TEACHERS' INSTITUTES FOR 1859. Tax undersigned will appoint, in different parts of the State, as many Institutes to continue in session from Monday evening to Friday evening of the weeks named, in the months of September, October and November, as he thinks he will be able to attend in person and secure the requisite help in conducting, as soon as he receives reasonable assurances on the following points: 1. A desire on the part of at least thirty teachers to have such an appointmert made. 2. A pledge on the part of the committee of the individuals asking for the appointment, to give due notice through the press of the neighboring towɔs and counties, and make all lo:al arrangements. 3. Accommodations for as many teachers, or persons intending to teach public schools the ensuing winter, as may report themselves on Monday evening or Tuesday morning. No favors are asked for the loiterers. 4. The free use of a suitable place for the sessions of the institutes during the day, and for public addresses in the evenings. Applicants will promote the success of the proposed meetings if they will designate their first, second and third choice as to the week in the month preferred, and also name the week in which, for any reason, the Institute should not be in their respective counties. HENRY BARNARD, Ag’t of Regents of Normal Schools. Mathematical Ocpartment. ! Editor JOURNAL:-Dear Sir:-In the April number of the Journal, I pointed out some errors in Mr. Whitcom's solution of problem 6, very briefly, and without attempting to be "profound," as the errors alluded to are so apparent that I supposed a bare reference to them would be sufficient : but as Mr. Whitcom has attempted, in the June number, to fortify himself in the false position which he had taken, I will, with your permission, point out more explicitly, and at length, wherein his solution is erroneous. I stated, in my article alluded to above, that if the equation y'=(x2—a?) tan. A tan. A' be the equation of a hyperbola, tan. A tan. Al will be equal to the sqare of the ratio of the semi-axes; that is, constant. Mr. Whitcom thinks it sufficient proof that the factor tan. A tan. A' is constant that the equation already contains two variables, viz: x and y, and asks if I am “ignorant of the fact that every equation between three variables must refer to, and only to, a surface ?” Every equation between three variables, two of which are independent, referred to three = 2, we axes, is the equation of a surface; but an equation of a line referred to two co-ordinate axes, may have any number of variables, provided they are all functions of x or y. For instance, the equation of the parabola 1 =y= Making x:=w and shall have for the equation of the parabola y'=2puz, an equation containing three variables, all of which are functions of x. In the circle, also, if the origin of co-ordinates be taken at the extremity of any diameer, and distances equal to the radius be measured off from the origin in opposite directions on the axis of X; and if from these points lines be drawn to any point of the circumference, we may have, representing the angles made by these two lines with the diameter by A and A', Mr. Whitcom's equation of the hyperbola, viz.: y'=(ix? — R') tan. A tan. A' 2Rx-X2 V2Rx -22 (1). In this case tan. A= ; and tan A'= ; there-R 2Rc-.ca fore tan. A tan. A'= **--R-. Substituting this in equation (1) we have the equation of the circle, viz.: y8=2Rx—«*. In a similar manner we might find the equations of other curves by first finding the value of tan. A tan. A' in terms of x and constants and substituting them in equation (1); but generally the equation of curves may be obtained more easily by other processes. The second objection which I made to Mr. Whitcom's solution, viz.: the confounding the co-ordinates of a particular point with the general co-ordinates of the curve is a valid one, and Mr. Whitcom's numerous references to Prof. Davies’ Analytical Geometry prove it to be so. That distinguished mathematician always represents particular co-ordinates by particular symbols: either by x or y, accented, or otherwise marked, or by other characters, and this is the universal custom of all writers on Analytical Geometry. Had there been no other error, however, in Mr. Whitcom’s solution, this would not have affected the numerical result. In the course of Mr. Whitcom’s investigation, he gave this formula for 64 +boya + a'y finding the length of an arc of a hyperbola, viz.; 2=x =a. a’ya With reference to this expression, I remarked that if we make y=0, the value of 2 will be infinite, instead of 0, which is its true value; and if we make x=0, 2=-a. Mr Whitcom says that these objections are not well founded. If he means by this that the