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 keep them very clear of confounding the isosceles with the equilateral, but use the English terms as often as the Latin and Greek, for the vernacular keeps the mind awake, while the foreign technical puts it into a passiveness more or less sleepy. Then give all the children octagons, and bring out from them its description by sides and angles; and then fold it so as to make eight isosceles triangles.
Another thing that can be taught by paper-folding is to divide polygons, regular or irregular, into triangles, and thus let them learn that every polygon contains as many triangles as it has sides, less two.
Proportions can also be taught by letting them cut off triangles, similar in shape to the wholes, by creases parallel to the base. Grund's "Plane Geometry" will help a teacher to lessons on proportion, and can be almost wholly taught by this paper-folding. Also Professor Davies's "Descriptive Geometry," and Hay's "Symmetrical Drawing."
Of course it will take a teacher who is familiar with geometry to do all that may be done by this amusement, to habituate the mind to consider and compare forms, and their relations to each other. Exercises on folding circles can be added. It would take a volume to exhaust the subject. Enough has been said to give an idea to a capable teacher. Care must be taken that the consideration should be always of concrete not of abstract forms. Mr. Hill says his "First Lessons in Geometry" were the amusements of his son of five years old. Pascal and Professor Pierce found out such amusements for themselves, which had the high end of preparing them for their great attainments in logical geometry.
Sometimes surprising applications of Geometry, thus practically appreciated, will be made by very small people. A boy of eight years of age, with whom I read over Mr. Hill's "Geometry for Beginners" for his amusement, in two months after invented a self-moving carriage for his sister's dolly, that would give it a ride of ten feet! A neighboring carpenter made it from his drafted model.
CHAPTER X.
READING.
This art should be taught simultaneously with writing, or, more properly, printing; and I should certainly advise that it do not come till children are hard upon seven years old, if they have entered the Kindergarten at three. For it properly belongs to the second stage of education, after the Kindergarten exercises on the blocks, sticks, peas, &c., are entirely exhausted; and the children have become very expert in sewing, weaving, pricking, and drawing. They will then have received a certain cultivation of intellect which will make it possible to teach Reading on a philosophical method, which will make the acquisition a real cultivation of mind, instead of the distraction it now is to those whose vernacular is English, the pot pourri of languages, and whose orthography should be called Kakography, it is so lawless.