8. ήβάσκω, 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. τεθνήξω (I shall be dead). ἱλασθην. 10. θρώσκω, I spring, leap, a. 2. εθορον, f. θοροῦμαι, pf. τεθορα. 13. πασχω (formed from παθσκω), I receive an impression (Lat. 14. πιπισκω, I drink, f. πίσω, a. 1, επῖσα, 15. πιπράσκω, I sell (the fut. and aor. in ordinary speech are expressed by αποδωσομαι and απεδόμην), pf. πεπράκα, pf. m. or p. πεπρᾶμαι, inf. πεπρᾶσθαι, a. επράθην, f. 3. πεπρασομαι (in the sense of the simple unused f. πρα. θησομαι). 16. στερίσκω (and στερεω), I deprive, rob, f. στερησω, a. 1. εστέρησα; mid, and pass. στερισκομαι, στερούμαι, f. στερησομαι, pf. εστερημαι, a. εστερήθην. 17. τιτρώσκω, I wound, f, τρωσω, a. 1. έτρωσα, pf. m. or τέτρωμαι, a. ετρώθην, f. τρωθήσομαι and τρωσομαι. p. 18. φάσκω, I am of opinion, I give an opinion, affirm the indicative and imperative are very rare), impf. έφασκον, f. φήσω, a. 1. εφησα. 19. χασκω, I open the mouth (ΧΑΝ), a. 2. εχάνον, f. χανουμαι, pf. κεχηνα, I stand open. μασιν. EXERCISES. GREEK-ENGLISH. Ολιγους εύρήσεις ανδρας έταιρους πιστούς εν χαλεποις πραγ Πασιν ανθρωποις μόρσιμον εστιν αποθανεῖν. Πενθοῦμεν τους τεθνηκότας. Ηδέως των παλαίων πράξεων μεμνηνται οἱ ανθρωποι. Ουκ αν εύροις ανθρωπον παντα ολβιώτατον. Η καλώς ζην, η καλως τεθνηκεναι, ὁ ευγενης βουλεται. Ει δεινα δι' ὑμετέραν κακοτητα πεπονθατε, μη τι θεοις τούτων μοιραν επαναφέρετε. Τα αλλα και πολεμος και μεταβολη τυχης αναλωσεν, ἡ τεχνη δε σώζεται. Παντ εστιν εξευρεῖν, εαν μη τον πονον φευγη τις. Ει τις γηράσας ζην ευχεται, άξιος εστι γηράσκειν πολλας εις ετῶν δεκαδας. Μεμνησο ότι θνητος υπαρχεις. Μεμνιο (μεμνῷο) αει ὁ ὑπ' αλλων εν επαθες. Τυχν τεχνην εύρηκας, ου τεχνῃ τυχην. Ουκ εστι βιον εύρειν άλυπον ουδενι. Αύχαριστος όστις εν παθων αμνημονεῖ. Δικαιον ευ πραττοντα μεμνῆσθαι των ατυχών. ENGLISH-GREEK. wilt find few faithful friends. He has found a faithful com 1 have found no companion faithful in difficulties. Thou panion in misfortune. It is fated for thee to die. I bewail my deceased father. They will bewail the deceased general. I gladly call to mind the great men of old (παλαι). I found no man very happy in all respects. I wish to live honourably or to die honourably. He has suffered dreadful things through his baseness. Through thy fellow thou wilt suffer much. War wastes men's substance. It is possible to discover many things, but not all. He has discovered many things. I hope to discover many things. Having grown old he prays to live, and is foolish. He will grow old for many decades of years. Remember that thou art my son. Even the wise have not discovered a life devoid of grief. He has received a benefit and forgotten it (in Greek, having received a beneft, he has forgotten it). Being in good circumstances myself (αυτος ευ πράττων) Ι' will remember the unfortunate. That man has received many benefits from me, yet he reviles me. Observe that διδάσκω, I teach, retains the k sound in f. διδάξω, 5. Verbs whose pure stem is strengthened by a reduplication at the a. 1. εδιδαξα, pf. δεδιδαχα, a. p. εδιδαχθην. 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 1. γιγνομαι (instead of γιγενομαι), I become (ΓΕΝ), a. 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. πεπτωκαι Here also belong several of the fourth class, as γιγνώσκω. 6. Verbs whose pure stem receives an ɛ in the Present and γαμεω, I marry (used of the man), pf. γεγαμηκα; but f. 2. δοκέω, I appear (in Lat. videor), I think, f. δόξω, a. 1. έδοξα, pf. p. δεδεγμαι (Lat. visus sum), a. p. εδόχθην. 3. 4. Ευρέω, I shear, cut the hair, mid. ξυρομαι, a. εξυραμην, but pf. εξυρημαι. ωθεω, I push, impf. εωθουν, f. ωσω and ωθήσω, a. 1. έωσα and ωσα, pf. εωκα, mid, f, ώσομαι, a, εωσάμην, pf. εωσμαι, a. p εωσθην. 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. 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." To describe an isosceles "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. LAMBDA (Princes-street): His exercise is correct. The most elegant exercises are correct.-NEW STUDENT (Chancery-lane): His division of his treatise on the conic sections is by Hymers.-MARSHALL (Dundyvan): 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.-I. 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 avocation; 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. -W. THEO. WATTS (St. Ives): In answer to his first question, considered to be the most valuable work as a French Dictionary?" we say Bescherelle's, which was employed in the preparation of Cassell's French Dictionary;" we presume we need not say that it is all in French. In 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 auge mentation by the editors is most likely to be valuable.-G. M. Y. (e augton): By Practical Geometry is merely meant the solving of problems by the rule and compasses, and the casting up of sums in lensuration, 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. "Which is 4 ་ a cu à boards, 38. 6d. FELICAL EDUCATOR. the pezzup with such information relating hatte & kamerary History of the Sacred Books * " 1 R 2 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 в, fig. 267, be placed before a convex mirror, D к, 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, HK, 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. Formule 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 E, the distance LA of the object L from the mirror AM by p, the distance A of the image from the mirror by p. Then, in the triangle L M 7, the normal м c divides the angle L M into two equal parts; whence, by Prop. iii. Book VI. of Euclid, we have LC: Cl:: LM: M7; therefore, by Prop. xvi, Book VI. of Euclid, we have clXLM LCX MI. 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 p': R (4.) R 0 p' = = ∞; that is, the image is placed at an infinite 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 forcomes negative; this indicates that the distance p' between mula No. 4 becomes then negative, and consequently p' bethe 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 introduequa-cing into formula No. 2, the condition that p' is negative,this Now, if the arc A M 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 CLAL-AC=p—R. Whence, substituting these quantities in the preceding tion, 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 ZA, which is true only at the limit, that is, when the angle 125 |