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EVOLUTION OF RADICAL QUANTITIES.

The operation for finding the root of a quantity which is already a root, is the same as in other cases of evolution. Hence we derive the following

RULE FOR THE EVOLUTION OF RADICALS.

Divide the fractional index of the quantity by the number expressing the root to be found. Or,

Place the radical sign belonging to the required root over the given quantity.

If the quantities have rational co-efficients, the root of these must be extracted and placed before the radical sign or quantity. EXAMPLE. Thus the square root of a3, is a3÷2 — að.

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From the preceding rules it will be perceived that powers and

roots may be brought promiscuously together, and subjected to the same modes of operation.

EXERCISE 58.

1. Find the cube root of a(xy). 2. Find the nth root of a√by. 3. Find the 4th root of a x ® √b. 4. Find the 7th root of 1237√d. 5. Find the 4th root of 81a3. 6. Find the 6th root of (a + b)3. 7. Find the nth root of (x-y). 8. Find the cube root of -125a3. 4a+ 9. Find the square root of 9x2y2* 10. Find the square root of -Gbx+963.

32a5x10 243 12. Find the square root of a + ay +

11. Find the 5th root of

13. Reduce ax2 to the form of the 6th root.

14. Reduce -3y to the form of the cube root.

15. Reduce a2 and a to a common index.

16. Reduce 43 and

index.

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to a common

17. Reduce a and b to the com. mon index .

18. Reduce 21 and 4 to the com

mon index.

19. Remove a factor from /294. 20. Remove a factor from 3—a3x3 21. Find the sum and difference of 16ax and 4a2x. 22. Find the sum and difference of 3/192 and 3/24.

23. Multiply 73√18 into 53 √4.
24. Multiply 4+2√2 into 2−√2.
25. Multiply 3+ √−2 by 2−3√−1.
26. Multiply a-b√-c by a+b√-c.
27. Divide 2 by 3√2.
28. Divide 6 by 3√2.
29. Divide 5/12 by *√2.
30. Divide 4*√3 by √2.
31. Find the cube of 3/2.
32. Find the square of 2 + √3.
33. Find the 4th power of 1 + √2.
34. Find the cube of 2 + √5.

REDUCTION OF EQUATIONS BY INVOLUTION.

In an equation, the letter which expresses the unknown quantity is sometimes found under a radical sign. We may have √xa.

To clear this of the radical sign, let each member of the equation be squared, that is, multiplied into itself. We shall then have x × √ x =aa. Or, x = a2.

The equality of the sides is not affected by this operation, because each is only multiplied into itself, that is, equal quantities are multiplied into equal quantities.

The same principle is applicable to any root whatever. If √xa, then a". For a root is raised to a power of the same name, by removing the index or radical sign.

Hence, to reduce an equation, when the unknown quantity is under a radical sign,

Involve both sides to a power of the same name as the root expressed by the radical sign.

N.B. It will generally be expedient to make the necessary transpositions, and to clear the equation of fractions, before involving the quantities; so that all those which are not under the radical sign may stand on one side of the equation.

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1. Reduce the equation 3x + 1 = 4.

2. Reduce the equation 4+3x-4 = 6 + 4.

3. Reduce the equation 3(4x+7) +4= 18.

4. Reduce 2x-10+4=14.

5. Reduce 8.

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10. Reduce 11. Reduce

(ax-bx) √2 + 28 √x+4 12. Reduce x + √ä

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13. Reducer+ √a2+x2=
14. Reduce x + a = √ a2 + x √ { b2+x2).
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15. Reduce √2 + x + √x = √(2 + x
16. Reduce √2-216-√√x.
17. Reduce √4x+17=2√x+1.
18. Reduce (6x) −2__4 / (6x) −9
(6x)+2 4√(6x)+6°

6. Reduce (2x+3) 3 + 4 = 8.
7. Reduce 12 + x = 2 + √x.
8. Reduce (3x + 1) + 5 = 10.
9. Reduce (x + a) = c − √(x+b).
* Key to Exercises 53, 54 will be given in next lesson.

