LESSONS IN ALGEBRA.-No. XXIV.
(Continued from page 454.)

REDUCTION OF EQUATIONS BY INVOLUTION.

IN an equation, the letter which expresses the unknown quantity is sometimes found under a radical sign. We may have xa.

To clear this of the radical sign, let each member of the equation be squared, that is, multiplied into itself. We shall then have × √ x = aa. Or, x = a2.

The equality of the sides is not affected by this operation, because each is only multiplied into itself, that is, equal quantities are multiplied into equal quantities.

The same principle is applicable to any root whatever. If "√xa; then x=a. For a root is raised to a power of the same name, by removing the index or radical sign. Hence,

To reduce an equation when the unknown quantity is under a radical sign.

Involve both sides to a power of the same name, as the root expressed by the radical sign.

N.B. It will generally be expedient to make the necessary transpositions, and to clear the equation of fractions, before involving the quantities; so that all those which are not under the radical sign may stand on one side of the equation.

=

4V (6x)+6°

REDUCTION OF EQUATIONS BY EVOLUTION.

In many equations the letter which expresses the unknown quantity is involved to some power. Thus,

In the equation x2 = 16,

We have the value of the square of x, but not of x itself.
If the square root of both sides be extracted,
We shall have x=4.

The equality of the members is not affected by this reduction. For if two quantities or sets of quantities are equal, their roots are also equal.

If (x+a)n=b+h, then x+a="√b+h. Hence,

To reduce an equation when the unknown quantity is a power. Extract the root of both sides which corresponds with the power expressed by the index of the unknown quantity.

1. Reduce the equation 6+x2-8=7

By transposition,

By evolution

The signs and

x2=7+8-6=9

x=±√9=+3. Ans.

are both placed before 9, because an even root of an affirmative quantity is ambiguous. 2. Reduce the equation 5.2 Transposing, etc., By evolution,

30= x2+34 x2= 16 x=4.

Ans.

5. Reduce the equation

6. Reduce 3 + 2√√x −

a2+ √x = = 6.

3+ d V(a2+vx)

=

8. Reduce the equation (x − 1) =

9. Reduce the equation 10. Reduce the equation 11. Reduce the equation 12. Reduce the equation

a + b

√x2-11 = 5. y2 — 4ab — a — (13+ √23 + y2)2 = 5. (3+ 3√ 329+x2) * = 144.

3

PROBLEMS.

2

And

Restoring the numbers,

That is

Proof.

x= = (c + b)2 — a x=(237163)2 - 22577 160000-22577137423. 137423+22577-163=237. When an equation is reduced by extracting an even root of a quantity, the solution does not always determine whether the answer is positive or negative. But what is thus left ambiguous by the algebraic process, is frequently settled by the statement of the problem.

Prob. 3. A merchant gains in trade a sum to which 320 pounds bears the same proportion as five times this sum does to 2500. What is the amount gained?

Prob. 4. The distance to a certain place is such, that if 96 be subtracted from the square of the number of miles, the remainder will be 48. What is the distance?

Prob. 5. If three times the square of a certain number be divided by 4, and if the quotient be diminished by 12, the remainder will be 180. What is the number?

Prob. 6. What number is that, the fourth part of whose square being subtracted from 8, leaves a remainder equal to 4? Prob. 7. What two numbers are those, whose sum is to the greater as 10 to 7; and whose sum multiplied into the less produces 270?

Prob. 8. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128?

Prob. 9. It is required to divide the number 18 into two such parts, that the squares of those parts may be to each

other as 25 to 16.

Prob. 11. What two numbers are as 5 to 4, the sum of whose cubes is 5103.

Prob. 12. Two travellers, A and B, set out to meet each other, A leaving the town C at the same time that B left D. They travelled the direct road between C and D; and on meeting, it appeared that A had travelled 18 miles more than B, and that A could have gone B's distance in 15 days, but B would have been 28 days in going A's distance. Required the distance between C and D.

Prob. 13. Find two numbers which are to each other as 8 to 5, and whose product is 360.

Prob. 14. A gentleman bought two pieces of silk, which together measured 36 yards. Each of them cost as many shillings per yard as there were yards in the piece, and their whole prices were as 4 to 1. What were the lengths of the pieces?

Prob. 15. Find two numbers which are to each other as 3 to 2; and the difference of whose fourth powers is to the sum

of their cubes as 26 to 7.

Prob. 16. Several gentlemen made an excursion, each taking the same sum of money. Each had as many servants attending him as there were gentlemen; the number of crowns which each had was double the number of all the servants, and the whole sum of money taken out was 3456 crowns. How many gentlemen were there?

