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LESSONS IN ALGEBRA.

453 I may exhibit these facts to you thus

In these instances, the radical sign or index is separately

applied to the numerator and denominator. But if the divisor Subjeci. Verb.

Object.

and dividend are reduced to the same index or radical sign, Primary youn

τον υιον

this may be applied to the whole quotient., δει ondary η Ασπασια

εν τω οικω

| 3. Thus "Va="vb=
καλουμενη

ένεκα της απειθειας.
.

For the root of a

fraction is equal to the root of the numerator divided by the You will readily see how each of the principal parts may be root of the denominator. extended; thus you may prefix an adjective to yuvn; you may qualify the verb by an adverb, and you may qualify the 1

4. Again, "V ab - Vb="va. For the product of this obj-ct by an epithet. As the primary, so also the secondary : quotient into the divisor is equal to the dividend; that is, subject and object may be qualified; the verb may also be va XoVb = 'vub. Hence-doubled, or a participle may be introduced instead of a second verb, or together with a second verb; and thus the simple Quantities under the same radical sign or index, may be divided sentence may be expanded into a compound sentence, and like rational quantities, the quotient being placed under the common give you an instance of what the Grammarianis meant when radical sign or index. they termed a sentence or the expression of a thought, a period or a circle (Trepiodus). I here give the sentence in an expanded 5. Divide (x3y2) by y. form, as a period or compound sentence :

These reduced to the same index are

*x2)+ Η κακη γυνη, η Ασπασια καλουμενη, προσκαλεσασα τον And the quotient is (23)= = x3 =gt. Ans.

. απειθη υιον, εν τω οικω, ένεκα της απειθειας, λοιδορεί και τυπτει

6. Divide v6a 3g by V 3x. ισχυρώς. .

7. Divide v dhx? by v dx.
This period may be presented in a state of analysis thus :

8. Divide (a3 + ax)" by a
Subject.
Verb.
Object.

9. Divide (h)" by (ax)m.
Primary η γυνη
λοιδορεί

10. Divide (a’y?)? by (ay)*.
Secondary η Ασπ. καλ.
. (και) τυπτει

εν τω οικω, ένεκα

A root is divided by another root of the same letter or quantity,
Qualifiers pockaledada oxupūs

απειθη.
.

by subtracting the index of the divisor from that of the dividend.

11. Thus al
By " qualifiers," is meant the words which qualify the import
or extent of the subject, the verb or the object. Among the

For a?=a3 = ab xam xam and this divided by ad is qualifying words, I might have placed ń and pov; the kal merely a xa

Xa x a unites the two verbs; which are the representatives of two

=a xa

at xa? =* = at.
sentences, for the whole is equivalent to

ao
First Sentence oj
yuun λοιδορεί

viov
12. In the same manner, amah = am

am
Second Sentence in
yuvn

vioy

13. Divide (3a)** by (3a) and may be made into three sentences, thus :

14. Divide (ar) by (az). Third Sentence yuun προσκαλεί

υίον. .

15. Divide a um by a m.
The sentence thus enlarged brings permanently into view 16. Divide (6+y)ã by (6+y).
what in grammar is called government. One word is said to

17. Divide (ry") by (ry).
govern another, when the second is dependent on the first,
and undergoes some inflectional change under the influence Powers and roots of the same letter may also be divided by
of the first

. Thus the nominative vios is changed into the each other, according to the preceding article.
accusative viov by the verb TurTEL, on which it depends, and
by which therefore it is said to be governed. In the same way

18. Thus a??-= c++= af. For af xat = q*=a?.

.
απειθης becomes απειθη under the operation of the verb τυπτει, When radical quantities which are reduced to the same index,
and in order to agree with viov. So the prepositiou ev changes have rational co-efficients, the rational parts may be divided
OIKOS into olky, and olky is said to be governed in the dative separateiy, and their quotient prefixed to the quotient of the radical
case by the preposition ev.
For the same reason aradea

parts.
appears in the genitive case, atsiltas, because it depends on
ivera, which requires its object to be in the genitive.

