinate is 3 units
greater than the corresponding ordinate in Example 1, the graph
of y = x + 3 may be obtained from that of y = x by simply producing
each ordinate 3 units in the positive direction.

In like manner the equations

represent two parallel lines on opposite sides of y=x and equi-
distant from it, as the student may easily verify for himself.

LINEAR GRAPHS. 7

Example 3. Plot the graphs represented by the following equa-
tions :

(i)y=2ar; (ii)y=2x + 4; (iii) y = 2x-5.

ZQ

Fig.

Here we only give the diagram which the student should verify
in detail for himself, following the method explained in the two
preceding examples.

EXAMPLES II.

\In the following examples Nos. 1-18 are arranged in groups of
three ; each group should be represented on the same diagram
so as to exhibit clearly the position of the three graphs rela-
to each other J\

Plot the graphs represented by the following equations :

y = 5x.

y= -3

=5x-4.

1.

4.

7.
10.
13.
16.

19. Shew by careful drawing that the three last graphs have a
common point whose coordinates are 2, 1.

20. Shew by careful drawing that the equations

37-5 =

2.
5.

8. y + x = S.

11. 3y=4.r + 6.

14. y-6 = 0.
17.

3.

6.

9.
12.
15.
18.

represent two straight lines at right angles.

8 GRAPHICAL ALGEBRA.

21. Draw on the same axes the graphs of a; = 5, a; = 9, ?/ = 3, y = 1 1 .
Find the number of square units enclosed by these lines.

22. Taking one-tenth of an inch as the unit of length, find the
area included between the graphs of a: =7, x= -3, y= -2, y = 8.

23. Find the area included by the graphs of

y=x + Q, y = x-6, y=-x + G, y=-x-6.

24. With one millimetre as linear unit, find in square centimetres
the area of the figure enclosed by the graphs of

y=-2x-8.

9. The student should now be prepared for the following
statements :

(i) For all numerical values of a the equation y = ax re-
presents a straight line through the origin.

(ii) For all numerical values of a and b the equation
y = ax+b represents a line parallel to y=ax, and
cutting off an intercept 6 from the axis of y.

10. Conversely, since every equation involving x and ?/
only in the first degree can be reduced to one of the forms
y=ax,y = ax + b, it follows that every simple equation connecting
two variables represents a straight line. For this reason an
expression of the form ax+b is said to be a linear function of x,
and an equation such as y = ax+b, or az+by + c = 0, is said to be
a linear equation.

Example. Shew that the points (3, -4), (9, 4), (12, 8) lie on a
straight line, and find its equation.

Assume y = ax + b as the equation of the line. If it passes through
the first two points given, their coordinates must satisfy the above
equation. Hence

These equations give a = -, 6= -8.
Hence y = ^a?-8, or 4a?-3y=24,

is the equation of the line passing through the first two points.
Since x= 12, y = 8 satisfies this equation, the line also passes through
(12, 8). This example may be verified graphically by plotting the
line which joins nny two of the points and shewing 'that it
through the third.

APPLICATION TO SIMULTANEOUS EQUATIONS.

Application to Simultaneous Equations.

11. It is shewn [E. A., Art. 100] that in the case of a simple
equation between x and y, it is possible to find as many pairs of
values of x and y as we please which satisfy the given equation.
We now see that this is equivalent to saying that we may find
as many points as we please on any given straight line. If,
however, we have two simultaneous equations between x and y,
there can only be one pair of values which will satisfy both
equations. This is equivalent to saying that two straight lines
can have only one common point.

Example. Solve graphically the equations :

x¥

If carefully plotted it will be found that these two equations
represent the lines in the annexed diagram. On measuring the
coordinates of the point at which they intersect it will be found that
x = 2, y = 3, thus verifying the solution given in E. A. Art. 103, Ex. 1.

12. It will now be seen that the process of solving two
simultaneous equations is equivalent to finding the coordinates
of the point (or points) at which their graphs meet.

13. Since a straight line can always be drawn by joining
any two points on it, in solving linear simultaneous equations
graphically, it is only necessary to plot two points on each line.
The points where the lines meet the axes will usually be the
most convenient to select.

10 GRAPHICAL ALGEBRA.

14. Two simultaneous equations lead to no finite solution
if they are inconsistent with each other. For example, the
equations

are inconsistent, for the second equation can be written
.r-f 3// = 2§, which is clearly inconsistent with .r+3# = 2. The
graphs of these two equations will be found to be two parallel
straight lines which have no finite point of intersection.

Again, two simultaneous equations must be independent.
The equations

are not independent, for the second can be deduced from the
first by dividing throughout by 4. Thus Mny pair of values
which will satisfy one equation will satisfy the other. Graphi-
cally these two equations represent two coincident straight lines
which of course have an unlimited number of common points.

EXAMPLES III.

Solve the following equations, in each case verifying the solution
graphically :

4. 2x-y=8, 5. 3# + 2y=16, 6. 6y-5x=18,

7. 2x + y=0, 8. 2#-y = 3, 9.

^y-^x=S.

10. Prove by graphical representation that the three points (3, 0),
(2, 7), (4, - 7) lie on a straight line. Where does this line cut the
axis of y ?

11. Prove that the three points (1, 1), (-3, 4), (5, -2) lie on a
straight line. Find its equation. Draw the graph of this equation,
shewing that it passes through the given points.

12. Shew that the three points (3, 2), (8, 8), ( -2, -4) lie on a
straight line. Prove algebraically and graphically that it cuts the
axis of .r at a distance 1 j from the origin.

GRAPHS 01* QUADRATIC FUNCTIONS.

11

1 5. We shall now give some graphs of functions of higher
degree than the first.

Example 1. Plot the graph of 2y = x2.

Corresponding values of x and y may be tabulated as follows :

X

3

2-5

2

1-5

1

0

-1

-2

-3

y

4-5

3-125

2

1-125

•5

0

•5

2

4-5

...

Here, in order to obtain a figure on a sufficiently large scale, it
will be found convenient to take two divisions on the paper for our
unit.

Fi

.6.

If the above points are plotted and connected by a line drawn
freehand, we shall obtain the curve shewn in Fig. 6. This curve
is called a parabola.

There are two facts to be specially noted in this example.

(i) Since from the equation we have «=±\/2y, it follows that
for every value of the ordinate we have two values of the abscissa,
equal in magnitude and opposite in sign. Hence the graph is sym-
metrical with respect to the axis of y ; so that after plotting with
care enough points to determine the form of the graph in the first
quadrant, its form in the second quadrant can be inferred without
actually plotting any points in this quadrant. At the same time, in
this and similar cases beginners are recommended to plot a few
points in each quadrant through which the graph passes.

GRAPHICAL ALGEBRA.

(ii) We observe that all the plotted points lie above the axis of x.
This is evident from the equation ; for since x2 must be positive for

y£

all values of x, every ordinate obtained from the equation y = ~n
must be positive.

In like manner the student may shew that the graph of 2y = - x*
is a curve similar in every respect to that in Fig. 6, but lying entirely
below the axis of x.

Note. Some further remarks on the graph of this and the next
example will be found in Art. 21.

x2
Example 2. Find the graph of y = 2x + — .

Here the following arrangement will be found convenient :

X

3

2

1

0

-1

_2

-3

-4

-5

-6

-7

-8

2x

6

4

2

0

-2

-4

-6

-8

-10

-12

-14

-16

x2
4

y

2-25

1
5

•25
2-25

0

•25

1

2-25

4

6-25

9

12-25

16
0

8-25

0

-1-75

— 3

-3-75

-4

-3-75

-3

-1-75

Y

j

/

j

/

(

/

1

\

\

\

/

*

X

\

/o

<

V

J

•

,

/

x,

/

v

k

B

7

From the form of the equation it is evident that every positive
value of x will yield a positive value of j/, and that as x increases y
also increases. Hence the portion of the curve in the first quadrant
lies as in Fig. 7, and can be extended indefinitely in this quadrant.
In the present case only two or three positive values of x and y rieed
be plotted, but more attention must be paid to the results arising
out of negative values of x.

MAXIMA AND MINIMA.

