to the floor. The floor of this room is level or horizontal; the wall of the room is vertical, and stands per

pendicular to the level floor. The surface of still water is level or horizontal; a plummet line hangs vertically; the plummet line is perpendicular to the surface of the water. It will be seen that the plummet line neither inclines to the one side nor to the other, that is to say, the openings or angles, which it forms with the horizontal line, on each side, are equal to each other.

Is the line C D perpendicular to A B? To which side does it incline? On which side does it form the greater angle or opening? To which side does CE incline? On which side does C E make the greater angle? Whether does C D or C E approach nearer to the perpendicular position? To which side does the line C F incline? To neither the one side nor the other. Then the angles on each side are equal to each other, and they are called right angles.

A lesson on Practical Geometry. Subject To erect a perpendicular. Mixed method. Illustrative, constructive, &c.

I want to show you how to draw one line perpen

dicular to another. From the given point, or mark, D, in the straight line A B, I want to erect a perpendicular, that is, a line which will neither incline to the one side nor to the other.

On each side of D, I take D F, of the same length as D E. (The teacher is supposed to construct the figure as

he describes it.) I open the legs of my compasses so that the opening between the points shall be greater than D E or D F ; I place one point of my compasses on the mark or point E, and sweep a portion of a circle; I now place the point of the compasses on the mark or point F, and sweep a portion of another circle, cutting the former in a point which we shall call c; I join D and C, and the line D c will be perpendicular to the line a B.

I shall now explain to you why this mode of construction causes CD to be perpendicular to ▲ B.

There are two things in the construction which cause the line D C to be perpendicular to A B: First, D E is of the same length as D F; second, the two circles were swept with the same radius or opening of the compasses. These two things cause the point c to lie directly over, or perpendicularly over the point D. If the second radius or opening of the compasses be taken less than

the first opening (here the teacher must describe the

figure), how will the line C D be inclined? It will be inclined towards the side F. Why? For the point where the two circles cut each other must lie nearer to

F than to E. But when the openings of the compasses are the same, the point where the circles cut each other lies neither more towards F than towards E, and therefore the line D C is equally inclined to a B, that is to say, D C is perpendicular to A B.

Observations. Although this may not be 'what is called a logical demonstration, yet it most certainly gives the pupil A SUFFICIENT REASON for concluding that the line CD is perpendicular to A B. It is further worthy of observation, that such familiar expositions prepare the mind of the pupil for following more strictly logical demonstrations. I am well aware, that some persons are disposed to say that the shortest course is to carry the pupil through Euclid's Elements; but, after the experience of a quarter of a century as a mathematical lecturer, I have no hesitation in saying, that it is quite impracticable to teach young persons the elements of Euclid until they have gone over some initiatory course of demonstrative geometry, by which the mind of the pupil is led to pass from the concrete to the abstract. It is true that this initiatory process of demonstration is always lengthy; but it acts like a mechanical power, for what we lose in time we gain in force.

ALGEBRA.

This subject should be taught by a demonstrative method,-proceeding from the concrete to the abstract. The leading simple elementary operations of quantities should be first taught in connection with the solution of problems.

A lesson on Algebra. Subject - Equations, &c.

Problem. A man bought a cow and a horse for 281. ; now the horse cost twice as much as the cow and 4l. more; what did he pay for the cow?

Here the problem tells us that the value of the cow

and the horse equals twenty-eight pounds. I may then write this down in the form of an equation, thus

one cow + one horse = 281.

Now I must put the horse into cows.

question tell us about the value of the horse? That a horse is worth two cows and 4l. more.

write down.

What does the

Then I may

We shall now write down, or substitute, this value of the horse in the first equation; thus

one cow +2 cows +47. 281.

=

What have I here substituted for one horse?
Putting the cows together, we have

If I take away the 47. from the left side of this equation, what must I take away from the other side to keep up the equality? Let us do this, — then

3 cows= 241.,

one cow= of 241. = 81.

that is to say, the value of the cow is 81.

Or thus more symbolically. Proceeding as before, we have one cow + one horse = 281.

Let us, for the sake of convenience, put a for the value of the cow in pounds, that is, let

Be

Why do I put 2 x for the value of "2 cows?" cause x pounds is the value of one cow, and therefore 2 x pounds will be the value of two cows.

Now let us put these values for the cow and the horse in our first equation. First writing x for the cow, and 2x+41. for the horse, we get

x+2x+41.= 281.,

3x+4l. = 281.

What have I done here? Exactly, an x added to 2 x will make 3 x, in the same manner as one cow added to two cows will make three cows.

In order to leave nothing but x's on the left side of this equation, what must I do?

.. x = 1 of 24l. = 81.,

that is to say, the value of the cow is 81.

MECHANICAL AND PHYSICAL SCIENCE.

All our instructions in these sciences should be based on observation and experiment. The methods of interrogation and ellipses are best adapted for giving familiar lectures on these subjects.

A lesson on Chemistry. Subject. To distinguish iron from copper. Mixed method - Experimental, Interrogative, Elliptical, &c.

PROPERTIES DERIVED FROM OBSERVATION.

What are

the names of these metals? The one is called iron, the other copper. The colour of the copper is reddish-yellow, that of the iron dark grey. They have some properties in common. They both have a peculiar glitter or lustre, called the metallic lustre, or the lustre common to all metals. Polished wood has a lustre, but it is not the metallic lustre. I can readily scratch the copper with my knife, but I cannot so easily scratch the iron; what inference do you draw from this? Copper is softer than iron. They may be both hammered out, they are both malleable. It takes a very intense heat to melt them,— they are not easily melted or fused. They are both drawn out into wires, they are both ductile.

USES. Name the uses of iron? Name the uses of copper? On what properties do these uses depend?

ORES OF IRON AND COPPER. This is a specimen of iron ore, that of copper ore; the one is called iron pyrites, the other copper pyrites; the one is called sulphuret of iron, being composed of sulphur and iron, the other is called sulphuret of copper, being composed of sulphur and copper. Compare their colours! The