« ForrigeFortsett »
(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.
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. 5 Two Theorems are said to be converse, each of the
other, when the hypothesis of each is the conclusion of the other.
If C is D, then A is B, (iii)
If A is not B, then C 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
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. OBS. This rule may be frequently applied with great ad
vantage in the demonstration of the converse of an established Theorem.
THE STRAIGHT LINE.
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
The boundaries of a solid are surfaces. Der. 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. Der. 8. A circle is a plane figure contained by one line,
which is called the circumference, and is such that all
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. DEF. 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. 11. When two straight lines are drawn from the same
point, they are said to contain, or to make with each