Although not so many solutions, as usual, were sent in this month, there are quite as many communications. Several problems have been received lately. There are now on hand several short communications on different subjects in Mathematics, which will appear in due season. One from Bowlder on the problem of lights will appear next month if space will permit.

Correspondents furnishing questions will please accompany them with their own solutions when they can do so.

All communications for this department should be addressed "Math. Dep't. O. J. Ed'n., care of F. W. Hurtt, Cincinnati, O.," until further notice; and, to be in time, should be mailed by the first of the month preceding that on which they are to appear.

THE USE OF GEOMETRY IN THE SOLUTION OF NUMERICAL

PROBLEMS.

EXAMPLE 9.-The difference between the first and second of three numbers is 7; the difference between the second and third is 2; and the numbers are such that the sum of the squares of the first and second is equal to the square of the third. What are the numbers?

GEOMETRICAL EQUIVALENT.-Given the difference between the base and perpendicular, and the difference between the perpendicular and hypothenuse of a right-angled triangle, to find the sides.

SOLUTION.-Let ABC be the required triangle; and let CD be the square of the hypothenuse, CE the square of the perpendicular, and

=

ML = 2BG, a given line. Therefore, since the area of LN, and the difference of the adjacent sides are given, the side CN, which equal to the required base, may be found as in Example 4.

B represent the difference between the second and third of those numbers. Take CD twice the sum of A and B, and about CD as a diameter describe a semi-circumference. Take CE = B, and draw EF perpendicular to CD, and let it meet the circumference at F. From C as a center, and with a radius equal to CE describe a circumference, and through C draw the secant FG terminating in the concave arc; then will FG represent the least of the required numbers. The reason of this construction will be obvious from an inspection of Fig. 1, where DR is equal to GM, and RT is equal to BG.

REMARK 1.—If from a scale of equal parts the lines A and B be laid off in the ratio of the numbers 7 and 2, and if the foregoing diagram be carefully constructed, the line FG will measure 8 by the same scale on which A and B respectively measure 7 and 2; and the calculation will verify this value of the line FG.

REMARK 2.-All problems, whether numerical or geometrical, producing equations of the first or second degree, are as susceptible as the examples already given, of being analyzed by the geometrical method.

REMARK 3.--In any problem producing an equation of the first or second degree, the answer may be constructed geometrically, as has been done in the present instance. Therefore in regard to the two classes of problems just specified not only may the geometrical mode of analysis be employed as a substitute for the algebraic, but geometrical construc-* tion may be employed as a substitute for numerical computation.

LANCASTER, O.

JOHN WILLIAMS.

Having visited this Institution since Mr. W. P. KERR, the present Principal took charge of it, we believe that the accommodations correspond with the external appearance of the building. Its Catalogue

shows it to have been highly prosperous during the past year.

The annual examinations commenced on the 30th ult., and the closing exercises are to occur on the 4th of July: Rev. H. L. Hitchcock, of Columbus, delivers the Anniversary Address on the P. M. of the day. The next term will commence on the 13th of September.

Answers to Questions on the School Law.

BY THE COMMISSIONER OF COMMON SCHOOLS.

QUESTION 101.-Since the passage of the present School act, a new schoolhouse has been erected in sub-district No. 6, in township, and its entire cost assessed upon the taxable property in said sub-district. A few months subsequent to its erection, two of the inhabitants were set off to an adjoining sub. district; and it is now proposed by the Board of Education of said township to include in their annual estimate of the amount of money necessary to be expended for school purposes, other than for the payment of teachers, the sum of $38, to be assessed upon the taxable property in said sub-district No. 6, for the purpose of refunding to the inhabitants set off as aforesaid, the tax which they paid for building said school-house. Can a tax for such a purpose be legally assessed as proposed?

