_n2 (n+1)2

4

2

+4n=2n(n+1)

+n2=

n2 (n+1)

2

+ } n (n+1)]

in which n represents the number of terms in each series.

The quantity S is evidently a perfect square, since Sn(n+1). Put

n=1, 2, 3, 4, &c. Then S=1, 9, 36, 100, &c.

S:

2 L. CAMPBELL.

Solution of Problem 44.-Suppose m denotes the number of gallons of mixture constantly in the cask, and n the quantity that flows out each minute. Let a represent any number of minutes and y the number of gallons of wine remaining in the cask at the end of a minutes. Then at the end of the time x+dx the quantity of wine in the cask will be denoted by y-dy. dy represents the wine and ndx the mixture that escapes from the cask during the infinitely short space of time dx.

Hence m ndx::y : dy.·. mdy=nydx. Now y decreases while x in

+ C. When x=o the quantity of wine in the cask

m

=m... y=m and Clog m. Hence log y=log m

n=2 and x=60. Then y=4.53+. See Naperian logarithms. Cottage Grove.

L. CAMPBELL.

Solution of Problem 45.-20 ft. is the sum of the base and hypothe(80)2=6400 is the difference of the squares of the same quantities. The difference of the squares of two numbers is equal to the pro

nuse.

Problem 48-Suppose a man has a cow, which at the end of three years begins to breed, and afterwards brings a female calf every year, and that each calf begins to breed in like manner, bringing forth a cow calf every year, and these last begin to breed in the same manner; it is required to determine the owner's whole stock at the end of 20 years. [No allowances being made, we presume, for sales, gifts, thefts or losses by death or otherwise.-MATH. ED. W. L.

Oak Creek.

Problem 49.-According to the present system of calculating leap years, what will the difference between the civil and solar years amount to, at the end of the 20th century? Also, what year divisible by four hundred should not be a leap year? M. C.

Problem 51.-Suppose that in a country 1-60 of the population die in a year, that the number of births is 1-45, and that the immigration and emigration exactly balance each other : in how many years will the population double itself?

Waupaca.

M. McG.

Problem 52.-Two hunters, A and B, kill a deer whose weight they are desirous of knowing. For this purpose they rest a stick across the limb of a tree, then suspending the deer at the shorter extremity, they find that its weight is just counterpoised by the weight of A, who suspends himself by his hands at the other extremity. Without changing the point of support of the stick, they take the deer from the shorter extremity and suspend it at the longer extremity of the stick, when it is found to be exactly balanced by B's weight, when he is suspended at the shorter extremity of the stick. Now supposing A to weigh 147 pounds, and B to weigh 192 pounds, what must have been the weight of the deer? Good Hope, Mil. Co. M. C. SULLIVAN.

EDITORIAL MISCELLANY.

THE CONVENTION AND ASSOCIATION.

The Circular of the State. Supt., calling a Convention of the Co. Supts, will be found in his Department. The move is an excellent one.

As will be seen by the Programme, the Meeting of the Association at Janesville promises to be unusually attractive. In addition to such exercises as our teachers may present, the State Superintendents of Illinois and Michigan Messrs. BATEMAN and GREGORY, are expected to be with us; also, Mr. SILL, Prof. of the English Language in the Michigan Normal School. We trust that as many as possible of our best teachers will be on the ground early enough to be present at the Primary Convention of County Superintendents, called by Mr. PICKARD, and that those officers will all be in attendance, and be able to remain during the sessions of the Association. Great good may be expected to result from such a joint meeting.

INSTITUTE AT FOX LAKE.

A four weeks Institute was held in connection with the Wis. Fem. College during the month or April last. Between 50 and 60 experienced teachers were present, and the exercises conducted by Messrs. Allen, Goldthwait and Gaylord, were interesting and profitable. A portion of the session was spent in drawing out the "best methods" of imparting instruction, the teachers themselves being required to exhibit their plans; a part of the time was given to the "lecture system "in conducting the exercises, while the important subject of "Theory and Practice" held a prominent place in every day's session.

A religious revival, in which many of the Teachers found a hope in the Saviour, added greatly to the interest and benefits of the occasion. Superintendent Pickard and Judge McArthur, each gave an evening Lecture, attended by the members of the Institute and by a large concourse of the citizens of the place. We have seldom or never attended an Institute, the immediate and prospective results of which were more important.

We understand Mr. Goldthwait intends to make the Normal Department in the Institution at Fox Lake, specially prominent. X. X.

CONNECTICUT.