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PAINTING IN WATER-COLOURS.-II.

SEMI-TONES.

THERE are one or two remarks to be made respecting the pencil to be used in drawing the outline. It must not be too hard or too soft: if it is too hard, an impression is made upon the paper which interferes with the surface, and is difficult to remove; if it is too soft, the lead is apt to dirty the tints; therefore a middle tone, as HB, used lightly, will prevent both dangers; and no attempt must be made to shade with the pencil, as this will also destroy the purity of the tints. One of the most important regulations necessary to observe in the process of outline is that it be decisive, not black, but free

attention will be required in studying their gradation and harmony. It is a fact which beginners can scarcely appreciate, as they are not in a position to comprehend its great impor tance, that it requires considerable experience to gain a just estimate of intermediate tones. Continual observation and much practice in using the brush together prepare the pupil to discriminate tones and tints, as they lay side by side, which an uneducated eye cannot perceive; to acquire this constitutes the course of study we wish our pupils to follow. In pursuit of this they must especially notice that they will discern two conditions under which they will find the various changes that arise in connection with shade and colour. With the former we may associate tone, with the latter tint; therefore, we trust

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from scratchy trials, which betray either a want of confidence in comprehending the exact form, or an imperfect ability to represent it. Besides bestowing especial care upon the general outline, the attention must be directed to the masses of light and shade, which must be lightly and very correctly made out, together with the positions and extent of all the most prominent semi-tones, which evidently assist in giving character and expression to the subject.

As the excellence of a picture essentially depends upon the proper management of the semi-tones and half-tints, we propose to make our first essays in sepia only; it is a warm and agree able colour, transparent, flows freely, and is capable of producing every gradation of tone that may be required. Our motive in recommending the use of sepia before attempting colours is that our pupils may more easily acquire the power of distinguishing and the practice of representing semi-tones. By restricting our first essays to this one simple colour we break the difficulties attending the execution, and we shall be better prepared to take up colours afterwards with more confidence when our whole

there will be no difficulty in understanding why, at the commencement of our lessons, we prefer to separate the difficulties attending these two conditions respectively; consequently, tone will be our first consideration in this lesson.

In all the broad and general masses of light and shade there will invariably be found several degrees of brilliancy or obscurity, sometimes arising from one part being more strongly illuminated than another; for if the face of an object is perpendicular or at right angles with the source or rays of light, it then receives the greatest amount of illumination; and as it is gradually removed from the light the brilliancy decreases until, when it has been turned altogether out of the course of the light, it falls into shadow. Thus, when the surface is not very even, it is evident that some parts receive more rays from the light than others. These accidental causes will break up the uniformity of the light or shade into a thousand different and distinctive gradations of tone. So also in the shadows; reflected light may strike more forcibly upon one part than another. If, then, we connect these countless varieties of tone

with the various colours of the object and their several degrees, we are led into a course of study and reflection that has no limit. But we must observe, however numerous these changes may be, they do not interfere with the masses of light and shade in their unity as a whole. Let the pupil half close his eyes when looking on an object upon which there is no restriction in the number of tones and tints; he will perceive that all the perceptible degrees of light or shade, including all the changes of colour found within their respective limits, blend together or are absorbed into one definable mass of light on the one hand, and shadow on the other; therefore, the result of our observations teaches us this: that our difficulties will not be so much with positive light and shade, or with positive colour, but with

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graduated outwardly. Be particular in having plenty of colour in the brush, and see that every part is covered as the brush proceeds, so that it may not be necessary to re-touch it whilst wet, for if the tint is interfered with by trying to fill up vacant spaces, cut shades will appear when the picture is dry. For the broad and cast shadows add some more sepia to that which remains in the saucer to produce a middle tone, darker than that of the background, and equal to the lightest parts of the shadows; with this paint in all the broad and cast shadows upon the walls in one uniform tone, as well as those parts which are of a similar depth of colour, viz., the tiles, windows, and the grass and stones upon the ground; then, when dry, and with the same colour, make out those portions of the shaded walls,

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the infinite variety and accidental changes that are to be found in combination with them.

