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Apollonius's Conics Book IV
A companion volume to our edition of Conics Books I-III |
The Green Lion announces a first English translation of Book IV of Conics,
translated and annotated by Michael N. Fried, as a companion volume to
our edition of Conics Books I-III. Conics IV deals with the way pairs
of conic sections can intersect or touch each other.
In his Introduction to the translation, Fried shows that this book has
been misappraised by scholars too much inclined to see Apollonius's work
merely as a precursor to the analytic geometry of the seventeenth century.
He writes, "Playfulness is one of the real delights of Book IV. One can
see in this playfulness the artful way Apollonius contends with the main
challenge of the book---the problem of how the opposite sections, specifically,
meet other sections of a cone and other opposite sections--- how he gives
this problem both foundation and context."
7 x 10", 104 pages.
Clothbound Library Binding, ISBN 1888009-20-9, $22.
Also available in a deluxe set with Conics
I-III.
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10% Discount
for Online Purchases
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From Michael Fried's Introduction
In reading and translating Book IV, I have tried to give Apollonius a fair
chance, to keep modern algebraic ideas about conics at a distance, and
to view the text with eyes trained only on the mathematical and philosophical
concerns of Apollonius's contemporaries and on the geometrical character
of the previous three books of the Conics. Approaching the
text this way allows one to see that Book IV, far from being dull, reveals
fundamental difficulties in Apollonius's treatment of conic sections.
First and foremost of these is, of course, the troubling nature of the
opposite sections. But besides that, the book also raises questions
as to Apollonius's basic understanding of how conic sections may be present
and related to one another in a single plane and this understanding
is crucial in reading the whole of the Conics.
. . .
The truth is, both in the letter introducing Book I, which also introduces
the whole of the Conics, and in the letter introducing Book IV itself,
Apollonius tells us quite plainly what Book IV is about: "This book treats
of the greatest number of points at which sections of a cone can meet one
another or meet a circumference of a circle...and, moreover, the greatest
number of points at which a section of a cone or a circumference of a circle
can meet opposite sections. Besides these questions, there are more than
a few others of a similar character." Among the these "few others,"
clearly, are all those concerning the greatest number of points opposite
sections can meet opposite sections (that this is one of the principal
subjects of Book IV is said explicitly in the introduction to Book I).
Now, Apollonius reports that Conon of Samos treated the case where a conic
section or circumference of a circle meets another conic section, but that
Conon's demonstrations were incorrect. Apollonius further reports
that, in connection with Conon's flawed proofs, Nicoteles remarked that
the case in which a conic section meets opposite sections could be solved,
but, as Apollonius makes sure to say, neither Nicoteles nor anyone else
provided a demonstration. As for the greatest number of points in
which opposite sections can meet opposite sections, Apollonius says that
no one has ever noticed this question, let alone treated it. Thus,
as in the introduction to the first book, Apollonius promises his readers
that in this book they can expect a fuller and more rigorous treatment
(and, therefore, to his mind, a more correct treatment) of familiar
questions, but also completely new material, which, as in so much of the
Conics, is precisely that concerning the opposite sections.
The importance of the opposite sections in the Conics cannot
be overemphasized. The existence of opposite sections may have been
known before Apollonius, as the references to Conon and Nicoteles in Apollonius's
prefatory letter to Book IV suggest, however, it is highly doubtful that,
before the Conics, there was anywhere a more than a perfunctory
treatment of them. The opposite sections are peculiar, and, in the
enunciations to propositions in the Conics, Apollonius usually separates
them from the other conic sections. This peculiarity, in part, has
to do with their number, for while there is a sense in which the opposite
sections are one curve, as a visual object they are clearly two.
Thus, for one, like Apollonius, whose work with curves is always governed
by a fundamentally geometric outlook, the plural-singular nature
of the opposite sections makes them an object of fascination, but it also
duly gives rise to a certain uneasiness with them, which one senses already
in the first book of the Conics.
Publication date, November 2002.
Now shipping.
New! Now available in a deluxe set
with Conics I-III.
Clothbound, ISBN 1-888009-20-9, List price $22.00.
10% discount for online orders.
For domestic orders we add a flat shipping charge of $7 regardless of
the number of items. For international shipping we add a flat shipping charge
of $14. International shipments of this book go by air when possible (depends
on country).
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Santa Fe NM 87505
505-983-3675
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