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a le bas de coton? 25. Le médecin a le bas de coton. 26. La a-t-elle une montre d'or? 8. Elle a une montre d'or et un de dame a le soulier de satin de la sœur du boulanger.

EXERCISE 8.

TO BE TRANSLATED INTO FRENCH.

d'argent. 9. Le boulanger a-t-il du vin ou de la bière? 10. Le boulanger a du thé et du café. 11. Votre frère a-t-il du fromage? 12. Il a du fromage et du beurre. 13. La dame a-t-elle une cuiller d'argent? 14. La dame a une cuiller et une fourchette d'argent. 12. Le boucher a-t-il de la viande aujourd'hui? 16. Oui, Monsieur, il a un morceau de bœuf. 17. Le charpentier a-t-il une table? 18. Oui, Monsieur, il a une table d'acajou. 19. Avez vous le livre du médecin ? 20. Non, Madame, mais j'ai le livre de votre sœur. 21. Qui a du café et du sucre? 22. L'épicier a du café et du sucre. 23. La elle a un livre. 25. A-t-elle une plume d'acier? 26. Non, Monsieur, elle a une plume d'or. 27. Vous avez le portecrayon du médecin.

1. Have you the tailor's book? 2. No, Sir, I have the physician's watch. 3. Who has the gold watch? 4. The lady has the gold watch and the silver pencil-case. 5. Have you the tailor's shoe? 6. I have the tailor's cloth shoe. 7. Have we the wooden table? 8. Yes, Sir, you have the wooden table. 9. Have they the silver knife? 10. They have the silver knife. 11. The lady has the silver knife and the gold pencil-sœur du libraire a-t-elle un gant? 24. Non, Monsieur, mais case. 12. Has she the satin dress? 13. The physician's sister has the satin dress. 14. Who as the wood? 15. The carpenter's brother has the wood. 16. Have you the woollen stockings? 17. No, Si-, but I have the cotton stockings. 18. Who has the baker's bread? 19. We have the baker's bread and the miller's flour. 20. Have we the horse's hay? 21. You have the horse's oats. 22. Have we the tailor's silk hat? 23. Yes, Sir, you have the tailor's silk hat and the shoemaker's leather shoe. 24. Have you the cloth shoe of the physician's sister? 25. No, Mɛ dam, I have the lady's silk dress.

SECTION IV.

EXERCISE 10.

1. Have you any tea? 2. Yes, Madam, I have a pound of tea. 3. Who has bread? 4. The baker has bread, butter, and cheese. 5. Has the tailor cloth? 6. The tailor has piece of cloth. 7. Has the physician gold? 8. Yes, Sir, the physician has gold and silver. 9. Has the lady a silver watch? 10. Yes, Miss, the lady has a silver watch and a gold pen. 11. Has your sister silk? 12. Yes, Sir, she has silk and cotton. 13. Have you a knife? 14. Yes, Sir, I have a steel knife and a silver fork. 15. Have you meat to-day, Sir? 16. Yes, Sir, I have a piece of beef. 17. Has your carpenter a mahogany table? 18. Yes, Sir, he has a mahogany table. 19. Has your sister a glove? 20. Yes, Sir, my sister has a silk glove. 21. Has the bookseller's son a gold pencil-case? 22. Yes, Sir, he has a gold pencil-case and a steel pen. 23. Who has your sister's watch? 24. Your brother has the gold watch and the un, m., une, f., answers to silk hat. 25. We have gold, silver, and steel (See Rule 5).

1. The article, pieceded by or contracted with the preposition de, according to rules 1 and 2 of Section III., is placed in French before words used in a partitive sense. Such words may generally be known in English when some or any is or may be prefixed to them [§13, (10), § 78, (1)].

Du pain,
De la viande,

De l'argent,

2. The French numeral adjective.

Bread, or some bread.

Meat, or some meat.

Money, or some money.

the English indefinite article, a or an [§ 13, (4), (11)].

Un homme,
Une femme,

A man.

A woman.

PROBLEMS AND QUERIES.

