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to be considered whether men are more at liberty in point of morals, to make themselves miserable without reason, than to make others so; or dissolutely to neglect their own greater good for the sake of a present lesser gratification, than they are to neglect the good of others whom nature has committed to their care. It should seem that a due concern about our own interest or happiness, and a reasonable endeavour to secure and promote it, is, I think, very much the meaning of the word prudence in our language-it should seem that this is virtue, and the contrary behaviour faulty and blameable; since in the calmest way of reflection, we approve of the first and condemn the other conduct, both in ourselves and others. This approbation and disapprobation are altogether different from mere desires of our own and their happiness, and from sorrow in missing it."

Again: "Without inquiring how far and in what sense virtue is resolvable into benevolence, and vice into the want of it, it may be proper to observe that benevolence and the want of it, singly considered, are in no sort the whole of virtue and vice. For if this were the case, in the review of one's own character, or that of others, our moral understanding and moral sense, it would be indifferent to everything but the degrees in which benevolence prevailed, and the degrees in which it was wanting. That is, we should neither approve of benevolence to some persons rather than others, nor disapprove of injustice and falsehood, upon any other account, than merely as an overbalance of happiness was foreseen likely to be produced by the first, and misery by the second. But now, on the contrary, suppose two men competitors for anything whatever which would be of equal advantage to each of them; though nothing indeed would be more impertinent than for a stranger to busy himself to get one of them preferred to the other, yet such endeavour would be virtue, in behalf of a friend or benefactor, abstracted from all consideration of distant consequences; as that examples of gratitude and friendship would be of general good to the world. Again, suppose one man should by fraud or violence take from another the fruit of his labour, with intent to give it to a third, who, he thought, would have as much pleasure from it as would balance the pleasure which the first possessor would have had in the enjoyment and his vexation in the loss; suppose that no bad consequences would follow, yet such an action would surely be vicious. Nay further, were treachery, violence and injustice not otherwise vicious than as foreseen likely to produce an overbalance of misery to society, then, if in any case, a man could procure to himself as great advantage by an act of injustice as the whole foreseen inconvenience likely to be brought upon others by it would amount to, such a piece of injustice would not be faulty or vicious at all." "The fact then appears to be, that we are constituted so as to condemn falsehood, unprovoked violence, and injustice, and to approve of benevolence to some rather than others, abstracted from all consideration of which conduct is likely to produce an overbalance of happiness or misery."

as a radical principle of his whole system. It will not therefore be necessary to make any distinct remarks on President Edward's theory.

LESSONS IN ARITHMETIC.-No. XXXII.
PERIODICAL, OR CIRCULATING DECIMAL.

DECIMALS which consist of the same figures or set of figures
repeated, are called PERIODICAL, OR CIRCULATING DECIMALS.

The repeating figures are called periods, or repetends. If one figure only repeats, it is called a single period, or repetend; as 11111, etc.; 33333, etc.

When the same set of figures recurs at equal intervals, it is called a compound period, or repetend; as 01010101, etc.;

123123123, etc.

If other figures arise before the period commences, the decimal is said to be a mixed periodical; all others are called pure, or simple periodicals. Thus 42631631, etc., is a mixed periodical; and 33333, etc., is a pure periodical decimal.

The danger of this theory is not by any means so great as that of the selfish scheme, because it comprehends a large part of actions which are truly virtuous. But all definitions of virtue which are not so comprehensive as to embrace the whole of moral excellence, are injurious; not only by leaving out of the catalogue of virtues such actions as properly belong to it, but by leaving men to form wrong conceptions of what is right and wrong, by applying a general rule, which is not correct, to practical cases. When it is received as a maxim that all virtue consists in seeking the happiness of the whole, and when a particular act seems to have that tendency, men are in danger of overlooking those moral distinctions by which our duty should be regulated. This effect has been observed in persons much given to theorise upon the general good as the end to be

aimed at in all actions.

