In affirming any manner, situation, position, or location of | persons or things, estar is used; as,

Juan está de moda, John is in |
the fashion.
Está de rodillas, he-is on (his);
knees.

Estoy de prisa, I-am in haste.
Estoy de tu parecer, I-am of
thy opinion.

Clara está en la iglésia, Clara is
in the church.
Estaba en la calle, he-was in
the street.

Estan á la puerta, they-are at
the door.

La

comida está en la mesa, the dinner is on the table. Donde está mi padre? where is my father? Está en la cama, he-is in bed. Estos quisantes ya están en flor, these peas are already in blossom. Está entre dos soldados, he-is between two soldiers. Estaba en Paris, he-was in Paris.

Hence, sin. (a + b) =

=

cos. (a + b) =

sin, a sin. b (2)

R

2. Expressions for the sine and cosine of a double arc. If, in the formulas of the preceding article, we make b = a, the first and second will become,

3. Expressions for the sine and cosine of half a given arc. If we put a for a in the preceding equations, we obtain,

We may also find the sine and cosine of a in terms of a. Since the sum of the squares of the sine and cosine is equal to the square of radius, we have,

cos. 2a + sin. 2}a = R2. And, from the preceding equation,

cos. a―sin. 2a = E cos. a.

If we subtract one of these from the other, we have, 2 sin. 2a R2 –

And adding the same equations,

E

Hence,

Also, CB CF::CH: CI; or, R: cos. b:: cos. : CI. cos, a cos. b

Therefore,

CI=

B

-B COS. a.

The cosine of 1′ = √/ 1 — sin.20.9999999577. The sines of very small angles are nearly proportional to the angles themselves. We might then obtain several other sines by direct proportion. This method will give the sines correct to five decimal places, as far as two degrees. By the following method they may be obtained with greater accuracy for the entire quadrant.

By Art. 4, we have, by transposition,

sin. (a + b) = 2 sin. a cos. b — sin. (a cos. (a+b) = 2 cos. a cos. b

1', and a = 1, 2, 3',

b),

cos. (a - b).

one-third of what he had in the other. How many were there in the other?"

Teacher. Well, Charles, you must first get one-fourth of eight, must you not?

Charles. Yes, sir.

Teacher. Well, one-fourth of eight is two, isn't it?

Charles. Yes, sir; one-fourth of eight is two.

Teacher. Well, then, three-fourths will be three times two, won't it?

Charles. Yes, sir.

Teacher. Well, three times two are six, eh?

Charles. Yes, sir.

If we put b= shall have,

THIS consists in asking what the lawyers call leading questions. It is practised, usually, whenever the teacher desires to help along the pupil. "John," says the teacher when conducting a recitation in Long Division, "John, what is the number to be divided called" John hesitates. "Is it the dividend?" says the teacher. "Yes, sir-the dividend." "Well, John, what is that which is left after dividing called? - is it?" 66 the remainder Yes, sir." A visitor now enters the room, and the teacher desires to show off John's talents. Well, John, of what denomination is the remainder?"

66

John looks upon the floor.

"Isn't it always the same as the dividend, John?" "Yes, sir."

"Very well, John," says the teacher, soothingly, "what denomination is this dividend?" pointing to the work upon the board. "Shillings, is it not?

"Yes, sir; shillings."

"Very well; now what is this remainder?" John hesitates.

"Why shillings too, isn't it?" says the teacher.

"Oh yes, sir, shillings!" says John, energetically, while the teacher complacently looks at the visitor to see if he has noticed how correctly John has answered!

We may further illustrate this drawing-out process, by describing an occurrence which, in company with a friend, we once witnessed. A teacher whose school we visited called upon a class in Colburn's First Lessons. They rose, and in single file marched to the usual place, with their books in hand, and stood erect. It was a very good-looking class.

"Were do you begin?" said the teacher, taking the book.

Pupils. On the 80th page, 3rd question.
Teacher. Read it, Charles.

Charles. (Reads.) "A man being asked how many sheep he had, said that he had them in two pastures; in one

At this point we interposed, and asked the teacher if he would request Charles to go through it alone. "Oh, yes,' said the teacher, "Charles, you may do it again." Charles again read the question, and-looked up. "Well," said the teacher, "you must first get one-fourth of eight, musn't you?" "Yes, sir." "And one-fourth of eight is two, isn't it?" "Yes, sir." And so the process went on as before, till the final eighteen sheep were drawn out as before. The teacher now looked round, with an air which seemed to say, "Now I suppose you are satisfied."

