(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. 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 which is asserted to follow therefrom. Thus in the typical Theorem If A is B, then C is D, (i) If C is not D, then A is not B, (ii) contrapositive, each of the other. Two Theorems are said to be converse, each of the 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 of the hypotheses of a group of demonstrated Theorems it can be said that one must be true, and of the conclusions that no two 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 demonstrated, and the validity of the rule in this instance is obvious from the circumstance that the converse of each of two such Theorems is the contrapositive of the other. Another example, of frequent occurrence in the elements of Geometry, is of the following type : 6 A SYLLABUS OF PLANE GEOMETRY. If A is greater than B, C is greater than D. If A is equal to B, C is equal to D. If A is less than B, C is less than D. Three such Theorems having been demonstrated geometrically, the converse of each is always and necessarily true. Rule of Identity. If there is but one A, and but one B; then from the fact that A is B it necessarily follows that B is A. OBS. This rule may be frequently applied with great ad vantage in the demonstration of the converse of an established Theorem. IO. THE STRAIGHT LINE. DEFINITIONS. DEF. 4. DEF. I. A point has position, but it has no magnitude. DEF. 2. A line has position, and it has length, but neither breadth nor thickness. The extremities of a line are points, and the intersection 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 are made to fall on that other part. Der. 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 straight DEF. 5. 8 A SYLLABUS OF DEF. 9. lines drawn from a certain point within the figure to 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 circum- 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 |