in the same plane, are not represented in a figure on a plane either in their true positions or their true dimensions; and the same is the case with respect to the planes and lines required in the construction: the requisite operations cannot be performed on paper, that is, on a plane. The one is a real construction which can be effected by means of a ruler and a pair of compasses; the other is an ideal construction which can only thus be really effected in space, where the solid, the several planes and their intersections exist in their true positions and dimensions. Descriptive geometry furnishes the means of actually effecting such constructions by referring points, lines, and planes as they exist in space to two fixed planes, by means of perpendiculars to these planes.

2. The most simple manner of determining the position of a point in a plane is by referring it to two fixed straight lines in the plane, at right angles to each other, by means of perpendiculars let fall from it upon these lines: the position of the point will be determined when the lengths of these perpendiculars are known.

3. In the same manner, if a point above the plane in which are the two fixed straight lines, be referred to that plane by a perpendicular let fall from it on the plane, and the foot of the perpendicular be referred to the fixed lines by perpendiculars, the position of the point in space will be determined when the lengths of the perpendiculars on the fixed lines, and that of the perpendicular on the plane are known. If every point whose position in space is required, is thus referred to a plane and to two fixed lines in it, the positions of these points would be determined.

If, for example, all the points to be determined in a geometrical magnitude which is above a horizontal plane, be referred to the plane by perpendiculars to it, that is, by vertical lines, and the intersections of these vertical lines with the horizontal plane be referred to two fixed lines in it, by perpendiculars on them, the positions of the points in space will be determined by the lengths of the perpendiculars on the fixed lines, and of the perpendiculars on the plane, drawn from the several points; the positions of the points in the horizontal plane giving what is termed the Plan of the points in the body, and the perpendiculars giving the heights of these points above the plane. In this mode of representation, which is called Horizontal Projection, in order to give a clear idea of the positions in space of the original points, it is necessary that the height of each point above the horizontal plane should be indicated in connexion with its projection or representation on that plane. When the number of points to be thus represented is considerable, the plan may become complicated by the figures or lines indicating the heights; but still this method, to which we shall hereafter recur, will be advantageously employed when most of the data and the results to be obtained from them are numerical. An equivalent method is, how

ever, better adapted for representing magnitudes in their true dimensions, and also in their geometrical relations.

4. A point in space being, as before, referred, by a perpendicular, to the plane in which the two fixed straight lines are, which we will suppose to be the plane of the paper, and, for the sake of illustration, we will in the first instance consider this to be vertical-so that the figure is viewed as drawings usually are-one of the fixed lines being horizontal, and the other vertical; and from the foot of this perpendicular there being drawn a perpendicular to the horizontal fixed line; if, in the same manner, the point be referred, by a perpendicular, to another plane passing through the horizontal fixed line, at right angles to the paper, which plane will therefore be horizontal, and from the foot of this perpendicular in the horizontal plane a perpendicular be let fall on the intersection of the two planes; the position of the point will be determined by three lines, the lengths of the perpendiculars let fall in the two planes on their intersection, and the distance of these perpendiculars from the intersection of the two fixed lines. We have supposed the vertical plane to be represented on the paper, and therefore the lines in the vertical plane will be represented on the paper in their true dimensions and positions: if now we conceive the plane of the paper to revolve about the intersection of the vertical and horizontal planes, until it coincides with the latter, the lines in the horizontal plane will likewise be represented on the paper in their true dimensions and positions. This is the fundamental principle of Descriptive Geometry, and if well understood little difficulty will be found in its practical application. We will therefore endeavour to render what has been here stated more clear, by means of figures.

5. The point A (Plate II. fig. 1), in the plane xu, being referred to two fixed lines xy, xv, in the plane, by perpendiculars Am, An, let fall from A upon these lines, the position of the point A will be determined when the lengths of the perpendiculars Am, An are known.

6. If the point A (fig. 2), supposed to be in front of the plané xu, be referred to that plane, by a perpendicular Ad, let fall from A upon au, and the point a' be referred to xy, av, by the perpendiculars a'm, dn, the position of the point A in space is determined when the lengths of the perpendiculars Aa', a'm, a'n are known. The length of the line Aa' is not however correctly represented in the figure, nor is its position with reference to a'm and a'n so represented.

7. If the point A, besides being referred, by the perpendicular Aa', to the plane xu, which we will suppose vertical, be referred, by a perpendicular Aa, to the plane xz at right angles to the plane xu, and therefore horizontal; and if from a, am be drawn perpendicular to xy, the position of the point A will be determined by the two perpendiculars on xy, ma' and ma, and mx, or a'n perpendicular to xv. The line Aa, or its equal ma', will represent correctly the distance of the

VOL. II.

D

point A from the plane xz, but then according to the ideal view o the lines in the figures, Aa' or ma will not so represent the distance of A from the plane. If, however, we conceive the plane of the paper to revolve about the line xy, the intersection of the vertical and horizontal planes, until it coincides with the latter, and the foot of the perpendicular Aa on the horizontal plane to be represented on the paper in its true position, and the line ma in its true direction, that is, at right angles to the line xy, as in fig. 3, without any regard to the ideal representation in fig. 2, the lines xm, ma', ma will be represented in their true dimensions.

We have here supposed that the plane of the paper revolves about the horizontal line xy until it become horizontal, and that in this position of the paper the point a, where the perpendicular on the horizontal plane, from the point A, falls on it, is marked; but it is evident that the result will be the same, if we suppose that the paper is horizontal, that the horizontal plane is represented on it by xz, that the point A above it is referred to this plane by a perpendicular to it from A, meeting the plane in a; and the point A being referred, by a perpendicular, to the vertical plane, this plane be conceived to revolve about the horizontal line ay until it coincides with the plane of the paper. This last is the usual way of presenting the principle on which are founded the methods of Descriptive Geometry. We now proceed to the details of these methods.

