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11, COMPOUND VOWELS.

FRENCH ACCENTS.
They are thus called, because, being united together, each 17. The constant use of certain marks called ACCENTS in the
vowel loses its own simple sound, and helps to form another | French language, constitutes a marked peculiarity which can-
new sound. They form but one syllable, and are consequentiy not escape the attention of the student. Rarely, except in
pronounced by one emission of the voice.

elementary works of the English Language, is the syllable of
There are seven Compound Vowels, viz. :-

any given word which requires an emphasis, marked.

18. But it is not so in the French Language ; here, accents
ai

ei
oi and

of various kinds are constantly meeting the eye on every page.

One thing, however, must be observed, viz. :--the position of 12. DIPHTHONGS.

the Accent does not always and infallibly mark the syllable of a

word, which must receive the stress of voice in common pro-
They are thus called, because, though pronounced as one nunciation.
byllable, yet the sound of both vowel is distinctly heard,

19. Modern Grammarians have established the following
There are sit Diphthoogs, viz. :-

rule, viz.: To PLACE THE STRESS OF Voice ON THE LAST

PRONOUNCED SYLLABLE OF EVERY WORD. ia ie io

and ui

20, A slight inspection only of the following examples will
The following ten combinations of THREE SUCCESSIVE VOWELS illustrate the above remarks.
are also called Diphthongs, viz. :-

The first syllable of this word is marked with

an accent; must the stress of voice therefore be iai iau ieu

oui uai uei Dé-ro-rer placed upon the syllable de: No :-if the rule uie and

be applied to this word, the stress of voice falls

on the last syllable, HER. These Diphthongs are thus divided into syllables, viz. :

It will then be asked, What is the use of this accent: We i-ai i-au i-eu

ou-i u-ai

answer, It modifies the sound of the rouel over which it is placed. u-ei u-ie and u-eu,

Again :- the word used now as an ex

ample, has the same kind of an accent as the They must, however, be pronounced quickly, and as one

word used in the previous example had; syllable.

and also, it is placed over the same vowel. Sometimes, also, we find FOUR SUCCESSIVE Vowels in the Lé-gère-ment But it has another different accent over the game word, riz:

first vowel of the second syllable; and,

according to the rule, the stress of rcice is
ouai in the word jou-ai,

not placed either upon the first or second
oueu
jou-eur, and

syllable, but upon the last.
ouée
bou-ée.

This second accent (observe its form and position) only

serves to modify the sound of the vowel over which it is placed.
The first example-ouai, is composed of two compound Sometimes, however, an accent is placed over a vowel of the
vowels, viz. : ou and ai.

syllable, which, according to the rule, receives the stress of
The second exampleoueu, is also composed of two com- voice, viz. :- :-Cé-lé-bri-té.
pound vowels, viz. : ou and eu.

Again, in the word used here as an example,
In the last example-ouée, the final E is silent, and the

a third, and still different accent is placed over
three vowels are thus divided, viz.; ou and é.

Ba-ti-ment

the vowel A. Its presence affects the sound of

that vowel only. It has nothing whatever to do 13. Y.

with the proper accent of that word, as the term Accent is The vowel Y is frequently found combined with other understood when applied to words in the English language. vowels, but in such combinations it is never used as a diph. As a general rule, the stress of voice is not so strong in the

French as in the English language. thong. Its use in combination is peculiar, and will be fully

21. Accents, as used in the French language, are certain explained hereafter,

marks differing from each other, and placed over certain vouels

only, for specific purposes. 14. NASAL VOWEL SOUNDS.

22. There are three Accents, viz. :

called the Acute Accent, These are certain sounds produced by the combination of the

Grave
vowels, with the consonants M and N, viz. :

Circumflex,
im
in

23. 'The acute accent, is used only over the vowel E, and
yn;

serves two purposes : whose sounds will be explained hereafter.

First, - to modify its sound.

Secondly,–10 mark the existenie of a dietinct and final 15. NASAL DIPHTHONGAL SOUNDS.

syllable, viz. :These are certain sounds produced by the combination of

Dé,

Trom-ré, nasal vowel sounds with a vowel, not nasal, before them,

Pe-tar-dé,

Cér-é-mo-nic.

