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7.

GEOMETRICAL CONSTRUCTIONS.

The drawing of tangents to circles, under various conditions.

8. The inscription and circumscription of figures in and about circles; and of circles in and about figures.

7 and 8 may be deferred till the Straight Line and Triangles have been studied theoretically, but should in all cases precede the study of the Circle in Geometry. The above constructions are to be taught generally, and illustrated by one or more of the following classes of problems: (a) The making of constructions involving various combinations of the above in accordance with general (i.e. not numerical) conditions, and exhibiting some of the more remarkable results of Geometry, such as the circumstances under which more than two straight lines pass through a point, or more than two points lie on a straight line.

(B) The making of the above constructions and combinations of them to scale (but without the protractor). (y) The application of the above constructions to the indirect measurement of distances.

(8) The use of the protractor and scale of chords, and the application of these to the laying off of angles, and the indirect measurement of angles.

SYLLABUS

OF

PLANE GEOMETRY.

INTRODUCTION.

[NOTE. The Association have prefaced their Syllabus by a Logical Introduction, but they do not wish to imply by this that the study of Geometry ought to be preceded by a study of the logical interdependence of associated theorems. They 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.]

I. Propositions admitted without demonstration are

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called Axioms.

Of the Axioms used in Geometry those are termed General which are applicable to magnitudes of all kinds: the following is a list of the general axioms more 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 another.

(d) If equals are added to equals the sums are equal.

(e) If equals are taken from equals the remainders are equal.

(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. 3. A Theorem is the formal statement of a proposition that may be demonstrated from known propositions. These known propositions may themselves be Theorems or Axioms.

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A Theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom.

typical Theorem

If A is B, then C is D, (i)

Thus in the

the hypothesis is that A is B, and the conclusion, that C is D.

From the truth conveyed in this Theorem it necessarily 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.

Thus,

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 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:

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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.

IO. 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 advantage in the demonstration of the converse of an established Theorem.

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