Plane and solid geometry

504 Pages·1925·69.723 MB·English

by  Arthur Schultze and Frank L. Sevenoak, revised by Elmer Schuyler

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1925

PLANE AND SOLID GEOMETRY
THE MACMILLAN COMPANY
NEWYORK•BOSTON•CHICAGO•DALLAS
ATLANTA•SANFRANCISCO
MACMILLAN & CO., LIMITED
LONDON· BOMBAY•CALCUTTA
MELBOURNE
THEMACMILLAN CO. OF CANADA , LTD.
TORONTO
PLANE AND SOLID
GEOMETRY
BY
ARTHUR SCHULTZE, PH.D.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS , NEW YORK
UNIVERSITY , AND HEAD OF THE MATHEMATICAL DEPART
MENT, HIGH SCHOOL OF COMMERCE,NEW YORK CITY
AND
FRANK . SEVENOAK
L
FORMERLY PRINCIPAL OF THE ACADEMIC DEPARTMENT
STEVENS INSTITUTE OF TECHNOLOGY
REVISED BY
ELMER SCHUYLER
HEAD OF THE DEPARTMENT OF MATHEMATICS
BAY RIDGE HIGH SCHOOL, BROOKLYN , NEW YORK
; viss
New York
THE MACMILLAN COMPANY
1925
All rights reserved
PRINTEDIN THEUNITEDSTATESOF AMERICA
COPYRIGHT, 1901, 1913,1925,
BY THE MACMILLAN COMPANY.
Setupandelectrotyped. PublishedMay,192.5
Norwood Press
J. S. Cushing Co. —Berwick &Smith Co.
Norwood,Mass., U.S.A.
1.Семии.
2-4-43
PREFACE TO THE FIRST EDITION
It is generally conceded that the final aim ofmathemati
cal teaching should benot only the acquisition ofpractical
knowledge but that training of the studen't s mind which
givesa distinct gain inmental power. In recognition ofthis
principle nearly all college entrance examinations n geometry
i
require some original work,and most textbooks devote con
siderable space to exercises. Comparatively little,however,
hasbeen done to introduce the student systematically toorigi
nal geometrical work. No teacher of physics or chemistry
would ask a student todiscover a law without soguiding his
work as to enable him toreach the desired result;many text
books and teachers expect the pupil to invent geometrical
proofs and to solve problems,entirely new to him,without
offering any assistancefurther than aknowledge ofthe well
established theorems ofall textbooks. Some writers give a
description of the analysis ofpropositions,which is entirely
logical and ofgreat advantage toaperson ofsomemathemati
calknowledge,butwhich isusually too abstract to be ofany
practicalvalue to thebeginner. In this book the attempt is
made o introduce the student systematically othe solution
t t
ofgeometrical exercises. Inthe beginning theexercises given
inacertain group are ofsimilar kind and related to the pre
ceding proposition; later some general principles are devel
oped which are offundamental importance fororiginal work,
r
as, foexample,the method ofproving the equality oflines
bymeans ofequal triangles; themethod ofproving thepro
portionality oflines bymeans ofsimilar triangles,etc.;and
vi PREFACE TO THE FIRST EDITION
finally the analyses oftheorems and problemsare introduced ,
but in amore concrete form than usual.
The propositions are arranged with the view to obtaining
a perfect logical and pedagogical order. An unusually large
number of exercises is given , selected with care for the pur
pose of securing increased mentalpower.
The general plan and the preparation of the greater part
of the book are the work of Dr. Schultze, while that of Dr.
Sevenoak has been chiefly editorial.
PREFACE THE FIRST REVISED
TO
EDITION
The main purpose of the revision of this book has been to
emphasize still further and to elaborate in greater detail the
principalaim of the original edition, viz.,tointroduce the stu
dent systematically to original geometric work. Tomake the
teaching ofgeometry both disciplinary and informational; to
give to the student mental training instead ofteaching him
mere facts; to develop his power instead ofmaking him
memorize,— these are the fundamental aims ofthis book.
Themeans employed for this purpose are similar tothose
used inthe first edition. Still greateremphasis,howeve,r has
been placed upon the generalmethods which may beused for
the solution oforiginal exercises. The grading and theselec
tion ofexercises have been carefully revised. All originals
that appeared unfit or too difficult have been eliminated or
y
replaced bsimpler and better ones. Topics offundamental
importance,e.g.themethods ofdemonstrating the equality
oflines,are represented n greater detail and illustrated by
i
a greater number ofexercises than in the first edition.
In addition tothese fundamental tendencies,a number of
minor improvements have been introduced, among which
may bementioned: