given to them under these circumstances, as their being left without demonstration would in initio violate the general rule upon which the system was developed.

4. The construction of the problems in space that are requisite as subordinate to demonstration are always given by Euclid; and, indeed, an adherence to this practice is in some respects desirable. Those which occur are however so simple, and the processes little more than corollaries from theorems actually given, that I have yielded to the wishes of the officer to whom the Board has committed the control of these volumes, and omitted them altogether. A few of them will, however, inevitably make their appearance in the "Descriptive Geometry" hereafter.

DEFINITIONS.

1. Parallel planes are such as, however far produced in all possible directions, they will never meet.

2. A line and plane are parallel when, however far produced in all possible directions, they will never meet.

3. A dihedral angle is formed by two planes which intersect. The line of intersection is called the edge of the angle, and the planes themselves the faces of the angle.

4. If from any point in the edge of a dihedral angle two straight lines be drawn, one in each of the faces perpendicular to the edge, they form the profile angle of the planes. Their inclination to one another is called the inclination of the planes. If the profile angle be a right angle, the planes are said to be perpendicular to one another.

5. If a straight line be perpendicular to each of two straight lines at their point of intersection, it is said to be perpendicular to the plane passing through those two lines. Conversely, the plane is said to be perpendicular to the line.

6. A line which is neither parallel nor perpendicular to a plane is said to be oblique to the plane. The inclination of an oblique line to a plane is the acute angle contained by that line and another drawn from the point in which it meets the plane, to the point in which a perpendicular from any point in the line meets the plane.

7. The trace of a line or plane (or any surface, indeed) is the point or line in which that line or plane meets some specified plane.

8. A solid angle (or more descriptively, a polyhedral angle), is formed by three or more planes which meet in a point, and each plane limited in its expansion by the two adjacent ones.

It is called trihedral, tetrahedral, pentahedral, etc., according as there are three, four, five, etc., planes.

The planes are called faces; their linear intersections, edges; and the point in which they all meet, the vertex of the solid angle.

9. If any number of lines be parallel and intercepted between two parallel planes, and planes join these two and two consecutively, the figure produced is a prism.

The prism is, for geometrical purposes, considered as capable of indefinite prolongation both ways from the parallel planes, or in the direction of the lines.

The lines are called the edges of the prism; the parallel planes, the ends or bases of the prism; and the planes joining the parallel lines, the faces.

10. If planes passing through a point without a plane, pass also through the sides of a polygon in that plane, and be limited by their adjacent intersections, they form a pyramid.

The pyramid takes the name of triangular, tetrangular, pentangular, etc., according to the figure of the polygon.

That polygon is the base, the point is the vertex, the planes are the faces, and their intersections the edges of the pyramid.

The pyramid is, geometrically, capable of indefinite extension, both below the base and beyond the vertex.

11. If a line making any angle with the plane of a circle move continually over the circumference, and always keep parallel to its first position, it generates a cylindrical surface.

If this surface be cut by two planes parallel to the circle, the intercepted portion is a cylinder.

These two planes are called the ends or bases of the cylinder; the original circle the directrix; the moving line the generatrix; and a line through the centre parallel to the generatrix, the axis of the cylinder.

12. If about a point taken without the plane of a circle a straight line move always touching the circumference of the circle, it generates a conical surface. The portion of it intercepted between the point and circle is called a cone. The point is the vertex of the cone, the circle is its base, and the generatrix in any position is called an edge of the

cone.

*

13. If a circle revolve about any diameter till its plane takes a reversed position, it will generate a spherical surface or a sphere.

The terms centre, diameter, etc., are the same as in a circle.

14. Polyhedron is a figure bounded by plane faces, and takes its name from the number of those faces.

Thus the tetrahedron has four faces, the hexahedron six, the octohedron eight, the dodecahedron twelve, and the icosahedron twenty. These names, however, are only applied when all the faces of the figure are equal to one another, forming subjects of much interest to the ancients; but, except from their being amongst the most frequent forms assumed by crystals (both artificially and naturally produced), they would be mere curiosities in modern science.

In modern science the polyhedron is always described, but seldom named, with one or two exceptions.

15. When three pairs of parallel planes mutually intersect, they enclose a figure called the parallelopiped.

It is oblique or rectangular, according as the containing pairs of planes are oblique or rectangular. The cube is the most confined instance of it.

16. Similar polyhedrons are such as have their angular points or vertices similarly situated, each with respect to the others.†

* The terms cone and cylinder are generally employed to signify not only the parts here defined by them, but the surfaces generally extended as far as may be

necessary.

†This definition is complete; all others yet given involve superfluous conditions, just as the definition of similar plane polygons does.

CHAPTER I.

PARALLELISM OF LINES AND PLANES.

PROPOSITION I.

If two parallel planes be cut by a third plane, the sections will be parallel.

Let the parallel planes MN, PQ be cut by the plane AD in the lines AB, CD: these lines, AB, CD, will be parallel.

For since AB, CD are in one plane, if they be not parallel, they will meet in some point, as E.

Then since AB is in the plane MN, the point E is in that plane; and similarly, E being in the line CD, it is also in the plane PQ. The planes MN, PQ therefore meet in E, which is impossible (Def. 1.) since they are by hypothesis parallel to one another.

