that its tendency is to lead the pupil to rely on his own resources, to systematize his discoveries in order that he may use them, and to gradually induce such a degree of self-reliance as enables him to prosecute his subsequent studies with satisfaction: especially if they sould happen to be such studies as Euclid's "Elements," the use of the globes, or perspective.

A word or two as to using the definitions and questions. Whether they relate to the mensuration of solids, or surfaces, or of lines; whether they belong to common square measure, or to duodecimals; or whether they appertain to the canon of trigonometry; it is not the author's intention that the definitions should be learned by rote; but he recommends that the pupil should give an appropriate illustration of each as a proof that he understands it.

Again, instead of dictating to the pupil how to construct a geometrical figure—say a square -and letting him rest satisfied with being able to construct one from that dictation, the author has so organized these questions that by doing justice to each in its turn, the pupil finds that,

when he comes to it, he can construct a square without aid.

The greater part of the questions accompanying the definitions require for their answers goometrical figures and diagrams, accurately constructed by means of a pair of compasses, a scale of equal parts, and a protractor, while others require a verbal answer merely. In order to place the pupil as much as possible in the state in which Nature places him, some questions have been asked that involve an impossibility.

Whenever a departure from the scientific order of the questions occurs, such departure has been preferred for the sake of allowing time for the pupil to solve some difficult problem; inasmuch as it tends far more to the formation of a self-reliant character, that the pupil should be allowed time to solve such difficult problem, than that he should be either hurried or assisted.

The inventive power grows best in the sun shine of encouragement. Its first shoots are tender. Upbraiding a pupil with his want of skill, acts like a frost upon them, and materially

checks their growth. It is partly on account of the dormant state in which the inventive power is found in most persons, and partly that very young beginners may not feel intimidated, that the introductory questions have been made so very simple.

TO THE PUPIL

WHEN it is found desirable to save time, omit copying the definitions; but when time can be spared, copy them into the trial-book, to impress the terms on the memory.

In constructing a figure that you know, use arcs if you prefer them; but, in all your attempts to solve a problem, prefer whole circles to arcs. Circles are suggestive, arcs are not.

Always have a reason for the method you adopt, although you may not be able to express it satisfactorily to another. Such, for example, as this: If from one end of a line, as a centre, I describe a circle of a certain size, and then from the other end of the line, as another centre, describe another circle of the same size, the points where those circles intersect each other, if they intersect at all, must have the same rela

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 cir cles 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

construction of your own.

have discovered a 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 as sured that no effort is lost, though at the time it may appear so You may improve more