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thoroughly to understand, as bearing upon many other questions besides those on interest, as will be seen from the examples given; also what is meant by so much in the shilling, so much in the pound, &c.,-that if a person spends twopence in the shilling in a particular way, and lays out two, three, ten shillings, he spends 4d., 6d., 20d., &c., in that particular thing.

A penny in the shilling is twenty pence in the pound, twenty pence in one pound is one hundred times that in a hundred pounds, and would be called so much per cent. The same in the common rule of three; they get into the way of stating their questions mechanically; but what the teacher should do is, instead of saying as 1 yard: 2s. 6d. : : 50 yards to the answer; he should say, if one yard cost 2s. 6d. two yards will cost twice as much; three yards three times; 50 yards 50 times as much, having recourse to the common-sense principle as much as possible.

The following questions, with those at the end of this section, may be useful to the teacher, as bearing upon the economic purposes of life, and will suggest others of a like kind :

The population of the parish in 1831 was 1040, at the census of 1841 it had increased 7 per cent., what is it at present?

In the population of the parish 20 per cent. of them ought to be at school; in this parish, containing 1040, only 12 per cent. are at school; how many are at school? and how many absent who ought to be there?

The population of the county in 1841 was 355,004;-82.8 numbers is equal to the sum of their squares, increased by twice their product.

(3.) That (ab)2 = a2 — 2ab + b2 = a2 + b2 — 2ab, or that the square of the difference of the numbers is equal, or the same thing as adding the squares of each separate number together, and then diminishing this by twice their product.

In each of these cases let a = 6, and b=4; then (a + b) (a - b) would become (6+4) x (6-4), or 10 x 2 = the square of 6 or 36, diminished by the square of 4 or 16, or (62 — 42) = 20.

(2.) (6+4)2 or 102=62 + 42 + 2 × 6 × 4.

or 36+16+48 = 100.

That is, it is the same thing if you add the two numbers together, and square the sum, or square each number separately, add them, and to this add twice their product.

(3.) (6-4) = 22 or 462 + 42—2 × 6 × 4.

or 36+16 48=4.

per cent. were born in the county, 1409 in other parts of England, 0.5 in Scotland, and 0.9 in Ireland; what number were born in each country?-how many in number, and what per cent. are unaccounted for ?

Give the average of the parish, how many to the square acre; number of the houses, how many to a house, &c. These questions ought also to be the vehicle of a good deal of instruction on the part of the teacher.

A sheet containing the names of the towns in each county, arranged by counties, and giving in a tabular form the popu lation in adjoining columns, according to the census of 1831 and of 1841, is to be had for a shilling, and offers great facility to a master for making questions of this kind; as well as affording useful statistical information.

In teaching them superficial and solid measure the following mode is adopted.

They are first shown, by means of the black board, what a square inch, foot, yard, &c., is, by proofs which meet the eye; that a square of two inches on a side contains four square inches; of three inches on a side nine square inches, and so on; or, in other words, that a square of one inch on a side could be so placed on a square of two inches as to occupy different ground four times, and in doing this it would exactly have occupied the whole square, one of three inches, nine times; thus showing clearly what is meant by a surface containing a certain number of square feet, &c.

The same illustration with an oblong, say nine inches by two, three, &c., two or three figures of figures so divided are painted on the walls.

Solid Measure.

The teacher takes a cube of four inches on a side, divided into four slices of one inch thick, and one of the surfaces divided into sixteen superficial inches; to this slice of one inch thick, containing sixteen solid inches, add a second, that will make 32, and so to the fourth, making 64; so that they now have ocular proof so simple that they must understand that the superficial inches in a square, or rectangle, is found by multiplying together the number in each side; the contents of a regular cube by multiplying the number of linear inches on one side by the number of slices.

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To apply this:

The master tells one of the boys to take the two-foot rule (a necessary thing in a village school), measure the length and breadth of the schoolroom. Yes, sir.

Length 26 feet, breadth 16 feet. What is the figure? An oblong-sides at right angles to each other. Multiply length and breadth-what is the area?

To another-Look at the boards of the floor; are they uniform in width? How are they laid? Parallel to each other. The breadth of the room you have got, and, as the boards are laid that way, you have the length of each board; measure the width of a board. Nine inches. Reckon the number of boards. What is the area of the room? Does it agree with your first measurement? If not, what is the source of error; the boards will turn out to be unequal in width.

The door what is the shape of the opening? An oblong, with one side a good deal longer than the other. Measure the height-the width: now what number of inches of surface on the door?

The rule again. Measure the thickness. Now how many solid inches?

The door-posts. Measure the height, width; now the depth. How many solid inches of wood in one post? How many in the whole door-posts? How many solid inches in a foot? Turn it into feet.

In the same way they may apply the rule-to find out the surface of a table, a sheet of paper, surface of a map, a page of a book, &c., but always making them do the actual measurement, first taking one child, then another.

Again the room-we have got the area-tell us how much water it would hold, if we could fill it as high as the walls; we have got two dimensions, what is wanting?-The height. We cannot reach up, sir.-Take your rule. Measure the thickness of a brick with the mortar.-About four inches.-Measure the first three courses.-A foot, sir.-Reckon the courses of the wall.-Thirty-six.-Then the height is what?-Twelve feet. Now find out the solid contents of the room.

Find the surface and solid contents of a brick.

In fact, the two-foot rule is to the village school what Liebig says the balance is to the chemist.

Another practical application, which works well in giving fixed ideas of linear measure is the following:

Take a hoop, say of two feet diameter; apply a string to the circumference; measure it.-Rather over six feet.-Another of three will be found to be nine, and, by a sort of inductive process, you prove that the circumference is three times the diameter: when farther advanced, give them the exact ratio, 3.14159, which they will work from with great facility.

Boys! make a mark on the hoop: let it rest on the floor, the mark being directly opposite the point which touches the floor;* trundle it, stopping every time when the mark rests upon the floor, and let another boy make a chalk mark where it touches; now take your two-foot rule and measure between each mark. What is it?-Six feet, twelve feet, eighteen feet, &c.-And the hoop has been round how many times at each mark?-Once at the first, twice at the second, three times at the third, &c.-Now, you see, if you trundle your hoop over a piece of level ground, and reckon the number of times it has gone round, you can tell the length of space it has gone over. How many miles to Winchester?-Nine, sir.-Measure the height of your father's cart-wheel, and tell him how often it will go round in going to market. Tell him he must not zigzag. The teacher should point out the sources of error. The philosophy of common life and everyday things is most attractive to them, and a book of this kind, if well done, would be a most useful one for our village schools.

This two-foot rule, and other appliances, setting to work both hands and head, amuses at the same time that it instructs, and gives a sort of certainty to their knowledge, fixing it in a way that learning things by mere rote never can.

In order that they may get correct ideas of what is meant by lines parallel and inclined to each other, and of a square, a circle, a triangle, &c., I have had painted on the upper part of the walls, above the maps, four series of simple figures marked, Series A, No. 1, 2, 3, angles and triangles. Series B, No. 1, squares and parallelograms. Series C, circles, &c.

*The teacher who has sufficient mathematical knowledge may exercise himself in trying to make out the nature of the curve traced out, by any given point in the surface of the hoop between two successive contacts with the floor. A curve of very curious properties, which interested mathematicians very much about 200 years ago, and was made out by the famous Pascal when labouring under a fit of toothache: it is the curve in which the pendulum keeping true time vibrates.

a square and a rectangular parallelogram, divided into linear inches. These figures are easily referred to, extremely useful, occupying no space which is wanted for other things, and cost nothing.

Of the simple solids the school is also provided with models, and these, with the figures on the wall, may be called into use in almost numberless ways.

What is the shape of the room-of the door-of a brickof a book-table, &c.? -a square or parallelogram on Series B, No. 1, No. 2. Look at the beam running between the walls, what are the figures of the two surfaces? What of a section perpendicular to either surface?-what slantwise?

The stove in the room, what is its figure?-A hollow cylinder. -The pipe carrying away the smoke?-The same. What would the figure of a section of the stove parallel to the floor be?-of the pipe?-A circle, No. 2, series C.-What of a section perpendicular to the floor, &c. The different sections of a cube-or any solids which may be about the room-but always referring to the exact figure on the wall. These figures will often supply the place of the black board.

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Again, tell a boy to turn the door on its hinges as far as he can-to find out what solid it would trace out if he could turn it entirely round.—A cylinder like the stove, but much larger. -What is the section of the solid part of the stove?—A ring inclosed between two concentric circles.-Concentric, what?If the door were a right-angled triangle, what figure would it generate by going quite round on the hinges?-A cone, like a sugarloaf. What if a semicircle, the line between the hinges the diameter?-A globe: and so on. Then again, the outer edge of the door and a line parallel to it, at 2, 3, &c., inches apart, would trace out a solid ring. What would the door trace out if, instead of revolving round its hinges, it revolved round one of its ends; and to illustrate this still further, fasten two pieces of string of unequal lengths to the top of a stick, which place perpendicular to the floor, then let two boys taking hold one at each end, walk round the stick, they will clearly see, that the finger of the short-stringed boy describes the inner surface, and of the long-stringed the outer surface-that every point in a circle is equally distant from the centreexplain what is meant by circles being in different planeswhat by concentric circles-and then the teacher will ask

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