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The first part names or defines the figure to which the theorem relates; the last part contains an additional property. The theorem is first stated in general terms, but in the proof we usually help the mind by a particular figure actually drawn on the page; so that, before beginning the demonstration, the theorem is restated with special reference to the figure to be used.
29. Type. — Beginners in geometry sometimes find it difficult to distinguish clearly between what is assumed and what has to be proved in a theorem.
It has been found to help them here, if the special enunciation of what is given is printed in one kind of type; the special statement of what is required, in another sort of type; and the demonstration, in still another. In the course of the proof, the reason for any step may be indicated in smaller type between that step and the next.
30. When the hypothesis of a theorem is composite, that is, consists of several distinct hypotheses, every theorem formed by interchanging the conclusion and one of the hypotheses is an inverse of the original theorem.
VI. On Proving Inverses.
31. Often in geometry when the inverse, or its equivalent, the obverse, of a theorem, is true, it has to be proved geometrically quite apart from the original theorem. But if we have proved that every x is y, and also that there is but one individual in the class y, then we infer that y is x. The extra-logical proof required to establish an inverse is here contained in the proof that there is but one y.
RULE OF IDENTITY.
32. If it has been proved that x is y, that no two x's are the same y, and that there are as many individuals in class x as in class y, then we infer y is x.
RULE OF INVERSION.
33. If the hypotheses of a group of demonstrated theorems exhaustively divide the universe of discourse into contradictories, so that one must be true, though we do not know which, and the conclusions are also contradictories, then the inverse of every theorem of the group will necessarily be true.
Examples occur in geometry of the following type: -
Three such theorems having been demonstrated geometrically, the inverse of each is always and necessarily true.
Take, for instance, the inverse of the first; namely, when c is greater than d, then a is greater than b.
This must be true; for the second and third theorems imply that if a is not greater than b, then c is not greater than d.
THE PRIMARY CONCEPTS OF GEOMETRY.
I. Definitions of Geometric Magnitudes.
34. Geometry is the science which treats of the properties
35. A part of space occupied by a physical body, or marked out in any other way, is called a Solid.
36. The common boundary of two parts of a solid, or of a solid and the remainder of space, is a Surface.
37. The common boundary of two parts of a surface is a Line.
38. The common boundary of two parts of a line is a Point.
39. A Magnitude is any thing which can be added to itself so as to double.
40. A point has position without magnitude.
41. A line may be conceived of as traced or generated by a point on a moving body. The intersection of two lines is a point.
42. A line on a moving body may generate a surface. The intersection of two surfaces is a line.
43. A surface on a moving body may generate a solid.
44. We cannot picture any motion of a solid which will generate any thing else than a solid.
Thus, in our space experience, we have three steps down from a solid to a point which has no magnitude, or three steps up from a point to a solid; so our space is said to have three dimensions.
45. A Straight Line is a line which pierces space evenly, so that a piece of space from along one side of it will fit any side of any other portion.
46. A Curve is a line no part of which is straight.
47. Take notice: the word “line," unqualified, will henceforth mean “straight line.”
48. A Sect is the part of a line between two definite points.
49. A Plane Surface, or a Plane, is a surface which divides space evenly, so that a piece of space from along one side of it will fit either side of any other portion.
50. A plane is generated by the motion of a line always passing through a fixed point and leaning on a fixed line.
51. A Figure is any definite combination of points, lines, curves, or surfaces.
52. A Plane Figure is in one plane.
53. If a sect turns about one of its end points, the other end point describes a curve which is called a Circle. The fixed end point is called the Center of the Circle.
54. The Radius of a Circle is a sect drawn from the center to the circle.
55. A Diameter of a Circle is a sect drawn through the center, and terminated both ways by the circle.
56. An Arc is a part of a circle.
57. Parallel Lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.
58. When two lines are drawn from the same point, they are said to contain, or to make with each other, an Angle.