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.

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 DECIMALS which consist of the same figures or set of figures the other conduct, both in ourselves and others. This appro-repeated, are called PERIODICAL, OR CIRCULATING DECIMALS. The repeating figures are called periods, or repetends. If one bation and disapprobation are altogether different from mere desires of our own and their happiness, and from sorrow in figure only repeats, it is called a single period, or repetend; as missing it."

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

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

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

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.

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 tions. likely to be brought upon others by it would amount to, such a piece of injustice would not be faulty or vicious at all." 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.”

"The

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.

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.

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

Similar periods are such as begin at the same place before or after the decimal point; as i and ·3, or 2·34 and 3·76, Dissimilar periods are such as begin at different places; as 123 and 42325.

etc.

Similar and conterminous periods are such as begin and end in the same places; as 2321 and 1634.

REDUCTION OF CIRCULATING DECIMALS.

CASE I. To reduce pure circulating decimals to common frac

To investigate this problem, let us recur to the origin of circulating decimals, or the manner of obtaining them. For example, 11111, etc., or i; therefore the true value of 1.1111, etc., or 1, 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 = 1; •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. So also 001001001, etc., or .001; therefore '001001, etc., or ·001=535i 002 = 5; etc.

999999

In like manner =·142857; and 142857 =88855; for, multiplying the numerator and denominator of by 142857, we have 3. 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 98 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 Is 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. 3, or 1.
Ans., or

LESSONS IN ARITHMETIC.

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.

13. Reduce 138 to a common fraction. Ans. 15, or 's

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 conter

minous ones.

Analysis.-Separating the mixed decimal into its terminate and periodical part, we have 16·1·06. Now·1=; In changing dissimilar periods, or repetends, to similar and 06; for, the pure period 6, and since the and conterminous ones, the following particulars require attenmixed period 06, begins in hundredths' place, its value is tion. 1. Any terminate decimal may be considered as interminate Therefore evidently only as much; but ÷ 10. by annexing ciphers continually to the numerator. Thus 46 161+5%. Now and, reduced to a common deno-460000, etc. = 460. minator and added together, make 8, or . Ans.

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 = €; ·0610;

0061008, etc.

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 4646+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 Hence, the denominator of the periodical part of a mixed cir- order, regarding its higher orders as terminate decimals. Thus culating decimal, is always as many 9s as there are figures in.3 may be made 33, or 3333, etc. the period with as many ciphers annexed as there are decimals in the terminate part.

12. Reduce ⚫8567923 to a common fraction. + 5899980. Solution.-Reasoning as before 8567923 = Reducing these two fractions to the least common denominator, 90992 8422218 whose denominator is the same as x 8567839 Ans. that of the other. Now $99938+ 585

85

Contraction.

8500000

85

9999900

8499915 1st Nu. 67923 2nd Nu. 8567838

9999900 Ans.

8499915

67923 9999900

=

9999900.

To multiply by 99999, annex as many ciphers to the multiplicand as there are 9s in the multiplier, etc. This gives the numerator of the first fraction or terminate part, to which add the numerator of the second or periodical part, and the sum will be the numerator of the answer. The denominator is the same as that of

the second or periodical part. Second Method.

8567923 the given circulating decimal.

85 the terminate part which is subtracted. 8567838 the numerator of the answer.

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 9s 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 right.

Having made the given periods similar, the next step is to make them 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. But the least common multiple of 3, 2, and 1 is 6; therefore 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.

ADDITION OF CIRCULATING DECIMALS.

next figure in the lower line, or, what is the same in effect,

Ex. 1 What is the sum of 17-23 + 41-2476 +86 +1.5 diminish the right hand figure of the remainder by 1. +35-423?

Operation.

Dissimilar. Sim. & Conterminous. 17-23 17-2323232 41.247641.2476476

8.61

= 8.6161616

1.5 = 1.5000000 35-42335-4232323 Ans. 104-0193648

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,

We derive the following general

RULE FOR ADDING CIRCULATING DECIMALS. First make the periods similar and conterminous, and find their sum as in Simple Addition. Divide this sum by as many 9s 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+8392+027? 5. What is the sum of 462·34+60-82+71·164+‍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+78-3176+735·3+375+ 27+ 187-4?

9. What is the sum of 5391·357+72·38+187·2i+4·2965+ 217.8496+42-176·523+58·30048 ?

10. What is the sum of 162+134·09+2·93+97·26+3·769230 +99-083+15+814?.

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

Operation. 52.8652.86868 8.37235 8.37235°

=

But

We first make the given decimals similar and conterminous, then subtract as in whole numbers. 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

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.

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.

Operation. 234.6234 = 204

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

10 X = 0316
3 301-714285 Ans.
Hence, we derive the following general

And

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 denomi

nators have factors common to both, the operation may be
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.