8. ἡβασκω, I grow to maturity, f. ήβησω, a. 1. ήβησα, pf. ήβηκα (ήβαω, I am young, but ανηβαω, I become young again, rejuvenesco). 9. θνησκω, commonly αποθνήσκω, I die (ΘΑΝ), a. 2. απέθανον, f. αποθανοῦμαι, pf. τεθνηκα (not αποτεθνηκα), fut. 3. τεθνηξω (I shall be dead). 10. θρώσκω, I spring, leap, a. 2. εθορον, f. θοροῦμαι, pf. τεθορα. 11. ἱλασκομαι, I propitiate, f. ἱλάσομαι, a. ἱλάσαμην, a. Ρ· ἱλασθην. 12. μιμνησκω (with gen.), I remember (ΜΝΑ), f. μνησω, a. 1. εμνησα ; mid. I remind myself, think, make mention, pf. m. μεμνημαι (Lat. memini), I am reminded, I call to mind, subj. · μεμνωμαι, ᾖ, ῆται, imper. μεμνησο, plpf. εμεμνημην, opt. μεμνημην, το, ἒτο, οι μεμνωμην, ᾠο, φτο; fut. 3. μεμνησομαι, I shall be reminded, a. εμνησθην, I am re. minded, f. μνησθήσομαι, I shall remember. 13. πασχω (formed from παθσκω), I receive an impression (Lat. patior), I suffer, undergo, a. 2. επᾶθον (ΠΕΝΘ), f. πεισομαι, pί. πεπονθα. 14. πιπισκω, I drink, f. πίσω, a. 1. επίσα. 15. πιπράσκω, I sell (the fut. and aor. in ordinary speech are expressed by αποδωσομαι and απεδόμην), pf. πεπρᾶκα, pf. m. or p. πεπρᾶμαι, inf. πεπρᾶσθαι, a. επράθην, f. 3. πεπρασομαι (in the sense of the simple unused f. πραθησομαι). 16. στερισκω (and στερεω), I deprive, rob, f. στερησω, a. 1. εστερησα; mid. and pass. στερισκομαι, στεροῦμαι, f. στερησομαι, pf. εστερημαι, a. εστερήθην. 17. τιτρωσκω, I wound, f. τρωσω, a. 1. ετρωσα, pf. m. or p. τέτρωμαι, a. ετρώθην, f. τρωθησομαι and τρωσομαι. 18. parkw, I am of opinion, I give an opinion, affirm (the indicative and imperative are very rare), impf. εφασκον, f. φήσω, a. 1. εφησα. 19. χάσκω, Ι open the mouth (ΧΑΝ), a. 2. εχάνον, f. χανουμαι, pf. κεχηνα, I stand open. EXERCISES.--GREEK-ENGLISH. 313 Ολιγους εύρήσεις ανδρας έταιρους πιστούς εν χαλεποις πραγμασιν. Πασιν ανθρωποις μόρσιμον εστιν αποθανεῖν. Πενθοῦμεν τους τεθνηκότας. Ηδεως των παλαιων πράξεων μεμνηνται οἱ ανθρωποι. Ουκ αν εύροις ανθρωπον παντα ολβιώτατον. δι' υμετεραν κακοτητα πεπόνθατε, μη τι θεοις τουτων μοιραν Η καλως ζην, η καλως τεθνηκεναι, ὁ ευγενης βουλεται. Ει δεινα επαναφέρετε. Τα αλλα και πολεμος και μεταβολη τυχης αναλωσεν, ἡ τέχνη δε σώζεται. Παντ εστιν εξευρεῖν, εαν μη τον πονον φευγῇ τις. Ει τις γηρασας ζην ευχεται, άξιος εστι υπαρχεις. Μεμνο (μεμνῷο) αει ὁ ὑπ' αλλων εν επαθες. Τυχν γηράσκειν πολλας εις ετῶν δεκαδας. Μεμνησο ότι θνητος τεχνην εύρηκας, ου τεχνῃ τυχην. Ουκ εστι βιον εύρειν άλυπον ουδενι. Αύχαριστος όστις ευ παθων αμνημονεῖ. Δικαιον πράττοντα μεμνῆσθαι των ατυχων. ENGLISH-GREEK. ευ wilt find few faithful friends. He has found a faithful com Observe that διδασκω, I teach, retains the & sound in f. διδαξω, 5. Verbs whose pure stem is strengthened by a reduplication at the a. 1. εδιδαξα, pf. δεδιδαχα, a. p. εδιδαχθην. VOCABULARY. Παντα, accus. in every respect. Μη τι, not at all, in no way. Αναλωσεν ; the aor. indicates the repetition ; the force may be given by usually. 'Α ὑπ' αλλων εν επαθες, literally, the things which you have well experienced from others, that is, the benefits you have received from others. Ουκ εστι, etc., take the Greek thus, ουκ εστι ουδενι (it is not possible for any one) εύρειν βιον άλυπον. Whence come the following forms? In what mood, voice, and person are they each? Write down their several significations apart from the connected terms ; namely, μεμνῆσθαι ; εὑρειν; εὑρηκας; μεμνησο; γηρασας; αναλωσεν; πεπονθατε; αποθανειν; πενθοῦμεν ; τεθνηκεναι ; επαθες ; αμνημόνει. beginning. This reduplication consists in the repetition of the first consonant of the stem in union with the connecting vowel. Only in a few verbs does the reduplication remain in the formation of the tenses. To this class belong γιγνομαι (instead of γιγενομαι), I become (ΓΕΝ), 2. εγενομην (ΓΕΝΕ-), pf. γεγενημαι, I have become, or γεγονα with a present meaning, as I am (but γεγονως χρονος, time past), f. γενησομαι. πιπτω (instead of πιπετω), I fall, imper. πιπτε (ΠΕΤ-), f. πεσοῦμαι, a. 2. επεσον, pf, πεπτωκα. Here also belong several of the fourth class, as γιγνωσκω. 6. Verbs whose pure stem receives an ɛ in the Present and Imperfect. 