# 1911 Encyclopædia Britannica/Equation/Theory of Equations

*Theory of Equations*.

1. In the subject “Theory of Equations” the term *equation* is
used to denote an equation of the form *x*^{n} − *p*_{1}*x*^{n−1} ... ± *p*_{n} = 0,
where *p*_{1}, *p*_{2} ... *p*_{n} are regarded as known, and *x* as a quantity
to be determined; for shortness the equation is written ƒ(*x*) = 0.

The equation may be *numerical*; that is, the coefficients
*p*_{1}, *p*_{2}^{n}, ... *p*_{n} are then numbers—understanding by number a
quantity of the form α + βi (α and β having any positive or
negative real values whatever, or say each of these is regarded
as susceptible of continuous variation from an indefinitely large
negative to an indefinitely large positive value), and *i* denoting
√−1.

Or the equation may be *algebraical*; that is, the coefficients
are not then restricted to denote, or are not explicitly considered
as denoting, numbers.

1. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.)

*Real Theory*.

2. Postponing all consideration of imaginaries, we take in the
first instance the coefficients to be real, and attend only to the
real roots (if any); that is, *p*_{1}, *p*_{2}, ... *p*_{n} are real positive or
negative quantities, and a root *a*, if it exists, is a positive or
negative quantity such that *a*^{n} − *p*_{1}*a*^{n−1} ... ± *p*_{n} = 0, or say,
ƒ(*a*) = 0.

It is very useful to consider the curve *y* = ƒ(*x*),—or, what
would come to the same, the curve Ay = ƒ(*x*),—but it is better
to retain the first-mentioned form of equation, drawing, if need
be, the ordinate *y* on a reduced scale. For instance, if the
given equation be *x*^{3} − 6*x*^{2} + 11*x* − 6.06 = 0,^{[1]} then the curve
*y* = *x*^{3} − 6*x*^{2} + 11*x* − 6.06 is as shown in fig. 1, without any
reduction of scale for the ordinate.

It is clear that, in general, *y* is a continuous one-valued
function of *x*, finite for every finite value of *x*, but becoming
infinite when *x* is infinite; *i.e.*, assuming throughout that the
coefficient of *x*^{n} is +1, then when *x* = ∞, *y* = +∞; but when
*x* = −∞, then *y* = +∞ or −∞, according as *n* is even or
odd; the curve cuts any line whatever, and in particular it cuts
the axis (of *x*) in at most *n* points; and the value of *x*, at any
point of intersection with the axis, is a root of the equation
ƒ(*x*) = 0.

If β, α are any two values of *x* (α > β, that is, α nearer +∞),
then if ƒ(β), ƒ(α) have opposite signs, the curve cuts the axis an
odd number of times, and therefore at least once, between the
points *x* = β, *x* = α; but if ƒ(β), ƒ(α) have the same sign, then
between these points the curve cuts the axis an even number of
times, or it may be not at all. That is, ƒ(β), ƒ(α) having opposite
signs, there are between the limits β, α an odd number of real
roots, and therefore at least one real root; but ƒ(β), ƒ(α) having
the same sign, there are between these limits an even number of
real roots, or it may be there is no real root. In particular, by
giving to β, α the values −∞, +∞ (or, what is the same thing,
any two values sufficiently near to these values respectively) it
appears that an equation of an odd order has always an odd
number of real roots, and therefore at least one real root; but
that an equation of an even order has an even number of real
roots, or it may be no real root.

If α be such that for *x* = or > a (that is, *x* nearer to +∞) ƒ(*x*)
is always +, and β be such that for *x* = or < β (that is, *x*
nearer to −∞) ƒ(*x*) is always −, then the real roots (if any)
lie between these limits *x* = β, *x* = α; and it is easy to find by
trial such two limits including between them all the real roots
(if any).

3. Suppose that the positive value δ is an inferior limit to the
difference between two real roots of the equation; or rather
(since the foregoing expression would imply the existence of real
roots) suppose that there are not two real roots such that their
difference taken positively is = or < δ; then, γ being any value
whatever, there is clearly at most one real root between the
limits γ and γ + δ; and by what precedes there is such real root
or there is not such real root, according as ƒ(γ), ƒ(γ + δ) have
opposite signs or have the same sign. And by dividing in this
manner the interval β to α into intervals each of which is = or
<δ, we should not only ascertain the number of the real roots
(if any), but we should also *separate* the real roots, that is, find
for each of them limits γ, γ + δ between which there lies this one,
and only this one, real root.

In particular cases it is frequently possible to ascertain the number
of the real roots, and to effect their separation by trial or otherwise,
without much difficulty; but the foregoing was the general process
as employed by Joseph Louis Lagrange even in the second edition
(1808) of the *Traité de la résolution des équations numériques*;^{[2]} the
determination of the limit δ had to be effected by means of the
“equation of differences” or equation of the order 12 *n*(*n* − 1), the roots
of which are the squares of the differences of the roots of the given
equation, and the process is a cumbrous and unsatisfactory one.

4. The great step was effected by the theorem of J. C. F.
Sturm (1835)—viz. here starting from the function ƒ(*x*), and its
first derived function ƒ′(*x*), we have (by a process which is a slight
modification of that for obtaining the greatest common measure
of these two functions) to form a series of functions

*x*), ƒ′(

*x*), ƒ

_{2}(

*x*), ... ƒ

_{n}(

*x*)

of the degrees *n*, *n* − 1, *n* − 2 ... 0 respectively,—the last term
ƒ_{n}(*x*) being thus an absolute constant. These lead to the immediate
determination of the number of real roots (if any)
between any two given limits β, α; viz. supposing α > β (that is,
α nearer to +∞), then substituting successively these two values
in the series of functions, and attending only to the signs of the
resulting values, the number of the changes of sign lost in passing
from β to α is the required number of real roots between the two
limits. In particular, taking β, α = −∞, +∞ respectively, the
signs of the several functions depend merely on the signs of the
terms which contain the highest powers of *x*, and are seen by
inspection, and the theorem thus gives at once the whole number
of real roots.

And although theoretically, in order to complete by a finite
number of operations the separation of the real roots, we still
need to know the value of the before-mentioned limit δ; yet
in any given case the separation may be effected by a limited
number of repetitions of the process. The practical difficulty
is when two or more roots are very near to each other. Suppose,
for instance, that the theorem shows that there are two roots
between 0 and 10; by giving to *x* the values 1, 2, 3, ... successively,
it might appear that the two roots were between 5 and 6;
then again that they were between 5.3 and 5.4, then between
5.34 and 5.35, and so on until we arrive at a separation; say it
appears that between 5.346 and 5.347 there is one root, and
between 5.348 and 5.349 the other root. But in the case in
question δ would have a very small value, such as .002, and even
supposing this value known, the direct application of the first-mentioned
process would be still more laborious.

5. Supposing the separation once effected, the determination of the single real root which lies between the two given limits may be effected to any required degree of approximation either by the processes of W. G. Horner and Lagrange (which are in principle a carrying out of the method of Sturm’s theorem), or by the process of Sir Isaac Newton, as perfected by Joseph Fourier (which requires to be separately considered).

First as to Horner and Lagrange. We know that between the
limits β, α there lies one, and only one, real root of the equation;
ƒ(β) and ƒ(α) have therefore opposite signs. Suppose any intermediate
value is θ; in order to determine by Sturm’s theorem
whether the root lies between β, θ, or between θ, α, it would be quite
unnecessary to calculate the signs of ƒ(θ),ƒ′(θ), ƒ_{2}(θ) ...; only the
sign of ƒ(θ) is required; for, if this has the same sign as ƒ(β), then
the root is between β, θ; if the same sign as ƒ(α), then the root is
between θ, α. We want to make θ increase from the inferior limit
β, at which ƒ(θ) has the sign of ƒ(β), so long as ƒ(θ) retains this sign,
and then to a value for which it assumes the opposite sign; we have
thus two nearer limits of the required root, and the process may
be repeated indefinitely.

Horner’s method (1819) gives the root as a decimal, figure by figure; thus if the equation be known to have one real root between 0 and 10, it is in effect shown say that 5 is too small (that is, the root is between 5 and 6); next that 5.4 is too small (that is, the root is between 5.4 and 5.5); and so on to any number of decimals. Each figure is obtained, not by the successive trial of all the figures which precede it, but (as in the ordinary process of the extraction of a square root, which is in fact Horner’s process applied to this particular case) it is given presumptively as the first figure of a quotient; such value may be too large, and then the next inferior integer must be tried instead of it, or it may require to be further diminished. And it is to be remarked that the process not only gives the approximate value α of the root, but (as in the extraction of a square root) it includes the calculation of the function ƒ(α), which should be, and approximately is, = 0. The arrangement of the calculations is very elegant, and forms an integral part of the actual method. It is to be observed that after a certain number of decimal places have been obtained, a good many more can be found by a mere division. It is in the progress tacitly assumed that the roots have been first separated.

Lagrange’s method (1767) gives the root as a continued fraction
*a* + 1/*b* + 1/*c* + ..., where a is a positive or negative integer (which
may be = 0), but *b*, *c*, ... are positive integers. Suppose the roots
have been separated; then (by trial if need be of consecutive integer
values) the limits may be made to be consecutive integer numbers:
say they are *a*, *a* + 1; the value of *x* is therefore = *a* + 1/*y*, where *y*
is positive and greater than 1; from the given equation for *x*,
writing therein *x* = *a* + 1/*y*, we form an equation of the same order for
*y*, and this equation will have one, and only one, positive root greater
than 1; hence finding for it the limits *b*, *b* + 1 (where *b* is = or > 1),
we have *y* = *b* + 1/*z*, where *z* is positive and greater than 1; and so on—that
is, we thus obtain the successive denominators *b*, *c*, *d* ...
of the continued fraction. The method is theoretically very elegant,
but the disadvantage is that it gives the result in the form of a
continued fraction, which for the most part must ultimately be converted
into a decimal. There is one advantage in the method, that
a commensurable root (that is, a root equal to a rational fraction)
is found accurately, since, when such root exists, the continued
fraction terminates.

