Доклад: Eratosthenes of Cyrene
Eratosthenes of Cyrene
Born: 276 BC in Cyrene, North Africa (now Shahhat,
Libya)
Died: 194 BC in Alexandria, Egypt
Eratosthenes was born in Cyrene which is now in Libya
in North Africa. His teachers included the scholar Lysanias of Cyrene and the
philosopher Ariston of Chios who had studied under Zeno, the founder of
the Stoic school of philosophy.
Eratosthenes also studied under the poet and scholar Callimachus who had also
been born in Cyrene. Eratosthenes then spent some years studying in Athens.
The library at Alexandria was planned by Ptolemy I
Soter and the project came to fruition under his son Ptolemy II Philadelphus.
The library was based on copies of the works in the library of Aristotle. Ptolemy II Philadelphus appointed
one of Eratosthenes' teachers Callimachus as the second librarian. When Ptolemy
III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to
go to Alexandria as the tutor of his son Philopator. On the death of
Callimachus in about 240 BC, Eratosthenes became the third librarian at
Alexandria, in the library in a temple of the Muses called the Mouseion. The
library is said to have contained hundreds of thousands of papyrus and vellum
scrolls.
Despite being a leading allround scholar,
Eratosthenes was considered to fall short of the highest rank. Heath writes:
[Eratosthenes] was, indeed, recognised by his
contemporaries as a man of great distinction in all branches of knowledge,
though in each subject he just fell short of the highest place. On the latter
ground he was called Beta, and another nickname applied to him, Pentathlos, has
the same implication, representing as it does an allround athlete who was not
the first runner or wrestler but took the second prize in these contests as
well as others.
Certainly this is a harsh nickname to give to a man
whose accomplishments in many different areas are remembered today not only as
historically important but, remarkably in many cases, still providing a basis
for modern scientific methods.
One of the important works of Eratosthenes was
Platonicus which dealt with the mathematics which underlie Plato's philosophy. This work was heavily
used by Theon of Smyrna when he wrote
Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes'
work studied the basic definitions of geometry and arithmetic, as well as
covering such topics as music.
One rather surprising source of information concerning
Eratosthenes is from a forged letter. In his commentary on Proposition 1
of Archimedes' Sphere and cylinder Book
II, Eutocius reproduces a letter
reputed to have been written by Eratosthenes to Ptolemy III Euergetes. The
letter describes the history of the problem of the duplication of the cube and, in particular, it describes a
mechanical device invented by Eratosthenes to find line segments x and y so
that, for given segments a and b,
a : x = x : y = y : b.
By the famous result of Hippocrates it was known that solving the problem of finding two
mean proportionals between a number and its double was equivalent to solving
the problem of duplicating the cube. Although the letter is a forgery, parts of
it are taken from Eratosthenes' own writing. The letter, which occupies an
important place in the history of mathematics, is discussed in detail in . An
original Arabic text of this letter was once kept in the library of the St
Joseph University in Beirut. However it has now vanished and the details given
in come from photographs taken of the
letter before its disappearance.
Other details of what Eratosthenes wrote in Platonicus
are given by Theon of Smyrna. In
particular he described there the history of the problem of duplicating the
cube (see Heath):
... when the god proclaimed to the Delians through the
oracle that, in order to get rid of a plague, they should construct an alter
double that of the existing one, their craftsmen fell into great perplexity in
their efforts to discover how a solid could be made the double of a similar
solid; they therefore went to ask Plato
about it, and he replied that the oracle meant, not that the god wanted an
alter of double the size, but that he wished, in setting them the task, to
shame the Greeks for their neglect of mathematics and their contempt of
geometry.
Eratosthenes erected a column at Alexandria with an
epigram inscribed on it relating to his own mechanical solution to the problem
of doubling the cube:
If, good friend, thou mindest to obtain from any small
cube a cube the double of it, and duly to change any solid figure into another,
this is in thy power; thou canst find the measure of a fold, a pit, or the
broad basin of a hollow well, by this method, that is, if thou thus catch
between two rulers two means with their extreme ends converging. Do not thou
seek to do the difficult business of
Archytas's cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form
of lines as is described by the godfearing
Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of
means, beginning from a small base. Happy art thou, Ptolemy, in that, as a
father the equal of his son in youthful vigour, thou hast thyself given him all
that is dear to muses and Kings, and may be in the future, O Zeus, god of
heaven, also receive the sceptre at thy hands. Thus may it be, and let any one
who sees this offering say "This is the gift of Eratosthenes of Cyrene".
Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the
'Sieve of Eratosthenes' which, in modified form, is still an important tool
in number theory research. The sieve
appears in the Introduction to arithmetic by
Nicomedes.
Another book written by Eratosthenes was On means and,
although it is now lost, it is mentioned by
Pappus as one of the great books of geometry. In the field of geodesy,
however, Eratosthenes will always be remembered for his measurements of the
Earth.
Eratosthenes made a surprisingly accurate measurement
of the circumference of the Earth. Details were given in his treatise On the
measurement of the Earth which is now lost. However, some details of these
calculations appear in works by other authors such as Cleomedes, Theon of
Smyrna and Strabo. Eratosthenes
compared the noon shadow at midsummer between Syene (now Aswan on the Nile in
Egypt) and Alexandria. He assumed that the sun was so far away that its rays
were essentially parallel, and then with a knowledge of the distance between
Syene and Alexandria, he gave the length of the circumference of the Earth as
250,000 stadia.
Of course how accurate this value is depends on the
length of the stadium and scholars have argued over this for a long time. The
article discusses the various values
scholars have given for the stadium. It is certainly true that Eratosthenes
obtained a good result, even a remarkable result if one takes 157.2 metres for
the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by
Eratosthenes as Gulbekian suggests in .
Several of the papers referenced, for example , and , discuss the accuracy of Eratosthenes'
result. The paper is particularly
interesting. In it Rawlins argues convincingly that the only measurement which
Eratosthenes made himself in his calculations was the zenith distance on the
summer solstice at Alexandria, and that
he obtained the value of 712'. Rawlins argues that this is in error by 16'
while other data which Eratosthenes used, from unknown sources, was
considerably more accurate.
Eratosthenes also measured the distance to the sun as
804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed
these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured
the tilt of the Earth's axis with great accuracy obtaining the value of ^{11}/_{83}
of 180, namely 23 51' 15".
The value ^{11}/_{83} has fascinated
historians of mathematics, for example the papers and are written just to examine the source of this value. Perhaps the
most commonly held view is that the value ^{11}/_{83} is due
to Ptolemy and not to
Eratosthenes. Heath argues that
Eratosthenes used 24 and that ^{11}/_{83} of 180 was a
refinement due to Ptolemy. Taisbak agrees with attributing ^{11}/_{83}
to Ptolemy although he believes that
Eratosthenes used the value ^{2}/_{15} of 180. However
Rawlins believes that a continued fraction method was used to
calculate the value ^{11}/_{83} while Fowler proposes that the
anthyphairesis (or Euclidean algorithm)
method was used (see also).
Eratosthenes made many other major contributions to
the progress of science. He worked out a calendar that included leap years, and
he laid the foundations of a systematic chronography of the world when he tried
to give the dates of literary and political events from the time of the siege
of Troy. He is also said to have compiled a star catalogue containing 675
stars.
Eratosthenes made major contributions to geography. He
sketched, quite accurately, the route of the Nile to Khartoum, showing the two
Ethiopian tributaries. He also suggested that lakes were the source of the
river. A study of the Nile had been made by many scholars before Eratosthenes
and they had attempted to explain the rather strange behaviour of the river,
but most like Thales were quite wrong
in their explanations. Eratosthenes was the first to give what is essentially
the correct answer when he suggested that heavy rains sometimes fell in regions
near the source of the river and that these would explain the flooding lower
down the river. Another contribution that Eratosthenes made to geography was
his description of the region "Eudaimon Arabia", now the Yemen, as
inhabited by four different races. The situation was somewhat more complicated
than that proposed by Eratosthenes, but today the names for the races proposed
by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are
still used.
Eratosthenes writings include the poem Hermes,
inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the
Greeks. Eratosthenes is said to have became blind in old age and it has been
claimed that he committed suicide by starvation.
J J O'Connor and E F Robertson
Список литературы
Для подготовки
данной работы были использованы материалы с сайта http://wwwhistory.mcs.standrews.ac.uk/