1. When a pure circulating decimal contains as many figures as there are units in the denominator less one, it is sometimes

President Edwards has a treatise on Virtue, in which he enters very deeply into speculation on the principles of moral conduct. His definition of virtue has surprised all his admirers: it is, "the love of being as such." When, however, this strange definition comes to be explained, by himself and his followers, it amounts to the same as that which we have been considering, which makes all virtue to consist in disinterested benevolence.

called a perfect period, or repetend. Thus, 142857, etc.,
and is a perfect period.

called the terminats part of the fraction.
2. The decimal figures which precede the period, are often

Circulating decimals are expressed by writing the period once with a dot over its first and last figure when compound; and when single by writing the repeating figure only once with a dot over it. Thus 46135135, etc., is written 46135 and 33, etc., 3.

Dr. Samuel Hopkins, who was his pupil, and well understood his principles, gives this as his definition of virtue, and has it

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To investigate this problem, let us recur to the origin of circulating decimals, or the manner of obtaining them. For example, 11111, etc., or 1; therefore the true value of 1-1111, etc., or 'i, must be from which it arose. For the same reason ⚫22222, etc., or 2, must be twice as much or ; ·33333, etc., or 3; 4; 5: =, etc.

Again, 010101, etc., or oi; consequently 010101» etc., or 01; 020202, etc., or '02 = ; ·030303, etc., or 03; 070707, etc., or 07, etc. 001001001, etc., or 001; therefore 001001, etc., or ··001

.002

=; etc.

So also =

multiplying the numerator and denominator of by 142857, In like manner =·142857; and 1428571888; for, we have 1531. So is twice as much as; three times as much, etc. Thus it will be seen that the value of a pure periodical decimal is expressed by the common fraction whose numerator is the given period, and whose denominator is as many 9s as there are figures in the period. Hence,

be

To reduce a pure circulating decimal to a common fraction.
Make the given period the numerator, and the denominator will
as many 9s as there are figures in the period.
Ex. 1. Reduce 3 to a common fraction.

2. Reduce 6 to a common fraction.

3. Reduce 18 to a common fraction. 4. Reduce 123 to a common fraction.

Ans. 1, or .

Ans. 8, or 3.

5. Reduce 297 to a common fraction.
6. Reduce 72 to a common fraction.
7. Reduce '09 to a common fraction.
8. Reduce 045 to a common fraction.
9. Reduce 142857 to a common fraction.

10. Reduce 076923 to a common fraction.

CASE II. To reduce mixed circulating decimals to common frac tions.

11. Reduce 16 to a common fraction.

Analysis.-Separating the mixed decimal into its terminate and periodical part, we have ·16·1+06. Now •1=; and 06; for, the pure period 6, and since the mixed period 06, begins in hundredths' place, its value is 6 Therefore evidently only as much; but §÷10= 161+%. Now and %, reduced to a common denominator and added together, make 18, or . Ans.

90.

In mixed circulating decimals, if the period begins in hundredths' place, it is evident from the preceding analysis that the value of the periodical part is only as much as it would be if the period were pure or begun in tenths' place; when the period begins in thousandths' place, its value is only To part as much, etc. Thus ·6 = §; ·06 = 10;

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8567838

0999000 Ans.

The

Note.-1. The reason of this operation may be shown thus: 85679238500000+ 67923. Now 85000008567923 is equal to 8567923-85,

2. It is evident that the required denominator is the same as that of the periodical part; for, the denominator of the periodical part is the least common multiple of the two denominators. Hence,

To reduce a mixed circulating decimal fraction.

Change both the terminate and periodical part to common fractions separately, and their sum will be the answer required.

Or, from the given mixed periodical, subtract the terminate part, and the remainder will be the numerator required. The denominator is always as many Is as there are figures in the period, with as many ciphers annexed as there are decimals in the terminate part.

PROOF.-Change the common fraction back to a decimal, and if the result is the same as the given circulating decimal, the work is

13. Reduce 138 to a common fraction. Ans. 13, or 38

14. Reduce 53 to a common fraction.

15 Reduce 5925 to a common fraction.

16. Reduce 583 to a common fraction. 17. Reduce 0227 to a common fraction. 18. Reduce 4745 to a common fraction. 19. Reduce 5925 to a common fraction.

20. Reduce 008497133 to a common fraction.

CASE III.-Dissimilar periodicals reduced to similar and conterminous ones.

In changing dissimilar periods, or repetends, to similar and conterminous ones, the following particulars require attention. 1. Any terminate decimal may be considered as interminate by annexing ciphers continually to the numerator. Thus 46 460000, etc. = 460.

etc.

