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SCHOOL EXAMINATIONS.

UCH has been said, of late, about the test of examination being introduced into the different departments of State. Indeed, a great deal has been done by the State in this respect. More, I hope, will yet be done. The Society of Arts, too, deserves the thanks of every true educator for what it has done to promote the cause of education, by means of its examinations; for I believe that, after all, the way to make education general is to create a want of educated men, by throwing open situations to those only who have been proved to be qualified to fill them. The time, I trust, is not far distant when the examination test will be applied to all candidates for employment, not in the State only, but in the Church also.

My present object is to call attention to Elementary School Examinations, and to draw from your correspondents different plans for conducting them. By school examinations, I do not mean those examinations by inspectors of schools sent by Government, or any other body; but examinations instituted and conducted by the managers and teachers of the schools. Such examinations, I am inclined to think, are not so general as they ought to be. The benefits to be obtained from thorough examination are threefold. First, it is an incitement to study. For when a boy knows that a certain amount of knowledge will be required of him at the end of a certain period, he will have some immediate inducement to apply himself vigorously to work, in order to satisfy the demand. Second. It is a test of work done. This consists, or should consist, not so much in finding out what a boy knows, as in showing to him and his examiner his weak points. Boys, as well as men, often give themselves credit for knowing many things which they find they do not know when they come to be tested. The discovery of the weak points will be likely to lead the scholars to see the necessity of paying more attention to the lessons of their teachers and to their own private studies; and the teachers to see the necessity of carefully preparing and giving their lessons. Third. Regular school examination will prepare the scholars for future State or other examinations which they may, some day, be called upon to undergo. Many, no doubt, whose knowledge is not so deficient, fail to pass well, through want of experience in definitely and immediately expressing what they know. Repeated school examinations would give this experience.

In order to make examinations effectual, they must be periodical, and founded on work previously appointed to be done. The periods should not be too long, otherwise the interest of the scholars will flag. There should be a "great go' once a quarter, and a "little go' at the end of every month. A week would not be too long for the quarterly examination, whereas a day will be long enough for the monthly. The work for every day during the quarter should be previously appointed, so that the scholars might know what they have to do, and the examination should be founded upon it. This will induce both teachers and pupils to prepare thoroughly.

The results of the examinations should be faithfully recorded in marks or otherwise; and quarterly reports, containing the results of the examinations, and other useful information, should be made and sent to the parents, who will be interested in the matter. If small prizes in books be offered, so much the better.

It would be presumptuous in me to suggest any mode of conducting these examinations as a rule. Every school and teacher will have their own modes. But there are a few general principles which must regulate every mode. The school would, of course, be divided into classes. But as there are great differences in the attainments of the higher and lower boys of the same class, it should be still further divided into three or four forms, each of which should contain boys of attainments as nearly equal as possible. The boys of the same form should be compared in examination. If prizes are offered, one should be given to each form, so that the lowest would have a chance as well as the highest.

The examination should be conducted orally and in writing on slates and on paper. The oral part should predominate in the lower, and the written in the higher classes. Too much dignity cannot be given to the examination.

T.

[This is a capital suggestion. We trust that the examinations will be thoroughly searching, and honestly bring to light all defects. The quarterly examinations only should, we think, be both oral and written: the papers adapted to each class. Ed. J. E.]

ADVICE TO PARENTS.-Do not grudge the expense of Education provided the object be attained. Parsimony here is truly "penny wise and pound foolish." The cheap article is to be had; but it is superficial, and too often deleterious. Spending money prudently and liberally on the Education of your children is only putting it out to the highest interest. * Would you save for them? That may be a curse. Give them a good sound Education, and you give them the noblest fortune; for you give them, as you launch them on the sea of life, the elements of safety, happiness, and wealth. "A father came one day to Aristippus, the philosopher, and asked him to undertake the education of his son. The philosopher demanded as a fee 500 drachms. The father, a covetous man, was frightened at the price, and told the philosopher that he could buy a slave for less money. * 'Do so,' said Aristippus, and then you will have two.' Look well to the educator. See that he is a clear-headed, large-hearted, conscientious earnest man. Be sure that his example in most things is such as you would choose your child to follow. The precept of the teacher may be forgotten: the example of the man never. -Dr. Cornish on Education.

