at c and D, twice in the same direction, while approaching the secondary axis, which it strikes in a. The other rays emitted from the point A, according to what has been stated in the preceding paragraph, converging also in a, this is the conjugal focus of the point. If we now draw the secondary axis from the point B, we find, in like manner, that the rays emitted from this point form their focus in b, and as the points situated between A and B have evidently their focus between a and b, a real image is formed at ab, the reverse of the object A B. To see this image it is necessary to receive it on a white screen, or to place the eye in the direction of the rays of emergence.

the principal focus distance. This ray, prolonged in a manner different from its direction, proceeds to strike the axis o a at the point a, which is the principal focus of the point A. Bringing the secondary axis to the point в, we find likewise that the virtual focus of this point is formed in B. We have then at ab the image of A B. This image is erect, virtual, and larger than the object.,

The magnitude is more or less considerable as the lens is more convex and the object nearer to the principal focus. Biconvex lenses thus employed, as magnifying glasses, take the name of microscopes.

Formation of Images in Bi-concave Lenses.-Bi-concave lenses, like convex mirrors, give only virtual images, whatever may be the position of the object. Let an object AB (fig. 300) be placed before such a lens. If we first draw the secondary axis from the point A, the rays A CAI emitted from this point are refracted twice in the same manner, departing from the axis A 0; so that the eye which receives the rays of emergence DE Fig. 300.

Reciprocally, if a b was the luminous object which emitted the rays, its image would be produced at AB. From this follows a consequence important to be borne in mind in connexion with the theory of optical apparatus to be described and GH sees them departing from the point where their prohereafter: If a very large object be sufficiently distant from a bi-longations meet at a, the secondary axis, o A. Likewise, when emitted from this point form a pencil of rays of divergence we conduct the secondary axis from the point B, the rays whose prolonged directions meet in b. The eye sees then in a b a virtual image of A B, which is always erect and smaller than the object.

convex lens, the real and reversed image which we obtain is very small, nearly approaching to the principal focus, and a little beyond this point, in relation to the lens; and if a very small object is placed near the principal focus, a little in advance of this point, the image which is formed at a great distance is very much magnified, and is more so the nearer the object is placed to the principal focus. This may be easily verified by experience, if we admit through a screen the image of a wax candle placed successively at unequal distances beyond a bi-convex lens.

2. Virtual Images.-Suppose a case where the object A B (fig. 299) is placed between the lens and its principal focus. Fig. 299.

If we draw the secondary axis o a from the point a, the ray AC, after being refracted twice, issues divergent with relation to the axis, since the point A is placed at a distance less than

Formula relative to Lenses.-In every lens we can reduce to that of the object, the ray of curve, and the index of the suban equation, the relation that exists between the focal distance, stance of which the lens is formed. Let us first suppose the case of a bi-convex lens: P being a luminous point situated in the axis (fig 301), let r I be a ray of incidence, IE its direction in the interior of the lens, E P' the ray of emergence, so that P' is the conjugal form of P. Again, let d' I and CE be the normals at the points of incidence and emergence, and granting IPA =a, EP' A'ẞ, ECAY, IC'A=0, NIP=i, E10=r, 120 =i.

The angles and r being external to the one to the triangle PIC, the other to the triangle CEP' we have i=a+8 and ry+B. Whence, i+r2 = a + B +y+d (1). Now, at the point I we have sin in sin r, and at the point E, sin rn sin ; but in supposing the arc AI of a very small number of degrees, it is the same with the angles i, r, and r', and we can replace, in the above formula, the sines by their arcs, which will give inr and r'ni, whence in (+). Besides the two triangles 10 E and coc' having the angle o equal, we have r+r=y+ô, whence in (y+). Carrying this value into the equation (1), we have

Fig. 301.

n(y+8)=a+B+y+8, or (n-1) (y+d) a +ẞ (2).

334

This granted, if we conceive that the arcs a and y are described from the points P and c as centres, with a ray equal to a unity, and if from the point p we describe the arc da with the ray PA, we have the proportions

whence we draw

a

1

= —, and — == ;

PA

AB

of domestic life, or that tend to its elevation and refinement. This state of things cannot last. The time is not far distant when many social tendencies must at length necessitate a cheap, efficient, and systematic education of the people. Till such is based upon system, and that of an enlarged and unsectarian character, no effectual power will be brought to bear upon much of the ignorance, and many of the social demoralisations, which are justly complained of. For considered in its noblest and most essential principles, in all that constitutes a foundation for what is enlarged and progressive in the individual, enlightened education will recognise no difference of sects or social position. It will give principles and afford enlightenment-leaving the individual to advance therefrom in such social direction of duty as shall best suit his position and aptitude.

