must be known*. No two of these planes must be parallel, in this system. In general they are so taken that each is perpendicular to the other two; and the projecting lines to one plane respectively parallel to the intersection of the other two.

6. Two of these planes are usually taken, the one parallel to the horizon, and the other perpendicular to it. In practice, however, it will be shown that one of these three vertical planes may be dispensed with; and that one is always employed, whose intersection with the horizontal plane is parallel to the top and bottom of the drawing. They are the same planes which, in practical works, are employed for the plan and elevation, or section. The line of intersection is called the ground-line.

7. In the application of algebra to the projective system, the line of intersection above spoken of is usually marked x x', the line of intersection perpendicular to it, in the horizontal plane, is marked Y Y'; and the line of intersection of the two vertical planes, is marked z z/; and the three projecting lines of the point parallel to these three, are respectively denoted by x, y, z. We shall de

x

Y

| figure different from itself, in any of these particular respects.

A point, having position but not magnitude, is given, when such conditions are given as shall fix its position absolutely.

A straight-line is said to be given in position when two points through which it passes are given, and in magnitude when its length is given; and in magnitude and position when the two extremities of it are given.

* We have said known. The word given is employed by geometers in two senses, which do not always precisely agree in signi fication: viz., 1. to signify that which is actually given amongst the conditions of the proposition; and 2. to signify those necessary consequences of the conditions which are actually given, and follow from them by processes of construction, or reasoning, which have been previously established. In the general processes of geometrical analysis, this difficulty is not felt by expe- A curve-line is said to be given when its rienced reasoners; but it creates great mode of genesis is given, and the position embarrassments to young students. In and magnitude of the quantities upon which the present case two projections must be it depends, are also given. given: but the third can at once be deduced from them, and is also, in common with them, therefore known. As, however, it is sometimes difficult to avoid the use of terms which are become part of the current language of science, we may probably inadvertently employ the word given in its usual signification on some occasions. We, therefore, annex the usual meanings of the word amongst writers in general.

The position of a point in space is a relative idea. Till some other data be furnished in relation to which its position is to be considered, we cannot say that its position is given: but when those are given, the point is given in position.

A geometrical magnitude is said to be given when such conditions respecting its figure, magnitude, and position are given, as shall be incompatible with any other

A surface is given when its genesis is given, and the position and magnitude of the quantities on which they depend, are given.

In all these cases, it is obvious that by given is meant determinable rather than determined—that is, fixed, rather than actually known. Some service would be rendered to geometry by a little change in this language; especially as the same word, given, is also applied to those points, lines, or surfaces of reference, which are assumed as those in respect to which the others are compared.

+We have said "in this system," lest the reader should suppose that the point was indeterminable when two of the planes are parallel, which would be an erroneous supposition, as they are then perfectly determinate. However the projections must be upon the two which are not parallel.

note the co-ordinate axes in the same manner in our descriptive geometry.

The three lines, x x', Y Y', z z/, are in this case called axes of coordinates, axes of reference, or co-ordinate axes; the three planes are called co-ordinate planes; and the lengths, x, y, z, are called the co-ordinates of the point. In the former case, planes of projection.

8. The polar, perspective, or scenographic projection is often advantageously employed. In this case the point is projected by a line drawn through a given point, or pole, upon a given plane, and is the foundation of perspective representations. Of this we shall speak more at large hereafter, in showing the foundation of that art, and the modes of calculation by which it may also be effected.

9. It is evident that if we can project a point, we can project a second upon any planes whatever: and hence that we can project the line which joins them.

It is, moreover, easy to perceive that the projection of a straight-line upon any plane, is the same with the line which joins the projection of its extremities. For, conceive two points, situated above the plane represented by the paper, and PP', QQ', the projecting lines, and p', q', their projections themselves. Now the two lines, PP/ and qa', being perpendicular to the same (Hutton's Course*, vol. i., cor. pr. 103,

But being parallel, they are in the same plane, by the definition of parallels. Hence that plane will intersect the plane of projection in a straight line, (Course i., pr. 89,) or the line joining P'q' is the intersection of the projecting plane with the plane passing through the projecting lines of the extremities of that line. Now the plane P P Q Q passes through the perpendicular P P', and is hence perpendicular to the plane of projection. (Course, pr. 110.) Hence we deduce the fact, that the projection of a given line upon a given plane is the intersection of that plane made by another plane which passes through the given line perpendicularly to the former one.

10. When the projections of a point upon two planes are given, the position of the point itself is known.

For when the projections are given, the projecting lines are also known; and as they both pass through the projected point, their intersection determines that point.

11. When two projections of a line are given, the position of the line itself is known.

For when the projections are given, the projecting planes are known, and as they both pass through the projected line, their intersection determines that line.

12. Straight-lines and planes admit of indefinite extension. They are said to be given in position in this case; but in magnitude and position, when their extremities are known, given, or determinable. When the lengths or boundaries only are known, they are said to be given in magnitude. (See note on Art. 5.)

13. When a line, whose projection

on a plane is given, intersects that plane, the part before the plane, if a vertical one, or above the plane, if a horizontal one, is visible; and the parts of the projection which represent the visible is

R

traced in full line; whilst the parts of the projection which represent the concealed parts of the line, are marked in dotted or broken lines. Thus, PR or P'R' represents the projection of the visible, and PQ or P'q' that of the hidden, part of the line, whose projection is Q R or QR'.

14. We have shown in Art. 10 that two projections are sufficient for the determination of a point or straight-line, and from this it may be inferred at once, that two projections are sufficient for the determination of any curve-line, however it be situated in space, since it is sufficient for the determination of any point taken arbitrarily in that curve, and hence for all of them, or for the whole curve which is their path or locus.

