Method of instruction, geometrical song—Anecdotes—Size—Song measure—Observations.

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"Geometry is eminently serviceable to improve and strengthen the intellectual faculties."—Jones.

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Among the novel features of the Infant School System, that of geometrical lessons is the most peculiar. How it happened that a mode of instruction so evidently calculated for the infant mind was so long overlooked, I cannot imagine; and it is still more surprising that, having been once thought of, there should be any doubt as to its utility. Certain it is that the various forms of bodies is one of the first items of natural education, and we cannot err when treading in the steps of Nature. It is undeniable that geometrical knowledge is of great service in many of the mechanic arts, and, therefore, proper to be taught children who are likely to be employed in some of those arts; but, independently of this, we cannot adopt a better method of exciting and strengthening their powers of observation. I have seen a thousand instances, moreover, in the conduct of the children, which have assured me, that it is a very pleasing as well as useful branch of instruction. The children, being taught the first elements of form, and the terms used to express the various figures of bodies, find in its application to objects around them an inexhaustible source of amusement. Streets, houses, rooms, fields, ponds, plates, dishes, tables; in short, every thing they see calls for observation, and affords an opportunity for the application of their geometrical knowledge. Let it not, then, be said that it is beyond their capacity, for it is the simplest and most comprehensible to them of all knowledge;—let it not be said that it is useless, since its application to the useful arts is great and indisputable; nor is it to be asserted that it is unpleasing to them, since it has been shewn to add greatly to their happiness.

It is essential in this, as in every other branch of education, to begin with the first principles, and proceed slowly to their application, and the complicated forms arising therefrom. The next thing is to promote that application of which we have before spoken, to the various objects around them. It is this, and this alone, which forms the distinction between a school lesson and practical knowledge; and so far will the children be found from being averse from this exertion, that it makes the acquirement of knowledge a pleasure instead of a task. With these prefatory remarks I shall introduce a description of the method I have pursued, and a few examples of geometrical lessons.

We will suppose that the whole of the children are seated in the gallery, and that the teacher (provided with a brass instrument formed for the purpose, which is merely a series of joints like those to a counting-house candlestick, from which I borrowed the idea,[A] and which may be altered as required, in a moment,) points to a straight line, asking, What is this? A. A straight line. Q. Why did you not call it a crooked line? A. Because it is not crooked, but straight. Q. What are these? A. Curved lines. Q. What do curved lines mean? A. When they are bent or crooked. Q. What are these? A. Parallel straight lines. Q. What does parallel mean? A. Parallel means when they are equally distant from each other in every part. Q. If any of you children were reading a book. that gave an account of some town which had twelve streets, and it is said that the streets were parallel, would you understand what it meant? A. Yes; it would mean that the streets were all the same way, side by side, like the lines which we now see. Q. What are those? A. Diverging or converging straight lines. Q. What is the difference between diverging and converging lines and parallel lines? A. Diverging or converging lines are not at an equal distance from each other, in every part, but parallel lines are. Q. What does diverge mean? A. Diverge means when they go from each other, and they diverge at one end and converge at the other.[B] Q. What does converge mean? A. Converge means when they come towards each other. Q. Suppose the lines were longer, what would be the consequence? A. Please, sir, if they were longer, they would meet together at the end they converge. Q. What would they form by meeting together? A. By meeting together they would form an angle. Q. What kind of an angle? A. An acute angle? Q. Would they form an angle at the other end? A. No; they would go further from each other. Q. What is this? A. A perpendicular line. Q. What does perpendicular mean? A. A line up straight, like the stem of some trees. Q. If you look, you will see that one end of the line comes on the middle of another line; what does it form? A. The one which we now see forms two right angles. Q. I will make a straight line, and one end of it shall lean on another straight line, but instead of being upright like the perpendicular line, you see that it is sloping. What does it form? A. One side of it is an acute angle, and the other side is an obtuse angle. Q. Which side is the obtuse angle? A. That which is the most open. Q. And which is the acute angle? A. That which is the least open. Q. What does acute mean? A. When the angle is sharp. Q. What does obtuse mean? A. When the angle is less sharp than the right angle. Q. If I were to call any one of you an acute child, would you know what I meant? A. Yes, sir; one that looks out sharp, and tries to think, and pays attention to what is said to him; and then you would say he was an acute child.