2. Any pure periodical may be considered as mixed, by taking the given period for the terminate part, and making the given period the interminate part. Thus ·46 = ·46 + ·0046, 3. A single period may be regarded as a compound periodical. Thus 3 may become 33, or 333; so '63 may be made 6333, or '63333, etc.

4. A single period may also be made to begin at a lower order, regarding its higher orders as terminate decimals. Thus .3 may be made 33, or 3333, etc.

5. Compound periods may also be made to begin at a lower order. Thus 36 may be changed to 363, or 36363, etc.; or by extending the number of places 479 may be made 47979, or 4797979, etc.; or making both changes at once 532 may be changed to 5325325, etc. Hence,

To make any number of dissimilar periodical decimals similar.

Move the points, so that each period shall begin at the same order as the period which has the most figures in its terminate

part.

21. Change 6814, 3.26, and 083 to similar and conterminous periods.

Operation. 6.8146.81481481

3.263.26262626

*083008333333

Having made the given perio ls similar, the next step is to make the m conterminous. Now as one of the given periods contains 3 figures, another 2, and the other 1, it is evident the new periodical must contain a number of figures which is some multiple of the number of figures in the different periods; viz: 3, 2, and 1. therefore But the least common multiple of 3, 2, and 1 is 6; the new periods must at least contain 6 figures. Hence, To make any number of dissimilar periodical decimals similar and conterminous.

First make the periods similar; then extend the figures of each to as many places as there are units in the least common multiple of the NUMBER of periodical figures contained in each of the given

decimals.

22. Change 46 162, 5·26, 73-423, ·486, and 12-5, to similar and conterminous periodicals. Operation.

46 16246-10216216 5.26 = 5.26262626 63-42363 42342342 •486 = 0·486666666 12-5 = 12·50000000

The numbers of periodical figures in the given decimals are 3, 2, 3, and 1; and the least common multiple of them is 6. Therefore the new periods must each have 6 figures.

23. Make 27, 3, and 015 similar and conterminous. 24. Make 4-321, 6·4263, and 6 similar and conterminous.

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First make the given decimals similar and conterminous. Then add the periodical parts as in simple addition, and since there are six figures in the period, divide their sum by 999999; for this would be its denominator, if the sum of the periodicals were expressed by a common fraction. Setting down the remainder for the repeating decimals, carry the quotient 1 to the next column, and proceed as in addition to whole numbers. Hence,

Ans. 104-0193648

We derive the following general

RULE FOR ADDING CIRCULATING DECIMALS. First make the periods similar and conterminous, and find their cum as in Simple Addition. Divide this sum by as many 93 as there are figures in the period, set the remainder under the figures added for the period of the sum, carry the quotient to the next column, and proceed with the rest as in Simple Addition,

If the remainder has not so many figures as the period, ciphers must be prefixed to make up the deficiency.

Another method of adding circulating decimals is, to reduce each to its equivalent vulgar fraction, and then add, reducing the result to a circulating decimal, if required.

2. What is the sum of 24·132+2·23+85·24+67·6' ? 3. What is the sum of 328 126+81-23+5·624+61·6? 4. What is the sum of 31-62+7-824+8·392+027 ? 5. What is the sum of 462·34+60-82+71·161+35 ? 6. What is the sum of 60-25+34+6-435‍+·45+45·24 ? 7. What is the sum of 9-814+1 5+87-26+083+124·09? 8. What is the sum of 3-6-478-3176+735·8+375+·27+ 187-4?

9. What is the sum of 5391-357+72·38+187·2i+4·2965+ 217·8196+42176·523+58·30018?

10. What is the sum of •162+131·05+2·95+97·26+3·769230 +99083+15+814?

SUBTRACTION OF CIRCULATING DECIMALS.
Ex. 1. From 52.86 take 8.37235.

Operation. 52.8652.86868 8.37235 8.37235

=

We first make the given decimals eimilar and conterminous, then subtract as in whole numbers. But since the period in the lower line is larger than that above it, we must borrow 1 from the next higher order. This will make the right hand figure of the remainder one less than if it was a terminate Decimal. Hence,

44:49632

We derive the following general

RULE FOR SUBTRACTING CIRCULATING DECIMALS Make the periods similar and conterminous, and subtract as in whole numbers. If the period in the lower line is larger than that above it, diminish the right hand figure of the remainder by 1.