13

x2
When y = Q, we have — + 2x = Q; thus the two values of x in the

graph which correspond to y = Q furnish the roots of the equation

16. If f(x) represent a function of x, an approximate
solution of the equation f(x) = 0 may be obtained by plotting
the graph of y=f(x\ and then measuring the intercepts made
on the axis of x. These intercepts are values of x which make
y equal to zero, and are therefore roots of /(#) = 0.

17. If f(x) gradually increases till it reaches a value a,
which is algebraically greater than neighbouring values on
either side, a is said to be a maximum value of /(#).

If f(x) gradually decreases till it reaches a value 6, which is
algebraically less than neighbouring values on either side, b is
said to be a minimum value of f(x).

When y=f(x) is treated graphically, it is now evident that
maximum and minimum values of f(x) occur at points where
the ordinates are algebraically greatest and least in the im-
mediate vicinity of such points.

Example. Solve the equation xz - 7x + 1 1 = 0 graphically, and find
the minimum value of the function xz - "tx + 11.

Put y = x2 - 7x + 11, and find the graph of this equation.

y 11

3-5

_i _i-25 -1 1

11

The values of x which make the
function #2-7#+ll vanish are
those which correspond to y = 0.
By careful measurement it will be
found that the intercepts OM and
ON are approximately equal to
2-38 and 4 '62.

The algebraical solution of
y? -7^+11=0

gives

If we take 2 '236 as the approximate
value of /y/5, the values of x will
be found to agree with those ob-
tained from the graph.

Hig

14

GRAPHICAL ALGEBRA.

/ 7\2 5 / 7\2

Again, a;2 - 7x + 1 1 = ( # - 5 J - £. Now f x - ^ J . must be positive

7
for all real values of x except z = ^, in which case it vanishes, and

the value of the function reduces to - ^, which is the least value it
can have.

The graph shews that when x = 3 '5, y= -1*25, and that this is
the algebraically least ordinate in the plotted curve.

18. The following example shews that points selected for
graphical representation must sometimes be restricted within
certain limits.

Example. Find the graph of x2 + y2 = 36.

The equation may be written in either of the following forms :
(i) y=±v/36^2; (ii) x=

In or<l« r that ?/ may be a real Quantity we see from (i) that 36 -x*
must be positive. Thus x can only have values between - 6 and + 6.
Similarly from (ii) it is evident that y must also lie between -6 and
+ 6. Between these limits it will be found that all plotted points
will lie At a distance 6 from the origin. Hence the graph is a circle
whose centre is (J and whose radius is 6.

This is otherwise c-vi.l.-nt. for the distance of any point P(x, y)
from the origin is given by OP = <Jx* + y*. [Art. 8.] Hence the
equation xa + y*=86 asserts that the graph consists of a series of
points all of which are at a distance 6 from the origin.

EXAMPLES IV. , 15

Note. To plot the curve from equation (ii), we should select a
succession of values for y and then find corresponding values of x.
In other words we make y the independent and x the dependent
variable. The student should be prepared to do this in some of the
examples which follow.

EXAMPLES IV.

1. Draw the graphs of y = x2, and x = y2, and shew that they
have only one common chord. Find its equation.

x2

2. From the graphs, and also by calculation, shew that y — -%

cuts x = - yz in only two points, and find their coordinates.

3. Draw the graphs of

(i) y*=-tx; (ii) y = 2x-^>, (hi) y=^ + x-2.

4. Draw the graph of y = x + x2. Shew also that it may be
deduced from that of y = x2, obtained in Example 1.

5. Shew (i) graphically, (ii) algebraically, that the line y = 2x - 3
meets the curve y = -j + # - 2 in one point only. Find its coordinates.

6. Find graphically the roots of the following equations to 2
places of decimals :

(i) ^ + x-2 = 0; (ii) #2-2*=4; (iii) 4a:2- 16# + 9 = 0;

and verify the solutions algebraically.

7. Find the minimum* value of x2 - 2x - 4, and the maximum
value of 5 + 4# - 2x2.

8. Draw the graph of y = (x- l)(x-2) and find the minimum
value of (x - l)(x - 2). Measure, as accurately as you can, the values
of x for which (x - 1) (x - 2) is equal to 5 and 9 respectively. Verify
algebraically.

9. Solve the simultaneous equations

and verify the solution by plotting the graphs of the equations and
measuring the coordinates of their common points.

10. Plot the graphs of #2 + y2=25, 3# + 4y = 25, and examine
their relation to each other where they intersect. Verify the result
algebraically.

16 GRAPHICAL ALGEBRA.

19. Infinite and zero values. Consider the fraction -

x

in which the numerator a has a certain fixed value, and the
denominator is a quantity subject to change ; then it is clear that
the smaller x becomes the larger does the value of the fraction

- become. For instance

1000000a.

By making the denominator x sufficiently small the value of
the fraction - can be made as large as we please ; that is, if x is

made less than any quantity that can be named, the value of -

will become greater than any quantity that can be named.

A quantity less than any assignable quantity is called zero
and is denoted by the symbol 0.

A quantity greater than any assignable quantity is called
infinity and is denoted by the symbol oo .

We may now say briefly

when x = 0, the value of - is oo .

Again if x is a quantity which gradually increases and finally
becomes greater than any assignable quantity the fraction becomes
smaller than any assignable quantity. Or more briefly

when x = oo , the value of - is 0.

20. It should be observed that when the symbols for zero
and infinity are used in the sense above explained, they are
subject to the rules of signs which affect other algebraical
symbols. Thus we shall find it convenient to use a concise
statement such as "when x= +0, y= -f oo " to indicate that when
a very small and positive value is given to x, the corresponding
value of y is very large and positive.

21. If we now return to the examples worked out in Art.
15, in Example 1, we see that when #=±oc, ?/=+oo; hence
the curve extends upwards to infinity in both the first and
second quadrants. In Example 2, when .r=-foo, ?/= + «>.
Again y is negative between the values 0 and - 8 of x. For all

INFINITE AND ZERO VALUES.

17

negative values of x numerically greater than 8, y is positive,
and when x= — oo , ?/= +QO. Hence the curve extends to infinity
in both the first and second quadrants.

The student should now examine the nature of the graphs in
Examples IV. when x and y are infinite.

Example. Find the graph of xy = 4.
The equation may be written in the form

4

from which it appears that when x = 0, y = oo and when x = cc, t/ = 0.
Also y is positive when x is positive, and negative when x is negative.
Hence the graph must lie entirely in the first and third quadrants.

It will be convenient in this case to take the positive and negative
values of the variables separately.

(1) Positive values :

X

0

00

1

o

2

3
14

4

5

•8

6

§

00

0

y

1

Fig- l

Graphically these values shew that as we recede further and
further from the origin on the x-axis in the positive direction, the
values of y are positive and become smaller and smaller. That is

18

GRAPHICAL ALGEBRA.

the graph is continually approaching the a>axis in such a way that
by taking a sufficiently great positive value of x we obtain a point
on the graph as near as we please to the x-axis but never actually
reaching it until x = <x>. Similarly, as x becomes smaller and smaller
the graph approaches more and more nearly to the positive end of
the y-axis, never actually reaching it as long as x has any finite
positive value, however small.

(2) Negative values :

X

-0

-1

-2

-3

4

-5

- 00

y

- GO

-4

-2

-1*

-1

-•8

-0

23. Every equation of the form# = -, or

The portion of the graph obtained from these values is in the third
quadrant as shewn in Fig. 10, and exactly similar to the portion
already traced in the first quadrant. It should be noticed that as
x passes from +0 to -0 the value of y changes from +00 to -co.
Thus the graph, which in the first quadrant has run away to an
infinite distance on the positive side of the y-axis, reappears in the
third quadrant coming from an infinite distance on the negative side
of that axis. Similar remarks apply to the graph in its relation to
the x-axis.

22. When a curve continually approaches more and more
nearly to a line without actually meeting it until an infinite
distance is reached, such a line is said to be an asymptote to
the curve. In the above case each of the axes is an asymptote.

= c, where c is

constant, will give a graph similar to that exhibited in the
example of Art. 21. The resulting curve is known as a
rectangular hyperbola, and has many interesting properties.
In particular we may mention that from the form of the
equation it is evident that for every point (x, y} on the curve
there is a corresponding point ( — x, —y) which satisfies the
equation. Graphically this amounts to saying that any line
through the origin meeting the two branches of the curve in
P and P' is bisected at 0.

24. In the simpler cases of graphs, sufficient accuracy can
usually be obtained by plotting a few .points, and there is little
difficulty in selecting points with suitable coordinates. But in
other cases, and especially when the graph has infinite branches,
more care is needed. The most important things to observe
are (1) the values for which the function J(x) becomes zero or

INFINITE VALUES. ASYMPTOTES. . 19

infinite ; and (2) the values which the function assumes for
zero and infinite values of x. In other words, we determine
the general character of the curve in the neighbourhood of the
origin, the axes, and infinity. Greater accuracy of detail can
then be secured by plotting points at discretion. The selection
of such points will usually be suggested by the earlier stages of
our work.

The existence of symmetry about either of the axes should
also be noted. When an equation contains no odd powers of x,
the graph is symmetrical with regard to the axis of y. Similarly
the absence of odd powers of y indicates symmetry about the
axis of x. Compare Art. 15, Ex. 1.

,. Draw the graph of y = ^ — j-. [See fig. on next page.]

We have y = -= ,, the latter form being convenient for

x-4 j _ 4'

infinite values of x. x

(i) When «/ = 0, «?=-«,!

/. the curve cuts the axis of a? at a distance - 3*5 from the origin,
and meets the line x = 4 at an infinite distance.

If x is positive and very little greater than 4, y is very great and
positive. If x is positive and very little less than 4, y is very great
and negative. Thus the infinite points on the graph near to the line
# = 4 have positive ordinates to the right, and negative ordinates to
the left of this line.

(ii) When a: = 0, y=-175, \

/. the curve cuts the axis of y at a distance - 1 "75 from the origin,
and meets the line y = 2 at an infinite distance.

By taking positive values of y very little greater and very little
less than 2, it appears that the curve lies above the line y = 2 when
x= + GO, and below this line when x= - GO.

The general character of the curve is now determined : the lines
PO'P' (x = 4) and QO'Q' (?/ = 2) are asymptotes ; the two branches of
the curve lie in the compartments PO'Q, P'O'Q', and the lower
branch cuts the axes at distances - 3 '5 an4 - 1*75 from the origin.

20

GRAPHICAL ALGEBRA.

To examine the lower branch in detail values of x may be selected
between - oo and - 3 '5 and between - 3 '5 and 4.

X

-00

...

-16

-8

-6

-3-5

-1

0

2

3

4

y

2

1-25

•75

•5

0

-1

-1-75

-5-5

-13

...

-00

II.

The upper branch may now be dealt with in the same way,
selecting values of x between 4 and oo. The graph will be found to
be as represented in Fig. 11.

25. When the equation of a curve contains the square or
higher power of ?/, the calculation of the values of y correspond-
ing to selected values of x will have to be obtained by evolution,
or else by the aid of logarithms. We give one example to
illustrate the way in which a table of four-figure logarithm*
may be employed in such cases,

USE OF LOGARITHMS. 21

Example. Draw the graph of ys = x(9 - x2).

For the sake of brevity we shall confine our attention to that part
of the curve which lies to the right of the axis of y, leaving the other
half to be traced in like manner by the student.

When x = Q, y = 0: therefore the curve passes through the origin.
Again, y is positive for all values of x between 0 and 3, and vanishes
when cc = 3; for values of x greater than 3, y is negative and con-
tinually increases numerically.

X

0

1

2

3

4

5

6

x2-

0

1

4

9

16

25

36

9-x*

9

8

5

0
0

_ n

-16

-27

y3

0

8

10

-28

-80

-162

logy3

1

1-4472

1-9031

2-2095

logy

•3333

•4824

•6344

•7365

...

y

0

2

2-15

0

-3-04

-4-31

-5-45

These points will be suf-
ficient to give a rough ap-
proximation to the curve.
For greater accuracy a few
intermediate values such as
ic=l-5, 2-5, 3-5 ... should
be taken, and the resulting
curve will be as in Fig. 12,
in which we have taken
two-tenths of an inch as our
linear unit.

Fig-. 12.

* In taking logarithms of the successive values of y3, the negative
sign is disregarded, but care must be taken to insert the proper
signs in the last line which gives the successive values of y.

22 GRAPHICAL ALGEBRA.

26. For convenience on the printed page we have supppsed
the squared paper to be ruled to tenths of an inch, generally
using one of the divisions on the paper as our linear unit. In
practice, however, it will often be advisable to choose a unit
much larger than this, especially in cases where one of the
variables increases or decreases much more rapidly than the
other. Attention is directed to this point in the examples which
follow. The student will find it difficult to get a satisfactory
graph unless a suitable unit of measurement is chosen.

EXAMPLES V.

1. Plot the graph of y = x3. Shew that it consists of a con-
tinuous curve lying in the first and third quadrants, crossing the
axis of x at the origin. Deduce the graphs of

(i) y=-x*-, (ii) y=\x*.

2. Plot the graph of y = x-x3. Verify it from the graphs of
y=x, and y=x?.

3. Plot the graph of y = —y> shewing that it consists of two

branches lying entirely in the first and second quadrants. Examine
and compare the nature and position of the graph as it approaches
the axes.

4. Discuss the general character of the graph of y=-2 where a

has some constant integral value. Distinguish between two cases
in which a has numerical values, equal in magnitude but opposite in
sign.

5. Plot the graphs of

(i) V=l+\, (ii) 2/ = 2+™.

Verify by deducing them from the graphs of t/ = -, and y=-^-

30 *£

6. Plot the graph of y = xs-3x. Examine the character of the
curve at the points (1, -2), ( - 1, 2), and shew graphically that the
roots of the equation x?-3x = Q are approximately -1732, 0, and
1-732.

7. Solve the equations :

3* + 2y=16, xy=10,

and verify the solution by finding the coordinates of the points
where their graphs intersect.

EXAMPLES V. 23

Plot the graphs of

15 -a;2 10-

and thus verify the algebraical solution of the equations

9. Trace the curve whose equation is y = ^- — , shewing that it

has two branches, one lying in the first and third quadrants, and the
other entirely in the fourth. Find the equations of its asymptotes.

Plot the graphs of

n. ,=±* i, ,-»

16. y = o^-
18 v=
21. y

.

23. 2/2=x2-5a; + 4. 24.

25. yl=«(3-»)(.-8)- ^

27. ,,-*^. ag.

29. 5ys = x(x*-64). 30.

31. Plot the graphs of y-xz, and of y = x2 + llx-3. Hence find
the roots of the equation a^-#2-lla; + 3 = Oto two decimal places.

32. Find graphically the roots of the equation

x3-4x
to three significant figures.

24 GRAPHICAL ALGEBRA.

Measurement on Different Scales.

27. Attention has already been drawn to the necessity for
care in selecting suitable units of measurement in graphical
work. In some of the practical applications we are about to
give this consideration is of special importance.

Although for the sake of simplicity we have hitherto measured
abscissae and ordinates on the same scale, there is no necessity
for so doing, and it will often be found convenient to measure
the variables on different scales suggested by the particular
conditions of the question.

As an illustration let us take the graph of # = ir, given in

Art. 15. If with the same unit as before we plot the graph
of y =#2, it will be found to be a curve similar to that drawn on
page 11, but elongated in the direction of the axis of y. In fac.t,
it will be the same as if the former graph were stretched to
twice its length in the direction of the y-axis.

28. Any equation of the form y — ax2, where a is constant, will
represent a parabola elongated more or less according to the
value of a ; and the larger the value of a the more rapidly will
y increase in comparison with x. We might have very large
ordinates corresponding to very small abscissae, and the graph
might prove quite unsuitable for practical applications. In
such a case the inconvenience is obviated by measuring the
values of y on a considerably smaller scale than those of x.

Speaking generally, whenever one variable increases much
more rapidly than the other, a small unit should be chosen for
the rapidly increasing variable and a large one for the other.
Further modifications will be suggested in the examples which
follow.

On the opposite page we give for comparison the graphs of
y=x* (Fig. 13), and y = 8x2 (Fig. 14).

In Fig. 13 the unit for x is twice as great as that for y.
In Fig. 14 the #-unit is ten times the y-unit.

