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.

It must be remembered that in a Kindergarten these lessons should not be more than a quarter of an hour long, perhaps not more than ten minutes; and that the making of paper windmills and boats, fly-boxes, and other toys out of folded paper, should occasionally intervene, prompting the children to inventions of their own. Sometimes surprising applications of their geometry to these little mechanical

efforts, will be made by very small people. A child of eight years of age, with whom I read over Mr. Hill's "Geometry for Beginners," for his amusement, with practical provings, in two months after invented a self-moving carriage for his sister's dolly, that would give it a ride of ten feet.

CHAPTER X.

ARITHMETIC.

NOT only a good deal of geometrical knowledge can be given to children by comparing the forms of blocks and folding paper, before they know how to read, but they can learn to count also. Blocks, melon-seeds, and sticks can be used. The first point is to prevent the error of their supposing that the several units of a number have different numerical names. Put down one block and say, that is one. Then take two blocks and say, there are two. Then take three and say, there are three. Tell the child to bring you two sticks; then to go and get three sticks. For a considerable time let the exercise be for a child to take out and bring to you certain numbers of blocks. You can then say 1 and 1 are 2; 1 and 1 and 1 are 3; 1 and 1 and 1 and 1 are 4; and when the number comes to be 8 or 9, you can help the child by telling him first to make the pile 2, and then to make it 3, and then to make it 4, and so on. Thus he will learn that 2 and 1 are 3, 3 and 1 are 4, &c. It will sometimes be wise to take something else than blocks to count; melon-seeds, or little sticks; and by and by they can be asked to think of one apple in one hand, and two apples in the other hand, and say what number of apples there would be if they were put into one pile. If there is no hurry at all, even the slowest child can be carried along in this gradual manner, without painful confusion of mind, and the life-long aversion that sometimes arises to arithmetical calculation be prevented. After children have learnt to count as far as 100, it is well to introduce multiplication, which they must see to be

addition of equal numbers; and I advise that the multiplication table should be learnt perpendicularly, but without the use of figures. Thus let them say twice 1, three times 1, 4 times 1, &c.; twice 2, three times 2, 4 times 2, &c., up to 10 times 2, before it is learnt in the usual way, (twice 1, twice 2, twice 3, &c.) Let them have objects to help along at first. Thus children will learn the substance of the multiplication, addition, and subtraction tables before they learn. to read. Sometimes it is necessary to postpone it, children's minds unfold so differently. We can exercise memory by repetition. But reason cannot be hurried. We should take care that the memory of results should not take the place of numerical apprehension, which is an act of reason. And written arithmetic should be postponed till the habit of mental calculation is fully formed. Warren Colburn discovered and established the method of nature in his primary book, and no variation from his principle is to be thought of. The teacher should consult Mr. Sheldon's book, which has a fine series of the earliest exercises.

Another thing that children can learn practically is the tables of measurement. Let the teacher have gill, pint, quart, and gallon measures; and let the children themselves fill up the gallon with the quart measure, the quart with the pint measure, the pint with the gill measure, till they have the table well by heart. Then let them have other vessels, of various capacity, and guess how much they will hold, and then measure and see. This is very entertaining, and educates a power. So they can have an inch measure divided into its three barleycorns; a nail divided into its inches; a quarter of a yard divided into nails; a yard divided into its feet; and learn to measure the furniture and judge of sizes. Again there can be the weights of troy weight, and of avoirdupois weight, and a pair of scales, and the children learn to weigh in their hands.

By the blocks which are divided into halves and quarters, and by these weights and measures, some idea of fractions

may be given, and by means of eagles, dollars, dimes, cents, and a piece of paper representing a mill, the foundation of decimal numeration can be laid in the mind.

But as I have so frequently said, let the teacher beware of premature abstraction with children, and be careful, especially in geometry, of inadvertencies of expression herself. I would suggest she should always say curve and not curved line. A line is length in one direction; a curve always changes the direction of the instrument making it. This verbal discrimination prevents a great deal of verbiage.