[Footnote b: Mr. Chambers has been good enough to call the instrument referred to, a gonograph; to that name I have no objection.]

[Footnote B: Desire the children to hold up two fingers, keeping them apart, and they will perceive they diverge at top and converge at bottom.]

Equi-lateral Triangle.

Q. What is this? A. An equi-lateral triangle. Q. Why is it called equi-lateral? A. Because its sides are all equal. Q. How many sides has it? A. Three sides. Q. How many angles has it? A. Three angles. Q. What do you mean by angles? A. The space between two right lines, drawn gradually nearer to each other, till they meet in a point. Q. And what do you call the point where the two lines meet? A. The angular point. Q. Tell me why you call it a tri-angle. A. We call it a tri-angle because it has three angles. Q. What do you mean by equal? A. When the three sides are of the same length. Q. Have you any thing else to observe upon this? A. Yes, all its angles are acute.

Isoceles Triangle.

Q. What is this? A. An acute-angled isoceles triangle. Q. What does acute mean? A. When the angles are sharp. Q. Why is it called an isoceles triangle? A. Because only two of its sides are equal. Q. How many sides has it? A. Three, the same as the other. Q. Are there any other kind of isoceles triangles? A. Yes, there are right-angled and obtuse-angled.

[Here the other triangles are to be shewn, and the master must explain to the children the meaning of right-angled and obtuse-angled.]

Scalene Triangle.

Q. What is this? A. An acute-angled scalene triangle. Q. Why is it called an acute-angled scalene triangle? A. Because all its angles are acute, and its sides are not equal. Q. Why is it called scalene? A. Because it has all its sides unequal. Q. Are there any other kind of scalene triangles? A. Yes, there is a right-angled scalene triangle, which has one right angle. Q. What else? A. An obtuse-angled scalene triangle, which has one obtuse angle. Q. Can an acute triangle be an equi-lateral triangle? A. Yes, it may be equilateral, isoceles, or scalene. Q. Can a right-angled triangle, or an obtuse-angled triangle, be an equilateral? A. No; it must be either an isoceles or a scalene triangle.

Square.

Q. What is this? A. A square. Q. Why is it called a square? A. Because all its angles are right angles, and its sides are equal. Q. How many angles has it? A. Four angles. Q. What would it make if we draw a line from one angle to the opposite one? A. Two right-angled isoceles triangles. Q. What would you call the line that we drew from one angle to the other? A. A diagonal. Q. Suppose we draw another line from the other two angles. A. Then it would make four triangles.

Pent-agon.

Q. What is this? A. A regular pentagon. Q. Why is it called a pentagon? A. Because it has five sides and five angles. Q. Why is it called regular? A. Because its sides and angles are equal. Q. What does pentagon mean? A. A five-sided figure. Q. Are there any other kinds of pentagons? A. Yes, irregular pentagons? Q. What does irregular mean? A. When the sides and angles are not equal.

Hex-agon.

Q. What is this? A. A hexagon. Q. Why is it called a hexagon? A.
Because it has six sides and six angles. Q. What does hexagon mean? A.
A six-sided figure. Q. Are there more than one sort of hexagons? A.
Yes, there are regular and irregular. Q. What is a regular hexagon?
A. When the sides and angles are all equal. Q. What is an irregular
hexagon? A. When the sides and angles are not equal.

Hept-agon.

Q. What is this? A. A regular heptagon. Q. Why is it called a heptagon? A. Because it has seven sides and seven angles. Q. Why is it called a regular heptagon? A. Because its sides and angles are equal. Q. What does a heptagon mean? A. A seven-sided figure. Q. What is an irregular heptagon? A. A seven-sided figure, whose sides are not equal.

Oct-agon.

Q. What is this? A. A regular octagon. Q. Why is it called a regular octagon? A. Because it has eight sides and eight angles, and they are all equal. Q. What does an octagon mean? A. An eight-sided figure. Q. What is an irregular octagon? A. An eight-sided figure, whose sides and angles are not all equal. Q. What does an octave mean? A. Eight notes in music.

Non-agon.

Q. What is this? A. A nonagon. Q. Why is it called a nonagon? A. Because it has nine sides and nine angles. Q. What does a nonagon mean? A. A nine-sided figure. Q. What is an irregular nonagon? A. A nine-sided figure whose sides and angles are not equal.

Dec-agon.

