History of Analytic Geometry

367 Pages·2012·2.8 MB·English

by  Boyer

Bibliographical Note This Dover edition, first published in 2004, is an
unabridged republication of the work originally published by Yeshiva
University, New York, in 1956 as numbers six and seven of the Scripta
Mathematica Studies.
9780486154510

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Table of Contents
Title Page
Copyright Page
Preface
CHAPTER I - The Earliest Contributions
CHAPTER II - The Alexandrian Age
CHAPTER III - The Medieval Period
CHAPTER IV - The Early Modern Prelude
CHAPTER V - Fermat and Descartes,
CHAPTER VI - The Age of Commentaries
CHAPTER VII - From Newton to Euler
CHAPTER VIII - The Definitive Formulation
CHAPTER IX - The Golden Age
ANALYTICAL BIBLIOGRAPHY
Index
Preface
THE history of analytic geometry is by no means an uncharted sea. Every
history of mathematics touches upon it to some extent; and numerous scholarly
papers have been devoted to special aspects of the subject. What is chiefly
wanting is an integrated survey of the historical development of analytic
geometry as a whole. The closest approach to such a treatment is found in two
articles by Gino Loria. One of these is in Italian and appeared in 1924 in a
periodical (the Memorie of the Reale Accademia dei Lincei for 1923) which is
not easily accessible; the other is in French and was published, in several
installments, from 1942 to 1945, in a Roumanian journal (Mathematica) which
is still less readily available. These two articles together constitute perhaps the
most extensive and dependable account of the history of analytic geometry.
Somewhat less inclusive treatments are found in German as parts of the works of
Heinrich Wieleitner (Geschichte der Mathematik, part II, vol. 2, 1921) and
Johannes Tropfke Geschichte der Elementar-Mathematik, 2nd ed., vol. VI,
1924). Had convenient translations of any of the above works been at hand—or
had J. L. Coolidge collated and amplified those portions of his admirable History
of Geometric Methods (1940) which pertain to analytic geometry—the present
work might never have been written. As it was, there seemed to be room for an
historical volume of modest size devoted solely to coordinate geometry. It soon
became evident that, in view of the amount of material available, some
limitations would have to be imposed if the work were to remain within
reasonable proportions. Not all relevant details could be included, for H. G.
Zeuthen (in Die Lehre von den Kegelschnitte im Altertum, 1886) had devoted
more than 500 pages to one specific aspect of a single chronological subdivision.
It was therefore decided that the present history should cover only such parts of
analytic geometry as might reasonably be included in an elementary general
college course. Consequently developments of the last hundred years or so are
largely omitted, for they are of a more advanced and highly specialized nature.
Even within this self-imposed limitation, the account is not intended to be
exhaustive with respect to detail. Factual information is presented largely to the
extent to which it is suggestive of the general development of ideas.
Biographical details in the main have been overlooked, not for want of
attractiveness, but because often they have little bearing upon the growth of
concepts. For similar reasons peculiarities of terminology and notation have
been accorded very limited space. Some attention has been given to the status of
analytic geometry vis-à-vis other branches of mathematics; but the impact of the
wider intellectual milieu has been referred to only where it was regarded as of
particular significance. It is of interest to note in this connection that the
development of coordinate geometry was not to any great extent bound up with
general philosophical problems. The discoveries of Descartes and Fermat in
particular are relatively free of any metaphysical background. Indeed, La
géométrie was in many respects an isolated episode in the career of Descartes—
one suggested by a classical problem of Greek geometry. It was the natural
outcome of historical tendencies; and had Descartes not lived, mathematical
history—in sharp contrast to philosophical—probably would have been much
the same, by virtue of Fermat’s simultaneous discovery. The work of Fermat is
practically devoid of philosophical interest, his discoveries being the result of a
close study of the achievements of his predecessors. Perhaps nowhere does one
find a better example of the value of historical knowledge for mathematicians
than in the case of Fermat, for it is safe to say that, had he not been intimately
acquainted with the geometry of Apollonius and Viète, he would not have
invented analytic geometry.
It is frequently held that mathematics develops most effectively when it is
closely associated with the world of practical affairs—when scholars and
artisans work together. However, to this general rule there seem to be more
exceptions than there are instances of it; and the discovery of analytic geometry
certainly seems to be one of the exceptions. For this reason the sociological
background has, in the present account, gone unemphasized. On the other hand,
bibliographical references to the source material have been granted a place of
some prominence in order to enable the reader to pursue the subject further in
directions which he may find especially attractive. Not all works cited in the
footnotes are included in the “Analytical Bibliography,” with which the volume
closes. It is felt that the usefulness of the bibliography is enhanced by limiting it
to items which are directly pertinent to the history of algebraic geometry and by
incorporating in each case a very brief indication of the nature of the material.
A conscientious effort has been made to see that the information presented is
substantially correct in detail, although perfect accuracy in this respect is rarely
achieved. However, it is the broad general picture which represents the principal
object of the book. Here there undoubtedly are further points which the reader
could wish to have seen included, but it is hoped that there are few portions of
could wish to have seen included, but it is hoped that there are few portions of
the work which he would prefer deleted. For both inspiration and information
the author is heavily indebted to the works of Loria and Wieleitner, and a special
measure of credit is due also to the books of Coolidge and Tropfke. To all the
scholars whose studies have served as the basis for the present volume, the
author would express his appreciation.
The manuscript of this work was completed about half a dozen years ago, and
major portions of it have appeared from time to time in Scripta Mathematica.
The bibliography contains a few items published since the completion of the
manuscript, but in most cases it was not feasible to make use of these in the body
of the work. There have, however, been few recent developments which would
lead one to alter materially judgments on the history of analytic geometry
expressed some six years ago.
The appearance of this book is due to the suggestion and encouragement of
Professor Jekuthiel Ginsburg of Yeshiva University, and to him, for continued
inspiration and assistance in the completion of the project, we extend our
warmest gratitude.
January 3, 1956
CARL B. BOYER
CHAPTER I
The Earliest Contributions
Mighty are numbers, joined with art resistless.
—EURIPIDES

