As the value of a fraction depends only on the relation of the numerator to the denominator, to find the numeral value of x, it will be sufficient to seek the relation of m to M, of R to r, and of D to d.

m 43

M=4000 miles nearly, and m=2150; hence, M 80

That is, in round numbers, the light of the full moon is twentyseven thousand six hundred times the light of Mars, when that planet is brightest, in its opposition to the sun.

We will add one more example by the way of farther illustration.

What comparative amount of solar light is reflected to the earth by Jupiter and Saturn, when those planets are in opposition to the sun;-the relative diameter of Jupiter being to that of Saturn as 111 to 83, and the relative distances of the Earth, Jupiter and Saturn, from the sun, being as 10, 52 and 95, respectively?

Ans. Taking the light reflected by Saturn for unity, that by Jupiter will be expressed by 24.45 nearly.

The philosophical student will readily perceive a more extended application of these principles to computing the relative light reflected to us by the different planets; but we have gone to the utmost limit of propriety, in an elementary work like this.

From Art. 94th to the end of this chapter can hardly be said to be algebra; it is natural philosophy, in which the science of algebra is used; however, we would offer no apology for thus giving a glimpse of the utility, the cui bono, and the application of algebraic science.

SECTION IV.

QUADRATIC EQUATIONS.

CHAPTER I.

(Art. 96.) Quadratic equations are either simple or compound. A simple quadratic is that which involves the square of the unknown quantity only, as ax2-b; which is one form of pure equations, such as have been exhibited in the preceding chapter.

Compound quadratics, or, as most authors designate them, adfected quadratics, contain both the square and the first power of the unknown quantity, and of course cannot be resolved as simple equations.

All compound quadratic equations, when properly reduced, may fall under one of the four following forms:

(1) x2+2ax=b

(2) x2-2ax-b

(3) x2-2ax=-b

(4) x2+2ax=-b

If we take x+a and square the sum, we shall have x2+2ax+a2

If we take x-a and square, we shall have

x2-2ax+a2

If we reject the 3rd terms of these squares, we have

x+2ax, and x2-2ax

The same expressions that we find in the first members of the four preceding theoretical equations.

It is therefore obvious that by adding a to both sides of the preceding equations, the first members become complete squares. But in numeral quantities how shall we find the quantity corresponding to a2? We may obtain a2 by the formal process of taking half the coefficient of the first power of x, or the half of 2a or -2a, which is a or -a, the square of either being a2.

Hence, when any equation appears in the form of x2±2ax= b,we may render the first member a complete square, and effect a solution by the following

RULE. Add the square of half the coefficient of the lowest power of the unknown quantity to the first member to complete its square; add the same to the second member to preserve the equality.

Then extract the square root of both members, and we shall have equations in the form of

x±a=±√b+a2

Transposing the known quantity a and the solution is accomplished.

In this manner we find the values of x in the four preceding equations, as follows:

(1) x=−a±√o+a2

(2) x= a±√b+a2

(3) x= a±√a2—b

(4) x=—a±√ a2 — t

When b is greater than a2 equations (3) and (4) require the square root of a negative quantity, and there being no roots to negative quantities, the values of x in such cases are said to be imaginary.

The double sign is given to the root, as both plus and minus will give the same power, and this gives rise to two values of the unknown quantity; either of which substituted in the original equation will verify it.

After we reduce an equation to one of the preceding forms, the solution is only substituting particular values for a and b; but in many cases it is more easy to resolve the equation as an original one, than to refer and substitute from the formula.

(Art. 97.) We may meet with many quadratic equations that would be very inconvenient to reduce to the form of x2+2ax=b; for when reduced to that form 2a and b may both be troublesome fractions.

Such equations may be left in the form of

ax2+bx=c

An equation in which the known quantities, a, b, and c, are all whole numbers, and at least a and b prime to each other.

There

We now desire to find some method of making the first member of this equation a square, without making fractions. We therefore cannot divide by a, because b will not be divided by a, the two letters being prime to each other by hypothesis. But the first term of a binomial square is always a square. fore, if we desire the first member of our equation to be converted into a binomial square, we must render the first term a square, and we can accomplish that by multiplying every term by a.

The equation then becomes

a2x2+bax=ca

Put y=ax. Then y2+by=ca

Complete the square by the preceding rule, and we have

We are sure the first member is a square; but one of the terms is fractional, a condition we wished to avoid; but the denominator of the fraction is 4, a square, and a square multiplied by a square produces a square.

Therefore, multiply by 4, and we have the equation

4y2+4by+b2=4ca+b2

An equation in which the first member is a binomial square and not fractional.

If we return the values of y and y2 this last equation becomes

4a2x2+4abx+b2=4ac+b2

Compare this with the primitive equation

ax2+bx=c.

We multiplied this equation first by a, then by 4, and in addition to this we find b2 on both sides of the rectified equation, b being the coefficient of the first power of the unknown quantity. From this it is obvious that to convert the expression ax2+bx into a binomial square, we may use the following

RULE 2. Multiply by four times the coefficient of x2, and add the square of the coefficient of x.

To preserve equality, both sides of an equation must be mul

tiplied by the same factors, and the same additions to both sides. We operate on the first member of an adfected equation to make it a square, we operate on the second member to preserve equality.

(Art. 98.) For the following method of avoiding fractions in completing the square, the author is indebted to Professor T. J. Matthews, of Ohio.

Resume the general equation ax2+bx=c

Now when b is even, we can complete the square by the first rule without making a fraction. In such cases this transformation is very advantageous.

When b is not even, multiply the general equation by 2, and the coefficient of x becomes even, and we have

Complete the square by the first rule, and we have

u2+2bu+b2=4ac+b2

An equation essentially the same as that obtained by completing the square by the rule under (Art. 97.); for we perceive the second member is the same as would result from that rule; hence this method has no superior advantage except when b is even, in the first instance.

(Art. 99.) The foregoing rules are all that are usually given for the resolution of quadratic equations; but there are some