Since those times, what wonderful accessions have been made to the general mass of information; what changes in educational systems; how has wealth increased and altered all the old relations and long-established ways of business! The boy of to-day has the culture of the man of yesterday; the girl surprises the woman with her wonderful advancement, and knowledge of things unknown in her girlhood, until the mother doubts. that this is a child of her begetting. But we are sometimes led to think that this is not all improvement, and that what we are pleased to style culture is, to a certain extent, mechanical expertness. We sometimes think that we may have a vast deal more learning, but, at the same time, less earnest thought; and earnest, well-directed thought, makes the man. Those were the days of Jay, of Livingston, of Josiah Ogden Hoffman, of Ambrose Spencer, of Samuel Jones, of Stephen Van Rensselaer, of Yates, of Philip Schuyler, of Rufus King, of Alexander Hamilton, and of many others whose integrity and patriotism should put to shame the time-servers and trimmers of to-day. They were men who endeavored earnestly and truly to carry out those two precepts of that greatest of men, Plato; "first, to make the safety and interest of their fellow-citizens the great aim and design of all their thoughts and actions, without ever considering their own personal advantage; secondly, so to take care of the whole collective body of the republic, as not to serve the interests of any one party to the prejudice or neglect of all the rest; for the government of a State is much like the office of a guardian or trustee, which should always be managed for the good of the pupil, and not of the persons to whom he is intrusted; and those men who, whilst they take care of one, neglect or disregard another part of the citizens, do but occasion sedition and discord." But though men have grown mechanical; though individual endeavor has lost its potency; though processes have taken the place of thought; though men are more guided by selfish interests—yet a great work has been effected. Information has become as necessary as the air we breathe; and every day the feeling is growing stronger, that ignorance and selfgovernment cannot go hand in hand. Woman is reaching forward to the higher functions of her nature, and man is every year gaining a deeper insight into the laws that should govern human actions and relations. We honor thee, first governor of the Empire State, among the great names of those times-and they are great for all time; not least on the scroll of honor shall thy name be written. EDITORIAL MISCELLANY. WE purpose from this time to devote a few pages of our Magazine to the consider ation of mathematical problems, questions in English analysis, and such other kindred matter as may be of use to the teacher. Teachers having difficult questions to propose, or elegant and peculiar solutions of problems, will please direct to MATHEMATICAL EDITOR, AMERICAN EDUCATIONAL MONTHLY, PROBLEM I. 430 Broome Street, New York city. Demonstrate that, if upon the three sides of a right-triangle, any three similar figures be described, the figure described upon the hypothenuse is equal to the sum of the other two. PROBLEM II. Demonstrate mathematically that the minute-hand of a clock must overtake the hour-hand. PROPERTIES OF NUMBERS. Every square number is either divisible by 5, or will leave for a remainder plus or minus 1. Every cube number is either divisible by 7, or will leave for a remainder plus or minus 1. Geometrical construction for a very close approximation to the cir cumference of the circle. A: B Da E Let BD be the diameter of a circle; erect at its extremity B, the perpendicular BC, and make it equal to the radius; prolong BD, and make Da=ab bc one-fifth of the radius; draw aC and cC, and make BA= aC. Now if we draw from A the line AE parallel to cC, BE will be only one-half millionth part smaller than 7. Ca2=Ba2×BC2=+1% and Ca=146 BC: Be :: BA: BE or : 13:14: BE from which = EDITORIAL CORRESPONDENCE. GERMAN GEOGRAPHICAL PUBLICATIONS. NUREMBERG, August 1, 1866. ERMANY, though pre-eminently a land of books, is not a land of school-books, for, as I have said in previous letters, the method of giving instruction is so different from that employed in American schools, that books are almost unneeded. Whatever can be taught by familiar lectures, is communicated to the child's mind in that way, and the Germans prefer this mode of teaching to that in vogue in America. There is an abundance of reading-books, because that is a department which can not be taught by oral communication. But the great variety of school histories, geographies, arithmetics, algebras, and grammars, which form a conspicuous part of the stock of an American bookstore, is not found in Germany; or if it be too strong an expression to say not found, let me word it, not thrust into the foreground as with us. Little text-books, in pamphlet form, there are indeed; but they are as unpretentious as possible, and bear the humble name of Leitfadens, or Rudimentary Hints. The so-called Hand-books which the German press issues are not works for the pupils of the schools, but for the use of the teachers, and contain the materials which are drawn upon to furnish the familiar lectures of the class-room. In one word, the method of instruction employed in our theological seminaries is adopted in all, or nearly all, the schools of Germany; a lecture is given, the pupils take notes, and answer questions the next day upon the instruction imparted. On some accounts this is a good system; it is better at any rate than that of committing the words of a text-book to memory, and repeating them by rote, in the manner prescribed in some schools. There is one class of text-books known in which Germany excels, and it is to them that I propose to devote a part at least of this letter. It is the Atlases which are used. The teacher may dispense with manuals of descriptive geography, but he can not dispense with the aid of good maps. The method of instructing in this department has called out two classes of works, which are in their respective ways among the most perfect that are known-the Geographical Hand-books, which are to be used by the teachers, and the Atlases, which are to be used by the scholars. Of the Hand-books I do not propose to speak at any length, excepting to say that there is not in the United States or in England, so far as I am aware, any thing so perfect in its way, as the Hand-books of Daniel and Klöden. They are each in three volumes, of about eight hundred pages to the volume, and form an admirable and thoroughly digested summary of all matters connected with geography-not executed in the manner of a gazetteer, but thoroughly worked out in a natural and not an alphabetical order. The authors are thorough geographers. I am personally acquainted with both Klöden and Daniel, and know that they are conscientiously giving the best years of their lives to those admirable manuals, and are keeping the new editions level with the advance of geographical science. The reader is probably not aware