24. 'The Grave accent, is used only oper the vowels A, E and
ian ien ion

uin
and ouin. U, viz. :-

છે, Père, and
16. LIQUIDS.
The following combinations of the consonants are called

and serves two purposes :
Liquids, riz. :

First,-to modify the sound of the vowel E.

Secondly,-to distinguish one part of speech from another,
ll and
gn,

viz, :-
The sounds of these liquids are very common in the French a is a Preposition, a is a Verb, is an Adverb, la is an Article,
language, and will be explained hereafter,

qu is an Adverb, ou is a Conjunction,

am

em

om

um

ym and

an

un

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viz. :

uan

ouan

Où,

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Âge,

accent,

T

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26. ^ This character, which represents the Circumflex | words commencing with a vowel or H mute, and is much
accent, is the union of the Acute and Grave accents, and is used in the French language, viz. :-
placed over each of the Vowels except Y. It indicates that

L'ami instead of Le ami,
the letter over which it is placed, has a sound twice as long as it

L'église

La église.
has without it, viz. :-

L'homme Le homme.
Bête, Bûche, Cóte,

S'il

Si il,
Gîte, Mêlée, and Tête.

30. EUPHONIC T.
This accent also indicates the suppression of the letter S,
after the vowel over which it is placed ; thus

This letter is thus called on account of its peculiar position

between two parts of speech, viz., the Verb and Pronoun. It Bête, Fête, and Tête

does not sound agreeably to the French ear to say were formerly written

a elle, a il, demande on, etc. Beste, Feste and Teste.

Therefore, to prevent the hiatus of sound between the vowels

at the end of the first and the beginning of the second words
The S was not sounded, but gave to the preceding vowel in the examples, this Euphonic T is inserted. It is used only
that prolonged sound now represented by the circumflex in asking questions, and then a hyphen is placed both before
“Cassel's Lessons in French."

and after it, viz. :--
This accent also serves to distinguish parts of speech from a-t-elle ? a-t-il ? ira-t-on ? demande-t-on
each other, viz. :

parle-t-il ? va-t-on:

prouve-t-il :
Crû is a Participle from the Verb CROÎTRE,

This letter cannot be translated, because it has no meaning.
Cru is a Noun and Adjective.

It is thus used, merely for the sake of euphony.
Dû is a Participle from the Verb Devoir.
Du is an Article and Noun.

31. PARENTHESIS AND PUNCTUATION.
Redû a Participle from the Verb REDEVOIR.
Sur is an Adjective.

In the French language, the
Sur is a Preposition.

Parenthesis ()

Comma,

Semicolon ; Colon :
TO is a Participle from the Verb TAIRE.

Period. Mark of Interrogation ? Mark of Excla-
Tu is a Pronoun.

mation ! Dash and Quotation
26. CEDILLA.

are the same, and used for the săme purposes, as in the English

language. B-sides the three kinds of accents just enumerated, certain

32. ASTERISKS.
other marks or signs are used, called

The ASTERIAS in the French language, are also the same,
Cedilla, Dieresis, Hyphen, and Apostrophe. and used for the same purposes, as in the English language,

viz. :--
ç The Cedilla is a peculiar mark, somewhat resembling a
figure 5 inverted, and placed only under the letter C, before the

+

$ 1 vowels A, 0, and U, viz. : Ç.

It indicates that the leiter C under which it is placed, has
the soft sound of the Letter S, viz. :

LESSONS IN MORAL SCIENCE.-No. I.
çà pronounced as if printed Sà.
Deça

Desà.

CONSCIENCE, OR THIE MORAL FACULTY.
Façade

Fasade.
Façonner

Fasonner.

