Here the natural process, in constructing the figure, would be to draw the lines A B and BC (Fig. 2), and taking P as the given point, to draw PDE, so that PD and DE might be as nearly equal as possible. The proper way, however, is to draw a straight line, P E, bisect it in D, and through the points D E to draw the lines AD B, C E B, meeting in B.

Again, suppose a problem spoke of a circle touching a given line in a given point, and passing

B

E

FIG. 2.

B

FIG. 3.

through another given point. Then we should not draw a straight line, and taking the points P and Q (Fig. 3), attempt to draw a circle through Q to touch AB in P. We should first draw the circle, then draw a tangent, A P B, and take a convenient point, Q, upon the circumference of the circle.

In like manner if, in a deduction, mention is made of a circle inscribed within, or circumscribed without, a triangle, we shall obtain a far more satisfactory figure by drawing the circle first, and then

forming a triangle round it or within it, respectively, than by drawing the triangle first.

These instances suffice to exhibit the necessity of considering the order of the constructions needed in our figure. There are some considerations to be attended to, also, respecting the shapes to be given to different figures, that an examination of the properties they are meant to illustrate may be made as easy to us as possible.

It is very important that the different parts of a figure should not exhibit apparent relations not really

involved in the problem illustrated. Lines should not seem to be equal, or to be at right angles to each other, when they are not necessarily so. Triangles should not seem to be isosceles or right-angled when the problem does not involve such relations. It is well to notice that, in general, the most convenient form of triangle for illustrating general properties is that shown in Fig. 4; here the angle A is one of about 75°, the angle B one of about 60°, and the angle C one of about 45°. When a quadrilateral figure is not necessarily either a parallelogram or a trapezium, it is well to construct it of such a

figure as ABCD (Fig. 5), in which the four sides are unequal, neither pair of opposite sides parallel, and the diagonals AC, BD do not make equal angles with any side. It will be noticed, also, that neither diagonal bisects the other. If we had a problem in which the bisections E and F of the diagonals were concerned, all that would be necessary, in order that neither diagonal might bisect the other, would be to draw the diagonals AC and BD first, so that their point of intersection, G, should be well removed from the bisections E and F; then join A B, BC, CD, and D A.

It is sometimes convenient to draw a part of the figure in darker lines than the rest. We may distinguish in this way, for instance, between the lines or circles belonging to the enunciation and those belonging to the construction. When we are in doubt as to the necessity of any construction, it may be lightly dotted in. In very complex figures, dark, light, broken, and dotted lines may be conveniently employed together.

Always letter every point of the figure which may have to be referred to as you proceed. It is often as well, when a result has been established which seems to promise to be useful towards the solution of a problem, to re-draw the figure, omitting all lines except those which have served to guide you to this result. But, except in such instances, or where the figure seems obviously unsuited to your requirements, or has become overcrowded with constructions, it is

well to keep to the same figure as long as possible. The habit of repeatedly re-drawing figures interferes with the concentration of the attention and the steady progress from result to result, which alone avail toward the solution of difficult problems.

III. ANALYSIS AND SYNTHESIS.

There are two general modes of treatment applicable to problems, termed, conventionally, the synthetical and the analytical, or synthesis and analysis. In the former, we study what is given and work up to what is sought; in the latter, we examine what is sought and work back to what is given. I am not concerned here with the correct applicability of the names 'synthesis' and ' analysis' to these processes, and shall therefore content myself with discussing the processes themselves under the names usually given to them.

It is a mistake to suppose that, as some have asserted, analysis is the method always employed— consciously or unconsciously-in the solution of problems. Of course, we are compelled to consider what it is we have to do or prove, and thus far the analytical method cannot but enter into our processes. But in the solution of a problem, we may proceed, as may be most convenient, by either the synthetical or the analytical process, or—which in complex problems is far more commonly the case-by an alternation of both methods. As an illustration of my

meaning, I may compare geometrical problems to those examples in algebra, trigonometry, &c., in which we have to establish the identity of two expressions. In such cases we may either take one expression, and try to work it into the same form as the other, or vice versa, we may select the latter to work upon, or—which is the surer process—we may work both down to a common form.

However, it will be better to select a few examples of geometrical problems, and to exhibit the application of different processes to them, than to discuss general rules. I begin with very simple examples.

Suppose we have to deal with the following deduction :

Ex. 1.—The line AB (Fig. 6) is bisected in C, and CD is drawn at right angles to AB. From any

FIG. 6.

point E in CD lines are drawn to A and B. Show that E A is equal to E B.

Having constructed a figure in accordance with these data, we go over the data thus: we have A C equal to C B, and CE at right angles to A B. We