508

jar with a neck wide enough to enable one to stick tin-foil all round the inside for the internal armature. The rod which goes through the cork is straight, and terminates below in a metallic chain, which puts it in communication with the tin-foil that forms the internal armature. A battery is a collection of several jars in a wooden box, fig. 408, communicating together, Fig. 408.

external armatures of the jars. The tin-foil also lines the sides of the box as far as the two metallic handles. The battery is charged, as seen in the figure, by bringing the internal armatures into communication with an electrical machine, and the external armatures with the ground by the wood of the box and the table on which it rests, or, still better, by a metallic chain attached to one of the metallic handles. An electrometer with a dial fixed to one of the jars, serves to indicate the charge of the battery. In spite of the large quantity of electricity collected in the apparatus, the electrometer diverges only slowly and a few degrees, which ought not to occasion surprise, since the divergence arises solely from the difference in the tension of the two armatures. The number of jars is generally four, six, or eight. The larger and more numerous they are, the more time is required to charge the battery. To discharge it, bring the two armatures into communication by means of the exciter, taking care to touch the external armature first. We must in this case use the glass-handled exciter, and take every precaution to avoid a shock, otherwise with a strong battery serious consequences might result. When we wish to strike down an animal, or any object whatever, we make use of the universal exciter represented in fig. 409. It is a small wooden box with two glass columns, on which copper rods are hinged. Between these columns is a glass support with a small plate, on which is placed the animal or object upon which we wish to experiment. The two copper rods being directed towards this object, one of them is made to communicate with the external armature of the battery, and the other with one of the balls of the glass-handled exciter. Then bringing the other ball of the exciter near the internal armature, a spark appears between this ball and the armature, and another between the branches of the universal exciter. It is the latter spark that strikes the object. Fig. 409.

internally by means of metal rods, and externally by tin-foil which lines the bottom of the box, and is in contact with the

is to consider habits and habitual actions as they partake of a moral character, or as they are the object of moral approbation or disapprobation. If we should remove from the list of moral actions all those which are prompted by habit, we should cut off the larger number of those which men have agreed in judging to be of a moral nature.

But the whole plausibility of their arguments depends on the pre-established connexion between happiness and a virtuous course of life. That true happiness is the natural effect of virtue, falls entirely short of proving that the essence of virtue consists in the tendency of certain actions to the person's true interest; whereas, when we perceive an action to be virtuous, we are conscious that it is not from any view of the connexion of the action with our own happiness that we approve of it; but our judgment is immediate, founded on a moral character perceived in the act itself. And in many cases virtue requires us to deny ourselves personal gratification for the sake of others. A man supremely governed by a regard to his own interest, is never esteemed a virtuous man by the impartial judgment of mankind. According to this theory, the only thing censurable in the greatest crimes is, that the guilty person has mistaken the best method of promoting his own happiness. Upon this principle, a man is at liberty to pursue his own interest at the expense of the happiness of thousands, and if he is persuaded that any action will tend to his own interest, he is at liberty to do it, whatever may be the consequences to others.

Dr. Paley adopts the principle that all virtue consists in a regard to our own happiness, taking into view the whole of our existence. His definition is, however, a very complicated one, and deserves to be analysed.

That there are virtuous habits and vicious habits, will scarcely be denied by any considerate person. A habit of lying, of swearing, of slandering, of cheating, of irreverence, of indolence, of vainglory, with many others, are, alas! too common. There are also virtuous habits, such as of industry, temperance, kindness, veracity, diligence, honesty, etc. To be sure, these virtues commonly flow from principle, but the practice of them is greatly facilitated by correct habits. Two considerations will show that men are properly accountable for those actions which proceed from habit. The first is, that in the formation of his habits man is voluntary. The acts by which they are formed are free acts, and the agent is responsible for all their consequences. The other consideration is, that habits may be counteracted and even changed by the force of virtuous resolutions and perseverance. Where habit has become inveterate, it may be difficult to oppose or eradicate it; but the strength of moral principle has often been found sufficient to counteract the most confirmed habits. When it is asserted that men long enslaved by evil habits cannot make a change, it is on the ground that no principle of sufficient power exists in the mind of the agent; but for that deficiency the man is responsible. Yet a power from without may introduce a new principle potent enough to overcome evil habits. The importance of possessing good habits is admitted by all moralists. Aristotle makes the essence of virtue to consist in "practical habits, voluntary in their origin," and agreeable to right reason. Dr. Thomas Reid, in his "Essay on the Active Powers," defines virtue to be "the fixed purpose to act according to a sense of duty," which definition Dugald Stewart Now every man desires to promote his own happiness, and modifies, by observing, "It is the fixed purpose to do what is according to Dr. Paley's theory, the only difference between an right, which evidently constitutes what we call a virtuous dispo-eminently good man and one of the opposite character is, that sition. But it appears to me that virtue, considered as an the one pursues a wiser course than the other; but they are attribute of character, is more properly defined by the habit both actuated by the same motives. which the fixed purpose gradually forms than by the fixed purpose itself." Dr. Paley lays it down as an aphorism, that "mankind act more from habit than reflection." "We are," says he, "for the most part, determined at once, and by an impulse which has the effect and energy of a pre-established habit." To the objection, "If we are in so great a degree passive under our habits, where is the exercise of virtue, or the guilt of vice?" he answers, "in the forming and contracting of these habits." "And hence," says he, "results a rule of considerable importance, viz. that many things are to be done and abstained from solely for the sake of habit."