As all men, when reason is developed, have a faculty by
Leçon

Leson.

which they can discern a difference between objects of sight Reçu

Resu.

which are beautiful and those which are deformed, so all men

possess the power of discerning a difference between actions,
27. DIERESIS.

as to their moral quality. The judgment thus formed is imme-
The l'IERESIS consists of two dots, placed over the vowels diate, and has no relation to the usefulness or injuriousness to
E, I, and U. It shows that the vowel over which it is placed human happiness, of the objects contemplated.
is pronounced separately from the preceding vowel; thus

Whatever difference of opinion may exist respecting the
indicating, in reality, a distinct syllable, viz. :

origin of this faculty, it is universally admitted that men, in

all ages and countries, have judged some actions to be good Naïveté pronounced as if printed Na-ive-té.

and deserving of approbation, while they have judged others Ouïr

Ou-ir.

to be bad and of ill desert. Poëte

Po-ete.

In all languages, we find words expressive of the ideas of

moral excellence, and moral evil. In the laws and penalties 28. HYPHEN,

established in all ages throughout the world, it is evidently The Hyphen is a short horizontal mark, which is used to In cases of Hagrant injustice or ingratitude, all men, of every

implied that some actions ought to be done, and others avoided. connect words and syllables, viz. :

country and of every age, agree in their judgment of their moral A-t-il, Belles-Lettres, Celui-ci, Demi-kilomètre,

evil. There is, in regard to such actions, no more difference Fait-on, Suis-je, und Très-rarement.

in the judgment of men, than respecting the colour of grass,

or the taste of honey. If any man does not perceive grass to Its use in connecting syllables is precisely the same as in be green, or honey to be sweet, we do not thence conclude the English language; that is, when a word is divided, so that that men's bodily senses are not similarly constituted, but that a part of it is at the extreme right-hand of a page, and the the organs of the individual who does not see and taste as rest at the extreme lett.

other men do, are defective, or depraved by disease.

To determine whether all men have one original moral 29. APOSTROPHE.

faculty, tł:e case proposed for their moral judgment should be

siraply good or evil. For a complex act, in which there is · The APOSTRVAE is like a comma placed at the upper end something good and something evil, or rather where there of letters, instead of at the lower end, or at the bottom on a must be an accurate weighing of motives in order to ascertain line with the lower end.

the quality of the action, is not a proper test as to the existIts use is, to show the elision or cutting off a vowel before ence of a uniformity of moral judgment in men. Therefore,

66

UP

the historical fact adduced by Dr. Paley,* from the history of good parent. If a case like this were presented to a thousand Valerius Maximus, is not at all suited to his purpose; because persons, from as many different parts of the world, there would the case is very complex, and one on which it is difficult to be but one judgment and one feeling; all would judge the determine at first view, what the true inoral character of the conduct of the son to be blameable. Different degrees Os action is. The facts, as related by him, are as follows: The moral disapprobation would be felt by those whose moral father of Caius Toranius had been proscribed by the Trium- faculty was in a cultivated state; but there would be no differvirate. Caius Toranius-coming over to the interests of that sence in the opinion entertained of his conduct. All would party_discovered his father's place of concealment to the feel disapprobation, accompanied by a desire for the punishment officers who were in pursuit of him, and gave them, withal, a of the offender. It is found that savages appear to have but description of his person by which they might distinguish him. an obscure exercise of conscience, but in proportion as their The old man, more anxious for the safety and fortunes of his minds are cultivated, this faculty becomes more manifest, and son than for the little that might remain of his own life, began operates more forcibly. immediately to inquire of the officers whether his son were well, and whether he had done his duty to the satisfaction of THE MORAL FACULTY, ORIGINAL AND UNIVERSAL. the generals. “That son (replied one of the officers), so dear

If conscience were not an original faculty, enabling us to to thy affections, has betrayed thee to us; by his information forin a conception of moral qualities, man could never acquire thou art apprehended, and diest.' With this, the officer struck such an idea by any other means. The opinion, therefore, a poniard to his heart, and the unhappy parent fell, affected that moral feelings are merely the effect of instruction and not so much by his fate, as by the means to which he owed it.” leducation, is erroneous. For every class of simple ideas there Now, the question is, if this story were related to the wild boy must be 'an appropriate faculty, without which these ideas caught some years ago in the woous of Hanover, or to a savage can never be acquired. In regard to the bodily senses, this is without experience and without instruction, cut off in his too evident to be called in question. Without the organ of infancy from all intercourse with his species, and consequentùy vision, the simple idea of light and colours could never be under no possible influence of example, authority, education, communicated by any instructions; without the organ of sympathy, or habit, whether or not such a one would feel hearing, no idea of sound can be conveyed; and so of the upon the relation any degree of that sentiment of disapproba- other senses. And it is equally true of that knowledge which tion of Toranius's conduct which we feel.