3. The e of the preposition de is elided before un and une As some of our young readers may be already acquainted with [$146].

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arithmetic and geometry, we propose the following problems and queries for exercise :

1. How could the first nine letters of any alphabet be made to represent numbers to any extent, say millions, without making their values dependent on their position?

2. In such a system of notation as we have supposed in the preceding question, what would be the use of the cipher? greater than one expressed by five figures, although the latter is 3. How is it that a number expressed by six figures must be composed of large digits, and the former of small ones?

N.B. The nine characters or figures employed in arithmetic are called digits, because men were, and still are, accustomed to count by the digiti or fingers. Of course, 9 is the largest digit, and 1 is the smallest.

4. Express by figures the number twelve thousand twelve hundred and twelve.

5. Arrange three points, so that they shall be equally distant from each other.

6. Arrange four points so that they shall be equally distant from each other.

7. Construct an isosceles triangle such, that each of its legs shall be double the base or third side.

8. The celebrated theorem of Pythagoras, which now forms the 47th proposition of the 1st book of Euclid's Elements of Geometry is, that "in any right-angled triangle the square described on the hypotenuse (see fig. 11, Lessons in Geometry, No. I.) is equal to the sum of the squares described on the other sides." In one of the latest and best editions of Euclid, it is asserted that "it is not indifferent on which sides of the lines which form the sides of the triangle the squares are described. If they were described upon the inner instead of the outer sides of the lines, the construction would be found to fail." Is this true?

ANSWERS TO CORRESPONDENTS.

G. P.-We are greatly obliged by his kind offer.

R. W. L.-We do not belong to the school he supposes.

M. N. We inform him that the " Series of Lessons in French," reprinted from the Working Man's Friend, will, "by special permission of her Majesty's Postmaster-General," te transmitted free to any person's address on the receipt of Seven Penta Stamps, at this office.

Printed and Published by JOHN CASSELL, 335, Strand, and Ludgate-hill,
London.-April 10, 1859.

THE INFLUENCE OF MORALITY OR IMMORALITY ON THE COUNTENANCE.

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We have taken the above engraving from a popular French publication, as it exhibits to the life what it was meant to illustrate. But what need have we to go to a literary institution or a French faubourg, to become acquainted with the growth and development of humanity? We have only to look around us to discover that everywhere man is the same. Much has been said of the science of phrenology; but without depreciating the facts on which it is professedly based, we confess that we have a more profound faith in the doctrine of physiognomy. No one can deny that the "human face divine" has in it something expressive of that which enters into and constitutes the character of the man. It may come out in the eye, or the lip, or the nose, or the general contour of the countenance; but there it is, and no one can give himself to the closer and deeper study of this subject without being able, more or less correctly, to read the mysterious symbols of human character and destiny.

evil would otherwise rear her temple of darkness and impurity. Look at the eye, nose, and mouth of the boy as he is at school, or as he is located in one of the faubourgs of Paris, and who does not perceive, from the very contour of the countenance, that his destiny will very much depend on the influences by which he may be surrounded? In the one case, you see him pass into the higher and more polite circles of the educated classes, yielding himself to all the softening, subduing, refining elements of pure female society; and in the other, you see him trained and educated in a rebellious and revolutionary faubourg, rushing into the scenes of dissipation, and surrendering himself to all the worst agencies of a wicked world. In the one instance, you see him choosing his profession, and contemplating a settlement in life:wedding himself to a virtuous, loving, and devoted woman, and in course of time becoming surrounded by a lovely and growing family, over which his presence sheds a warm Carefully examine the above engraving. Look at the head and sunny cheerfulness;-in the other instance, you see and face of the child represented in the first figure. Who can the man emerging from the scenes of brutal intoxication to divine what that young intelligence will become in the future plunge into deeper, darker vices, till his conscience is burof his life? Is there anything in his features to indicate dened with guilt and misery, and life becomes a burden, that he will act a conspicuous part on the great wide stage of from which he perhaps seeks relief in suicide; or it may this world? Or is he to sink in the scale of intelligent being, be, that his conduct renders him obnoxious to law, and till he takes on the mere animal nature, or, what is still worse, he comes to a premature death. If he be spared this tretill he become the very personification of vice and sin? Even mendous fate, he comes to beggary, and goes down to the in the outlines of the infant countenance there may be the grave unlamented and unwept. How different this from the index of the future man. These outlines will become more career of the man whose happiest days are spent in the bosom marked and definite in the boy amid the studies and pur- of his loving family, and who grows old amid the most genial suits of the school. The period of boyhood is one of wondrous influences, honoured, revered, beloved-who sees his development; and if this were but carefully watched, the foundation might in many cases be laid for the erection of a true manly nobility; and that undermined, on which moral

