Seventh Century A.D. Armenian Math Problems
David Davidian
From: _Quantum_ Magazine, March/April 1993 pages 42-46
The Problem Book of Anania of Shirak
------------------------------------
"On the ancient peak of Ararat
The centuries have come like seconds
And passed on."
-Avetik Issahakian
by Yuri Danilov
Some years ago Journalists interviewing celebrities liked to ask them: "What
books would you take with you if you were to go off on a space flight?" And
though the number of books allowed on the trip varied from 10 to 30,
depending on the type of spacecraft and the generosity of the interviewer,
and celebrities are people of the most varied tastes, ages, and professions,
not one of them dared to say that he or she would want to take with them at
least one book of arithmetic problems.
Some of these people certainly excluded this kind of literature because they
were trained in the humanities and had nothing but scorn for "numbers" (though
secretly afraid of them). Others steered clear of such puzzle books because
they were masters of incomparably more difficult branches of modern
mathematics and didn't mind saying for all the world to hear that they didn't
know how to solve mere arithmetic problems. Professional mathematicians were
no exception. Here's what the Russian mathematician Alexander Khinchin, a
specialist in statistics, wrote about arithmetic: "I willingly confess that
any time a fifth-grader asked me to help solve an arithmetic problem, it was
a hard work for me, and sometimes I failed completely. Of course, like most of
my friends, I could easily solve the problem by the natural algebraic route
--constructing equations or sets of equations. But we were supposed to avoid
using algebraic analysis at all costs! . . . By the way, it's a fact that is
well known and oft repeated that, as a rule, neither high school graduates,
nor students at teaching colleges, nor teachers beginning their careers
(nor, I must add, scientific researchers) can solve arithmetic problems. It
seems the only people in the world who are able to solve them are fifth-grade
teachers."
Now, I'm not insisting that a book of arithmetic problems be included in the
bookbag of anyone flying into space. But a sense of justice induces me to
recommend one particular problem book, one that will satisfy the most
fastidious taste and supply food for thought sufficient not only for a
relatively short flight to the Moon but for a extended space voyage--say, to
Venus and back.
One for the "road"
They both took out the books they brought for the road. Kingsley
glanced at the Royal Astronomer's book and saw a bright cover with
a group of cutthroats shooting at each other with revolvers. "God
knows what this kind of stuff leads to," thought Kingsley.
The Royal Astronomer looked at Kingsley's book and saw the History
of Herodotus. "Good Lord, next he'll be reading Thucydides," thought
the Royal Astronomer.
--Fred Hoyle, The Black Cloud
The book I'm talking about isn't very big, but its 24 problems constitute 24
elegant miniatures from seventh-century Armenia. Naive and wise at the same
time, rich in striking detail and the bright coloration of the period, these
problems are reminiscent of the reliefs on the famous monument of Armenian
architecture, the church on the island of Akhtamar in Lake Van (in what is
now Turkey_. They are as inseparable from the image of Armenia as the elegant
letters of the Armenian alphabet, invented by Mesrop Mashtots, or the songs of
Komitas, or the paintings of Saryan.
An edition of these incredibly beautiful problems has long been a
bibliographic rarity. It was published under the title Problems and Solutions
of Vardapet [1] Anania of Shirak, Armenian Mathematician of the Seventh
Century (translated and published by I. A. Orbeli, Petrograd, 1918).
The abundance of close observations and wide-ranging information about the way
of life and customs of that remote epoch when Anania of Shirak lived and
worked have actually rendered a disservice to his problem book. For many years
the book was known only to researchers in the humanities--specialists in
Armenian history who jealously guarded their treasure and wouldn't let just
anyone see it. Even now, after research by K. P. Patkanov, the learned monk
Father Kaloust, J.I. Orbeli, A. Abramyan, V. K. Chaloyan, and others has
brought the works of Anania of Shirak to light in scholarly circles, the
general reader remains ignorant of the very existence of this remarkable
problem book.
Vardapet Anania of Shirak
Once fell in love with the art of calculation, I thought that no
philosophical notion can be constructed without number, considering
it the mother of all wisdom.
--Anania of Shirak
Among ancient Armenian thinkers, Vardapet Anania of Shirak stands out because
of the breadth of his interests and the unique mathematical orientation of his
work. Some of his works have been preserved. In addition to the Problems and
Solutions, the following tracts have found a special place in the estimation
of scholars: On Weights and Measures, Cosmography and Calendrical Theory, and
Armenian Geography of the Seventh Century A.D. (the authorship of the last
work was long attributed to another outstanding thinker of ancient Armenia,
Movses of Khoren).
In his autobiography, Anania of Shirak has this to say about himself:
I, Anania of Shirak, having studied all the science of our
Armenian land and having learned the Holy Scripture intimately,
in the expression of the psalmist, "every day I illuminated the
eyes of my mind." Feeling myself lacking in the art of calculation,
I came to the conclusion that it is fruitless to study philosophy,
the mother of all sciences, without number. I could find in Armenia
neither a man versed in philosophy nor books that explained the
sciences. I therefore went to Greece and met in Theodosiople a man
named Iliazar who was well versed in ecclesiastical works. He told me
that in Forth Armenia [2] there lived a famous mathematician,
Christosatur. I went this person and spent six months with him. But
soon I noticed that Christosatur was a master not of all science but
only of certain fragmentary facts.