values of z given above are not its true values under the suppositions there made, why does he not point out the error and give the true values? I also remarked that the above formula was obtained by considering the co-efficient of dx in the z = dx differential expression, constant, when in fact it contains y, and is therefore variable. Mr. Whitcom admits that this objection is well founded, and attributes the error to carelessness on his part. He therefore gives this revised formula for rectifying the hyperbola, viz: aʼæo—62x-a The absurdity of this is apparent at a glance, for every finite value of x gives z a value infinitely small. The formula for the length of an elementary arc of a plane curve, is da= Ndx4 + dy”. To employ this formula in any case, we differentiate the equation of the curve, and from the given equation and its differential equation, find the value of dy in terms of x and dx, and substitute it in the formnla. We then integrate the result between the proper limits, and the result obtained will express the length required. But it frequently happens that the differential expression refuses to be integrated, except by series, and this is the case with the hyperbola. This curve has never been rectified other than by series ; for example of which see Davies’ Mathematical Dictionary, pages 298 and 493. În conclusion I beg leave to propose the following modification of problem 6 for Mr. Whitcom to solve according to his method : There are three foxes forty rods due north of a grey-hound. They all start to run at the same time with the same speed. The first runs due cast, the second due north, and the third due west. The grey-hound, directing his course towards the first fox, runs until he overtakes him; he then directs his course towards the second fox which he also overtakes, and also the third in the same manner. If the grey-hound runs twice as fast as the foxes, how many rods will he have run when he overtakes the third fox? MADISON, June 14. Jas. M. INGALLS. Editorial Miscellany . A considerable portion of the present number is occupied with the proceedings of the annual meeting of the State Teachers' Association. It was a pleasant and profitable gathcring. was largely attended by teachers, school officers, and friends of education, and we believe will be productive of much good in uniting together the various parts of our educational system, in giving point and efficiency to the efforts hereafter to be made in the direction of progress and reform, and in establishing and promulgating correct views of the teachers' profession, proper estimates of the nature of his work, and an ac quaintance with the best means to be employed in its successful prosecution. The weather was all that could be desired, the citizens of Madison extended a generous welcome to the members of the Association, and the exercises were generally very interesting and profitable. The inauguration of Chancellor Barnard was a special feature of the occasion, and was witnessed by an audience which filled the City Hall to its utmost capacity. Dr. Barnard's inaugural address was characteristic of the man, giving in earnest, eloquent language, broad and comprehensive views of education and educa. tional systems, pervaded all through by strong common sense, and a practical adaptation of general views to the particular circumstances surrounding the field in which he expects to work. The address on behalf of the Regents of the University by Carl Schurz, Esq., was worthy of the reputation this gentleman has acquired as a profound thinker and finished scholar, and that of J. T. Clark, Esq., on behalf of the Normal Regents, was well written, in good taste, and appropriate to the occasion. We hope hereafter to give the addresses in full. The orations of the graduating class evinced some ability, but were too long, and, with two or three exceptions, seemed to lack somewhat in vigor and purity. We have no space to notice the efforts of the different young gentlemen, but must give our hearty commendation of the oration on philosophy and the valedictory addresses by Samuel Fallows. The oration was a forcible, logical, and eloquent presentation of the claims of philosophy as an important part of a liberal education, and the valedictories were conceived in good taste, were very appropriate, and were delivered in a graceful and effective manner. The address before the literary societies by Rev. Dr. Smith, of Lane Seminary Cincinnati, the subject of which was "Decision of Character," secured the uns qualified approbation of all who heard it, as a powerful and scholarly production, worthy of the reputation of the speaker, and abounding in valuable thought. The week will be remembered by many a teacher as an