19. Thus acv bd - avb=cd. For this quotient multiplied into the divisor is equal to the dividend.

20. Divide 24x7 ay by 6v a

21. Divide 18dhv bx by 2hv x. LESSONS IN ALGEBRA.-No. XXIII.

22. Divide by (aox?) by y'ax).

23. Divide 16V 32 by 8V 4. (Continued from page 436.)

24. Divide bvxy by vy.

25. Divide ab(x?bj* by a(x). DIVISION OF RADICAL QUANTITIES.

These reduced to the same indes are ab(x27;+ and a(z3)a?.
The division of radical quantities may be expressed by
writing the divisor under the dividend,' in the form of a

The quotient then is 6(8+= 100,7
To save the trouble of reducing to a common index, the

division may be expressed in the form of a fraction.
1. Thus the quotient of a divided by v b is

The quotient will then be ab(x?b) 2. And (a+h) divided by (64x)" is

Hence we deduce the following

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EXAMPLES FOR PRAOTICE,

17. Required the cube of 2d +Vx.
GENERAL RULE POR DIVIDING RADICALS.

18. Required the 4th power of v d.
If the radıcals consist of the same letter or quantity, subtract 19. Required the 4th power of — V2 -
the index of the divisor from that of the dividend, and place the

20. Required the 6th power of va tb.
remainder orer the common radical part or rout.
If the radicals have cu-efficients, the co-eficient of the dividend

EVOLUTION OF RADICAL QUANTITIES
must be divided by that of the divisor.

If the quantities have the same radical sign or index, divide The operation for finding the root of a quantity which is
them as rational quantities, and place the quotient under the common already a root, is the same as in other cases of evolution. Hence
radical sign.

we derive the following

RULY FOR THE EVOLUTION OF RADICALS.

Divide the fractional indor of the quantity by the number express 1. Divide 23 // be by 3V ac.

ing the root to be found. Or, 2. Divide 10%/108 by 5 3/4.

Place the radical sign belonging to the required root over the 3. Divide 10/27 by 2/3.

given quantity.

If the quantities have rational co-efficients, the root of these
4. Divide 8V 108 by 2 v 6.

must be extracted and placed before the radical sign or quantity.
5. Divide (a2b2d3) & by
24.

1. Thus the square root of al, is a:=2=27
6. Divide (16a3 - 12a2x)" by 2a.

2. Required the cube root of a(xy)?. 7. Divide 6v 138 by 2V 6.

3. Required the nth root of aby. 8. Divide 8 3V 512 by 4 V 2.

4. Required the 4th root of va x® vb. 9. Divide iv 5 by ! V 2.

5. Required the 7th root of 128 vd. 10. Divide v 7 by ’V7.

From the preceding rules it will be perceived that powers 11. Divide 6V 54 by 3V/2.

and roots may be brought promiscuously together, and sub12. Divide 4 3772 by 2 v 18.

jected to the same modes of operation.

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EXAMPLES FOR PRACTICE.

INVOLUTION OF RADICAL QUANTITIES.

1

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To involve a radical quantity to any required power.

Multiply the inder of the root into the index of the power to
shub i w to be raised.
1. Tia the square of a

razalx2 = a1. For al xat=a.
2. Bequired the cube of ab.
3. Pacuared the nth power of a

1
4. Bequire the fifth power of a'y
6. Bequired the cube of a'ri".
6, Buquired the square of a
7. Required the cube of a

al
8. Bequired the nth power of an.

Arut is raised to a power of the same name by removing the
indu vi radial sign.

X.B. When the radical quantities have rational co-efficients,
there must be involved by actual multiplication.

9. Thus the cube ofV6 + , is 6+*.
10. And the nth power of (a - y)", is a y.
11. The square of a "Vx, is alty.

Por a"? Xavi= a?" **.
12. Required the nth power of an rin.
13. Required the square of av x-
14. Required the cube of 3ay.

But if the radical quantities are connected with others by
the signs + and --, they must be involved by a multiplication
of the several terms.
15. Pequired the square of a tvy and of a --V'y.
atvy

a-Vy
vy

a-Vy

az-avy "vyty

-avyty 4 wyty

as-2av yty. 16. Kad tie cube of a -vb.

1. Find the 4th root of 81a?.
2. Find the 6th root of (a + b) 3.
3. Find the oth root of (x ——-y).
4. Find the cube root of -125a3x.

