(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
which is asserted to follow therefrom. Thus in the
typical Theorem
If A is B, then C is D, ()

the hypothesis is that A is B, and the conclusion,
that C is D.
From the truth conveyed in this Theorem it neces-
sarily follows:

If C is not D, then A is not B, (ii).
Two such Theorems as (i) and (ii) are said to be

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)
is the converse of the typical Theorem (i).
The contrapositive of the last Theorem, viz.:

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



strated, 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 occur-
rence in the elements of Geometry, is of the follow-
ing type:

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.


[ocr errors]





DEF. 1. 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 inter-

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. 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


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
[An angle is a simple concept incapable of definition,
properly so called, but the nature of the concept may
be explained as follows, and for convenience of

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
other, a plane angle. The point is called the vertex,
and the straight lines are called the arms, of the
A line drawn from the vertex and turning about the
vertex in the plane of the angle from the position of
coincidence with one arm to that of coincidence with
the other is said to turn through the angle: and the
angle is greater as the quantity of turning is greater.
Since the line may turn from the one position to the
other in either of two ways, two angles are formed by
two straight lines drawn from a point. These angles
(which have a common vertex and common arms)
are said to be conjugate. The greater of the two is
called the major conjugate, and the smaller the minor
conjugate, angle.
When the angle contained by two lines is spoken of
without qualification, the minor conjugate angle is to

« ForrigeFortsett »