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OF

PLANE GEOMETRY,

ON THE HEURISTIC PLAN,

WITH NUMEROUS EXTRA EXERCISES, BOTH
T'HEOREMS AND PROBLEMS, FOR

ADVANCE WORK.

BY

G. IRVING HOPKINS,

INSTRUCTOR IN MATHEMATICS AND PHYSICS IN High

SCHOOL, MANCHESTER, N.H.

BOSTON, U.S. A.:
D. C. HEATH & COMPANY.

1891.

COPYRIGHT, 1891,
By G. I. HOPKINS.

TYPOGRAPHY BY J. S. CUSHING & Co., Boston, U.8.A.
PRESSWORK BY BERWICK & SMITH, Boston, U.S.A.

PREFACE.

THIs book is published primarily for the author's pupils, and secondarily for that constantly increasing number of teachers who are getting more and more dissatisfied with the old methods of teaching geometry, but who have hitherto found no manual suited to their needs. That the reasoning faculties of a child or youth are developed by use, goes without saying; and consequently the method of teaching geometry whereby the pupil originates the demonstration, or simply demonstrates, rather than memorizes the demonstrations of another, needs no defence at my hands. This manual has stood the test of three years' work in the class-room, and differs from other geometries that provide for original work, in that the original demonstrations and constructions are not side issues, so to speak, but are required of the pupil in that sequence of theorems and problems which may be called the regular required work, and of which complete demonstrations are usually given in other manuals for the pupil to memorize. In this work demonstrations are given only where the average pupil would be at a loss to know how to proceed, and generally, as illustrative of methods, while others are partially given and left for the pupil to complete. In other cases, wherever the author has found that his own pupils were not working to advantage, he has introduced suggestions, of which pupils may or may not avail themselves.

iii 18:26 GS

The old method of division into books has been abandoned, as it serves no practical use, since different authors have made different divisions. It serves to make the subject a continuous one in the mind of the pupil, without the artificial breaks of the old way.

The arrangement of the theorems whereby the essential ones, together with several simple additional ones for giving the pupil more drill, followed by a set of non-essential but more difficult ones for advance work, the author has found advantageous.

He has also found it more useful to the pupil to be compelled to construct his own diagram, and state the converse of the theorems he is to prove. Practical problems of computation have also been given, immediately following any given subject, so that the pupil can immediately see the practical application of the theorems he has demonstrated.

All the problems of construction have been placed together 1 after the theorems, for the sake of uniformity, as they form no part of the logical sequence of geometrical truths as embodied in the theorems. Many of them are of practical, as well as disciplinary, importance, however, and so it is left to the skilful and judicious teacher to take them up as the interests of his pupils demand.

The author has received valuable suggestions from Professor E. L. Richards of Yale College, and also from Professor T. H. Safford of Williams College. He also desires to publicly express his indebtedness to Hon. J. W. Patterson, Superintendent of Public Instruction for New Hampshire, and E. R. Goodwin, Principal of the High School at Lawrence, Mass., for their hearty and outspoken indorsement of his

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work, and encouragement to persevere in elaborating the method. Finally, thanks are due the publishers and printers for the excellence and beauty of the mechanical work. In conclusion, the author would be glad to receive suggestions from those teachers into whose hands this manual may chance to fall, with a view to its improvement as a regular

class text-book.
G. I. H.

MANCHESTER, N.H.,
May, 1891.

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