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them, if the strings were 2, 3, 4 feet, &c. long, what the circumferences would be; at first some of them would say six feet, nine feet, &c., not seeing that their piece of string was the radius and not the diameter; difference to be pointed out, and that the circumferences of circles are in proportion to their diameters.

4

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Here they may be shown that the area of a circle is the cirradius diameter cumference x or the circumference x and 2 since 3.14159 is the circumference of a circle whose diameter is unity, 3.14159 × =78539 is the area, and that the area of circles are to each other as the squares of their diameters; this expression they can work with practically afterwards, in measuring timber, &c.*

The contents of a cylinder :

The teacher should not be content with merely showing them how to find the contents of a cylinder, or any other regular figure, but should point out to them, in this case, for instance, anything in the room of a cylindrical form, such as the stove if round, the pipe which carries off the smoke, &c.; and taking the diameter of a section, and from this finding the area of it, and multiplying into the height or length would give the solid contents; that for an iron roller, or any other roller hollow in the middle, they must take the diameter of the outer and inner surface, get the area of these sections, and subtracting them from each other, would give the area of a section or ring which, multiplied into the length of the roller, would give the quantity of solid matter in it; thus calling their attention, and actually measuring vessels, &c., the shape of which they are familiar with.

This, of course, applies to other regular solids than the cylinder.

In the case of the cylinder, let d the outer diameter, ď the inner, then

(*78539) d2 = area of outer circle,
(78539) d'2 = area of inner circle;

and (78539) (d2 — d12)

area of section of the ring; and if h denote the height, solid contents will be (78539) (d2 — d12) h; then to give particular values to d, d' and h, and work out the results.

* See Appendix (B).

Examples for Practice.

A boy at the age of 15 begins to save 7 d. per week, what will he have saved at the end of one, two, three, &c., years.

of

What will his savings amount to when he reaches the age twenty-one? And what would it be if put into the savings' bank at the end of each year, interest three per cent.

Supposing at the age of 21 he begins to save 18. per week, and at the end of each year puts it into the bank, what would he have when he is 31 years of age ?*

A goes to the village shop and lays out 10s. per week, on an average, for necessaries for his family, every week in the year; but, for want of thought and of understanding his own interests, has got into the habit of running a bill, and having his things booked, as it is called; for this the shopkeeper is obliged to charge 10 per cent. more than for ready money. How much does A lose by this in the year?-or how much more does he pay than the ready-money customer?†

Such questions not to be drily set, but to have their bearings explained.

The following extract from An Educational Tour in Germany,' &c. affords a very useful and practical hint to the schoolmaster:

"In Holland I saw what I have never seen elsewhere, but that which ought to be in every school—the actual weights and measures of the country. These were used not only as a means of conveying useful knowledge, but of mental exercise and cultivation.

“There were seven different liquid measures, graduated according to the standard measures of the kingdom. The teacher took one in his hand, held it up before the class, and displayed it in all its dimensions. Sometimes he would allow it to be passed along by the members of the class, that each one might have an opportunity to handle it, aud to form an idea of its capacity. Then he would take another, and either tell the class how many measures of one kind would be equivalent to one measure of the other, or, if he thought them prepared for the question, he would obtain their judgment upon the relative capacity of the respective measures. this way he would go through with the whole series, referring from one to another, until all had been examined, and their relative capacities understood. Then followed arithmetical questions, founded upon the facts they had learned, such as, if one measure full of anything costs so much, what would another measure full (designating the measure) cost, or seven other measures full? The same thing was then done with the weights. "In the public schools of Holland, too, large sheets or cards were hung upon the walls of the room, containing fac-similes of the inscription and

In

Supposing the whole expenditure of a parish in rates to be £920 10s. in the year, and the whole property rated at £5276 98. 4d., what is that in the pound?

Supposing the number of acres in the parish to be 7000, what would that be per acre?

A spends £250 10s. 6d. per annum, of this 38. in the pound is paid in house rent, 98. 8d. in food, 3s. 4d. in clothing, the rest in sundries; how much in the pound is paid in sundries; and what is his absolute expenditure in each of the above things?

Supposing him to save £80 per annum out of the above income, and his proportionate expenditure in each article as above, what would be the sum spent for each?

The whole amount of taxation in this country is upwards of 50 millions, supposing it is this sum, and that every twenty shillings paid in taxes is disposed of as follows:

Expenses of the army and navy

King's judges, &c., and other departments of state
Interest of the national debt

What is the exact sum paid to each?

s. d.

7 2

0 10

12 0

What would be the expense of digging three acres, two roods, and 20 perches of ground at 4d. per pole? What of double trenching it for the purpose of planting, at 10d. per pole?

How many trees to plant an acre at such and such distances, &c.?

A pole or perch of land is 16 feet square the usual measure, but here they have a measure for underwood called woodmeasure, a pole of which is 18 feet square. How much is the wood-acre larger than the ordinary acre?

A labourer agrees to move a piece of earth 25 feet long, 15 feet wide, and 10 feet high, a certain distance at 1s. 6d. per cubic yard, what would his work come to?

A pair of horses plough of an acre in one day, the width of each furrow is one foot. How many miles will the boy walk who drives the plough?

relief-face and reverse-of all the current coins of the kingdom. The representatives of gold coins were yellow, of the silver white, and of the copper, copper colour."-Mann's Educational Tour, with Preface by W. B. Hodgson, LL.D.

Supposing the furrows were only nine inches or six inches. broad, how far would he have to walk? Work this out, and reduce the difference into yards.

A window is five feet nine inches high, four feet six inches broad. How many square feet of glass for a house of ten windows?

How many panes, each nine inches by twelve inches, and what would the cost be at per foot.

GEOMETRY.

A knowledge of some of the more simple parts of geometry is quite necessary for any schoolmaster who wishes to be thought competent to his work, or to stand in what may be looked upon as the first class of teachers in our elementary schools. For this purpose it is highly desirable that they should at least know so much of the subject as would enable them to teach the first three books of Euclid, with a few propositions out of the other books. Many of the propositions in the first three books are of easy application to the mechanic arts, particularly to the carpenter's shop, to the principles of land-measuring, &c., and an edition of these, pointing out such propositions and their application, with a few practical deductions from each, would be of great use in our elementary schools.

There are many of the appliances of the carpenter with his tools, and of other mechanic trades, so strictly geometrical and so easy of proof, as to be easily learned, and the workman who knows them, instead of being a machine, becomes an intelligent being, and has sources of enjoyment opened out to him which many of them would turn to a good purpose.

Even a knowledge of the axioms of Euclid, such as "things which are equal to the same are equal to one another." If equals be added to equals the wholes are equal.

If equals be added to unequals, the wholes are unequal, &c.: suggest modes of reasoning, which are extremely useful; and a thorough knowledge of the kind of reasoning in the propositions of the three books gives a man a habit and a power of drawing proper conclusions from given data, which he would scarcely be able to acquire with so little trouble in any other way.

Children may easily be made to understand what is meant by the terms perpendicular, horizontal, right angle, and lines parallel to each other, by referring to the things in the

room.

Thus the walls are perpendicular, or at right angles to the floor-the boards are horizontal and parallel to each other— the courses of bricks are parallel-the door-posts perpendicular to the floor, &c.; the beams, rafters, &c., of the roof, all might be referred to as illustrating things of this kind.

The way in which the circle is divided ought to be understood; the number of degrees in a quadrant, &c.; that the three angles of a triangle are equal to two right angles; and therefore if a triangle is right-angled, or has one right angle, the remaining two must be equal to a right angle.

The proposition that if two sides of a triangle are equal, the angles opposite are equal, and the converse.

To bisect a given rectilineal angle.*

To draw a perpendicular from a given point in a line, or let one fall on a line from a point without it.

The one that either of two exterior angles is greater than the interior and opposite angle-showing from this how the angle under which an object is seen diminishes as you recede from, and increases as you advance towards it.

The proposition about the areas of triangles and parallelograms, as applying to the superficial measurement of rectilineal figures.

The 47th in the first book, that the square of the side opposite the right angle is equal to the sum of the squares on

* The following is a very interesting and useful application of this proposition in showing how a meridian line may be laid down by it:

Tell the boys to stick in the ground, and in the direction of the plumbline, a straight rod, to observe and mark out the direction and length of its shadow on a sunny morning before twelve o'clock, say at eleven: to observe in the afternoon when the shadow has exactly the same length; join the two extremities of the shadows, and on the line which joins them, which is the base of an isosceles triangle, describe an equilateral triangle; a line drawn from the point where the staff goes into the ground to the vertex of this triangle will be the true meridian, or by simply drawing a line from the stick to the middle of the line joining the extremities of the shadows.

Place the compass on the line and let them observe how much the two meridians differ: that the length of the shadow, at equal intervals from noon, will be the same both in the morning and in the afternoon, &c.

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