Prob. 17. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished four times as many men as there were companies in the whole regiment; but these being found insufficient, each company furnished three men more; when their number was found to be increased in the ratio of 17 to 16. How many companies were there in the regiment?

ADFECTED QUADRATIC EQUATIONS Equations are divided into classes, which are distinguished from each other by the power of the letter that expresses the unknown quantity. Those which contain only the first power of the unknown quantity are called simple equations, or equations of the first degree. Those in which the highest power of the unknown quantity is a square, are called quadratie, or equations of the second degree; those in which the highest power is a cube, are called cubic, or equations of the third degree, etc.

Thus xab, is an equation of the first degree.
x2= c,
and x2 +ax=d,

are quadratic equations, or equations of the second degree.
x3 = h, and x3+ax2 + bx
=d,
are cubic equations, or equations of the third degree.
Equations are also divided into pure and adfected equations. A
pure equation contains only one power of the unknown quantity.
This may be the first, second, third, or any other power. An
adfected equation contains different powers of the unknown
quantity. Thus,

Sx2= d. - b, is a pure quadratic equation.

x2 + bx= d, an adfected quadratic equation. c, a pure cubic equation.

23

x3 + ax2 + bxh, an adfected cubic equation.

In a pure equation, all the terms which contain the unknown quantity may be united in one, and the equation, however complicated in other respects, may be reduced by the rules which have already been given. But in an adfected equation, as the unknown quantity is raised to different powers, the terms containing these powers cannot be united.

An adfected quadratic equation is one which contains the unknown quantity in one term, and the square of that quantity in another

term.

The unknown quantity may be originally in several terms of the equation. But all these can be reduced to two, one containing the unknown quantity, and the other its square.

It has already been shown that a pure quadratic is solved by extracting the root of both sides of the equation. An adfected quadratic may be solved in the same way, if the member which contains the unknown quantity is an exact square.

Thus the equation x2+2ax + a2 = bh, may be reduced by evolution. For the first member is the square of a binomial quantity. And its root is x+a. Therefore,

x+a=√b+h, and by transposing a,

x = √b + h -f.

But it is not often the case, that the member of an adfected quadratic containing the unknown quantity, is an exact square, till an additional term is applied, for the purpose of making the required reduction.

In the equation x2+2ax= b, the side containing the unknown quantity is not a complete square. The two terms of which it is composed are indeed such as might belong to the square of a binomial quantity. But one term is wanting. We have then to inquire, in what way this may be supplied. From having two terms of the square of a binomial given, how shall we find the third?

Of the three terms, two are complete powers, and the other is twice the product of the roots of these powers, or, which is the same thing, the product of one of the roots into twice the other.

In the expression +2ax, the term 2ax consists of the factors 2a and x. The latter is the unknown quantity. The other factor 2a may be considered the co-efficient of the unknown quantity; a co-efficient being another name for a factor. As x is the root of the first term 2, the other factor 2a is twice the root of the third term, which is wanted to complete the square. Therefore half of 2a is the root of the deficient term, and a2 is the term itself.

The square completed is x2+2ax+a2, where it will be seen that the last term a' is the square of half of 2a, and 2a is the co-efficient of x, the root of the first term.

In the same manner it may be proved that the last term of the square of any binomial quantity is equal to the square of half the co-efficient of the root of the first term.

1 46 2.

In these, and similar instances, the root of the third term of the completed square is easily found, because this root is the same half co-efficient from which the term has just been derived. Thus, in the last example, half the co-efficient of 1 x is and this is the root of the third term 26, When the first power of the unknown quantity is in several terms, these should be united in one, if this can be done by the rules for reduction in addition. But if there are literal coefficients, these may be considered as constituting, together, a compound co-efficient or factor, into which the unknown quantity is multiplied.

Thus, ax + bx+ dx= :=(a+b+d) ×x. The square of half this compound co-efficient is to be added to both sides of the equation.

9. Reduce the equation

Uniting terms
Completing the square
And

10. Reduce the equation

Therefore

x2+3x+2x+x=d x2+6x=d x2+6x+9=9+ d x=- ·3±√9+d. Ans. x2+ax+bx} x2+(a+b) x x = h

2

x2 + (a + b) x x + ( " + "') = ( ~ +13) " + %.

11. Reduce the equation x2+ax-x=b.

LESSONS IN GREEK.-No. LIII,
BY JOHN R. BEARD, D.D.
MOODS.

THE moods represent the circumstances under which the subject is united with the verb, or the manner in which the affirmation of the verb is made. In simple propositions the indicative, the subjunctive, and the optative moods are employed; the imperative is a form by itself, since the imperative does not make a simple statement.

