T.-Just so. But let us try to prove it in a more business-like way, by our machinery of similar triangles. What lines shall we draw?

V.—BH and AH. We have the triangles ABH and BCH.

T. We could not make any thing out of these, I fear. Try the triangles CAH and BCH.

V. The angle C is common; but I do not see any other equal angles.

L.-The angle CHB is equal to BAH, because it is formed by the chord BH and a tangent CH, and the other angle is subtended by the same arc BH, therefore they are both measured by half that arc; therefore, the two triangles are similar; and

AC: CH = CH: BC, or BC X AC = CH2.

T.-Very good. This truth may then be appended as a corollary to our theorem about the intersection of chords; for the two truths or properties are very closely related. In many books of geometry, you will find these three truths demonstrated independently, as three distinct theorems. But this mode of presenting them, effectually conceals from the learner the beautiful analogy which binds them all into one fundamental truth, of which these three theorems are only different cases. In all your future geometrical studies, be then on the look out for such analogies, by considering whatever truth you are investigating under all the possible varieties of such circumstances as are only accidental, and do not affect the essential or fundamental conditions of the question. In short, learn to discuss a geometrical question as you discussed the problem of the "couriers" or of the "lights" in algebra.

You may go now; we have had a good time.

NOTE. This way of drawing out from the scholars themselves what you wish them to know, may be slower, more laborious to the teacher; but will it not stimulate and develop in them the inventive faculty, help to make them origi nal, not merely second-hand mathematicians?

FREEDMEN'S BUREAUS and Civil Rights bills are valuable expedients for the hour; but the vital wants of the South are a new plow, a new pulpit, and a new school-house.

UP-NORTH LETTERS.

No. X.

CLEVELAND, September, 1866.

FRIEND WHITE: Here we are, at the autumnal equinox; and almost a year has gone by since my last letter to the MONTHLY was written, though not many weeks intervene between my letters to its blonde editor. I use the word blonde as applicable to his name and character, rather more than to his complexion. And even at this late day I should not appropriate the requisite time for writing a letter for your readers, were it not for your frequent appeals to me for such service. I should be very far from certain that my letters could interest your readers, were it not for certain decided intimations which you give me, from time to time, to that effect. If they do any body any good, I am glad of it. From the beginning, their intention was not to be argumentative, solid and elaborate, but light, rambling and off-hand.

The third week of our current school year closes to-day. We now employ 118 teachers, including two of the German language, both graduates of the University of Berlin; one of French, a graduate of the University of France, and whose diploma bears the signature of Guizot; one of Penmanship, and one of Vocal Music. Of these teachers, eighteen are gentleman and one hundred ladies. Twenty of them are new teachers; that is, they are new to us, though most of them have had experience in other places. Very greatly to our regret, Mr. Charles H. Adams, Principal of the West St. Clair St. School, was obliged, on account of ill health, to resign that position near the close of our last school year. His place has been filled by the election of Mr. Levi Rodgers, a graduate of Dartmouth. Mr. A. G. Hopkinson, after a year's absence, has resumed the principalship of our West High School. During my connection with the Cleveland schools, about 170 different teachers have been employed; and of them all, not one has died while thus employed, and but two since their resignation and removal from our city.

Our schools still suffer from a chronic want of additional buildings, though in this respect we are somewhat better off than we

were one year ago. We now hope for greatly additional improvements in this regard; for the contract for our new Hudson Street School was this week let to responsible builders. The building, alone, will cost $46,000. The site, fencing, furniture, and so forth, will probably raise the amount to about $70,000. It is designed for seventeen teachers, and pupils to match. I think it will be one of the best district school buildings of my acquaintance in Ohio. Would that it were ready for present occupation. We could fill every room, and then need three additional houses of the same capacity.

You inquire why I did not attend the educational meetings at Indianapolis last month; and I reply that my absence therefrom was not intentional, but unavoidable; perhaps I should say, providential. While spending a few days at the pleasant town of Marquette, on Lake Superior, I, like our distinguished ancestor, "fell from the estate wherein I was created," or rather from the planks over which I was running, and inflicted a painful wound. upon my face, which so marred my good looks that for two weeks I regarded myself as in a condition which forbade that I should rush into the presence of the handsome men and women who are always found greatly in the majority at all national meetings of teachers.