VOL. I.

The low suburbs of a city, such as Paris.

children's children under the third or the fourth genera tion, and who goes down to his last resting-place amid the prayers and tears of those he loved, and whose dying moments are cheered by the prospect of a happy reunion in a world where life is perfect, and joy complete.

3

LESSONS IN LATIN.--No. II.

By JOHN R. BEARD, D.D.

PRELIMINARY INSTRUCTIONS IN THE VERBS.

In the former lesson I had to employ the word curro, and in doing so I used particularly the form curr. This form is called the stem of the word. The stem of a word is that which contains its essential letters, or the letters which are necessary to make it what it is. Thus, curr is found in every form into which the verb curro passes. Observe that the second ris added merely to strengthen the word, or give force in pronunciation. You find this stem, cur or curs, for instance, in the English words current, incur, concourse, concurrence, discourse. Observe again, that many of our Latin words have come to us through the French. They have, therefore, entered the English in the form which they had received in the French.

Now curro, of which I have spoken before, is of the third conjugation. The person-endings in it will not therefore be the same as they are in the verb amo, I love. The personendings in amo are o, as, at; amus, atis, ant. In the tables or paradigms (patterns), which I am about to give, the persontherefore easily recognise them, and ought in all cases to repeat endings are printed in italics, as amo, amas, amat; you will them until you have imprinted them on your memory.

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EXAMPLE.-Amare, to love; stem, am.
This is exemplified in concourse and discourse, where

an o has been introduced by the French pronunciation, as
these words come to us immediately from the French concours
and discours.

1st per. Amo,

PRESENT ACTIVE INDICATIVE.
Singular.
I love.
2nd Amas, thou lovest
3rd Amat, he loves
Plural

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1st per. Amamus, we love
2nd Amátis, you love
3rd Amant, they love

-antur, they

PRESENT PASSIVE INTICTATIVE.
Singular.

1st per. Amor, I am loved
2nd Amáris, thou art loved
3rd Amatur, he is loved
Plural.

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1st per. Amémur, we are loved
2nd
Amamini, you are loved
3rd Amantur, they are loved

Observe then that in order to form any person, you must first get the stem, by cutting off the last syllable. Then to the stem thus obtained, add the proper person-ending. Suppose you have to deal with the verb laudo, I praise; and suppose you want to express in Latin, the English they praise; the way to proceed is-throw away the o in laudo; by so doing, you get laud; now, they praise is in the third person plural; the personending of the third person plural is ant, as shown above; subjoin ant to laud, and you have laudant, which means they praise. Or if you have to put laudas into English, by looking at the table you find that its termination, namely, as, is the person

The stem of a word is found generally by cutting off the final vowel or syllable. In curro you obtain the stem curr by The taking away the o. in reality is the sign of the first person singular, or I. The word for I is not prefixed in Latin, except when it is required for emphasis, because the terminations of the verb show clearly what person is meant-that is, whether it is the first person, I, or the second person, thou, or the third person, he. In the English there is a necessity for the constant use of the personal pronoun, because the endings of the verb are not so different from each other as in the Latin. Thus, in English, we say, I run, thou runnest, he runs, we run, you run, they run. Here, out of six persons, the verb has the same termination for four, namely, I run, we run, you run, they run. But for the pronouns I, we, you, they, the reader or auditor would not be aware from the use of the verb which person was intended. In the Latin, however, the verb has a peculiar ending for every person. After this explanation, we will call these terminations person-endings. These person-ending of the second person singular, and consequently iaudas endings vary with the tense and the voice. That is, they are different in the past tense from what they are in the present tense; and they are different in the passive voice from what they are in the active voice. At present we will confine ourselves to the present tense and the active voice. In curro, the person-endings of the present tense, indicative mood, active voice, are as follow:

