a annum. ed in the creation of a very large debt, which must speedily be paid. Who, I ask, is in duty, and honor, and law, bound to pay this debt ? It may be well briefly to review the action of the Association in this matter. At the close of the year 1856, the former Editor of the Jour. nal resigned bis office. Certain responsible parties in Cincinnati, offered to conduct the publication of the Journal at an expense of $500 per annum. This would have been a reduction of more than $1,000 of the expenses incurred in former years; and in this way the Journal would not only have paid its way, but, doubtless, become a source of pecuniary profit to the Association. But this favorable proposition was rejected. In the last number of the Journal, publisbed prior to that meeting, the editor bad advanced the following statement and proposition : We do not believe that any new man, unacquainted with the business, can take charge of the Journal under such restrictions as the Association impose, and make it pay all expenses the first year. The printers' bill is about $2,500 per Office rent, fuel, postage, etc., about $100. Editor's salary $1,500. Total, $4,100. That a more economical plan for conducting the Journal should be adopted, none can dispute. '“We will not recommend a course for the Association to pursue, but it seems to us that the plan adopted by the N. Y. State Association might well be pursued by their Ohio brethren. The New York Teacher is the educational organ of three States, -New York, New Jersey, and California. It has a paying circulation twice as large as that of the Ohio Journal. But at the close of the last volume it found itself in debt to the amount of $2,400. Mr. Cruikshanks offered to take the entire responsibility of the concern for a term of three years, to pay all expenses, and to make it pay him what he could. They accepted his proposition. The Teacher is atill the organ of the Association, as heretofore. Associate Editors are appointed as formerly. But the Association is relieved from all pecu. niary obligations. Mr. C. acts as Resident Editor, and its fiscal management is his own personal concern. “ There is a gentleman residing in Columbus who, probably, could be induced to take charge of the Journal on the same conditions. The Association might appoint Associate Editors as heretofore, and continue to control its character, and at the same time be relieved from all care and expense as to its publication. “The gentleman of whom we speak, is Col. S. D. Harris, for many years & Teacher, and still deeply interested in the cause of education. He has had much experience as an Editor and Publisher, is possessed of eminent qualifications for taking charge of the Journal. We know of no man more competent to occupy this position, and we feel the utmost confidence that he would make the Journal quite equal to the best educational paper in the country.” And what kind of treatment did this proposition receive? The Executive Committee thought so meanly of it that they would not even present it to the Association for consideration. Had this plan been adopted, the Association would now be free from debt. Its members could come up to our next meeting without any fear that their time was to be chiefly occupied in listening to the piteous cry, "give! give!” Having rejected these two most excellent offers, it was determined to pursue a course in regard to the Journal which all might have known would result as it has. A gentleman who held an office of high respectability and usefulness, was induced to resign that post and take the editorial charge of the Journal ; and who will pretend that he should not receive his promised salary? A heavy balance is also due for paper, printing, etc., which must be provided for without delay. If any, who last winter voted with the majority, think it a hard case that they should be required to foot these bills, let them draw consolation from the reflection that they can blame nobody but themselves. And if our next meeting shall be disturbed, and rendered unpleasant and unprofitable by the necessity of meeting this subject, let the respon sibility be placed where it justly belongs. Truth. а Mathematical Department. PROF. W. H. YOUNG, ATHENS, EDITOR (All communications for this Department should be addressed to the Editor, Ohio University, Athens, O.; and to be in time, must be mailed by the first of the month preceding that in which they are expected to appear.] SOLUTIONS OF QUESTIONS PUBLISHED IN OCTOBER. No. 18. ARITHMETICAL QUESTION.-Divide $400 between A., B. and C., so that A. shall have one-fourth of the whole more than B., and B. one-third of the remainder more than C. SOLUTION BY J. L. CLARK. If A. gets £ of the whole more than B. he will get $100, leaving $300 still to be divided, and of which A. and B. are to have equal parts. But after A.'s share is taken out, B. is to have f of the remainder more than C., or twice as much as C.: hence A.'s share of the $300 must also be twice as much as C.'s, and we must divide it into five equal parts, giving C. 1, B. 2, and A. 2; or, C. $60, B. $120, and C. $120. Adding the $100 to A.'s share, we have $60+. $120 + $220 $100. [Some correspondents erred in referring the remainder, spoken of in the question, to what is left after taking out the "one-fourth of the whole ;" whereas by a mathematical interpretation of the language, it can only mean what is left after taking out A.'s entire share.—Editor.] No. 19. Having given the perpendicular (a) between two equal parallel chords, including one-third the area of the circle, it is required to find the radius of that circle. SOLUTION BY A. SCHUYLER. If we draw diameters joining the extremities of the chords, the area between the chords will be divided into two equal triangles and two equal sectors. Let x=radius, A = angle formed by the diameters, and a= the given perpendicular between the chords. Then x? Sin. A = area of the two triangles, = area of the two sectors, A Хтх2 T22 : and = area between the chords. 