A.D. 1380 and prechid (i.e. boded) seyinge, A strengere than I schal come aftir me.

It will not be necessary to call the young pupil's attention to the differences of inflections. But the mere reading aloud of these sentences will suffice to make him realize how little the English language changed from 1606 A.D. to 1150 A.D. (during which period the English-speaking and French-speaking classes had comparatively little intercourse with each other) as compared with the change between 1150 A.D. and 1380 A.D., when the two classes had learned to co-operate against the crown, and to recognize a community of interests. Thus he will be prepared for hearing that in 1362 A.D. French had become so unintelligible that it was supplanted by order of an Act of Parliament, which enacted that all pleadings in the law-courts should be conducted in "English, not French," inasmuch as French had become "much unknown in the realm"; and thus he will more easily realize the importance and remember the date of one of the most important events in the history of his native country.

For those who live in London, or other places of historical interest, it is scarcely necessary to say how valuable a stimulus for the study of history may be derived from a visit to Westminster Abbey or the Tower of London, especially after a little preliminary reading and study has prepared the pupil for what he is to see.

Geometry is so much more easy and interesting for the young than Algebra, that it may be properly included in the present treatise, although Algebra will be excluded.

The principal reason why so many young pupils fail in Geometry is that they are left to rely upon a book, instead of following and retaining the oral instruction of their teacher.

"I remember," writes Preceptor, "that after I had taken a young pupil successfully through the first six or seven Propositions of Euclid without the aid of a book, one day when I found myself obliged to go out, and unable to give the usual lesson, I ventured to place the book in the boy's hands, telling him to study the seventh or eighth Proposition by himself. To my horror, as I passed the door soon afterwards, I heard him singing the Proposition. I immediately anticipated the worst. My anticipations were fulfilled when I returned and found that he had learned the Proposition by heart, and could readily repeat it, without being able to understand a word of it. My neglect on this occasion caused a week's retrogression, and I felt that it would have been much better if the half-hour had been spent in play."

Very similar is the Author's experience. Boys have so strong a memory and so great an aversion to think-where memory can serve as a substitute for thought-that it is most important, in teaching Geometry, not to leave the pupil to himself till he has formed the

Geometry is so much more easy and interesting for the young than Algebra, that it may be properly included in the present treatise, although Algebra will be excluded.

The principal reason why so many young pupils fail in Geometry is that they are left to rely upon a book, instead of following and retaining the oral instruction of their teacher.

"I remember," writes Preceptor, "that after I had taken a young pupil successfully through the first six or seven Propositions of Euclid without the aid of a book, one day when I found myself obliged to go out, and unable to give the usual lesson, I ventured to place the book in the boy's hands, telling him to study the seventh or eighth Proposi tion by himself. To my horror, as I passed the door soon afterwards, I heard him singing the Proposition. I immediately anticipated the worst. My anticipations were fulfilled when I returned and found that he had learned the Proposition by heart, and could readily repeat it, without being able to understand a word of it. My neglect on this occasion caused a week's retrogression, and I felt that it would have been much better if the half-hour had been spent in play."

Very similar is the Author's experience. Boys have so strong a memory and so great an aversion to think-where memory can serve as a substitute for thought-that it is most important, in teaching Geometry, not to leave the pupil to himself till he has formed the habit of reasoning, and has learned enough to make him understand that it pays, in Euclid, not to try to remember, but to reason.

Many excellent teachers object to Euclid as being cumbrous, circui tous, and artificial. But until some other text-book is uniformly or generally adopted, it seems likely that he will maintain his present position. Leaving, therefore, to specialists the task of suggesting better methods or text-books, the Author will merely mention two or three expedients which he has found useful in teaching Euclid, pure and simple, to young children.

1. The Definitions and Axioms.-To begin by learning all the definítions and axioms is both tedious and bewildering.

Begin by doing or proving something definite; and then, in the course of your theorems or problems, introduce your definitions, axioms, and postulates, as you need them. Let the pupil collect them as they arise, and write them down for himself in a book.

They must undoubtedly be finally learned by heart; but before learning them, let the pupil understand their utility; and, so to speak, instead of regarding them as Euclid's axioms, let him be led to feel that they are his own axioms, which he has himself seen to be selfevident, and of which he himself demands the concession.

Thus, instead of beginning with Euclid's definition of a point, as "that which hath no parts and no magnitude," and a line as "length without breadth," we may for a long time appeal to common sense,

and take for granted that, in drawing figures the lines are to be as thin and even as possible, and the points no larger than is necessary to make them clearly visible.

But after the boy has been learning Geometry some time, you may draw with a thick piece of chalk upon a blackboard a straight line AC, of perceptible and uneven breadth, and point out (proving it by measurement, if you like) that in the triangle ABC, of which the sides AB, AC are called equal, it is not exactly true to say that AB = AC; for although AB is equal to one side of the thick line AC, it is not equal to the other side. Similarly, if AB, AC meet in a large point A, which "hath parts and magnitude," it will depend upon the part of the point from which you begin to measure, whether AB is really equal to AC.

Hence the pupil may be able to perceive that Euclid's Propositions could not always be exactly true of points and lines, unless points were "without magnitude" and lines "without breadth." But until he is able to perceive this, it will be best not to trouble him with Euclid's definitions of a point and line, but to leave him to "common sense."

2. The use of Rules and Compass.-Before proceeding to the Propositions, he should be taught how to draw triangles and circles neatly, so as to familiarize him with the use of the ruler and compasses.