In the meanwhile young women, of the classes referred to, might do much for themselves. Presuming an ability to read, or at least the wish to learn, an incalculable power of self-improvement lies open before them. They want, I think, but to Anderstand the immense amount of social good that depends apen their individual culture, of the refinement that is its contingent, of the enlightenment and morality they may be producers of, of the lofty principles they may infuse into the generation to come, to be at once aware of their great indiviiua duty, and to prepare for it by self-culture and discipline. In thus speaking, I do not refer to book-knowledge only, or to any special class amongst the great industrial masses, but to woman generally, however poor and lowly, and to every act and duty which can improve her in person, in manners, in behaviour, and in the conduct of her home. These matters, nience trivial as they may seem, are amongst the greatest needs of the age, and whilst men are so immensely progressing as they are, and showing the sterling character of this progress by their thirst for science, it will shame woman if she be behindhand in the improvement of her mind, the culture of her person, her better knowledge of domestic duties, and the adornment of her If we DA. = lume. For let it be recollected that refinement is but a thing it degree, and all that is worthiest in it, and most essential, is dan asana ing the is possible for the poorest homes as the richest, and that there women may be as much gentlewomen in habits, manners, and nutes, is in the proudest homes of the land. Granting that they have to work for daily bread-so much the better; the I & rednement that adorns labour is the noblest of its kind.

This aspiring young women of the industrial classes in that positive course di duty which is and must be theirs, these papers will refer more to the culture of personal and domestic life, s thu than to book-knowledge actually as such, though, situated as they are, it is mostly from books alone that any true informaten on these points can be derived. Manuals and cheap books the sign, and the on cocking, household-work, needlework, the moral management and education of infancy, and other relative subjects, exist in abundance; but it is to these subjects as allied, and to their effects in educating woman, that I shall direct my attention, as well as to what relates to the decoration and management mysire the same sign, but of homes-the latter a somewhat novel feature, as refinement in alliance with narrow means has been little understood, or even considered, by our most advanced thinkers.

So much has been done by Mr. Cassell for the self-education of the masses, in his admirable series of the POPULAR EDUCATOR,

mob.ined by the same that it will be only necessary for me to refer to some few points

SPUCATION.

No, L.

ས་

connected therewith. The first important one is that of perseverance. Patient endeavour will win a victory over the hardest educational difficulties; and let what is done, be done well. Scholarship, even when relating to merely rudimentary matters, admits of nothing which is slovenly. Another thing of importance is, that when the power of reading is acquired, and there exists a taste for it, let what is low and trashy in periodical literature or books be passed by, as unworthy the countenance of ally young woman who aspires towards some degree of selfcultivation and improvement. Tales relating to seductions, murders, improbable marriages, and profligate courses of life, to have no charms for her. The eye, and ear, and mind of woman, of whatever degree, cannot be kept too pure and unsonhisticated; and all publications which refer to these sets ought to be avoided as so much moral poison. Their reigns but for a day-once educate the great industrial masses, give them he noble training which geometry and mathematics, gua, grammar geography, languages and history will afford,

and all these corrupt and sensual excrescences of a cheap press
will be heard of no more, simply for the reason that there will
be no market for them. Not but what fiction, in its best kinds,
performs a noble educative duty, besides affording healthy and
innocent amusement. Perhaps, as far as woman is concerned,
even when all classes shall be generally and efficiently edu-influence all the future incidents of individual life.
cated, fiction will always preponderate as the section of litera-
ture most favoured; but this much may be observed, because
a thing worth remembrance, that proportionately as the mental
faculties become strengthened by study and thought, and as
soon as education gives discipline to the mind, the taste for
fiction diminishes. For the period of youth it will always
have a charm, and in many senses an educative effect-just as
nations in their first beginning are amused and instructed by
ballads and legends. Of those branches of fiction destined to
instruct and delight, the novels of Sir Walter Scott will always
stand pre-eminent. Exquisitely pure in their moral tendency,
picturesque as Creswick's and Gainsborough's landscapes in
their delineation of scenery, fresh and spirited in depicting
character, they are destined to enchant unborn millions, and
the broadest sign of mental advance and moral purity will be
on those days when they pass from hand to hand as readily
amidst the masses, as they do at present amongst the pro-
visionally educated. One thing has alone to be guarded against
in their perusal. They underrate, and in some senses ridicule,
the Puritanism of the seventeenth century. But this defect
the popular knowledge of history begot by education will
rectify; for the time will come when English men and women
will look back with as much pride on this phase of their
national history, as educated Americans to their War of Inde-
pendence and Washington. Dickens, too, and Thackeray, are
eminently moral teachers; and there is yet another class of
fiction worthy of all attention, though but in its germ: that
which gives vitality to many abstract questions, and to points
of social and industrial reform, painting pictures, as it were, of
life under better and approaching aspects.

