SCIENTIFIC.

The use of Geometry in the Solution of Numerical Problems. All problems, whether numerical or geometrical, which, when stated algebraically, form equations of the first or second degree, may be solved independently of algebra, by means of the principles embraced in the first four books of Davies' Geometry. Problems which form equations of a higher degree than the second, may have their conditions represented by the higher orders of curves.

The simplest class of problems are those whose conditions may be represented by the divisions of a straight line. The following is an example of this class:

"A man has two horses, and a saddle worth fifty dollars. If the saddle be put on the first horse it will make his value double that of the second; but if the saddle be put on the second horse, it will make his value treble that of the first. What is the value of each horse ?"

SOLUTION.

BDF

1 I 1 I I

C Let AB represent the value

I 4 E.

of the first horse, and BC the value of the saddle. Then AC=AB +BC twice the value of the first horse. Bisect AC at D; then AD the value of the second horse. Make DE=BC= the value of the saddle; then AE-AD+DE= three times the value of the first horse. Take CFAB= value of first horse; then since the whole line AE contains three values of the first horse, and since AB and CF each represent one value, the sum of BD, DF and CE must also represent one value. Now, since AD=CD, and AB=CF, it follows that BD=FD. Also, since BC-DE, take away from each the common part DC, and there remains BD=CE. Hence BD, FD and CE are equal to each other, and each represents the third part of the value of the first horse. Divide AB and CF each into three equal parts; then will the whole line AE be divided into nine equal parts; and BC, the value of the saddle, will contain five of these parts. But the saddle is worth fifty dollars. Therefore, one of these parts must represent ten dollars. AB, the value of the first horse, contains three of these parts. The first horse is therefore worth thirty dollars. And AD, the value of the second horse, contains four of these parts. The second horse is therefore worth forty dollars.

Another class of problems may be solved by means of rectangular diagrams. All problems which give equations of the first degree may be solved either by means of the straight line or of the rectangle.

Problems which give equations of the second degree, require for their solution the employment of the circle.

Occasion will be taken hereafter to present examples of these two classes of problems.

LANCASTER, O.

Exchange and Union Schools.

METHODS OF EXCHANGE.

JOHN WILLIAMS.

The methods of exchange are so numerous, and may become so varied in the operations of an ingenious, zealous teacher, that the attempt to present the subject commensurately with its merits, or its ever-widening extent, would prove altogether futile.

I shall content myself with giving in a few successive articles, some of the most prominent and useful methods, as practiced in many of our best Union Schools, adding or interpolating some such variations as have developed themselves in my own attention to the subject.

From the different forms in which exchange presents itself as applicable to the exercises in one and the same school, I shall first select the exchange plan of conducting recitations..

Though equally valuable when applied to the common branches, as geography, grammar, arithmetic or orthography, I shall present, as briefly as consistent with a clear development of the subject, the exchange plan of teaching chemistry, as this branch will afford as good a subject for a specimen exercise as any other, involving as it does as many difficulties, and hence giving as good a field for the exercise of ingenuity in bringing the exchange plan to bear upon it as any other.

The exchange plan takes for granted not only a systematic arrangement in outlines of each particular subject, as heat, or affinity, or ponderable bodies, but a classified and comprehensive view of the whole subject of chemistry. It is farther desirable, if not indispensable to the highest success of the plan, that the teacher is not only perfectly famil iar with such a series of classified outlines, and so saturated with his subject that he can carry them into detail to any indefinite extent beyond the possibilities of the time daily given to this branch, but that he shall have prepared the outlines himself, with reference to all the text and reference books to which his class may have access, and with due regard, too, to the capacities of the class.

As an introduction to this science, as well as to many or all others, a comprehensive view of all knowledge may be presented.

I have found the following to serve my purpose as well as any other:

Definition-That which is or may be known.

KNOWLEDGE.

GRAND DIVISIONS.

LITERATURE-Investigation of mind by mind, and the com

munication of mind to mind.

SCIENCES-Applications of mind to matter and quantity.

ARTS-Application of mind and hand to matter.

This, or some similar classification, may be presented on the blackboard by the teacher, with definitions and explanations, and copied into note books by scholars, who will afterward, from day to day, give such portions of it as may have been assigned to them, without notes, from memory, on the black-board, with definitions and explanations, subject to the criticism of the class and teacher.

Two or more divisions of this outline, or other outlines, may be assigned to as many scholars, for each day's recitation, until every scholar shall have given the whole series satisfactorily.

In order to economize time, those scholars to whom subjects were assigned for reports in connection with the outlines, may be engaged in writing them out on the black-board while the remainder of the class are attending to the subject assigned to the class generally. Free opportunity is given to those who take particular subjects for investigation and reports, to make new or modified outlines, to extend or fill them up with details on the board, or orally, at pleasure, subject always to correction or criticism from the other members of the class.

The general recitation may be conducted as it ordinarily is, under the common plan of question and answer, or it may be better managed, at least for the most part, by an outline presented at the previous recitation, which will enable the class to proceed with their subject during recitation with little questioning or prompting from the teacher. He may thus act as chairman in sustaining the order of the recitation, in restricting or exciting criticism in the class, and in adding any trations or explanations necessary to the proper development of the subject.

MARLBORO UNION SCHOOL

illus

A. H.

To Teachers and Friends of Science.

Thanks are due to the many who have so promptly responded to the circular published in the March number of the Journal, calling for aid in the great work which our State Association has recently undertaken, An apology is also due for the delay of the promised blanks. This was the fault of no one, but was caused by the demand exceeding the supply.

A large number of blanks for recording both Meteorological and Periodical Phenomena, has at length been received; and copies have been forwarded to all who have sent to the undersigned their names and

address, indicating their willingness to coöperate in the proposed work. Copies have also been sent to some others, and many more will be sent to such, with the hope of enlisting their sympathy and aid.

All who desire, or are willing to engage in this enterprise, are earnestly requested, if they have not already done so, to send their names, indicating to the undersigned, the particular branch in which they will labor. To all such, blanks will be immediately sent, without charge. All who receive these blanks, are requested, if they find it impossible to use them, to hand them to some one who will. This is especially desired in the case of the blanks for recording Periodical Phenomena.

Feeling the great importance of circulating among the teachers of Ohio that most valuable little work, published by the Smithsonian Institution, entitled "Directions for Collecting, Preserving and Transporting Specimens of Natural History," the undersigned has procured at his own expense, two hundred copies (all that could be had at the time); and he will mail one copy, post-paid, to any one who will send him his address and four letter stamps, or three copies for eight letter stamps. Its value to one engaged in the study of Natural History, or in collecting specimens, can hardly be estimated; and it is believed that no one thing will do more to promote the study of this interesting branch of science than the free circulation of this little work. Address S. N. Sanford, Granville, Ohio.

SUGGESTIONS TO OBSERVERS.

The attention of those who are interested in the work of observation and investigation which our "State Association" has undertaken, is respectfully invited to a few suggestions bearing upon this important subject.

The report, to carry out the recommendations of which a committee has been raised, and much preliminary work been done, could of course only touch in a very general way, upon many points which need much more attention.

It should be remembered that the great and primary object in all this work of observation and investigation, is self-improvement and the improvement of pupils; first, by the acquirement of knowledge, from God's own text-book-Nature; and secondly, by the formation of habits of observation-habits most valuable, and which tell most powerfully upon the whole being. In all our efforts in this direction, we should keep this primary object distinctly, if not exclusively, in view: