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GENERAL RULE FOR MULTIPLYING RADICALS.

Radicals of the same root are multiplied by adding their fractional exponents.

If the quantities have the same radical sign, or index, multiply them together as you multiply rational quantities, place the product under the common radical sign, and to this prefix the product of their co-efficients.

If the radicals are compound quantities, each term in the multiplier must be multiplied into each term of the multiplicand by writing the terms one after another, either with or without the sign of multiplication between them.

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One of our students has sent us the following creditable metrical version of

HORACE, Book 3, Ode 7.

Austeria, why bemoan the youth

Whom favouring winds and love of truth,

Laden with precious things will bring

Back to thy side in early spring?

Thy Gyges long by tempests tossed,
And cast on some deserted coast,
Full many a cold and sleepless night
Will pass before he greets thy sight.
Though his admiring hostess tries

To catch him with her young bright eyes,
Declares her love, around him plays,
And tempts him in a thousand ways;
Relates how simple Prætus nigh,
Had raised the assassin's knife on high,
And stabbed Bellerophon the good
To please a woman's spiteful mood,-
How Peleus feared the assassin's arms
For slighting Hypolita's charms,
And cites him history to prove
That faith is seldom kept in love;-
Her arts are vain. More deaf is he
Than rocks in the Icarian Sea;
Yet fears he lest Enipeus win
Thy heart ere he returns again.

What youth like Gyges e'er could guide
A ship on the Etruscan tide?

What youth like Gyges could restrain

A fiery courser on the plain?

Then close thy door at night betimes,

Nor heed the serenader's ryhmes;

To Gyges ever constant prove,

And Gyges will requite thy love.-J. T. P.

SON.

BY J. R. LOWELL.

The rich man's son inherits lands,
And piles of brick, and stone, and gold;
And he inherits soft, white hands,
And tender flesh that fears the cold;
Nor dares to wear a garment old:

A heritage, it seems to me,
One would not care to hold in fee.

The rich man's son inherits cares;
The bank may break, the factory burn;
Some breath may burst his bubble shares.
And soft, white hands would hardly earn
A living that would suit his turn:

A heritage, it seems to me,

One would not care to hold in fee.

What does the poor man's soon inherit?
Stout muscles and a sinewy heart;

A hardy frame, a hardier spirit;
King of two hands; he does his part,
In every useful toil and art:
A heritage, it seems to me,

A king might wish to hold in fee.

What does the poor man's son inherit?Wishes o'erjoyed with humble things;

A rank adjudged by toil-worn merit;
Content that from employment springs;
A heart that in his labour sings;

A heritage, it seems to me,

A king might wish to hold in fee.

What does the poor man's son inherit?

A patience learned by being poor,
Courage, if sorrow come, to bear it,

A fellow feeling that is sure

To make the outcast bless his door:

A heritage, it seems to me,

A king might wish to hold in fee.

Oh! rich man's son, there is a toil
That with all others level stands:
Large charity doth never soil,
But only whitens, soft, white hands:
This is the best crop from the lands:
A heritage, it seems to me,

Worth being rich to hold in fee.

Oh! poor man's son, scorn not thy state;-
There is worse wearines than thine,
In merely being rich and great;
Work only makes the soul to shine,
And makes rest fragrant and benign:
A heritage, it seems to me,
Worth being poor to hold in fee.

Both heirs to some six feet of sod,
Are equal in the earth at last;
Both children of the same dear GOD;
Prove title to your heirship vast,
By record of a well-filled past:
A heritage, it seems to me,
Well worth a life to hold in fee.

Quand vous aurez donné à une femme le nom sacré d'épouse, vous devez vous consacrer à son bonheur, comme elle doit se consacrer au vôtre. Mais l'obligation qui pèse sur vous est d'autant plus grande que votre femme est une créature plus faible, et que vous lui devez, comme étant le plus fort, le bon exemple et toutes sortes de secours.-Silvio Pellico.

Le méchant fuit sans être poursuivi de personne; mais le juste est hardi comme un lion, et ne craint rien,-Salomon.

LESSONS IN GREEK.-No. XLVIII.

By JOHN R. BEARD, D.D.

INVARIABLE WORDS.

THE words which we have hitherto studied are susceptible of certain changes. With these changes I have endeavoured to make the student familiar. There are other words which do not undergo change, or undergo change only to a small extent. Many of these have occurred in the course of the instructions I have given. Nevertheless, invariable words must be put together, and spoken of specifically.

Relations to Place.

