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. 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, To catch him with her young bright eyes, Had raised the assassin's knife on high, What youth like Gyges e'er could guide 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. THE RICH MAN'S SON AND THE POOR MANS SON. BY J. R. LOWELL. The rich man's son inherits lands, A heritage, it seems to me, The rich man's son inherits cares; A heritage, it seems to me, What does the poor man's soon inherit? A hardy frame, a hardier spirit; 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; 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, 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 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; Both heirs to some six feet of sod, 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 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. Prepositions have a relation to place, and denote the direction of an action in regard to place. Thus, say "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. THE words which we have hitherto studied are susceptible of 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 εê and ex; so, also, rep and super, and To and sub. The following six words may also be considered as prepositions, namely:: { with over, above on against instead of 7. ανα 8. κατα 9. παρα 10. μετα 11. συν 12. ὑπερ { παρα τον ποταμον, to walk along the river. μετὰ τοῦ πατρος, to be with your father. OVY TY OTρаTY, to follow the army. ύπερ των ἐπομενων, to be above those who follow. oi vо Tivi, those who are under some one, πро τмν πνλшν, before the gates. φᾶρος βαλεῖν αμφι τινα, to throw a cloak around a person. Tа TEρI TIVOS, the things around a person, circumstances. εp' iππwv εivai, to be on horseback. αντι βασιλεως ( to make one who was a king δοῦλον ποιειν αναβαινω αναλαμβάνω καταβαίνω παραγω παραβαίνω μεταμορφόω μεταλαμβάνω συλλαμβάνω ὑπερβαίνω ὑποβαλλω 13. υπο υπογελαω 14. προ 15. αμφι 16. περὶ 17. επι 18. αι τι προβαίνω αμφιβαίνω περιβαίνω επιβαλλω αντιβαίνω αντιδιδωμι a slave. χαριν (προς), in favour of. πроîка (катα), gratuitously. Sometimes the preposition is expressed and united to the noun, as: παραχρῆμα (παρα, αἱ ; χρῆμα, the thing), at the moment. πрсνруоν (πро, for; epyov, the deed), usefully, beforehand. εкπodшV (εK, from; πove, the foot), at a distance, far from. Adverbs formed from adjectives imply a substantive :Dat. ίδια (εν ιδια χωρα), in particular ; πεζῇ εν πεζῇ όδῳ), on foot. Acc. μακραν (εις μακραν όδον), a long way, at a distance. The neuter of the adjective is often employed as an adverb; as the dat. wow, much, by much; nov, agreeably; devov and deva, terribly; TOTEρa, whether? επiτηdes, on purpose. 6. Affirmation. uny, yes, certainly, in truth. åpa, in the poets ει μη ει και sive nisi, si non et si, etiam si et si quod ut ut ne, ut non quia quia itaque cum or quum postquam 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. γοῦν επειδη επειδαν quum όπως ὡς, ώςπερ dum quomodo sicut when. until, how? as, as if Of these conjunctions some are simple, as Kaι, TE, ws; others are compound, as ούτε (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, τοι), now then ; τοιγαροῦν (τοι, γαρ, οῦν), wherefore, on that account; or, again, several words in a separate state, ov μnv aλλa, however; πλny eɩ μn, if only. One or two others deserve notice, as ȧre, seeing that, as being, e.g. ȧre ayabos, as being good (Latin, utpote bonus); πεр, although, e.g. ayaboç TEр, although good. There are certain words employed as adverbs, in the composition of which there is a conjunction, e.g. dnλovori, evidently; that is, dŋλov eσriv ori, it is evident that; εviore, sometimes, made up of evi, for eσriv, 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 rendered into English, and which can be appreciated only in by plane geometry, when there is no occasion to apply any other principles than those of that science.-ALFRED C. HILARY will find the work on the original, by the fine sense of a superior scholar. Chemistry, published in Chambers's Educational Course, very good and cheap. Interjections. 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, though not perhaps absolutely necessary. Dr. Beard's Latin Made Easy, 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 be taken of the transaction in the sugar account. Whether the money te An usher is an paid at once or at a future time, makes no difference. assistant teacher, and the qualifications for the office are obviously, knowledge, patience, good temper, and power of command, not to mention others. The advantage of matriculation is, that it enables you to take a degree. matriculate at the University of London; either of the two is sufficient. Private students cannot take degrees there without matriculating and join We cannot find room for lessons in civil law. D. HORNBY has solved Problems 56 to 69 of the Second Centenary of Αιζεταιl Problems. His solutions and occasional remarks do him great credit. Thos. Boncock has solved sixteen of the Second Centenary and Αγησίλαος περι ανδρειας και δικαιοσυνης ερωτηθείς, πότερα βελτιων, Ουδεν ανδρειας, εφη, χρήζομεν, εαν TAVTEC μEV δικαιοι. Ου μόνον δε το μη αποδιδοναι χαριτας αδικον εκρινεν, thecessary to know both French and German in order to αλλα και το μη πολυ μείζους τον μείζω δυναμενον. Ala To φιλοπονος είναι, πᾶν μεν το παρον ἡδέως επινε, πᾶν δε το συν-ing one of the colleges or institutions in connection with the university. τυχον 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 Αγησιλαος, Ο Τιθραῦστα, νομίζεται παρ' ήμιν τῳ αρχοντι καλλίον είναι την στρατιαν η ἑαυτον πλουτίζειν, και παρα των πολεμίων λαφυρα μᾶλλον πειρᾶσθαι η δῶρα λαμβανειν. Αποθνησκων δε τους φίλους εκέλευσε, Μηδεμιαν εικόνα ποιησασθαι ει γαρ τι καλον εργον πεποίηκα, τουτο μοι μνημειον εστιν' ει δε μηδεν, ουδ' οἱ παντες ανδριάντες. VOCABULARY AND REMARKS. Το μη πολυ μειζους, supply αποδιδοναι F. II. BIRTWHISTLE all of them, except only No. 35. ERGATES may obtain the books he mentions at any bookseller's, either new or second-hand. There are several religious newspapers which are known to every body. The English Churchman, the Guardian, and the 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 Konovaι (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). 2 Δίδοντος, offering to give ; λαφύρον, ου, το, booty, spoil. Avepiavres, all images would not (no image would) be a manent) memorial of me. 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. (per-German letters, provided you make yourself familiar with the German style G. ARCHBOLD: Our Lessons in German will prepare you for translating What are these parts (what is their mood, tense, person, etc.)? namely: ερωτηθείς ; χρηζομεν; αποδιδοναι; έκρινεν ; ήσθιεν; κοιμηθῆναι; απελθοι; απεκρίνετο; πειρᾶσθαι; πεποίηκα. Decline the following:—ανδρειας; χαριτας; μείζους; τοπος; δῶρα ; εικονα ; μνημείον ; ανδριαντες. 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 discontinuance in the present publication. Our correspondents may be 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.-Tav: 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 of writing. It would also be desirable to get a work containing some Ger- L. G. LAMBE: We find no promise given in the place you mention. 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 handwriting, and quickness and accuracy in accounts. Our lessons in 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. 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