Greek
Mathematician 275 – 195 B.C.
Dr. Jill
Raiguel
TED 446
Winter 2004
Cal Poly,
Pomona
Prepared By: Farhod Azarbaydjani, a.k.a. Mr. A
E-mail: farout@verizon.net
Date: February 26, 2004
Grade Levels: Ninth through twelfth grade.
Subject(s): Mathematics (Interdisciplinary:
Algebra, Geometry, Earth Science, History)
Standard: Satisfies 1997 California State Standards for Math (Algebra, and Geometry)
Anticipatory Set: This lesson can be anchored in the student’s prior knowledge of measurement, shapes, Algebra, Geometry, Greek History and/or Science.
Resources/Materials: Bibliography of Greek Mathematician 275 – 195 B.C. The transparency of Eratosthenes calculation. Quiz handout & 9-dot handout
Duration/Time: 20 minutes
Goals:
Short-term:
ü Demonstrate problem-solving skills.
ü Promote fluency- reading and writing skills.
ü
Prepare students for a geometry, or inductive reasoning lesson.
ü
Participate in effective group working.
Long-term:
ü To instill an appreciation of science, encourage student interest in the sciences.
Objectives: As the conclusion of this activity, students will be able to:
ü Think outside the box using lateral cognitive processing.
ü Reference and cite the work of Greek mathematician Eratosthenes and recall his contribution to science.
ü Define, replicate, and create from memory a table of prime numbers from 1 to 20.
ü Convert stadia into feet.
ü Conceptualize mathematical error.
Procedure:
q At the begining of the lesson handout 9-dot puzzle: Allow 3to5 minutes to solve. Reveal solution if necessary (right).
q Provide students with a copy of the attached Eratosthenes biography to be included in their mathematician’s journal. 10-minute Journal topic – Some theorist have suggested that the Universe is spherical and others believe it to be triangular, based upon your individual educated opinion, how do you perceive the shape of the Universe? or Think of a problem in the world today that seems impossible to solve and how you would solve it.
q Have students begin group work by reading the biography on Eratosthenes.
q Using the numbers 1 through 20 contained on the quiz, have the students reconstruct Eratosthenes table of prime numbers by:
§ Eliminating multiples of 2’s
§ Eliminating multiples of 3’s
§ These are the prime numbers 2,3,5,7,11,13,17, and 19.
q T & F Twenty (20) Question Quiz
q Convert stadia into feet (i.e. stadia = 600 feet so 1’=1/600 stadia.)
q Introduce 9-dot/ Thinking outside the box puzzle. If neither group has solved after 5 minutes move-on.
X600
/5,280
250,000 stadia = 15,000,000 feet = 28,409 miles
q Show students transparency on how Eratosthenes measured the circumference of the Earth.
q Discuss the accuracy of Eratosthenes calculations, the sources of error and the fact that your results depend upon other people making accurate measurements.
It’s 1633, you’re Sir
Isaac Newton and you are about to be expelled from school for speaking foolishly
and contradicting your professors view that the World is flat. Write a letter to
the Headmaster supporting your claim that the world is round.
Support your reasons by explaining what you know.
Check for Understanding: In addition to the quiz, feedback, and in-class direct-observation, the writing assignment is perhaps the best method to validate understanding.
Independent Study: Have students who are interested in Eratosthenes research what other mathematical principles he is responsible for. If interested in Earth Science: research the methods of estimating the size of Jupiter and/or other planets within the solar system.
Technology: Microsoft Excel spreadsheet to convert stadia into miles, feet. Show Earth.mpg movie of Earth.
Greek
Mathematician 275 – 195 B.C.
Eratosthenes
Born in Cyrene which is now in Libya in North
Africa in 275 B.C, 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 all-round scholar,
Eratosthenes was considered to fall short of the highest rank. Heath
writes [4]: -
[Eratosthenes]
was, indeed, recognized 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
all-round 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 [14]. 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 [14] 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
[4]):-
...
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 [4]:-
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 god-fearing 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 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 [11] 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 meters for the stadium as some have deduced from
values given by Pliny.
It is less good if 166.7 meters was the value used by Eratosthenes as Gulbekian
suggests in [11].
Several of the papers referenced, for example [10],
[15] and [16], discuss the accuracy of Eratosthenes' result. The paper [15] 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 7
12'.
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 [9] and [17] 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
[4] argues that Eratosthenes used 24
and
that 11/83 of 180
was a
refinement due to Ptolemy.
Taisbak [17] agrees with attributing 11/83 to Ptolemy
although he believes that Eratosthenes used the value 2/15
of 180
.
However Rawlins [15] believes that a continued
fraction method was used to calculate the value 11/83
while Fowler [9] proposes that the anthyphairesis (or Euclidean
algorithm) method was used (see also [3]).
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 is said to have became blind in old
age and it has been claimed that he committed suicide by starvation.
T F
1.
Eratosthenes was born
in Cyrene, which is near Greece.
T F
2.
On the death of
Callimachus in about 240 BC, Eratosthenes became the third librarian at
Alexandria.
T F
3.
Eratosthenes spent
some years studying in Rome.
T F
4.
Eratosthenes was often
considered a bright scholar.
T F
5.
One source of
information concerning Eratosthenes is from a forged letter.
T F
6.
Eratosthenes was
obsessed with writing letters.
T F
7.
Other details of what
Eratosthenes wrote in Platonicus are given by Plato’s writings.
T F
8.
Eratosthenes invented
rational and irrational numbers.
T F
9.
Eratosthenes was so
obsessed with math he pulled his hair out.
T F
10. Another
book written by Eratosthenes was On Beans and, although it is now lost,
it is mentioned by Pappus as one of the great books of geometry.
T F
11. In
the field of Geodesy know called Geometry, however, Eratosthenes will always be
remembered for his measurements of the Earth.
T F
12. Eratosthenes
accurate measurement of the diameter of his head inspired him to measure the
Earth.
T F
13. 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 7 degrees 12'.
T F
14. Eratosthenes
also accurately measured the distance to the sun as 804,000,000 stadia and the
distance to the Moon as 780,000 stadia.
T F
15. Ptolemy
tells us that Eratosthenes measured the tilt of the Earth's axis with great
accuracy obtaining the value 33 degrees 51' 15".
T F
16. Eratosthenes
made major contributions to physical geography.
T F
17. Eratosthenes
committed suicide by starving himself to death.
T F
18. Eratosthenes went blind from staring at the Sun.
T F
19. The
Moon features a crater named after Eratosthenes.
T F
20. The
Prime number sieve is no longer a useful tool in geometry.
BONUS-
By process of elimination
crosses out the multiples of 2’s then 3’s, and circles the remaining Prime
numbers.