404
5. Find the square root of

9.r yo

32arlo
6. Find the 5th root of

213
7. Find the square root of x? — 66x + 962.
8. Find the square root of a: tayt

4
9. Reduce ax? to the form of the 6th root
10. Reduce - 3y to the form of the cube root.
11. Reduce a’ and af to a common index.

at
12. Reduce 45

and

to a common index.
13. Reduce a? and 6 to the common index ..
14. Reduce 24 and 4to the common index 5.
16. Remove a factor from v 294.
16. Remove a factor from Vm-a*r?.
17. Find the sum and difference of v 16 a'd and v4a-r.
18. Find the sum and difference of //192 and 8v 24.
19. Multiply 73/18 into 5% V4.
20. Multiply 4+2V2 into 2 -- v'2.
21. Multiply ala tv cinto (a-VC,H.
22. Multiply 2(a + b)ā into 3(a + b)in
23. Divide 6v 54 by 3//2.
24. Divide 4 V 72 by 23/18,
25. Divide v 7 by v7.
26. Divide 8 3 7 512 by 4 372.
27. Find the cube of 17 v 21.
28. Find the square of 5 + V 2.
29. Find the 4th power of .v 6.
30. Find the cube of vx-vb.

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FRENCH READING S.-No. XXVI.

COLLOQUIAL EXERCISE.
LA MARGUERITE ET L'E'PI DE BLE'.

1. Où était la marguerite, lors- | 13. Que se passait-il au village ?

que les enfants repassèrent le | 14. Où était alors Marie? SECTION II.

soir ?

15. Que faisait Louise?

2. Que fit Louise alors ? 16. Qu'arriva-t-il dix années Lorsque les deux enfants repassèrent le soir, la marguerite 3. Que dit le père de l'auteur ?

plus tard ? était fanée sur la tête de Marie, qui n'y songeaite déjà plus 4. Que fit-il ensuite ? 17. Que rappela à son fils, le et s'en revenait en faisant l'école buissonnièreb.... Louise, ' 5. Où se trouve la pâque. père de l'auteur ? chargée d'une récolte d'herbe, éleva sa figure blonde au rette?

18. Où le conduisit-il ? dessus de la haie, et jeta un regard d'espoir à son épi floris- 6. Que fait la reine des prés ? 19. Que lui montra-t-il ? sant.

7. Qu'offre-t-elle après la mois. 20. Que dit le jeune homme ? --Souviens-toit de ces petites filles, me dit mon père avec son

21. Que répondit le pere ? gravité. Je soupçonne qu'elles ressemblent l'une et l'autre, 8. Et la marguerite des par- | 22. Où allérent-ils de là ? aux objets si différents de leurs væux.

terres ?

23. Que virent-ils dans Puis il me fit l'histoire de la margueritet et celle du 9. Qu'est-ce que l'histoire du champ de blé mûr ? froment.

froment ?

24. Que faisait la jeune fille ?

25. Que dit le père ? Dans les champs et dans les jardins, la marguerite est la 10. Que fait le froment ? même; jolie, coquette et inutile. Celle qu'on nomme la 11. Que continua de faire 26. Qu'avait distrait de ses propâquerette, s'établit dans l'herbe courte, pour l'étoiler de son 12. Que fit Louise quand l'épi | 27. Par quel souhait finit lo

duits la jeune fille? bouton d'or et de son blanc diadème. Celle qui prend le

de blé fut mûr ?

morceau ? nom de reine des prés, se dresse et se balance sur sa tige au dessus de tout ce lui l'environne. Mais quand la faux du moissonneur vient l'abattre, elle n'offre au sol qu'une faire l'école buissonnière ; this familliar expression may be ren.

NOTES AND REFERENCES.--a. L. part ii., $ 49, R. (1).-6. graine funeste et à la dent des bestiauxd qu’un fourrage dered in English by: to play the truant, to stay away from school. détesté. Enfin, celle qui brille dans les parterres, n'a que -c. L.S. 36, R. 2-d.the singular of bestiaux, is betail.

