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Mansfield Merriman and Robert S. Woodward.

Octavo, Cloth, $i.oo each.


By David Eugene Smith.


By George IJruce Halsted.


By Laenas Gifford Weld.


By James McMahon.


By William E. Bverly.


By Edward W. Hyde.


By Robert S. Woodward.

By Ale.xa.nder Macfarlane.


By Willi A.M Woolsey Johnson.
By Mansfield

By Thomas S. Fiske.






No. 10.




Professor of Civil Engineering in Lehigh University.




London- CHAPMAN & HALL, Limited.


Copyright, 1896,





First Edition, September, 1896.
Second Edition, January, 1898.
Third Edition, August, 1900.
Fourth Edition, January, 1906.

' « « ,

.* c t



The volume called Higher Mathematics, the first edition
of which was published in 1896, contained eleven chapters by
eleven authors, each chapter being independent of the others,
but all supposing the reader to have at least a mathematical
training equivalent to that given in classical and engineering
colleges. The publication of that volume is now discontinued
and the chapters are issued in separate form. In these reissues
it will generally be found that the monographs are enlarged
by additional articles or appendices which either amplify the
former presentation or record recent advances. This plan of
publication has been arranged in order to meet the demand of
teachers and the convenience of classes, but it is also thought
that it may prove advantageous to readers in special lines of
mathematical literature.

It is the intention of the publishers and editors to add other
monographs to the series from time to time, if the call for the
same seems to warrant it. Among the topics which are under
consideration are those of elUptic functions, the theory of num-
bers, the group theory, the calculus of variations, and non-
EucUdean geometry; possibly also monographs on branches of
astronomy, mechanics, and mathematical physics may be included.
It is the hope of the editors that this form of pubHcation may
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.

December, 1905.




The following pages are designed as supplementary to the
discussions of equations in college text-books, and several methods
of solution not commonly given in such works are presented
and exemplified. The aim kept in view has been that of the
determination of the numerical values of the roots of numerical
equations, and algebraic analysis has been used only to further
this end. Historical references are given, problems stated as
exercises for the student, and the attempt has everywhere been
made to present the subject clearly and concisely. The volume
has not been written for those thoroughly conversant with the
theory of equations, but rather for students of mathematics,
computers, and engineers.

This edition has been enlarged by the addition of five articles
which render the former treatment more complete and also give
recent investigations regarding the expression of roots in series.
While not designed for college classes, it is hoped that the book
may prove useful to postgraduate students in mathematics,
physics and engineering, and also tend to promote general interest
in mathematical science.

South Bethlehem, Pa.,
December, 1905.


Art. I. Introduction Page i

2. Graphic Solutions 3

3. The Regula Falsi 5

4. Newton's Approximation Rule 6

5. Separation of the Roots 8

6. Nutvlerical Algebraic Equations 10

7. Transcendental Equations 13

8. Algebraic Solutions 15

9. The Cubic Equation 17

10. The Quartic Equation 19

11. QuEsiTic Equations 21

12. Trigonometric Solutions 24

13. Real Roots by Series 27

14. Computation of All Roots .... 28

15. Roots of Unity 31

16. Solutions by Maclaurin's Series 33

17. Symmetric Functions of Roots 37

18. Logarithmic Solutions 39

19. Infinite Equations 43

20. Notes and Problems 45

Index 47


Art. 1. Introduction.

The science of algebra arose in the efforts to solve equations.
Indeed algebra may be called the science of the equation, since
the discussion of equalities and the transformation of forms into
simpler equivalent ones have been its main objects. The solu-
tion of an equation containing one unknown quantity consists
in the determination of its value or values, these being called
roots. An algebraic equation of degree n has n roots, while tran-
scendental equations often have an infinite number of roots. The
object of the following pages is to present and exemplify convenient
methods for the determination of the numerical values of the
roots of both kinds of equations, the real roots receiving special
attention because these are mainly required in the solution of
problems in physical science.

