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positive, and vice versa, and hence if either is proved the other is proved also.
6o. Two theorems are converse to one another when the hypothesis and conclusion of the one are respectively the conclusion and hypothesis of the other.
Ex. If an animal is a horse it has four legs.
As is readily seen from the foregoing example, the truth of a theorem does not necessarily establish the truth of its converse, and hence a theorem and its converse have in general to be proved separately. But on account of the peculiar relation existing between the two, a relation exists also between the modes of proof for the two. These are known as the direct and indirect modes of proof. And if any theorem which admits of a converse can be proved directly its converse can usually be proved indirectly. Examples will occur hereafter.
7°. Many geometric theorems are so connected with their converses that the truth of the theorems establishes that of the converses, and vice versa.
The necessary connection is expressed in the Rule of Identity, its statement being :
If there is but one X and one Y, and if it is proved
that X is Y, then it follows that Y is X. Where X and Y stand for phrases such as may form the hypotheses or conclusions of theorems, and the “is” between them is to be variously interpreted as “equal to," sponds to," etc.
Ex. Of two sides of a triangle only one can be the greater, and of the two angles opposite these sides only one can be the greater. Then, if it is proved that the greater side is opposite the greater angle it follows that the greater angle is opposite the greater side.
In this example there is but one X (the greater side) and one Y (the greater angle), and as X is (corresponds to or is opposite) Y, therefore Y is (corresponds to or is opposite) X.
8o. A Corollary is a theorem deduced from some other theorem, usually by some qualification or restriction, and occasionally by some amplification of the hypothesis. Or a corollary may be derived directly from an axiom or from a definition.
As a matter of course no sharp distinction can be drawn between theorems and corollaries.
Ex. From the theorem, “ The product of two odd numbers is an odd number,” by making the two numbers equal we obtain as a corollary, “The square of an odd number is an odd number.”
State the contrapositives and the converses of the following theorems :
1. The sum of two odd numbers is an even number. 2. A diameter is the longest chord in a circle. 3. Parallel lines never meet. 4. Every point equidistant from the end-points of a line
segment is on the right bisector of that segment.
THE LINE AND POINT.
9°. Space may be defined to be that which admits of length or distance in every direction; so that length and direction are fundamental ideas in studying the geometric properties of space.
Every material object exists in, and is surrounded by space. The limit which separates a material object from the space which surrounds it, or which separates the space occupied by the object from the space not occupied by it, is surface.
The surface of a black-board is the limit which separates the black-board from the space lying without it. This surface can have no thickness, as in such a case it would include a part of the board or of the space without or of both, and would not be the dividing limit.
10°. A flat surface, as that of a black-board, is a plane surface, or a Plane.
Pictures of geometric relations drawn on a plane surface as that of a black-board are usually called Plane Geometric Figures, because these figures lie in or on a plane.
Some such figures are known to every person under such names as “triangle,” “square," "circle,” etc.
n°. That part of mathematics which treats of the properties and relations of plane geometric figures is Plane Geometry. Such is the subject of this work.
The plane upon which the figures are supposed to lie will be referred to as the plane, and unless otherwise stated all figures will be supposed to lie in or on the same plane.
12°. The Line. When the crayon is drawn along the black-board it leaves a visible mark. This mark has breadth and occupies some of the surface upon which it is drawn, and by way of distinction is called a physical line. By continually diminishing the breadth of the physical line we make it approximate to the geometric line. Hence we may consider the geometric line as being the limit towards which a physical line approaches as its breadth is continually diminished. We may consequently consider a geometric line as length abstracted from every other consideration.
This theoretic relation of a geometric line to a physical one is of some importance, as whatever is true for the physical line, independently of its breadth, is true for the geometric line. And hence arguments in regard to geometric lines may be replaced by arguments in regard to physical lines, if from such arguments we exclude everything that would involve the idea of breadth.
The diagrams employed to direct and assist us in geometric investigations are formed of physical lines, but they may equally well be supposed to be formed of threads, wires or light rods, if we do not involve in our arguments any idea of the breadth or thickness of the lines, threads, wires or rods employed.
In the practical applications of Geometry the diagrams frequently become material or represent material objects. Thus in Mechanics we consider such things as levers, wedges, wheels, cords, etc., and our diagrams become representations of these things.
A pulley or wheel becomes a circle, its arms become radii of the circle, and its centre the centre of the circle ; stretched cords become straight lines, etc.
13°. The Point. A point marks position, but has no size. The intersection of one line by another gives a point, called the point of intersection.
If the lines are physical, the point is physical and has some size, but when the lines are geometric the point is also geometric.
14°. Straight Line. For want of a better definition we may say that a straight line is one of which every part has the same direction. For every part of a line must have some direction, and when this direction is common to all the parts of the line, the line is straight.
The word “ direction” is not in itself definable, and when applied to a line in the absolute it is not intelligible. But every person knows what is meant by such expressions as “the same direction,” “opposite direction," etc., for these express relations between directions, and such relations are as readily comprehended as relations between lengths or other magnitudes.
The most prominent property, and in fact the distinctive property of a straight line, is the absolute sameness which characterizes all its parts, so that two portions of the same straight line can differ from one another in no respect except in length.
Def.-A plane figure made up of straight lines only is called a rectilinear figure.
15°. A Curve is a line of which no part is straight; or a curve is a line of which no two adjacent parts have the same direction.
The most common example of a curve is a circle or portion of a circle.
Henceforward, the word “line,” unless otherwise qualified, will mean a straight line.
16o. The “rule” or “straight-edge” is a strip of wood, metal, or other solid with one edge made straight. Its common use is to guide the pen or pencil in drawing lines in Practical Geometry.
17o. A Plane is a surface such that the line joining any two arbitrary points in it coincides wholly with the surface.
The planarity of a surface may be tested by applying the rule to it. If the rule touches the surface at some points and not at others the surface is not a plane. But if the rule touches the surface throughout its whole length, and in every position and direction in which it can be applied, the surface is a plane.
The most accurately plane artificial surface known is probably that of a well-formed plane mirror. Examina