6. Newton’s method (1711), as perfected by Fourier(1831), may be
roughly stated as follows. If *x* = γ be an approximate value of any
root, and γ + *h* the correct value, then ƒ(γ + *h*) = 0, that is,

ƒ(γ) + | h |
ƒ′(γ) + | h^{2} |
ƒ″(γ) + ... = 0; |

1 | 1·2 |

and then, if *h* be so small that the terms after the second may be
neglected, ƒ(γ) + *h*ƒ′(γ) = 0, that is, *h* = {−ƒ(γ)/ƒ′(γ) }, or the new approximate
value is *x* = γ − {ƒ(γ)/ƒ′(γ) }; and so on, as often as we please.
It will be observed that so far nothing has been assumed as to the
separation of the roots, or even as to the existence of a real root;
γ has been taken as the approximate value of a root, but no precise
meaning has been attached to this expression. The question arises,
What are the conditions to be satisfied by γ in order that the process
may by successive repetitions actually lead to a certain real root of the
equation; or that, γ being an approximate value of a certain real
root, the new value γ − {ƒ(γ)/ƒ′(γ) } may be a more approximate value.

Fig. 1. |

Referring to fig. 1, it is easy to see that if OC represent the assumed
value γ, then, drawing the ordinate CP to meet the curve in P, and
the tangent PC′ to meet the axis in C′, we shall have OC′ as the new
approximate value of the root. But observe that there is here a
real root OX, and that the curve beyond X is convex to the axis;
under these conditions the point C′ is nearer to X than was C; and,
starting with C′ instead of C, and proceeding in like manner to draw
a new ordinate and tangent, and so on as often as we please, we
approximate continually, and that with great rapidity, to the true
value OX. But if C had been taken on the other side of X, where the
curve is concave to the axis, the new point C′ might or might not
be nearer to X than was the point C; and in this case the method,
if it succeeds at all, does so by accident only, *i.e.* it may happen
that C′ or some subsequent point comes to be a point C, such that
CO is a *proper* approximate value of the root, and then the subsequent
approximations proceed in the same manner as if this value had been
assumed in the first instance, all the preceding work being wasted.
It thus appears that for the proper application of the method we
require *more* than the mere separation of the roots. In order to be
able to approximate to a certain root α, = OX, we require to know
that, between OX and some value ON, the curve is always convex
to the axis (analytically, between the two values, ƒ(*x*) and ƒ″(*x*) must
have always the same sign). When this is so, the point C may be
taken anywhere on the proper side of X, and within the portion XN
of the axis; and the process is then the one already explained.
The approximation is in general a very rapid one. If we know for the
required root OX the two limits OM, ON such that from M to X the
curve is always *concave* to the axis, while from X to N it is always
convex to the axis,—then, taking D anywhere in the portion MX
and (as before) C in the portion XN, drawing the ordinates DQ,
CP, and joining the points P, Q by a line which meets the axis in D′,
also constructing the point C′ by means of the tangent at P as before,
we have for the required root the new limits OD′, OC′; and proceeding
in like manner with the points D′, C′, and so on as often as
we please, we obtain at each step two limits approximating more and
more nearly to the required root OX. The process as to the point D′,
translated into analysis, is the ordinate process of interpolation.
Suppose OD = β, OC = α, we have approximately ƒ(β + *h*) = ƒ(β) +
h{ƒ(α) − ƒ(β) } / (α − β), whence if the root is β + *h* then *h* = − (α − β)ƒ(β) / {ƒ(α) − ƒ(β) }.

Returning for a moment to Horner’s method, it may be remarked that the correction h, to an approximate value α, is therein found as a quotient the same or such as the quotient ƒ(α) ÷ ƒ′(α) which presents itself in Newton’s method. The difference is that with Horner the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h; if too many decimals are taken, there would be a waste of work; but the error would correct itself at the next step. Of course the calculation should be conducted without any such waste of work.

*Imaginary Theory*.

7. It will be recollected that the expression *number* and the
correlative epithet *numerical* were at the outset used in a wide
sense, as extending to imaginaries. This extension arises out
of the theory of equations by a process analogous to that by which
number, in its original most restricted sense of positive integer
number, was extended to have the meaning of a real positive
or negative magnitude susceptible of continuous variation.

If for a moment number is understood in its most restricted
sense as meaning positive integer number, the solution of a simple
equation leads to an extension; *ax* − *b* = 0 gives *x* = *b*/*a*, a
positive fraction, and we can in this manner represent, not
accurately, but as nearly as we please, any positive magnitude
whatever; so an equation *ax* + *b* = 0 gives *x* = −*b*/*a*, which
(approximately as before) represents any negative magnitude.
We thus arrive at the extended signification of number as a
continuously varying positive or negative magnitude. Such
numbers may be added or subtracted, multiplied or divided
one by another, and the result is always a number. Now from
a quadric equation we derive, in like manner, the notion of a
complex or imaginary number such as is spoken of above. The
equation *x*^{2} + 1 = 0 is not (in the foregoing sense, number = real
number) satisfied by any numerical value whatever of *x*; but
we assume that there is a number which we call i, satisfying the
equation *i* ^{2} + 1 = 0, and then taking a and *b* any real numbers,
we form an expression such as *a* + *bi*, and use the expression
number in this extended sense: any two such numbers may be
added or subtracted, multiplied or divided one by the other,
and the result is always a number. And if we consider first
a quadric equation *x*^{2} + *px* + *q* = 0 where *p* and *q* are real numbers,
and next the like equation, where *p* and *q* are any numbers
whatever, it can be shown that there exists for *x* a numerical
value which satisfies the equation; or, in other words, it can
be shown that the equation has a numerical root. The like
theorem, in fact, holds good for an equation of any order whatever;
but suppose for a moment that this was not the case; say that
there was a cubic equation *x*^{3} + *px*^{2} + *qx* + *r* = 0, with numerical
coefficients, not satisfied by any numerical value of *x*, we should
have to establish a new imaginary *j* satisfying some such equation,
and should then have to consider numbers of the form *a* + *bj*, or
perhaps *a* + *bj* + *cj* ^{2} (*a*, *b*, *c* numbers α + β*i* of the kind heretofore
considered),—first we should be thrown back on the quadric
equation *x*^{2} + *px* + *q* = 0, *p* and *q* being now numbers of the last-mentioned
extended form—*non constat* that every such equation
has a numerical root—and if not, we might be led to *other*
imaginaries *k*, *l*, &c., and so on *ad infinitum* in inextricable
confusion.

But in fact a numerical equation of any order whatever has
always a numerical root, and thus numbers (in the foregoing
sense, number = quantity of the form α + β*i*) form (*what real*
*numbers do not*) a universe complete in itself, such that starting
in it we are never led out of it. There may very well be, and
perhaps are, numbers in a more general sense of the term
(quaternions are not a case in point, as the ordinary laws of
combination are not adhered to), but in order to have to do with
such numbers (if any) we must start with them.

8. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness (the meaning being as before), every equation has a root. Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition.

A number a such that substituted for *x* it makes the function
*x*_{1}^{n} − *p*_{1}*x*^{n−1} ... ± *p*_{n} to be = 0, or say such that it satisfies the
equation ƒ(*x*) = 0, is said to be a root of the equation; that is, a
being a root, we have

*a*

^{n}−

*p*

_{1}

*a*

^{n−1}... ±

*p*

_{n}= 0, or say ƒ(

*a*) = 0;

and it is then easily shown that *x* − a is a factor of the function ƒ(*x*),
viz. that we have ƒ(*x*) = (*x* − *a*)ƒ_{1}(*x*), where ƒ_{1}(*x*) is a function
*x*^{n−1} − *q*_{1}*x*^{n−2} ... ± *q*_{n−1} of the order *n* − 1, with numerical coefficients
*q*_{1}, *q*_{2} ... *q*_{n−1}.

In general a is not a root of the equation ƒ_{1}(*x*) = 0, but it may be so—*i.e.*
ƒ_{1}(*x*) may contain the factor *x* − a; when this is so, ƒ(*x*) will
contain the factor (*x* − *a*)^{2}; writing then ƒ(*x*) = (*x* − *a*)^{2}ƒ_{2}(*x*), and assuming
that a is not a root of the equation ƒ_{2}(*x*) = 0, *x* = *a* is then said to
be a double root of the equation ƒ(*x*) = 0; and similarly ƒ(*x*) may
contain the factor (*x* − *a*)^{3} and no higher power, and *x* = *a* is then a
triple root; and so on.