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1st person
2nd
3rd

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ACTIVE VOICE.-INDICATIVE MOOD.
PRESENT TENSE of the verb curro, I run.
curro, I run
1st person
curris, thou runnest
currit, he runs

2nd 3rd

currimus, we run
curritis, you run
currunt, they run

means thou praisest. I have entered into this full and minute explanation once for all. If you take pains to make yourself master of it, you will not require its repetition. But take care not merely to consult the tables I give; you must commit them to memory; and never pass on until you have made them in each case your own. Having learnt the form or example, learn the vocabulary, and then put the Latin exercise into English, and the English exercise into Latin. Do this also from memory; but after you have done it, compare your translation with the table or example, and correct it accordingly.

Discover and write down the English representatives of the Latin words here used; that is to say, the English words derived from these Latin words. For instance, from delecto, I delight, we have delight, delightful, delightfully; from orno, I adorn, we have ornament, ornamentally, adorn, adornment; from educo (which properly means I draw out), we have to educate, educator, education. Do the same after every separate exercise.

What I have called "the characteristic" of the verb, may be These person-endings vary also in another way, which I called the sign of the conjugations. Thus, of the first conjuproceed to explain. Latin verbs are commonly divided into gation a long is the sign, and i is the sign of the third. These four classes, which bear the technical name of conjugations. are Latin signs. Of the corresponding part of the English This division may not be the best, but it is that which is cus-verb, to is the sign; that is, the preposition to is in general the tomary, and therefore I retain it. These four classes or conju- English sign of the infinitive mood. gations are determined or characterised by the vowel which precedes the termination re in the infinitive mood; thus:The first conjugation ends in äre, as amare, to love. The second conjugation ere, as docere, to teach. The third conjugation

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ère, as regere, to rule,
ire, as audire, to hear.

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The fourth conjugation We say then that the first conjugation is known by having a long before re of the infinitive; the second by having e long; the third by having e short; the fourth by having i long. The same fact may be put before you in a different way; thus, a long is characteristic of the first conjugation; e long, of the second; e short, of the third; long, of the fourth. In general it may be remarked, that in the first conjugation a long prevails; in the second, e long prevails; in the third, e short prevails; and in the fourth, i long prevails.

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LESSONS IN LATIN.

you are blamed; we try; you are tried; he is wounded; I am educated.

Now, before you go forward in this exercise, and in every other, ask yourself, and ascertain that you give the right answers to the following or similar questions; namely, of what conjugation is the verb amo? of what tense is amo? of what person is amo? of what number is amo? of what mood is amo? of what voice is amo? Do the same with all the rest.

SECOND CONJUGATION.

ACTIVE VOICE.

PRESENT INDICATIVE.

PERSON-ENDINGS.

PASSIVE VOICE. PRESENT INDICATIVE.

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Plural.

-emur, we

PERSON-ENDINGS.

Singular.

Plural.

Singular.

-e0, I

-emus, we

-eor,

-es,

-etis, thou

ye

-eris,

-et, he

-ent,

they

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thou -emini, ye

Plural.

we

-imur, -imini, ye

-tuntur, they

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PRESENT PASSIVE INDICATIVE.

Singular.

-entur, they

1st per. Audio,

I hear

1st per. Audior,

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2nd Audis,

3rd "

thou hearest he hears Audit, Plural.

EXAMPLE. Monére, to remind; stem, mon.

PRESENT ACTIVE INDICATIVE.

Singular.

1st per. Moneo, I remind

2nd

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3rd

1st per.

Mones, thou remindest

Monět, he reminds

Plural.