DEFINITIONS.

8. When a point without a plane is referred to the plane by a perpendicular to it, the point where the perpendicular from the original point meets the plane is called the Orthographic Projection of the original point.

Thus the point a (fig. 4), where the perpendicular Aa, from the point A, to the plane az meets the plane, is called the orthographic projection of the point A on the plane xz.

9. The line Aa which determines the projection a of the point A is called the Projecting straight line, or simply, the Projecting Line of the point A.

10. If from the several points A, M, N, O... B of a straight line AB (fig. 5), perpendiculars Aa, Mm, Nn, Oa... Bb be let fall upon the plane xz, the line amnob will be the orthographic projection of AB on the plane xz.

Since the plane passing through Aa and AB will contain all the perpendiculars Mm, Nn, Oo ... Bb (Geometry of Planes, Prop. 33), the orthographic projection of a straight line AB is a straight line ab determined by the projections of its extremities.

11. The plane ABba passing through the straight line AB, perpendicular to the plane az, is called the Projecting Plane of that line.

12. If from the several points A, M, N, O... B (fig. 6) of any line AMNOB perpendiculars Aa, Mm, Nn, Oo... Bb be let fall upon the plane az, the line amnob passing through all the points a, m, n, o,... b will be the orthographic projection of the line AMNOB on the plane xy.

13. The surface which contains all the lines Aa, Mm, Nn, &c., and which coincides with the curve AMNOB, is the projecting surface of that curve upon the plane xz.

14. The plane upon which points and lines are projected is called the Plane of Projection.

Thus, xz in the foregoing cases is the plane of projection.

15. The point in which a straight line, produced if necessary, meets a plane is called its Trace upon that plane.

Thus, let the straight line AB (fig. 7), when produced, meet the plane az in the point a; the point a is the trace of AB on the plane xz.

16. The straight line in which a plane, or the line in which any surface, produced if necessary, meets a plane is called its Trace upon that plane.

Thus, ap (fig. 8) being the line in which the plane ABpa meets the plane az, ap is the trace of ABpa on the plane xz.

17. It is evident that points and lines may, in the foregoing manner, be projected on two or more planes by perpendiculars let fall upon these planes. As for our present purpose more than two planes are not necessary, we shall restrict ourselves to these.

Aa (fig. 9) being perpendicular to the plane az, a is the projection of the point A upon xz; and if Aa' be perpendicular to the plane xu which cuts the plane az in xy, a' will be the projection of A upon xu.

Aa and Bb (fig. 10) being perpendicular to the plane xz, the straight line ab is the projection of the straight line AB upon the plane az; and if Aa' and Bb' be perpendicular to the plane au, the straight line a'b' will be the projection of the straight line AB upon the plane xu.

When two planes xz, xu are thus employed, they are called Coordinate Planes.

18. A point is completely determined when its projections on two planes which cut one another are given.

a and a' (fig. 9) being the projections of a point upon the planes xz and xu, the point of which they are the projections must be A, the intersection of the perpendiculars aA, a'A to the plane xz, xu drawn from the points a, a'.

It is to be remarked here, that two points, taken at will on two planes which cut one another, may not be the projections upon those planes of any single point in space.

19. In order that two points, which are respectively on two planes that cut one another, may be the projections of the same point in space, it is necessary that the perpendiculars drawn from these points upon the intersection of the two planes, should fall upon the same point of this intersection, and this condition is sufficient.

Let a and a' (fig. 9) be two points on the planes xz and xu respectively; in order that they may be the projections of the same point in space, the perpendiculars am, a'm from them, on the intersection xy of the planes, must fall at the same point m.

For if A be any point in space, the plane Aama' passing through its orthographic projections on the planes xz, xu is perpendicular to each of these planes (Prop. 32); consequently, each of the planes xz, xu being perpendicular to the plane Aama', their intersection xy is perpendicular to the plane Aama' (Prop. 34); and therefore each of the angles ama, xma' is a right angle: that is, the perpendiculars to the intersection xy drawn from the projections a, a' of the point A fall on the same point m of the intersection.

And this condition is sufficient.

Let am and a'm, drawn from the points a and a', in the planes zz and au, perpendicular to xy, fall on the same point, m, of xy; a and a' are the orthographic projections of the same point in space.

For ay being perpendicular to ma and ma', it is perpendicular to the plane ama' (Prop. 4); therefore each of the planes xz, xu is perpendicular to the plane ama' (Prop. 32); and consequently the plane ama' is perpendicular to each of the planes az, xu. If, therefore, straight lines be drawn from the points a, a' perpendicular to the planes xz, xu respectively, they will be in the plane ama' (Prop. 33), and will meet in a point A in that plane.

20. A straight line is determined in position and magnitude when its projections on two planes which cut each other are given. Let ab, a'b' (fig. 10) be the projections of a straight line on the planes xz, xu; the position and magnitude of the line of which ab, a'b' are the projections, are determined.

Let a plane perpendicular to xz pass through ab, and be terminated by perpendiculars to xz, from a and b and another plane perpendicular to xu pass through a'b', and be terminated by perpendiculars to au from a' and b': the intersection AB of the planes thus determined is the straight line of which ab, a'b' are the projections.

21. In order that ab, a'b' may be the projections of the same determinate straight line in space, it is necessary that a and a' should be the projections of one point in space, and also that b and b' should be the projections of another point in space; and, therefore, that the perpendiculars drawn on xy from a and a' should fall on the same point in it, and likewise that the perpendiculars on xy from b and should fall on the same point.

22. A straight line in space is, however, given in position, when