Q. What is this? A. A regular decagon. Q. What does a decagon mean? A. A ten-sided figure. Q. Why is it called a decagon? A. Because it has ten sides and ten angles, and there are both regular and irregular decagons.

Rect-angle or Oblong.

Q. What is this? A. A rectangle or oblong. Q. How many sides and angles has it? A. Four, the same as a square. Q. What is the difference between a rectangle and a square? A. A rectangle has two long sides, and the other two are much shorter, but a square has its sides equal.

Rhomb.

Q. What is this? A. A rhomb. Q. What is the difference between a rhomb and a rectangle? A. The sides of the rhomb are equal, but the sides of the rectangle are not all equal. Q. Is there any other difference? A. Yes, the angles of the rectangle are equal, but the rhomb has only its opposite angles equal.

Rhomboid.

Q. What is this? A. A rhomboid. Q. What is the difference between a rhomb and a rhomboid? A. The sides of the rhomboid are not equal, nor yet its angles, but the sides of the rhomb are equal.

Trapezoid.

Q. What is this. A. A trapezoid. Q. How many sides has it? A. Four sides and four angles, it has only two of its angles equal, which are opposite to each other.

Tetragon.

Q. What do we call these figures that have four sides. A. Tetragons, tetra meaning four. Q. Are they called by another name? A. Yes, they are called quadrilaterals, or quadrangles. Q. How many regular tetragons are among those we have mentioned? A. One, that is the square, all the others are irregular tetragons, because their sides and angles are not all equal. Q. By what name would you call the whole of the figures on this board? A. Polygons; those that have their sides and angles equal we would call regular polygons. Q. What would you call those angles whose sides were not equal? A. Irregular polygons, and the smallest number of sides a polygon can have is three, and the number of corners are always equal to the number of sides.

Ellipse or Oval.

Q. What is this? A. An ellipse or an oval. Q. What shape is the top or crown of my bat? A. Circular. Q. What shape is that part which comes on my forehead and the back part of my head? A. Oval.

The other polygons are taught the children in rotation, in the same simple manner, all tending to please and edify them.

The following is sung:—

Horizontal, perpendicular,
Horizontal, perpendicular,
Parallel, parallel,
Parallel, lines,
Diverging, converging, diverging lines,
Diverging, converging, diverging lines.

Spreading wider, or expansion,
Drawing nearer, or contraction,
Falling, rising,
Slanting, crossing,
Convex, concave, curved lines,
Convex, concave, curved lines.

Here's a wave line, there's an angle,
Here's a wave line, there's an angle;
An ellipsis,
Or an oval,
A semicircle half way round,
Then a circle wheeling round.

Some amusing circumstances have occured from the knowledge of form thus acquired.

"D'ye ken, Mr. Wilderspin," said a child at Glasgow one day, "that we have an oblong table: it's made o' deal; four sides, four corners, twa lang sides, and twa short anes; corners mean angles, and angles mean corners. My brother ga'ed himsel sic a clink o' the eye against ane at hame; but ye ken there was nane that could tell the shape o' the thing that did it!"

A little boy was watching his mother making pan-cakes and wishing they were all done; when, after various observations as to their comparative goodness with and without sugar, he exclaimed, "I wonder which are best, elliptical pan-cakes or circular ones!" As this was Greek to the mother she turned round with "What d'ye say?" When the child repeated the observation. "Bless the child!" said the astonished parent, "what odd things ye are always saying; what can you mean by liptical pancakes? Why, you little fool, don't you know they are made of flour and eggs, and did you not see me put the milk into the large pan and stir all up together?" "Yes," said the little fellow, "I know what they are made of, and I know what bread is made of, but that is'nt the shape; indeed, indeed, mother, they are elliptical pan-cakes, because they are made in an elliptical frying-pan." An old soldier who lodged in the house, was now called down by the mother, and he decided that the child was right, and far from being what, in her surprize and alarm, she took him to be.

On another occasion a little girl had been taken to market by her mother, where she was struck by the sight of the carcasses of six sheep recently killed, and said, "Mother, what are these?" The reply was, "Dead sheep, dead sheep, don't bother." "They are suspended, perpendicular, and parallels," rejoined the child. "What? What?" was then the question. "Why, mother," was the child's answer, "don't you see they hang up, that's suspended; they are straight up, that's perpendicular; and they are at equal distances, that's parallel."