ANSWER.-The School law does not authorize the assessment of a tax upon the property of a sub-district, or of a township, for the purpose of refunding to any inhabitant a tax paid by him for building a school-house in the sub-district from which he may have been transferred. The reason is obvious. By sec. 22, it is provided that the amount of money annually needed in any of the sub-districts of a township, for purchasing school-house sites, building, furnishing or reparing school-houses, providing fuel, and for all other school purposes, other than the payment of teachers, shall be assessed by the county auditor on all the taxable property of the township, not included in any city or incorporated village or territory annexed thereto, forming any special district. Under the operation of this provision of the law, the transfer of an inhabitant from one sub-district to another, can neither increase nor diminish the amount of his school tax. The fact that sec. 23 provides that a portion of the cost of purchasing a school-house site, and erecting or reparing a school-house thereon, in any particular sub-district, wherein the inhabitants had not, prior to the enactment of the present School law, borne a reasonable share of the burden of taxation for such purpose, in comparison with other sub-districts in the township, does not require, in order that justice and equity may be done, that a tax assessed against a person for building a school-house should be refunded to him when set off to another sub-district; because the assessment of a portion of the cost of the school-house upon the property of a particular sub-district is authorized solely on the ground that its inhabitants have not borne their share of such burdens in comparison with the inhabitants of other sub-districts in the township.

The legality of the tax to refund the money in question, is not established by the circumstance, that the Board of Education, contrary to justice and equity, caused the whole cost of the school-house to be imposed on the property of subdistrict No. 6. A transgression of one provision of a law, does not authorize the violation of another.

In the School laws of those States where the cost of a school-house is required to be assessed upon the property of the district in which it is erected, some provision is generally made for the relief of those set off to other districts shortly after being taxed for building school-houses. The School law of the State of New York, for instance, contains the following provision :-"Every taxable inhabitant of a district, who shall have been, within four years, set off from any

other district without his consent, and shall, within that period have actually paid in such other district, under a lawful assessment therein, a district tax for building a school-house. shall be excepted by the trustees of the district where he shall reside from the payment of any tax for building a school-house therein. QUESTION 102.-When a school examiner vacates his office by resignation refusal to serve, removal out of the county, or otherwise, should the vacancy be filled by appointment for the full term of two years, or only for the unexpired term of the examiner who thus vacates his office?

ANSWER.-The appointment should evidently be made for the full term of two years. This is obvious from the language employed in the last clause of sec. 44 of the school act: All vacancies in said board, which may thereafter occur, whether from expiration of the term of office, refusal to serve, or otherwise, shall be filled by like appointment by said judge."

It requires no argument to show that vacancies resulting from expiration of the term of office, should be filled by appointment for the full term of two years. But the two classes of vacancies are required to be filled by like appointment by said judge. The term "like" must either be construed as referring to the length of time which the school examiners first appointed under the present school law, were entitled to hold their office, or to the fact that appointments to fill vacancies resulting from any of the causes stated in sec. 44, are required to be made for the same length of time, viz.: two years.

That the foregoing is a proper interpretation of sec. 44, will appear the more manifest by observing the language employed on the subject of vacancies, in sec. 3:-"In case a vacancy shall occur in the office of director, by death, resignation, refusal to serve or otherwise, it shall be the duty of the township clerk to fill such vacancy by appointment for the unexpired term." In this section of the law, vacancies which are occasioned by expiration of the term of office are not coupled with those resulting from other causes, and for the obvious reason that the former are required to be filled by an election, as provided in secs. 2 and 4. When, therefore, the term of office of a local director has expired, the vacancy resulting therefrom can not be filled by an appointment.

By sec. 2, it is provided that each director shall continue in office, until his successor is elected and qualified.

The general rule on this subject is, that, where the law does not expressly declare that the appointment shall be for the unexpired term, and where it does not provide that one part of the board of officers shall be elected, and another go out of office at regular and stated intervals, all vacancies shall be filled by appointments for the full term.

H. H. BARNEY, Commissioner of Common Schools.

CIRCULATION OF THE JOURNAL.

The first, second and third columns in the following Table, exhibit the circulation of the Journal, in the several counties in Ohio, at the close of the corresponding volumes, and the fourth, the present circulation of the current volume. In addition to these, we have more than 180 subscribers in other States, making the whole number over 2100. The number of subscribers in Ohio reported last year at this time was 2000.