It will be useful practice for the student to plot other similar
graphs on the same or a larger scale. For example, in Fig. 14
the graphs of y = 16.r2 andy = 2#2 may be drawn and compared
with that of y = &r2.

GRAPHS OF = a;2 AND =

25

CM

u«

in

<N &

£

oo co -t

26

GRAPHICAL ALGEBRA.

29. Besides the instances already given there are several of
the ordinary processes of Arithmetic and Algebra which lend
themselves readily to graphical illustration.

For example, the graph of y=xL may be used to furnish
numerical square roots. For since x=*]y, each ordinate and
corresponding abscissa give a number and its square root.
Similarly cube roots may be found from the graph o>iy=y?.

Example 1. Find graphically the cube root of 10 to 3 places of
decimals.

The required root is clearly a little greater than 2. Hence it will
be enough to plot the graph of y = x* taking # = 2*1, 2*2, ... The
corresponding ordinates are 9*26, 10*65, ...

When x = 2, y = 8. Take the axes through this point and let the
units for x and y be 10 inches and '5 inch respectively. On this
scale the portion of the graph differs but little from a straight line,
and yields results to a high degree of accuracy.

11

10

9

8
I

Y

y

~7

^

/

.

/

/•

y

/

7

•

j

>

"

/

;

/

i

/

/

!

^

1

2-1 2-154 2-2

Fi£- IS-
When y=10, the measured value of x will be found to be 2*154.

Example, 2. Shew graphically that the expression 4a;2 + 4x-3 is
negative for all real values of x between '5 and - 1 *5, and positive
for all real values of x outside these limits. [Fig. 16.]

Put y — 4#2 + 4x - 3, and proceed as in the example given in Art. 16,
taking the unit for x four times as great as that for y. It will be
found that the graph cuts the axis of x at points whose abscissae
are *5 and - 1 "5 ; and that it lies below the axis of x between these
points. That is, the value of y is negative so long as x lies between
•5 and - 1 *5, and positive for all other values of x.

ILLUSTRATIVE EXAMPLES.

27

Or we may proceed as follows :

Put yj=:4a;2, and y.2= -4x + 3, and plot the graphs of these two
equations. At their points of intersection y1 = y2> an(^ *ne values
of x at these points are found to be '5 and - 1-5. Hence for these
values of x we have

= - 4a: + 3,

- 3 = 0.

Thus the roots of the equation 4o;2 + 4x - 3 = 0 are furnished by the
abscissae of the common points of the graphs of 4#2 and - 4# + 3.

Again, between the values "5 and - T5 for x it will be found
graphically that y^ is less than y2» hence y\-y^ or 4a;2 + 4a;-3 is
negative.

\\

P

Y

^

/

\

\

/

\

\

\

\

5

/

I

V

\

i

t

\

\

\

a

\

\

\

//

\

\

5

yi

\

x

^

/

5

X

-

•5

\

-1

O

1

\

1

1-5

\

/

x

\

/

x.

S

Yf

Fig. 16.

Both solutions are here exhibited.

The upper curve is the graph of y = 4#2 ; PQ is the graph of
y — - 4# + 3 ; and the lower curve is the graph of y - 4#2 + 4x - 3.

30. Of the two methods in the last Example the first is the
more direct and instructive ; but the second has this advantage :

If a number of equations of the form x2=px+q have to be
solved graphically, y=x>" can be plotted once for all on a con-
venient scale, and y=px + q can then be readily drawn for
different values of jt/and q.

Equations of higher degree may be treated similarly.

28

GRAPHICAL ALGEBRA.

For example, the solution of such equations as

can be made to depend on the intersection of y = x* with
other graphs.

Example. Find the real roots of the equations

(i) x3- 2-5* -3 = 0; (ii) x3-3x + 2=0.
Here we have to find the points of intersection of
(i) y = x3, (ii) y = x3,

Plot the graphs of these equations, choosing the unit for x five
times as great as that for y.

X

•5** 3

Fig. 17.

It will be seen that y = 2'5a; + 3 meets y = x? only at the point for
which x = 2. Thus 2 is the only real root of equation (i).

Again y = 3x-2 touches y = x3 at the point for which a?=l, and
cuts it where x= -2.

Corresponding to the former point the equation xs-3x + 2 = Q has
two equal roots. Thus the roots of (ii) are 1, 1, -2.

TRIGONOMETRICAL FUNCTIONS.

29

31. Apart from questions of convenience with regard to any
particular graph, we may observe that in many cases the
variables whose values are plotted on the two axes denote
magnitudes of different kinds, so that there is no necessary
relation between the units in which they are measured.

A good illustration of this kind is furnished by tracing the
variations of the Trigonometrical functions graphically.

Example. Trace the graph of sin x.

In any work on Trigonometry it is shewn that as the angle x
increases from 0° to 90°, the value of sin x is positive, and increasing
gradually from 0 to 1. From 90° to 180°, sin a: is positive, and
decreasing from 1 to 0. From 180° to 270°, sin x is negative, and
increasing numerically from 0 to — 1 . And from 270° to 360°, sin x
is negative, and decreasing numerically from - 1 to 0.

(See Hall and Knight's Elementary Trigonometry, Art. 86.)

We shall here exhibit these variations independently by putting

2/ = sin x, and plotting the values of y corresponding to values of x

differing by 30°.

By the aid of a table of sines we have :

X

0°

30°

60°

90°

120°

150°

180°

210°

240°

270°

...

y

or

0

•5

•866

1

•866

•5

0

-•5

-•866

-1

The graph is represented by the continuous waving line shewn in
Fig. 18.

T5C

360'

Fig. 18.

On the a;-axis each division represents 6° and on the y-axis ten

divisions have been taken as the unit.

30 GRAPHICAL ALGEBRA.

EXAMPLES VI.

1. Draw the graph of y = #2 on a scale twice as large as that in
Fig. 13, and employ it to find the squares of 72, 17, 3 '4; and the
square roots of 7'56, 5'29, 9'61.

2. Draw the graph of y = >Jx taking the unit for y five times aa
great as that for x.

By means of this curve check the values of the square roots found
in Example 1.

3. From the graph of y = x3 (on the scale of the diagram of
Art. 29) find the values of %9 and ^9 -8 to 4 significant figures.

4. A boy who was ignorant of the rule for cube root required
the value of ^/1471. He plotted the graph of y = x3, using for x
the values 2-2, 2 '3, 2 '4, 2 '5, and found 2 '45 as the value of the cube
root. Verify this process in detail. From the same graph find the
value of vAOTS.

5. Find graphically the values of x for which the expression
.r2 - 2x - 8 vanishes. Shew that for values of x between these limits
the expression is negative and for all other values positive. Find
the least value of the expression.

6. From the graph in the preceding example shew that for any
value of a greater than 1 the equation x2 - 2x + a = 0 cannot have
real roots.

7. Shew graphically that the expression #2-4# + 7 is positive
for all real values of x.

8. On the same axes draw the graphs of

y = x\ y = x + 6, y = x-G, y=-x + 6, y=-x-Q.
Hence discuss the roots of the four equations

9. If x is real, prove graphically that 5-4x- xz is not greater
than 9 ; and that 4xa - 4x + 3 is not less than 2. Between what
values of x is the first expression positive ?

10. Solve the equation x3 = 3xz + 6x - 8 graphically, and shew
that the function x3 - Sx2 - Qx + 8 is positive for all values of x
between -2 and 1, and negative for all values of x between 1 and 4.

11. Shew graphically that the equation x3 + px + q = 0 Jias only
pne real root when p is positive.

EXAMPLES VI. 31

12. Trace the curve whose equation is y = 2*. Find the
approximate values of 24'75 and 25'25. Express 12 as a power of 2
approximately.

Prove also that Iog2 26'9 + log2 38 = 10.

13. By repeated evolution find the values of 10^, 10*, 10*, 10TX

Thence by multiplication by 10 find the values of 10^, 10*, 10*, 10^.
Use these values to plot a portion of the curve y=10* on a large
scale. Find correct to three places of decimals the values of log 3,
log 5, Iog3'25, log!5'36. Also, by choosing numerical values for
a and b, verify the laws

log ab = log a + log b ; log -r = log a - log b.