When he has drawn several circles he should be told to measure the distance from the center to two or three points in the circumference, and to ascertain whether they differ in length; and an oval, or ellipse, having been drawn for him by the teacher, the pupil should be shown how this equality of the radii distinguishes the circle from the ellipse. But I should not as yet trouble him with the definition of a circle.

3. The First Proposition.-We now tell the child that we allow him compass and a ruler, but not a measuring-rule; and he is to try to describe, on a given straight line, a triangle with three equal sides.

After criticising his hap-hazard attempts, and pointing out that, even when the pupil is near the mark, he is proceeding by "guesswork," we offer to describe one in which the sides shall be exactly equal. It will be found that the First Proposition, thus introduced, will present no difficulty, and the child ought to be able speedily to work the Problem himself.

In the course of this Proposition, call attention to the fact that we have assumed that "things that are equal to the same thing are equal to one another." Tell the boy that an assumption of this kind is called an Axiom (which means "assumption") and bid him write it down in a manuscript book as the first of Euclid's Axioms.

4. The Second Proposition.-The Second Proposition presents more difficulty. For when the child learns that he is "from a given point to draw a straight line equal to a given straight line," he naturally replies that he can do it at once, by measuring with his pencil from the given point a distance equal to the given straight line.

We must therefore introduce this Proposition as a test of ingenuity, by saying that "of course any one can do this with the use of a measur

ing-rule, but we are expected to show our ingenuity by doing it without a measuring-rule, and with the aid of a compass used merely for the purpose of describing a circle."

When the construction is completed, before proceeding to Euclid's proof, it will simplify matters to go backwards from the conclusion, and to say, "Now you see that BG is equal to BC, do you not?" Yes.

"Then if we can show that AL is equal to BG,' the thing required will be done; for AL will have been drawn from A, equal to CB, will it not?" Yes. "Well, then, we shall show that AL and BG are equal in the following way: First, we shall show that DG and DL, the radii of the large circle, are equal, and then that DB and DA, sides of the equilateral triangle, are also equal; and subtracting the small equal lines from the large equal lines, we shall show that the remainder BG is equal to the remainder AL."

Probably the boy will find no difficulty at all in this reasoning; but it should now be pointed out to him that here we are assuming that "if equals be taken from equals, the remainders are equal"; and this statement having been illustrated from the subtraction of numbers, lines, and spaces, must be written down as another of Euclid's Assumptions, or Axioms.

This Proposition will need to be repeated perhaps two or three times by the teacher before the pupil can easily work it himself. But he may be helped by being accustomed to a summary of it in dialogue, thus: After he has completed the construction, he may be asked, "What are you going to prove?" That AL is equal to CB. "How are you going to prove it?" By proving that AL is equal to BG, and that BG is equal to BC.

When the teacher is repeating this proposition the second time, he must occasionally stop in the midst of a sentence and let the pupil complete it, to see whether he can take up the reasoning; nor must

1 AL and BG may be drawn in red ink, or otherwise distinguished from the rest of the figure, so as to call special attention to them.

the pupil be asked to work the problem himself till the teacher hás good reason to think that the task can be successfully accomplished..

5. Tests of Understanding.—When the first three Problems are mastered, they should be drawn upside down; or numbers should be substituted for letters, so that the pupil may be habituated to recognize the truth of the process in all circumstances, and to depend entirely on the reason, and in no respect on memory.

It should be needless to add that the pupil must not have the figure before him to begin with, nor should he be allowed to draw the figures in silence, and then state what he has been doing. Before he is allowed to draw a line, or join two points, or describe a circle, he must state precisely what he is intending to do.

Very often it is a good plan that the pupil should dictate the construction while the teacher executes it. The advantage of this is, that, if the pupil dictates inaccurately, the teacher can correct him silently by carrying into effect the inaccurate instructions, and showing their absurdity. For example, in the First Proposition, the pupil perhaps says, "Let AB be the given line" (omitting the word straight), upon which the teacher will draw a curved line. Or again, instead of saying, "from the center A, at the distance AB, describe the circle BCD," the pupil may say, "from the center A describe the circle BCD"; upon which the teacher will proceed from center A to describe an absurdly large circle BCD, passing through a second B, with a radius of eight or nine inches, or (if on a blackboard) of one or two feet.

6. Angles. Before proceeding to the Fourth Proposition, we must now introduce the pupil to angles. Great care is here needed to prevent him from falling into the error of confusing angles with triangles. For this purpose, Euclid's definition is of little use. "A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line." For, replies the boy, "What is the meaning of 'the inclination of two straight lines to one another?' I thought I knew what an angle was like; but I am sure I do not understand what 'inclination' is." To such a boy, at the present stage, Euclid's definition conveys no meaning, and tends rather to confuse him.

The best means of introducing a boy to the notion of an angle is to lay a stick AB upon another CAD, and then gradually make AB revolve upon the pivot A, so that it passes from a position (AB1) of coincidence with AD to a position (AB) where it is in a straight line with AD. Point out, as you move the stick away from AD, that the moving line is sloped or inclined more and more to the fixed line AD; and that when the moving line is half way (AB2) it is sloped equally with respect to AD and AC. Placing the moving line in different positions, e.g. AB, (half way between AD and AB) elicit from the pupil that AB2 is more inclined to AD than AB, is, and that AB is more inclined to AD than AB, is. Then show him that, if the line (AB) is half way between AD and AB, AB2 is twice as much inclined