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. Bytogether 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.
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