1. Place where you are

3. Place whence you come

4. Place through which you pass

5. Place at which you stop

down which you go

6. Different relations of position

Place by the side of

together with

The Prepositions.

The prepositions require careful study, as on them, as well as on other invariable verbs, the sense very much depends, and you will be ignorant of some of the most delicate shades of meaning, and unaware of many an elegance, if you do not familiarise your mind with the import and the usage of the prepositions and the conjunctions particularly.

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Prepositions have a relation to place, and denote the direction of an action in regard to place. Thus, I say 44 you go from home;" "you go to home;""you go round the house;" "you go over the wall." In order, therefore, to your possessing an exact knowledge of the prepositions, of which there are in Greek eighteen, you must study them in their relation to place. In the ensuing table, they are arranged as they stand in regard to place.

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2. Place whither you go

3. προς

ad

to

4. εκ οι εξ

ex, e

5. απο

ab, a

6. δια

per

7. ανα

per

out of from through up

8. ката

ad, de

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11. συν and ξυν cum

with

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over

12. ὑπερ

super

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In this general view it is only the general meanings that I have been able to give. It is the office of Syntax to set forth details, especially as the import of the prepositions is modified by the case which they govern, for, as you already know, most of them take more cases than one.

A little close attention will show you that the Latin prepositions are intimately connected with the Greek in origin as well as signification: thus, ev and in are the same; so are εK and ex; so, also, rep and super, and vño and sub.

The following six words may also be considered as prepositions, namely:

ατερ

ανευ

over, above

under, by before

around

on

against instead of

7. ανα

8. κατα

9. παρα

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ἕνεκα αχρι μέχρι πλην

on account of up to, until but, except

ανευ τῆς δικαιοσυνης, without justice ένεκα τούτου, on that account μεxoi 'Pwμns, to (as far as), up to Rome πλŋν ¿vog, one excepted.

17. επι

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αναβαινω

αναλαμβάνω καταβαίνω

παραγω παραβαίνω μεταμορφόω

μεταλαμβάνω συλλαμβάνω ὑπερβαίνω ὑποβαλλω

υποτελ

προβαίνω αμφιβαίνω περιβαίνω

a slave.

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επιβαλλω αντιβαίνω αντιδιδωμι

I go against.

I give instead, in exchange for.

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18. αιτι

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Sometimes the preposition is expressed and united to the noun, as:

παραχρῆμα (παρα, αἱ ; χρῆμα, the thing), at the moment. πроνруоν (πро, for; spyov, the deed), usefully, beforehand. εKπodшV (εK, from; πoνs, the foot), at a distance, far from.

Adverbs formed from adjectives imply a substantive :Dat. ιδια (εν ιδια χωρα), in particular ; πεζῇ εν πεζῇ όδῳ), on foot. Acc. μακραν (εις μακραν ὁδον), a long way, at a distance.

The neuter of the adjective is often employed as an adverb; as the dat. Toλw, much, by much; ǹov, agreeably; devov and deva, terribly; TоTEрa, whether? εñiτηdes, on purpose.

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6. Affirmation.

uny, yes, certainly, in truth.

ápa, in the poets

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ει μη

ει και

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sive

nisi, si non et si, etiam si

et si

quod ut

ut

ne, ut non

quia

quia

itaque

cum or quum postquam

quum

dum quomodo sicut

or, whether. unless, if not.

and if.

and if.

that.

so as, so that.

in order that.

in order that not, lest..

because, since,

because, that. therefore.

since.

after that.

when.

until.

how?

as, as if

Of these conjunctions some are simple, as raι, TE, wc; others are compound, as ουτέ (ου and τε), μεντοι μεν and του), και τοι (και and τοι), τοινυν (του and νυν), ώςτε (ὡς and τε), διοτι (δια οτι, neuter of όςτις), γοῦν (γε and οὖν), επειδη (επει and δη), επειδαν (επει, δε, and ar), όταν (ότε and ar); and others are two separate words, as ει μη, ίνα μη.

There are other conjunctions, whether a single word, as ἡνικα, when, or several words united, as τοιγαρτοι (τοι, γαρ, Toi), now then; Tоiyapovv (Toι, yap, ovv), wherefore, on that account; or, again, several words in a separate state, ov μyv alla, however; \ηy & μn, if only.

One or two others deserve notice, as ȧre, seeing that, as being, e.g. are aуalog, as being good (Latin, utpote bonus); πεр, although, e.g. ayabos πεр, although good.