.-e from son éclat d'un jours et ne présente à l'homme ni parfum ni naître ; L. part ii., p. 96---f. from suivre; L part ii., p. 106.-9. produit.

from mourir ; L. part ii., p. 96.--h. L. S. 36, R. 2.-i. from L'histoire du froment, au contraire, est l'histoire de la civi- reconnaître; L. part ii, p. 102.--;. travail is the subject of the lisation' même. Il naît avec elle, il en développe et en verb.-k. distrait, taken.--1. from valoir; L. part ii., p. 108. suit' les progrès, et il ne meurts que sous les latitudes où elle expire. Jo Ceci est la plus admirable harmonie de la nature. Les jour suivants, Marie continua de faire appelli à

LE CHIEN DU LOUVRE. nos marguerites, de s'en parer une heure et de les perdre à travers ses jeux.—Quand l'épi fut mûr, Louise vint le

SECTION 1. cueillir, et l'emporta soigneusement. Ce jour-là, il y avait fête au village.13 Marie passa les heures à danser, 14

Passant que ton front se découvre !! sa marguerite aux cheveux. “Louise, faisant sa moisson

Là plus d'un brave est endormi, dans les règles, 15 égrena l'épi et en

conserva la riche

Des fleurs pour le martyr du Louvre,

Un peu de painé pour son ami ! Dix années plus tard, malgré la recommandation de mon père, 16 j'avais oublié les deux sœurs.

C'était le jour de la bataille ; Il me rappela" leur histoire et celle de la marguerite et de l'épi. Puis il, me

Il s'élança sous la mitraille ;5

Le chien suivit. conduisit dans le village, 18 et me montra, à la fenêtre d'une chaumière encadrée de feuillage et de fleurs, une jeune

Le plomb tous deux vint les atteindre, e

Est-ce le maître qu'il faut plaindre ? fille qui achevait sa toilette et dont les longs cheveux

Le chien survit.d
noirs étaient ornés d'un cercle de perles et d'une margue-
rite....
-C'est Marie ! m'écriai-je ?20

Morne, vers le brave il se penche?
-Oui, c'est Marie, 21 toujours vaine comme la première

L'appelle, et de sa tête blanche

Le caressant, fleur que nous lui donnâmes. Elle a passé sa jeunesse oisive dans les fêtes ; et elle attend un mari qui ne viendra jamais,

Sur le corps de son frère d'armes,

Laisse rouler ses grosses larmes® car elle n'a d'autre dot que les marguerites desséchées de

Avec son sang. De là, nous allâmes dans un champ de blé mûr,22 et nous

Des morts voilà le char qui roule ; vimes une autre jeune fille, 23 d'une beauté calme et grave,

Le chien, respecté par la foule, dont les vêtements annonçaient l'aisance, et qui

, les mains

A pris son rang, posées sur un tronc d'arbre, contemplait24 doucement la

L'ail abattu, l'oreille basse,10 moisson dorée.

En tête du convoi qui passe,
Je reconnus' Louise, et mon père me dit :

Comme un parent."
Cette moisson est à elles et vient, sillon par sillon, année
par année, du premier épi qu'elle a reçu de nous et qu'a

Au bord de fossell avec peine, multiplié son travail. Elle n'a distraitk de ses produits, à

Blessé de juillet,' il se traîne chaque saison, que la part des pauvres,26 multipliée aussi

Tout en boitant; par sa charité ; et elle obtiendra sa récompense après la

Et la gloire y jette son maitre, récolte, en épousant le plus sage et le plus riche fermier du

Sans le nommer,1% sans le connaître :

Ils étaient tant !13. Une telle leçon ne vaut-elle' pas un petit coin de terre, et n'ai-je pas raison d'avoir un champ de blé dan mon jardin ?

Gardien du tertre funéraire,14 Heureux si mes épis trouvaient autant de Louises?' que mes

Nul plaisir ne le peut distraire15 marguerites rencontrent de Maries!... JARDINEUR.

De son ennui;

semence.

notre jardin.

canton.

En fuyant la main qui l'attirel6

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guages. By Professor De LOLM & and HENRY BRIDGEMAN, Esq. Ce n'est pas lui !!

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de son maître ?