An algebraic equation is one that involves only the opera-
tions of arithmetic. It is to be first freed from radicals so as
to make the exponents of the unknown quantity all integers;
the degree of the equation is then indicated by the highest ex-
ponent of the unknown quantity. The algebraic solution of an
algebraic equation is the expression of its roots in terms of
the literal coefficients ; this is possible, in general, only for linear,
quadratic, cubic, and quartic equations, that is, for equations
of the first, second, third, and fourth degrees. A numerical
equation is an algebraic equation having all its coefficients real
numbers, either positive or negative. For the four degrees


above mentioned the roots of numerical equations may be
computed from the formulas for the algebraic solutions, unless
they fall under the so-called irreducible case wherein real
quantities are expressed in imaginary forms.

An algebraic equation of the n^^ degree may be written
with all its terms transposed to the first member, thus :

X" + a,x"-' + a,x'"-' + . . . + a„_,x + a„ = o,

and, for brevity, the first member will be called /{x) and the
equation be referred to as/{x) = o. The roots of this equa-
tion are the values of x which satisfy it, that is, those values of
X that reduce /(x) to o. When all the coefficients a^, a^, . . .a„
are real, as will always be supposed to be the case, Sturm's
theorem gives the number of real roots, provided they are un-
equal, as also the number of real roots lying between two
assumed values of x, while Horner's method furnishes a con-
venient process for obtaining the values of the roots to any
required degree of precision.

A transcendental equation is one involving the operations
of trigonometry or of logarithms, as, for example, x -\- cos;i; = o,
or a^"" -\- xir' = O. No general method for the literal solution
of these equations exists ; but when all known quantities are
expressed as real numbers, the real roots may be located and
computed by tentative methods. Here also the equation may
be designated as/(;f) = o, and the discussions in Arts. 2-5 will
apply equally well to both algebraic and transcendental forms.
The methods to be given are thus, in a sense, more valuable
than Sturm's theorem and Horner's process, although for
algebraic equations they may be somewhat longer. It should
be remembered, however, that algebraic equations higher than
the fourth degree do not often occur in physical problems, and
that the value of a method of solution is to be measured not
merely by the rapidity of computation, but also by the ease
with which it can be kept in mind and applied.

Prob. I. Reduce the equation (a + x)^ -{- (a — x)^ = 2b to an
equation having the exponents of the unknown quantity all integers.



c'', from which the two

Art. 2. Graphic Solutions.

Approximate values of the real roots of two simultaneous
algebraic equations may be found by the methods of plane
analytic geometry when the coefficients are numerically
expressed. For example, let the given equations be

x" -{-y" = a\ x' — bx =y — cy,

the first representing a circle and the second a hyperbola.
Drawing two rectangular axes OX and OY, the circle is de-
scribed from O with the radius a. The coordinates of the
center of the hyperbola are found to be OA = \b and AC ^\c,

while its diameter BD = ^ b' —

branches may be described.

The intersections of the circle

with the hyperbola give the

real values of x and y. % If

a=i I, b ^ ^, and ^ = 3, there

are but two real values for x

and two real values for j/,

since the circle intersects but

one branch of the hyperbola ;

here Om is the positive and

Op the negative value of x, while 7nn is the positive and pq

the negative value oi y. When the radius a is so large that

the circle intersects both branches of the hyperbola there are

four real values of both x and y.

By a similar method approximate values of the real roots of
an algebraic equation containing but one unknown quantity may
be graphically found. For instance, let the cubic equation
x^ -^ ax — b ^ o be required to be solved.* This may be
written as the two simultaneous equations

y ^=z x"", y = — ax -\- b,

* See Proceedings of the Engineers' Club of Philadelphia, 1884, V'.l. IV,
pp. 47-49


and the graph of each being plotted, the abscissas of their
points of intersection give the real roots of the cubic. The

curve J/ = x^ should be plotted upon
cross-section paper by the help of a
table of cubes ; then OB is laid off
equal to d, and OC equal to a/d, tak-
ing care to observe the signs of a and
d. The line joining B and C cuts
the curve at p, and hence g/> is the
real root oi x^ -\- ax — b = o. If the
cubic equation have three real roots the straight line BC will
intersect the curve in three points.