is acquired by what some have called the internal senses. If In our judgment, such a case would afford no criterion by there were in man no such faculty as taste, by which beauty which to determine whether men possess constitutionally a is perceived, no idea of the beautiful could possibly be commoral sense. For, in the first place, the trial would be no better municated. A horse has no perception of the beauty of a than if the question were proposed to a child two years old, scene which perhaps enchants his rider, even though the aniin whose mind the moral faculty is not yet developed. A mal sees all the objects with equal distinctness. So it is in human being, arrived at adult age without instruction or com- regard to moral qualities. There must be an original faculty munication with others, would be-as it regards to the mind to give us the simple idea which we have of morality; otherin a state differing very little froin that of infancy. It is not wise the idea of virtue or vice could never have entered the held that the moral sense will be exercised without the usual human inind, and the feelings of moral obligation, of which means by which human faculties are developed. If an all men are conscious, would never have been felt. organical defect in the brain should prevent the intellectual We are aware that those who advocate the utilitarian scheme, faculties from coming into exercise, the unhappy individual resolve all our ideas of morality and moral obligation into the thus deprived of reason would prove nothing in regard to the mere principles of benefit or 112jury, apprehended to be conoperations of reason where it is developed. So, also, if a nected with each action. Dr. Paley inturins us, that the human being were brought up from early infancy in a dark subject continued to be involved in impenetrable mystery, dungeon, and if no information were coinmunicated to him, until he took this view of it. the mental faculties would not be developed, and it would be It is deemed useless to argue this point; it cannot be absurd to have recourse to such a one to ascertain what decided by reasoning. The appeal must be made to the confaculties belong to the human mind. The same remark will sciousness of every man. apply to the case of the wild boy, referred to by Dr. Paley ; If any one persists in declaring that he sees no evil in any and also, though in an inferior degree, to savages of the action but as it is evidently detrimental to human happiness, most degraded class.

nothing can be said in the way of argument to alter convicLet it then be fairly understood what it is which is asserted tions derived from his own consciousness. All that is proper in regard to conscience, as an original, universal faculty. It to be said is, that the mind of such a person is differently is, that every human mind, when its faculties have been constituted from that of most men; cr rather that an impartial developed, and have arrived at some degree of maturity, dis- examination of this subject has not been made. It is recomcerns a quality in certain actions which is termed moral; that mended to such persons carefully to scrutinise the exercises of is, it intuitively perceives that soine actions are right and their own minds; they will perceive that the idea of virtue or

moral good is entirely distinct from that of mere utility. Another objection to the historical fact adduced by Dr. There is, indeed, a connexion between these two things which Paley, is, that it presents to the mind, not a case of simple, is very intimate, and this seems to have misled many in their unmixed good or evil, but a complex case, in which-before a judgments. Virtuous conduct leads to happiness, and is judgment can be formed of the action of the son--it must be always beneficial; yet our idea of its moral character is not decided whether a man ought to be governed by a regard to derived from this consideration, but from the nature of the llie welfare of a parent, or to the public good.

If the son action itself, believed that the party in pursuit of his father was promoting the public good, he might feel that he ought to be governed A MORAL FACULTY BEING SUPPOSED, WHETHER by this rather than by filial affection. Here, then, we have

ITS DICTATES ARE UNIFORM? presented a complex and difficult case in morals, about which men would be very apt to differ; and we are to determine

One of the strongest objections which has been brought whether all men-even those totally uneducated—would view against the doctrine luid down is, that among men of different

countries, and of entirely different education, there is no To render the case a suitable one to be a test of the question immorality of the same actions. Whereas, it is alleged, that

azreement in their judgments respecting the morality or under consideration, it should be supposed that the father if such a faculty were originally a part of man's constitution, was acting in conformity with the strictest principles of recti- there would as certainly re uniformity, as in the perception of the good of the commonwealth but its destruction; and that objects by the external senses. Now, if the dictates of cont by inercenary motives, or by unjust and unnatural dislike to a tion makes them ? And what is gained by maintaining the the sun, in betraying the place of his concealment, was actuated science in men of different countries do so much differ

, does it

not show that the moral feelings of men are just what educaCome the chapter of his Mural Philosophy, under the head " The Moral existence of a moral faculty, as part of man's original consti

some wrong

it in the same light.

tutions

Bonse.