[By using paper ruled to tenths of an inch, if 10 in. and 1 in. be
taken as units for x and y respectively, a diagonal scale will give values
of x correct to three decimal places and values of y correct to tivo.]

14. Calculate the values of x(Q - x)2 for the values 0, 1, 2, 3, ... 9
of x. Draw the graph of x(Q - x)2 from x = 0 to x = Q.

If a very thin elastic rod, 9 inches in length, fixed at one end,
swings like a pendulum, the expression x(9-x)z measures the
tendency of the rod to break at a place x inches from the point of
suspension. From the graph find where the rod is most likely to
break.

15. If a man spends 22s. a year on tea whatever the price of tea
is, what amounts will he receive when the price is 12, 16, 18, 20, 24,
28, 33, and 36 pence respectively ? Give your results to the nearest
quarter of a pound. Draw a curve to the scale of 4 Ibs. to the inch
and 10 pence to the inch, to shew the number of pounds that he
would receive at intermediate prices.

16. Draw the graphs of cos x and tan x, on a scale twice as large
as that in Art. 31.

17. Draw the graph of sin x from the following values of x :

5°, 15°, 30°, 45°, 60°, 75°, 85°, 90°.
Find the value of sin 37°, and the angle whose sine is '8.

18. Find from the tables the value of cos x when

x = Q°, 10°, 20°, 30°, 40°, 50°, 60°.

Draw a curve on a large scale shewing how cos x varies as x
increases from 0° to 60°.

Find from the curve the values of cos 25° and cos 45°. Verify by
means of the tables.

19. Draw on the same diagram the graphs of the functions sin x,
cos x, and sin x + cos x.

Derive from the figure the general solution of sin x + cos x = 0.

GRAPHICAL ALGEBRA.

gui

is the elevation of the gun. Find from the tables the value of
1000 sin 2A when A has the values

10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°,

and draw a curve shewing how the range varies as A increases from
10° to 50°.

21. From the tables find the values of tan IQx - 2 tan 9x + 1 for
the following values of x: 0°, 1°, 2°, ... 9°. Draw a curve shewing
ho\v tan lOx - 2 tan 9x + 1 varies with x when x lies between 0° and
9°. Find to the nearest tenth of a degree a value of x for which the
given expression vanishes.

Practical Applications.

32. In all the cases hitherto considered the equation of the
curve has been given, and its graph has been drawn by first
selecting values of x and y which satisfy the equation, and then
drawing a line so as to pass through the plotted points. We
thus determine accurately the position of as many points as we
please, and the process employed assures us that they all lie on
the graph we are seeking. We could obtain the same result
without knowing the equation of the curve provided that we
were furnished with a sufficient number of corresponding values
of the variables accurately calculated.

Sometimes from the nature of the case the form of the equa-
tion which connects two variables is known. For example, if
a quantity y is directly proportional to another quantity x it
is evident that we may put y = ax^ where a is some constant
quantity. Hence in all cases of direct proportionality between
two quantities the graph which exhibits their variations is a
straight line through the origin. Also since two points are
sufficient to determine a straight line, it follows that in the
cases under consideration we only require to know the position
of one point besides the origin, and this will be furnished by
any pair of simultaneous values of the variables.

Example 1. Given that 5 '5 kilograms are roughly equal to 12*125
pounds, shew graphically how to express any number of pounds in
kilograms. Express 7| Ibs. in kilograms, and 4£ kilograms in pounds,

PRACTICAL APPLICATIONS.

33

Here measuring pounds horizontally and kilograms vertically, the
required graph is obtained at once by joining the origin to the
point whose coordinates are 12*125 and 5 '5.

CO

,

, "

I

i-

*"

1

r"

L-T

r—

--"

6

t —

\^

-

—

_^_

—

—

— •

~-~

PC

?W

fld

s

Fig. 19.

10 12-125

By measurement it will be found that 7J Ibs. = 3 '4 kilograms, and
4£ kilograms = 9 -37 Ibs.

Example 2. The expenses of a school are partly constant and
partly proportional to the number of boys. The expenses were
£650 for 105 boys, and £742 for 128. Draw a graph to represent
the expenses for any number of boys ; find the expenses for 115 boys,
and the number of boys that can be maintained at a cost of £710.

If the expenses for x boys are represented by £y, it is evident
that x and y satisfy a linear equation y = ax + b, where a and b are
constants. Hence the graph is a straight line.

800

700

600

100

105

110

120

128 X

115
Fig. 20.

As the numbers are large, it will be convenient if we begin
measuring ordinates at 600, and abscissae at 100. This enables us
to bring the requisite portion of the graph into a smaller compass.
The points P and Q are determined by the data of the question, and
the line PQ is the graph required.

By measurement we find that when #=115, y = 690; and that
when y = 710, # = 120. Thus the required answers are £690, and
120 boys.

34

GRAPHICAL ALGEBRA.

33. Sometimes corresponding values of two variables are
obtained by observation or experiment. In such cases the data
cannot be regarded as free from error ; the position of the
plotted points cannot be absolutely relied on ; and we cannot
correct irregularities in the graph by plotting other points selected
at discretion. All we can do is to draw a curve to lie as evenly
as possible among the plotted points, passing through some
perhaps, and with the rest fairly distributed on either side of
the curve. As an aid to drawing an even continuous curve a
thin piece of wood or other flexible material may be bent into
the requisite curve, and held in position while the line is drawn.*
When the plotted points lie approximately on a straight line,
the simplest plan is to use a piece of tracing paper or celluloid
on which a straight line has Ibeen drawn. When this has been
placed in the right position the extremities can be marked on
the squared paper, and by joining these points the approximate
graph is obtained.

Example 1. The following table gives statistics of the population
of a certain country, where P is the number of millions at the
beginning of each of the years specified.

Year

1830

1835

1840

1850

1860

1865

1870

1880

P

20

22-1

23-5

29-0

34-2

38-2

41-0

49-4

Let t be the time in years from 1830. Plot the values of P
vertically and those of t horizontally and exhibit the relation between
P and t by a simple curve passing fairly evenly among the plotted
points. Find what the population was at the beginning of the
years 1848 and 1875.

The graph is given in Fig. 21 on the opposite page. The popula-
tions in 1848 and 1875, at the points A and B respectively, will be
found to be 27 '8 millions and 45 '3 millions.

Example 2. Corresponding values of x and y are given in the
following table :

X

i

4

6-8

8

9-5

12

14-4

y

4

8

1-2-2

13

14-8

20

24-8

Supposing these values to involve errors of observation, draw the
graph approximately and determine the most probable equation
between x and y. [See Fig. 22 on p. 36.]

* One of " Brooks' Flexible Curves " will be found very useful.

PRACTICAL APPLICATIONS.

35

CO

36

GRAPHICAL ALGEBRA.

After carefully plotting the given points we see that a straight
line can be drawn passing through three of them and lying evenly
among the others. This is the required graph.

20

10

Fig. 22.

10

Assuming y=ax + b for its equation, we find the values of a and b
by selecting two pairs of simultaneous values of x and ?/.

Thus substituting x = 4, y = 8, and x= 12, y = 2Q in the equation, we
obtain a=T5, 6 = 2. Thus the equation of the graph is y=l'5x + 2.

34. In the last Example as the graph is linear it can be
produced to any extent within the limits of the paper, and so
any value of one of the variables being determined, the corre-
sponding value of the other can be read off. When large values
are in question this method is not only inconvenient but unsafe,
owing to the fact that any divergence from accuracy in the
portion of the graph drawn is increased when the curve is
produced beyond the limits of the plotted points. The follow-
ing Example illustrates the method of procedure in such cases.

Example. In a certain machine P is the force in pounds required
to raise a weight of W pounds. The following corresponding values
of P and W were obtained experimentally :

P

2-48

3-9

6'8

8-8

9-2

1*1

13-3

W

21

36-25

66-2

87-5

103-75

120

1.V2--)

By plotting these values on squared paper draw the graph con-
necting P and W, and read off the value of P when W= 70. Also
determine a linear law connecting P and W ; find the force necessary
to raise a weight of 310 Ibs., and also the weight which could be
raised by a force of 180 '6 Ibs.

PRACTICAL APPLICATIONS. 37

As the page is too small to exhibit the graphical work on a
convenient scale we shall merely indicate the steps of the solution,
which is similar in detail to that of the last example.

