[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 2? Sin. A = area of the two triangles, A x #x2 = area of the two sectors, = area between the chords. A 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 A = 30°, 43', 30". Also, by Natural Sines, Sin. 14:1:: fa: x; or, x= 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. 66 X = πα 180; 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 The expression for the X length of x will be and we shall have h: a :: 1 180' cos. 8: which, 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= Substituting for – and 180 and 2, R = 399.73, and D= 799.46. Ans. X 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. 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 beyond 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. We think there is such a power of a quantity as the Zero Power, for1st. 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 I 름 only by an infinitessimal, and we get a = 1+0; or al = 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 aaaaa by aa, we suppress the factors aa; or, since exponents are symbols of powers, aaaaa = aa = 25 • a? = a3 X a- a2, 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 af, a, a’, al and 1, successively, by a?, we get 1 1 ax, a', 1, a az and a?, a?, ao, a-1, a-3, 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 equivalence to (1)". Or, to bring out more clearly the subtractive nature of the negative sign, we may take the expression ab-", where the positive exponent points out a factor which increases the result, while the negative exponent points out a factor which diminishes the result. 3d. It is asked, if there be a zero power, what is the zero root ? } I Let us see. It is manifest that Vā=a?. So, ara, and, whatever be the value of x, this must still be true. Hence when x becomes 1 greater than any assignable quantity, we have Va=, or Pa=a«. The principle is the same as is applied to any other exponent, viz: Any root of a quantity is equal to a power of that same quantity whose exponent is the index of the root inverted. 4th. It is held, that if (1000)=1, and 1° = 1, therefore (1000) 1°, and 1000 =1. And so it does, so long as we consider the quantities with reference to their infinitessimal exponents, but no longer. Some of our correspondents object to the reasoning. But we admit the conclusion, and yet see no reductio ad absurdum. The difficulty, we think, is in forgetting that all finite quantities are equal when measured by an infinitessimal scale, as is done above, or by a scale of infinites, as is sometimes done. This may seem paradoxical, but not more so than the necessary doctrine, that there is no distance in infinite space and no succession of dates in eternity. Our earth at every point of her orbit is the same distance from the confines of space, and the nineteenth century is as early a date, estimated from the Beginning, as the first cooling of the primitive rocks. So, measured by a similar scale, 1000 = 1; but, detached from their zero exponents, it does not follow. - In education, as in everything else, causes will produce effects; if, therefore, we want good effects, let us combine the causes that will produce them. - The province of Education opens a wide field for the knavery of quacks and charlatans, who make a practice of plundering the unwary and the ignorant. The wretch who, by his bold and interested presumption, puts to hazard the health of the body, is a subject of mental detestation and reproach ; but he is still more detestable, who tampers with the health of the youthful mind. There is a large amount of valuable geographical information derivable daily from newspapers, that may be presented to pupils by Teachers. |