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It will, we think, be admitted, that in all countries and con- fundamental truths, there has never been any difference of
ditions in which men have been found, there exists a percep- opinion. It is not meant that all men distinctly think of
tion of a difference in the moral character of actions ; that is, these primary truths in morals; for many are so inattentive,
some things are accounted wrong, which ought not to be done, or so much occupied with sensible objects, that they can
and some right which ought to be done.

scarcely be said ever to reflect on the subject of moral duty.
Again, it has never been pretended as being a matter of But let an act of manifest injustice be performed before their
fact, that between men of different countries there is a total eyes, and among a thousand spectators there will be but one
difference in the opinions entertained respecting what is right opinion, and but one feeling. "If a strong man, for example

,
and what is wrong. A few cases only of difference are alleged, violently takes away the property of one weaker than himself

,
in which this discrepancy is observed; but in regard to those and for no other reason than because he covets it, all men will
actions which are reckoned good or evil, there is a general condemn the act. So, if any one who has received from
agreement. As to those in which there seems to be a funda- another great benefits, not only refuses to make any grateful
mental difference, an explanation will be given hereafter. No return, but, on the contrary, returns evil for good, all men will
nation, or tribe, or class of mankind has ever held that it is a agree in judging his conduct to be wrong. All intuitively
virtuous and proper thing to do injury to men, or that there is discern that for a ruler to punish the innocent and spare the
no more harm in taking away life than in preserving it. It guilty, is morally wrong. It is not true, in fact, that there is
has never been held that ingratitude-though everywhere no agreement among men as to the fundamental principles of
common in practice—is a commendable thing; or that deceit morals. Their judgments on these points are as uniform as
and fraud are as praiseworthy as honesty and fair dealing. on the axioms of mathematics; as in their agreement that the
,: There is in every country a difference made in the estima-starry firmament is grand and beautiful; yea, as uniform as
tion of the character of men, derived from the course of their concerning the greenness of the grass, or the varied colours of
conduct. Some men are reckoned good in the public estima- the rainbow.
tion, while others are considered wicked; the for!ner obtain Mr. Locke, in his zeal to disprove the existence of innate
esteem, the latter are despised. That course of conduct which truths, attempts to render uncertain some of these first truths
secures a good reputation, does not in any country consist of of morals.
actions which we consider wicked, but of actions which in all When we go beyond these first principles, we may expect to
countries are considered praiseworthy; and men have never find men falling into grievous error respecting moral duty;
obtained a bad character by a course of goou behaviour. and this often appears in their application of general prin.
It is also important to observe, that the conduct of a people ciples to particular cases, Most men either do not reason at all

,
is not a fair test of the internal state of the mind, as it relates or reason badly, and draw from sound principles incorrect
to morals. We know that individuals often pursue a course conclusions. For the most part, they receive implicitly what
of conduct, which in their serious moments they condemn. they have been taught; or they are governed in their opinions
Yet the power of temptation, and the habit of indulgence are by the common sentiment; or they adopt as true what is most
such, that notwithstanding the convictions of conscience, they for their interest, or most agreeable to iheir feelings. And as
continue in a course of evil-doing. It would be a very incon- men are often under the influence of feelings or passions
clusive inference to determine from their habitual conduct, which produce perturbation of mind, and so bias the judgment,
that they acted in accordance with the dictates of conscience, it is easy to see how errors of judgment respecting moral con-
And what is true of individuals, may be true of nations and duct, in many cases, may spring up. And yet it is true, that
tribes. Those customs which they have received from their there are primary truths in morals, in which all men agree, ko
forefathers, may not meet with the approbation of their moral soon as they are presented to the mind. As in other cases, by
sense, and yet such is the force of an established custom, that pursuing a course of sophistical reasonings, conclusions may
they go on in the way in which they were brought up. be arrived at which are contradictory to these first principles

