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.

314

ETITTW, I fall in.

VOCABULARY.

Пpovoεw, I think beforehand.

EUTINTW, I fall with; ouμ- Arn, ns, n, confusion of mind,
perplexity, folly, misfortune.
πιπτει, it happens.
Κεφαλη, ης, ή, the head.
ATWOEW, I push away.
Povos, ou,ò, slaughter, murder.
Eigwoew, I push in.
Zvμpeow, I bear with, I am Mevednuos, ov, o, Menedémos.
Axaios, ou, d, an Achæan.
Anpoolens, ous, o, Demos-

useful.

Evvdiarpıßw, I live with some

one.

Evdokiuεw, I am famous.

thenés. Πωποτε, ever.

REMARKS ETC.

EK 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. Inpai, 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 Tw orρarny, 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 έδοξε τῳ 'Αγιῳ Πνευματι και ἡμιν, etc. ; 30 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, opekov; yeyovāσi; yeyote' (why not γενοιτοΐ); γενη; επεσεν; συμπεση, μνησθητι; ευδαιμονεῖ; γεγαμηκα; τεθνηκεν; στρατεύσασθαι εξυρατο; απεώσαντο. Why is πεσοντι in the dative case? Why is σοι in the dative case? Why is Eupenidov in the genitive case? Why is syw employed with yeyaunka? Write out in full the following nouns-xεipa; Kivovvov; arηy; amp; Ovyarno ἡμεραν; στρατηγῳ, πολέμιους; κεφαλην, πολιν ; Δημοσθενης; Μενέδημος; Ευριπίδου.

EXERCISES.-GREEK-ENGLISH.

Πολλακις εκ κακου εσθλον εγενετο, και κακον εξ αγαθού. Xeipa πEGOVTι opežov. 'O ayaπwv Kivduvov EμTEGELтai aury. Οι ανθρωποι προς αρετην γεγονᾶσιν. Μη μοι γενοιθ ̓ ἡ βουλο- | μαι, αλλ' ὁ συμφέρει. Μη σπευδε πλουτεῖν, μη ταχυ πενης | γενη. Πολλακις ὁ ευδοκιμεῖν πειρωμενος, ου προνοησας, εις μεγαλην και χαλεπην ατην επεσεν. Όταν ατυχειν σοι συμπεση τι, Ευριπίδου μνησθητι, Ουκ εστιν όστις παντ' ανηρ ευδαιμονεί. Μενέδημος προς τον ερωτήσαντα ει γημαι ὁ σπουδαιος, έλεξεν, Εγω γεγαμηκα. Η του φιλου θυγατηρ ενατην ημεραν γεγαμημένη, τεθνηκεν. Έδοξε τη στρατηγῳ επι τους πολεμίους στρατεύσασθαι. Δημοσθενης εξυρατο την κεφαλήν. Οἱ στρατιωται εις την πολιν ειςεωσθησαν. Οἱ στρατιῶται απεώσαντο τους πολεμίους.

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 no doubt, many of our students are already acquainted-we use of Colleges, Schools, and Private Students; a book with which, forth the following correspondence; and in order that all may introduced an anecdote into the preface, which has called 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 not be its only use; for all the self-educating students of Students apart from the Text-Book itself. This, however, will Cassell's Euclid will find it a valuable and essential auxiliary for the purpose of self-examination. By this means they will satisfactory progress in Geometry or not; and they will thus be enabled at once to determine whether they are making 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 exercise of memory and judgment, the demonstration of each succession, let them endeavour to write out, by the double own minds, that they are right in the demonstration of each Proposition in its proper order. If they feel satisfied in their 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 dissatisfed 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 quested some one to write out for him the first Proposition of understood the defnitions, postulates, and axioms. He rethe first Book, and give it to him to see if he could discover the construction and demonstration of the Problem without looking at Euclid's; this request was complied with, and he succeeded, producing identically the solution given by Euclid. successive Propositions of the first Book in proper and regular The same thing was done with the second, third, fourth, and order, and the gentleman almost always succeeded in discovering 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 I cut my brother's hair (sheared his head). The citizens Elements! On this singular experiment in the history of the have determined to go against the enemy. The general deter- human mind, two remarkable things cannot fail to be observed mined to besiege the city. From a good (man) I have become by the reader: first, the extraordinary perseverance and bad. They have become bad from (being) good. They fell success of the student; and, second, the wonderful adaptation into danger. I shall fall into danger. They will fall into of the logical order and method of the Propositions for their We recollect being danger. Do not ye hasten to be rich, lest you become poor, sure and rapid acquisition by all; for this student was premy children. My brother marries thy sister. Thy sister is viously wholly unacquainted with the manner in which married to my brother. I shall marry thy sister. My chil-Euclid had executed his great work. dren, extend your hands to those who are falling. Man is born for virtue.