3 A 3 * By the Table of Natural Sines, we find that A is between 30° 43' and 30° 44'. By Double Position, we find the correction, and that A = 30°, 43', 30". Also, by Natural Sines, Sin. A:1::a: ; or, 3= 1.8873a. No. 20. We have received several interesting papers upon this question, but have not room to notice them this month. We would remark, however, that the peculiar properties possessed by the numbers 9 and 11, are in virtue of the relations of these numbers to the unit of the scale. If we form a number on some other scale than 10, for instance, on a scale of 14, the properties of 11 will be transferred to 15, and those of 9 to 13, and so for others; making a difference, of course, when scale unit is odd and when it is even. No. 13. Published in August. A bridge when measured on the floor is 80 ft. in length, and, by looking across from one end to the other, it is found that the middle is two feet higher than the ends. If the floor of the bridge be the arc of of a circle, what is the diameter of that circle ? The solution is substantially that of James McClung. = 40 ft X = COS X. TX X Let h = height of the arc = 2 = ver. sin. a. the number of degress in an arc similar to a, with unity as radius, and its ver. sin. will be 1 - cos x. The expression for the . . length of x will be 180, and we shall have h:a::1 — 208.8 : 180; 1which, by reducing, gives 1 - cos x = .000872x. From the table of Natural sines and cosines, we find x in this equation must be 5° 44', nearly. 7200 Also, we have : 40 ::1:R= 180 Substituting for s and and X, R= 399.73, and D= 799.46. Ans. TX ACKNOWLEDGMENTS.—All the questions were solved by A. Schuyler; Nos. 13 and 18, by J. L. C.; No. 13, by Joseph Turnbull, S. Harvey and James McClung; No. 18, by James Brown, J. W. Duffield, H. T. R. and E. Adamson ; and No. 20, by E. T. T. The solution of No. 13, by the "Salineville” correspondents, was entirely legitimate. We have received some interesting Questions for solution, but think it better to dispose of what is already before our readers in this volume, that there may be a clean drawer with which to begin the next. Several correspondents have referred to the article on the “Zero Power,” published in October; but as a full reply has not been elicited, we submit some remarks with the view of removing the objections seriatim. a PERSONAL.- The interest manifested by many of the Teachers in the Mathematical Department, during the past year, has been duly appreciated by ourself, as manager of this Department, and we can only wish they had been better repaid. We have heard no complaints, it is true, but this we attribute to the forbearance of our readers, upwards of thirty of whom have favored us with their correspondence. There have been some causes operating to weaken the interest, and thus hinder the success of the Mathematical Department, but they were beyoņd our control, and we pass them by. We have, however, long been convinced that this part of the Journal might be made more profitable to a larger number; but the inclination of correspondents has led in another direction, and we have not been willing to dictate. It may be well to suggest, that fewer problems and more frequent discussions of the principles and methods—the theory and practice—of mathematical instruction will better meet the purposes of an educational periodical. A a = = We think there is such a power of a quantity as the Zero Power, for 1st. It agrees with the definition of an exponent, viz: An exponent shows how many times a quantity must be taken as a factor to produce a given power. Zero (0) is a symbol of an infinitessimal, and is equivalent to A or 1. Take the expression a', where, whether a be integral or fractional, it must be admitted that the smaller we make x, the more nearly will a approximate to unity. Also, it will be admitted that, however small x becomes, it performs the function of an exponent; for if not, at what point does it fail ?—whence we conclude that, when x becomes less than any assignable quantity, at differs from unity only by an infinitessimal, and we get a=1+0; or all = 1. 1. And this result admits of an interpretation as satisfactory as any of the conclusions based upon infinitessimals and infinites. 2d. It is the necessary result oflegitimate reasoning. Division is the suppression of factors. To divide ab by a, we suppress the factor a. To divide aaaau by aa, we suppress the factors aa; or, since exponents are symbols of powers, aaaaa • aa= a5 ; a? = a3X a 는 a, where, as before, we suppress the factor a?. But suppressing the root factors of any power, diminishes the exponent of that power. For, exponents show how many times the root factor is taken, and as suppressing this factor any number of times is taking it that number of times less, we must diminish the exponent accordingly. Hence, dividing a4, ao, a’, al and 1, successively, by aż, we get 1 1 and a?, ał, a', a-1, a-?, where it is manifest that the corresponding terms must be equal. Nor do we find any difficulty in the negative exponents. + and — point out some kind of opposition between two classes of quantities, as may be illustrated by the liabilities and assets of an estate, as is seen in reducing a ship's traverse, in computing the plot of a survey, and as appears in the analytic discussions of magnitudes involving positive and negative abscissas and ordinates. Similarly with exponents. They always show how many times a factor is taken ; but while the + exponent refers that factor to the numerator, the exponent refers it to the denominator. The difference between a2 and a 2 is, that the minus sign in the latter expression is a qualifying symbol, showing its ' 2 equivalence to (a)? Or, to bring out more clearly the subtractive |