Monēmus, we remind you remind they remind

Monétis,

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Monent,

3rd 99

PRESENT PASSIVE INDICATIVE. Singular.

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1st per. Moneor, I am reminded Moneris, thou art reminded 2nd 3rd, Monetur, he is reminded Plural.

3rd VOCABULARY.

1st per. Audimus, we hear 2nd, Auditis, you hear 3rd Audiunt, they hear

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VOCABULARY. Fulcio, 4 Nutrio, 4 Punio, 4 Reperio, EXERCISES.-LATIN-ENGLISH.

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1st per. Monemur, we are reminded 2nd Monemini, you are reminded Monentur,they are reminded

Custodio, 4
Dormio, 4
Erudio, 4
Ferio, 4

I guard
I sleep
I instruct

I strike

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I fear Timèo, 2 Et means and

Si means i

Debes; docet; exercétur; florémur; gaudémini; mordentur; movémus; movétis; movent; times; timet; terrétur; terrémini; debeo; parere; si paretis laudámini; si tacemus laudámur; docéris et educaris; tacent et laudantur; mordeor et vulneror; si vulneras vituperáris.

ENGLISH-LATIN.

Thou fearest and art frightened: if I am silent I am blamed; he rejoices; we rejoice; they rejoice; he tries to bite; we try to educate; thou obeyest and art praised; we bite; if we bite we are blamed; they exercise; you are moved; he dances; they are delighted; you are adorned.

N.B. In this exercise, and in those which follow, words and forms previously given are repeated for the sake of practice.

ACTIVE VOICE.

THIRD CONJUGATION,

PASSIVE VOICE. PRESENT INDICATIVE.

Custódis; fulcitur; venit; cur dormis? bene dormit; eruditur ; valde dormis puniris; pungis; occidit; valde fallis; auditur; reperit; si bene erudis laudáris; vincitur; cur taces? tacet et punítur; reperiuntur; vestiris; bene vestiuntur; si bene vestimini delectamini; male erudiuntur; si vinceris vinciris.

ENGLISH-LATIN.

Why do you slay? he is guarded; they guard; if you are guarded instructed; you are well educated; thou sleepest much; they read; you are conquered; he blames and punishes; he hears and is if you dance you are delighted; he is supported; why are they punished? they are heard; I am clothed ill; they are struck and

reminded.

RECAPITULATION-TERMINATIONS OR PERSON-ENDINGS OF
THE FOUR CONJUGATIONS.
INDICATIVE MOOD, PRESENT TENSE.

ACTIVE VOICE. Singular.

1.

2.

3.

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1st per.-or -10

PASSIVE VOICE. Singular. 8. 2. -eor -or

4.

-ior

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-28 -et -at Plural.

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-is

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-it

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2nd -āris eris -ĕris iris -aturētur -Itur -itur 3rd -it

Plural.

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1st per. -amus -ēmus -Imus-īmus 2nd -atis -ētis -Itis -ītis -unt -iunt 3rd -ant -ent

PRESENT INDICATIVE.

PERSON-ENDINGS.

PERSON-ENDINGS.

EXAMPLE.-Rěgère, to rule or guide; stem, reg.

PRESENT ACTIVE INDICATIVE.

Singular.

1st per. Rego,

2nd Regis,

"

3rd Regit,

Plural.

I rule thou rulest he rules

1st per. Regimus, ce rule 2nd Regitis, you rule Srd Regunt, they rule

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99

PRESENT PASSIVE INDICATIVE.
Singular.
I am ruled
1st per. Regor,
2nd Regĕris, thou art ruled
3rd
Regitur, he is ruled

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Plural. 1st per. Regimur, we are ruled Regimini, you are ruled Reguntur, they are ruled

2nd,

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3rd VOCABULARY. I read Lego, 3 Occido, 3 I slay I fall Decido, 3 I paint Pingo, 3 I prick Pungo, 3 EXERCISES.-LATIN-ENGLISH.