1. γαμεω, I marry (used of the man), pf. γεγαμηκα; but f. γαμώ, a. 1. εγημα, mid. γαμουμαι, I am married (of the woman, in Lat. nubo), a. εγημαμην, pf. pass. γεγαμημαι, (Lat. in matrimonium ducor), a. εγαμηθην, etc. 2. δοκέω, I appear (in Lat. videor, I think, f. δόξω, a. 1. εδοξα, pf. p. δεδεγμαι (Lat. visus sum), a. p. εδόχθην. VOCABULARY. Εμπίπτω, I fall in. Evvdiaтpıßw, I live with some one. Evdokiuεw, I am famous. IIpovoew, I think beforehand. thenés. Πωποτε, ever. REMARKS ETC. Ex Kakov, etc., literally, often from bad good became, and bad from good; the sentence is very elliptical and may be rendered, a thing that was bad has often become good, and a thing that was good has become bad. Evpiridov, etc., remember the saying of Euripides. Tnual, the opt. aor. 3rd pers. sing.; the optative is required because the verb is dependent and past. Evarny, etc., literally, having been married the ninth day, that is, for nine days; time during which, in Greek as in Latin, is put in accusative case. Edožε Ty σтрarηyw, it seemed good to the general, that is, the general resolved; this is the usual form to express any deliberate determination; thus the apostolic council (Acts xv. 25) said, εδοξεν ἡμιν γενομενοις ὁμοθυμαδόν, etc., and εδοξε τῳ 'Αγιῳ Πνευματι και ήμιν, etc. ; so έδοξε μοι, έδοξε σοι, έδοξε τῳ πατρί, etc., I resolved, thou didst resolve, father resolved. Explain, with reference to what has preceded, the formation of these verbs, assigning the root, the voice, the mood, the tense, and the person; namely, opežov; yeyovāow; yero' (why not γενοιτο?); γενη; επεσέν; συμπεση, μνήσθητι ; ευδαιμονεῖ; γεγαμηκα, τεθνηκεν, στρατεύσασθαι, εξυρατο; аTEσaνTO. Why is eGovT in the dative case? Why is oot in the dative case? Why is Eugenidov in the genitive case Why is yw employed with yeyaunka? Write out in full the following nouns-χειρα, κινδυνον, ατην, ανηρ, θυγατηρ ; ἡμεραν; στρατηγῳ; πολεμίους; κεφαλην, πολιν ; Δημοσθενης; Μενέδημος ; Ευριπίδου. EXERCISES. GREEK-ENGLISH. Πολλακις εκ κακού εσθλον εγενετο, και κακον εξ αγαθού. Χειρα πεσοντι ορεξον. ̔Ο αγαπῶν κινδυνον εμπεσείται αυτῷ. Οι άνθρωποι προς αρετην γεγονᾶσιν. Μη μοι γενοιθ' ά βουλομαι, αλλ' ὁ συμφερει. Μη σπευδε πλουτεῖν, μη ταχυ πενης γενη. Πολλακις ὁ ευδοκιμεῖν πειρωμενος, ου προνοήσας, εις μεγαλην και χαλεπην ατην επεσεν. Όταν ατυχειν σοι συμπεση τι, Ευριπίδου μνησθητι, Ουκ εστιν ὁστις παντ' ανηρ ευδαιμονεί. Μενέδημος προς τον ερωτήσαντα ει γημαι ὁ σπουδαίος, έλεξεν, Εγω γεγαμηκα. Η του φίλου θυγατηρ ενατην ήμεραν γεγαμημένη, τεθνηκεν. Έδοξε τη στρατηγῳ επι τους πολεμίους στρατεύσασθαι. Δημοσθενης εξυρατο την κεφαλήν. Οἱ στρατιωται εις την πολιν ειςεωσθησαν. Οἱ στρατιῶται απεώσαντο τους πολεμίους. ENGLISH-GREEK. I cut my brother's hair (sheared his head). The citizens have determined to go against the enemy. The general determined to besiege the city. From a good (man) I have become bad. They have become bad from (being) good. They fell into danger. I shall fall into danger. They will fall into danger. Do not ye hasten to be rich, lest you become poor, my children. My brother marries thy sister. Thy sister is married to my brother. I shall marry thy sister. My children, extend your hands to those who are falling. Man is born for virtue. | | CORRESPONDENCE. THE STUDY OF EUCLID. IN a small book published by Mr. Cassell, and entitled The Self and Class Examiner in Euclid; containing the Enunciations of all the Propositions and Corollaries in Cassell's Edition; for the use of Colleges, Schools, and Private Students; a book with which, no doubt, many of our students are already acquainted-we introduced an anecdote into the preface, which has called forth the following correspondence; and in order that all may have the subject completely before them, we first quote that preface : "At the request of several teachers, who make use of Cassell's Euclid as a text-book, the Enunciations of the Propositions in the different Books of Euclid, and the Corollaries to these Propositions, are reprinted in this little book, in order to serve the important purpose of the Examination of the Students apart from the Text-Book itself. This, however, will not be its only use; for all the self-educating students of Cassell's Euclid will find it a valuable and