when he begins an untried exercise of mind; he would study the several parts of a dissected map before he begins to put them together.

Our preliminary method would depend more upon palpable objects than even in the preceding articles: for, whereas, in arithmetic, the tangible instruments were only helps to the acquirement of a difficult abstract notion,-in geometry, according to our preliminary system, they are the objects whose properties are to be studied.

The first thing to be done is, not to give the notions attached to the words point, line, straight line, surface, and plane surface, for they exist already, but to take care that the ideas are attached to the right words. About the term point there is no difficulty; we need hardly warn the instructor not to use the definition of Euclid, but to proceed as follows. Instead of digging a pencil, or the end of a pair of compasses into the paper, and calling the visible surface so produced a point, let all the first points shown be made by drawing two intersecting lines slightly upon the paper with a hard and well-pointed pencil, using the hand only, and not the ruler or compasses. Having made several of these, the learner should be required to find the points in which they cut one another, by showing them with a fine needle. When he can do this, he should be allowed to try to draw a line through two or more points, either straight or curved, or composed of both species. A flat ruler should then be given to him, with which he should be shown how to draw a straight line. And here we must observe, that his notion of a straight line will probably be, one which is parallel to the upper and under edges of the paper. Thus he has been told that he

VOL. II.

K

cannot write straight without lines, and so on. This misconception must be corrected by drawing lines over the paper in all directions with the same ruler, and applying the term straight to them all. The learner must be made to understand, that the line which comes off the ruler is of the same form in whatever position the ruler may be held, and that a line which is straight in any one position is so in every other; that what he has been accustomed to call a straight line, means a straight line in the same direction as the upper edge of the paper. As to defining a straight line or a plane surface, we think it had better be let alone, unless perhaps the definition ascribed to Plato, or one of his school, be called in as an illustration only, which is, that a straight line is that which can be so held before the eye, that nothing but a point shall be visible; and a plane surface, that which can be made to appear as a straight line in the same manner. But the plane surface may be illustrated by the definition of Euclid. Taking a plane, and also a piece of a cone, cylinder, sphere, or any other which may be at hand, the child may be shown that the edge of the straight ruler may be made to rest entirely upon the first in every direction, which is not true of any of the others. We may remark, that the usual idea of the word plane, is that which ought to be attached to the term level, or horizontal plane, a misconception similar to that just mentioned with regard to the straight line.

The most serious difficulty in the definitions is that of the word angle, because it contains a notion hitherto almost unconsidered. The best substitute is the word opening. Several intersecting straight lines may be drawn, making acute, obtuse, and right angles, care

being taken that some of the longest lines shall contain very small angles, and some of the shortest very obtuse angles. Two of these sets being pointed out, the learner is required to say which lines open widest. Most probably he will fix upon those which appear to contain most space, or which have the longest lines. If this happen, cut out the angles from the paper, making the incision opposite to the angle curvilinear, as in the following diagram :—

A

C B

D

Placing the larger angle undermost, lay the other upon it, so that the angular points B and A may be one over the other, and the line A C, as far as it goes, may lie under B D. This will be best done by using pasteboard, so that the under edges may be placed side by side on the table, the angles being held upright. There will then be no further difficulty as to which two lines open widest, or contain the greatest angle; the notion of equal angles may be established in the same way. For schools, such angles might be cut in wood, by a common carpenter, and also some triangles, with moveable angles, as in the following diagram:

Before proceeding further, the following propositions should be verified :-The greater angle of a triangle is opposite to the greater side; a triangle, which has two

sides equal has the angles opposite to those sides equal; the exterior angle of a triangle is equal to the sum of the interior and opposite angles. Any one who is acquainted with the subject will see the ease with which these propositions may be submitted to ocular demonstration, by cutting off the angles from paper triangles.

The definition, given by Euclid, of a right angle, is the one which we should prefer. Translated into language fit for a child, it is this: the line A makes a right angle with the line B, when it does not lean to either side, or make the opening on one side greater than the other. This may be verified, as in the preceding page, when two lines, perpendicular to one another, have been drawn. There is what we consider an omission in the work of Euclid, though we are aware that it will not be looked upon in this light by many-it is that of angles. which are equal to, and greater than, two right angles. It is desirable that the learner should be made to see the distinction between the part of a line and its continuation. For example, the line A B, and its part A C,

D

B

E

contain no angle, or coincide, while A B, and its continuation A D, make a greater opening than any angle considered by Euclid, and one which is evidently two right angles. This may be illustrated by taking a common pair of compasses, and opening them from the position in which the two legs coincide to that in which they form the continuation of one another. For fear, however, of having more to answer for than we intended, we must warn all those who teach, that a sharp pair of

compasses is a dangerous tool, particularly when opened as we have described, and in the hands of children. This instrument should always be handled with the most delicate touch, both on this account, and because no accuracy can ever be obtained by using it roughly. To return to our subject: any other line, A E, should be pointed out as making two openings with A C; one, most commonly known by the name of the angle CA E, and less than two right angles; the other, EA C, greater than two right angles. In illustration, it may be noticed, that the line A E, in our diagram, may revolve into the position A C, either from left to right, or from right to left. In the first case it describes the smaller angle just mentioned; in the second case, the larger. The beauty and generality of Euclid, Book III., prop. 20, are materially diminished by the absence of this convention.

We hold it essential, that the accurate use of the pencil and ruler should be one of the first things taught, in drawing straight lines only. As an exercise of this process, we propose the following verification of a simple proposition:-After the learner has been made to practise drawing straight lines from one given point (see page 97) to another, the lines being as thin as is consistent with their being distinctly seen, let him choose two points, A and B, and draw any number of lines in any directions through A, and two lines only through B, cutting all those which were drawn through A. The pair of lines through B will, therefore, form a four-sided figure with every pair that can be chosen out of those drawn through A, and the opposite corners of every such four-sided figure should be joined, giving the diagonals of them all. The diagonals of each figure in