At one of our recent teachers' meetings here, we discussed, not for the first time, the subject of school punishments; what, other than corporal, are appropriate. We do not forbid the infliction of bodily chastisement, neither do we encourage its frequent and indiscriminate practice. The rules established by our Board of Education touching this subject, are as follows:

"In inflicting corporal punishment-which should be resorted to only in cases of extreme necessity, arising from flagrant and persistent disobedience-no other instrument than a common rod or whip shall be employed.

"2. Teachers shall make to the superintendent monthly reports of all cases of corporal punishment in their respective schools."

The object of the former of these rules is to prevent the use of ferules, and other like instruments.

During our last school year, we had an enrolment of 9,270 pupils; and the cases of corporal punishment inflicted were 1,372. It thus appears that during the 200 days of schools, one child in every seven enjoyed a whipping; that is, provided no one child

experienced this happiness but for a single time. This, however, was not the case; for I judge that not more than one in twentyfive of the children in our schools received this chastisement. In our High Schools, which enrolled 277 pupils, there was not one case of corporal punishment. The same is true of thirty of our schools of lower grades. Nearly all these punishments were inflicted upon boys, which is good evidence that mankind are everso-much worse than womankind; or if not worse, certainly more unfortunate.

I do not regard the number of corporal punishments inflicted in our schools the last year, as unreasonably large, as schools go nowadays. Our average daily attendance was 5,333; school days 200; days of schooling, 1,066,600; which divided by the number of punishments (1,372), equals 777. Suppose a single pupil had received all the schooling and all the punishments, he would have been whipped but once in almost four school years.

I am not at all prepared to say that corporal punishment should never be inflicted in schools. I doubt not that in some cases it is productive of greater good than could be secured by other means. But these instances are, in my opinion, not very numerous, and I fully believe that half of all the punishments of this kind which are inflcted in families and elsewhere, result only in injury, moral as well as physical, to those for whose good they are designed. Here I come back to the question already stated: What other forms of punishment are proper in the administration of school government?

It is almost dark now, and it is time for me to go home. I can not, therefore, attempt an answer to this inquiry at present. I may resume the subject at a future time, but be that as it may, I wish that some of your able correspondents would take the matter in hand, and give us their opinion on the subject. I believe that the discussion of the question may be made useful to many of the younger class of teachers in our State, and elsewhere. YOURS TRULY.

POSTSCRIPT.-As my letter of a month ago does not appear in the October number of the MONTHLY, I conclude that it reached you too late. I will add a few items of local interest.

Mr. Sidney A. Norton, for nine years teacher of the sciences

in our Central High School, has resigned that position for a similar post in the Mt. Auburn Seminary, in or near Cincinnati. We greatly regret his leaving, for he is a most successful teacher. We can but admire the wisdom of our Cincinnati friends in coming to Cleveland for teachers of the best quality; but we object to their taking them on such short notice. We have scores of teachers that would do much to improve the schools of Porkopolis, but we seriously object to parting with them, even for a purpose so necessary.

Mr. Norton's place has been supplied by the appointment of Mr. Theodore W. Hopkins of this city, who two years ago graduated at Yale College.

Some two hundred of the students of Oberlin College are, through the autumn, making special preparation for teaching next winter. They are under the training of Rev. W. N. Bartlett, Principal of the Preparatory Department. A course of lectures by gentlemen from abroad has helped on the work. Messrs. Ingersoll and Cowdery, of Sandusky, Thome and Smyth, of Cleveland, have lectured within the fortnight past. Y. T.

SCHOOLS OF MIAMI COUNTY.

MR. EDITOR: Perhaps a letter pertaining to the schools of this county may not prove uninteresting to your readers. We have good school-houses as a general thing, but they are poorly supplied with apparatus.

The sub-district schools continue in session from six to nine months each year. The summer terms are taught almost exclusively by women; and they constitute about one-third of the teachers employed in the winter school. The wages for this year will average about as follows: Male teachers, per month, $42; female, $33.333 winter term, and $22 summer term.

Although our school examiners have raised the "standard of qualifications," I am satisfied it would be better for both schools and teachers if they would raise it still higher. Good teachers and good wages are certainly the result of strict examinations. When a high standard is maintained by the examiners, and our local directors adopt that sound maxim-"Good teachers at the price of good teachers, but, at any price, good teachers"-then and only then, may we expect the rising generation to become educated men and educated women. C. D. WRIGHT.

TROY, O., Sept. 21, 1866.