"Shall we ask Charles to do it again?" said we. The teacher assented. Charles again read the question, and again -looked up. We waited, and he waited;-but the teacher could not wait. "Why, Charles," said he impatiently: "you want one-fourth of eight, don't you?" "Yes, sir," said Charles, promptly; and we thought best not to insist further at this time upon a repetition of "yes, sir," and the class were allowed to proceed in their own way.

The

This is, indeed, an extreme case, and yet it is but a fair sample of that teacher's method of stupefying mind. This habit of assisting the pupil to some extent, is, however, a very common one, and as deleterious to mind as it is common. teacher should at once abandon this practice, and require the scholar to do the talking at recitation. We need hardly suggest that such a course of extraction at recitation, aside from the waste of time by both parties, and the waste of strength by the teacher, has a direct tendency to make the scholar miserably superficial. For why should he study, if he knows from constant experience that the teacher, by a leading question, will relieve him from all embarrassment? It has often been remarked, that "the teacher makes the school." Perhaps in no way can he more effectually make an inefficient school than by this drawing-out process.

We look upon the two processes just described as very prominent and prevalent faults in our modern teaching; and if by describing them thus fully, we shall induce any to set a guard upon their practice in this particular, we shall feel amply rewarded.

THE MORE EXCELLENT WAY.

It is always a very difficult question for the teacher to settle, "How far shall I help the pupil, and how far shall the pupil be required to help himself?" The teaching of nature would seem to indicate that the pupil should be taught mainly to depend on his own resources. This, too, we think is the teaching of common sense. Whatever is learned should be so thoroughly learned, that the next and higher step may be comparatively easy. And the teacher should always inquire, when he is about to dismiss one subject, whether the class understand it so well that they can go on to the next. He may, indeed, sometimes give a word of suggestion during the preparation of a lesson and by a seasonable hint, save the

scholar the needless loss of much time. But it is a very great evil if the pupils acquire the habit of running to the teacher as soon as a slight difficulty presents itself, to request him to remove it. Some teachers, when this happens, will send the scholar to his seat with a reproof perhaps, while others, with a mistaken kindness, will answer the question or solve the problem themselves, as the shortest way to get rid of it. Both these courses are in general wrong. The inquirer should never be frowned upon; this may discourage him. He should not be relieved from labour, as this will diminish his self-reliance without enlightening him; for whatever is done for a scholar without his having studied closely upon it himself, makes but a feeble impression upon him, and is soon forgotten. The true way is, neither to discourage inquiry nor answer the question. Converse with the scholar a little as to the principles involved in the question; refer him to principles which he has before learned, or has now lost sight of; perhaps call his attention to some rule or explanation before given to the class; go just so far as to enlighten him a little, and put him on the scent, then leave him to achieve the victory himself. There is a great satisfaction in discovering a difficult thing for one's self, and the teacher does the scholar a lasting injury who takes this pleasure from him. The teacher should be simply suggestive, but should never take the glory of a victory from the scholar by doing his work for him,-at least, not until he has given it a thorough trial himself.

The skill of the teacher, then, will be best manifested, if he can contrive to awaken such a spirit in the pupil that he shall be very unwilling to be assisted; if he can kindle up such a zeal, that the pupil will prefer to try again and again before he will consent that the teacher shall interpose. We shall never forget a class of boys, some fourteen or fifteen years of age, who in the study of algebra had imbibed this spirit. A difficult question had been before the class a day or two, when we suggested giving them some assistance. "Not to-day, sir," was the spontaneous exclamation of nearly every one. Nor shall we forget the expression that beamed from the countenance of one of them, when, elated with his success, he forgot the proprieties of the school and audibly exclaimed, "I've got it! I've got it!" It was a great day for him; he felt, as he never before had felt, his own might. Nor was it less gratifying to us to find that his fellows were still unwilling to know his method of solution. The next day a large number brought a solution of their own, each showing evidence of originality. A class that has once attained to a feeling like this, will go on to educate themselves when they shall have left the school and the living teacher.