To give the above remarks some practical meaning we propose to make Figs. 1 and 2 the subject for a sepia drawing. After the whole of the outline and details of form have been completed, prepare a light wash of sepia in a saucer, and commence as in Fig. 1 to put in the background for the purpose of relieving or throwing up the whole of the subject, after the manner explained in the first lesson of washing in a flat tint (we will not call this background sky-skies will be the subject of a future lesson), and proceed as follows. Let the paper be inclined, and pass a brushful of water across the upper part of the picture from a to a; then fill the brush with the previously prepared light tone of sepia, and continue from, and including the lower edge of the water, and spread it across from side to side and around the outline of the building. Before the edges b, b become dry, wash them off with the water brush moderately charged, so that the background, when finished, may be

roof, and ground which are darker, according to Fig. 2. Many of these semi-tones may be partly passed over a second time, and in some cases a third, especially the cast shadows on the roof that lies under the wall of the highest part of the building. For the more delicate tones upon the light side of the building prepare a tint weaker than any yet used, with which, by careful management, the stones and bricks which compose the wall can be shown as well as those differences of colour known as weather stains, to be found on every wall, but especially old ones, besides many degrees of tone that other accidental circumstances, such as damp or decay, may produce. Last of all, the finishing touches may be added with some darker tint, carefully making out the forms of the windows, stones, weeds, etc., without painting dark lines about the edges. All objects and parts of objects should as far as possible be made out by tones carried strictly to their edges; thus everything will be properly relieved and understood, whether it appears light upon a dark ground, or dark upon a light one. Beginners frequently draw a

which it refers in gender, number, and case, both being in the masculine gender, the singular number, and the accusative case. In the second sentence the noun Quirites agrees with the pronoun vos in number and person, both being in the second person plural; they do not agree in case, for vos is in the nominative, while Quirites is in the vocative case. In the third sentence the possessive pronoun meus, like other adjectives, agrees with its noun frater in gender, number, and case, both being in the masculine gender, singular number, and nominative case. In the fourth sentence the noun consul agrees with the pronoun ego in gender, number, and case, both being in the masculine gender, singular number, and nominative case. From these instances comes forth the rule that

line of a darker colour around the edges of objects, thinking | In the first sentence the pronoun eum agrees with the noun to they are adding to the effect by making everything more distinct the fact is, the effect is destroyed by dark lines; they make the picture flat; and as these dark edges are never seen in Nature, we are not justified in using them. We have yet a few observations to make upon Fig. 2. There are not many cast shadows, and it must be remembered that, as a general rule, cast shadows are darker than broad shadows. The reason that the cast shadow on the tiles, caused by the intervention of the higher portion of the building, is darker than the broad shadow on the side of the building, is that the reflected light from the roof lowers the tone of the broad shadow without making any difference to the cast shadow as it falls upon the same plane, that is, the roof. By a proper attention to semi-tones in lights, and to reflections in shadows, we do more to give relief and effect to the picture than by any other means. Colours, however forcible their contrasts may be, will only produce a map-like flatness, without the indispensable addition of semi-tones and reflections. Therefore, we advise our pupils to continue the use of sepia only for some time, until they have acquired a sufficient power of execution and an insight into some of the mysteries that lie between the two extremes of light and shade.

LESSONS IN LATIN.-XLIX.

SYNTAX-AGREEMENT.

IN regard to the sentence mors omnium rerum est extremum
(death is the end of all things), which occurs in the Latin-English
Exercise given in the last lesson, a remark may be desirable.
Here mors, the subject, is in the feminine gender, and extremum,
the attribute, in the neuter. The rule, therefore, seems to be
broken. But the fact is, that in this and similar cases the
neuter adjective denoting a being or thing of a certain class
may be looked upon as a substantive, and is to be construed
as such. This construction, it may also be noticed, is very
frequent in Greek. Other examples are given here :-

Turpe senex miles, turpe senilis amor.-Ovid.
Varium et mutabile semper (est) femina.-Virgil.

