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ELEMENTS OF SOLID GEOMETRY.
(1.) The first six books of Euclid's Elements, to which we have directed the attention of the student in the preceding part of this volume, are confined to the investigation of the properties of rectilinear figures and circles which are all described upon the same plane. It is evident that it may be, and very frequently is, necessary to consider the mutual relations and properties of right lines and circles which are in different planes, and also the various circumstances which regulate the relative position of planes themselves. Besides this, there are numerous other surfaces on which, as well as on planes, right lines or circles, or both may be drawn. The properties of such surfaces, and the various lines which may be described upon them, form an important part of geometrical science. But even this gives a very inadequate notion of the infinite extent of the field which geometry presents to our contemplation. The right line and circle are the most simple of all lines, and those which perhaps most frequently become the subjects of examination ; but they are far from including the whole even of that class of lines which are described upon a plane, not to mention innumerable curves which are drawn upon other surfaces, but which are not all in the same plane. The variety of surfaces is as infinite as that of lines. They are divided into plane and curved surfaces. All plane surfaces are perfectly alike in their properties, but curved surfaces admit of endless variety
Of this extensive field for geometrical investigation, long-established usage has assigned a certain part to the elements of geometry.' The 'elements of plane geometry' are confined to the properties of right lines and circles described upon the same plane, excluding ellipses, hyperbolas, and innumerable other lines, which in common with the right line and circle admit of being drawn upon a plane, and are thence called 'plane curves. These are generally assigned to the province of the sublimer geometry, and their properties are investigated most easily and effectually by the powerful instruments of analysis. The student who desires to prosecute such investigations is referred to our treatise on Analytic Geometry.
The elements of solid geometry are confined to the investigation of the circumstances which determine the mutual position of right lines and circles which are not in the same plane, the properties of solid figures, which are bounded by planes, and those of the surfaces denominated spheres, cylinders, and cones, and the solids bounded by these alone, or by these conjointly with planes. The unlimited variety of curved surfaces which do not come under these denominations are resigned to the province of the higher geometry, and like the plane curves' already mentioned are brought under the dominion of analysis. The student who desires to penetrate to the depths of this department of the science, will find ample information and assistance in the beautiful work of Monge, entitled Application d'Algèbre à la Géométrie.
Conformably to what we have now stated, we shall devote the present treatise to the investigation of the conditions which determine the mutual position of right lines which are not in the same plane, of different planes with respect to each other and to right lines, the properties of figures or spaces bounded by planes, and the principal properties of spheres, cylinders, and cones.
Of the Relative Position of Right Lines and Planes.
(2) DEF.—A plane is a surface such that a right line cannot be drawn through two points in it without having all its points in the surface.
There are other surfaces besides a plane in which it is possible to assume two points such that if a right line be drawn through them, and be indefinitely produced, it will lie entirely in the surface, but in a plane surface it is impossible to assume two points with which this will not happen. This is not true of any other surface. (3) Cor.-Hence it follows, that one part of a right line cannot be in a plane while another part of it is above or below it.
If two planes cut each other, their common inter
section will be a right line.
For if any two points of their common intersection be assumed, and a right line be drawn through them, this right line must lie entirely in each of the planes (2), and must therefore be their common intersection.
(5) DEF.—A plane is said to be drawn through a right line when it is drawn through two points of that line. The whole line will in this case be in the plane.
(6) It is evident that innumerable planes may be drawn through the same right line, or what is the same, any number of planes may intersect each other in the same right line. This will easily be perceived if any plane drawn through the right line be conceived to revolve round that right line. The different positions which it will assume in different parts of its revolution will be those of different planes drawn through the right line.
(7) Two planes can have only one line of intersection.
For suppose that they had a second. Through any two points on those lines of intersection let a right line be drawn. By (2) every part of this line is in each of the planes. Therefore it is a third line of intersection; and the same being true of right lines drawn through every two points on the lines of intersection, it follows that every right line which is drawn in one plane is also in the other, and therefore the two planes are identical.
Hence two distinct planes cannot have more than one line of intersection,
This proposition is analogous to that in virtue of which two right lines can intersect only in one point.
(8) If a point be given, and also a right line not pass
ing through the given point, a plane may be drawn through them, and but one such plane can be drawn.
Let a plane be drawn through the given right line, and being indefinitely produced, let it be conceived to revolve round that right line
In its revolution it must sweep through all the surrounding space, and must therefore pass through the given point.
There is but one plane passing through the given right line which also passes through the given point ; for if we were to suppose a second plane it would evidently have two intersections with the first, viz. the given right line and another intersection passing through the given point.
The student should recollect that planes are considered as indefinitely produced. (9) Cor.—Hence a right line and a point, provided the point be not on the right line, are sufficient to determine a plane.
(10) A plane, and but one plane, can be drawn through
three points which are not on the same right line.
Let a right line be drawn through any two of the points, and a plane, and but one plane, can be drawn through this line and the third point (8), (11) . Cor.--Hence three points, not placed in the same right line, are sufficient to determine a plane.
(12) A plane, and but one plane, can be drawn through
two intersecting right lines.
For let a point be assumed on each of them different from their point of intersection. A plane, and but one plane, can be drawn through the two assumed points and the point of intersection (10), and the two intersecting lines will be in this plane (2). (13) Cor.—Hence two right lines which intersect are sufficient to determine a plane. (14) Def.The plane which is drawn through two intersecting lines is usually called the plane of those lines,' or 'the plane of the angle' which those lines contain.
(15) A plane, and but one plane, can be drawn through
two parallel lines. A plane may be drawn through them because by their definition they are in the same plane; and but one plane can be drawn through them, because but one plane can be drawn through either of them, and any point assumed upon the other (8).
(16) If two right lines intersect, a third right line
may be drawn through their point of intersection
perpendicular to each of them. Let A B and AC be the right lines intersecting at A. Take equal parts A B, AC from A and draw BC. Bisect B C at D, and draw a line D E perpendicular to BC, and making any angle with AD, and let the acute angle be AD E. Through Á and in the plane of the lines A DE (XIII) draw A E perpendicular to AD. The line A E will then be also perpendicular to AB and AC. For draw EC and EB.