The indicative denotes that in consequence of an apprehension of the mind, whether the object apprehended be material or intellectual, the subject is simply applied to the verb, and therefore serves to express historical events and absolute facts; its office being merely to indicate or declare a reality. In Greek the use of the indicative generally resembles its use in other languages. Some peculiarities, however, have to be set forth.

The indicative, in union with the particle av (KE, KEV), presents a condition whose realisation depends on circumstances, to which reference is made by the conjunction. This form of speech occurs partly in the historical tenses and partly in the future.

The indicative of the historical tenses in connection with av, denotes that a condition takes place as often as the requisite circumstances occur, consequently neither always nor merely

...

would or could... if; would or could have This usage may be represented by these English forms-but for, indicating that as the conditional circumstances did not (or do not) exist, so the contingency did not take place.

Frequently are these forms used-ɛideç av, you would have seen; nynow av, you would have thought; εyvw av rig, one would observe.

Sometimes for the expression of this sense you find the indicative without av, when the condition which, under certain circumstances, would have taken place, is represented as actually taking place. In most verbs this mode of expression is admissible only when a subordinate hypothetical sentence is subjoined to the principal sentence; e. g.

Ηισχυνόμην μεντοι, ει ὑπο πολεμιου εξηπατηθην

I should be ashamed indeed, if I were deceived by an enemy.
In a somewhat similar manner we use the indicative had; e.g.
He had gone, if the ship had arrived in time.
With the ideas of duty, will, and power, this use of the
indicative is common, without the hypothetical clause; as

Μενειν εξην τῳ κατηγοροῦντι τῶν αλλων
The accuser of others ought to remain.

So wishes, which are not fulfilled or cannot be fulfilled, take the indicative without av; e.g.

Σωκρατης πιστευων θεοις πως ουκ εἶναι θεους ενόμιζεν ; As Socrates believed in the gods, how did (could) he deny their existence?

The subjunctive and the optative moods denote that the predicate is applied to the subject according to a conception; consequently the relation here is one of dependence, the dependence may exist exclusively in the mind, or it may exist in the mind as represented by a word, that is, a verb or a conjunction. The subjunctive and the optative are in their essence very much alike, differing merely in regard to time; for the subjunctive sets forth the conception as at present lying in the mind of the speaker; the optative places that conception in the past. Thus, the subjunctive present in Greek corresponds with the subjunctive present in Latin, while the optative in Greek corresponds with the imperfect subjunctive in Latin.

In simple sentences the use of the subjunctive is very limited, and extends only to these instances :

1. Homer, and the Epic poets in general, put positive as well as negative propositions sometimes in the subjunctive. By this mode of expression it is intimated that the speaker has not a certain but only a doubtful idea of the matter in question; and consequently the form serves to set forth an opinion, or an undecided notion; e.g.

In this way certainty is intimated, and a positive assurance given; thus the example may be rendered

By no means will I leave thee; or,

Thou hast no need to fear I will leave thee;

where the statement is exemplified, that the subjunctive depends on a state of mind. The subjunctive is the mood of conception or idea, as the indicative is the mood of fact; as the latter implies independence, so the former implies dependence.

In sentences with un ou you are to understand or supply before the un, opa, see, or σкожε, consider. In this way you as good as say, "You must take care that this or that does not take place." The phrase, therefore, serves to express an undecided or doubtful denial.

Αλλα μη ουκ ή διδακτον αρετη
But virtue can scarcely be taught.
Μη ου τοῦτ' η χαλεπον, θανατον εκφυγεῖν
To avoid death may not be difficult.

That is, oкOTEL μn ov xaλeñor, etc., consider whether it is not easy to avoid death.

The optative, which represents a conception as lying to the speaker beyond the present, finds its proper application in the expression of a wish. Of the use of the optative in simple propositions there are two different forms, namely, the optative in and by itself (without av), the simple optative; and the optative with av. The optative without av represents the expression as the free act of the mind, and without any stands in a sentence subordinate to another, which is hyporeference to the province of reality. The optative with av thetical, and which arises from the nature of the case; consequently the conception appears to ensue from certain preof an eventual reality, since, if the required circumstances vailing circumstances. But the conception assumes the form occurred, then the consequent result would take place. Instances of these usages are the following:

Easily, I think, can a god, if willing, even at a distance save a man.

Hesitation ; opt. with ar: Ουκ αν εγω ταυτα φήσαιμι.

I could not, I think, affirm these things.

In the refined tone of Attic conversation the optative with av was a polite form of expression, by which convictions and determinations were set forth in a hesitating or qualified manner, and by which the harshness of direct requests was softened down; e. g.

Ωρα αν ειη πραττειν τα δέοντα
This is the time, I would submit, to do one's duty.
Λεγοις αν ά δει λεγειν

Be so good as to say what ought to be said.