This equation must give for a at least a commensurable value, without which the transformation (1) will be impossible.

If, instead of VA+ √B, we should have to reduce A-VB, it would suffice to change throughout in the preceding method the sign of √b.

3

For example, let the expression be 141 √200. We shall have A=14, B=200, A2-B=−4; hence (A2—B)c=—4c; we shall then have the perfect cube -8, by taking c=2. Consequently, M=-1, b=a2+1, and equation (4) becomes 8a3+6a-14-0. It can be satisfied by the commensurable value a=1, which gives b=2. Again, we have already obtained c=2; hence, finally,

Again, let the expression be V-11±2√1. We will pass 2 under the radical of the second degree; we shall then have A=-11, B=—4, A2—B 125. As 125 is already the cube of 5, it will suffice to make c=1. Consequently, we have M=5, b=a2-5, and equation (4) becomes 4a3—15a +11=0. But this equation is satisfied by the value a=1; hence b=-4, and, consequently,

382. Let us consider the more general expression VA± √B, and take

The problem, again, is to determine rational numbers for a, b, c, if it be possible.

Raising (5) to the power n, and equating separately the rational parts, we obtain

We can, as in the case of the cubic radical, square these two equalities, and subtract the one from the other; but the reductions will be immediately perceived by observing that we ought to have, at the same time,

A+ √B=c(a+√b)", A— √B=c(a— √b)" ;

We see from this that it will be necessary to take c of such a value that the second member of this last equation shall be rational. Calling this second member M, we shall have a2-b=M, whence b=a2-M; substituting this

value of b in (6), the resulting equation in a will have a commensurable root every time that the transformation (5) is possible.

383. In the resolution of equations of the third degree, what renders the irreducible case so remarkable is, that although we are assured that the three roots are real, it is, nevertheless, impossible to make the imaginary quantities disappear otherwise than by means of series. This difficulty is not confined to the equation of the third degree; it will be encountered equally in the general formula

n

n

A+B√−1+√A−B√ −1

which formula I shall stop to consider for a moment.

To consider this expression in its most general sense, we ought to combine the n determinations of the first part with the ʼn determinations of the second, so that we shall have, in all, n2 values. But the expression is rarely taken in so general a sense, and I proceed to define that which we ordinarily attach to it.

As the two radicals which have the index n represent the roots of the binomial equation, their determinations are equal in number to the quantities which have the form ƒ+g√1. Moreover, it is manifest that to each determination of the first radical there corresponds one of the second, which only differs by the sign of √-1. But we suppose that these corresponding values are those which ought to be added in formula (8); and, with these restrictions, the values of x are all real, and only n in number.

The product of these two radical values, thus taken in a same pair, is real and positive; but for the product of the two radicals we have, in general,

n

A+B√ 1XVA-BV-1-VA2+B2,

and the radical which expresses this product can only have a single real and positive value; hence, if we represent it by K2, we ought to be able to characterize the conjugate values, which must be added in formula (8), by the condition that their product be equal to K3.

Formula (8) can be regarded as a general expression of the roots of an equation whose degree is marked by the number of values of which the equation is susceptible; hence, provided that it be taken in its greatest extension, or with the restriction which we have just mentioned, the degree of the equation must be either n2 or n.

This last remark leads us to explain how we form an equation, when we know the expression for its root; that is to say, that an equation being given, susceptible of taking different values, by reason of the multiple values of the radicals which it contains, it is required to find an equation free from radicals which has these values for roots. I will take, for example, the same expression (8).

To abridge, let us make

A+B√1=a, A-B√1=b;

the problem reduces itself to eliminating y and z between the three equations

y+z=x, y"=a, z"=b.

But here the elimination can be conducted according to a very simple pro

cess, analogous to that which has been employed for reciprocal equations. By the rules of multiplication we have

(y+zm)(y+z)=ym+1+zm+1+yz(ym¬1+zm−1).

But y+zr and yz=Vab; hence, making ab=c, the equation will become

ym+1+2m+1=x(yTM+zTM)—c(y-1+2-1).

By means of this formula we express, in function of x and c, successively all the quantities y2+z2, y3+z3, &c. When we have arrived at y"+z", we replace y+2" by a+b, and then we shall have the required equation, which will be of the degree n in x.

This equation contains c; but we have c=ab=VA+B2; hence, c is, in general, susceptible of n different values. By putting in the equation each of these n values in its turn, we shall have n equations, and, consequently, nxn, or no values of x. This, in fact, ought to be the case, from what has been said at the close of the preceding article. If we should wish to have a single equation which has all these values for roots, it would be still necessary to eliminate c between the equation of the degree n in x and the equation cab.

But if in formula (8) we only wish to associate the radical values whose product is real, it is this real value solely which we must choose for c, and we shall only have a single equation of the degree n for determining all the values of x.

RESOLUTION OF THE EQUATION OF THE FOURTH DEGREE.