Supposing in general that ƒ(*x*) = (*x* − *a*)^{α}F(*x*) (α being a positive
integer which may be = 1, (*x* − *a*)^{α} the highest power of *x* − *a* which
divides ƒ(*x*), and F(*x*) being of course of the order *n* − α), then the
equation F(*x*) = 0 will have a root *b* which will be different from *a*;
*x* − *b* will be a factor, in general a simple one, but it may be a multiple
one, of F(*x*), and ƒ(*x*) will in this case be = (*x* − *a*)^{α} (*x* − *b*)^{β} Φ(*x*) (β a
positive integer which may be = 1, (*x* − *b*)^{β} the highest power of
*x* − *b* in F(*x*) or ƒ(*x*), and Φ(*x*) being of course of the order *n* − α − β).
The original equation ƒ(*x*) = 0 is in this case said to have α roots each
= *a*, β roots each = b; and so on for any other factors (*x* − *c*)^{γ}, &c.

We have thus the *theorem*—A numerical equation of the order n
has in every case *n* roots, viz. there exist *n* numbers, *a*, *b*, ... (in
general all distinct, but which may arrange themselves in any sets
of equal values), such that ƒ(*x*) = (*x* − *a*)(*x* − *b*)(*x* − *c*) ... identically.

If the equation has equal roots, these can in general be determined,
and the case is at any rate a special one which may be in the first
instance excluded from consideration. It is, therefore, in general
assumed that the equation ƒ(*x*) = 0 has all its roots unequal.

If the coefficients *p*_{1}, *p*_{2}, ... are all or any one or more of them
imaginary, then the equation ƒ(*x*) = 0, separating the real and imaginary
parts thereof, may be written F(*x*) + *i*Φ(*x*) = 0, where F(*x*),
Φ(*x*) are each of them a function with real coefficients; and it thus
appears that the equation ƒ(*x*) = 0, with imaginary coefficients, has
not in general any real root; supposing it to have a real root *a*, this
must be at once a root of each of the equations F(*x*) = 0 and Φ(*x*) = 0.

But an equation with real coefficients may have as well imaginary
as real roots, and we have further the *theorem* that for any such
equation the imaginary roots enter in pairs, viz. α + β*i* being a root,
then α − β*i* will be also a root. It follows that if the order be odd,
there is always an odd number of real roots, and therefore at least one
real root.

9. In the case of an equation with real coefficients, the question
of the existence of real roots, and of their separation, has been
already considered. In the general case of an equation with
imaginary (it may be real) coefficients, the like question arises
as to the situation of the (real or imaginary) roots; thus, if
for facility of conception we regard the constituents α, β of a
root α + β*i* as the co-ordinates of a point *in plano*, and accordingly
represent the root by such point, then drawing in the plane any
closed curve or “contour,” the question is how many roots lie
within such contour.

This is solved theoretically by means of a theorem of A. L. Cauchy
(1837), viz. writing in the original equation *x* + *iy* in place of *x*, the
function ƒ(*x* + *iy*) becomes = P + *i*Q, where P and Q are each of them
a rational and integral function (with real coefficients) of (*x*, *y*).
Imagining the point (*x*, *y*) to travel along the contour, and considering
the number of changes of sign from − to + and from + to − of
the fraction corresponding to passages of the fraction through
zero (that is, to values for which P becomes = 0, disregarding those
for which Q becomes = 0), the difference of these numbers gives the
number of roots within the contour.

It is important to remark that the demonstration does not presuppose
the existence of any root; the contour may be the infinity
of the plane (such infinity regarded as a contour, or closed curve),
and in this case it can be shown (and that very easily) that the difference
of the numbers of changes of sign is = *n*; that is, there are within
the infinite contour, or (what is the same thing) there are in all *n* roots;
thus Cauchy’s theorem contains really the proof of the fundamental
theorem that a numerical equation of the nth order (not only has
a numerical root, but) has precisely *n* roots. It would appear that
this proof of the fundamental theorem in its most complete form is
in principle identical with the last proof of K. F. Gauss (1849) of
the theorem, in the form—A numerical equation of the nth order
has always a root.^{[3]}

But in the case of a finite contour, the actual determination of the difference which gives the number of real roots can be effected only in the case of a rectangular contour, by applying to each of its sides separately a method such as that of Sturm’s theorem; and thus the actual determination ultimately depends on a method such as that of Sturm’s theorem.

Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.

10. A class of numerical equations which needs to be considered
is that of the binomial equations *x*^{n} − *a* = 0 (*a* = α + β*i*,
a complex number).

The foregoing conclusions apply, viz. there are always *n* roots,
which, it may be shown, are all unequal. And these can be found
numerically by the extraction of the square root, and of an nth root,
of *real* numbers, and by the aid of a table of natural sines and
cosines.^{[4]} For writing

α + βi = √(α^{2} + β^{2}) { | α | + | β | i }, |

√(α^{2} + β^{2}) | √(α^{2} + β^{2}) |

there is always a real angle λ (positive and less than 2π), such that
its cosine and sine are = α / √(α^{2} + β^{2}) and β / √(α^{2} + β^{2}) respectively; that
is, writing for shortness √(α^{2} + β^{2}) = ρ, we have α + β*i* = ρ (cos λ + *i* sin λ),
or the equation is *x*^{n} = ρ (cos λ + *i* sin λ); hence observing that
(cos λ/*n* + *i* sin λ/*n* )^{n} = cos λ + *i* sin λ, a value of *x* is = *n**√*ρ (cos λ/*n* + *i* sin λ/*n*).
The formula really gives all the roots, for instead of λ we may write
λ + 2*s*π, *s* a positive or negative integer, and then we have

x = n√ρ ( cos | λ + 2sπ |
+ i sin | λ + 2sπ |
), |

n | n |

which has the *n* values obtained by giving to *s* the values 0, 1, 2 ...
n − 1 in succession; the roots are, it is clear, represented by points
lying at equal intervals on a circle. But it is more convenient to proceed
somewhat differently; taking one of the roots to be θ, so that
θ^{n} = *a*, then assuming *x* = θ*y*, the equation becomes *y*^{n} − 1 = 0, which
equation, like the original equation, has precisely *n* roots (one of them
being of course = 1). And the original equation *x*^{n} − *a* = 0 is thus
reduced to the more simple equation *x*^{n} − 1 = 0; and although the
theory of this equation is included in the preceding one, yet it is
proper to state it separately.

The equation *x*^{n} − 1 = 0 has its several roots expressed in the form
1, ω, ω^{2}, ... ω^{n−1}, where ω may be taken = cos 2π/*n* + *i* sin 2π/*n*; in fact,
ω having this value, any integer power ω^{k} is = cos 2π*k*/*n* + *i* sin 2π*k*/*n*, and
we thence have (ω^{k})^{n} = cos 2π*k* + *i* sin 2π*k*, = 1, that is, ω^{k} is a root of
the equation. The theory will be resumed further on.

By what precedes, we are led to the notion (a numerical) of the
radical *a*^{1/n} regarded as an n-valued function; any one of these being
denoted by *n**√*a, then the series of values is *n**√*a, ω*n**√*a, ... ω^{n−1} *n**√*a;
or we may, if we please, use *n**√**a* instead of *a*^{1/n} as a symbol to denote
the *n*-valued function.

As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations.

*Algebraical Equations*.

11. The equation is

*x*

^{n}−

*p*

_{1}

*x*

^{n−1}+ ... ±

*p*

_{n}= 0,

and we here *assume* the existence of roots, viz. we assume that
there are *n* quantities *a*, *b*, *c* ... (in general all of them different,
but which in particular cases may become equal in sets in any
manner), such that

*x*

^{n}−

*p*

_{1}

*x*

^{n−1}+ ... ±

*p*

_{n}= 0;

or looking at the question in a different point of view, and
starting with the roots *a*, *b*, *c* ... as given, we express the product
of the *n* factors *x* − *a*, *x* − *b*, ... in the foregoing form, and thus
arrive at an equation of the order *n* having the *n* roots *a*, *b*, *c*....
In either case we have

*p*

_{1}= Σ

*a*,

*p*

_{2}= Σ

*ab*, ...

*p*

_{n}=

*abc*...;

*i.e.* regarding the coefficients *p*_{1}, *p*_{2} ... *p*_{n} as given, then we
assume the existence of roots *a*, *b*, *c*, ... such that *p*_{1} = Σ*a*, &c.;
or, regarding the roots as given, then we write *p*_{1}, *p*_{2}, &c., to
denote the functions Σ*a*, Σ*ab*, &c.

As already explained, the epithet algebraical is not used in opposition
to numerical; an algebraical equation is merely an equation
wherein the coefficients are not restricted to denote, or are not explicitly
considered as denoting, numbers. That the abstraction is
legitimate, appears by the simplest example; in saying that the
equation *x*^{2} − *px* + *q* = 0 has a root *x* = 12 {*p* + √(*p*^{2} − 4*q*) }, we mean that
writing this value for *x* the equation becomes an identity, [12 {*p* +
√(*p*^{2} − 4*q*) }]^{2} − *p*[12 {*p* + √(*p*^{2} − 4*q*) }] + *q* = 0; and the verification of
this identity in nowise depends upon *p* and *q* meaning numbers.
But if it be asked what there is beyond numerical equations included
in the term algebraical equation, or, again, what is the full extent
of the meaning attributed to the term—the latter question at any rate it would be very difficult to answer; as to the former one, it
may be said that the coefficients may, for instance, be symbols of
operation. As regards such equations, there is certainly no proof
that every equation has a root, or that an equation of the *n*th order
has *n* roots; nor is it in any wise clear what the precise signification
of the statement is. But it is found that the assumption of the
existence of the *n* roots can be made without contradictory results;
conclusions derived from it, if they involve the roots, rest on the
same ground as the original assumption; but the conclusion may
be independent of the roots altogether, and in this case it is
undoubtedly valid; the reasoning, although actually conducted by
aid of the assumption (and, it may be, most easily and elegantly
in this manner), is really independent of the assumption. In illustration,
we observe that it is allowable to express a function of *p* and *q*
as follows,—that is, by means of a rational symmetrical function of
*a* and *b*, this can, as a fact, be expressed as a rational function of
*a* + *b* and ab; and if we prescribe that a + *b* and *ab* shall then be
changed into *p* and *q* respectively, we have the required function of
*p*, *q*. That is, we have F(α, β) as a representation of ƒ(*p*, *q*), obtained
as if we had *p* = *a* + *b*, *q* = *ab*, but without in any wise assuming the
existence of the *a*, *b* of these equations.