, But a more satisfactory explanation of those facts, in which and this will produce perplexity ;'or even a kind of specumen seem conscientiously to go contrary to the fundamental lative assent may be yielded to such conclusions of ratiocina, principles of morals, is, that the principle on which they act tion; but whenever it is necessary to form a practical is correct, but through'ignorance or error they make an erro- judgment, the belief of intuitive truths must prevail. Our neous application of it.

assent in these cases is not a matter of choice, but of necessity.
When parents murder their own female children-a thing Bishop Berkeley thought he had demonstrated that there was
very customary in China—it is on the principle that they will no external world; and many others thought there was no
be subject to more misery than happiness in the world; and daw in his reasoning : but all these speculative sceptics were:
therefore it is doing them a favour. Here, the general prin. nevertheless, practical believers in the real existence of
ciple is correct-that parents should consult the best interests external objects. Atheistical and infidel philosophers have
of their offspring-but the mistake is in the application. The often endeavoured to prove that there is no intrinsic difference
same may be said of the practice of exposing aged parents, between right and wrong, and some of them probably per-
when they become incapable of enjoying the world.

suaded themselves that this opinion was true; but these very
As to those acts of cruelty which the Pagans perform in men, when an act of great injustice towards themselves or
their religious services (the wife committing herself to the friends was committed, could not but feel that it was morally
fames with the body of her deceased husband, children evil; and when they saw an act of disinterested benevolenco
voluntarily thrown into the Ganges, or persons devoting their performed, they could not but approve it as morally good.
own lives by falling under the car of Juggernaut), they are
performed on the principle that what God requires, or what
pleases him, or what will secure happiness for ourselves or
friends, should be done. It is true that the will of God should

LESSONS IN ALGEBRA.-No. XXI.
be obeyed, whatever sacrifice he may require; their error
is in thinking that such sacrifices are required by Hinn.

(Continued from page 391.)
HOW FAR ALL MEN ARE AGREED IN THEIR MORAL

POWERS OF ROOTS.
JUDGMENTS.
As the subject of morals is very extensive, and particular fractional index has been a unit. There is another class of

In the preceding examples of roots, the numerator of the
cases may be complicated, and as men are not only ignorant, quantities, the numerators of whose indices are greater
but prejudiced by the errors received in their education, it is
hut here wonderful that they should adopt different opinioers either 'as powers of roots

, or roots of povrers.
than 1, as 01, c#, etc.

These quantities may be considered
on these subjects than on other matters.
which is true in regard to every department of human know-
ledge, is doubtless true in regard to the science of morali; denoted by a fractional index, the denominator

, like the fine

N.B.- In all instances, when the root of a quantity is
There are certain self-evident truths, which are intuitively
perceived by every one who has the exercise of reason, as over the radical sign, expresses the root, and the numerator

}
soon as they are presented to the mind. In regard to these the power. Thus, a denotes the cuhe root of the first power

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of a, i.e. that a is to be resolved into three equal factors ; Obs. From the manner of performing evolution it is
for a' xay xa
at xat }

On the other hand, co denotes the evident, that the plan of denoting roots by fractional indices,
third power of the fourth root of c, or the fourth root of the is derived from the mode of expressing powers by integral

indices. third power. One expression is equivalent to the other.

1. Required the cube root of as. Ans. a'. 1. What is as equal to ? 2. What is ao equal to ?

2. Required the cube root of a or a'. Ans. a), or iv a. 3. What is yo equal to ?

4. What is 83

equal to 6. Write the fifth root of the fourth power of a.

For at xet x at, or 3/a XVax Va=a. 6. Write the seventh power of the ninth root of d.

3. Required the Afth root of ab.

4. Required the nth root of a”.
The value of a quantity is not altered, by applying to it a 5. Required the seventh root of 20--.
fractional index whose numerator and denominator are equal. 6. Required the fifth root of (a—x)3.