Some algebraic equations of higher degrees may be graphic-
ally solved in a similar manner. For the quartic equation
s" -\- Az' -\- Bz — C ^ o, \\. is best to put s = A^x, and thus
reduce it to the form x* -{- x'' -\- dx — c = o ; then the two
equations to be plotted are

J/ ^ X* -{- x'', y ■= — bx -\- c,

the first of which may be drawn once for all upon cross-section
paper, while the straight line represented by the second may
be drawn for each particular case, as described above.*

This method is also applicable to many transcendental equa-
tions ; thus for the equation .^;f — ^sin;f = o it is best to
write ax — sin;r = o ; then y := smx is readily plotted by help
of a table of sines, while y ^= ax is a straight line passing
through the origin. In the same way a"^ — x'^ ^^ o gives the
curve represented byj:=rt'*and the parabola represented by
y = x"^, the intersections of which determine the real roots of
the given equation.

Prob. 2. Devise a graphic solution for finding approximate
values of the real roots of the equation x''-\- ax^-\- bx''-{- ex + d =o.

Prob. 3. Determine graphically the number and the approximate
values of the real roots of the equation arc x — 2> sin x = o.
(Ans. — Six real roots, x = ± 159°, ± 430°, and ± 456°.)

* For an extension of this method to the determination of imaginary roots,
see Phillips and Beebe's Graphic Algebra, New York, 1S82.



Art. 3. The Regula Falsi.

One of the oldest methods for computing the real root of
an equation is the rule known as " regula falsi," often called
the method of double position.* It depends upon the princi-
ple that if two numbers x^ and x^ be substituted in the expres-
sion y"(;r), and if one of these renders f{x) positive and the other
renders it negative, then at least one real root of the equation
f[x) = o lies between x^ and x^. Let the figure represent a
part of the real graph of the equation y =y(,r). The point X,
where the curve crosses the axis of abscissas, gives a real root
OX of the equation f{x) = o. Let OA and OB be inferior and
superior limits of the root OX which are determined either by
trial or by the method of Art, 5.
Let Aa and Bb be the values of
/{x) corresponding to these limits.
Join ad, then the intersection C of
the straight line al? with the axis
OB gives an approximate value
OC for the root. Now compute
Cc and join ac, then the intersection D gives a value OD which
is closer still to the root OX.

Let ,r, and x^ be the assumed values OA and OB, and let
/{x^) and/(-rj) be the corresponding values o( /(x) represented
by Aa and Bd, these values being with contrary signs. Then
from the similar triangle AaC and BdC the abscissa OC is

'''~ A^.)-A^\) ■*'""^At'J-/(^,) ""'^"^ At-.) -/K) •

By a second application of the rule to x^ and x^, another value

x^ is computed, and by continuing the process the value of x

can b^ obtained to any required degree of precision.

As an example \e\. f{x) =1 x^ -\- ^x'' -|- 7 = o. Here it may

be found by trial that a real root lies between —2 and — 1.8.

* This originated in India, and its first publication in Europe was by Abra-
ham ben Esra, in 1130. See Matthiesen, Grundziige der antiken und moder-
nen Algebra der litteralen Gieichungen, Leipzig, 1878.


For ^, = — 2,f{x^ = — 5, and for x^ = —i.S,/{x,) = +4.304;
then by the regula falsi there is found x^ = — 1.90 nearly.
Again, for x^ = — 1.90, /{x^) = -{- 0.290, and these combined
with x^ and /{x^) give x^ = — 1.906, which is correct to the
third decimal.

As a second example let /{x) = arc;r — sin;ir — 0.5 = o.
Here a graphic solution shows that there is but one real root,
and that the value of it lies between 85° and 86°. For x^= 85°,
/(•^i) = — 0.01266, and for x^ = 86°, /(,r,) — -f 0.00342 ; then
by the .rule ;ir3 = 85" 44', which gives /(;tr3) = — 0.00090. Again,
combining the values for x^ and x^ there is found x^ = 85° 47'.
which gives /{x^) = — 0.00009. Lastly, combining the values
for x^ and x^ there is found ;irj, = 85 " 47'.4, /vhich is as close an
approximation as can be made with five-place tables.

In the application of this method it is to be observed that
the signs of the values of x and /(x) are to be carefully re-
garded, and also that the values of /(x) to be combined in one
operation should have opposite signs. For the quickest
approximation the values of /(:r) to be selected should be those
having the smallest numerical values.