Plot the values of P vertically and the values of W horizontally.
It will be found that a straight line can be drawn through the points
corresponding to the results marked with an asterisk, and lying
evenly among the other points. From this graph we find that when
FF=70, P=7.

Assume P = aW + b, and substitute for P and JFfrom the values
corresponding to the two points through which the line passes.
By solving the resulting equations we obtain a = '08, & = 1'4. Thus
the linear equation connecting P and W is P= '08 W + 1'4.

This is called the Law of the Machine.

From this equation, when JF=310, P = 26'2, and when P=180'6,

Thus a force of 26 '2 Ibs. will raise a weight of 310 Ibs.; and when
a force of 180 '6 Ibs. is applied the weight raised is 2240 Ibs. or 1 ton.

Note. The equation of the graph is not only useful for determin-
ing results difficult to obtain graphically, but it can always be used
to check results found by measurement.

35. The example in the last article is a simple illustration of
a method of procedure which is common in the laboratory or
workshop, the object being to determine the law connecting two
variables when a certain number of simultaneous values have
been determined by experiment or observation.

Though we can always draw a graph to lie fairly among the
plotted points corresponding to the observed values, unless
the graph is a straight line it may be difficult to find its
equation except by some indirect method.

For example, suppose x and y are quantities which satisfy an
equation of the form xy — ax-\-l>y^ and that this law has to be
discovered.

By writing the equation in the form

- + - = 1, or au + bv = l ;
y x

where u = -, y = -, it is clear that u, v satisfy the equation of a

straight line. In other words, if we were to plot the points
corresponding to the reciprocals of the given values, their linear
connection would be at once apparent. Hence the values of
a and b could be found as in previous examples, and the required
law in the form xy=ax+~by could be determined.

38

GRAPHICAL ALGEBRA.

Again, suppose x and y satisfy an equation of the form xny = c,
where n ana c are constants.

By taking logarithms, we have

n log x + log y = log c.

The form of this equation shews that log# and logy satisfy
the equation to a straight line. If, therefore, the values of log.r
and logy are plotted, a linear graph can be drawn, and the
constants n and c can be found as before.

Example, The weight, y grammes, necessary to produce a given
deflection in the middle of a beam supported at two points, x centi-
metres apart, is determined experimentally for a number of values
of x with results given in the following table :

X

50

60

70

80

90

100

y

270

150

100

60

47

32

1-778
1-845
1-903
1-954

2-000

2-431
2-J7<>
2-000
1-778
1-672
1-519

Assuming that x and y are connected by the equation xny = c,

find n and c.

log x | log y

From a book of tables we obtain the annexed —
values of log a: and logy corresponding to the
observed values of x and y. By plotting these
we obtain the graph given in Fig. 23, and its
equation is of the form

n log x + log y — log c.

To obtain n and c, choose tivo extreme points through which the line
passes. It will be found that when

log x =1-642, logy = 2-6

and when log# = 2'l, Iogy=r21.

Substituting these values, we have

2-6 -fwx l-642 = logc (i),

l-21-fnx2-l =logc (ii) ;

.-. l-39-0'458w = 0;
whence n = 3'04.

/. from (ii) log c = 6'38+ 1'21

= 7-59;

.-. c = 39x 106, from the tables.
Thus the required equation is rr3y = 39x 108.

The student should work through this example in detail on a
larger scale. The adjoining figure was drawn on paper ruled to
tenths of an inch and then reduced to half the original scale.

PRACTICAL APPLICATIONS.

39

vo

1 6

*

40

GRAPHICAL ALGEBRA.

EXAMPLES VII.

1. Given that 6 '01 yards = 5 '5 metres, draw the graph shewing
the equivalent of any number of yards when expressed in metres.

Shew that 22 '2 yards = 20 '3 metres approximately.

2. Draw a graph shewing the relation between equal weights in
grains and grammes, having given that 10'8 grains = 1 '17 grammes.

Express (i) 3 '5 grammes in grains.

(ii) 3 '09 grains as a decimal of a gramme.

3. If 3 '26 inches are equivalent to 8 '28 centimetres, shew how
to determine graphically the number of inches corresponding to
a given number of centimetres. Obtain the number of inches in
a metre, and the number of centimetres in a yard. What is the
equation of the graph ?

4. The following table gives approximately the circumferences
of circles corresponding to different radii :

a 1 15-7

20-1

31-4

44

52-2

r

2-5

3-2

5

7

8-3

Plot the values on squared paper, and from the graph determine
the diameter of a circle whose circumference is 12 '1 inches and the
circumference of a circle whose radius is 2 '8 inches.

5, For a given temperature, C degrees on a Centigrade are equal
to F degrees on a Fahrenheit thermometer. The following table
gives a series of corresponding values of F and C :

0

-10

-5

0

5

10

15

25

40

F

14

23

32

41

50

59

77

104

Draw a graph to shew the Fahrenheit reading corresponding to
a given Centigrade temperature, and find the Fahrenheit readings
corresponding to 12-5° C. and 31° C.

By observing the form of the graph find the algebraical relation
between F and C.

6. For a certain book it costs a publisher £100 to prepare the
type and 2s. to print each copy. Find an expression for the total
cost in pounds of x copies. Make a diagram on a scale of 1 inch to
1000 copies, and 1 inch to £100 to shew the total cost of any number
of copies up to 5000. Read off the cost of 2500 copies, and the
number of copies costing £525.

EXAMPLES VII.

41

7. At different ages the mean after-lifetime ("expectation of
life") of males, calculated on the death rates of 1871-1880, was
given by the following table :

Age

6

10

14

18

22

26

27

Expectation

50-38

47-60

44-26

40-96

37-89

34-96

34-24

Draw a graph to shew the expectation of any male between the
ages of 6 and 27, and from it determine the expectation of persons
aged 12 and 20.

8. In the Clergy Mutual Assurance Society the premium (£P) to
insure £100 at different ages is given approximately by the following
table :

Age

20

22

25

30

35

40

45

50

55

P

1-8

1-9

2-0

2-3

2-7

3'1

3-6

4-4

5-5

Illustrate the same statistics graphically, and estimate to the
nearest shilling the premiums for persons aged 34 and 43.

9. If W is the weight in ounces required to stretch an elastic
string till its length is I inches, plot the following values of W and I :

W

2-5

3-75

6-25

7-5

10

11-25

I

8-5

8-7

9-1

9-3

9-7

9-9

From the graph determine the unstretched length of the string,
and the weight the string will support when its length is 1 foot.

10. In the following table P and A (expressed in hundreds of
pounds) represent the Principal and corresponding Amount for 1
year at 3 per cent, simple interest.

P

2-3

2-7

3-0 | 3-5

3-9

5-2

7-6

A

2-369

2-781

3-090

3-605

4-017

5-356

7-828

Plot the values of P and A on- a large scale, and from the graph
determine the Principal which will amount to (i) £329. 12s. ;
(ii) £587. 8s.

42

GRAPHICAL ALGEBRA.

11. The highest and lowest marks gained in an examination are
297 and 132 respectively. These have to be reduced in such a way
that the maximum for the paper (200) shall be given to the first
candidate, and that there shall be a range of 150 marks between the
first and last. Find the equation between x, the actual marks
gained, and y, the corresponding marks when reduced.

Draw the graph of this equation, and read off the marks which
should be given to candidates who gained 200, 262, 163 marks in the
examination.

12. A body starting with an initial velocity, and subject to an
acceleration in the direction of motion, has a velocity of v feet per
second after t seconds. If corresponding values of v and t are given
by the annexed table,

V

9

13

17

21

25

29

33

37

41

45

t

1

2

3

4

5

6

7

8

9

10

Elot the graph exhibiting the velocity at any given time. Find
•oin it (i) the initial velocity, (ii) the time which has elapsed when
the velocity is 28 feet per second. Also find the equation between
v and /.

13, The connection between the areas of equilateral triangles and
their bases (in corresponding units) is given by the following table :

Illustra
equilatera

Area

•43

1-73

3-90

6-93

10-82

15-59

Base

1

2

3

4

5

6

te these results graphically, and determine the area of an
1 triangle on a base of 2 '4 ft.