THE NATURE OF VIRTUE.

The theories on this subject have been numerous, and contrary to one another. It is now proposed to mention some of the principal of them. We shall first mention the theory of Mr. Hobbes and his followers, who deny that there is any natural distinction between virtue and vice, and maintain that by nature all actions are indifferent, and that our ideas and feelings on the subject of morality are altogether the effect of education and association. Mr. Hobbes did indeed maintain that men are bound to obey the civil government under which they may happen to live, and to conform to the religion established by law, however contrary to their own private judgment. All moral duty, according to this theory, was resolved into the authority of the law of the land. As no natural moral rule existed, it was held that, except so far as a man was restrained by civil authority, he had a right to do what he pleased; and while he confined himself within these bounds, he need feel no concern about the consequences of his conduct.

Perhaps the most extraordinary system of virtue every promulgated was that of Mandeville, who maintained that all pretensions to virtue were mere hypocrisy, which men assumed from the love of praise. This writer forgot that hypocrisy assumes it as true that that which is counterfeited is an object of esteem and approbation among men. That virtue consists in the mere pursuit of pleasure, or of our own interest, is a system as old as Epicurus, and has had many abettors up to this time.

"Virtue," says he, "is the doing good to mankind, in obedience to the will of God, for the sake of everlasting happiness," according to which definition the good of mankind is the object, the will of God the rule, and everlasting happiness the motive of human virtue. If the question be asked, why we should seek the good of mankind, the answer is, from a regard to our everlasting happiness; and if the question be, why should we make the will of God the rule of our conduct, the answer must be the same; so that really all virtue is resolved into a regard to our own happiness.

This theory loses sight of all intrinsic difference between moral good and evil, and admits the principle that happiness is the only conceivable good, and that anything is virtuous the tendency of which is to promote our greatest happiness.

A theory the opposite of that which makes a regard to private interest the ground of virtue, is the one which makes all virtue to consist in a regard to the public good. This is the theory of Bishop Cumberland in his work, De Legibus, and is not essentially different from the scheme of those who make all virtue to consist in disinterested benevolence. No doubt, much that deserves the name of virtue consists in good will to others, and in contributing to their welfare; but it is not correct to confine all virtuous actions to the exercise of benevolence. We can conceive of benevolence in a being who has no moral constitution. Something of this kind is observable in brute animals, and atheists may exercise benevolence to their friends. The indiscriminate exercise of benevolence to creatures, without any respect to their moral character, might appear to be an amiable attribute, but it could not properly be called a moral attribute. A prudent regard to our own welfare and happiness is undoubtedly a virtue. It has been considered so by the wisest of men, and we know that prudence was one of the four cardinal virtues of the heathen. As the whole is made up of parts, it is evident that if it is a virtue to promote the well-being of the whole, it must be so of each of the parts. The pursuit of our own happiness where it does not infringe on the rights of others, has nothing evil in it, but is approved by every impartial mind. Some who maintain that all virtue consists in benevolence, admit that we may seek our own happiness just as we seek that of our neighbour; but the human constitution is not formed to exercise that abstract impartiality. While we are bound to promote the welfare of our neighbour and of strangers, our obligation is still stronger to endeavour to secure our own happiness; and if a friend and a stranger stand in equal need of a benefit which we have it in our power to bestow, it is evidently our duty to consult first the welfare of our friend, other things being equal.

What Bishop Butler has said on this subject in his short treatise on "Virtue," is worthy of consideration: "It deserves

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. bation and disapprobation are altogether different from mere The repeating figures are called periods, or repetends. If one 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.

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 false-
hood, 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 exam-
ples 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 over-
balance of happiness or misery."

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

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

123123123, etc.

decimal is said to be a mixed periodical; all others are called If other figures arise before the period commences, the 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.

called the terminals 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.

Similar periods are such as begin at the same place before or after the decimal point; as i and 3, or 2:34 and 376, 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 tions.

To investigate this problem, let us recur to the origin circulating decimals, or the manner of obtaining them. For example, 11111, etc., or 'i; 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 = 8, etc.

=

Again, 010101, etc., or òi; consequently 010101 etc., or 01; 020202, etc., or '02; 030303, etc., or 03; 070707, etc., or 07, etc. So also y .001001001, etc., or .001; therefore '001001, etc., or ·001=;

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 161+06. Now 1=; and 06; for, the pure period 6, and since the mixed period 06, begins in hundredths' place, its value is evidently only as much; but 10%. Therefore •1616+5%. Now and, reduced to a common denominator 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;

006 100=58, etc.

Note.-1. The reason of this operation may be shown thus: 85679238500000 + 67923. Now 8500000-85 + 67923 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 right.

511

Ans. 15, or

20. Reduce 008497133 to a common fraction.

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

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

etc.

taking the given period for the terminate part, and making the 2. Any pure periodical may be considered as mixed, by 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.

order, regarding its higher orders as terminate decimals. Thus 4. A single period may also be made to begin at a lower 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 6-814, 3.26, and 083 to similar and conterminous periods.

Operation. 6.8146.81481481 3.263.26262626 0830 08333333

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. 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 125, to similar and conterminous periodicals. Operation.

46 16246-10216216
5·26 = 5.26262626
63-42363 42342342
4860.486666666
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 045 similar and conterminous.
24. Make 4-321, 6·4263, and 6 similar and conterminous.