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Scribo, 3 I write
Vinco, 3
I conquer
Bene means Well
ru
Male
Valde
Much

Fallis; fallitur; fallimur; fallo et vituperor; cedit; legis; scribit; bene legit; valde fallis; si diligitur gaudet; pungimur; vincis; vincimur; vincuntur; decidit; occidis; si occidis peráris; bene monet; male educáris; movémur valde; saltámus et gaudémus; laeditur: laedimini; defenditis; defenduntur; diligor.

1st per.-amur-emur-imur-imur

2nd,,-amini-mini-imini-îmini 3rd,, -antur-entur-untur-iuntur

RECAPITULATORY EXERCISES-LATIN-ENGLISH.

Cedo; legis; movémus; exereéris; mordent; florent fallor; legere tentat; cur male legis? dormit male; valde diligĕris; vincimini; bene scribunt; si bene pingitis laudámini; defendimur; ferimus; cur punítis? vestímur; vincímus; vincimur; vincímur; vincitis; custodíris; ornátur; laudantur; timémur; valde times; mordémini; educámus; malé saltant.

ENGLISH-LATIN.

They yield; if you yield you are conquered; if you are conquered you are bound; I am supported; they sleep; why do they punish? why are they punished? you are clothed ill; thou conquerest; thou art conquered; thou bindest; thou art bound; they prick; they are pricked; why dost thou move?

As in the exercises which are immediately to follow, we shall have occasion for parts of the verb esse to be, I shall here lay before you so much of that verb as may be necessary for my purpose.

THE VERB ESSE, to be.
INDICATIVE MOOD-PRESENT TENSE.
Singular.
Sum,

Plural,

vitu

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Those adjectives which in the singular and in we, form the plural in 6, thus "I am good" is bonus sum but "we are good" in homi sumus. In order to form the plural, out off the termination us, you thus get the stem; to the atem add f

Remark conn. In Latin the order of the words is not so

LESSONS IN ARITHMETIC.-No. III.
DEFINITIONS AND SIGNS.

calculations with numbers, and these are usually called the
In arithmetic, there are four different methods of performing
four common rules, namely, Addition, Subtraction, Multipisca-
from, and Division.

Addition is the process by which we collect several numbers into one, which is called their sum.

has been invented which is called the sign of addition; it is in the form of an upright cross, thus +; and its name is plus or more, signifying added to. It is placed between two or more numbers to denote that their sum is to be found: thus, 6+7 make 13; +6 + 10 make 20; and so on. In these exand the number 20 the sum of the numbers 4, 6, and 10. amples, the number 13 is the sum of the numbers 6 and 7;

In order to shorten the language of arithmetic, a character

Figidly Read as it is in English. In English we say I am good, and do not commonly say good 1. But in Latin we may any either aum bonus, I am good, or bonus sum, good am I. This change in the relative position of the words of a sentence, is called inversion The Latin language has great capability of inversion. The inversions which it employs are neither Henatural nur elitrary The invisions depend on the sense, If we wish to throw emphasis in the epithet good, then we Another character has been invented, which is called the must place bunus Beat Fue peample, suppose you wish to say that I am good hub mod agh, you do ao by putting bonus sign of equality; it is in the form of two small parallel strokes, before um But if you wished to say I am good, in opposition is made, equal to. It is placed between numbers, or combinathus; and its name is equation, signifying that which is, or to some one who maya yum uia uul good; then, you say, sumtions of numbers, to denote that the expressions or numbers hamus, and nuk konus sum. As then these inversions were a moans by which the Homana gave expression to their own feelings and opinions, they were with them perfectly natural; and if they have the appearance of being unnatural to us, it is merely because we expresa emphasis differently; that is, we express by certain tones of the voice the emphasis which they expressed by the position of words. In saying this, however, 1 do not mean to assert that the Romans never gave emphasis by intonation. The beginning of a sentence is the place of chiet emphasis, neat to this stands the end; an intermediate position has least emphasia.