14. A body falling freely under gravity drops 8 feet in t seconds
from the time of starting. If corresponding values of s and t at
intervals of half a second are as follows :

t

•5

1

1-6

2

2-5

3

3-5

4

8

4

16

36

64

100

144

196

256

draw the curve connecting 8 and t, and find from it

(i) the distance through whi«h the body has fallen after 1 min.
48 sees.

(ii) the distance through which it drops in the 4th second.

EXAMPLES VII.

43

15. A body is projected with a given velocity at a given angle
to the horizon, and the height in feet reached after t seconds is
given by the equation h = 6±t- 16£2. Find the values of h at
intervals of Jth of a second and draw the path described by the
body. Find the maximum value of h, and the time after projection
before the body reaches the ground.

16. The keeper of a hotel finds that when he has G guests a day
his total daily profit is P pounds. If the following numbers are
averages obtained by comparison of many days' accounts determine
a simple relation between P and G.

G

21

27

29

32

35

P

-1-8

Ml

3-2

4-5

6-6

For what number of guests would he just have no profit ?

17. A man wishes to place in his catalogue a list of a certain class
of fishing rods varying from 9 ft. to 16 ft. in length. Four sizes have
been made at prices given in the following table :

9ft.

11 ft. 9 in.

14 ft. 4 in.

16ft.

15s.

22s

31s.

38s.

Draw a graph to exhibit prices for rods of intermediate lengths,
and from it determine the probable prices for rods of 13 ft. and
15 ft. 8 in.

18. The following table gives the sun's position at 7 A.M. on
different dates :

Mar. 23

Ap. 3

Ap. 20

May8

May 27

June 22

July 18

Aug. 5

Aug. 25

80° E.

82° E.

85° E.

89° E.

92° E.

95° E.

94° E.

91° E.

85° E.

Shew these results graphically, and estimate approximately the
sun's position at the same hour on June 8th.

19. At a given temperature p Ibs. per square inch ^represents the
pressure of a gas which occupies a volume of v cubic inches. Draw
a curve connecting p and v from the following table of corresponding
values :

P

36

30

257

22-5

20

18

16-4

15

V

5

6

7

8

9

10

11

12

44

GRAPHICAL ALGEBRA.

20. Plot on squared paper the following measured values of x and
y, and determine the most probable equation between x and y :

X

3

5

8-3

11

13

15'5

18-6

23

28

y

2

2-2

3-4

3-8

4

4-6

5-4

6-2

7-25

21. The following table refers to aqueous solution of ammonia at
a given temperature ; x represents the specific gravity of the solu-
tion, and y the percentage of ammonia :

X

•996

•992

•988

•984

•980

•976

•968

y

•91

1-84

2-80

3-80

4-80

5-80

7-82

Draw a graph shewing the variations of x and y, and find its
equation.

22. Corresponding values of x and y are given in the following
table :

X

i

3-1

6

9-5

12-5

16

19

23

y

2

2-8

4-2

5-3

6-6

8-3

9

10-8

Supposing these values to involve errors of observation, draw the
graph approximately, and determine the most probable equation
between x and y. Find the correct value of y when x= 19, and the
correct value of x when = 2'8.

23. The following corresponding values of x and y were obtained
experimentally :

X

0-5

1-7

3-0

4-7

5-7

7-1

8-7

9-9

10-6

11-8

y

148

186

265

326

436

.VJU

502

611

(I.VJ

It is known that they are connected by an equation of the form

involv

, but the values of x and y involve errors of measurement.
Find the most probable values of a and b, and estimate the error in
the measured value of y when x = 9 '9.

EXAMPLES VII.

45

24. In a certain machine P is the force in pounds required to
raise a weight of W pounds. The following corresponding values of
P and W were obtained experimentally :

P

2-8

3-7

4-8

5-5

6-5

7-3

8

9-5

10-4

11-75

W

20

25

31-7

35-6

45

52-4

57-5

65

71

82-5

Draw the graph connecting P and W, and read off the value of P
when JF=60. Also determine the law of the machine, and find from
it the weight which could be raised by a force of 31 '7 Ibs.

25. The following values of x and y, some of which are slightly
inaccurate, are connected by an equation of the form y = ax'2 + b.

X

i

1*6

3

3-7

4

5

5-7

6

6-3

7

y

3-25

4

5

6-5

7-4

9-25

10-5

11-6

14

15-25

By plotting these values draw the graph, and find the most
probable values of a and b.

Find the true value of x when y = 4, and the true value of y
when x = 6.

26. The following table gives corresponding values of two variables
a; and y :

X

2-75

3

3-2

3-5

4-3

4-5

5-3

6

7

8

10

y

11

9-8

8

6-5

6-1

5-4

5

4-3

4-1

4

3-9

These values involve errors of observation, but the true values are
known to satisfy an equation of the form xy = ax + by. Draw the
graph by plotting the points determined by the above table, and
find the most probable values of a and &. Find the correct values of
y corresponding to # = 3-5, and x = 7.

27. Observed values of x and y are given as follows :

X

100

90

70

60

50

40

y

30

31-08

33-5

35-56

37-8

40-7

Assuming that x and y are connected by an equation of the form
xyn=c, find n and c.

46 GRAPHICAL ALGEBRA.

28. The following values of x and y involve errors of observation

X

66-83

63-10

58-88

51-52

48-53

44-16

40-36

y

144-5

158-5

177-8

208-9

236-0

264-9

309-0

If x and y satisfy an equation of the form x"y = c, find n and c.

29. In the following table the values of C and C' represent the
calculated and observed amounts of water, in cubic feet per second,
flowing through a circular orifice for different heads of water repre-
sented by H feet.

H

60

69-12

82

92-16

106

115-2

134

C

•0133

•0141

•0154

•0163

•0175

•0182

•0197

C'

•0133

•0141

•0153

•0162

•0173

•0180

•0194

Plot the graph of C and H and also that of C' and H, and deduce
the probable error in the observed flow for a head of 120 feet.

30. The following table gives the pressures (in Ibs. per sq. in.)
and corresponding Fahrenheit temperatures at which water boils :

P

29-7

14-7

12-25

9-80

7-84

6-86

t

249-6

212-0

203-0

192-3

182-0

176-0

Shew graphically the relation between temperature and pressure
of boiling water.

31. It is known that the relation of pressure to volume in satu-
rated steam under certain conditions is of the form pvn = constant.
Find the value of the index n from the following data :

P

10-2

14-7

20-8

24-5

33-7

39-2

45-5

V

37-5

26-6

19-2

16-4

12-2

10-6

9-2

where p is measured in Ibs. per sq. in. , and v is the volume of 1 Ib.
of steam in cub. ft.

EXAMPLES VII.

47

32. The following table gives the speed and corresponding indi-
cated horse-power of the engines of a ship :

Speed in knots

11

124

13-3

14-25

14-8

15-5

LH.R

1000

1500

2000

2500

3000

3500

At what speed will she go when she develops 4000 I.H.P. ?

33. In testing a steam-engine when steam was expanded to 4'8
times its original volume, the following quantities of steam per
indicated horse-power per hour were used :

Steam per I.H.P.
per hr. in Ibs.

16-9

17

17-2

18

20-3

I.H.P.

40-5

33

25-5

19

11

When the ratio of expansion in the engine was 10 instead of 4 '8,
the steam used was as follows :

Steam per I.H.P.
per hr. in Ibs.

15

15-5

16

18

26-5

I.H.P.

33

27-2

23

15

5

At what H.P. will the consumption of steam be the same in the
two cases, and what is the consumption of steam at that H.P. ?

34. The power required to produce a given speed in the case of
each of two ships is given in the following tables :

(i)

Speed

8

10-7

12-7

14

16

16-2

I.H.P.

500

1000

1500

1950

2800

3000

Speed

8

10

12

12-5

13-5

14-5

16-1

16-7

I.H.P.

200

400

920

1100

1500

2000

3000

3500

At what speed will they generate the same H.P. ?

48 GRAPHICAL ALGEBRA.

ANSWERS.

I. PAGE 4.

7. 36. 8. 32. 9. 25. 11. l'2sq. cm.

12. y = 3x. Any point whose ordinate is equal to three times its
abscissa.