There are certain words employed as adverbs, in the composition of which there is a conjunction, e.g. dnλovori, evidently; that is, onλov torv ori, it is evident that; Eviore, sometimes, made up of evi, for εoriv, and ore (in Latin, est quando). The Greek is rich in conjunctions. Some of its conjunctions express shades of meaning which can scarcely, if at all, be

440

than those of that science.-ALFRED C. HILARY will find the work on rendered into English, and which can be appreciated only in by plane geometry, when there is no occasion to apply any other principles Chemistry, published in Chambers's Educational Course, very good and the original, by the fine sense of a superior scholar. cheap.

Interjections.

Interjections, as expressing almost inarticulately the passions and emotions of the mind, are also numerous in the language The of the Greeks, who were a people of strong feelings. principal interjections are these:

, O! (sign of the vocative, &, 0! expressing pain or surprise.) Lov, ah! alas!

ε, eheu! ho!

φεν, αλ!

Baßai, ah! oh! (Latin, papæ !)

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HISTORICAL ANECDOTES.

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Αγησίλαος περι ανδρειας και δικαιοσυνης ερωτηθείς, πότερα βελτιων, Ουδεν ανδρειας, εφη, χρήζομεν, εαν παντες ὦμεν δικαιοι. Ου μόνον δε το μη αποδιδοναι χαριτας αδικόν έκρινεν, Ala To αλλα και το μη πολυ μείζους τον μείζω δυναμενον. φιλοπονος είναι, πᾶν μεν το παρον ἡδέως επινε, πᾶν δε το συντυχον ήδεως ησθιεν' εις δε το ασμένως κοιμηθῆναι πᾶς τόπος ην ίκανος αυτῳ. Διδοντος δε αυτῷ παμπολλα δῶρα Τιθραυστον, ει απελθοι εκ της χώρας, απεκρινατο ὁ Αγησίλαος, Ο Τιθραῦστα, νομίζεται παρ' ἡμῖν τῷ αρχοντι καλλίον ειναι την στρατιαν η ἑαυτον πλουτίζειν, και παρα των πολεμίων λαφυρα μᾶλλον πειρᾶσθαι η δῶρα λαμβανειν. «Αποθνησκων δε τους φιλους εκέλευσε, Μηδεμιαν εικόνα ποιησασθαι· ει γαρ τι καλον εργον πεποιηκα, τουτο μοι μνημειον εστιν' ει δε μηδεν, ουδ' οἱ παντες ανδριάντες.

VOCABULARY AND REMARKS.

Το μη πολυ μείζους, supply αποδιδοναι
OvvτUXOV, whatever he met with;

W. W. SNELLING will see one of his questions answered in the last paraThe graph. As to the other, we can only find room for a partial answer. following is the order in which mathematics may be studied with advantage.

Arithmetic-Cassell's, Colenso, or De Morgan.
Algebra-Cassell's, Lund's Wood, or Colenso.

Euclid-Cassell's.

Trigonometry-Hymers, Colenso, Hemming, or Snowball.

Mechanics (including Statics and Dynamics)-Snowball, Pratt, Whewell, Moseley, Earnshaw, or Tate.

Conic Sections-Hymers or Puckle.

Analytical Geometry-Hymers, O' Brien, or Young.

Newton's Principia-Evans, or Whewell.

Differential Calculus-De Morgan, Todhunter, Hemming, or Hall.

Integral Calculus-Hymers, De Morgan, or Hall.

Optics-Griffin.

Airy's Tracts, etc. etc.

JAMES BURLEY: The scales appended to the maps in the POPULAR EDUCATOR explain themselves. The numbers affixed to different points of the line, show how many miles are represented by the lengths of the line between the zero point (or farthest point to the left) and the above points. A WOULD-BE PHOTOGRAPHER: The instrument in question is an ordinary camera-obscura, and may be obtained of Horne, Thornthwaite and Co., Newgate-street, or any optical instrument maker.

PERSEVERANTIA: Professor De Lolme's Manual will be found very useful, Dr. Beard's Latin Made Easy, though not perhaps absolutely necessary. is scarcely required by one who has his lessons in the POPULAR EDUCATOR. Andrews and Stoddard's Latin Grammar, is indispensable to a thorough knowledge of Latin.. Dr. Beard's History of English Literature is not published separately. The Historical Educator is now complete in two volumes. You are mistaken in supposing that if a trader keeping his books by double entry, sold £30 worth of sugar for ready money, no notice would Whether the money te be taken of the transaction in the sugar account. An usher is an assistant teacher, and the qualifications for the office are obviously, knowpaid at once or at a future time, makes no difference. ledge, patience, good temper, and power of command, not to mention others. The advantage of matriculation is, that it enables you to take a degree. heart know both French and German in order to matriculate at the University of London; either of the two is sufficient. Private students cannot take degrees there without matriculating and join game or the colleges or institutions in connection with the university. We cannot find room for lessons in civil law.

