tion to one end of such line which they have to

the other.

The most improving method of entering the solutions is to show, in a first figure, all the circles in full by which you have arrived at the solution, and to draw a second figure in ink, without the circles.

It is not so much the problems which you are assisted in performing, as the problems you perform yourself, that will improve your talents and benefit your character. Refrain, then, from looking at the constructions invented by other persons-at least till you have discovered a construction of your own. The less assistance you seek the less you will require, and the less you will desire.

As the power to invent is ever varying in the same person, and as no two persons have that power equally, it is better not to be anxious about keeping pace with others. Indeed, all your efforts should be free from anxiety. Pleasurable efforts are the most effective. Be assured that no effort is lost, though at the time it may appear so. You may improve more

while studying one problem that is rather intricate to you, than while performing several that are easy. Dwell upon what the immortal Newton said of his own habit of study. "I keep," says he, "the subject constantly before me, and wait till the first dawnings open by little and little into a full and clear light."

2

INVENTIONAL GEOMETRY.

THE science of relative quantity, solid, superficial, and linear, is called Geometry, and the practical application of it, Mensuration. Thus we have mensuration of solids, mensuration of surfaces, and mensuration of lines; and to ascertain these quantities it is requisite that we should have dimensions.

The top, bottom, and sides of a solid body, as a cube,1 are called its faces or surfaces, and the edges of these surfaces are called lines.

The distance between the top and bottom of the cube is a dimension called the height, depth, or thickness of the cube; the distance between the left face and the right face is anoth

1 The most convenient form for illustration is that of the cubic inch, which is a solid, having equal rectangular surfaces. 2 A surface is sometimes called a superficies.

er dimension, called the breadth or width; and the distance between the front face and the back face is the third dimension, called the length of the cube.

Thus a cube is called a magnitude of three dimensions.

The three terms most commonly applied to the dimensions of a cube are length, breadth, and thickness.

1. Place a cube with one face flat on a table, and with another face toward you, and say which dimension you consider to be the thickness, which the breadth, and which the length.

2. Show to what objects the word height is more appropriate, and to what objects the word depth, and to what the word thickness.

As a surface has no thickness, it has two dimensions only, length and breadth. Thus a surface is called a magnitude of two dimensions.

3. Show how many faces a cube has.'

1 The surfaces of a cube are considered to be plane surfaces.