14. The lines are x = 5, y = 8. The point (5, 8).

15. A circle of radius 13 whose centre is at the origin.

II. PAGE 7.

21. 32 sq. units. 22. 1 sq. in.

23. 72 sq. units. 24. '64 sq. cm.

III. PAGE 10.

1.

x =

i, y=

5.

2.

*=2,

y=lO.

3.

* = 3,

y = Yl

4.

x =

3, y=

-2

5.

x = 4,

y = 2.

6.

x = Q,

y = 8.

7.

x =

-2,3,

' = 4

8.

x = Q,

y=-3.

9.

x= -

3,0.

.0.

At

the pc

>int

(0,

21).

11.

3x + 4

•y=7.

IV. PAGE 15.
1. y = x. 2. (0,0), (-4,2). 6. (2,1).

6. (i) 1-46, -5-46; (ii) 3'24, -1-24; (iii) 3'32, '68.

7. -5; 7. 8> ~1; 3'79' " '79; 4'62) ~1<62>
9. x = 8, or 6; y = 6, or 8.

10. The straight line 3# + 2y = 25 touches the circle #2 + t/2=25 at
the point (3, 4).

V. PAGE 22.

3. Each axis is an asymptote to the curve, which approaches
the axis of y much less rapidly than it does the axis of x.

7. x = 2, y; y = 5, 3. 8. x=±3',y=±2.

9. x = 2 :y=-l. 31. -3, 373, -27. 32. -2, 4-41, 1 -W.

ANSWERS. 49

VI. PAGE 30.

1. -52, 2-9, 11-6; 2*75, 2-3, 3-1. 3. 2 '080, 2-140. 4. 2-4.
5. -2,4; -9. 9. -5andl. 10. -2,1,4.

12. 26-9, 38, 3-58. 13. '477, '699, '512, 1-86.

14. 3 in. from the point of suspension.

15. 22 Ibs., 16i Ibs., 14f Ibs., 13£ Ibs., 11 Ibs., 9£ Ibs., 8 Ibs.,
7J Ibs. The curve is a rectangular hyperbola whose equation is
xy = 22x12.

17. -602. 53°. 18. -906. '707. 19. x=mr + — .

4

20. The range varies from 342 yards, when A = 10°, to 984 '8 yards
when A =40°. It reaches its maximum of 1000 yards when A =45°,
and is again equal to 984*8 yards, when A =50°.

21. 5-9°, 7'7°.

VII. PAGE 40.

2. (i) 53-7 grains ; (ii) -2 3. 39 '3 ; 91 '6; y='393#.

4. 3-85 in.; 17 '6 in. 5. 54'5°F. 86'9°F. F =

5

6. y=100 + ; £350; 4250. 7. 45 '96; 39 '40.

8. £2. 12s.; £3. 8s. 9. 8'1 in.; 24 '375 oz.

10. (i) £320 ; (ii) £580. 11. y = ^x-f20. 112; 168; 78.

12. 5 ft. per sec.; 5 min. 45 sees.; v = 5 + 4t. 13. 2 -49 sq. ft.

14. (i) 52ft.; (ii) 112ft. 15. max. height = 64 ft. ; 4 sees.

16. P='6#-14-4; 24. 17. 26s. ; 36s. Qd.

18. 93'5°E. 20. y=-21a + l-37.

21. y = 249-8 -250x. 22. y= -4# + l-6; 9'2; 3.

23. a = 45-7, 6=119. Error = 9 '43 in defect.

24. 8-6; P=-14JF+-2; 225 Ibs.

25. a = i; 6 = 3. 2; 12. 26. a = 3, 6 = 2. 7; 4'25.
27. n = 3, c = 27 x 105. 28. n = 1 "5, c = 79500.

29. 1 -3 per cent, in defect. 31. w=j^.

32. -16 knots. 33. H.P.=6'9. 23 Ibs

34. 15-15 knots; 2420 H. P.

WORKS by H. S. HALL, M.A., and S. R. KNIGHT, B.A.

ELEMENTARY ALGEBRA FOR SCHOOLS. Seventh
Edition, Revised and Enlarged. Globe 8vo. (bound in
maroon-coloured cloth). 35. 6d. With Answers (bound
in green-coloured cloth). 45. 6d. Answers separately, is.

SOLUTIONS OF THE EXAMPLES IN ELEMENTARY
ALGEBRA FOR SCHOOLS. Crown 8vo. 8s. 6d.

HIGHER ALGEBRA. A Sequel to Elementary Algebra for
Schools. Fifth Edition, Revised and Enlarged. Crown 8vo.
7s. 6d.

SOLUTIONS OF THE EXAMPLES IN HIGHER AL-
GEBRA. Crown 8vo. los. 6d.

ALGEBRA FOR BEGINNERS. Globe 8vo. Without
Answers. 2s. With Answers. 2s. 6d.

ALGEBRAICAL EXERCISES AND EXAMINATION
PAPERS. With or without Answers. Third Edition,
Revised and Enlarged. Globe 8vo. 2s. 6d.

ARITHMETICAL EXERCISES AND EXAMINATION
PAPERS. With or without Answers. Third Edition,
Revised and Enlarged. Globe 8vo. 2s. 6d.

ELEMENTARY TRIGONOMETRY. Globe 8vo. 45. 6d.

SOLUTIONS OF THE EXAMPLES IN ELEMENTARY
TRIGONOMETRY. Crown 8vo. 8s. 6d.

ALGEBRAICAL EXAMPLES Supplementary to Hall and
Knighfs "Algebra for Beginners" and "Elementary
Algebra" (Chaps. I.-XXVII.). By H. S. Hall, M.A. With
or without Answers. Globe 8vo. 2s.

MACMILLAN AND CO., LIMITED, LONDON.

WORKS BY
H. 8. HALL, M.A., AND F. H. STEVENS, M.A.

A TEXT-BOOK OF EUCLID'S ELEMENTS, including
Alternative Proofs, together with additional Theorems and
Exercises Classified and Arranged. Third Edition, Enlarged.
Books I.-VL, XL (and XII. 1 and 3). Globe 8vo. 4s. 6d.

Also in parts, separately, as follows —

Book I. - - Is.
Books I. and II. Is. 6d.
Books II. and III. 2s.
Books I.— III. - 2s. 6d.

Sewed, 2s.

Books III. and IV. - 2s.
Books III.— VI. - 3s.
Books IV.— VI. - 2s. 6d.
Books V., VI., XI. and XII.
1 and 3. - - 2s. 6d.
Book XI. - - Is.

Books I.— IV. -

Sewed, 2s. 6d.

Book I. covers the First Stage of the South Kensington

Syllabus.
Books I. to IV. cover the First and Second Stages of the

Syllabus.

CAMBRIDGE REVIEW.— "fo teachers and students alike we can heartily
recommend this little edition of Euclid's Elements. The proofs of Euclid are

written out. Useful notes and easy examples are scattered throughout each
book, and sets of hard examples are given at the end. The whole is so evidently
the work of practical teachers, that we feel sure it must soon displace ever;
other Euclid.''

JOURNAL OF EDUCATION.— "The most complete introduction to plane
geometry based on Euclid's Elements that we have yet seen."

SCHOOLMASTER.— "The whole work shows evidence of deep mathematical
thought and of the authors' experience as practical teachers. It is an edition of
Euclid that can be highly recommended."

KEY TO A TEXT-BOOK OF EUCLID'S ELEMENTS.
Crown 8vo. Books I.— IV. 6s. 6d. Books VI. and XI.
3s. 6d. Books I.— VI. and XI. 8a. 6d.

AN ELEMENTARY COURSE OF MATHEMATICS, com-
prising Arithmetic, Algebra, and Euclid. Adapted to the
First Stage of the South Kensington Syllabus. Globe 8vo.

SCHOOL GUARDIAN.—" For those students who intend offering themselves
for the first stage Mathematics Examination of South Kensington, and who like
all their work between a single pair of covers, no book can be better than
the volume now before us. ... The volume is in every way worthy of
praise."

MACMILLAN AND CO., LTD., LONDON.

QA Hall, Henry Sinclair
219 A short introduction to

H35 graphical algebra

1903 2d ed., rev. and enl.

Physicals
Applied Sci.

PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET

UNIVERSITY OF TORONTO LIBRARY