MATHEMATICS originally was the science of number and magnitude. At first
it was limited to the natural numbers and rectilinear configurations; but even
from the early primitive stages mankind presumably was concerned with the
problem from which analytic geometry arose—the correlation of number with
geometrical magnitude. The beginnings of the association of numerical
relationships with spatial configurations are prehistoric, as are also the first
connections between number and time. The harpedonaptae (“rope-stretchers” or
surveyors) of Egypt and the astronomers of Chaldea bear witness to the early
concern of mathematics with such associations. The very oldest written
documents from Mesopotamia, Egypt, China, and India give evidence of the
concern with mensuration. Pre-Hellenic papyri and cuneiform texts abound in
elaborate problems involving the concepts of length, area, and volume.1 So
highly developed was this aspect of the Egyptian and Babylonian civilizations
that one finds there, among other things, the correct result for the volume of the
frustum of a pyramid with a square base.
It is indeed possible to have an analytic geometry of points and straight lines
alone, a direction toward which ancient mensuration pointed; but historically the
subject arose instead from the comparison of curvilinear with rectilinear
magnitudes. Here also the Egyptians and Babylonians, in their geometry of the
circle, took the first steps. The former made a remarkably accurate estimate of
the ratio of the area of the circle to the area of the square on the diameter, taking
this ratio to be (1− )2, equivalent to taking a value of about 3.16 for π. The
Babylonians adopted the cruder approximation 3 for π (although an instance is
known in which the value is taken as 3 ), but their geometry of the circle
nevertheless surpassed that of the contemporary Egyptians. They recognized that
the angle inscribed in a semicircle is right, anticipating Thales by well over a
thousand years. Moreover, they were familiar at about the same time with the
Pythagorean theorem. Combining these two famous propositions, they found—
for a given circle of radius r—the relationship between the length of a chord c
and its sagitta s. This property, when symbolically expressed in such a form as
4r2 = c2 + 4(r — s)2, may in a sense be regarded as an equation of the circle in
terms of the rectangular coordinates c and s. The Babylonians never reached this
point of view, for such essential elements of analytic geometry as coordinates
and equations of curves arose considerably later; but it is well to bear in mind
how closely certain aspects of ancient mathematics approach their modern
counterparts. Primitive systems of coordinates were used by Nilotic surveyors as
early as 1400 B.C., and probably also by Mesopotamian star-gazers;2 but there is
no evidence that Egyptian or Babylonian geometers ever explicitly developed a
formal geometric coordinate system.
The nascent state of the idea of coordinates was not the only difficulty in the
way of the development of analytic geometry. Deficiencies in arithmetic were
possibly just as serious. The systems of numeration used in the Nile and
Mesopotamian valleys were not so well adapted to calculation as is ours. The
hieratic script of the Egyptians made use of the principle of cipherization in
connection with the ten-scale, but did not apply the idea of local value or
position; the Babylonian sexagesimal notation, on the other hand, employed the
positional principle, but cipherization was impracticable in conjunction with a
base or radix of that size. Granted that these systems of numeration were
imperfect, it is nevertheless open to doubt that difficulties in methods of
computation operated as seriously to obstruct the growth of algebra as did other
factors. After all, the Babylonians calculated the diagonal of a square to the
equivalent of half a dozen decimal figures! The shortcomings were probably
more in number concepts than in number symbols. Algebra calls for a higher
degree of abstraction than does geometry, and this element seems to have been
lacking in pre-Hellenic mathematics. Number referred essentially to concrete
whole numbers, and the idea of general fractions was missing in Egyptian
writings. Much time was spent in finding ways of avoiding all but unit fractions,
so that the ratio of 2 to 43 would be written as or as
. Whether the Babylonians achieved the concept of general
rational number is open to question because of ambiguities in the interpretation
of tables, the use of which was greatly emphasized. Elaborate tables give pairs of