12. Starting from the equation

*x*−

^{n}*p*

_{1}

*x*

^{n−1}+ ... =

*x*−

*a.x*−

*b*. &c.

or the equivalent equations *p*_{1} = Σ*a*, &c., we find

*a*

^{n}−

*p*

_{1}

*a*

^{n−1}+ ... = 0,

b

^{n}−

*p*

_{1}

*b*

^{n−1}+ ... = 0;

· · ·

· · ·

· · ·

(it is as satisfying these equations that *a*, *b* ... are said to be
the roots of *x ^{n}* −

*p*

_{1}

*x*

^{n−1}+ ... = 0); and conversely from the last-mentioned equations, assuming that

*a*,

*b*... are all different, we deduce

*p*

_{1}= Σ

*a*,

*p*

_{2}= Σ

*ab*, &c.

and

*x*−

^{n}*p*

_{1}

*x*

^{n−1}+ ... =

*x*−

*a.x*−

*b*. &c.

Observe that if, for instance, a = b, then the equations
a^{n} − *p*_{1}*a*^{n−1} + ... = 0, b^{n} − *p*_{1}*b*^{n−1} + ... = 0 would reduce themselves
to a single relation, which would not of itself express
that a was a double root,—that is, that (*x* − *a*)^{2} was a factor of
x^{n} − *p*_{1}*x*^{n−1} +, &c; but by considering *b* as the limit of *a* + *h*,
*h* indefinitely small, we obtain a second equation

*na*

^{n−1}− (

*n*− 1)

*p*

_{1}

*a*

^{n−2}+ ... = 0,

which, with the first, expresses that a is a double root; and then
the whole system of equations leads as before to the equations
*p*_{1} = Σ*a*, &c. But the existence of a double root implies a certain
relation between the coefficients; the general case is when the
roots are all unequal.

We have then the *theorem* that every rational symmetrical
function of the roots is a rational function of the coefficients.
This is an easy consequence from the less general theorem, every
rational and integral symmetrical function of the roots is a
rational and integral function of the coefficients.

In particular, the sums of the powers Σ*a*^{2}, Σ*a*^{3}, &c., are rational
and integral functions of the coefficients.

The process originally employed for the expression of other functions
Σ*a*^{α}*b*^{β}, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance,Σ*a*^{α}*b*^{β} = Σ*a*^{α}Σ*a*^{β} − Σ*a*^{α+β}; but this is very objectionable; the true theory consists in showing that we have systems of equations

p_{1} |
= Σa, | |

p_{2} |
=Σab, | |

p_{1}^{2} |
= Σa^{2} + 2Σab, | |

p_{3} | = Σabc, | |

p_{1}p_{2} | = Σa^{2}b + 3Σabc, | |

p_{1}^{3} | = Σa^{3} + 3Σa^{2}b + 6Σabc, |

where in each system there are precisely as many equations as there are root-functions on the right-hand side—*e.g.* 3 equations and 3 functions Σ*abc*, Σ*a*^{2}*b* Σ*a*^{3}. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:

Σab | = p_{2}, | |

Σa^{2} | = p_{1}^{2} − 2p_{2}, | |

Σabc | = p_{3}, | |

Σa^{2}b | =p_{1}p_{2} − 3p_{3}, | |

Σa^{3} | = p_{1}^{3} − 3p_{1}p_{2} + 3p_{3}, |

and so on. The other process, if applied consistently, would
derive the originally assumed value Σ*ab* = *p*_{2}, from the two equations
Σ*a* = *p*, Σ*a*^{2} = *p*_{1}^{2} − 2*p*_{2}; *i.e.* we have 2Σ*ab* = Σ*a*·Σ*a* − Σ*a*^{2},=
*p*_{1}^{2} − (*p*_{1}^{2} − 2*p*_{2}), = 2*p*_{2}.

13. It is convenient to mention here the theorem that, *x*
being determined as above by an equation of the order n, any
rational and integral function whatever of *x*, or more generally
any rational function which does not become infinite in virtue
of the equation itself, can be expressed as a rational and integral
function of *x*, of the order *n* − 1, the coefficients being rational
functions of the coefficients of the equation. Thus the equation
gives *x ^{n}* a function of the form in question; multiplying each
side by

*x*, and on the right-hand side writing for

*x*its foregoing value, we have

^{n}*x*

^{n+1}, a function of the form in question; and the like for any higher power of

*x*, and therefore also for any rational and integral function of

*x*. The proof in the case of a rational non-integral function is somewhat more complicated. The final result is of the form φ(

*x*)/ψ(

*x*) = I(

*x*), or say φ(

*x*) − ψ(

*x*)I(

*x*) = 0, where φ, ψ, I are rational and integral functions; in other words, this equation, being true if only ƒ(

*x*) = 0, can only be so by reason that the left-hand side contains ƒ(

*x*) as a factor, or we must have identically φ(

*x*) − ψ(

*x*)I(

*x*) = M(

*x*)ƒ(

*x*). And it is, moreover, clear that the equation φ(

*x*)/ψ(

*x*) = I(

*x*), being satisfied if only ƒ(

*x*) = 0, must be satisfied by each root of the equation.

From the theorem that a rational symmetrical function of the roots
is expressible in terms of the coefficients, it at once follows that it is
possible to determine an equation (of an assignable order) having
for its roots the several values of any given (unsymmetrical) function
of the roots of the given equation. For example, in the case of a
quartic equation, roots (*a*, *b*, *c*, *d* ), it is possible to find an equation
having the roots *ab*, *ac*, *ad*, *bc*, *bd*, *cd* (being therefore a sextic equation):
viz. in the product

*y*−

*ab*) (

*y*−

*ac*) (

*y*−

*ad*) (

*y*−

*bc*) (

*y*−

*bd*) (

*y*−

*cd*)

the coefficients of the several powers of y will be symmetrical functions
of *a*, *b*, *c*, *d* and therefore rational and integral functions of the coefficients
of the quartic equation; hence, supposing the product so
expressed, and equating it to zero, we have the required sextic
equation. In the same manner can be found the sextic equation
having the roots (*a* − *b*)^{2}, (*a* − *c*)^{2}, (*a* − *d* )^{2}, (*b* − *c*)^{2}, (*b* − *d* )^{2}, (*c* − *d* )^{2}, which
is the equation of differences previously referred to; and similarly
we obtain the equation of differences for a given equation of any
order. Again, the equation sought for may be that having for its
n roots the given rational functions φ(*a*), φ(*b*), ... of the several
roots of the given equation. Any such rational function can (as
was shown) be expressed as a rational and integral function of the
order *n* − 1; and, retaining *x* in place of any one of the roots, the
problem is to find y from the equations *x ^{n}* −

*p*

_{1}

*x*

^{n−1}... = 0, and y = M

_{0}

*x*

^{n−1}+ M

_{1}

*x*

^{n−2}+ ..., or, what is the same thing, from these two equations to eliminate

*x*. This is in fact E. W. Tschirnhausen’s transformation (1683).

14. In connexion with what precedes, the question arises as to
the number of values (obtained by permutations of the roots) of
given unsymmetrical functions of the roots, or say of a given set
of letters: for instance, with roots or letters (*a*, *b*, *c*, *d* ) as before,
how many values are there of the function *ab* + *cd*, or better,
how many functions are there of this form? The answer is 3,
viz. *ab* + *cd*, *ac* + *bd*, *ad* + *bc*; or again we may ask whether, in
the case of a given number of letters, there exist functions with
a given number of values, 3-valued, 4-valued functions, &c.

It is at once seen that for any given number of letters there exist 2–valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its sign; or say the two values are Δ and −Δ. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + QΔ; this has only the two values P + QΔ, P − QΔ.

In the case of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals.

The theory is an extensive and important one, depending on
the notions of *substitutions* and of *groups* (*q.v.*).

15. Returning to equations, we have the very important
theorem that, given the value of any unsymmetrical function of
the roots, *e.g.* in the case of a quartic equation, the function
*ab* + *cd*, it is in general possible to determine rationally the value
of any similar function, such as (*a* + *b*)^{3} + (*c* + *d* )^{3}.

The *a priori* ground of this theorem may be illustrated by means of
a numerical equation. Suppose that the roots of a quartic equation
are 1, 2, 3, 4, then if it is given that *ab* + *cd* = 14, this in effect determines
*a*, *b* to be 1, 2 and *c*, *d* to be 3, 4 (viz. *a* = 1, *b* = 2 or *a* = 2, *b* = 1,
and *c* = 3, *d* = 4 or *c* = 3, *d* = 4) or else *a*, *b* to be 3, 4 and *c*, *d* to be 1, 2;
and it therefore in effect determines (*a* + *b*)^{3} + (*c* + *d*)^{3} to be = 370,
and not any other value; that is, (*a* + *b*)^{3} + (*c* + *d*)^{3}, as having a
single value, must be determinable rationally. And we can in the
same way account for cases of failure as regards particular equations;
thus, the roots being 1, 2, 3, 4 as before, *a*^{2}*b* = 2 determines a to be
= 1 and *b* to be = 2, but if the roots had been 1, 2, 4, 16 then *a*^{2}*b* = 16
does not uniquely determine *a*, *b* but only makes them to be 1, 16 or
2, 4 respectively.