ENGLISH-GREEK.

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, 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 "East Dereham, Norfolk, March 29, 1854. myself, containing all the Definitions, Postulates, Axioms, "Dear Sir,-I feel much pleasure in forwarding to you the Enunciations of the propositions and corollaries of Euclid's enclosed, which I received this morning from Mr. Charles Elements. The diagrams were omitted, as well as the solutions Chalmers, and I hope that this confirmation of your anec- of the problems and the demonstrations of the theorems. The class came prepared with two or three of the enunciadote' will be ample satisfaction to you for my having questioned tions, which had been prescribed to them at the previous its accuracy. "After expressing doubt as to the correctness of your state-meeting. The young gentlemen were required to draw for ment, I felt bound to use my best endeavours to solve the diffi- themselves the diagrams upon a board, and to make any culty. I am not acquainted with Mr. Charles Chalmers; I additional construction that might be required. The result therefore inquired of his brother, Mr. Patrick, if Mr. Charles was, that with very little assistance from me, and no assistance Chalmers was still alive, and in his answer he gave me his from Euclid, they performed, in the course of a session, what I, brother's address. I wrote to him, without, of course, men- without any assistance from either master or Euclid, required tioning my correspondence with you upon the subject. You three years to accomplish. Mr. Murray, who was some time have his very kind and explicit answer.-I am, dear sir, yours teacher of Mathematics in the Merchiston Academy, was so much pleased with the system, that when appointed to the respectfully, 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, "Castle Bank, Edinburgh, March 26, 1854. "CHARLES CHALMERS.

"Professor Wallace, Dalston."

"H. J. WARIN.

"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.

"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 proposi tions 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 suc-trict; but we fear that if once we open our pages to this subject, our P. E. ceeded. 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, "CHARLES CHALMERS.

"H. J. Warin, Esq."

LAMBDA (Princes-street): His exercise is correct. The most elegant treatise on the conic sections is by Hymers.-MARSHALL (Dundyvan): His 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 his diswould 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. -W. THEO. WATTS (St. Ives): In answer to his first question, "Which is 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 second question, we say that Noel and Chapsal's "Grammaire Française" is to be relied upon as an excellent work; and that the aug mentation by the editora 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 Mensuration, 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, bus 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. 8. (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.

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p=

1

R

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 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 B will form at b, a virtual image of this point. The eye, which receives the divergent rays DE, H K, etc., then sees at ab an image of a B.

Fig. 267.

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

p'=
2 R
Р

(4.)

B

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 of the image from the mirror by p'. Then, in the triangle L M 7, the normal мc divides the angle LM into two equal parts; whence, by Prop. iii. Book VI. of Euclid, we have

LC: C:: LM: MI;

therefore, by Prop. xvi. Book VI. of Euclid, we have

C/XLM LCX MI.

Now, if the arc Aм does not exceed 5 or 6 degrees, the straight lines M L and M / are sensibly equal to AL and ▲, that is, top and p'. Also, we have

cl=cA-Al=R-p', and CL-AL-AC=p—R.

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.

4th. If the object passes between the centre and the principal focus, we have p less than R; and we conclude from formula No. 4, that p' 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 p=R, which gives p' = ∞; that is, the image is placed at an infinite distance; or, in other words, the reflected rays are parallel to the axis.

R
0

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

Whence, substituting these quantities in the preceding equa-cing into formula No. 2, the condition that p' is negative,this tion, we have

2 =R

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

p

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