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axis, and supporting by means of two columns a second graON PHYSICS, OR NATURAL PHILOSOPHY. duated circle m, which measures the inclination. A frame No. LVI.

r supports the needle a b, and a level n serves by means of the

three screws to give a horizontal position to the diameter (Continued from page 444.)

passing through the two zeros of the circle m,

To observe the inclination, we begin by determining the TERRESTRIAL MAGNETISM—THE COMPASS.

magnetic meridian, which is done by turning the plate a upon Inclination or Dip; Magnetio Equator. - From the fact that the circle m till the needle becomes vertical, which takes place the compass affects the direction towards the north, it might when the needle is in a plane perpendicular to the magnetic be supposed that the force which acts upon it comes from a meridian. Then turning the plate a through 90 degrees on point in the horizon ; but this is not the case, for if we arrange the circle m, we bring the vertical circle m to the magnetic the needle in such a manner that it can move freely in a verti- meridian: The angle dca which the magnetised needle then cal plane about a horizontal axis, we shall find that though makes with the horizontal diameter, is the angle of inclination. the centre of gravity of the needle exactly coincides with the But there are two causes of error which must be taken into axis of suspension, its south pole inclines constantly to the account. First, the magnetic axis of the needle may perhaps north pole of the earth in this quarter of the globe. In the not coincide with the axis of the needle itself, whence arises southern hemisphere, on the contrary, it is the north pole of an error which is corrected by the method of turning already the needle which inclines towards the south pole of the earth. described. Secondly, the centre of gravity of the needle may When the vertical plane in which the needle moves coincides possibly not coincide with the axis

of suspension, and then with the magnetic meridian, the angle it makes with the the angle de a is too small or too large, according as the centre horizon is called the inclination or dip. In any other plane of gravity is above or below the centre of suspension, for in the than the magnetic meridian, the dip increases, and is ninety first case the action of gravitation is contrary to that of terresdegrees in a plane pei pendicular to the magnetic meridian. trial magnetism in turning the needle, while in the second it In fact, the magnetic force of the earth being then resolved is in the same direction. This error is corrected by reversing into two forces, the one vertical and the other horizontal, the the poles of the needle, which is accomplished by rubbing it former gives the needle its vertical position, while the latter, with the contrary poles of two bars in such a manner, that acting in the direction of the axis of suspension, cannot turn each pole of the magnet is rubbed by a pole of the same name the needle.

as its own. The direction of the needle being then changed, The inclination, like the declination, varies with the latitude, if its centre of gravity was before above the point of suspenbut according to a more fixed law. Near the north pole it is sion, it is now below, and the angle of inclination which nearly ninety degrees, and proceeding thence towards the was before too small is now too large. The true value is found equator, it diminishes till it becomes nothing at the equator, by taking the mean between the two. In the southern hemisphere the inclination reappears, but in

Astatic Needle and System.--A needle withdrawn from the a contrary direction, that is to say, it is the north pole of the magnetic action of the earth is said to be astatic. This is the needle which sinks below the horizon.

case with a needle moveable about an axis in the plane of the The curre which passes through all the points at which the magnetic meridian in a direction parallel to the inclination. inclination is nothing is called the magnetic equator, and the An astatic system is the combination of two needles of the same points at which it is 90° are called magnetic poles. According power placed parallel, with their contrary poles opposite each to the observations of M. Duperrey, the magnetic equator other, as seen in fig. 371. If the two needles are of exactly appears to intersect the terrestrial equator in two diametrically opposite points, one in the Pacific and the other in the Atlantic

Fig. 371. Ocean. The inclination varies at the same place from one period to another.

The Inclination Compass is the name given to an instrument used for measuring the magnetic inclination. This compass, which is made entirely of copper, is composed first of a horizontal circle m, graduated, and supported by three screws, fig; 370. Above this circle is a plate X, moveable about a vertical

Fig. 370.

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the same power, the contrary action of the earth upon the
poles a' and b, as well as on a and b', is self-destructive, and
the system is completely astatic.
MAGNETISATION AND LAWS OF MAGNETIC ACTION.

Sources of Aagnetisation. The various sources of magnetisa-
tion are the influence of powerful magnets, terrestrial magnet-
ism, and electricity. There are three methods of magnetising
by magnets, simple touch, separate touch, and double touch.
Whichever of these methods be employed, there is a limit to
the degree of acquired magnetic power, depending upon the
temper and intensity of the magnets employed. When this
limit is attained, the bar is said to be magnetised to saturation.
When it has been passed, the bar soon returns to it, and
even tends to sink below it, if the magnetic power is not
retained by means of a process to le described presently,

Method of Simple Touch.-- This consists in sliding the pole of a powerful magnet from one end of the bar we wish to magnetise to the oiher, and repeating the operation several times, but always in the same direction. This process is incapable of producing any great effect, and consequently is only applicable to small bars. It has the further inconvenience of frequently developing consequent points.

134

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VOT. V.

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