the parts are better seen than in a mere pictorial representation.

In this motion every point in XZ describes the quadrant of a circle; and all the figures traced upon that plane retain their original magnitudes and relation of their several parts to one another. In the figure, SB' represents SB the vertical trace; and it is easy to see that however it may change its position with respect to the other trace SA, it does not change with respect to the axis OX; and hence, so long as

Z

B

the relation of SB to SA is independent of the angle formed by them in space, no constructive change results from this supposed revolution. DEF. 7. The following terms are used in reference to this revolution of the coordinate planes :

(1) To plan the elevation, is to bring the elevation into continuity with the plan;

(2) To elevate the plan, is to bring the plan into continuity with the elevation;

(3) To plan the section, is to bring the section by revolution round OY, into continuity with the plan.

DEF. 8. We shall designate the geometry of Euclid by the term Theoretical Geometry, to distinguish it from the Descriptive Geometry of which we here treat.

DEF. 9. The model, or the representation of the figure as it exists, and as we contemplate it in Theoretical Geometry, we shall call the eidograph, or likeness; and the drawing which we execute as its substitute in Descriptive Geometry, from the one part being placed at right angles to its natural position, the orthograph.

DEF. 10. By the regions of space, we mean those portions of space which are separated by the coordinate planes from each other, and of which these planes are the limitations; these planes being indefinitely continued in their respective directions.

The first region is that above the H and before the V (H and V denoting the horizontal. and vertical planes); the second, above H and behind V; the third, below H and behind V ; Y and the fourth, below H and before V. The regions being in succession from the spectator in revolving round the axis.

One or two other definitions will be given

further on, when they will be better comprehended than at present. In Descriptive, as in Theoretical Geometry, the properties of the eidographic figure, as well as of the orthographic, must be investigated; for the eidographic properties are always the authorities for orthographic practice. This, of course, will often require us to give the figures of the problem in both kinds; and it will, for the most part, be found that those properties are not only most directly but most easily obtained from the eidograph.

It is important to get a clear notion of the positions of the traces for particular positions of the planes; and conversely, of the planes themselves from the positions of the traces. For this purpose we give, attached to a verbal description, the figures both in eidograph and orthograph, of each peculiarity of case.

We shall minutely describe them in reference to the first region of space; and then only give the general representation of the cases for the other three regions.

5. If the plane be parallel to one of the coordinate planes, there will be no trace upon that plane, and the trace upon the other will be parallel to the axis.

Though in these figures the planes are marked with boundary lines, in conformity with our practice in theoretical geometry, it is to be understood that in all cases they are capable of unlimited extension in all their directions. In the orthographs, it is not usual to make them at all, further than by the axis drawn horizontally on the paper, the boundaries of the paper being taken as the marked boundaries of the plane on which the orthograph is made. The traces, moreover, instead of being continued to the boundaries of the paper, are only drawn so far as the practical constructions in the individual case require them to be.

In all the figures which we have given, the traces of the plane separate the parts of it in front of the elevation and above the plan, from the parts of it which are concealed behind the one and below the other. We can generally choose the particular parts of the coordinate planes in reference to our drawing, as to fulfil this condition; but where it is for any reason desirable to have it otherwise, a system of notation expressive of position will enable us to express the parts of a figure behind the elevation, below the plan, or both.

It will be observed that we have marked the intersection of the delineated plane with the axis, both in eidograph and in orthograph, by the letter P; and any point in the horizontal trace by P1, and in the vertical trace by P. This plane we shall briefly term "the plane P." In like manner, if more planes be drawn, we shall mark them by other letters Q, R, etc., plain, and with subscribed numerals taken in the same way, Q, R, signifying any point of their respective horizontal traces, and Q, R, etc., points in the vertical traces of the planes, Q, R, etc. When, therefore, we mark by literal notation, the signification will be apparent at once.

But as we often draw our orthographs without letters, the method of using round small dots (instead of full line, as in the preceding diagrams), to represent the parts of the traces which lie behind the vertical plane of the eidograph and below the horizontal plane, is found to fully answer the purpose in view. It is only necessary to keep in memory, that when the vertical plane is turned about the axis on the horizontal, the part of it below the axis comes to the front of the axis in the orthograph. Hence the following figures will be intelligible and distinguishable, whether considered in full line and lettered, or in dotted line without letter:

All these, obviously, are perfectly distinct by means of either notation.

All the varieties delineated in the first region may thus have correspondent varieties delineated in any region, either by dotting or by literal designations.

THE ORTHOGRAPH DELINEATION OF LINES.

It has been already noticed that the most convenient method of considering a line, in reference to Descriptive Geometry, is as the intersection of two planes; and that when the traces of the two planes passing through it are given, those of the line itself (or the points in which it cuts the coordinate planes) are also given. We have, hence, a choice of two methods of representation :—

(1.) The traces of the two planes which pass through it ; (2.) The traces of the line itself.

The circumstance that the line itself whilst absolutely fixed, may have innumerable planes drawn through it, shows that the same line may be represented by the traces of the planes in innumerable different ways; the only condition to be fulfilled being that the vertical traces of these planes shall all pass through one given point on the elevation, and the horizontal through one given point on the plan. This has the advantage of allowing us to make any hypothesis we please respecting the positions of the defining planes, compatible with that twofold condition; and we shall presently see that the one most generally advantageous is, to take the planes perpendicular to the coordinate planes. The trace of the plane perpendicular to the plan upon the plan is then called the horizontal projection of the line; and that of the plane perpendicular to the elevation upon the elevation is called the vertical projection of the line. The planes themselves are called the projecting planes.

*

However, in the mere exhibition of a line which is given, it is usually most convenient to define it by its traces;' as any other proposed mode of connecting it with the other data is at once obtained from these. The following cases are enunciated in reference to their eidographs, from their being those which most usually occur in actual constructions for practical purposes. In passing from the eidograph, the line itself is of course omitted, its traces, or the traces of its defining planes, being all that our object requires us to use.

The following are the only possible positions of lines in reference to the coordinate planes :

1. Oblique to both coordinate planes;

2. Oblique to one and parallel to the other;

3. Parallel to both planes; and, consequently to the axis;

4. Perpendicular to one plane, and therefore parallel to the other. We shall take these four classes in succession, and give specimen eidographs as before; and in the figures it is to be understood that

When, however, the line is parallel to one of the coordinate planes, it makes no trace upon that plane; and when parallel to the axis, it makes no trace upon either plane. In this case we have no alternative but to define the line by planes through it; and the projecting planes are for the most part employed for this purpose.