essential auxiliary for the purpose of self-examination. By this means they will be enabled at once to determine whether they are making satisfactory progress in Geometry or not; and they will thus also be able to ascertain whether they can pass an Examination in Euclid, just as well as if they had gone up for this purpose to the University of London. All they have to do is this, after having carefully studied any given Book of Euclid, let them lay aside that work, in order that they may not be tempted to look into it for assistance; let them take up this little book and open it at the place in which they wish to examine themselves; and looking at, or writing out, each Enunciation in succession, let them endeavour to write out, by the double exercise of memory and judgment, the demonstration of each Proposition in its proper order. If they feel satisfied in their own minds, that they are right in the demonstration of each in succession, they may go on writing the rest of the demonstrations until they have finished the book in question, or until they have come to a complete stand-still. But if they feel dissatished with their own demonstrations, or can proceed no farther than a certain point, they may then compare what they have done with the demonstrations given by Euclid, and judge at once of their own progress, whether it be really satisfactory or not. If not, of course they must again resume the study of the Book in question with more care and attention than before; and they may rest assured that by continuing this mode of self-examination from time to time, they will ultimately succeed both to their own satisfaction and to that of others who may have occasion to examine them, or to apply to them for information. "For the encouragement of all, we may here mention an anecdote, which we had from good authority. A gentleman, who was a respectable bookseller in a town in the north, had his attention called to the Elements of Euclid. He read and understood the definitions, postulates, and axioms. He rethe first Book, and give it to him to see if he could discover quested some one to write out for him the first Proposition of the construction and demonstration of the Problem without Ιooking at Euclid's ; this request was complied with, and he succeeded, producing identically the s 1 tion given by Euclid. The same thing was done with the second, third, fourth, and order, and the gentleman almost always succeeded in dissuccessive Propositions of the first Book in proper and regular covering the solutions of the Problems and the demonstrations of the Theorems exactly as Euclid had done before him, at the distance of nearly two thousand years. This process was carried on regularly, we believe, until he had fully accomplished the demonstrations of the whole of the first six Books of the Elements! On this singular experiment in the history of the human mind, two remarkable things cannot fail to be observed by the reader: first, the extraordinary perseverance and success of the student; and, second, the wonderful adaptation of the logical order and method of the Propositions for their sure and rapid acquisition by all; for this student was previously wholly unacquainted with the manner in which Euclid had executed his great work. We recollect being told that it was within a space of two years that this laborious and ingenious task was accomplished by the gentleman alluded to, who was no other than a brother of the late celebrated Dr. Chalmers." A gentleman, who is one of our students of geometry, and a very ingenious correspondent to boot, "having," to use his own words, "been much struck with the improbability of the achievement" above stated respecting a brother of the late Dr. Chalmers, wrote to a friend on the subject, who might be very naturally supposed to be acquainted with it; but not receiving a satisfactory answer, he wrote to us, expressing his doubts and wishing for additional information and confirmation. As the anecdote was with us a matter of memory, and memory is sometimes treacherous, at least in the smaller circumstances of a case, we referred him to our original authority. The following letters, which