15. If a plane be drawn through the two projecting lines of any point, it will be perpendicular to the intersection of the planes of projection.

For these lines are, by the method of projection, perpendicular to the planes of projection, and hence the plane passing through them is perpendicular to those planes (Course, pr. 110); and hence again, perpendicular to their common section (pr. 110, cor. 3), or in other words, to the ground-line. (Art. 6.)

16. This line is called the projecting plane of the point, since it is the plane in which the projecting lines of that point are situated; and its intersection with the planes of projection are called the projecting parallels, since they are respectively parallel to the projecting lines, that on the vertical plane to the vertical projecting line, and that on the horizontal to the horizontal. They are called, therefore, the vertical projecting parallel, and the horizontal projecting parallel. Sometimes, simply the vertical parallel, and the horizontal parallel. They are both perpendicular to the ground-line. (Course, pr. 110, cor. 3.)

17. If the vertical plane of projection be made to revolve about the ground-line till it coincides with either the visible or hidden portion of the horizontal plane of projection; or the horizontal plane revolves about the ground-line till it coincides with the vertical one; then in both cases the vertical projecting parallel will coincide either by superposition or continuation, as the case may be, with the horizontal projecting parallel, and they will form one straight-line, perpendicular to the ground-line.

For both the projecting parallels being perpendicular to the groundline, they form with it two right angles; and when the planes come into coincidence, these two lines are in the same plane, and making angles with it on the same side equal to two right angles, they form one straightline.

18. If, therefore, we draw the ground-line x' o x, and in it take from o* (some fixed point from which the distances are estimated) the part o G, and draw P' P" perpendicular to the ground-line; if, also, we take GP the vertical projecting parallel, and G P" the horizontal one; we shall then have the point p perfectly defined by the lines o G, GP and GP". In this, OG is the distance of the point

from the plane Y Z, G P is its distance from the plane x Y, and GP" is its distance from the plane x Z. Of these we shall have rarely occasion to mention o G, and the other two are called the horizontal distance of the point P, and the vertical distance of the same point.

19. In conformity with the method of notation here employed, we shall hereafter designate the ground-line by x' x (or sometimes o x), the vertical projection of points (which in their own places in space are denoted by P, Q, R, &c,) by P', ', R', &c., and their horizontal projections by P", Q", R", &c.

20. The intersection of one plane with another, considered as being situated in the latter, is called the trace of the former plane upon the latter, or simply the trace of that plane.

21. When the traces of a plane upon the planes of projection are given, the plane itself is given; since no other plane different from this can make those traces.

22. Most of the mental and practical processes with respect to points and lines, are effected by means of these projections, whilst those respecting planes are effected by means of their traces upon the planes of projection.

23. The planes of projection, when brought into coincidence (as in art. 17,) being considered as one, we shall denominate it the compound plane, or picture plane.

Having now stated the objects we have in view, and explained the signification of the terms we employ, together with brief demonstrations of their propriety in a geometrical sense, we shall, in our next number, proceed to the solution of a few elementary problems respecting the straight-line, plane, and dihedral angle. We shall afterwards give a series of applications of these constructions to practical purposes before we enter upon the more intricate inquiries to which descriptive geometry is applicable.

*The line Z Z', drawn through O in this perpendicular to the ground-line, is identical with the trace (next def.) of the plane which has been omitted, or that which, in Art. 7, was called the plane, Y Z.

A POPULAR COURSE OF ASTRONOMY.

No. V.

ONE of the most involved and complicated problems ever proposed to the ingenuity of man, was the problem of the Heavens. A hollow concave above him, the whole of whose surface, go where he may, is apparently at the same comparatively small distance from him; the sun taking his journey across it, in a path which is not daily the same, returning day after day through some unknown region, to flood again the vast canopy of the heavens with light; stars seen in thousands at night, on this vast canopy, moving with one common motion slowly across it, between night-fall and day-break; this host of stars, different at different seasons of the year, but the same at the same season, preserving, in the general alteration of their position, their relative distances; except six of them, which wander about among the rest with a most devious motion, and are therefore called planets; the moon, too, moving with the common daily motion of the rest of the host of heaven, but, besides, revolving completely through it. every month; Winter, Spring, Summer, and Autumn, connecting themselves somehow with the variations of the daily path of the sun, and returning year after year at their appointed seasons; and eclipses of the sun and moon dependent by some inscrutable relation upon relative positions of the sun and moon;-all these things requiring, as they must have done and did, a great length of time, and much and patient observation to discover, constitute in their aggregate a relation of phenomena which as far surpasses any other yet unravelled by the higher researches of the human intellect, in its complication, and the vastness and dignity of the truths which it embraces, as in the simplicity of its results.

The sphere of the heavens has been hitherto spoken of as fixed and immoveable in space; and as having in its centre the EARTH,-of dimensions infinitely small and evanescent with regard to it-rolling perpetually round one of its own diameters, but never moving its centre from that of the great quiescent sphere of the visible heavens. The reader is now about to learn that this description of the position of the earth in space is incorrect:-that it does not occupy continually the same position in the centre of the sphere of the visible heavens-that its centre, and the axis within itself about which its revolution takes place, are not at rest-that these are in fact moving at the rate of about 19 miles in each second of time—that this motion is not directly forward in space, but continually round in a curve which returns into itself, and which is very nearly a circle, whose radius is ninety-five millions of miles -that nevertheless this enormous circle of the earth's revolution is itself as nothing in its dimensions compared with the dimensions of the great sphere of the visible heavens, so that the motion of the earth from the centre of that sphere, may be considered evanescent as compared with the radius of the sphere, and everything which occurs with regard to the fixed stars, as occurring precisely as it would occur if the earth's centre were really quiescent in space. Thus, then, whatever has been