epoch in his or her edu. cational life, and the knowledge gained and acquaintances made will exert a happy influence, inducing more rapid growth, a broader development, and an increase of zeal in the good work. Since our last issue we (the Editor and his better half) took a trip as far east as New York city. We knew something of the space-annihilating power of steam as applied upon our railways, but never realized it so fully before. Leaving Palmyra at 6 o'clock A. M. of Thursday, on Saturday at 10:30 A. M. we were in New York city, a distance of one thousand miles. We took the new route through Michigan, leaving Milwaukee at 12 M., crossing the lake in the fine steamer City of Cleveland, Captain Dougall, and arrived at Grand Haven at 8 o'clock P. M. The trip across the lake is exceedingly pleasant as a change from railway travel, and adds to the attractiveness of the route. The Detroit & Milwaukee Railway, though scarcely finished, has a smooth track, almost entirely free from dust, is stocked with fine cars and seems to be carefully and efficiently managed. It is 186 miles long from Grand Haven to Detroit, and is likely to secure a fair share of travel and business, Arriving at Detroit, we crossed the river in the company's ferryboat (for which no extra charge is made) to Windsor, the western terminus of the Great Western Railway of Cananda, in the spacious cars of which we took seats, and at 8 A. M. left for Suspension Bridge, This is a well stocked road, and the management is as systematic and regular as clock-work. The depots and grounds surrounding them are kept clean, no noise or confusion is heard or seen, the conductors are gentlemanly and attentive, and the cars are well supplied with pure iced water, a great comfort in warm weather. Arriving at Suspension Bridge we took seats in the cars of the New York Central, and in a little less than twelve hours found ourselves in the capital of the Empire State. This road (the Central) is well known to the traveling public, and the long and well filled trains show their appreciation of it as a safe, expeditious and well managed route. Taking the cars of the Hudson River Road at Albany, we arrived in New York at 10:30 A. M., in forty-six hours and thirty minutes from Milwaukee, without a single detention or accident, making close connections as advertised with every train on the entire route—and it is with confidence that we recommend the Detroit & Milwaukee, the Great Western, and the New York Central Railways to the patronage of those of our renders who may be induced by business or pleasure to visit the east. DEATH OF HORACE MANN.-Tbis venerable and distinguished educator ceparted this life at Yellow Springs, Ohio, on the 2d instant, aged 63 years. He will be widely mourned and long remembered. We shall give the particulars of his useful life next month. PRESCOTT.-The citizens of this place held a public meeting on the 22d ult., and after con. siderable discussion decided to levy a tax of $1500 for the purpose of making such an addition to their school house as would enable them to establish a graded school. The Great Republic Monthly for A ugust is an improvement or all preceeding numbers. It contains several very valuable articles, and gives a large amount of reading for the price, Eee terms in former number of this Journal, The Ladies' Home Magazine for August is as interesting as ever. It is the beet family magazine with which we are acquainted. We furnish the Journal and Magazine to new subscri. bers for $200 a year. NEW EXCHANGE8.— The lowa School Journal, Vol. 1, No.s 1 and two: Andrew J. Stevens, Editor, N, W. Mills & Co.. publishers, Des Moines, Iowa. This is a gond looking double column sixteen paged quarto, the scope of which is somewhat broader than that of other school journals, it having a space devoted to agriculture, to household economy, to general literature, news, etc. We give it a bearty welcome as a co-laborer in the field of progress, and commend it to all who wish an interesting and valuable monthly of the kind abo ve indicated. The Southern Teacher, a Journal of School and Home Education, edited by W. S. Barton ; Vol. 1, No. 1.; Montgomery, Alabama; Barrett & Wimbish, printers and binders. A well got up journal of 48 pages, about the size of the New York Teacher. It is edited with care and ability, and we trust the teachers of Alabama will give it a hearty support. A Journal of Education, and of Science, Art, Language and Literature, edited by A. Curtis, A.M. M.D., Vol. 1, No. 2.; Cinciapati; published monthly by Longley Brothers. A wide awake practical journal, containing articles on the different topics above mentioned, some very spicy criticisms on language, art, etc., models for teaching, etc. |