From the last Annual Report, by the State Superintendent, Hon. DAVID N. CAMP, we gather that "more than three-fourths of the school-houses in the State are now in good condition, a statement, that could not have been made made with truth before." To bring about this result, eighty-five thousand dollars have been expended during the past year; and this we consider good evidence of enduring progress. Of 1600 schools in the State, about one-third have a School Library, more than half have Outline Maps, and 646 have a set of Hol

brook's School Apparatus. In 1861, 10 Institutes of five days each were held, in the 8 counties of the State, with an average attendance of 68. Since 1839, about 150 Institutes have been held, in 103 of the 162 towns of the State, being an average of 7 each year, or nearly one each year for each county; thus showing that in the State where they originated, Institutes have not in 20 years outlived their usefulness. But Connecticut still has but one Normal School, and we are surprised at this fact, in a State where so much has been done for Educational progress. School-houses, Libraries, good text-books, and apparatus-and even Institutes, will never make our Schools what they ought to be, without adequate measures to train our teachers for their work.

MICHICAN.

The Twenty-Fifth Annual Report of the State Supt., is before us. We have extracted some important practical observations upon Moral Education, as the opening article of this number.

Compared with our own, Michigan is an old State, though a little less in population. As might be expected, her educational system is more advanced in some respects. We observe that there is somewhat of that comprehensiveness and unity in her educational work, the want of which among ourselves we have pointed out. The University, the Normal School, the Reform School, the State Board of Education, the Inspectors of the several Towns, &c., all report to the State Supt. His Report therefore, with appendices, presents in one pamphlet, something like a complete educational aspect of the State. Eight Institutes were held during the year. An office analagous to that of our County Superintendents is recommended.

FROM AND OF THE COUNTY SUPERINTENDENTS.

Through letters addressed to the State Supt., or to this office, or by gleaning a little from the newspapers, we are enabled to present further indications of the working of the County Superintendency. Most of the counties have now been heard from, and from every quarter come encouraging words. Though it may probably have happened that a bad selection has here and there been made, we believe the office is destined to work great good, and that it will be sustained by public sentiment.

KENOSHA CO.-We have had, this spring over one hundred and twenty appli cants for licenses. Perhaps the sieve has been too fine for the first chaffing, and a few have been dissatisfied; but the majority are highly pleased with the arrangement. It is sometimes remarked, "you have rejected good disciplinarians.” Admit it, but at the same time, if we have discarded good Teachers because they lacked only qualifications, they will make an effort to be better qualified, and will be successful at the next examination. If on the contrary, they do not exert themselves, they are not worthy the name, much less the situation, and should be rejected. Allowing us to be the judge, nine-tenths of the teachers are energetic, and are doing all in their power to bring our schools to a standard never

before attained. As an inducement and stimulus for our schools, we require a report from every Teacher once in four weeks, furnishing them with appropriate printed blanks. This system is working admirably, as it is breaking up the detestable habit of communicating, and the fashionable practice of being tardy. We publish these reports in the County papers, consequently the strife and competition will result in much good. We are making arrangements to celebrate the most glorious Fourth with a County School celebration, believing that we can in no other way, manifest a deeper feeling of patriotism.

B. L. DODGE, Co. Supt.

RACINE CO-Dear Sir:--I have as yet been unable to accomplish much towards extending the circulation of the Journal; but I hope to succeed better in the future. Of course every live teacher should read the Journal; and as Racine is now the "banner city," I trust the teachers throughout the county will try to achieve the same enviable distinction for the county.

The result of the examinations in this county is as follows:
Whole No. of applicants,

No. certificates granted,

149

.....104

There were five applicants for First, and ten for Second Grade, certificates; There will probably be a much larger proportion of higher grade applicants at the next examinations.

I am now in the midst of my second tour among the schools of the county; and have thus far been highly gratified by the evidence I have met with, not only of material progress on the part of the schools, but also of the more general prevalence, among the teachers, of a desire to improve, especially in the Theory and Practice of Teaching. I find fewer dull, lifeless schools, in which teachers and scholars seem to be principally concerned in "killing time," and more live teachers, who are making an effort to keep up with the times, are in terested in their work, and are trying to do their duty faithfully; and I hail this sentiment as the forerunner of speedy, substantial, and tangible improvement in the condition of our schools. C. W. LEVENS, Co. Supt.

WALWORTH CO.-Mr. Cheney has resumed the publication of his "Diary" in the Delavan Patriot, and we doubt not with beneficial effects.

GREEN CO.-The P. O. address of the Co. Supt. W. C. GREEN, is changed from Farmer's Grove to Monroe. Things are going on quietly in this county-at least we hear nothing but as a couple dozen of the "Gleanings " were ordered some time ago, we conclude that the Supt. wishes his applicants to have read at least one small book on the duties of their intended calling, before they meet for another examination.

JEFFERSON Co.-We extract a few sentences from a "Review of School Matters" in the Jefferson Republican: "220 applications have been presented, and 170 certificates granted; of which number 20 are for a shorter term than one year. During our examinations, five Teachers' Institutes have been held in the several Inspection Districts; at which very many of the most able and suc