we insert in the order of their receipt by us, will explain to our students the matter as it really stands; and we wish to interest them in the subject, because we think these letters convey most valuable hints both to teachers and learners, and especially to self-instructors; and we earnestly hope that the grand lesson of self-reliance, and of persevering, plodding industry, will not be lost upon them; but that many, hereafter, will date the period of their full determination to study and to instruct themselves, and also of their future success in learning and in life, to the perusal of this correspondence: "East Dereham, Norfolk, March 29, 1854. "Dear Sir, I feel much pleasure in forwarding to you the enclosed, which I received this morning from Mr. Charles Chalmers, and I hope that this confirmation of your anecdote' will be ample satisfaction to you for my having questioned its accuracy. After expressing doubt as to the correctness of your statement, I felt bound to use my best endeavours to solve the difficulty. I am not acquainted with Mr. Charles Chalmers; I therefore inquired of his brother, Mr. Patrick, if Mr. Charles Chalmers was still alive, and in his answer he gave me his brother's address. I wrote to him, without, of course, mentioning my correspondence with you upon the subject. You have his very kind and explicit answer.-I am, dear sir, yours respectfully, "H. J. WARIN, "Professor Wallace, Dalston." "Castle Bank, Edinburgh, March 26, 1854. "It is not the fact that I demonstrated the whole of the first six books of Euclid without some slight acquaintance with the Elements of Geometry. My brother, the late Dr. Chalmers, initiated me when about twelve years of age, in the study of Euclid's Elements. I had not proceeded, however, beyond the first eight propositions of the first book, when the study was discontinued in consequence of a severe illness, the effects of which rendered it necessary for me to forego all regular instruction. Upon my recovery, I thought of resuming, without the assistance of a master, the study of Geometry. Without referring to Euclid, I revised the first eight propositions which I previously demonstrated with my brother, the various steps in the demonstration of which I found that I had forgotten. I, however, after weeks of study, succeeded in the demonstration of all of them. It occurred to me to attempt the next proposition, which is a problem, and also to construct the diagram, without referring to Euclid. In this I also succeeded. You may well imagine the state of excitement, and enjoyment, and consciousness of power which I felt upon this achievement. I now resolved to prosecute the study of Geometry without having recourse to Euclid at all. I, of course, availed myself of the definitions, axioms, postulates, and of the corollaries as they occurred. I covered up all the diagrams. At the end of three years, or thereabouts, I succeeded in the demonstration of all the theorems, and in the solution of all the problems of the first six books, with the exception of the fifth. This last I studied in the usual way, but without the assistance of a master. To describe an isosceles triangle having each of the angles at the base double of the vertical angle, almost baffled me. 1, however, after three months of hard study, succeeded in solving this problem. These I think are mainly the particulars as regards my study of the first six books of Euclid.-I am, sir, yours truly, "H. J. Warin, Esq." "CHARLES CHALMERS. "East Dereham, April 6, 1854. "Dear Sir, I have improved your hint, and feel much pleasure in forwarding to you the consent of Mr. C Chalmers to make public the contents of his letter of 26th March. "Allow me to repeat, that I never for a moment supposed that the information you received was not correctly stated in the anecdote.' My doubt was as to the accuracy of that information, and was induced by the extraordinary character of the feat-a designation which I still hold to be no misnomer, notwithstanding the modest assertion of the person who achieved it. -I am, dear sir, yours respectfully, "H. J. WARIN. "Professor Wallace, Dalston." "Castle Bank, Edinburgh, April 