In such a phrase as the following, Excisa ferro est Pergamum
(Pergamum has been cut down by the sword), where excisa
is feminine to agree with urbs (city), though Pergamum is of
the neuter gender, the explanation is found in the fact that the
construction follows the sense of the passage rather than the
grammatical form. Explain thus, Eunuchus bis die acta est
-that is, fabula or comedia, Eunuchus being the name of one
of the favourite Roman comedies.

In the sentence Athenæ fuerunt urbs (Athens was a city), Athens is in the Latin of the plural number, and urbs of the singular. Here, again, the sense is regarded more than the sound; for Athena, though plural in form, is singular in meaning. The sense must always have predominance. In this sentence

Amantium iræ amoris integratio est ;

Lovers' quarrels are the restoration of love; where the plural subject ira is identified with the singular attribute integratio, the sense causes a departure from the strict grammatical rule; and an attempt to put the grammar right might issue in a change of the sense for instance, for ira read ira, a lovers' quarrel is the restoration of love, one quarrel might not have the alleged effect; besides, what was general nas now become particular. In these two sentences

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country.

2. Vos, Quirites, veneramini Jovem, You, O Quirites, venerate Jupiter.

My brother is industrious.

4. Quæ ego consul dixi vera sunt.

A noun agrees with a pronoun in gender, number, and case, or gender and number.

The pronoun may be implied, as

Hostis hostem occidere volui.

(I) an enemy wished to kill an enemy. Hostis is in the first person singular, in concord with the pronoun ego implied in volui, a verb of the first person singular, perfect The pronoun ille is used to mark out a person or thing emphatically, and generally in a good sense; iste, employed also for emphasis, conveys reproach, as—

tense.

Magno ILLI Alexandro est simillimus.

He is very like THE CELEBRATED Alexander the Great.
Non erit ISTA amicitia sed mercatura.
THAT would not be friendship but traffic.

Observe that in the last example ista agrees with amicitia,
though in a similar case the pronoun in English is in the neuter
gender. The general fact may be stated thus, that in Latin
pronouns referring to something gone before agree with the noun
to which they are prefixed; what has gone before may be a sen
tence, or a statement, or a fact, or even a noun, as—

Pompeio, QUOD populi Romani lumen fuit, extincto. Pompey, WHO was the LIGHT of the Roman people, being dead. We may invert our proposition; for, as a noun may agree with a pronoun, so may a pronoun agree with a noun, as1. HÆC est nobilis ad Trasimenum PUGNA. THIS is the famous BATTLE near Trasimenum. 2. QUI CANTUs dulcior inveniri potest? WHAT sweeter SONG can be found?

3. QUOD CARMEN est aptius?
WHAT VERSE is more suitable?
4. VIRGO, quæ patria est TUA?

O VIRGIN, what is THY country ?
5. COLONIAM QUAM Fregellas appellant.
The COLONY WHICH they call Fregellæ.
6. LIBRI QUOs ad te misi optimi sunt.
The BOOKS WHICH I have sent to thee are very good.
7. Est mihi FILIUS QUEM valde amo.

I have a SON WHOM I greatly love.

8. Optima est FILIA tua, mater, EAM diligas. Thy DAUGHTER is excellent, O mother, love her. By studying these examples, particularly the first, second, third, and fourth, you will see that in general

A pronoun agrees with its noun in gender, number, and case. A relative pronoun and a demonstrative pronoun may agree with its noun only in gender and in number, as appears from the sixth, seventh, and eighth. There are cases in which the relative agrees with its noun or pronoun in person only, as

Ille EGO, QUI quondam, gracili modulatus avena,
I, that well-known poet, WHO, etc.
Nunc horrentia Martis arma CANO.

Now sing of the frightful arms of Mars.