M

E

PROPOSITION II.

If a straight line be parallel to a plane, all planes through this line which cut that plane will have their sections with it parallel to the line parallel to it, and to one another.

B

M

D

Let the line AB be parallel to the plane MN, and the planes AD, AD' drawn through AB, cut the plane MN in CD, C'D': then CD, C'D' will be parallel to AB, and to one another.

(1.) Since AB, CD are in one plane, they are either parallel to one another, or they will meet in some point E.

If parallel, the proposition is admitted; but if not, let AB, CD meet in some point E. Then since E is in the line CD, which is itself in the plane MN, E is in the plane MN; and the line AB meets the plane MN in the point E. But this is impossible, since AB is parallel to

MN by the hypothesis. Wherefore since AB, CD are in one plane ABCD and never meet, they are parallel. Similarly, C'D' is parallel to AB.

(2.) The planes AD, AD' intersecting in the line AB, they cannot meet in any point out of that line; and hence the lines CD, C'D' in them, and parallel to AB, can never meet: but they are in one plane MN; and hence they are parallel.

PROPOSITION III.

If a line without a plane be parallel to a line in the plane, the first line will be parallel to the plane.

(Preceding figure.)

Let the line AB without the plane MN be parallel to the line CD in it: then AB will be parallel to the plane MN.

For since AB, CD are parallel, they are in one plane, and can never meet; and since AD and MN meet in the line CD, they cannot meet in any point without CD. Whence AB, lying wholly in the plane ACD, can never meet the plane MN; that is, AB is parallel to MN.

COR. If through CD one of two parallel lines AB, CD, a plane be drawn; this plane will either wholly include AB, or be parallel to it.

PROPOSITION IV.

If through two parallel lines planes be drawn, they will either coalesce, be parallel, or have their intersection parallel to those two lines.

(1.) The planes may each be drawn through the other line, since parallel lines are in one plane; and hence they may coalesce with the plane which contains the parallels, and therefore with one another.

(2.) They may never meet, since the lines through which they pass never meet; and in this case they would be parallel.

(3.) Let AB, CD be parallel lines, through which are drawn any two planes AF, ED which neither coalesce nor are parallel they will meet in some straight line EF; and this line is parallel to each of the lines AB, CD.

For since AB is parallel to the line CD lying in the plane ED, it is parallel to the

plane ED itself (Prop. III.); and hence it can never meet the line EF in that plane. But EF is also in the plane AF, since it is the intersection of that plane with the plane ED; and hence AB, EF which lie in the same plane and never meet, are parallel.

In the same way it is proved that EF is parallel to CD.

PROPOSITION V.

If two planes intersect, and lines be drawn in those planes parallel to the intersection of the planes, one in each, these lines will be parallel.

Let the two planes AD, DE intersect in CD; and in them respectively let AB, EF be drawn parallel to CD: then AB, EF will be parallel to one another.

(1.) If AB, EF be not in one plane, draw a plane through AB and E to cut the plane CF in EF'. Then since through AB and CD the planes AF', CF' are drawn, intersecting in EF', the line EF' is parallel to CD (Prop. Iv.). But (Hypoth.) EF is parallel to CD; and C hence through the point E two lines EF, EF' are drawn parallel to CD, which is impossible.

Wherefore AB, EF are in one plane.

Again, since CD is parallel to the line

AB in the plane ABEF, it is parallel to the plane itself (Prop. III.); and since through the line CD which is parallel to the plane ABEF planes AD, DE are drawn to cut it in the lines AB, EF, these lines are themselves parallel to one another (Prop. 11.).

PROPOSITION VI.

If each of two lines be parallel to a third line, they will be parallel to one another.

(Same figure.)

Let AB and EF be each parallel to CD; they will be parallel to one another.

For since AB, CD are parallel, they are in one plane; and similarly EF, CD are in one plane. These planes intersect in CD; and parallel to CD the lines AB, EF are drawn in those planes respectively. Wherefore AB, EF are parallel (Prop. v.).

COR. 1. If any number of lines be each of them parallel to one line, hey will be parallel each to all the others.

ČOR. 2. If any number of lines be parallel to one of two parallel lines, and any number parallel to the other, all the lines will be parallel, each to all the others.

For the proof may be extended to a fourth, a fifth, etc., parallel line without limitation as to number. Let the student apply it to an instance or two in detail.

PROPOSITION VII.

If two straight lines which meet be parallel to two others which also meet, but not in the same plane with them: then

(1.) The plane which contains the first two will be parallel to that which contains the other two; and

(2.) The angle contained by the first two will be equal to the angle contained by the two others.

(1.) Let the lines AB, BC be respectively parallel to DE, EF: then the plane MN through the former pair will be parallel to the plane PQ through the latter pair.

For if not, let the plane MN cut PQ in some line, as GH. Then, since AB is parallel to a line DE in the plane PQ, it is parallel to PQ (Prop. 111.); and since the plane MÑ is drawn through the line AB, which is parallel to PQ to meet PQ in GH, GH is parallel to AB (Prop. 11.); and therefore, again, to DE (Prop. vI.).