As to the *a posteriori* proof, assume, for instance,

*t*

_{1}=

*ab*+

*cd*,

*y*

_{1}= (

*a*+

*b*)

^{3}+ (

*c*+

*d*)

^{3},

*t*

_{2}=

*ac*+

*bd*,

*y*

_{2}= (

*a*+

*c*)

^{3}+ (

*b*+

*d*)

^{3},

*t*

_{3}=

*ad*+

*bc*,

*y*

_{3}= (

*a*+

*d*)

^{3}+ (

*b*+

*c*)

^{3};

then *y*_{1} + *y*_{2} + *y*_{3}, *t*_{1}*y*_{1} + *t*_{2}*y*_{2} + *t*_{3}*y*_{3}, *t*_{1}^{2}*y*_{1} + *t*_{2}^{2}*y*_{2} + *t*_{3}^{2}*y*_{3} will be respectively
symmetrical functions of the roots of the quartic, and therefore
rational and integral functions of the coefficients; that is, they
will be known.

Suppose for a moment that *t*_{1}, *t*_{2}, *t*_{3} are all known; then the
equations being linear in *y*_{1}, *y*_{2}, *y*_{3} these can be expressed rationally
in terms of the coefficients and of *t*_{1}, *t*_{2}, *t*_{3}; that is, *y*_{1}, *y*_{2}, *y*_{3} will be
known. But observe further that *y*_{1} is obtained as a function of
*t*_{1}, *t*_{2}, *t*_{3} symmetrical as regards *t*_{2}, *t*_{3}; it can therefore be expressed
as a rational function of *t*_{1} and of *t*_{2} + *t*_{3}, *t*_{2}*t*_{3}, and thence as a rational
function of *t*_{1} and of *t*_{1} + *t*_{2} + *t*_{3}, *t*_{1}*t*_{2} + *t*_{1}*t*_{3} + *t*_{2}*t*_{3}, *t*_{1}*t*_{2}*t*_{3}; but these last are
symmetrical functions of the roots, and as such they are expressible
rationally in terms of the coefficients; that is, *y*_{1} will be expressed
as a rational function of *t*_{1} and of the coefficients; or *t*_{1} (alone, not
*t*_{2} or *t*_{3}) being known, *y*_{1} will be rationally determined.

16. We now consider the question of the algebraical solution
of equations, or, more accurately, that of the *solution of equations*
*by radicals*.

In the case of a quadric equation *x*^{2} − *px* + *q* = 0, we can by the
assistance of the sign √( ) or ( )^{1/2} find an expression for *x* as a
2–valued function of the coefficients *p*, *q* such that substituting
this value in the equation, the equation is thereby identically
satisfied; it has been found that this expression is

*x*= 12 {

*p*± √(

*p*

^{2}− 4

*q*) },

and the equation is on this account said to be algebraically solvable,
or more accurately solvable by radicals. Or we may by writing
*x* = −12 *p* + *z* reduce the equation to *z*^{2} = 14 (*p*^{2} − 4*q*), viz. to an equation
of the form *x*^{2} = *a*; and in virtue of its being thus reducible we say
that the original equation is solvable by radicals. And the question
for an equation of any higher order, say of the order *n*, is, can we
by means of radicals (that is, by aid of the sign *m**√*( ) or ( )^{1/m}, using
as many as we please of such signs and with any values of m) find
an n-valued function (or any function) of the coefficients which
substituted for *x* in the equation shall satisfy it identically?

It will be observed that the coefficients *p*, *q* ... are not explicitly
considered as numbers, but even if they do denote numbers, the
question whether a numerical equation admits of solution by radicals
is wholly unconnected with the before-mentioned theorem of the
existence of the *n* roots of such an equation. It does not even
follow that in the case of a numerical equation solvable by radicals
the algebraical solution gives the numerical solution, but this requires
explanation. Consider first a numerical quadric equation with
imaginary coefficients. In the formula *x* = 12 {*p* ± √(*p*^{2} − 4*q*) }, substituting
for *p*, *q* their given numerical values, we obtain for *x* an
expression of the form *x* = α + βi ± √(γ + δi), where α, β, γ, δ are
real numbers. This expression substituted for *x* in the quadric
equation would satisfy it identically, and it is thus an algebraical
solution; but there is no obvious *a priori* reason why √(γ + δi)
should have a value = *c* + di, where *c* and *d* are real numbers calculable
by the extraction of a root or roots of real numbers; however
the case is (what there was no *a priori* right to expect) that √(γ + δi)
has such a value calculable by means of the radical expressions
√{√(γ^{2} + δ^{2}) ± γ}; and hence the algebraical solution of a numerical
quadric equation does in every case give the numerical solution. The
case of a numerical cubic equation will be considered presently.

17. A cubic equation can be solved by radicals.

Taking for greater simplicity the cubic in the reduced form
*x*^{3} + *qx* − *r* = 0, and assuming *x* = *a* + *b*, this will be a solution if only
3*ab* = *q* and *a*^{3} + *b*^{3} = *r*, equations which give (*a*^{3} − *b*^{3})^{2} = *r*^{2} − 427 *q*^{3}, a
quadric equation solvable by radicals, and giving *a*^{3} − *b*^{3} = √(*r*^{2} − 427 *q*^{3}),
a 2-valued function of the coefficients: combining this with *a*^{3} + *b*^{3}
= *r*, we have *a*^{3} = 12 {*r* + √(*r*^{2} − 427 *q*^{3}) }, a 2-valued function: we then
have a by means of a cube root, viz.

*a*= ∛[12 {

*r*+ √(

*r*

^{2}− 427

*q*

^{3}) }],

a 6-valued function of the coefficients; but then, writing *q* = b/3a, we
have, as may be shown, *a* + *b* a 3-valued function of the coefficients;
and *x* = *a* + *b* is the required solution by radicals. It would have
been wrong to complete the solution by writing

*b*= ∛[12 {

*r*− √(

*r*

^{2}− 427

*q*

^{3}) } ],

for then *a* + *b* would have been given as a 9-valued function having
only 3 of its values roots, and the other 6 values being irrelevant.
Observe that in this last process we make no use of the equation
3*ab* = *q*, in its original form, but use only the derived equation
27*a*^{3}*b*^{3} = *q*^{3}, implied in, but not implying, the original form.

An interesting variation of the solution is to write *x* = *ab*(*a* + *b*),
giving *a*^{3}*b*^{3} (*a*^{3} + *b*^{3}) = *r* and 3*a*^{3}*b*^{3} = *q*, or say *a*^{3} + *b*^{3} = 3*r*/*q*, *a*^{3}*b*^{3} = 13 *q*;
and consequently

a^{3} = | 32 | {r + √(r^{2} − 427 q^{3}) }, b^{3} = | 32 | {r − √(r^{2} − 427 q^{3}) }, |

q | q |

*i.e.* here *a*^{3}, *b*^{3} are each of them a 2-valued function, but as the only
effect of altering the sign of the quadric radical is to interchange
*a*^{3}, *b*^{3}, they may be regarded as each of them 1-valued; a and b
are each of them 3-valued (for observe that here only *a*^{3}*b*^{3}, not ab,
is given); and *ab*(*a* + *b*) thus is in appearance a 9-valued function;
but it can easily be shown that it is (as it ought to be) only 3-valued.

In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression

*x*= ∛[12 {

*r*+ √(

*r*

^{2}− 427

*q*

^{3}) }] + 13

*q*÷ ∛[12 {

*r*+ √(

*r*

^{2}− 427

*q*

^{3}) }],

this may depend on an expression of the form ∛(γ + δi) where
γ and δ are real numbers (it will do so if *r*^{2} − 427 *q*^{3} is a negative number),
and then we *cannot* by the extraction of any root or roots of
real positive numbers reduce ∛(γ + δi) to the form *c* + di, *c* and d
real numbers; hence here the algebraical solution does not give the
numerical solution, and we have here the so-called “irreducible
case” of a cubic equation. By what precedes there is nothing in
this that might not have been expected; the algebraical solution
makes the solution depend on the extraction of the cube root of
a number, and there was no reason for expecting this to be a real
number. It is well known that the case in question is that wherein
the three roots of the numerical cubic equation are all real; if the
roots are two imaginary, one real, then contrariwise the quantity
under the cube root is real; and the algebraical solution gives
the numerical one.

The irreducible case is solvable by a trigonometrical formula, but
this is not a solution by radicals: it consists in effect in reducing the
given numerical cubic (not to a cubic of the form *z*^{3} = *a*, solvable by
the extraction of a cube root, but) to a cubic of the form 4*x*^{3} − 3*x* = *a*,
corresponding to the equation 4 cos^{3} θ − 3 cos θ = cos 3θ which serves
to determine cosθ when cos 3θ is known. The theory is applicable
to an algebraical cubic equation; say that such an equation, if it
can be reduced to the form 4*x*^{3} − 3*x* = *a*, is solvable by “trisection”—then
the general cubic equation is solvable by trisection.

