MATHEMATICAL DEPARTMENT.

TO CORRESPONDENTS.-Several solutions and problems, which it would give us much pleasure to publish, we are obliged to lay aside on account of the difficulty of procuring the necessary diagrams for them. The cuts could not be götten up nearer than Milwaukee, and to send there for them involves no little trouble and expense. No similar Mathematical Department, with which we are acquainted, deems it profitable, at least for the present we judge from appearances to lay before its readers any diagrams more difficult than can be represented by the compositor with the materials of his case. Will our correspondents take friendly notice and govern themselves accordingly?

We had intended to say the above last month, but owing to delays in the mail the "copy" therefor did not reach our printer till it was too late. This delay will also account for typographical errors in our last number, our corrected proofsheet not getting back to the compositor's room in season.

We

PERSONAL-If we did not dislike calling names, we could mention many of our old friends and contributors that we should be very glad to hear from soon, should have a greater variety to select from.

Solution of Problem 56.-Let E be the

F

E

center of a sphere whose radius = r and D the center of a sphere whose radius = r. Take DE to represent the major axis of an o ellipse and let C be a point in the circumference from which the greatest portion of spherical surface is visible. Let x and y be the co-ordinates of the point C, the origin being at the center of the major axis. Put 2a, and denote the minor axis of the ellipse by 26. Then,

DE

DCV (a - x)2 + y2.

line DE from which the greatest porPlace z EF, then 2a·

2 =

DF.

EC = √ (α + x)2 + y* ;'` We will first find a point F in the tion of spherical surface is visible. Assume p 3.1416, then the visible surface as seen from F will be represented by

=

Now ECDC may be regarded as a straight line divided into two parts, EC and DC having a known ratio of m to n. Hence

The values of x and y can now be readily found from (1) and (2).
L. CAMPBELL.

Note.-J. T. Dodge, in attempting to give an answer to Problem 56, says, "The extremity of the minor axis will be the required point." This is true when the spheres are equal to each other; but in any other case it is entirely erroneous. L. C.

Solution of the Problem in Regard to the Hands of a Watch.-The solution of the problem, "The hour and minute hands of a watch are together at 12 o'clock at what time are they next together?" as given in Ray's Arithmetic, Part III., is very difficult for pupils to comprehend, at least for mine. My solution is founded on the following reasoning:

The face of a watch is divided into 12 equal spaces or parts, marked out by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. The minute hand moves over these 12 equal parts in 1 hour, while the hour hand moves in the same time over 1 of those 12 equal parts. Hence the minute hand gains on the hour hand in 1 hour of time 11 of those 12 equal parts into which the face of a watch is divided, consequently, in hour, 1 part.

In solving this problem, the first question which arises is, how much has the minute hand to gain on the hour hand to be next with it after 12 o'clock? The answer must evidently be, the minute hand has to gain a whole rotation or the 12 equal parts of the face. The second question then is, in how much time will the minute hand gain these 12 parts, when it gains 1 part in hour? The answer is in 12 times hour, which gives 1 hour 5,5 minutes after 12 o'clock, or 5,5 minutes past 1 o'clock.

Should the problem be, at what time after 12 o'clock will the minute and hour hands of a watch be together the 10th time? then the minute hand has evidently to gain 10 rotations, equal to 120 parts of the face. 120 times hour gives gives 10 hours 545 minutes after 12 o'clock, or 54 minutes past 10 o'clock.

In the problem, at what time after 12 o'clock will the minute and hour hands of a watch be opposite to each other? the minute hand has to gain on the hour hand half a rotation, or 6 of the equal parts of the face. 6 times hour 32 minutes past 12 o'clock.

In the problem, at what time after 12 o'clock, will the minute and hour hands of a watch be the 10th time opposite to each other? the minute hand has to gain on the hour hand 9 rotations, which are equal to 114 parts of the face. 114 times hour gives 10 hours 21 minutes after 12 o'clock, or 21 minutes past 10 o'clock. F. E. TENNIESEN.

As is usual with documents of the kind, the statistics take up the bulk of the volume. These are arranged on a plan somewhat different from that heretofore

adopted. We gave the usual summary last month.

DISTRICTS--TOWN ORGANIZATION RECOMMENDED.

There is a gratifying tendency, under the Superintendent's advice, to the enlarging rather than the ensmalling of districts, and to the giving up of the joint districts with their numerous inconveniences. In two years the joint districts have decreased from twenty to seventeen per cent. of the whole number; and although eight or ten thousand school-children have been added during the year, and many new districts have of course been formed, as population has spread, it appears that almost as many old ones have been given up. This is gratifying. But the most important and the only radical improvement in this direction, the Town Organization, is recommended in the following forcible terms: "It would be far better for our schools, were the present district system so modified as to have each township constitute one district with as many subdistricts as the wants of the people would require. The interests of the several sections of a town are generally identical, while their modes of advancing their interests may be and often are as varied as selfishness and jealousy can make them. Each section organized as an independent district must necessarily follow the will of its own majority. This majority may in some instances be inimical to the public school, and thus deprive a large minority of privileges which they would gladly secure for themselves, and which they might secure could they be united with the more liberal majorities of other sections of the town. If majorities were always with the right, or always acted for the public good, this plea need not be made. It may be urged on the other hand, that an enlightened majority of one section may be overborne by the less liberal yet larger majorities of other sections of the town. This is admitted, but it is believed that such cases would be comparatively rare, and the greatest good of the greatest number should be our motto. In point of economy, I doubt not the township system would be far preferable to the present system. More time of better officers would naturally be secured. It would secure facilities for gradation not now possessed, and give to each town a central high school of easy access to all who would be fitted to enter such a school. The finances could be more economically mauaged. Teachers could be thus furnished with permanent employment by a slight change of location from term to term, and in many instances the inhabitants would be better accommodated by such an arrangement. The management of the schools could thus be made more uniform, and their reports would be more reliable. Each sub-district might still maintain a separate organization, whose trustees should have the general care and supervision of the school-house and other property, and at the same time visit and inspect the school of their

sub-district.

