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in the afternoon. The primary was a small hot room in the rear of the building. I immediately proceeded thither and opened the windows and the door to let out the stifling air, shut up since last Friday. The walls, windows and floor looked dirty, the furniture was bad and oldfashioned-a promoter of doctors and undertakers' bills, of spinal diseases, goitre and consumption. God help the little ones who are thus crucified daily and hourly by stingy school-boards and blind parents.

The room had evidently been swept by some one, but the dust was swept into a heap beside the door, and an old broom lay beside it. Dust also lay plentifully on the furniture, but no duster was visible.

“Who does the sweeping here?” I inquired of one of the children. “I dunno; I guess the monitor does," was the reply. 6 And who is the monitor?”

“He was a big boy and belonged to the main school,” said one little girl. “He left when you were engaged.”

66 And is there no duster here?" I asked.

“Oh yes, here it is,” said my little informant, flourishing a dirty rag that once had been a towel, but was now a combination of duster, inkrag and black-board wiper. I hesitated to take the object presented, into

my hand, but seeing nothing else available, I took it and commenced dusting off the furniture. The children watched me curiously; at last the little girl I had spoken to, and whose name was Lizzie (she is the daughter of my patron, the portly baker), came up to me and asked shyly,

“Shall I sweep out that dirt ?"

“If you please, Lizzie," I replied. So she swept out the dirt in a very housewifely manner. Just then the bell rang, and oh, the noise that followed that signal! The children ran and jumped into their seats and banged their books and slates upon the desks as I had never seen it done before. Lizzie dropped the broom where she stood and plumped into the first seat with a mien that proclaimed her as belonging to the 6 first scholars."

After the noise had subsided a little, I picked up the broom and asked where its place was.

“Put it in the penance-corner,” said one child. “ Penance-corner! What is that?”

“ That's the closet under the stairs, ma'am, where the bad children are put,” was the answer.

I put the broom into the closet, which was partly filled with broken chairs and brooms and several bundles of kindling. A child might manage to double up in the remaining space, and even breathe there for a while, but certainly not long. I could not help thinking of the Black Hole of Calcutta-indeed, to a little child the latter could not have seemed more horrible than this dark, unwholesome closet, on a hot summer's or a cold winter's day. When I returned to my desk the last buzz of the noisy little crowd before me had subsided, and the children sat silent and expectant. No register had been kept before, but as I was expected to prevent the usually irregular attendance, I had procured a record and now proceeded to enroll the pupils. There were sixty-five this first morning; on inquiring I found that a dozen or more of the regular pupils were absent and in their stead I had several interlopers—children who came from curiosity to see the new teacher of their little friends.

I next examined the books, of which I found a great variety, nearly every child being provided with a different set. Thus I had five diferent primers in one class, and other readers in proportion. The majority had slates, or fragments of slates, but many were without pencils. The latter difficulty was soon overcome, as many were willing to lend from their abundance, and so we managed to be busily employed before recess arrived.

I have always found it easy to keep little children employed; they seem imbued with the true thirst for knowledge. It is also quite easy to overtax their strength and to ruin this finest of instruments, the infant mind, by a false ambition to produce prodigies of learning. Older scholars are less in danger of such perilous experiments; but the little ones seem to have a tempting adaptation for them. They are always anxious for action and ready for work. Those that I am teaching now, form no exception to this rule, and so the first forenoon passed quite pleasantly and without occasion for discipline, save a word now and then to one or the other whose thoughts wandered from the lesson. The children were evidently bent on making a favorable impression on me, and to exhibit proficiency of scholarship. I felt that whatever troubles might arise they would not come from this quarter, and I inwardly resolved to spare no pains in making “pennance corner” a thing of the past.

SPELLING.

BY A SUBSCRIBER.

In the December number of the JOURNAL Mr. C. H. Allen says, in the article on “Spelling:” " It would seem that the result shows something defective in our system of learning and teaching spelling.” Having been a school officer for some years, and having observed the way "spelling” is taught in our common country schools, it is my humble opinion that it is merely a waste of time and money. If teachers were compelled to have more written exercises it would create a vast improvement. Under the present system most of the time is occupied by oral spelling, and the majority of pupils forget how to spell a word within a few days after passing it. If all such lessons were written out, would it not leave a more lasting impression on their minds?

Another great error is committed by teachers of little experience, and that is to crowd all the different branches taught in our common schools through in one day; under this practice justice cannot be done to either. It might be different in graded schools, but in our common country schools, where pupils differ so much in age, which compels a teacher to have many clases, nothing is taught thoroughly. Why not take ample time for each branch taught, even if it should take all the school hours of one day? Please let us have something bearing on the above subject.Monroe, December 15.

BAD SPELLS.-Under this head we see a paragraph in many papers stating that at a recent convention of teachers in Orange county, N. Y., and also at other country conventions, a large number of words, something like eighty out of a hundred, were misspelled. And the comment is that teachers ought to go to school again, and learn before they try to teach. Examples of the words are given, and it appears that the most of them are words not in common use, some of exclusively technical, and scarcely any of them such as would be wanted in ordinary writing. The word cachinnation is said to have been spelled in twenty different ways and in the right way by very few.