tion with current events, such as emigration, colonisation, or
the existing war, the value is really great of knowing some-
thing objectively of such places as Melbourne, Canada, or
Sebastopol. With knowledge of this kind, vague generalities
become valuable facts, that in many cases may serve to

Of history a word must be said. Of that of her own country, every young woman of whatever rank should be ambitious to have some knowledge. The time will come, when she will prepare herself as though she were to be an open and wellwritten chronicle for her children to read, and think what homes will be, what domestic life-even where humble-when instead of scolding them, or telling idle tales, a mother can impart pretty facts and interesting points of history to her little listening children. There are hundreds of such alone in the history of our own country; and considering this history as a means, direct and indirect. of individual improvement, I know nothing better. The relation of the past to the present and the future can never, with advantage, be overlooked; and the knowledge of what noble men and women have done for the advance of civil and religious truth in this our country, should lead each individual to renewed efforts for self-improvement and culture. To such amount of historical knowledge there may be additions. The history of the Jews is a deeply interesting one: whilst some knowledge of both Greek and Roman history lays, as it were, the true foundation of much which is valuable and interesting in the history both civil and political of modern times--so much have the literature and fine arts of the Greeks, and the laws, the literature, and the conquests of the Romans, influenced all the circumstances of our modern civilisation and advance.

Passing from these matters connected with rudimentary learning and general self-culture, I have a still more important study to urge upon the attention of ALL sensible young women of the middle and operative classes. It is that of PHYSIOLOGY, or an acquaintance with the functions of the body, and how Pains should be taken to write a fair and open hand. This best disease is avoided and health ensured. It is a new study, is attainable by writing large-hand copies for a sufficient period at least as far as the general public are concerned; but it is till the hand has obtained facility and the eye accuracy. One not less a need or less true for all that-and it is especially a large-hand copy written well and carefully every day for some study befitting woman. On her depends so much the health months, would go far to ensure a handwriting of character and of children and the health of homes, that were this admirable, firmness. Nothing, next to grammar and spelling, betrays caste this priceless knowledge spread, as I earnestly believe and and breeding so much as handwriting. This ought not to be; trust it will be, two-thirds of the disease, the crime, the early and with the present facilities for self-improvement which deaths, would be prevented; whilst so far from there being exist, and with the leisure left from their hours of labour, most anything indelicate or improper in a knowledge of the human young women of the industrial classes might learn to write a body and its functions, that I think it is one which may be good hand and spell well. The acquirement of these advan-called emphatically, considering the office which is so especially tages is so mere a fact of simple industry and attention, as to hers-THE STUDY OF WOMAN! To meet this need of a general be scarcely excused when absent. When large hand is written understanding of the laws of life and health, Lessons in Phywell, and a smaller one undertaken, great advantage would be siology have been given in the pages of the POPULAR EDUCATOR. found in making short extracts in a copy-book from a book or An attentive perusal of these I would recommend, as well as newspaper this would improve style and spelling, and lead to that of two little books by the late Dr. Andrew Combe of the habit of expressing ideas on paper, not only with facility, Edinburgh. The first is entitled, "The Principles of Phybut without mannerism or vulgarity. Of all habits of self help, siology applied to the Preservation of Health," and the second, none exceeds this, especially when habit and facility are so far "A Treatise on the Medical Management of Infancy." They advanced as to enable the copyist to condense the material into are published at the price of half-a-crown each, by Simpkin a few lines and into her own language, so that it thus becomes and Marshall, London; and are thus placed within the reach what is called by lawyers and students "note taking.". In of the provident and industrious, who, for the sake of a noble this way grammar and correct spelling are not only acquired, self-respect and self-improvement, will themselves, or joining but a precision given to remembered facts and habits of thought with other young women, save an occasional sixpence from the which become invaluable. week's wages.

Lessons in Arithmetic are given in the POPULAR EDUCATOR, and my only duty here is to impress upon the minds of young women its two-fold value as an acquirement. It is obviously of much utility to those who have to make many small purchases, and to calculate matters connected with work or wages; but there is a higher point still, namely, that its acquirement and use have certain faculties in a woman's mind, which, by a curious physical law, are often reflected in the capacity of her children. And, as I shall presently show, no fact of this kind can be relevantly disregarded in woman's true culture of herself.