of the fact, that the most of the maps which he sees are mere copies of maps previously existing. Sometimes the copy is executed with much more skill than the original; sometimes the engraver adopts a kind of drawing and of lettering which would deceive any one who should not carefully compare the copy with the original; but the fact remains, that almost all maps which are familiarly known to the public are mere transcripts of others previously existing. The preparation of the first copy of a map is a task requiring the utmost patience and skill. The statistics in the possession of government bureaus, the records of travelers, and a thousand-and-one documents must be consulted, before a reliable original map can be wrought out. Item after item must be gathered where it can be found. Step by step it is filled in, one source yields one fact, and another source another, till at length all is done. It is in this way that the maps published by Justus Perthes, of Gotha, and Reimer, of Berlin, are prepared. Some of the most celebrated maps in the world are merely transferred from those brought out by these two houses. I was assured in Gotha last summer that even the celebrated Physical Atlas, published in Edinburgh by no less celebrated a geographer than Mr. Keith Johnston, was, in its first edition, a reproduction of Berghaus' original, with English instead of German names. This was some years ago, however, and now Mr. Johnston constructs original maps, some of them of equal excellence with the German ones. I gave the reader, a few months ago, a brief sketch of the gentlemen who are at the head of the great Gotha house, known throughout the world as Justus Perthes'. I hoped then to give an account of the system of works which they publish, and which are intended for the use of scholars of every grade of acquisition. I will not pretend to give a catalogue of all the books published by them, but only of their best atlases. These cover the whole field; there is hardly a single want which some of their works do not meet. These are all original; and the gentlemen who are engaged in preparing them are among the most competent geographers of our time. Let me briefly allude to some of the atlases published by Justus Perthes. First, because relating to the earth in its more general characteristics, and also because the most expensive and elaborate work, is Berghaus' Physical Atlas. This is the embodiment of a lifetime of scientific labor. Prof. Berghaus has always been the intimate friend of such men as Humboldt and Dove, and has incorporated the results of their labors in this great work. A very slight acquaintance with German makes the book available to American students, and no geographical library is complete without possessing it. Yet it is an expensive work, and would cost not much less than a hundred dollars in America. Prof. Petermann has, however, worked out and published a little volume, based on Berghaus', and which can be had for a small sum. It is published in the English language, and is a really valuable work-on the whole, the best small physical atlas that I know. The two works of Berghans and Peterman entirely cover this field of Physical Geography. The next one to be mentioned is Historical Geography. And in this department the Gotha house publishes a work utterly without a rival. This is the celebrated work of Spruner, known all over Europe, and found in the chief American libraries. The Historical Atlas allows the student to take any part of the world's history, and to have before him the configuration of the country about which he is reading, not as it is at present, but as it was then. Is he looking into the colonial history of North America? He spreads out before him, not a map of the Union as it is, but of the country as it was when possessed by the English, French, and Spanish colonies. Is he reading the history of the German Reformation? He turns to the map of Central Europe in the sixteenth, and not in the nineteenth century. Is he following the victorious steps of Genghis Khan, Charlemagne, or Frederick the Great? He turns to the maps of Asia and Europe which present the political configuration of their times. I have not Spruner before me as I write, but if I recollect correctly, there are above seventy-five maps in the whole series, commencing with the very dawn of history, and continuing down to the present time. They have all been carefully drawn up by Major Spruner, an accomplished Bavarian scholar, and cost in America not far from twentyfive dollars. Mr. Perthes has published two abridged editions of this work, one in English and one in German. The latter can be imported for about three dollars, the former for about five. The German one is much the best, for, in accordance with a hint given by an Englishman, the one intended for England leaves out, for the purpose of distinctness, the lines of mountains; and this makes it impossible for the reader to trace the intimate connection between the history and geography-a connection, I need not say, more markedly determined by the lines of mountains than by almost any other single element. Prof. Dittmar has published an admirable work of this character, small and yet excellent. The substance of this work will, ere many months, be in the possession of American scholars-the Messrs. Appletons having now in the hands of the engraver a selection made by myself from Dittmar's work, with the addition of some of Quin's and Smith's best maps. But no work will be likely to supersede the great work of Spruner; and, although too expensive to be generally owned by private readers, it ought to be in all public libraries. Next to a Physical and Historical Atlas comes a large and authentic Descriptive Atlas of the world at the present time. This should be to the readers of newspapers what Spruner's should be to the readers of history. And such a work is the admirable atlas known by the name of Stieler, and published also by Justus Perthes. Mr. Stieler, who began the work more than a quarter of a century ago, has been dead several years, but Major Stulpnagel, the younger Berghaus, and Petermann have risen to take his place, and a new edition is published every year. There are, I think, about eighty maps in the work, all of them worked out from original materials, and changed every year with the advancement of geographical discovery. The engraving of this work is by no means remarkable for elegance, and a reader not in the secret of the painstaking care which distinguishes it, would have'no conception of the value of the Stieler maps. They are stolen, however, in England, France, and America, as well as Germany; sometimes the copy is engraved with twice the elegance of the original, but without a trace of originality, excepting where the spelling must be adapted to the needs of a foreign tongue. There are several smaller editions of the Stieler maps, consisting of more or less of those in the larger one bound up together. I wish these were better known in America, for they are so thorough as to be far more valuable than the most of the atlases which have currency in the United |