CHAPTER IX.

GEOMETRY.

Rev. Dr. Hill, the present President of Harvard College, in his articles in Dr. Barnard's "Journal of Education,”has set forth the importance of Geometry in the earliest education, giving the Science of Form precedence to that of numbers. Of course he does not mean that logical demonstration is to form one of the exercises of little children! but that observation of differences and resemblances of shape, and the combination of forms, should be inwoven with the amusements of children. He invented a toy on the principle of the Chinese tanagram, (published by Hickling, Swant & Co., in Boston,) to further an exercise which begins in the cradle with the examination of the hands and feet.

The blocks are the first materials. Take the cube and ask how many faces it has; how many corners; and whether one face is larger than another or equal; and finally, lead the child to describe a cube as a solid figure with six equal sides, and eight corners. Then take a solid triangle from the box and draw out by questions that it has five sides and six corners, that three of its sides are equal, and two others equal; that the three larger sides are four-sided, and the two smaller sides are three-sided; and that the corners are sharper than those of a cube.

Make analogous use of all the blocks, and of the furniture of the room, of the sphere and its parts, the cylinder, &c. Do not require the definition-formulas at first, but content yourself with opening the children's eyes to the facts which the formula afterwards shall declare.

Paper-folding can be made subservient to another step, just short of abstraction.

Give each one of a class a square piece of paper, and proceed thus: What is the shape of this paper? How many sides has it? Which is the longest side? How many corners has it? Have in hand, already cut, several acute and obtuse angled triangles, and showing them, ask if the corners of the square are like these corners? If they are as sharp as some of them; or as blunt as some? Spreading out the triangle before them say, which is the sharpest corner, and which the bluntest? and let the children compare them with the corners of the square, by laying them upon the square. They will see that the square corners are neither blunt nor sharp, but as they will perhaps say, straight. Let them look round the room, and on the furniture and window-sashes, find these several kinds of corner. At least they can always find right angles in the furniture. Then tell them there is another word for corners, namely, angles, that a square corner is a right angle, a sharp corner a sharp angle, and a blunt corner a blunt angle.

If the teacher chooses she can go farther and tell them that acute is another word for sharp, and obtuse another word for blunt; (or these two Latin words may be deferred till by and by, one new word angle being enough to begin with.)

You can then say, "Now tell me how you describe a square, supposing somebody should ask you that did not know ;" and give them more or less help to say: “A square is a figure with four equal sides and four straight corners (or right angles)." To prove to them that it is necessary to mention the right angles in describing a square, you can make a rhombus, and show them its different shape with its acute and obtuse angles. Having thus exhausted the description of a square, let every one double up his square, and so get an oblong. Ask if this is a square? What is it? How does it differ from a square? Are all four sides different from each other? Which sides are alike? How are the corners (or

angles)? In what, then, is it like a square? In what does it differ? Bring out from the child at last the description of an oblong, as a four-sided figure with straight corners (or right angles), and its opposite sides equal. Contrast it with some parallelogram which is not a rectangle, and which you must have ready. Let them now fold their oblongs again, and crease the folds; then ask them to unfold and say what they have, and they will find four squares. Ask them if every square can be folded so as to make two oblongs, and then if every oblong can be so divided as to make two squares? If they say yes to this last question, give them a shorter oblong, which you must have ready, and having made them notice that it is an oblong, by asking them to tell whether its opposite sides are equal, and its angles right angles, ask them to fold it, and see if it will make two squares. They will see that it will not. Then ask them if all oblongs are of the same shape; and then if all squares are of the same shape?

The above foldings will be enough for å lesson, and if the children are small it will be enough for two lessons.

Beginning the next time, ask them what is the difference between an oblong and square? and if they have forgotten, do not tell them in words, but give them square papers and let them learn it over again as before, by their own observations. Then give them again square pieces of paper, and ask them to join the opposite corners, and crease a fold diagonally (but do not use the word diagonally). Then ask them what shape they have got? They will reply, a three-sided figure. Ask them how many corners or angles it has, and then tell them that, on account of its being three-cornered, it is called a triangle. Now let them compare the angles, and they will find that there is one straight corner (right angle) and two sharp corners (acute angles). Ask them if the sides are equal, and they will find that two sides are equal and the other side longer. Set up the triangle on its base, so that the equal sides may be in the attitude of the outstretched

legs of a man; call their attention to this by a question, and then say, on account of this shape this triangle is called equal-legged, as well as right-angled a right-angled equallegged triangle. By giving them examples to compare it with, you can demonstrate to them that all right-angled triangles are not equal-legged, and all equal-legged triangles are not right-angled. Show them an equal-legged rightangled triangle, an equal-legged acute-angled triangle, and an equal-legged obtuse-angled triangle, and this discrimination will be obvious. The word isosceles can be introduced, if the teacher thinks best; but I keep off the Greek and Latin terms as long as possible.

Now tell the children to put together the other two corners of their triangles, laying the sharp corners on each other, and crossing the fold; unfolding their papers they will find four right-angled equal-legged triangles creased upon their square paper. Are all Are all these of the same shape, and of the same size? Now fold the unfolded square into oblongs, and make a crease, and they will find, on unfolding again, that they have six isosceles triangles, two of them being twice as large as any one of the other four. Ask, are all these triangles of equal size? Are all of them similar in shape? leading them to discriminate the use in geometry of the words equal and similar. Can triangles be large and small without altering the shape? Then similar and equal mean differently? Are all squares similar? are all squares equal? are all triangles equal? are all triangles similar? What is the difference between a square and oblong? What is the difference between a square and a triangle? What is the difference between a square and a rhombus? What kind of corners has a rhombus? In what is a square like a rhombus? How do you describe a triangle? What is the name of the triangles you have learnt about? They will answer right-angled, equal-legged triangles. Then give them each a hexagon, and ask them what kind of corners it has? Whether any one is more blunt than another? Whether

any side is greater than another? How many sides has it? And then draw out from them that a hexagon is a figure of six equal sides, with six obtuse angles, just equal to each other in their obtuseness. Having done this, direct the folding till they have divided the hexagon into six triangles, meeting at the centre. Ask them if these are right-angled triangles, and if they hesitate, give them a square to measure with. Then ask them if they are equal-legged (isosceles) triangles. They may say yes, in which case reply yes, and more than equal-legged, they are equal-sided. All three sides are equal, and so they have a different name, - they are called equilateral. Ask, what is the difference between equilateral and isosceles, if you have given them these names, and help them, if necessary, to the answer, "equilateral triangles have all the sides equal, isosceles triangles have only two sides equal." Are equilateral triangles all similar, that is, of the same shape? Are isosceles triangles all similar? and if they hesitate or say yes, show two isosceles triangles, one with the third side shorter, and one with it longer than the other two sides.

Now give to each child a square, and tell them to fold it so as to make two equal triangles; then to unfold it, and fold it into two equal oblongs. Unfold it again, and there will be seen, beside the triangles, two other figures, which are neither squares, oblongs, or triangles, but a four-sided figure of which no two sides are equal, and only two sides are parallel, with two right angles, one obtuse and one acute angle. Let all this be brought out of the children by questions. As there is no common name for this figure, name it trapezoid at once. Then let them fold the paper to make two parallelograms at right angles with the first two, and they will have two equal squares, and four equal isosceles triangles, which are equal to the two squares. Now fold the paper into two triangles, and you will have eight triangles meeting in the centre by their vertices, all of which are right-angled and equal-legged. Ask them if they are equal-sided? so as to