4, 1854. "Dear Sir,-You are quite at liberty to make any use of the communication I made to you, respecting the method which I adopted in the study of the elements of Geometry. The fact is not so remarkable, however, as you seem to imagine. I would say that it requires mainly perseverance to accomplish it. I was so impressed with its practicability, and its superiority over the usual method, that I taught a class of young gentlemen in Merchiston Castle Academy the first six books of Enclid upon this method. I put into the hands of the young gentlemen a pamphlet, which was printed for myself, containing all the Definitions, Postulates, Axioms, Enunciations of the propositions and corollaries of Euclid's Elements. The diagrams were omitted, as well as the solutions of the problems and the demonstrations of the theorems. The class came prepared with two or three of the enunciations, which had been prescribed to them at the previous meeting. The young gentlemen were required to draw for themselves the diagrams upon a board, and to make any additional construction that might be required. The result was, that with very little assistance from me, and no assistance from Euclid, they performed, in the course of a session, what I, without any assistance from either master or Euclid, required three years to accomplish. Mr. Murray, who was some time teacher of Mathematics in the Merchiston Academy, was so much pleased with the system, that when appointed to the Hill-street Institution in Edinburgh, as head Master, adopted the system in that institution, and, I have been told, was very successful. I remain, Sir, yours truly, "CHARLES CHALMERS. "H. J. Warin, Esq." We intended to prefix the preceding correspondence to a new edition of the Self and Class Examiner, etc.; but we have thought it better to give the useful information which it contains a wider circulation in the P. E. ANSWERS TO CORRESPONDENTS. treatise on the conic sections is by Hymers.-MARSHALL (Dundyvan): His LAMBDA (Princes-street): His exercise is correct. The most elegant exercises are correct.-NEW STUDENT (Chancery-lane): His division of his time is very good; let him go on and prosper; any scheme is better than none.-A. LANGLEY (Steyning): His letter does him very great credit indeed, and so does his scheme for the improvement of the poor of bis district; but we fear that if once we open our pages to this subject, our P. E. would be greatly diverted from its original purpose.-T. Bocock (Great Warley): Received.-J. D. (Kilmarnock) will be kept in view.-H. HUDSON Westminster) should begin with the Lectures on Euclid in the P. E. and study them hard; then try to solve the problems.-J. SIMPSON (Winterbourne): We should be glad to advise him if we could. He should go to the Crystal Palace on a shilling day, and get into conversation with the parties there engaged in similar avocations; perhaps he might see Sir Joseph Paxton, and get a hint from him.-ISAK (Leicester): We are glad our old correspondent is alive. His solution is better than the one inserted. "Which is -W. THEO. WATTS (St. Ives): In answer to his first question, considered to be the most valuable work as a French Dictionary?" we say Dictionary; we presume we need not say that it is all in French. In Bescherelle's, which was employed in the preparation of "Cassell's French answer to his secon question, we say that Noel and Chapsal's" Grammaire Française" is to be relied upon as an excellent work; and that the augmentation by the editors is most likely to be valuable.-G. M. Y. (Thornton): By Practical Geometry is merely meant the solving of problems by the rule and compasses, and the casting up of sums in en-uration, etc. No objection can be made to Euclid's method and order, provided the student have sufficient time and taste for the subject, so as to wish the union of theory and practice. Other methods have been suggested, but we are not acquainted with their experimental working, and therefore cannot personally recommend them. ASPIRANT (Lower House) will oblige us by a little more patience.