Subject: [HM] The History of Horn Angles [4/4]
From: Ken Pringle (kenneth.pringle@studentmail.newcastle.edu.au)
Date: Tue Dec 21 1999 - 21:28:12 EST
( c ) Possible near-identity of Eudoxean and Menaechmean solutions.
The deliberations invite conclusion that the Eudoxean solution of the
problem of finding the mean proportionals must have been the same as
that traditionally called the Menaechmean. Perhaps Eudoxus provided a
two-parabola solution and Menaechmus subsequently one by means of either
parabola and the hyperbola. From the account of Menaechmus's solution in
Eutocius's anthology of the cube duplication we form the opinion that
Menaechmus definitely employed the intersection of a parabola and a
hyperbola. The accompanying two-parabola solution probably is not from
Menaechmus. On textual grounds (including considerations of the figures
used, as to which Knorr 1989, pp.96-7) the expositions by analysis,
which alone seem to be pertinent, probably come from different authors.
Menaechmus though must have been aware of the possibility of solution by
intersection of two parabolas. Otherwise we should not encounter
reference to the 'triads of Menaechmus'. He may have known some form of
two-parabola solution coming from a predecessor.
Proclus's remark concerning Eudoxus (Morrow op. cit., p.67), that he
'multiplied the number of the propositions concerning the "section"
which had their origin in Plato, employing the method of analysis for
their solution', has sometimes been interpreted as referring to
contribution to study of the conic sections. The reference to solution
by analysis is in keeping with a two-parabola solution of the problem of
the mean proportionals. The presumptive ultimate source for the remark
is Eudemus's history of geometry. The ascription to Plato probably is an
embellishment. If Eudemus stated that Eudoxus extended some part of the
theory of the conic sections, it is implied that someone preceded him in
the discipline. In this connection we have no stronger indication than
that pointing to Democritus.
( d ) A possible background to early stereometric consideration of the
conic curves.
Speculatively it is possible to provide a plausible background to
supposed involvement of Democritus in the genesis of study of the conic
sections. The conic sections as sections of the right circular cone
might have been noticed first in connection with the shapes of shadows
of a spherical globe as cast onto a flat surface by a light source in
various relative positions. The shapes or appearances of shadows likely
would be studied in connection with scene painting. We know from a
passage in Vitruvius's treatise De Architectura (VII. praefatio 11) that
Democritus of Abdera discussed problems arising in connection with scene
painting. There was developed in the second half of the fifth century B.
C. a technique called skiagraphia, literally "shadow painting". It seems
to have combined shadow effects and perspective (Keuls 1978, pp.72-5).
Anciently there was a legend that painting began with the drawing of
outlines around cast shadows (ibid., pp.75-6). Certainly, satisfactory
depiction depends on understanding the disposition of shadow for given
objects with a given disposition of light sources.
The case of the illuminated sphere lends itself readily to idealisation.
The scene painter comes to recognise that a sphere as distinct from a
circular representation of it appears circular in outline from whichever
vantage point viewed. This gives by idealisation using the notion of
visual rays the fundamental observation underlyng the use of focal
spheres in reasoning about the conic sections. Parmenides B 8.42-9
admits at least one interpretation (Mourelatos 1970, pp.120-30)
involving the notion of perspectival invariance of the sphere in the
sphere reference of the passage. Diogenes Laertius (Lives IX. 41)
asserts that Democritus somewhere alluded to the doctrine of the One
held by Parmenides and Zeno. We may suppose on the basis of this (though
anachronistic interpretation may have been involved) that Democritus
knew the writing of Parmenides. The notion of perspectival invariance of
the sphere thus may have been the more present to mind by reason of
Democritus having noted the sphere reference in the poem.
Conclusion.
The discussion brings into an order the separate testimonies that have
come down concerning the alternative titles of a writing of Democritus
and the solutions that issued from Eudoxus and Menaechmus of the problem
of the two mean proportionals between given extremes. The conjectural
order places focus-directrix treatment of conics as far back at least as
Eudoxus. A discrete proportion that easily might have been found by
Democritus if he had the notion of the curves as possible shapes of
boundary of geometrical shadow of a sphere is implicated as basis of the
development. An historical-cultural context that might have been
conducive to such notion has been indicated. The explanation offered of
the alternative titles perhaps has more to commend it than the one
proposed by Heath.
Bibliography
Durell, Clement V
Elementary Coordinate Geometry. G. Bell and Sons, Ltd. London. 1960.
Granger,Frank.
Vitruvius : On Architecture. 2 vol. Loeb Classical Library (text and
English translation), London & Cambridge, Mass. 1934.
Heath, Thomas L.
The Thirteen Books of Euclid's Elements. 2nd ed. 3 vol. Cambridge
University Press, Cambridge. 1926. (repr. Dover, New York. 1956.)
Hicks, R.D.
Diogenes Laertius. 2 vol. Loeb Classical Library (text and English
translation), London & New York. 1925.
Keuls, Eva C.
Plato and Greek Painting. E. J. Brill, Leiden. 1978.
Knorr, Wilbur Richard.
The Ancient Tradition of Geometric Problems. Birkhauser, Boston.
1986. Textual Studies in Ancient and Medieval Geometry.
Birkhauser, Boston. 1989.
Lynch, John Patrick.
Aristotle's School. A Study of a Greek Educational Institution.
University of California Press, Berkeley & Los Angeles. 1972.
Morrow, Glenn R.
Proclus : A Commentary on the First Book of Euclid's Elements.
Princeton University Press. Princeton, New Jersey. 1970.
Mourelatos, Alexander P. D.
The Route of Parmenides. A Study of Word, Image, and Argument in the
Fragments. Yale University Press, New Haven & London. 1970.
Shute, Richard.
On the History of the Process by which the Aristotelian Writings Arrived
at Their Present Form. An Essay. Clarendon Press, Oxford. 1888.
[ end of part 4 / 4 ]
Kenneth W. Pringle
This archive was generated by hypermail 2b28 : Tue Jan 04 2000 - 09:27:54 EST