In relation to Geography, much that is valuable relating thereto may be acquired by the study of maps. There are excellent ones to be found in the POPULAR EDUCATOR, and in these pages, which will serve every purpose of study and reference. Points in history have an added interest, when their localities are thus prefigured to the mind, and in connec

But this is such an important subject, and so intimately connected with improved personal habits and domestic duties, that I must resume it in my next paper.

Les hommes sont tourmentés par l'opinion qu'ils ont des choses, non par les choses mêmes.-Epictete.

L'homme n'a guère de maux que ceux qu'il s'est attirés luimême: c'est l'abus de ses facultés qui le rend malheureux. La nature lui fait payer cher le mépris qu'il fait de ses legons.-J. J. Rousseau.

Il n'est d'affreux que le commencement du malheur: au comble de l'adversité on trouve, en s'éloignant de la terre, des régions tranquilles et sereines.-Chateaubriand.

Le malheur, loin de dégrader l'homme, l'élève, s'il n'est pas un lâche.-Silvio Pellico.

LESSONS IN GREEK.-No. XLII.

By JOHN R. BEARD, D.D.

ΤΗΣ VERBS ΙΝ μι.

THE chief peculiarity of the conjugation in μ consists in this, that the verbs which belong to it, in the present, the imperfect, and several in the second aorist active and middle also, take special person-endings different from those of the conjugation in w, and in the indicative of the other tenses want the mood

vowel. The formation of all the other tenses, with a few

exceptions, coincides with the formation of the verbs in w. Several verbs in μι, which have a monosyllabic stem, take in the present and imperfect a reduplication, which consists in this, that when the stem begins with a single consonant or a mute and a liquid, the first consonant of the stem is repeated with a, or if the stem begins with στ, πτ, or an aspirated vowel, an aspirated precedes the stem ; as

The subjunctive, as in verbs in w, has the mood vowels w and n, which, however, blend into one with the characteristic vowel, which causes contractional deviations from the conjugation in w, as an and ay melt into and (not into a and, as in contracted verbs in a ω); oy melts into ᾧ (not into or, as in contracted verbs in ow).

ἱ-στα-ω=ἱστῶ ἱ-στα-ης=ι-στῆς ἱστα η ταιΞιστή-ται στασης = στῆς

στα ω = στῶ

τι θεω = τι-θῶ διδο-ω = διδῶ

According to the second aorist corny is formed the indicative of both aorists passive of all verbs, ε-τυφθ-ην, ε-τυπ-ην, ε-στα-θην,

ης, η, ητον, ητην, ημεν, ητε, ησαν.

The person-endings of the optative in the imperfect and second aorists, except the first person singular, difer from those of the optative of the historical tenses in the conjugation in w only in this, that it is preceded by an ŋ. as

τιθεσης = τιθῇς τι θεω-μαι= τιθῶ μαι ἱσται-ην σται ην τι-θει-ην θει την δι-δοι-ην δοι-ην

διάδοσης == δι-δος διδο η = διέφ.

This formation of the subjunctive of ίστημι and τιθημι is fol. lowed by the subjunctive aorist first and second passive of all verbs, as

In the dual and plural of the optative imperfect the n is commonly dropped and the termination of the third person plural, ησαν, is usually shortened into ev, as

The same obtains in the optative of the passive aorists of all verbs, as τυφθείημεν = τυφθεῖμεν, τυπείημεν, τυπείεν (quite like τιθείην).

In the optative second aorist of the verbs ίστημι, τιθημι, διδωμι, on the contrary the shortened forms are very rare, except the third person plural.

δι-δο-τον
The forms of the first aorist active, έθηκα, ήκα, and εδωκα,
δι-δο των
however, are used only in the indicative and especially in the
δι-δο-τε singular; in the other persons commonly, and always in the
δι-δο-τωσαν | other moods and the participle, the forms of the second aorist
are employed. So instead of the forms of the first aorist
διδοντων.
middle of τιθημι, ίημι, and διδωμι, those of the second aorist
middle are used. On the contrary the indicative forms of the
singular second aorist of τιθημι, ίημι, and διδωμι (εθ-ην, ήν, and
ow) are not to be employed.

In the tense-formation of the entire active, as well as of the τεθναμεν, τεθνατε, τεθνᾶσι(ν), inf. τεθνάναι froin τέθνηκα, middle future and Grst aorist, the short characteristic vowel | θνησκω (ΕΝΑ).