-W. E. FISHER should study the Palmerstonian Penmanship in the P. E., the Lessons in English, and in the Mathematics. If you read one whole hour, study two whole hours what you have read, and you will most likely master it. But take it sentence by sentence, and don't read it in the lump; by this we mean, that if a sentence takes five minutes to read it carefully, it will take ten minutes to master it and fix it in the memory; and this is in the proportion of one hour to two hours. Then as to observation for three hours, this means that you are to go out into the world, and look at the practice of what you have been studying; see how it works, and gather practical information to join with your theoretical. For instance, if you have read an account of the steam-engine in the P. E. and taken double the time for reflection upon it that you took in reading about it, then go and look at a steam-engine in actual operation, and take at least three times the same time to examine its working that you took to read the description of the machine, so as to understand the nature of it before you saw it.-J. C. ATKINSON (King's Cliff): Received.-A LOVER OF LEARNING (Holborn): The more the better.-C. S. (Tottenham-court-road): His letter should not have been anonymous when he wanted so great a favour.-J. T. M. is greatly mistaken.-H. K. W. (Brixton): We see nothing to hinder his success in matriculating next year. The classical subjects have been already mentioned. The poetry is not quite up to our mark yet. S. KERR (Straid): The question he sends us is too old and well known; it might interest him, but not the majority of our subscribers.-AYIN TSADI: Try Lunds Wood, Booth's Analytical Dictionary, Malte-Brun's Physical Geography, and Jameson's Dictionary of the Scotch Language; and go on with Hutton and Euclid.-W. C. HUNSLET (Leeds) should study arithmetic and penmanship, and especially English, so as to be able to spell well; and these are all in the P. E.-J. H. EASTWOOD (Middleton) removes the defect in Euclid's and in Thomson's demonstration of the 11th Prop. of the 2nd Book, alluded to in Cassell's edition, as follows, see the fig. to that Prop. in said edition: "Let A B be the given straight line, &c. Produce C A to F, and make E F equal to E B (I. 3). Now, because EA B is a triangle, the two sides EA and AB are together greater than the side EB (1. 20). But EP is equal to E B (Const.); therefore EA and A B together are greater than EF, that is, than E A and A F. From each of these unequals take away the common part E A, and the remainder AB is greater than the remainder AF (Ar. 5). Then AB is divided in H, etc. Complete the parallelogram FH, and produce G H to K. Because, etc."-ADA (Ballynahinch): Received, and under consideration.-G. C. (Guildford) had better learn the arithmetic and bookkeeping in the P. E.-T. Fox (Newport) had better stick to the French lessons in the P. E.-W. COTCHEIFER: Received.-A COMMON MINER (St. Austell): His question is a very good one; but it is rendered unnecessarily complex by decimals extended to the 6th place. AN UNHAPPY STUDENT is frightened at his own shadow. There is no necessity, in learning Italian or anything else, to be able to keep up with the lessons as they are published. Every one must learn as he can, and not make himself uneasy because he cannot keep pace with those that are before him and have more time than he has. This is an emulation which is injurious. Let him remember the fable of the hare and the tortoise. As to his not liking to get into arrears with the P. E., this is a very laudable feeling, but he must not be discouraged if he does; he must remember that there are other students, especially in Italian, who are driving on at a great rate, and they must have our attention and encouragement also. We had a letter from some of these t'other day, who urged us most manfully to give the Italian lessons a great deal faster; now he wishes them to be slower; what are we to do in this case? Simply this: to take our own way. C. A. R. and A. T. (Brompton): As to the pronunciation of the poet Byron's name, we can only say that we have heard it pronounced both ways by very good judges, and knowing that there is no rule for the pronunciation of proper names, we must depend on those whom we hear pronouncing them. We were taught by James Sheridan Knowles, Esq., Teacher of Elocution at Glasgow, to say Bir'-ron; but we have often heard it pronounced by others Byron.