The general rule may be given thus:-The relative pronoun
agrees with its antecedent in gender and number. The ante-
cedent is the word, noun, or pronoun which goes before, and to
which in sense the relative refers. In the fifth sentence above,
coloniam is the antecedent to quam; and in the sixth libri is the
A proposition may be the antecedent, as-
Ego cum Pompeio in sermonibus versatus sum; qua nec possunt
scribi nec scribenda sunt.
I have discoursed with Pompey; which things (that is, the things which
were then spoken) can neither be written nor must be written.

antecedent to quos.

The things are true which I said When the reference is made to a fact, the neuter quod is used
commonly, having id before it, as-

when I was consul.

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Timoleon, id quod difficilius putatur, multo sapientius tulit secundam quam adversam fortunam.

Timoleon bore good fortune much more wisely than bad, a thing (that is,

to bear good fortune wisely than bad fortune.

reflective pronoun, denoting one's self. In these distinctions
lies the root of the diverse usages of the two.
These usages

to do so) which is thought more difficult (that is, it is thought more difficult may find a classification if we call ejus a demonstrative and suus a reflective pronoun. As a demonstrative, ejus is used when a reference is made to some person or thing not being the subject of the immediately connected verb; but the reflective suus must be used when the reference is to the subject of the immediately connected verb: for example:

In regard to gender, the same rules prevail between the relative and antecedent as between the subject and the attribute. In point of place, the relative is often put before the noun to which it refers, as in the following very rare construction:

Cecidere manu, QUAS legerat HERBAS.

The HERBS WHICH she had gathered fell from her hands.

The demonstrative pronoun must sometimes be supplied, as— Quos cum Matio pueros miseram (ii) epistolam mihi attulerunt; Those boys whom I had sent with Matium brought the letter to me ; where observe the marked difference of idiom, for boys, which is in the nominative case, and forms a part of the subject in English, is in Latin (pueros) in the accusative case, and forms part of the object.

The demonstrative pronoun is added to the relative for the sake of emphasis, as

Quam quisque novit artem, in hac se exerceat.

Lit. Trans. -What art each one knows, in this himself let him exercise.

Id. Trans.-Let each exercise himself in that art with which he is acquainted.

:

Alcibiades quum desertus ab omnibus jaceret
(When Alcibiades lay deserted by all),
amica corpus ejus texit pallio suo,

(a female friend covered HIS body with her cloak).
Here ejus is requisite, because the demonstrative is needed to
point out that corpus refers to Alcibiades. With pallio, how-
ever, ejus would be wrong; suus is required there, because the
cloak belonged to the subject of the verb texit; her cloak;
whose cloak? the cloak of the female friend.
The distinction is marked in these two instances
1. Omitto Isocratem discipulosque ejus.
I emit Isocrates and his disciples.
2. Isocrates cum discipulis suis adfuit.
Isocrates with his disciples was present.

Both ejus and suus are rendered into English by his, but they are of different import. In the first sentence his refers to discipulos, and not to the subject of the sentence, and therefore its there the pronoun refers to Isocrates, the subject of the sentence, and, in consequence, you want not the demonstrative ejus, but the refective suus.

The antecedent noun is sometimes repeated with the relative. proper Latin representative is ejus; not so the his in the second; Casar is fond of this construction ::

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The rule may be stated thus :

Pronouns agree with pronouns in gender and number, or in gender, number, person, and case.

Idem, added to another pronoun, gives force to a proposition,

39

Ptolemæus quum vivus filio regnum tradidisset, ab eodem illo vitâ privatus dicitur.

Ptolemy is said to have been deprived of life by the very son to whom in his lifetime he had given up his kingdom.