As to the communication of knowledge, aside from that immediately connected with school-studies, there is a more excellent way than that of pouring it in by the process already described. It is but just that we should give a specimen of the method of doing this. We shall now proceed to do so under the head of

WAKING UP MIND.

The teacher of any experience knows, that if he will excite a deep and profitable interest in his school, he must teach many things besides book-studies. In our common schools there will always be a company of small children, who, not yet having learned to read understandingly, will have no means of interesting themselves, and must depend mainly upon the teacher for the interest they take in the school, This to them is perhaps the most critical period of their lives. Whatever imperssion is now made upon them will be enduring. If they become disgusted with the dullness and confinement of school, and associate the idea of pain and repulsiveness with that of learning, who can describe the injury done to their minds? If, on the other hand, the teacher is really skilful, and excites in them a spirit of inquiry, and leads them in suitable ways to observe, to think, and to feel that the school is a happy place even for children, it is one great point gained. We may suggest here, then, that it would be well to set apart a few minutes once a day for a general exercise in the school, when it should be required of all to lay by their studies, assume an erect attitude, and give their undivided attention to whatever the teacher may bring before them. Such a course would have its physiological advantages. It would relieve

the minds of all for a few minutes. The erect attitude is a healthful one. It would also serve as a short respite from duty, and thus refresh the older scholars for study. We may further add, that for the benefit of these small children every general exercise should be conducted with reference to them, and such topics should be introduced as they can understand. CONDUCTING RECITATIONS.

In considering a teacher's qualifications, the power of exciting an interest in the recitations of his school may not be overlooked. No man can be successful for any length of time without this. This comprises what is usually implied by APTNESS TO TEACH. All men have not this faculty by nature in an equal degree. Some may talk for an hour upon an interesting topic in the presence of children without commanding their attention; while there are others who can take even a common-place subject and secure for any length of time an all-absorbing interest in every word. This difference is seen in every grade of public speakers, and in all descriptions of writers: but perhaps more strikingly than anywhere else it is observable among teachers. Enter one school, and you may notice that the scholars are dull and listless; indifference sits undisturbed upon their brows; or perhaps they are driven by the activity of their own natures to some expedient to interest themselves, while the teacher is, with very commendable spirit, laboriously-perhaps learnedly-explaining some principle or fact designed for their edification. The secret is, he has not yet learned to awaken their attention; he fails to excite their interest.

Pass to another school. A breathless silence pervades the room; the countenances of the children, upturned towards the teacher, beam with delight. As he kindles into earnestness and eloquence, they kindle into responsive enthusiasm. Whenever his eye meets theirs, he sees-he feels, the glow radiated by the fire he is lighting in their souls, and his own gathers new warmth and enthusiasm in return. Such a man is apt to teach; and you could scarcely break the spell by which he holds his class, "though you should give them for playthings, shining fragments broken from off the sun."

He who possesses this gift naturally has very great advantage as a teacher to begin with. The ability to tell well what he knows is of more consequence to the teacher, than the greatest attainments without the power to communicate them. Combine high attainments with the ability to tell, and you have the accomplished teacher.

But this power to communicate is not necessarily a natural gift; it comes not always by intuition. It can be acquired. It is founded in philosophy; and he who can understand anything of the workings of his own mind, who can revert to the mental processes he went through in order to comprehend a principle, who can go back to that state of mind he was in before he comprehended it, and then by one step more can put himself in the place of the child he is teaching, realising exactly his perplexities and feeling his precise wants, can become the apt teacher. Those who fail in this are usually those who have forgotten the steps they took to acquire their own knowledge, or perhaps who never noticed what steps they did take.

To acquire this rare qualification should be the constan study of the teacher. To this end he should recall, as fa as possible, the operations of his own mind in childhood By studying his own mind, he learns, often most effectually, what he needs to know of others. Whenever he is preparing to teach any principle or fact to others, let him ask himself questions like the following:-What was the dark point in this when I studied it? Where did my mind labour most? What point did my teacher fail to explain? Such questions will frequently suggest the very difficulty which perplexes every mind in the same process. Again, the following inquiries may be very useful:-In studying this, what was the first point which appeared clear to me? After this, what was the second step, and how did that follow the first? The next in order? And the next? Was this the natural order? If not, what is the natural order? The right answers to these questions will suggest the course to be pursued in the instruction of a class.

The teacher can scarcely ask a more important question than this :- What is the natural order of presenting a given subjecte