For the purpose of bringing the important theorems ast near as possible to the definitions, postulates, etc., on which they rest, I have found it necessary to deviate somewhat from the usual sequence of propositions. Thus, I have grouped in the same section the prism and its limiting case, the cylinder, because they have so many properties in common. I have treated the pyramid and its limiting case, the cone, in like manner, etc. Always, I have aimed to give the most direct proof possible, and to save the student, by means of corollaries, the labor of reproducing constructions unnecessarily. An experience of twenty years in teaching mathematics leads me to think that the student who gets up the subject from this brief work, in the end will be at no disadvantage from not having used some one of our larger popular textbooks. Many of the diagrams used in illustration are, by permission, from Professor Wells' geometry. In thanking him for this act of courtesy, I desire also to acknowledge my indebtedness to him for valuable aid rendered me through the agency of his text-books, some of which I have had in class-room use from the date of their publication. BUCKNELL UNIVERSITY, LEWISBURG, PA., Aug., 1893. WILLIAM C. BARTOL. THE ELEMENTS OF SOLID GEOMETRY. SECTION I. LINES AND PLANES IN SPACE. DEFINITIONS. 1. A plane is a surface such that, if a straight line be passed through any two of its points, the line will lie wholly in the surface. When a line lies wholly in a plane, it may be said of the plane that it passes through the line. 2. The intersection of two surfaces is a line containing all the points which are common to the two surfaces. 3. The intersection of a line and a plane is the point where the line pierces the plane. This point is called the foot of the line. 4. A line is perpendicular to a plane when it is perpendicular to every line of the plane, passing through its foot. And the plane is then perpendicular to the line. 5. Planes are parallel if they do not meet, however far extended. 6. A line and a plane are parallel if they do not meet, however far extended. 1 PROPOSITION I. 7. THEOREM. Through a straight line an indefinite number of planes may be passed. B Let AB be a straight line lying in a plane (1). Now, we may conceive of the plane as rotating about the line as an axis, and thus occupying successively an indefinite number of positions. But the plane in any position is a plane through the line AB. Hence, through the line AB an indefinite number of planes may be passed. Q. E. D. COROLLARY 1. Through a straight line and a point without it, only one plane may be passed. For if the plane be rotated about the line until it includes the given point, any further rotation, in either direction, will cause the plane to no longer include the given point. 9. SCHOLIUM. Since only one plane may be passed through a straight line and a point without it, a plane is determined by a straight line and a point without it. 10. COROLLARY 2. A plane is determined by three points not in the same straight line. For two of the points may be joined by a straight line, and the plane rotated about the line, as in COR. 1, until the third point is included. 11. COROLLARY 3. A plane is determined by two straight lines which intersect, or by two parallel lines. 12. COROLLARY 4. The intersection of two planes is a straight line. |