there is undoubtedly a reason why ten units of the first order, rather than any other number, were taken to form one unit of the second order. It probably originated in the practice of counting by means of the fingers and thumbs.

The above will be sufficient for the second lesson.

Suppose now we have dots just sufficient to form ten rows, we would arrange them into a square, thus:

How many squares have we here? One. This one square is then a unit, but it is neither a unit of the first order (a dot), nor a unit of the second order (a row of ten dots), but ten such rows, forming one square. We therefore call it a unit of the third order, and denote it thus, 100. The naughts in the second and first places denote the absence of rows and dots in addition to those in the square. This gives a clear idea of one hundred.

Let now the scholars be required to express in dots the numbers denoted by the following expressions: 100, 101, 102, etc.; 110, 111, 112, etc.; 120, 121, 122, etc., etc.

Let the dots also be first written, and required to be denoted by the proper figures. This will be the proper limit for the third lesson.

If we have ten units of the third order, or ten squares of dots, we place the squares, side by side, thus forming a rectangle of dots, and this will be a unit of the fourth order, denoted thus, 1000. It will not be necessary actually to make this rectangle of dots on the board, for the imagination, which ought to be cultivated, will readily form the correct picture of it in the mind, and when this is done, the mind will have a true conception of one thousand.

Ten of these rectangles would form a larger square, or a unit

of the fifth order, denoted thus, 10,000, and of this, the imagination is to form the picture, which will be the true conception of one ten thousand.

Ten of these squares will form a larger rectangle, or a unit of the sixth order, denoted thus, 100,000, and of which the picture formed by the imagination will be the true conception of one hundred thousand.

Ten of these larger rectangles form a still larger square, or a unit of the seventh order, thus denoted, 1,000,000, of which the picture formed by the imagination will be the true conception of a million.

This process might be continued to any extent, and it will be found that every unit of an odd order is a square, and every unit of an even order a rectangle. The names of these orders of units should now be given, thus: Units, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, billions, etc.

It will be observed that ten units of the first order equal one of the second, ten of the second, one of the third, etc., and in general, ten units of any order equal one unit of the next higher order, and conversely, one unit of any order equals ten units of the next lower order.

Time will be required thus to teach, but the practice of hurrying classes over subjects so rapidly that it is impossible to understand them, can not be too severely reprehended.

COMPARATIVE GEOGRAPHY.

BY T. E. SULIOT.

'Geography can just as little be contented with being a mere description of the earth and a catalogue of its divisions, as a detailed account of the objects in nature can take the place of a thorough and real natural history."-Ritter's Comparative Geography.

Within the present generation, great improvements have been made in the teaching of geography in our schools. Many years ago, in his report of his educational tour through Europe, Horace

Mann gave a graphic description of the mode of teaching pursued in the schools of Germany, where teacher and pupil drew on the blackboard from recollection the principal physical features of each country, then located the chief towns, etc. The admirable works on Physical Geography by Prof. Trail, of Edinburgh, Sir John Herschell, and by Mary Somerville, the best of all, had made accessible to the general reader and to young students the leading points of a science that, with her sister Geology, is one of the daughters last adopted into the divine family of the Physical Sciences.

But Mary Somerville's work is too abstruse in some of its parts, presupposing in the reader a knowledge of kindred subjects, the want of which obscures the text.

The text book used in our schools-Warren's Atlas of Physical Geography-is too sketchy for our collegiate classes. He would confer a great benefit on education, who, from these and similar works (as Humboldt's Cosmos), should compile a treatise of moderate size, embodying their excellencies and leaving out all the. details which are too scientific for an elementary work. Who is more capable of performing that important task than that accomplished scholar, T. W. Harvey, who has made Physical Geography the special study of his life, and to whose lectures on that subject many of us have listened with delight? May he be spared to undertake and complete a work which, perhaps, he is even now silently preparing!

Mr. White's little work has revolutionized the study of Primary Geography, and converted into a lively, mind-awakening exercise what before was a lifeless and wearisome drudgery-the committing to memory of unintelligible abstractions.