Now this may all be true and yet the teachers may be good spellers and well fitted to be teachers. Such words the most of literary men would rarely if ever use, and when using, would consult

a dictionary. The true test of good spelling would be rather to read aloud to a company a sentence from some standard author, and and let the hearers write the sentence, and them compare the result with the original. Good spelling is shown by writing a letter or an essay without error. It would surprise most people to be told how large a proportion of educated men and women make blunders in writing their own language, and employing only the words of every day speech.- Observer.

That good man, the late Father Taylor of Boston, had little knowledge of grammar. On one occasion, when entangled in the exuberance of his own speech, he had got quite astray; he stopped and said, “ Brethren, my nominative has lost its verb, and can't find it; but I'm bound for the kingdom of heaven all the same!”

A NEWSPAPER AND A BIBLE in every house, and a good school in every district, are the principal supporters of virtue, morality and civil liberty.-Franklin.

PROPERTIES OF SQUARE NUMBERS.

BY L. CAMPBELL.

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2

If A, B, C and D represent any numbers such that A'=B-C2 and B-C=D, then C=B-D, and this value of c, placed in the equation, A'=B'-C gives A’=B*—B(-D), from which we find A? +D2

A2-D2 B= 2D

since C=B-D. ; consequently C=

2D Hence A’= { A' +D

(m). If in this equation we substitute for A and D any numbers whatevor; (A being greater or less than D), we shall have three square numbers, whole or fractional, one of which will be equal to the difference of the other two. Place D=1, then equation (m) becomes

A2+1)
A’—

A?-1
2 J

(n).
Placing D=2, equation (m) becomes
A’=
A2+4)
A2-4)

(p). i 4 Now if A be an odd number, A' +1 and A'-1 are even numbers, and therefore, divisible by 2. If A be an even number, A’ is divisible by 4, and hence, A2 +4 and A’—4 are divisible by 4; consequently when A is an odd

A? +1 A2-1 number

and

are whole numbers; and, when A is an even

2 A? +4 A2-4 number and are whole numbers. Hence 4

4 Eq. (n) becomes

Eq. (p) becomes When A= 1, 1= 1'- 0° When A= 2, 22:

= 2 - 02
A= 3, 3'= 52 — 4?

A= 4, 4= 52 — 32
A= 5, 5=:13-12

A= 6, 6'=10?– 8?
A= 7, 7=25-24

A= 8, 8=172 — 152
A= 9, 9=41' -40°

A=10, 10=26°-24°
A=11, 11'=61-60%

A=12, 12=379-35
A=13, 13'=859-84

A=14, 14=509-489
Etc., etc.

Etc., etc. It is evident now that the square of any term in the series, 3,5,7, etc., is equal to the square of an odd number diminished by that of an even number; the square of any term in the series, 4, 8, 12, etc., is equal to the difference of the squares of two odd numbers, and the square of any term in the series, 6, 10, 14, etc., is equal to the difference of the squares of two even numbers. It will also be seen, that the square

of any odd number, greater than unity, is equal to the difference of the

2

squares of two numbers whose difference = unity; and that the square of auy even number greater than 2 is equal to the difference of the squares of two numbers whose difference = 2. Since all numbers are either odd or even, the

square

of
any

whole number greater than 2 is, therefore, equal to the difference of the squares of two other numbers whose difference = 1 or 2.

In the equation A'=B2-C?, A, B and C may be taken to represent the sides of a right angled triangle, and the sides A, B and C, may always be expressed in whole numbers by placing A=any whole number, B=4941, and C=4'71, when A=an odd number; B= A?+4

4 and c=424, when A=an even number.

If in eq. (m) we assume D=0, the values B and C, expressed in that equation, become infinite, and therefore we have

A’=(infinity)?— (infinity)?; that is, when D=0, we have B=C=any quantity whatever, from zero to infinity, while A is constantly=0. Clearing eq. (m) of fractions, we obtain

(2AD)'=(A'+D)--(A-Do)'; and this equation, being true for all values of A and D, may be expressed in whole numbers by substituting for A and D any whole numbers whatever. For example: suppose A=7, and D=4, then (2x7x 4)!=(72 +4?)?—(72—4°)?, which gives 569=652—33”. When A=4, and D=7, we have 56=652—(-33).

SOME THOUGHTS ON WOMAN.

BY MRS. S. C. SIRRINE, PLAINFIELD, WAUSHARA CO.

Woman was created for companionship; to move in a sphere of high responsibilities; to smooth the rusticities of man, and give the polish of refinement to civil society. Female education and female influence, therefore, are interwoven with all the domestic sociabilities of every family, are intermingled with, and give the sweetest relish to every cup of human happiness.

The history of all ages has shown conclusively, that the proper education of females, is inseparably connected with all those refinements of rational intercourse which characterize good society; and experience has long since settled the question, that these refinements have their origin, and receive their perfection from high mental and moral culture, existing in the female portion of every community,

Their influence, although not obtrusive, is, nevertheless, all pervading, and strongly resembles a kind of secret inspiration, which moves

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