-W. H. W. (Hackney) will be able to answer his own questions by consulting the indexes to the first and second vols. of the P. E. LABORE VINCO (Soho) and Q. PRINGLE (Glasgow): Received.-T. L. B. NEWTON (Devon): His suggestions are very good; we wish that he would write them out more plainly and in better English, to save us the trouble. | We think he will require Latham, or some such book, in preparing for the matriculation examination in English, W. GRANT: Cassell's French Dictionary, just published, is the best we know. Riddle's Latin Dictionary is among the best; but we would recommend Cassell's, which will soon be completed. To mention all the translations of the classics would occupy too much space, and it would be a mere catalogue of books.-G. S. RUTHERFORD (Marylebone): The petition to the U. of L. has reference to all degrees whatever.-R. ROBINSON, JUN. (London): Cassell's Latin Dictionary, see Literary Notices in this page. There will be more lessons in history. The questions proposed must be deferred for the present.-G. FANCOURT (Melon): Received.~ J. W. HONO (Tidswell): His plan is good, but not sufficiently matured; the contributors must not be mixed up in a paper intending to confer honour on the publisher; the latter should stand alone in his glory. A. H. (Ashton U. L.) says "that mastic varnish is an excellent varnish for maps." The study of French and Latin would assist that of English very much.-W. C. (3rd West York Militia): His solution of the pine query is right.-J. A. CARSE (Laurieston): Received. LITERARY NOTICES. Now Ready, price 9s. 6d. strongly bound. CASSELL'S FRENCH AND ENGLISH DICTIONARY: Composed from the French Dictionaries of the French Academy, Bescherelle, Landais, etc.; from the English Dictionaries of Ogilvie, Johnson, Webster, etc.; and from the Technological and Scientific Dictionaries of both Languages. By Professors DE LOLME and WALLACE, and HENRY BRIDGEMAN, Esq. The following are the distinctive features which render this Work superior to any of the same class now extant. It has been compiled with unusual care from the very best authorities. It contains correct renderings of all the most modern words and phrases-including those of science, art, manufacture, commerce, law, politics, etc., as well as familiar conversation-which are indispensable to a knowledge of language, but yet are rarely, if ever, to be found properly translated in any Dictionary. 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CASSELL'S ECLECTIC GERMAN READER: containing choice Selections from the best German Authors, in Prose and Verse. Price 2s. paper covers, or 2s. 6d. cloth. CASSELL'S LESSONS IN GERMAN PRONUNCIATION: consisting of easy Extracts from German Writers. Price 1s. paper covers, or Is. 6d. cloth. Now ready, Vol. I., in cloth boards, 5s. 6d. THE POPULAR BIBLICAL EDUCATOR. This work is intended to supply the people with such information relating to the study of the Bible as the POPULAR EDUCATOR has given in reference to Secular Instruction. It contains a Literary History of the Sacred Books -Accounts of their Original Text-Canonical Authority, and most Ancient Versions The Principle and Laws of Interpretations, and the Methods of Discovering the Literal or Symbolical Meaning of Inspired WritingaIllustrations of the Geography and Natural History of Palestine-The Manners and Customs, the Laws and Worship of its People-The Antiquities of the Four Great Monarchies-The Fulfilment of Prophecy concerning them and other ancient nations-and the Fruits of modern Travel and Discovery in the East, etc. The work is written in a popular style, and is therefore specially adapted to supply Families, Sunday-school Teachers, and others, with that amount of information respecting the Holy Bible which they.need in order to meet the charges of Infidels and the subtleties of Romanists, and to confirm and establish their own minds in the genuineness and authenticity of Holy Writ. Wherever the subject requires Pictorial Illustrations they re introduced. p' = 1 that is, the image is formed at the principal focus, as shown in our last lesson. Convex Mirrors.