18. A quartic equation is solvable by radicals, and it is to be
remarked that the existence of such a solution depends on the
existence of 3-valued functions such as *ab* + *cd* of the four roots
(*a*, *b*, *c*, *d*): by what precedes *ab* + *cd* is the root of a cubic
equation, which equation is solvable by radicals: hence *ab* + *cd*
can be found by radicals; and since *abcd* is a given function, ab
and *cd* can then be found by radicals. But by what precedes,
if *ab* be known then any similar function, say *a* + *b*, is obtainable
rationally; and then from the values of *a* + *b* and *ab* we may by
radicals obtain the value of *a* or *b*, that is, an expression for the
root of the given quartic equation: the expression ultimately
obtained is 4-valued, corresponding to the different values of the
several radicals which enter therein, and we have thus the expression
by radicals of each of the four roots of the quartic
equation. But when the quartic is numerical the same thing
happens as in the cubic, and the algebraical solution does not in
every case give the numerical one.

It will be understood from the foregoing explanation as to the
quartic how in the next following case, that of the quintic, the question
of the solvability by radicals depends on the existence or non-existence
of *k*-valued functions of the five roots (*a*, *b*, *c*, *d*, *e*); the
fundamental theorem is the one already stated, a rational function
of five letters, if it has less than 5, cannot have more than 2 values,
that is, there are no 3-valued or 4-valued functions of 5 letters: and
by reasoning depending in part upon this theorem, N. H. Abel (1824)
showed that a general quintic equation is not solvable by radicals;
and *a fortiori* the general equation of any order higher than 5 is not
solvable by radicals.

19. The general theory of the solvability of an equation by radicals
depends fundamentally on A. T. Vandermonde’s remark (1770)
that, supposing an equation is solvable by radicals, and that we have
therefore an algebraical expression of *x* in terms of the coefficients,
then substituting for the coefficients their values in terms of the roots,
the resulting expression must reduce itself to any one at pleasure of
the roots *a*, *b*, *c* ...; thus in the case of the quadric equation, in the
expression *x* = 12 {*p* + √(*p*^{2} − 4*q*) }, substituting for *p* and *q* their values,
and observing that (*a* + *b*)^{2} − 4*ab* = (*a* − *b*)^{2}, this becomes *x* = 12 {*a* + *b* +
√(*a* − *b*)^{2}}, the value being *a* or *b* according as the radical is taken
to be +(*a* − *b*) or −(*a* − *b*).

So in the cubic equation *x*^{3} − *px*^{2} + *qx* − *r* = 0, if the roots are *a*, *b*, *c*,
and if ω is used to denote an imaginary cube root of unity, ω^{2} + ω +
1 = 0, then writing for shortness *p* = *a* + *b* + *c*, L = *a* + ω*b* + ω^{2}*c*, M =
*a* + ω^{2}*b* + ω*c*, it is at once seen that LM, L^{3} + M^{3}, and therefore also
(L^{3} − M^{3})^{2} are symmetrical functions of the roots, and consequently
rational functions of the coefficients; hence

^{3}+ M

^{3}+ √(L

^{3}− M

^{3})

^{2}}

is a rational function of the coefficients, which when these are
replaced by their values as functions of the roots becomes, according
to the sign given to the quadric radical, = L^{3} or M^{3}; taking it = L^{3},
the cube root of the expression has the three values L, ωL, ω^{2}L;
and LM divided by the same cube root has therefore the values
M, ω^{2}M, ωM; whence finally the expression

*p*+ ∛{12 (L

^{3}+ M

^{3}+ √(L

^{3}− M

^{3})

^{2}) } + LM ÷ ∛{12 L

^{3}+ M

^{3}+ √(L

^{3}− M

^{3})

^{2}) }]

has the three values

*p*+ L + M), 13 (

*p*+ ωL + ω

^{2}M), 13 (

*p*+ ω

^{2}L + ωM);

that is, these are = *a*, *b*, *c* respectively. If the value M^{3} had been
taken instead of L^{3}, then the expression would have had the same
three values *a*, *b*, *c*. Comparing the solution given for the cubic
*x*^{3} + *qx* − *r* = 0, it will readily be seen that the two solutions are
identical, and that the function *r* ^{2} − 427 *q*^{3} under the radical sign must
(by aid of the relation *p* = 0 which subsists in this case) reduce itself
to (L^{3} − M^{3})^{2}; it is only by each radical being equal to a rational
function of the roots that the final expression *can* become equal to
the roots *a*, *b*, *c* respectively.

20. The formulae for the cubic were obtained by J. L. Lagrange
(1770–1771) from a different point of view. Upon examining
and comparing the principal known methods for the solution of
algebraical equations, he found that they all ultimately depended
upon finding a “resolvent” equation of which the root is
*a* + ω*b* + ω^{2}*c* + ω^{3}*d* + ..., ω being an imaginary root of unity,
of the same order as the equation; *e.g.* for the cubic the root is
*a* + ω*b* + ω^{2}*c*, ω an imaginary cube root of unity. Evidently the
method gives for L^{3} a quadric equation, which is the “resolvent”
equation in this particular case.

For a quartic the formulae present themselves in a somewhat
different form, by reason that 4 is not a prime number. Attempting
to apply it to a quintic, we seek for the equation of which the
root is (*a* + ω*b* + ω^{2}*c* + ω^{3}*d* + ω^{4}*e*), ω an imaginary fifth root of
unity, or rather the fifth power thereof (*a* + ω*b* + ω^{2}*c* + ω^{3}*d* + ω^{4}*e*)^{5};
this is a 24-valued function, but if we consider the four values
corresponding to the roots of unity ω, ω^{2}, ω^{3}, ω^{4}, viz. the values

( |

any symmetrical function of these, for instance their sum, is a 6–valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange’s resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory.

21. It is to be remarked, in regard to the question of solvability
by radicals, that not only the coefficients are taken to
be arbitrary, but it is assumed that they are represented each
by a single letter, or say rather that they are not so expressed
in terms of other arbitrary quantities as to make a solution
possible. If the coefficients are not all arbitrary, for instance,
if some of them are zero, a sextic equation might be of the
form *x*^{6} + b*x*^{4} + *cx*^{2} + *d* = 0, and so be solvable as a cubic; or
if the coefficients of the sextic are given functions of the six
arbitrary quantities *a*, *b*, *c*, *d*, *e*, *f*, such that the sextic is really
of the form (*x*^{2} + *ax* + *b*)(*x*^{4} + *cx*^{3} + *dx*^{2} + *ex* + *f*) = 0, then it breaks
up into the equations *x*^{2} + *ax* + *b* = 0, *x*^{4} + *cx*^{3} + *dx*^{2} + *ex* + *f* = 0,
and is consequently solvable by radicals; so also if the form
is (*x* − *a*) (*x* − *b*) (*x* − *c*) (*x* − *d*) (*x* − *e*) (*x* − *f*) = 0, then the equation
is solvable by radicals,—in this extreme case rationally. Such
cases of solvability are self-evident; but they are enough
to show that the general theorem of the non-solvability by
radicals of an equation of the fifth or any higher order does not
in any wise exclude for such orders the existence of particular
equations solvable by radicals, and there are, in fact, extensive
classes of equations which are thus solvable; the binomial
equations *x*^{n} − 1 = 0 present an instance.

22. It has already been shown how the several roots of the equation
*x*^{n} − 1 = 0 can be expressed in the form cos 2*s*π/*n* + *i* sin 2*s*π/*n*, but the
question is now that of the algebraical solution (or solution by
radicals) of this equation. There is always a root = 1; if ω be any
other root, then obviously ω, ω^{2}, ... ω^{n−1} are all of them roots; *x*^{n} − 1
contains the factor*x*− 1, and it thus appears that ω, ω^{2}, ... ω^{n−1} are
the *n*−1 roots of the equation

*x*

^{n−1}+

*x*

^{n−2}+ ...

*x*+ 1 = 0;

we have, of course, ω^{n−1} + ω^{n−2} + ... + ω + 1 = 0.

It is proper to distinguish the cases *n* prime and *n* composite;
and in the latter case there is a distinction according as the prime
factors of *n* are simple or multiple. By way of illustration, suppose
successively *n* = 15 and *n* = 9; in the former case, if α be an imaginary
root of *x*^{3} − 1 = 0 (or root of *x*^{2} +*x*+ 1 = 0), and β an imaginary root
of *x*^{5} − 1 = 0 (or root of *x*^{4} + *x*^{3} + *x*^{2} +*x*+ 1 = 0), then ω may be taken
= αβ; the successive powers thereof, αβ, α^{2}β^{2}, β^{3}, αβ^{4}, α^{2}, β, αβ^{2},
α^{2}β^{3}, β^{4}, α, α^{2}β, β^{2}, αβ^{3}, α^{2}β^{4}, are the roots of *x*^{14} + *x*^{13} + ... +*x*+ 1 = 0;
the solution thus depends on the solution of the equations *x*^{3} − 1 = 0
and *x*^{5} − 1 = 0. In the latter case, if α be an imaginary root of
*x*^{3} − 1 = 0 (or root of *x*^{2} +*x*+ 1 = 0), then the equation *x*^{9} − 1 = 0 gives
*x*^{3} = 1, α, or α^{2}; *x*^{3} = 1 gives*x*= 1, α, or α^{2}; and the solution thus
depends on the solution of the equations *x*^{3} − 1 = 0, *x*^{3} − α = 0, *x*^{3} − α^{2} = 0.
The first equation has the roots 1, α, α^{2}; if β be a root of either of the
others, say if β^{3} = α, then assuming ω = β, the successive powers are
β, β^{2}, α, αβ, αβ^{2}, α^{2}, α^{2}β, α^{2}β^{2}, which are the roots of the equation
*x*^{8} + *x*^{7} + ... +*x*+ 1 = 0.

