tersect in one point, giving as many new points as there are four-sided figures, all which points, if the figure be correctly drawn, will lie in the same straight line, which straight line will pass through B. A construction of this sort, in practised hands, is a very good trial of the straightness of a ruler, and such as many of those sold in the shops will not bear.

The preceding example, which contains nothing but the construction of straight lines, will serve as an exercise in that particular operation. The following, taken from Mascheroni's Géométrie du Compas, in which the ruler does not appear at all, but only the compasses, will furnish a similar test of accuracy in the use of that instrument. As we are writing for the instructor, and not for the pupil, we do not think it necessary to add the diagram. Take two points, A and B; it is required to find the middle point of the line A B without drawing that line, or using a ruler in any way. With the centre A, and the opening of the compasses A B, describe a circle. On this circle cut off BC, CD, DE, with the same opening of the compasses B A. With the centre B, and the opening BA, describe a circle, and also another with the centre E and opening EB. Let the two last mentioned circles cut one another in P and P. On the circle whose centre is B, beginning from the point p, take the opening BA three times, cutting off arcs px, xy, yz. With the centre P, and opening P B, describe a circle, from which cut off BM towards A with the opening P z. The point M is the bisection of the line A B. Of course, the whole of every circle need not be drawn; the eye will point out how much is necessary. When this construction has been made, the points A B may be carefully joined with

the ruler, and the point M ought to be upon the line A B, and in the middle of it.

To ascertain with what degree of accuracy the compasses are used for the measurement of lines, the simpliest method is the verification of Euclid III. 35, which involves the equality of the products of the segments of lines which cut one another in a circle.

The best definition of parallel lines, for our present purpose, is the appearance they present when drawn. It must be observed, that our object now being to impress isolated facts upon the eyes of the learner, no definition is so good as a figure; and it is quite sufficient that no mistake should be made in applying the common phraseology. Thus, instead of defining parallel lines, as those which would never meet, though ever so far produced, a definition which it is impossible to verify, let parallel lines be drawn, and let the student be required to verify, by measurement, the property that any third line makes equal angles with the two parallels. Thus, having drawn a line at right angles to one, he will find that it is also at right angles to the other; and having drawn several such, he will find that the parts of them intercepted between the parallels are equal. In our next article we shall show that it is almost immaterial what definitions are adopted in this part of the course.

Our object being to convey the knowledge of the facts of geometry, and to form a perfect acquaintance with, and readiness in the use of, its language, we now recommend that the propositions of the first four and the sixth books of Euclid should be enunciated with ocular demonstrations. And here we must observe that some of the more elementary properties have disadvantages in this respect, on account of their simplicity. For

example, with two obviously equal triangles before our eyes, we tell the learner, that if two sides of the one are respectively equal to two sides of the other, and if the included angles are also equal, the triangles will be equal in every respect. In the meanwhile, he will have outstripped us; for seeing that the two triangles are equal, and hearing that, after all our words, we tell him nothing more than he knows, having also no distinct conception of the connexion between the hypothesis and its consequences, he will imagine we might just as well have said at once "Here are two equal triangles." To create the notion which we want to give proceed in the following manner:-Draw a straight line, and on it require the student to construct various triangles, which he will readily do. Continue this until he perfectly comprehends the proposition, "The triangles which have one given side in common are infinite in number." After this, with any radius, and one end of the given line as a centre, describe a circle. By taking any point of this circle as a vertex, the following may be established:-"There is an infinite number of triangles which have one side in common, and besides this a second side of the one equal to a second side of the other." Again, by drawing any other straight line from one extremity of the given line, and taking any point in this as a vertex, it follows that "an infinite number of triangles may be drawn which have one side and one angle in common." Now give the learner an angle formed in wood or pasteboard, so that the directions of two lines containing that angle may be found by drawing the pencil round its edge, and require him to draw two triangles of different sizes, having that angle, and also having a side of one equal to a side of the

other. This will be readily done; after which vary the question, and require him to draw two triangles which shall have that angle, and the two containing sides in one respectively equal to those of the other, and which shall have different third sides. This he will find, in a few trials, that he cannot do; and by similar steps, with regard to the remaining angles and the areas, he will come to a perception of the proposition of Euclid in the following somewhat more striking form:-" If two triangles agree in one angle and the two sides which contain it, it is impossible that they should differ in any respect." The same process may be followed in many of the more simple propositions, and it is for the instructor to consider, in every case, whether the proposition is more remarkable in what it affirms, or in what it denies, and to shape the enunciation accordingly. The following propositions, it appears to us, might be simplified by the preceding method:

1. Axiom 11, which is, in fact, a proposition admitting of demonstration.

2. Book I. Propositions 4, 7, 8, 11, 15, 26, 33. 3. Book III. Propositions 14, 24, 26, 27, 28, 29. Propositions 5, 6, 7, 26.

4. Book VI.

The greater part of the remaining propositions are such as admit of simple ocular demonstration by common measurement. Among those which are of minor importance, or of none at all, we may name,

Book I. Propositions 2, 3, 7, 16 (included in 32), 17 (included in the corollary of 32), 39, 40, 44, 45, 48. Book II. 7, 8, 9, 10, 11, 14.

Book IV. 10, 11, 12, 13, 14, 15, 16. structions are better made by other methods. Book VI. 25, 27, 28, 29, 30.

The con

Three most important preliminaries, which it is necessary to treat in a different manner from Euclid, are the definition of proportion, the measure of an angle, and the area of a square or rectangle. With regard to the first, there are difficulties in the way, unless the learner has a tolerably correct notion of arithmetical fractions. Those who have not must omit all propositions of the sixth book; to the rest, the following definition may be given as a first step:

Take a simple proposition out of the sixth book; the following, for example-that equiangular triangles have their sides proportional. Two equiangular triangles, A B C and a bc, are drawn, in which A B of the first is twice or three times, &c., the corresponding side ab of the other. By measurement, it will appear that A C is twice or three times, &c., the corresponding side ac. The pupil is then exercised in the following way :-If ac is contained three times in A C, how often is the half of ac contained in A C? How often is the third part of ac contained in A C? and so on. It will appear that, in the case under consideration, any part of ac, the fifth, for example, is contained in A C as often as the fifth part of ab is contained in A B; and also that ac is contained in twice A C, or three times A C, &c., as often as ab is contained in twice A B, three times A B, &c. Now take the same construction, drawing ab so as not to be contained an exact number of times in A B. It will still appear from measurement that any part of ab is contained as often in A B with a remainder, as the same part of ac is contained in A C with a remainder. For example, if A B contains the fifth part of ab seven times and something over, A C will also be found to contain the fifth part of ac seven