DIOCLES OF CARYSTUS
ON BURNING MIRRORS
LIVED ABOUT 240 BC IN CARYSTUS
(NOW KÁRISTOS), EUBOEA (NOW EVVOIA), GREECE
J. J. O'CONNOR AND E. F. ROBERTS
AUTHOR NAME HERE
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INFORMATIVE TEXT HERE |
Diocles of Carystus was a contemporary of
Apollonius. Practically all that was knew
about him until recently was through fragments
of his work preserved by Eutocius in his
commentary on the famous text by Archimedes
On the sphere and the cylinder. In this work
we are told that Diocles studied the cissoid
as part of an attempt to duplicate the cube.
It is also recorded that he studied the problem
of Archimedes to cut a sphere by a plane
in such a way that the volumes of the segments
shall have a given ratio.
The extracts quoted by Eutocius from Diocles'
On burning mirrors showed that he was the
first to prove the focal property of a parabolic
mirror. Although Diocles' text was largely
ignored by later Greeks, it had considerable
influence on the Arab mathematicians, in
particular on al-Haytham. Latin translations
from about 1200 of the writings of al-Haytham
brought the properties of parabolic mirrors
discovered by Diocles to European mathematicians.
Recently, however, some more information
about Diocles' life has come to us from the
Arabic translation of Diocles' On burning
mirrors whose discovery is described below.
From this work we learn that Zenodorus travelled
to Arcadia and entered into discussions with
Diocles, so that certainly Diocles was working
in Arcadia at the time. This may not seem
a very major centre of mathematical importance
at the time for such an outstanding scholar
as Diocles to be working in, but as Toomer
writes in [4]:-
It would be wrong to conclude from this that
Archadia was a cultural centre in this period
... : the whole of the introduction confirms
the impression we derive from other contemporary
sources, that mathematics during the Hellenistic
period was pursued, not in schools established
in cultural centres, but by individuals all
over the Greek world, who were in lively
communication with each other both by correspondence
and in their travels.
Let us quote from Diocles' introduction to
On burning mirrors in the translation by
Toomer [4]:-
Pythian the Thasian geometer wrote a letter
to Conon in which he asked him how to find
a mirror surface such that when it is placed
facing the sun the rays reflected from it
meet the circumference of a circle. And when
Zenodorus the astronomer came down to Arcadia
and was introduced to us, he asked us how
to find a mirror surface such that when it
is placed facing the sun the rays reflected
from it meet a point and thus cause burning.
Toomer notes that his translation of when
Zenodorus the astronomer came down to Arcadia
and was introduced to us could, perhaps,
be translated when Zenodorus the astronomer
came down to Arcadia and was appointed to
a teaching position there.
It is only recently that an Arabic translation
of Diocles On burning mirrors has been found
in the Shrine Library in Mashhad, Iran. No
writing of Diocles was known to Heath in
1921 when he wrote [3], but Toomer translated
and published the newly found Arabic translation
of the lost treatise On burning mirrors by
Diocles in 1976.
It has been noticed that On burning mirrors
is loosely in three parts, for three separate
topics are studied. These three topics are
burning mirrors, Archimedes' problem to cut
a sphere by a plane, and duplicating the
cube. Sesiano (see [4]) has suggested that
we may have three short works by Diocles
combined into one work and this would have
a certain attraction since the title On burning
mirrors fails to reflect properly the contents
of the whole. If Sesiano's suggestion is
correct then we know that the three were
combined early on since by the time of Eutocius
they formed a single work.
On burning mirrors is a collection of sixteen
propositions in geometry mostly proving results
on conics. It is thought that three of the
propositions are later additions to the text,
while the remaining ones give a remarkable
insight into the theory of conics in the
early second century BC.
The first of these propositions proves what
has long been known to have been first established
by Diocles, namely the focal property of
the parabola. The next two propositions give
properties of spherical mirrors and with
Propositions 4 and 5 giving the focus directrix
construction of the parabola. These constructions
are again properties of the parabola that
Diocles was the first to give.
The problem of Archimedes to cut a sphere
in a given ratio which was known to be in
the work through the writing of Eutocius
referred to above is studied in Propositions
7 and 8. The duplication of the cube problem,
again referred to by Eutocius, is studied
by Diocles in Proposition 10. The next two
propositions solve the problem of inserting
two mean proportions between a pair of magnitudes
using the cissoid curve which was invented
by Diocles. The final three propositions
solve generalisations of the duplication
of the cube problem using the cissoid, and
another problem of the two mean proportionals
type.
There are other fascinating deductions that
Toomer makes as editor of [4]. A study of
the work lead him to claim, contrary to what
has long been believed, that the terms "hyperbola",
"parabola", and "ellipse"
were introduced into the theory of conics
before the time of Apollonius.
In On burning mirrors Diocles also studies
the problem of finding a mirror such that
the envelope of reflected rays is a given
caustic curve or of finding a mirror such
that the focus traces a given curve as the
Sun moves across the sky. The solution of
this problem would, of course, have interesting
consequences for the construction of a sundial.
Neugebauer, in an appendix to [4] (see also
[6]), shows that this problem cannot be solved
exactly while in [5] Hogendijk shows that,
using methods available to Diocles, the problem
can be solved approximately. Hogendijk in
[5] then considers the interesting possibility
that Diocles gave arguments of this type
in the original text but that later copiers
of the text could not understand this part
so omitted it.
K H Dannenfeldt, G J Toomer, Biography in
Dictionary of Scientific Biography (New York
1970-1990). Biography in Encyclopaedia Britannica.
Books:
T L Heath, A History of Greek Mathematics
(2 Vols.) (Oxford, 1921). G J Toomer, Diocles
On Burning Mirrors, Sources in the History
of Mathematics and the Physical Sciences
1 (New York, 1976).
Articles:
J P Hogendijk, Diocles and the geometry of
curved surfaces, Centaurus 28 (3-4) (1985),
169-184. O Neugebauer, Note on Diocles' 'burning
mirror', in From ancient omens to statistical
mechanics, Acta Hist. Sci. Nat. Med. 39 (Copenhagen,
1987), 37-42.
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