7. Required the cube root of all.
Thus, a ==a =an. For the denominator shows that

8. Required the fourth root of a-1. a is resolved into a certain number of factors; and the nume

9. Required the cube root of ass. rator shows that all these factors are multiplied together in an.

10. Required the nth root of xm.

11. Required the third root of ah. On the other hand, when the numerator of a fractional in

12. Required the fourth root of .x8. dex becomes equal to the denominator, the expression may

13. Required the second root of zin. be rendered more simple by rejecting the index,

14. Required the fifth root of d3.

15. Required the cighth root of a3. Instead of a”, we may write a.

The rule in the preceding article may be applied to The index of a power or root may be exchanged for any every case in evolution. But when the quantity whose root other index of the same value.

is to be found, is composed of several factors, there will fre-
Instead of af we may put as.

quently be an advantage in taking the root of each of the
For in the latter of these expressions, a is supposed to be factors separately.
resolved into twice as many factors as in the former; and the

This is done upon the principle that the root of the product of
nulnerator shows that twice as many of these factors are to be several factors is equal to the product of their roots.
inultiplied together. Hence the value is not altered.

Thus Vab=Va XV b. For each member of the equation if
From the preceding article it will be easily seen, that a frac- raised to any power, will give the same result.
tional index may be expressed in decimals,

When, therefore, a quantity consists of several factors, we
7. Thus at a or 20-5; that is, the square root is equal find the root of the factors separately, and then multiply thein

may either extract the root of the whole together, or we may
to the fifth power of the tenth root.

into each other.
8. Express ał in decimals. 9. Express a in decimals.
10. Express ał in decimals. 11. Express a$ in decimals. 16. The cube root of xy is either (ry), or r!,).

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17. Required the fifth root of 3y.
11
12. Express a in decimals.

18. Required the sixth root of abh.
In many cases, however, the decimal can be only an approxi-

19. Required the cube root of 8). mation to the true index.

20. Required the nth root of any.

The root of a fraction is equal to the root of the numerator 13. Thus a* = an.3 nearly, or c0 33333 more nearly.

divided by the root of the denominator. In this manner, the approximation may be carried to any

at degree of exactness which is required.

21. Thus the square root of

For

6
14. Express aš in decimals. 15. Express a in decimals.
N.B. These decimal indices form a very important class of

22. Required the nth root of
numbers, called logarithms.

6

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EVOLUTION.

23. Required the square root of

ау
The process of resolving quantities into equal factors, is called
Evolution.

Signs.--(1.) An odd root of any quantity has the same sign
In subtraction, a quantity is resolved into two parts. as the quantity itself.
In division, a quantity is resolved into two factors.

(2.) An even root of a positive quantity is ambiguous.
In evolution, a quantity is resolved into equal factors.

i .
Evolution is the opposite of involution. The former is finding
a power of a quantity, by multiplying it into itself. The other But an even root of a positive quantity may be either posi-
is finding a root, by resolving a quantity into equal factors. tive or negative. For the quantity may be produced from the
A quantity is resolved into any number of equal factors, by one, as well as from the other.
dividing its index into as many equal parts.

Thus the square root of a' is ta, or- a.
From the foregoing principles we deduce the following An even root of a positive quantity is, therefore, said to

be ambiguous, and is inarked with the sign + Thus the
GENERAL RULE FOR EVOLUTION.

square root of 35 is +136. The 4th root of x is +21
Divide the index of the quantity by the number expressing the The ambiguity docs not exist, however, when from the
root to be found. Or,

nature of the case, or a previous multiplication, it is known
Place the radical sign belonging to the required root over the whether the power has actually been produced from a positive
giren quantity.

or from a negative quantity.
If ine quantities have co-efficients, the root of these must be But no even root of a negative quantity can be found.
extracted and placed before the radical sign or quantity. Thus, The square root of — a® is neither + a nor — a.

For taxta=ta?. And

a X
Х

-a=+aalso.
To find the square root of d', divide the index 4 by 2, i. e.

An even root of a negatire quantity is, therefore, said to be d=d. So the cube root of lø, is

impossible or imaginary.

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