It thus appears that the only case which need be considered is that
of *n* a prime number, and writing (as is more usual) *r* in place of ω,
we have *r*, *r*^{2}, *r*^{3},...*r*^{n−1} as the (*n* − 1) roots of the reduced equation

*x*

^{n−1}+

*x*

^{n−2}+ ... +

*x*+ 1 = 0;

then not only *r*^{n} − 1 = 0, but also *r*^{n−1} + *r*^{n−2} + ... + *r* + 1 = 0.

23. The process of solution due to Karl Friedrich Gauss (1801)
depends essentially on the arrangement of the roots in a certain
order, viz. not as above, with the indices of *r* in arithmetical
progression, but with their indices in geometrical progression;
the prime number *n* has a certain number of prime roots *g*,
which are such that *g*^{n−1} is the lowest power of *g*, which is ≡ 1
to the modulus *n*; or, what is the same thing, that the series of
powers 1, *g*, *g*^{2}, ... *g*^{n−2}, each divided by *n*, leave (in a different
order) the remainders 1, 2, 3, ...*n*− 1; hence giving to *r* in
succession the indices 1, *g*, *g*^{2}, . . . *g*^{n−2}, we have, in a different
order, the whole series of roots *r*, *r*^{2}, *r*^{3},...*r*^{n−1}.

In the most simple case, *n*= 5, the equation to be solved is *x*^{4} + *x*^{3} +
*x*^{2} +*x*+ 1 = 0; here 2 is a prime root of 5, and the order of the roots
is *r*, *r*^{2}, *r*^{4}, *r*^{3}. The Gaussian process consists in forming an equation
for determining the periods P_{1}, P_{2}, = *r* + *r*^{4} and *r*^{2} + *r*^{3} respectively;—these
being such that the symmetrical functions P_{1} + P_{2}, P_{1}P_{2} are
rationally determinable: in fact P_{1} + P_{2} = −1, P_{1}P_{2} = (*r* + *r*^{4}) (*r*^{2} + *r*^{3}),
= *r*^{3} + *r*^{4} + *r*^{6} + *r*^{7}, = *r*^{3} + *r*^{4} + *r* + *r*^{2}, = −1. P_{1}, P_{2} are thus the roots
of *u*^{2} + *u* − 1 = 0; and taking them to be known, they are themselves
broken up into subperiods, in the present case single terms, *r* and *r*^{4}
for P_{1}, *r*^{2} and *r*^{3} for P_{2}; the symmetrical functions of these are then
rationally determined in terms of P_{1} and P_{2}; thus *r* + *r*^{4} = P_{1}, *r·r*^{4} = 1,
or *r*, *r*^{4} are the roots of *u*^{2} − P_{1}*u* + 1 = 0. The mode of division is more
clearly seen for a larger value of *n*; thus, for *n*= 7 a prime root is
= 3, and the arrangement of the roots is *r*, *r*^{3}, *r*^{2}, *r*^{6}, *r*^{4}, *r*^{5}. We may
form either 3 periods each of 2 terms, P_{1}, P_{2}, P_{3} = *r* + *r*^{6}, *r*^{3} + *r*^{4}, *r*^{2} + *r*^{5}
respectively; or else 2 periods each of 3 terms, P_{1}, P_{2} = *r* + *r*^{2} + *r*^{4},
*r*^{3} + *r*^{6} + *r*^{5} respectively; in each case the symmetrical functions of
the periods are rationally determinable: thus in the case of the two
periods P_{1} + P_{2} = −1, P_{1}P_{2} = 3 + *r* + *r*^{2} + *r*^{3} + *r*^{4} + *r*^{5} + *r*^{6}, = 2; and the
periods being known the symmetrical functions of the several terms
of each period are rationally determined in terms of the periods, thus
*r* + *r*^{2} + *r*^{4} = P_{1}, *r·r*^{2} + *r·r*^{4} + *r*^{2}·*r*^{4} = P_{2}, *r·r*^{2}·*r*^{4} = 1.

The theory was further developed by Lagrange (1808), who,
applying his general process to the equation in question, *x*^{n−1} +
*x*^{n−2} + ... +*x*+ 1 = 0 (the roots *a*, *b*, *c*... being the several powers
of *r*, the indices in geometrical progression as above), showed
that the function (*a* + ω*b* + ω^{2}*c* + ...)^{n−1} was in this case a given
function of ω with integer coefficients.

Reverting to the before-mentioned particular equation *x*^{4} + *x*^{3} +
*x*^{2} +*x*+ 1 = 0, it is very interesting to compare the process of solution
with that for the solution of the general quartic the roots whereof are
*a*, *b*, *c*, *d*.

Take ω, a root of the equation ω^{4} − 1 = 0 (whence ω is = 1, −1, *i*,
or −*i*, at pleasure), and consider the expression

*a*+ ω

*b*+ ω

^{2}

*c*+ ω

^{3}

*d*)

^{4},

the developed value of this is

= | a^{4} + b^{4} + c^{4} + d ^{4} + 6 (a^{2}c^{2} + b^{2}d ^{2}) + 12 (a^{2}bd + b^{2}ca + c^{2}db + d ^{2}ac) |

+ω | {4 (a^{3}b + b^{3}c + c^{3} + d ^{3}a) + 12 (a^{2}cd + b^{2}da + c^{2}ab + d ^{2}bc) } |

+ω^{2} | {6 (a^{2}b^{2} + b^{2}c^{2} + c^{2}d ^{2} + d ^{2}a^{2}) + 4 (a^{3}c + b^{3}d + c^{3}a + d ^{3}b) + 24abcd} |

+ω^{3} | {4 (a^{3}d + b^{3}a + c^{3}b + d ^{3}c) + 12 (a^{2}bc + b^{2}cd + c^{2}da + d ^{2}ab) } |

that is, this is a 6-valued function of *a*, *b*, *c*, *d*, the root of a sextic
(which is, in fact, solvable by radicals; but this is not here material).

If, however, *a*, *b*, *c*, *d* denote the roots *r*, *r* ^{2}, *r*^{4}, *r*^{3} of the special
equation, then the expression becomes

r ^{4} | + r ^{3} + r + r ^{2} + 6 (1 + 1) | + 12 (r ^{2} + r ^{4} + r ^{3} + r ) |

+ ω{4 (1 + 1 + 1 + 1) | + 12 (r ^{4} + r ^{3} + r + r ^{2}) } | |

+ ω^{2}{6 (r + r ^{2} + r ^{4} + r ^{3}) | + 4 (r ^{2} + r ^{4} + r ^{3} + r ) } | |

+ ω^{3}{4 (r + r ^{2} + r ^{4} + r ^{3}) | + 12 (r ^{3} + r + r ^{2} + r ^{4}) } |

viz. this is

^{2}− 16ω

^{3},

a completely determined value. That is, we have

*r*+ ω

*r*

^{2}+ ω

^{2}

*r*

^{4}+ ω

^{3}

*r*

^{3}) = −1 + 4ω + 14ω

^{2}− 16ω

^{3},

which result contains the solution of the equation. If ω = 1, we have
(*r* + *r* ^{2} + *r* ^{4} + *r* ^{3})^{4} = 1, which is right; if ω = −1, then (*r* + *r* ^{4} − *r* ^{2} − *r*^{3})^{4} = 25;
if ω = *i*, then we have {*r* − *r*^{4} + *i*(*r* ^{2} − *r*^{3}) }^{4} = −15 + 20*i*; and if ω = −*i*,
then {*r* − *r* ^{4} − *i* (*r* ^{2} − *r* ^{3}) }^{4} = −15 − 20*i*; the solution may be completed
without difficulty.

The result is perfectly general, thus:—*n* being a prime number,
*r* a root of the equation *x*^{n−1} + *x*^{n−2} + ... + *x* + 1 = 0, ω a root of
ω^{n−1} − 1 = 0, and *g* a prime root of *g*^{n−1} ≡ 1 (mod. *n*), then

*r*+ ω

*r*

^{g}+ ... + ω

^{n − 2}

*r*

^{ g n−2})

^{n−1}

is a given function M_{0} + M_{1}ω ... + M_{n−2}ω^{n−2} with integer coefficients,
and by the extraction of (*n* − 1)th roots of this and
similar expressions we ultimately obtain *r* in terms of ω, which is
taken to be known; the equation *x*^{n} − 1 = 0, *n* a prime number,
is thus solvable by radicals. In particular, if *n* − 1 be a power of 2,
the solution (by either process) requires the extraction of square
roots only; and it was thus that Gauss discovered that it was
possible to construct geometrically the regular polygons of 17
sides and 257 sides respectively. Some interesting developments
in regard to the theory were obtained by C. G. J. Jacobi (1837);
see the memoir “Ueber die Kreistheilung, u.s.w.,” *Crelle*, t. xxx.
(1846).

The equation *x*^{n−1} + ... + *x* + 1 = 0 has been considered for its
own sake, but it also serves as a specimen of a class of equations
solvable by radicals, considered by N. H. Abel (1828), and since
called Abelian equations, viz. for the Abelian equation of the
order *n*, if *x* be any root, the roots are *x*, θ*x*, θ^{2}*x*, ... θ^{n−1}*x* (θ*x*
being a rational function of *x*, and θ^{n}*x* = *x*); the theory is, in fact,
very analogous to that of the above particular case.