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1. Esopus scriptor clarus, fuit gibbosus. 2. Scythæ, homines bellicosi, terribiles fuerunt. 3. Phonices erant nautæ peritissimi. 4. Græcia fuit patria multorum hominum illustrium. 5. Conscientia bene actæ vitæ jucunda est. 6. Murus et porta de cœlo tacta sunt. 7. Duo fulmina Romani imperii fuerunt Cneius et Publius Scipio. 8. Brutus et Cassius interfectores Cæsaris fuerunt. 9. Vespasianus quæstor Cretam et Cyrenas provinciam sorte cepit. 10. Pompeius, a militibus desertus, Ægyptum petiit. 11. Philosophia vitæ est dux, virtutis indagatrix, expultrixque vitiorum. 12. Quid dicam de thesauro omnium, memoria ?

KEY TO EXERCISES IN LESSONS IN LATIN.—XLVI.
EXERCISE 178.-ENGLISH-LATIN.

1. Meorum peccatorum me pœnitet. 2. Vitae eum tædet. 3. Vitæne eum tædet? 4. Illos vitæ non tædet. 5. Omnium hoc interest. 6. Tuâ ac meâ hoc interest. 7. Nostrâne interest ? 8. Sermonis nostri eos tædet. 9. Vesperascit. 10. Pluit. 11. Pluitne ?

12. Grandinat. 13. Fulgurat. 14. Tonat. 15. Pluet totum per diem. 16. Ningit. 17. Pluit sanguinem. 18. Abi domum, nam vesperascit. 19. Libidinum eos pœnitet. 20. Fratris tui me piget. 21.

Idem, in eonjunction with another pronoun, must be rendered Alexandrum interempti Cliti amici pœnituit. by also, or the same as, or yet, as

Ultimus cœli complexus, qui idem æther vocatur.
The great expanse of the sky, which is also called ether.
Ipse may be joined to all persons, whether they are subjects
or objects. Ipse adds force to the word with which it is con-
nected; as, ego ipse, I myself; ille ipse, he himself. Ipse some-
times stands without a personal pronoun. If ipse refers to the
subject, it is in the case of the subject; if it refers to the object,
it is in the case of the object; for example:-

Subject.-Non egeo medecinâ; me ipse consolor.
I do not need medicine; I console myself.

Object.-Fac ut diligentissime te ipsum custodias.

Take care to guard yourself, most attentively.

Et ipse is used with the force of our even; alone, that is, by itself, without going further, as

Virtutes et ipsa tædium pariunt.
Even virtues beget wearisomeness.

Ejus (from is, ea, id) differs from suus, though both are rendered his in English. Ejus is properly of that person or that person's. Suus is one's own. The former is the genitive singular of is, that man; the latter is connected with se, sui, sibi, the

KEY TO EXERCISES IN LESSONS IN LATIN.-XLVIII.
EXERCISE 179.-LATIN-ENGLISH.

1. Athens was a city. 2. Romulus was a king. 3. Bactra, the chief
place of the district, lay under Mount Parapamisus. 4. Leonidas was
king of the Spartans. 5. Jugurtha led an army which attended to
land and cattle rather than to war. 6. Money is a queen. 7. Plea-
sures are very winning mistresses. 8. Flatterers are a very base race
of men.
9. Twenty ships, the fleet of the Romans, were impelled by
Oars. 10. Pompey, our dear friend, will come. 11. Corinth, the sun
of all Greece, has been razed. 12. Death is the end of all things. 13.
Carthage and Corinth, most opulent cities, were overturned. 14. Not
the sword but gold conquered the Lacedæmonians, the bravest of
mortals. 15. Philosophy, the discoverer of truth, is also the guide of
character and discipline.

EXERCISE 180.-ENGLISH-LATIN.

1. Theba, Bootiæ caput, site sunt sub monte Citharone. 2. Lon. dinum Angliæ est caput. 3. Magistra Angliæ est Londinum. 4. Aliarum virtutum generatrix est frugalitas. 5. Scythi, bellicosissimi homines, carne vescuntur. 6. Omnium mortalium bellicosissimi sunt Scythæ. 7. Victor omnium nationum est Populus Romanus. 8. Totius mundi caput erat Roma. 9. Athenæ olim artium erat domicilium, et nutrix philosophiæ ac eloquentiæ.

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