The presidents of the several local boards of trustees might form a board for the general management of the schools of the town, designating some of their number as agents for the employment of teachers, and to act as trustees of the high school. The State may not be ready for the full operation of such a plan, but it can never enter upon it at a more favorable time. Each year of delay wears the ruts deeper in the old track, and makes a change more difficult."

The most noticeable fact disclosed by our yearly school-statistics, is the large non-attendance, amounting for the last year to 38 per cent. of the whole. If we add to this the large number who, though registered in the public schools, attend with great irregularity, we have much cause for disquietude. Fully one-half our children may be regarded as constantly absent from school, and that kind of irregularity which arises from truancy, connected as it invariably is with idle. nese, and dishonesty towards parents and teachers, and often with vice, is even worse, as Mr. Pickard remarks, than absenteeism. Gov. Salomon notices this evil, in his late Message, and suggests whether compulsory attendance may not be a justifiable remedy. We can see many reasons for and none against a stringent application of such a rule against all truancy of registered pupils, and all children in short of idle and vagrant habits. The cure however for much of the non-attendance and irregularity of attendance, that arises from apathy or want of interest in the schools, is to be found in making the schools themselves more attractive, and this principally by furnishing them with more competent teachers. The untidiness, disorder, profanity and obscenity, which, like seven unclean spirits, seem to find a congenial abode in and around some schools, and from the influence of which thoughtful parents may well wish to shield their children, is not unnaturally accompanied, nay, is often induced, by an undiscerning, unsympathizing mode of government and a dead, mechanical mode of teaching, which, failing to arouse and direct the better impulses of the children, and to inspire a love for order, industry and study, of course leave passion, intellect and energy to run riot. Non-attendance upon such schools and we fear their number is by no means small-is not an evil of the most positive character.

HIGH SCHOOLS.

n

In this connection we may appropriately call attention to the argument reprinted in the present number for schools of a more advanced grade than the ordinary mixed school. The argument is principally upon the legal and constitutional grounds of the question and is very satisfactory. We advert to the subject here to remark that any considerable increase in the number of such schools must be preceded by the town organization. It is true that an existing law allows a union of districts for high school purposes, but experience shows that aside from such towns and villages as adopt the union school system with no reference to the law in question, there is little disposition to enter into any such arrangement as that contemplated by the law. There is a vast difference between doing a thing and having it done for you. When the law provides for graded schools as an integral part of the school system, and makes them obliga

tory under suitable conditions of population and property valuation, on pain of forfeiture of the public school money, such schools will spring up everywhere without difficulty.

SCHOOL HOUSES.

Although some improvement is recognized by the Superintendent, a great and sore evil still remains to be removed. The truth is that in many localities no one can be found capable of planning a good school-house; and even if a good model is at hand, it is by no means certain that it will be followed. We have long believed that a small School Building Fund, affording a moderate aid, on condition of following approved models, would work an immense change in schoolhouses. Various plans, to meet varying wants and scales of intended expendi ture, with ample details as to seating, warming, ventilation, out-houses, etc., should be kept in this case at the State Department and furnished as needed. We quote a portion of Mr Pickard's remarks:

"The quality of material, of which school houses are built, is not always a criterion by which to judge of internal arrangements. I have found some of fair exterior that were full of all uncleanness. Some log school houses, unattractive, perhaps repulsive in their exterior, have been found the very dwelling place of neatness. Comfort is not found alone in costly dwellings. Many of our best appearing houses are little better than torture boxes. I have found seats in a primary school room two inches higher than an ordinary chair, and the desk proportionately high. Behind these desks were hidden children of six to ten years of age. Hardly a child in the school room could touch his feet to the floor. The seats were evidently arranged for the comfort of the parents, who would assemble once a week for religious services. I could but wonder at the size of the people who would be comfortable in such seats. In several new houses I have found an utter disregard of ventilation-low ceilings and very tight walls, with windows so arranged as to admit of opening only from the bottom, thus exposing children who, from the size of the room and the number of windows, were compelled to sit near them, to a draft of cold air, whenever the large stove had fairly commenced its work.

"In the construction of school houses, however, there has been, as a general thing, very commendable improvements. In almost every county good models are found. Old fashioned seats have, in many old school houses, been remodeled and suited to the comfort and health of the occupants. Some easy mode of making the upper as well as the lower sash movable has been discovered. Upon the whole, there has been very commendable progress in the right direction. Time will work a radical change in the apathetic neighborhoods, if not in the line of improvement, in the utter demolition of what are improperly called school houses.

"In efforts to make the school houses comfortable and attractive, we should not forget the important appendages alike demanded by our moral and physical

natures.

"Very many of our school houses are destitute of out-houses. I would here