WHERE THERE'S A WILL, THERE'S A WAY.-A glover's apprentice in Edinburgh resolved to qualify himself for a higher profession. The relation with whom he lived was very poor and could not afford a candle, and scarcely a fire at night, and as it was only after shop hours that this young man had leisure, he had no alternative but to go into the streets at night, and plant himself with his book near a shop-window, the lights of which enabled him to read it; and when they were put out, he used to climb a lamp-post and hold on with one hand while he read with the other. That person lived to be one of the greatest oriental scholars in the world, and the first book in Arabic printed in Scotland was his production.- Canadian Journal of Education.

DYNAMICAL DIFFICULTIES.

1.-A body of finite dimensions, in which all points retain an invariable position with respect to one another, is called a rigid body.

2.-Change of position is a necessary result of motion.

3.-Therefore if any kind of motion, whether simple or compound, be predicated of one point in a rigid body, the same must be predicated of every other point; or, in other words, if any motion be predicated of one point, motion parallel to that must be predicated of every other point, or the relative positions of the points will be changed.

4.-Therefore if the centre of gravity, or any other point in a rigid body, moves around any distant axis, and has no additional motion, every part of the body must move around that same axis.

5.—In order that every point in a body shall move about a common axis outside itself, the body must continually present the same side to this axis.

6. If any additional motion be given to any part of the body, every other part of the body must partake of this additional motion. Thus, if one point in a body have, in addition to its motion about the distant axis, another motion around an axis within the body, this additional motion will be indicated by the turning away of the sides of the body from the distant axis. In this case every point of the body will have two turning motions, one about the distant axis, and the other about the axis within the body.

According to the accepted theory of motion, a body that simply revolves without rotation always presents the same side to a point infinitely distant, and different sides to the centre of revolution. This is utterly inconsistant with the accepted definition of a rigid body-(prop. 1.)-for in this case no two points of the body situated in the plane of revolution have parallel motion, they are describing orbits about different centres, and their paths intersect each other twice during each revolution.

Other inconsistencies and unnecessary complications might be pointed out, but the above will be sufficient in the meantime, and I will now proceed to point out in a few more propositions what appear to me to indicate some of the fundamental fallacies upon which these contradictions are based.

7.—A curved line is a line sui generis; it is continually bending, and the smallest conceivable part of it bends, and therefore no part of it, however small, can be represented by a straight line. All reasoning based upon the assumption that it can, is fallacious.

From this it follows that a circle cannot be represented as a polygon with a great or an infinite number of sides. If these supposed sides have dimensions they are straight lines and cannot represent the characteristic of a curve, viz., the continual bending or turning, and if they have no dimensions they are points, and an infinite number of points cannot make a line. 8.-Motion of translation may be curvilinear or rectilinear.

9.-Curvilinear motion is motion sui generis, and as a curved line is a line that is continually bending or turning, so is curvilinear motion a continually bending or turning motion, and cannot therefore be represented by motion in a straight line, or any number of motions in straight lines.

Neither is curvilinear motion a resultant of any number of rectilinear motions, for the diagonal of any parallelogram of rectilinear forces is a straight line.

10.-The actual tangental or centrifugal force of a planet or any other revolving body cannot be dynamically represented by a straight line or any number of straight lines, neither can the centripetal force of gravitation be represented, in its action on a revolving body, by any number of straight lines, for the direction of these forces as exerted by, and on the body, is continually changing, while the characteristic of a straight line is that it never changes its direction.

It may be objected that these are mere quibbles, but I maintain that they are quite the contrary; they are fundamental points of dynamical theory; they are the basis of columns upon columns of formidable formulæ, and when a fallacy, however minute, is made the basis of a long calculation or chain or reasoning, it may become magnified into a formidable error as it reaches the conclusion. Thus by neglecting the minute turn made by a body in the infinitesimal parts of its curved path, which the mathematician theoretically represents by a non-turning or straight line, he is led on to an error of a whole rotation by when the circle is completed.