Guyot's charming book-"The Earth and Man”—traces out striking analogies which reveal systematic order, where hitherto we had seen nothing but accident and confusion. His schoolmaps now published, with the geographical text-books which are promised soon to follow, will triumphantly carry on in the higher classes the work so happily begun in our primary schools by Mr. White's book.

But a new book, fuller than Guyot's and still more interesting, has been lately published by J. B. Lippincott & Co., Philadelphia : Ritter's Comparative Geography, translated from the German by

William L. Gage. In this work, within the moderate compass of about 200 8vo. pages, we have presented to us in language always clear and terse, occasionally rising to poetical expression and to eloquence, those beautiful analogies between the various continents, oceans, &c., of the earth, the exposition of which, to use the words of the author, converts geography from a mere lifeless aggregate of inorganic parts into a science where system takes the place of what had seemed chaotic confusion, and exhibits the habitation of man as a true organization, whose several parts contribute to the harmony, use and beauty of the whole.

We earnestly recommend this book to the young students of our High Schools and Colleges as eminently calculated to invest with new and wondrous interest what perhaps they have been learning merely from a sense of duty and necessity, rather than from its own intrinsic attraction.

To all teachers of Geography, this work is nearly indispensable, until its principal features are embodied into our common geographical text-books. The attentive perusal of either Ritter or Guyot, is a necessary qualification for a teacher who aims to be something better than the wearied recipient of lifeless recitations of tasks lifelessly committed to memory from the dull pages of a catalogue of proper names and political divisions arbitrarily parceled out of the surface of the earth, so beautifully prepared for the habitation of man by a wise and benevolent Creator.

INTERESTING WRITING LESSONS.

BY MISS ELIZABETH HACKING.

Teachers of Penmanship, who give many lessons in succession in the copy-book, may enter the school-room feeling as much enthusiasm as an artist might do when first coming in sight of the Eternal City. But this ardor will soon be chilled, for as soon as the pleasure of a new copy-book has worn away, with the want of variety, and the monotony of writing one column after another, the scholars lose all interest, and it then becomes more difficult for the teacher of Penmanship to keep the minds of the scholars

upon what they are doing, than in those branches in which new ideas are presented daily.

For an interesting exercise in writing, the scholars being prepared with waste-book, and wishing each member of the class to do something which interests him, I show them a number of specimens of various kinds. Placing one before them to criticise, and observing them as they look at it, some by their expression say, "I like that, and can do it"; while others may look listless, though, perhaps, not wanting in either taste or ambition. Presenting a different one to them, those faces void of expression before may change, and sparkle as the darkened waters of the ocean do, when the sun suddenly gleams from behind a cloud. So, I imagine, is it with artists; for were a number of them shown a painting, however beautiful it might appear to some, others would view it with indifference, and, if making it a study, would attain little of its perfection. When each pupil is provided with something which interests him, at a signal given by the teacher they begin to imitate the models they have selected.

Verses of poetry, business notes, a letter, or ornamental penmanship, form an attractive lesson which may develop some latent ideal of beauty, and kindle anew the waning interest of the pupil.

In criticising the penmanship of exercises in other branches, it may be observed that the signature is written with less care than the other portion. If one lesson be devoted at the commencement of a term to practice in writing their own name, grade, and the district which they attend, it will soon be seen by comparison that in giving such lesson the time is well spent. At the close of each regular lesson, one pupil's name, with grade, district and date, written on the board by the teacher to be copied by that scholar and criticised by the class at the next lesson, will secure, if persevered in, a permanent care as to names.

There is not a more beneficial or pleasing exercise for scholars than to practice ovals and "the line of beauty," and while doing so, to keep precise time to the teacher counting. When teaching a well disciplined class, who obey with military exactness, much more rapid progress will be made by counting for every line; for by this means some acquire a better slope, and others get rid of a contracted manner of writing. Counting also assists by guiding