D. HORNBY has solved Problems 56 to 69 of the Second Centenary of

ευcal Problems. His solutions and occasional remarks do him great F. II. BIRTWHISTLE all of them, except only No. 35. credit. Thos. Boncock has solved sixteen of the Second Centenary and

ERGATES may obtain the books he mentions at any bookseller's, either known to every body. The English Churchman, the Guardian, and the new or second-hand. There are several religious newspapers which are Record are the principal church weekly newspapers. EDWARD WOOD may get all the information he requires from any opti

cian.

GULIELMUS TALBATUS: We cannot promise what you wish at present. ARITHMETIC: 81,000,000,000,000,000,000,000,000,000. To find the true discount of a given sum for a given time at a given rate, say-The amount of £100 for the given time at the given rate: £100: the interest of the given sum for the given time at the given rate: the discount required. In business discount is the same as interest.

W. M. WILLEY: We have been exceedingly gratified with your French communication, which does you infinite credit, considering how short a

кoμηvaι (from Royaw, I lie, I sleep), every place sufficed to time you have studied the language. It is satisfactory to know that our afford him a pleasant couch (or bed).

Audovros, offering to give; λapupov, ov, ro, booty, spoil. Avepiavres, all images would not (no image would) be a (permanent) memorial of me.

t

What are these parts (what is their mood, tense, person, etc.)? namely:

ερωτηθείς ; χρηζομεν ; αποδιδοναι; εκρινεν ; ησθιεν; κοιμη θῆναι; απελθοι; απεκρίνετο ; πειρᾶσθαι; πεποιηκα. Decline the following :-ανδρειας; χαριτας; μείζους; τοπος; δῶρα ; εικονα ; μνημείον ; ανδριάντες.

ANSWERS TO CORRESPONDENTS.

We have received numerous expressions of gratitude for our past efforts in the cause of Popular Education, and regret at the prospect of their disOur correspondents may be continuance in the present publication. assured that their communications will meet with all the consideration to which they are entitled. We hope still to promote the education of the people in other ways, and have repeatedly intimated our intention of giving articles on drawing early in the coming year.

C. R. The lessons on Chemistry are concluded. Electricity and galvanism will be discussed in our lessons in Physics.-Tau: The principles in physics, corresponding to axioms in Geometry or any other subject, are those which require no demonstration. For instance, it is an axiom that that the effect of a force is the same at whatever point in its line of action it be applied. Again, it is an axiom that two equal pressures applied at the extremities of equal arms of a lever balance each other, and that the pressure on the fulcrum is equal to their sum. Problems are capable of solution

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labours have been so successful and so highly appreciated.

A NEW SUBSCRIBER's handwriting is good, but must be improved still further before he can hope to get into a merchant's office.

G. ARCHBOLD: Our Lessons in German will prepare you for translating German letters, provided you make yourself familiar with the German style of writing. It would also be desirable to get a work containing some German mercantile letters.

L. G. LAMBE: We find no promise given in the place you mention. ICONOCLASTES may get the works he wants from Paternoster Row. R. H. STY had better study the subjects he mentions in the order he has given them. His hand-writing is hardly good enough for a clerk's situation in these improved times.

JOHN TENNANT: We cannot engage to enter upon the subjects of which you speak.

T. H. SOUTHELL: The chief qualifications for a clerkship are superior Our lessons in handwriting, and quickness and accuracy in accounts. Bookkeeping ought to be amply sufficient, but they must be thoroughly mastered, and our correspondent must get a facility in working arithmetical questions correctly, besides improving his handwriting.

H. K. L.: Our lessons in Chemistry are closed.

LITERARY NOTICES.

CASSELL'S LESSONS IN LATIN.-Price 2s. 6d. paper covers, or 3s. neat cloth.

A KEY TO CASSELL'S LESSONS IN LATIN. Containing Translations of all the Exercises. Price 1s. paper covers, or is. 6d. cloth.

CASSELL'S CLASSICAL LIBRARY.-The First Volume of this Work, price 1s. 6d. cloth, consists of a LATIN READER, adapted to "Cassell's First Lessons in Latin."-Volume II. comprises LATIN EXERCISES, price 2s. neat cloth.-Volume III. contains THE ACTS OF THE APOSTLES in the Original Greek, with copious Notes and a Lexicon, price 2s. 6d. neat cloth.

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