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and / BA C, respectively, equal D E, D F, and L EDF.

To prove: The triangles A B C and D E F equal.

III. The Construction, which consists of the drawing of aid lines, superposition of figures, etc. Here the authority for the work should be shown to rest upon the geometric postulates which are discussed in the text. In problems the Construction is given a prominent place. At all times the work should be done with the greatest care and accuracy.

IV. The Demonstration, which is shown to rest solidly upon definitions, axioms, and previously proved theorems and problems previously constructed and proved.

It is of vital importance that the pupil fully understand that the truth set to be proved in the Particular Enunciation is not established until the very best authority has been given, and then the pupil should be led to see clearly just how the conclusion of the General Enunciation follows the proof of the Particular Enunciation. The pupil must not be permitted to conclude that his work in Constructional Geometry is useless because he can not use it as authority for his present work. The definitions and axioms there given are authority in the present work. (The fundamental notions there developed will be of great value in the work in hand.) Truths which are there proved and are shown to depend solely upon the definition for authority or which followed from the application of axioms are in full force here. But he must understand once for all that the further truths which he discovered and carefully tested with instruments must be now formally established by the strictest of logical reasoning. He has studied Practical Geometry.

He can be shown that the fundamental notions there developed will be of the greatest value in the present science of reasoning, which deals with the truths there discov. ered and practically used.

When the study of Concrete Geometry, as recommended by the Committee of Ten, has been widely introduced into the grammar grades of our schools, this work can be begun in the beginning of the High School.

SYMBOLS AND ABBREVIATIONS.

2 = angle.
Ls= angles.
A = triangle.
As= triangles.

= rectangle. Os= rectangles.

circle.
circles.

= arc.

=

Cor. corollary.
Iden. identical.
Rt. right.
Ex. = exercise.
Auth.= give authority=why?
Hyp. = hypothesis.
St. = straight
Pt. point.
Pis.

= points.
P. P.= previous proposition
Prop.= proposition.
Def. definition.
Sug. = suggestion.
Sup. = supplementary.
Adj. : adjacent.
Ext. = exterior.

. Int. interior. Alt. = alternate. Ext.-int. = exterior-interior. Alt.-int. = alternate-interior. Q. E. D. = quod erat demon

strandumwhich was to

be proved. Q. E. F. =quod erat faciendumwhich was

to be done.

S =

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S = arcs.

perpendicular.
Is= perpendiculars.
11 parallel.
Il s parallels.

parallelogram.
parallelograms.
is, or are, greater than.
is, or are, less than.
therefore.
since, or because.
approaches as a limit.
foot or feet.

inch or inches.
Isos. isosceles.
Ax. = axiom.

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