Prob. 4. Compute by the regula falsi the real roots of Jf^ — 0.25=0.
Also those of jc^ + sin 2Jc = o.

Art. 4. Newton's Approximation Rule.

Another useful method for approximating to the value of
the real root of an equation is that devised by Newton in 1666.'^

^ct - -' If J/ =/{x) be the equation of a
curve, OX in the figure represents a
real root of the equation /(x) = o.
Let OA be an approximate value of
OX, and Aa the corresponding value
of/(;ir). At a let aB be drawn tangent
to the curve ; then OB is another approximate value of OX.

* See Analysis per equationes numero terminorum infinitas, p. 26g, Vol. I
of Horsely's edition of Newton's works (London, 1779), where the method is
given in a somewhat different form.


Let Bb be the value of f{x) corresponding to OB, and at b
let the tangent bC be drawn ; then OC is a closer approxima-
tion to OX, and thus the process may be continued.

Let/'(,i-) be the first derivative oi f{x)\ or,f'{x) — df{x)/dx.
For X ^^ x^ — OA in the figure, the value of /{x^) is the ordi-
nate Aa, and the value of /'(^,) is the tangent of the angle
aBA ; this tangent is also Aa/AB. Hence AB — f\x^/f\x^,
and accordingly OB and OC are found by

;,_;,_ Ml X -X - -^^"l^^-

which is Newton's approximation rule. By a third application
to x^ the closer value x^ is found, and the process may be con-
tinued to any degree of precision required.

For example, let f{x) = x^ -{- ^x'^ -|- 7 = O. The first deriv-
ative isy"'(^) = ^x* -f- \ox. Here it may be found by trial that
— 2 is an approximate value of the real root. For x^^= — 2
f{^\) =^ — 5' "^'""^ /X-*"!) = 60, whence by the rule x^ = — 1.92.
Now for x^ = — 1.92 are found f{x^ = — 0.6599 and
/'(x^) = 29 052, whence by the rule x^ = — I.906, which is
correct to the third decimal.

As a second example let /{x) = x^ -{- 4sin;ir = o. Here
the first derivative is /'(,r) = 2,i' -f-4cos;f. An approximate
value of X found either by trial or by a graphic solution is
;r= — 1.94, corresponding to about — 1 1 1°09'. For ;ir, = — 1.94,
/(x^) = 0.03304 and /\xj = — 5.323, whence by the rule
x^= — 1.934. By a second application x^ = — 1.9328, which
corresponds to an angle of — 110° 54i-'.

In the application of Newton's rule it is best that the
assumed value of ,r, should be such as to render y'(;r,) as small
as possible, and also/'(,r,) as large as possible. The method
will fail if the curve has a maximum or minimum between a
and b. It is seen that Newton's rule, like the regula falsi,
applies equally well to both transcendental and algebraic equa-
tions, and moreover that the rule itself is readily kept in mind
by help of the diagram.


Prob. 5. Compute by Newton's rule the real roots of the alge-
braic equation x* — "jx -\- 6 = o. Also the real roots of the trans-
cendental equation sin x -\- arc :*: — 2 = o.

Art. 5. Separation of the Roots.

The roots of an equation are of two kinds, real roots and
imaginary roots. Equal real roots may be regarded as a spe-
cial class, which he at the limit between the real and the imagi-
nary. If an equation has p equal roots of one value and g equal
roots of another value, then its first derivative equation has
p — I roots of the first value and g —i roots of the second
value, and thus all the equal roots are contained in a factor
common to both primitive and derivative. Equal roots may
hence always be readily detected and removed from the given
equation. For instance, let x* — gx" -\- 4.x -\- 12 = o, of which
the derivative equation is 4Jir' — i8;r-|-4 = o; as;ir — 2 is a
factor of these two equations, two of the roots of the primitive
equation are -f- 2.

