Plane and Solid Analytic Geometry: An Elementary Textbook

302 Pages·1905·15.29 MB·English

by  Charles Hamilton Ashton

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PLANE AND SOLID
ANALYTIC GEOMETRY
AN ELEMENTARY TEXT-BOOK
BY
CHARLES H. ASHTON, A.M.
ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY
OF KANSAS
REVISED EDITION
UNIVERSITY
iOf^TON
COLLEGE OF LiSERAL ARTS
yBKARY
NEW YORK
CHARLES SCRIBNER'S SONS
1905
nnaVaviv^.
COPYRIGHT, 1900, BY
CHARLES SCRIBNER'S SONS
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PREFACE
The present work is intended as a text-book for class
room use, and not as an exhaustive treatise on the sub-
ject. This object has been kept constantly in mind in
writing the book, and every subject has been treated from
A
this point of view. large part of the book was mimeo-
graphed and tested by use for several years in the author's
classes in Harvard University.
The author has tried to meet the needs of a class which
occupies from sixty to seventy recitation hours upon the
subject, and it is thought that the book ought to be com-
pleted by the average class in that time. Necessarily
some subjects which usually find a place in books on
Analytic Geometry have been omitted ; but it is thought
that nothing has been omitted which has an important
bearing on future mathematical study.
The conies have been treated from their ratio defini-
tion, and much space and time have been gained by not
repeating proofs wliich are identical, or very similar, for
the three forms of the conic. Analytic methods are used
throughout the book, and the author has attempted to
give proofs whicli are concise and easily understood by
the average student, but, at the same time, mathemati-
cally rigorous. In this connection he would call atten-
tion to the proofs in oblique coordinates (Arts. 12, 28),
which are usually given without reference to the direc-
tions of the lines, and, therefore, do not liohl if tlie
positions of the points are changed.
PREFACE
yi
Numerous problems, which have been selected with
great care, have been inserted after nearly every article.
In the early part of the book these are mainly numerical,
but later the student has been asked to prove a large
number of theorems. A considerable number of theorems
which are usually proved in the text are here inserted
as problems, and in many places the student has been
asked to derive formulas for two of the conies, after the
corresponding formulas for one of the conies have been
obtained. It is only by solving such problems that the
student can acquire any real grasp of the subject.
The attention of the teacher is also called to the two
chapters on loci (Chaps. VIII and XIV), in which a large
number of problems are given and the methods of solving
them discussed to the treatment of poles and polars by
;
the aid of harmonic division (Chap. XII); and to the
system of polar cocirdinates used in the Solid Geometry.
The author desires to acknowledge gratefully the assist-
ance of Mr. B. E. Carter of the Massachusetts Institute
of Technology, who has read with great care both manu-
script and proof of jNIr. E. Y. Huntington, who made
;
many valuable suggestions during the early stages of
the work and of Mr. W. R. Marsh, his colleague in the
;
preparation of the series, of which this volume is the first
to appear.
Cambridge,
November, 1900.