LIST OF PRINCIPAL AUTHORITIES.

EDMUND HALLEY, Apollonii Pergaei Conicorum libri octo et Sereni Antissensis de sectione cylindri et coni libri duo. (Oxford, 1710.)

EDMUND HALLEY, Apollonii Pergaei de Sectione Rationis libri duo, ex Arabico versi. (Oxford, 1706.)

J. L. HEIBERG, Apollonii Pergaei quae Graece exstant cum commentariis antiquis. (Leipzig, 1891-3.)

H. BALSAM, Des Apollonius von Perga sieben Bücher über Kegelschnitte nebst dem durch Halley wieder hergestellten achten Buche deutsch bearbeitet. (Berlin, 1861.)

J. L. HEIBERG, Litterargeschichtliche Studien über Euklid. (Leipzig, 1882.)

J. L. HEIBERG, Euclidis elementa. (Leipzig, 1883–8.)

G. FRIEDLEIN, Procli Diadochi in primum Euclidis elementorum librum commentarii. (Leipzig, 1873.)

J. L. HEIBERG, Quaestiones Archimedeae. (Copenhagen, 1879.)

J. L. HEIBERG, Archimedis opera omnia cum commentariis Eutocii. (Leipzig, 1880-1.)

F. HULTSCH, Pappi Alexandrini collectionis quae supersunt. (Berlin, 1876-8.)

C. A. BRETSCHNEIDER, Die Geometrie und die Geometer vor Euklides. (Leipzig, 1870.)

M. CANTOR, Vorlesungen über Geschichte der Mathematik. (Leipzig, 1880.) H. G. ZEUTHEN, Die Lehre von den Kegelschnitten im Altertum. Deutsche Ausgabe. (Copenhagen, 1886.)

CONTENTS.

INTRODUCTION.

PART I. THE EARLIER HISTORY OF CONIC SECTIONS

AMONG THE GREEKS.

CHAPTER I.

PART II. INTRODUCTION TO THE CONICS OF APOLLONIUS.

THE AUTHOR AND HIS OWN ACCOUNT OF THE

CONICS.

lxviii

.

§ 4.

$5.

CHAPTER IV.

§ 2. The use of coordinates

Method of finding two mean proportionals

Method of constructing normals passing through

a given point

THE CONSTRUCTION OF A CONIC BY MEANS OF

CXV

cxviii

CXXV

cxxvii

.

TANGENTS.

CXXX

PROPOSITIONS LEADING TO THE REFERENCE OF A CONIC

TO ANY NEW DIAMETER AND THE TANGENT AT ITS

EXTREMITY

CONSTRUCTION OF CONICS FROM CERTAIN DATA

ASYMPTOTES

TANGENTS, CONJUGATE DIAMETERS AND AXES.

EXTENSIONS OF PROPOSITIONS 17-19

HARMONIC PROPERTIES OF POLES AND POLARS

INTERCEPTS MADE ON TWO TANGENTS BY A THIRD

FOCAL PROPERTIES OF CENTRAL CONICS

102

109

113

THE LOCUS WITH RESPECT TO THREE LINES ETC.

INTERSECTING CONICS.

119

126

VALUES OF CERTAIN FUNCTIONS OF THE LENGTHS OF

CONJUGATE DIAMETERS

221

INTRODUCTION.

PART I.

THE EARLIER HISTORY OF CONIC SECTIONS
AMONG THE GREEKS.

CHAPTER I.

THE DISCOVERY OF CONIC SECTIONS: MENAECHMUS.

THERE is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius in his commentary on the second Book of Archimedes' treatise On the Sphere and Cylinder*; and the following is a translation of the letter as far as the point where we find mention of Menaechmus, with whom the present subject begins.

"Eratosthenes to King Ptolemy greeting.

"There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,

Too small thy plan to bound a royal tomb.

Let it be double; yet of its fair form

Fail not, but haste to double every side t.

In quotations from Archimedes or the commentaries of Eutocius on his works the references are throughout to Heiberg's edition (Archimedis opera omnia cum commentariis Eutocii. 3 vols. Leipzig, 1880-1). The reference here is III. p. 102.

+

μικρόν γ' ἔλεξας βασιλικοῦ σηκὸν τάφου

διπλάσιος ἔστω· τοῦ καλοῦ δὲ μὴ σφαλείς
δίπλαζ ̓ ἕκαστον κώλον ἐν τάχει τάφου.

Valckenaer (Diatribe de fragm. Eurip.) suggests that the verses are from the

H. C.

b