384. After having made the second term disappear, the general equation of the 4° degree is

x1+px2+qx+r=0

If we make x=a+b+c, squaring, there results

x2=a2+b2+c+2(ab+ac+bc),

or, transposing,

x2— (a2+b2+c2)=2(ab+ac+bc);

raising anew to the square, we have

xa—2(a2+b2+c2)x2+(a2+b2+c2)3=2(a2b2+a2c2+b2c2)+8abc(a+b+c);

then, replacing a+b+c by x, and transposing, we obtain

x1-2(a2+b2+c2).x2-8abcx+(a2+b2+c2)2

-4(a2b2+ac2+b2c2)=0.

This equation is without a second term, and by the manner in which it has been formed, we know that it admits of the root x=a+b+c. Thus, we resolve equation (1) in determining a, b, c, by the condition that it shall be identical with the preceding, which gives

−2(a2+b2+c2)=p
-8abe q

(a2+b2+c2)2 —4(a2b2+a2c2+b2c2)=r.

These equalities show that, by taking a2, b2, c2 for unknowns, these three quantities are the roots of an equation of the 3° degree, the coefficients of which are (see Art. 245)

Such is the reduced equation upon which the solution of equation (1) depends. Suppose that the three values of z have been determined, which designate by z', z", z'", we shall have

If the signs be combined in all possible ways, there will result eight values for a+b+c or x. But as the last term of the reduced equation (2) was

not only the roots of the proposed equation, but also those of an equation which would differ from it in the sign of q.

At the same time it may be perceived that, to have only the roots of the

1

89,

proposed, it is necessary to add only the values of a, b, c, for which abc= and the product of which has, consequently, the contrary sign to q. In each particular case it will be easy to determine for the radicals three values, A, B, C, which shall fulfill this condition; and afterward, with these values, we form the four roots of the proposed, to wit,

x=+A+B+C, x=+A-B-C,
x=-A+B-C, x=— -A−B+C.

Generally, instead of A, B, C, the three radicals are placed, and the values of x are written thus:

x=+√ï' +√z" — √z"", x=+√z' — √z"+ √ z′′,

But it is necessary to understand that in applying these formulas to particular cases there must be taken for √z', √√z", √z""' three determinations, the product of which shall be of the same sign as q. This observation is important; failing to have regard to it, we might find false roots.

385. The nature of the roots of the reduced equation will make known the nature of the roots of the proposed. But the reduced having its last term negative, has always one positive root (see Art. 248, Prop. VIII., Cor. 4), and the product of the other two roots should be positive; then, if these last are not imaginary, they will be both positive or both negative. I pass over the case in which q=0, because then the proposed would be solved by the rules for the second degree. Consequently, there are three cases only to be examined.*

1o. Case where the three roots of the reduced equation are positive. There the four values of r are evidently real, and if the radicals √z′, √√z", √z′′ be regarded as representing positive determinations, their product will be positive;

* This explains an operation in Art. 365.

then the preceding formulas will be specially applicable to the case of q>0. For q<0 it would be necessary to change the sign of one of the radicals.

2o. Case where the reduced has one root z' positive, and two z", z'"' negative. The radical √z' will be real, but the radicals √z" and √z"" will be imaginary; consequently, the four values of r will be imaginary also, unless z'z'''. When z'z'"', one of the two quantities √z"+ √z"" and √z” — √z”” will become zero, and supposing it to be the latter, the values of r will be simply

x=√7', x=√7′′, x=−√Z'+2√z", x=— √z' —2 √z′′.

The first two are real, since z' is positive, and the other two are imaginary, since z" is negative. Besides, as in the reduction, we have supposed √√z", we ought to have here √√√√"""; so that this product can only have the sign of q by choosing for a sign contrary to that of q, since, by hypothesis, z' is negative.

=2"

3°. Case in which the reduced has one root z' positive, and two roots z", z'" imaginary. The positive root z' being known, we can divide the reduced by x-z', and we shall have an equation of the second degree, which will give for z" and z'"' imaginary values of the form

2"=f+g√ −1, z""=f-g√ -1.

Consequently, two of the values of r will contain the sum

√s+8√=1+√5-8√=

f+g√−1+√ƒ—g√−1;

and the other two will contain the difference

ƒ + g√ 1-Vf-g√-1.

THE DIOPHANTINE ANALYSIS.

386. THIS branch of analysis derives its name from its inventor, Diophantus, of Alexandria, in Egypt,, who flourished about the year 360, A.D. It relates chiefly to the finding of square and cube numbers.

The solutions of the questions must frequently be left, notwithstanding the various rules that have been given for this purpose, to the talents and ingenuity of the learner, who, in pursuing these inquiries, will soon perceive that nothing less than the most refined algebra, applied with great skill and judgment, can surmount the various difficulties which attend them; and, in this respect, no one, perhaps, has ever excelled Diophantus, or discovered a greater knowledge of the extent and resources of the analytic art.

When we consider his work with attention, we are at a loss which to admire most, his singular sagacity, and the peculiar artifices he employs in forming such positions as the nature of the problems requires, or the more than ordinary subtilty of his reasoning upon them.

Every particular question puts us upon a new way of thinking, and furnishes a fresh vein of analytical treasure, which can not but prove highly useful to the mind in conducting it through other difficulties of this kind whenever they occur, and also in enabling it to encounter more readily those that may arise in subjects of a different nature.

The following directions for resolving questions in the Diophantine analysis