The reason for diminishing the right hand figure of the remainder by 1, if the period in the lower line is larger than that above it, may be explained thus:

When the period in the lower line is larger than that above it, we must evidently borrow 1 from the next higher order, Now if the given decimals were extended to a second period, in this period the lower number would also be larger than that above it, and therefore we must borrow 1. But having borrowed one in the second period, we must also carry one to the

next figure in the lower line, or, what is the same in effect, diminish the right hand figure of the remainder by 1.

Another method of subtracting circulating decimals is, to reduce each to its equivalent vulgar fraction, and then subtract, reducing the result to a circulating decimal, if required.

2. From 85-62 take 13.76432. Ans. 71-85193.

3. From 476 32 take 84-7697.
4. From 3.8564 take '0382.
5. From 46 123 take 41.3.

6. From 801-6 take 400-75.
7. From 4.7824 take ⚫87.

8. From 1419-6 take 1200.9.

9. From 634852 take '021.

10. From 8482-421 take 6031-035.

MULTIPLICATION OF CIRCULATING DECIMALS.

Ex. 1. What is the product of 36 into ⚫25?
Operation.
•36=33=14
•25= +6%= 38.
Now X 38=356

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92

092 Ans.

We first reduce the given periodicals to common fractions; then multiply them together. Finally, we reduce the product to a periodical decimal. Hence,

We derive the following general

RULE FOR MULTIPLYING CIRCULATING DECIMALS. First reduce the given periodicals to common fractions, and multiply them together as usual. Finally, reduce the product to decimals and it will be the answer required.

If the numerators and denominators have common factors, the operation may be contracted by cancelling those factors before the multiplication is performed.

2. What is the product of 37.23 into 26? Ans. 9 928.

3. What is the product of 123 into &?

4. What is the product of 245 into 7-3 ?

5. What is the product of 24 6 into 15·7 ?
6. What is the product of 48-23 into 16.13?
7. What is the product of 8574-3 into 87.5?
8. What is the product of 3·973 into 8 ?

9. What is the product of 49640 54 into 70503?
10. What is the product of 7-72 into 297 ?

DIVISION OF CIRCULATING DECIMALS.
Ex. 1. Divide 234-6 by 7.

Now 78 And 3

Operation. 23462343 794 •j = 1

= 2? 1 × ? = $329 301-714285 Ans.

We first reduce the divisor and dividend to common fractions, and divide one by the other; then reduce the quotient to a decimal.

Hence, we derive the following general

RULE FOR DIVIDING CIRCULATING DECIMALS. Reduce the divisor and dividend to common fractions; divide one fraction by the other, and reduce the quotient to decimals. After the divisor is inverted, if the numerators and denominators have factors common to both, the operation may contracted by cancelling those factors.

2. Divide 319-28007112 by 764-5. Ans, 0·4176325.
3. Divide 18-56 by 3.

4. Divide 6 by 123,

be

5. Divide 2-297 by 297.

6. Divide 750730-518 by 87.5. 7. Divide 54 by 15.

8. Divide 13-5169533 by 4.297.

9. Divide 24-081 by 386.

10. Divide 36 by 25.

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 quantitics are multiplied into equal quantities.

The same principle is applicable to any root whatever. If √xa; then an. 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.

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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+ab+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

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x=±√9=+3. Ans.

The signs and are both placed before 9, because an even root of an affirmative quantity is ambiguous.

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Proof. √137423+22577-163237. 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. 10. It is required to divide the number 14 into two such parts that the quotient of the greater divided by the less, may be to the quotient of the less divided by the greater as

16 to 9.

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 quadratic, 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 x = a + b, is an equation of the first degree.
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,

(x2= d — b, is a pure quadratic equation.
x2 + bx = d, an adfected quadratic equation.
√x3 = b - -c, a pure cubic equation.

2:3

+ ax2 + bx = h, 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 coll taining 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 + 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 − a.

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 is the root of the first term a, 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 terin, and a2 is the term itself,

The square completed is 2+2axa, 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.

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