Perhaps I may be asked to expound a better theory, as I object to the existing one. My answer to this is, that it is easier to find a fault than to remedy it, and that I am not competent to undertake the more difficult task. I suspect that if mathematicians who are competent to the task would carefully reconsider the grounds of the great controversy between Newton and Liebnitz, they will be led to the conclusion that it is desirable to return to the method of Fluxions.

As regards the description of the phenomena of curvilinear motion, I will venture to offer a suggestion, namely, that as there are different kinds of curvilinear motion, such as rotation and revolution, besides revolution of various kinds; and as these have some points of general resemblance and other points of specific variation, it would simplify our descriptions if we adopted generic and specific terms, as is done in natural history and other branches of science where description takes the leading place that in mathematics is occupied by definition.

I have no particular affection for the word "circumferation," proposed below, and am aware that it is open to etymological objections. A generic term is required, and that is the best term I can find in the mean time.

12.-If any point in a rigid body moves in a curvilinear path, the body circumferates.

13.-The axis of circumferation of a body is a line drawn through the centre or focus of the curve described by any point in the body, and perpendicular to the plane of its motion.

14. The radius of circumferation of any point in a body is a perpendicular drawn from that point to the axis of circumferation.

15. The plane of circumferation is generated by the motion of the radius of circumferation.

16.-The velocity of a circumferating body varies as the product of the length of the radius of circumferation multiplied by its angular velocity.

17. A body rotates when all its parts circumferate about an axis passing through its centre of gravity.

18.-A body revolves when all its parts circumferate about an axis that does not pass through its centre of gravity.

19.—A circumferating body will, during each complete circumferation, present all its sides to any other body situated outside the path of its circumferation.

In accordance with these definitions the moon's great motions would be described as thirteen circumferations in one year, these consisting of twelve revolutions about the common centre of gravity of herself and the earth, and one revolution about the sun; thus she must by proposition 19 present all her sides twelve times to the sun, and thirteen times to a fixed star during that period.

By this mode of regarding the moon's motion some important, dynamical facts, regarding which even able astronomers manifest considerable mental confusion, become simple and manifest even to those who have but elementary mathematical knowledge. It is continually stated by astronomers that in consequence of the moon's rotation she tends to assume the form of a spheroid of rotation. This is not the case. A spheroid of rotation results from the unequal velocities of the parts of the rotating body, when their velocities are proportioned to their distance from a line or axis passing through the centre of gravity of the body. The equatorial velocity in this case is the maximum, and the polar the minimum. Such is not the case with the moon, the velocity of those parts constituting the poles of the assumed axis of rotation being equal to the mean equatorial velocity. This follows from proposition 16, by which it will be seen that the mean velocity of the side of the moon furthest from the earth is to that of the side nearest the earth in the ratio of 239,160 to 237,000 nearly, while it exceeds the velocity of the centre of gravity in the ratio of 238,080 to 237,000; and therefore, that if the moon were fluid or imperfectly rigid, the greater centrifugal force of that outer portion would elongate it in that direction, and in like manner the part nearest the earth being attracted by the earth with a force exceeding the mean centripetal force in exactly the same ratio, it would be elongated to the same extent towards the earth, and thus form a prolate and not an oblate spheroid. Thus the movements of the earth would, according to my definitions, be described as follows :—

The earth makes-in round numbers-3661 circumferations in one year. Of these 3531 are rotations on her own axis, 12 are revolutions about the fulcrum or axis passing through the common centre of gravity of the earth and moon, and one revolution about the sun.

What will be the form of the earth by virtue of these circumferations? First, an oblate spheroid on account of the 353 rotations, this oblate spheroid will tend to the prolate form (which I suggest shall be called the spheroid of revolution,) by the varying velocity of its parts induced by the