(c) Things that are equal to the same thing are equal to one another. (d) If equals are added to equals the sums are equal. (e) If equals are taken from equals the remainders are equal. (f) If equals are added to unequals the sums are unequal, the greater sum being that which is obtained from the greater magnitude. (8) If equals are taken from unequals the remainders are unequal, the greater remainder being that which is obtained from the greater magnitude. (h) The halves of equals are equal. 3. A Theorem is the formal statement of a proposition that may be demonstrated from known propositions. These known propositions may themselves be Theo rems or Axioms. 4. A Theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that i If C is not D, then A is not B, (ii). contrapositive, each of the other. 5 Two Theorems are said to be converse, each of the other, when the hypothesis of each is the conclusion of the other. Thus, If C is D, then A is B, (iii) If A is not B, then C is not D, (iv) is termed the obverse of the typical Theorem (i). 6. Sometimes the hypothesis of a Theorem is complex, i.e. consists of several distinct hypotheses; in this case every Theorem formed by interchanging the conclusion and one of the hypotheses is a converse of the original Theorem. 7. The truth of a converse is not a logical consequence of the truth of the original Theorem, but requires independent investigation. 8. Hence the four associated Theorems (i) (ii) (iii) (iv) resolve themselves into two Theorems that are independent of one another, and two others that are always and necessarily true if the former are true; consequently it will never be necessary to demonstrate geometrically more than two of the four Theorems, care being taken that the two selected are not contrapositive each of the other. 9. Rule of Conversion. If the hypotheses of a group of demonstrated Theorems be exhaustive—that is, form a set of alternatives of which one must be true, and if the conclusions be mutually exclusive—that is be such that no two of them can be true at the same time, then the converse of every Theorem of the group will necessarily be true. OBs. The simplest example of such a group is presented when a Theorem and its obverse have been demon 6 A SYLLABUS OF PLANE GEOMETRY. strated, and the validity of the rule in this instance If A is greater than B, C is greater than D. If A is less than B, C is less than D. follows that B is A. vantage in the demonstration of the converse of an IO. BOOK I. THE STRAIGHT LINE. DEFINITIONS. a DEF. 1. A point has position, but it has no magnitude. breadth nor thickness. section of two lines is a point. DEF. 3. A surface has position, and it has length and breadth, but not thickness. The boundaries of a surface, and the intersection of two surfaces, are lines. DEF. 4. A solid has position, and it has length, breadth, and thickness. The boundaries of a solid are surfaces. DEF. 5. A straight line is such that any part will, however placed, lie wholly on any other part, if its extremi ties are made to fall on that other part. Def. 6. A plane surface, or plane, is a surface in which any two points being taken the straight line that joins them lies wholly in that surface. DEF. 7. A plane figure is a portion of a plane surface enclosed by a line or lines. DEF. 8. A circle is a plane figure contained by one line, which is called the circumference, and is such that all a straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle. Der. 9. A radius of a circle is a straight line drawn from the centre to the circumference. DEF. 10. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the reference it may be reckoned among the definitions.) DEF. II. When two straight lines are drawn from the same point, they are said to contain, or to make with each |