— The problem of determining the number of the real and
imaginary roots of an algebraic equation is completely solved
by Sturm's theorem. If, then, two values be assigned to x the
number of real roots between those limits is found by the same
theorem, and thus by a sufficient number of assumptions limits
may be found for each real root. As Sturm's theorem is known
to all who read these pages, no applications of it will be here
given, but instead an older method due to Hudde will be
presented which has the merit of giving a comprehensive view
of the subject, and which moreover applies to transcendental
as well as to algebraic equations.*

If any equation y -=1 f{x) be plotted with values of x as
abscissas and values oi y as ordinates, a real graph is obtained
whose intersections with the axis (^JTgive the real roots of the

* Devised by Hudde in 1659 and published by Rolie in 1690. See CEuvres
de Lagrange, Vol. VIII, p. 190.


equa/ion y"(;tr) = o. Thus in the figure the three points marlced
X gi> 'e three values OX for three real roots. The curve which
repr' sents j ^ f{x) has points of maxima and minima marked
A, 3. id inflection points marked B. Now let the first deriva-

tive equation dy/dx=^ f\x) be formed and be plotted in the
same manner on the axis O' X . The condition /'(-r)= o gives
the abscissas of the points A^ and thus the real roots OX' give
limits separating the real roots oi f{pc) == o. To ascertain if a
real root OX lies between two values of O' X' these two values
are to be substituted m fix): if the signs oi f{x) are unlike in
the two cases, a real root of fix) = o lies between the two
limits; if the signs are the same, a real root does not lie between
those limits.

In like manner if the second derivative equation, that is,
dy/dx'' = /"{x), be plotted on O'X", the intersections give
limits which separate the real roots of /"'(-^)=o. It is also
seen that the roots of the second derivative equation are the
abscissas of the points of inflection of the curve/ = /(x).

To illustrate this method let the given equation be the
quintic /{x) = x^ — 5^' -{-6x -\- 2 = o. The first derivativ^e
equation is f\x) = $x* —i$x^'}-6 = o, the roots of which are
•approximately — 1.59, —0.69, +0.69. -\- 1.59. Now let each
of these values be substituted for x in the given quintic, as also
the values — 00 , o, and -\- 00 , and let the corresponding values
of /(;r) be determined as follows :


;r = — 00, —1.59, —0.69, o, +0.69, +1.59, +00;
f{x)=-^, +2.4, -0.6, +2, +47. +1-6, +CO.

Since f{x) changes sign between x^=^ — 00 and x^ = — i-59>
one real root lies between these limits ; smce f{x) changes sign
between ;r, = — i .59 and x^^= — 0.69, one real root lies between
these limits ; since y"(;tr) changes sign between x^^ — 0.69 and
x^ = O, one real root lies between these limits; since /(;ir) does
not change sign between ^1:3 = and ;tr^ = 00 , a pair of imagi-
nary roots is indicated, the sum of which lies between -\- 0.69
and 00 .

As a second example let f{x) =^ e' — ^—4 = 0. The first
derivative equation is f'{x) =: r* — 2^* = O, which has two
roots ^ = |- and ,?^ = o, the latter corresponding to ;ir = — co .
For X := — <^ , f{x) is negative; for e^ =.^, f{x) is negative ; for
X ^ -\- CO , f{x) is negative. The equation e^ — ^ — 4 =
has, therefore, no real roots.

When the first derivative equation is not easily solved, the
second, third, and following derivatives may be taken until an
equation is found whose roots may be obtained. Then, by
working backward, limits may be found in succession for the
roots of the derivative equations until finally those of the
primative are ascertained. In many cases, it is true, this proc-
ess may prove lengthy and difficult, and in some it may fail
entirely; nevertheless the method is one of great theoretical
and practical value.

Prob. 6. Show that e" + e'^'' —4 = has two real roots, one
positive and one negative.

Prob. 7. Show that x^ -\- x -\- \ ^=- o has no real roots; also that
x^ — X — \ =0 has two real roots, one positive and one negative.

Art. 6. Numerical Algebraic Equations.

An algebraic equation of the ;?"' degree may be written
with all its terms transposed to the first member, thus:

;r" + a^x""-^ + a^x''-"- + . . . -f a„_^x -|- «„ = O ;


and if all the coefficients and the absolute term are real num-
bers, this is commonly called a numerical equation. The first
member may for brevity be denoted by/(^,r) and the equation
itself by/(^-) = o.

The following principles of the theory of algebraic equations
with real coefficients, deduced in text-books on algebra, are
here recapitulated for convenience of reference :

1 3 4 5 6 7

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