-In order to ascertain the formation of whence, by inversion, we have images in convex mirrors, let an object A B, fig. 267, be placed before a convex mirror, D K, at any distance. If we draw the secondary axes, a c and B c, it will follow from the construction of the foci in convex mirrors, explained in our last lesson, that all the rays emitted from the point a are divergent after reflection, and that their prolongations will meet in a point a, which is the virtual image of the point A. In like manner the rays emitted from the point в will form at b, a virtual image of this point. The eye, which receives the divergent rays DE, н K, etc., then sees at ab an image of a B. B H Fig. 267. It follows from this construction, that whatever be the position of an object before a convex mirror, the image is always virtual, erect, and less than the object. Formula for Spherical Mirrors.-The relation which exists between the relative position of an object and that of its image in spherical mirrors, may be represented by a very simple formula. Let us first consider the case of a concave mirror: in fig. 268, let the radius of curvature cм be denoted by R, the distance LA of the object L from the mirror A M by p, the distance 7A of the image from the mirror by p'. Then, in the triangle L M 7, the normal мc divides the angle LM7 into two equal parts; whence, by Prop. iii. Book VI. of Euclid, we have LC: C: LM: M/; therefore, by Prop. xvi. Book VI. of Euclid, we have clXLM LCX MI. Now, if the arc A м does not exceed 5 or 6 degrees, the straight lines M L and M / are sensibly equal to AL and A?, that is, to p and p'. Also, we have cl=cA-Al-R-p', and CL-AL-AC=p—R. Whence, substituting these quantities in the preceding equation, we have (R-P) p (p-R) p', or Rp-pp'—pp'-Rp'; which, by transposition and reduction, becomes Rp Rp 2 pp' (1). Dividing all the terms of this equation by pp' R, and suppressing common factors, we have 2nd. If the object be made to approach the centre c, the value of p diminishes and that of p' increases; for dividing both terms of the fraction in formula No. 3 by p, we have = R 2 -R (4.) 3rd. If the object be made to coincide with the centre, that is, if p = R, then p' R; and the image coincides with the object. cipal focus, we have p less than R; and we conclude from 4th. If the object passes between the centre and the prinformula No. 4, that is greater than R; that is, that the image is on the other side of the centre. When the object reaches the principal focus, we have pR, which gives p∞o; that is, the image is placed at an infinite R distance; or, in other words, the reflected rays are parallel to the axis. 5th. If the object passes between the principal focus and the mirror, we have p less than R; the denominator in formula No. 4 becomes then negative, and consequently p' becomes negative; this indicates that the distance p' between the image and the mirror must be reckoned along the axis in a direction contrary to p. The image then becomes virtual and is situated on the other side of the mirror. By introduformula becomes cing into formula No. 2, the condition that p' is negative,this under this form it comprehends the case of virtual images in concave mirrors. In the case of convex mirrors, the image being always virtual, p' and R have the same sign, since the image and the centre are on the same side of the mirror, whilst the object being on the other side of the mirror p has a contrary sign; now by introducing this condition into formula No. 2, we have for the formula relative to convex mirrrors. This could also be found geometrically by the same considerations that were employed in finding formula No. 2 in concave mirrors. It is of importance to observe that the different formula in the preceding are not strictly correct, since they rest on hypotheses which partake of the same character, namely, that the straight lines LH and M, fig. 268, are respectively equal to LA and A, which is true only at the limit, that is, when the angle 125 |