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QUESTIONS FOR Review.

of

What is the meaning of delecto? what person, tense, and mood ja delas fuel what is the stem of lego? what is the stem of vinco? What person, mood, and tense is eruditur? put the proper accent wwanemus on laudatur; on legitur; on vincimur, we are bound; he is conquered; on occiditur, he is slain; on fulciris; yes, in educamur; on vestiris; what is the person-ending of the thud person singular, passive voice, indicative mood, of amo? wivel of docent of punio? what is the person-ending of the and person plural, passive voice, indicative mood of moveo? of art of vituperat of ornot of diligo? what is the English for but educatur? for vituperdris? for laederis? for occidor? hit for diligimur for defenditur? what English words Lived from the Latin debeo; terreo; moveo; floreo; laudo; wiba vitupero; audio; punio; tento; erudio; vestio; Tallo Write down from memory the person-endings four conjugations, indicative mood, present tense, active tve voices, singular and plural number. Put on each part Soper accent also from memory. What is the Latin sign of jugation? of the third? of the second? of the fourth om memory a table of the four conjugations, with the caning appended to each. What is the English sign of Give mood? when are pronouns prefixed to verbs in Latin? hey not usually prefixed as in English? what is the perof the second person, present tense, indicative mood of jugation? the same in the passive voice? how do the hrough what language have some words of Latin to the English language? what is denoted by these , what is accent? what good may be exdying Latin? write down in your own words the which may accrue from studying Latin, not mentioned structions. What is the meaning of inversion? what atins express by inversion? what is the meaning of esse form the plural of bonus? from the plural docti form What is the meaning of salvus of caecus? of lifference between salvi sunt aud sunt salvi?

on each side of it are equal: thus, 6+7=13, means 6 added to 7 is equal to 13; and 4+6+10=20, means 4 added to 6 added to 10 is equal to 20; and so on.

between any two numbers; the difference signifies the number Subtraction is the process by which we find the difference which must be added to the smaller of two numbers to make the sum equal to the other number.

A character has also been invented, which is called the sign of subtraction; it is in the form of a small horizontal line, thus; and its name is minus or less, signifying made less by. It is placed between two numbers, of which the greater stands on the left, to denote that their difference is to be found: thus, 14-6-8, means 14 made less by 6 is equal to 8. In this example, the number 8 is the difference between the numbers 14 and 6.

Multiplication is the process by which we find the sum of one number repeated as many times as there are units in another number; the two numbers are called factors, the former being called the multiplicand, and the latter the multiplier; and the sum thus obtained is called the product of both numbers, or the multiple of either.

A character has been invented, which is called the sign of multiplication; it is in the form of an oblique cross, thus X and its name is into, or times, signifying multiplied by. It is placed between two or more numbers to denote that their pro duct is to be found: thus, 6X 7=42, means that 6 times 7 is equal to 42; and 4 × 6 × 10=240, means that 4 times 6 times 10 is equal to 240; and so on. In these examples, the number 42 is the product of the factors 6 and 7, or a multiple of 6 or 7; and the number 240, the product of the factors 4, 6, and 10, or a multiple of 4, 6, or 10. In the former example, the number 6 may be the multiplicand, and the number 7 the multiplier; or, conversely, the number 7 the multiplicand, and the number 6 the multiplier.

Division is the process by which we find how many times one number is contained in another; the former number being called the divisor, the latter the dividend, and the number of times the quotient.

division; it is like the sign of subtraction, with this difference, A character has been invented, which is called the sign of that when the dividend stands on the left, and the divisor on the right of it, the horizontal line has a dot above it and a dot below it, thus; and its name is upon or by, signifying divided by. It is placed between two numbers to denote that their quotient is to be found: thus, 124=3, means that 12 divided by 4 is equal to 3. In this example, the number 12 is the dividend, the number 4 the divisor, and the number 3 the quotient. In the use of this sign, the greater number is always placed on its left.

In the use of the sign of division, the dots may be dispensed with, if we place the dividend above and the divisor below the horizontal line, thus: 3. This mode of representing division is very useful in the doctrine of fractions; also in the theory of proportion, where the quotient 3 is called the ratio of 12 to 4.

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