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[NOTE. In the following Introduction are collected together certain general axioms which, though frequently used in Geometry, are not peculiar to that science, and also certain logical relations, the distinct apprehension of which is very desirable in connection with the demonstrations of the Propositions. They are brought together here for convenience of reference, but it is not intended to imply by this that the study of Geometry ought to be preceded by a study of the logical interdependence of associated theorems. The Association think that at first all the steps by which any theorem is demonstrated should be carefully gone through by the student, rather than that its truth should be inferred from the logical rules here laid down. At the same time they strongly recommend an early application of general logical principles.]
1. Propositions admitted without demonstration are called.
2. Of the Axioms used in Geometry those are termed General which are applicable to magnitudes of all kinds: the following is a list of certain general axioms frequently used.
(a) The whole is greater than its part.
(b) The whole is equal to the sum of its parts.
(c) Things that are equal to the same thing are equal to one
(d) If equals are added to equals the sums are equal.
(e) If equals are taken from equals the remainders are
(ƒ) If equals are added to unequals the sums are unequal, the greater sum being that which is obtained from the greater magnitude.
(g) 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 a proposition enunciating a fact whose truth is demonstrated from known propositions.
These known propositions may themselves be Theorems or Axioms.
4. The enunciation of 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),
the hypothesis is that A is B, and the conclusion, that C is D.
From this Theorem it necessarily follows that:
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).
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
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.
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:
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 the converse of each is necessarily true.
10. 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.
This rule may be frequently applied with great advantage in the demonstration of the converse of an established Theorem.
THE ELEMENTS OF PLANE GEOMETRY.
THE STRAIGHT LINE.
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 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 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 extremities 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.