A more general theorem obtained by Abel is as follows:—If the
roots of an equation of any order are connected together in such
wise that *all* the roots can be expressed rationally in terms of
any one of them, say *x*; if, moreover, θ*x*, θ_{1}*x* being any two of the
roots, we have θθ_{1}*x* = θ_{1}θ*x*, the equation will be solvable algebraically.
It is proper to refer also to Abel’s definition of an *irreducible* equation:—an
equation φ*x* = 0, the coefficients of which are rational functions
of a certain number of known quantities *a*, *b*, *c* ..., is called irreducible
when it is impossible to express its roots by an equation of an inferior
degree, the coefficients of which are also rational functions of *a*, *b*, *c* ...
(or, what is the same thing, when φ*x* does not break up into factors
which are rational functions of *a*, *b*, *c* ...). Abel applied his theory
to the equations which present themselves in the division of the
elliptic functions, but not to the modular equations.

24. But the theory of the algebraical solution of equations
in its most complete form was established by Evariste Galois
(born October 1811, killed in a duel May 1832; see his collected
works, *Liouville*, t. xl., 1846). The definition of an irreducible
equation resembles Abel’s,—an equation is reducible when it
admits of a rational divisor, irreducible in the contrary case;
only the word *rational* is used in this extended sense that, in
connexion with the coefficients of the given equation, or with the
irrational quantities (if any) whereof these are composed, he
considers any number of other irrational quantities called
“adjoint radicals,” and he terms rational any rational function
of the coefficients (or the irrationals whereof they are composed)
and of these adjoint radicals; the epithet irreducible is thus taken
either absolutely or in a relative sense, according to the system of
adjoint radicals which are taken into account. For instance,
the equation *x*^{4} + *x*^{3} + *x*^{2} + *x* + 1 = 0; the left hand side has here
no rational divisor, and the equation is irreducible; but this
function is = (*x*^{2} + 12 *x* + 1)^{2} − 54 *x*^{2}, and it has thus the irrational
divisors *x*^{2} + 12 (1 + √5)*x* + 1, *x*^{2} + 12 (1 − √5)*x* + 1; and these, if
we *adjoin* the radical √5, are rational, and the equation is no
longer irreducible. In the case of a given equation, assumed to be
irreducible, the problem to solve the equation is, in fact, that of
finding radicals by the adjunction of which the equation becomes
reducible; for instance, the general quadric equation *x*^{2} + *px* +
*q* = 0 is irreducible, but it becomes reducible, breaking up into
rational linear factors, when we adjoin the radical √(14 *p*^{2} − *q*).

The fundamental theorem is the Proposition I. of the “Mémoire
sur les conditions de résolubilité des équations par radicaux”;
viz. given an equation of which *a*, *b*, *c* . . . are the *m* roots, there is
always a group of permutations of the letters *a*, *b*, *c* . . . possessed
of the following properties:—

1. Every function of the roots invariable by the substitutions of the group is rationally known.

2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group.

Here by an invariable function is meant not only a function of
which the form is invariable by the substitutions of the group, but
further, one of which the value is invariable by these substitutions:
for instance, if the equation be φ(*x*) = 0, then φ(*x*) is a function of the
roots invariable by any substitution whatever. And in saying that
a function is rationally known, it is meant that its value is expressible
rationally in terms of the coefficients and of the adjoint quantities.

For instance in the case of a general equation, the group is simply
the system of the 1.2.3 . . . *n* permutations of all the roots, since,
in this case, the only rationally determinable functions are the symmetric
functions of the roots.

In the case of the equation *x*^{n−1} . . . + *x* + 1 = 0, *n* a prime number,
*a*, *b*, *c* . . . *k* = *r* , *r* ^{g}, *r* ^{g2} . . . *r* ^{gn−2}, where *g* is a prime root of *n*, then the
group is the cyclical group *abc . . . k*, *bc . . . ka*, . . . *kab . . . j*, that is,
in this particular case the number of the permutations of the group
is equal to the order of the equation.

This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois’s theory. But the problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to the transformation and classification of irrationals.

Returning to the question of solution by radicals, it will be readily
understood that by the adjunction of a radical the group may be
diminished; for instance, in the case of the general cubic, where the
group is that of the six permutations, by the adjunction of the square
root which enters into the solution, the group is reduced to abc,
bca, cab; that is, it becomes possible to express rationally, in terms
of the coefficients and of the adjoint square root, any function such
as *a*^{2}*b* + *b*^{2}*c* + *c*^{2}*a* which is not altered by the cyclical substitution
*a* into *b*, *b* into *c*, *c* into *a*. And hence, to determine whether an
equation of a given form is solvable by radicals, the course of investigation
is to inquire whether, by the successive adjunction of
radicals, it is possible to reduce the original group of the equation
so as to make it ultimately consist of a single permutation.

The condition in order that an equation of a given prime order n
may be solvable by radicals was in this way obtained—in the first
instance in the form (scarcely intelligible without further explanation)
that every function of the roots *x*_{1}, *x*_{2} . . . *x*_{n}, invariable by the
substitutions *x*_{ak + b} for *x*_{k}, must be rationally known; and then
in the equivalent form that the resolvent equation of the order
1.2. . . (*n* − 2) must have a rational root. In particular, the condition
in order that a quintic equation may be solvable is that Lagrange’s
resolvent of the order 6 may have a rational factor, a result obtained
from a direct investigation in a valuable memoir by E. Luther,
*Crelle*, t. xxxiv. (1847).

Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.

The general theory of Galois in regard to the solution of equations
was completed, and some of the demonstrations supplied by E.
Betti (1852). See also J. A. Serret’s *Cours d’algèbre supérieure*, 2nd
ed. (1854); 4th ed. (1877–1878).

25. Returning to quintic equations, George Birch Jerrard
(1835) established the theorem that the general quintic equation
is by the extraction of only square and cubic roots reducible to
the form *x*^{5} + *ax* + *b* = 0, or what is the same thing, to *x*^{5} + *x* + *b* = 0.
The actual reduction by means of Tschirnhausen’s theorem was
effected by Charles Hermite in connexion with his elliptic-function
solution of the quintic equation (1858) in a very elegant
manner. It was shown by Sir James Cockle and Robert Harley
(1858–1859) in connexion with the Jerrardian form, and by
Arthur Cayley (1861), that Lagrange’s resolvent equation of the
sixth order can be replaced by a more simple sextic equation
occupying a like place in the theory.

The theory of the modular equations, more particularly for the
case *n* = 5, has been studied by C. Hermite, L. Kronecker and
F. Brioschi. In the case *n* = 5, the modular equation of the order 6 depends, as already mentioned, on an equation of the order 5;
and conversely the general quintic equation may be made to
depend upon this modular equation of the order 6; that is,
assuming the solution of this modular equation, we can solve
(not by radicals) the general quintic equation; this is Hermite’s
solution of the general quintic equation by elliptic functions
(1858); it is analogous to the before-mentioned trigonometrical
solution of the cubic equation. The theory is reproduced and
developed in Brioschi’s memoir, “Über die Auflösung der
Gleichungen vom fünften Grade,” *Math. Annalen*, t. xiii. (1877–1878).

26. The modern work, reproducing the theories of Galois,
and exhibiting the theory of algebraic equations as a whole, is C.
Jordan’s *Traité des substitutions et des équations algébriques* (Paris,
1870). The work is divided into four books—book i., preliminary,
relating to the theory of congruences; book ii. is in two chapters,
the first relating to substitutions in general, the second to substitutions
defined analytically, and chiefly to linear substitutions; book
iii. has four chapters, the first discussing the principles of the general
theory, the other three containing applications to algebra, geometry,
and the theory of transcendents; lastly, book iv., divided into seven
chapters, contains a determination of the general types of equations
solvable by radicals, and a complete system of classification of these
types. A glance through the index will show the vast extent which
the theory has assumed, and the form of general conclusions arrived
at; thus, in book iii., the algebraical applications comprise Abelian
equations, equations of Galois; the geometrical ones comprise Q.
Hesse’s equation, R. F. A. Clebsch’s equations, lines on a quartic
surface having a nodal line, singular points of E. E. Kummer’s
surface, lines on a cubic surface, problems of contact; the applications
to the theory of transcendents comprise circular functions,
elliptic functions (including division and the modular equation),
hyperelliptic functions, solution of equations by transcendents.
And on this last subject, solution of equations by transcendents,
we may quote the result—“the solution of the general equation of
an order superior to five cannot be made to depend upon that of the
equations for the division of the circular or elliptic functions”;
and again (but with a reference to a possible case of exception),
“the general equation cannot be solved by aid of the equations which
give the division of the hyperelliptic functions into an odd number
of parts.” (See also Groups, Theory of.)
(A. Ca.)

Bibliography.—For the general theory see W. S. Burnside and
A. W. Panton, *The Theory of Equations* (4th ed., 1899–1901); the Galoisian theory is treated in G. B. Matthews, *Algebraic Equations* (1907). See also the *Ency. d. math. Wiss.* vol. ii.

- ↑ The coefficients were selected so that the roots might be nearly 1, 2, 3.
- ↑ The third edition (1826) is a reproduction of that of 1808; the first edition has the date 1798, but a large part of the contents is taken from memoirs of 1767–1768 and 1770–1771.
- ↑ The earlier demonstrations by Euler, Lagrange, &c, relate to the
case of a numerical equation with real coefficients; and they consist
in showing that such equation has always a real quadratic divisor, furnishing
two roots, which are either real or else conjugate imaginaries
α + β
*i*(see Lagrange’s*Équations numériques*). - ↑ The square root of α + β
*i*can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables.