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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 chalkmark where it touches; now take your two-foot rule and measure between each mark. What is it?-Six feet, twelve feet, eighteen feet, etc. — And the hoop has been round how many times at each mark? One at the first, twice at the second, three times at the third, etc. — 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 every-day things is most attractive to children, 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, and fixes 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, etc., 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, etc., a square and a rectangular parallelogram, divided into linear inches. These figures are easily re
* 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, is the curve in which the pendulum keeping true time vibrates.
ferred 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 brick --of a book -- table, etc. ? - 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 à section perpendicular to either surface ? - what slant-wise ?
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? etc. 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.
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 sugar-loaf.— 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, etc. inches apart, would trace out à solid ring. What figure would the door trace out, if, instead of revolving round its hinges, it were made to revolve 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 centre-explain what is meant by circles being in different planes — what by concentric circles—and then the teacher will ask them, if the strings weré 2, 3, 4 feet, etc. long, what the circumference would be; at first some of them would say six feet, nine feet, etc., 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. Here they may be shown that the area of a circle is the radius
diameter circumference + or the circumference to
' 4 and since 3•14159 is the circumference of a circle whosé diameter is unity, 3:14159 + i = •78539 is the area, and that the areas of circles are to each other as the squares of their diameters; this expression they can work with practically afterwards, in measuring timber, etc.
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, etc.; and taking the diameter of à 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, etc., 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, d' the inner, then
(-78539) d = area of outer circle,
(•78539) d'? = area of inner circle; and (•78539) (da-d2) = area of section of the ring; and if h denote the height, the solid contents will be
(•78539) (42—02) h; then to give particular values to d, d' and h, and work out the results.
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, etc. years.
What will his sayings amount to when he reaches the age of 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 1s. 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 ?
Such questions ought to have their bearings and application to every-day life explained to the children.
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 cbarge 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?
Supposing the whole expenditure of a parish in rates to be £920 10s. in the year, and the whole property rated ut £5276 9s, 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 3s. in the pound is paid for house rent, 9s. ed. 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:
s.' d. Expenses of the army and navy ........... ...... 7 2 King's judges, etc., and other departments of state.. 0 10
Interest of the national debt... ............ 12 0 What is the exact sum paid to each ?
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, etc.?
A pole or perch of land is 16 feet square, the usual measure, but here they have a measure for underwood called wood measure, 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 i 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?
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 houso of ten windows ?
How many panes, each nine inches by twelve inches, and what would the cost be at per foot.
The following extract from “An Educational Tour in Germany,” etc., 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