On another occasion a child came crying to school, at having been beaten for contradicting his father, and begged of me to go to his father and explain; which I did. The man received me kindly, and told me that he had beaten the child for insisting that the table which he pointed out was not round, which he repeated was against all evidence of the senses; that the child told him that if it was round, nothing would stand upon it, which so enraged him, that he thrashed him, as he deserved, and sent him off to school, adding, to be thus contradicted by a child so young, was too bad. The poor little fellow stood between us looking the picture of innocence combined with oppression, which his countenance fully developed, but said not a word. Under the said table there happened to be a ball left by a younger child. I took it up and kindly asked the man the shape of it? he instantly replied, "Round." "Then," said I, "is that table the same shape as the ball?" The man thought for a minute, and then said, "It is round-flat." I then explained the difference to him between the one and the other, more accurately, of course, than the infant could; and told him, as he himself saw a distinction, it was evident they were not both alike, and told him that the table was circular. "Ah!" said be, "that is just what the little one said! but I did not understand what circular meant; but now I see he is right." The little fellow was so pleased, that he ran to his father directly with delight. The other could not resist the parental impulse, but seized the boy and kissed him heartily.

The idea of size is necessary to a correct apprehension of objects. To talk of yards, feet, or inches, to a child, unless they are shown, is just as intelligible as miles, leagues, or degrees. Let there then be two five-feet rods, a black foot and a white foot alternately, the bottom foot marked in inches, and let there be a horizontal piece to slide up and down to make various heights. Thus, when the height of a lion, or elephant, &c. &c., is mentioned, it may be shown by the rod; while the girth may be exhibited by a piece of cord, which should always be ready. Long measure is taught as follows:

Take barley-corns of mod'rate length,
And three you'll find will make an inch;
Twelve inches make a foot;—if strength
Permit; I'll leap it and not flinch.
Three feet's a yard, as understood
By those possess'd with sense and soul;
Five feet and half will make a rood,
And also make a perch or pole.
Oh how pretty, wond'rously pretty,
Every rule
We learn at school
Is wondrously pretty.

Forty such poles a furlong make,
And eight such furlongs make a mile,
O'er hedge, or ditch, or seas, or lake;
O'er railing, fence, or gate, or stile.
Three miles a league, by sea or land,
And twenty leagues are one degree;
Just four times ninety degrees a band
Will make to girt the earth and sea.
Oh how pretty, &c.

But what's the girth of hell or heaven?
(No natural thought or eye can see,)
To neither girth or length is given;
'Tis without space—Immensity.
Still shall the good and truly wise,
The seat of heaven with safety find;
Because 'tis seen with inward eyes,
The first resides within their mind.
Oh how pretty, &c.

Whatever can be shewn by the rod should be, and I entreat teachers not to neglect this part of their duty. If the tables be merely learnt, the children will be no wiser than before.

Another anecdote may be added here, to shew that children even under punishment may think of their position with advantage. Doctor J., of Manchester, sent two of his children to an infant school, for the upper classes, and one of his little daughters had broken some rule in conjunction with two other little ladies in the same school; two of the little folks were placed, one in each corner of the room, and Miss J. was placed in the centre, when the child came home in the evening, Doctor J. enquired, "Well, Mary, how have you got on at school to day?" the reply was "Oh, papa, little Miss —— and Fanny ——, and I, were put out, they were put in the corners and I in the middle of the room, and there we all stood, papa, a complete triangle of dunces." The worthy doctor took great pleasure in mentioning this anecdote in company, as shewing the effect of a judicious cultivation of the thinking faculties.

In my peregrinations by sea and land, with infants, we have had some odd and amusing scenes. I sometimes have had infants at sea for several days and nights to the great amusement of the sailors: I have seen some of these fine fellows at times in fits of laughter at the odd words, as they called them, which the children used; at other times I have seen some of them in tears, at the want of knowledge, they saw in themselves; and when they heard the infants sing on deck, and explain the odd words by things in the ship, the sailors were delighted to have the youngsters in their berths, and no nurse could take better care of them than these noble fellows did.

I could relate anecdote after anecdote to prove the utility of this part of our system, but as it is now more generally in the training juvenile schools, and becoming better known, it may not be necessary, especially as the prejudice against it is giving way, and the public mind is better informed than it was on the subject, and moreover it must be given more in detail in the larger work on Juvenile Training or National Education.