I then went to Constantinople, where I met acquaintances who told me:
"Why did you go so far, when much closer to us, in Trebizon, on the
coast of Pontus [3] lives the Byzantine vardapet Tyukhik. He is full
of wisdom, is known to kings, and knows Armenian literature." I asked
them how they knew this. They answered: "We saw ourselves that many
people traveled long distances to become pupils of so learned a man.
Indeed the archdeacon of the patriarchate of Constantinople,
Philagrus, traveled with us, bringing many young persons to become
pupils of Tyukhik." When I heard this, I expressed my gratitude to
God, who had quenched the thirst of His slave.
I went to Tyukhik at the monastery of St. Eugene and explained why I
had come. He received me graciously and said: "I praise Our Lord that
He sent you to learn and to transplant science in the domain of St.
Gregory; I am glad that all your country will learn from me. I myself
lived in Armenia for many years as a youth. Ignorance reigned there."
Vardapet Tyukhik loved me as a son and shared all his thoughts with
me. The Lord bestowed upon me His blessing: I completely assimilated
the science of number, and with such success that my fellow students
at the king's court began to envy me.
I spent eight years with Tyukhik and studied many books that had not
been translated into our language. For the vardapet had an innumerable
collection of books: secret and explicit, ecclesiastical and pagan,
books on art, history, and medicine, books of chronologies. Why
enumerate them by title? In a word, there is no book that Tyukhik did
not have. And he had such a gift from the Holy Spirit for translating
that when he sat down to translate something from the Greek into
Armenian, he did not struggle as other translators did, and the
translation read as if the work were written in that language
originally.
Tyukhik told me how he had achieved such vast erudition and how he had
learned the Armenian language. "When I was young," he said, "I lived
in Trebizon, at the court of the military chief Ioannus Patricus, and
for a long time, up to the accession of Mauritius to the throne I
served as a military man in Armenia and learned your language and
literature. During one attack by Persian troops on the Greeks, I was
wounded and escaped to Antioch. I lost all my possessions. Praying to
the Lord to heal my wounds, I made a promise: "If You prolong my life,
I shall dedicate it not to accumulating perishable treasures but to
collecting treasures of knowledge." And the Lord heard my prayers.
After I recovered I went to Jerusalem, and from there to Alexandria
and Rome. Upon returning to Constantinople, I met a famous philosopher
from Athens and studied with him for many years. After that I returned
to my homeland and began to teach and instruct my people."
After some years that philosopher died. Not finding a replacement for
him, the king and his courtiers sent for Tyukhik and invited him to
assume the teacher's position. Tyukhik, citing the promise he made to
God not to move far from the city, turned down the offer. But because
of his wide leaning, people came streaming from all countries to study
with him.
And I, the most insignificant of all Armenians, having learned from
him this powerful science, desired by kings, brought it to our
country, supported by no one, obligated only to my own industry, God's
help, and the prayers of the Blessed Educator. And no one thanked me
for my efforts.
Problems and Solutions
A half and one sixth and one nine-ninth of all the books were printed
on verge'; one fifth and one two-hundred-eighty-fifth--on rag paper;
one forty-fifth and one eight-hundred-fifty-fifth--on vellum, and
forty-five inscribed copies--on Dutch paper. And so, find how many
copies were printed in all.
--Imitation of Anania of Shirak
A Latin proverb says habent sua fata libelli ("books have their own fate").
The fate of Problems and Solutions by Anania of Shirak is quite amazing. The
manuscripts of Anania's book were preserved only because, according to
Armenian historians, "in ancient and medieval Armenia manuscripts were guarded
from invaders, like weapons, and cherished, like one's own children." Biding
their time, the manuscripts lay in the Matenadaran, a renowned depository of
ancient manuscripts (now the Mesrop Mashtots Institute of Ancient Manuscripts).
And its hour finally arrived. In 1896 the learned monk Father Kaloust used two
manuscripts to publish the problem book, supplementing it with an introduction
and commentary. In 1918 the book was translated into Russian, edited,
annotated, and typeset by Iosef Orbeli, a prominent scholar (and later a
member of the Academy of Sciences of the USSR).
In the translator's words, the problems of Anania are "amusing, full of life,
and simple." Orbeli goes on to say: "The subjects of the problems are
generally taken from everyday life. The scene is predominantly his homeland
Shirak and the surrounding countryside, and the dramatis personae, if they are
named, are the local princes--the Kamsarakans, including Nersekh, who was a
contemporary of Anania." Like other ancient authors, Anania of Shirak used
only "aliquots" -- that is, fractions with a numerator of 1. When it is
necessary to write fractions with numerators other than 1, one has to
represent it as a sum of aliquots (see the epigraph above).
Like any true work of art, the problems of Anania suffer terribly in the
retelling. You have to read the originals (albeit in translation) in their
full glory. So let's open Anania's problem book--a gift from across the ages.
Problems 1 and 8 relate to the Armenian uprising against the Persians in A.D.
572.
Problem 1
My father told me the following story. During the famous wars between the
Armenians and the Persians, prince Zaurak Kamsarakan performed extraordinary
heroic deeds. Three times in a single month he attacked the Persian troops.
The first time, he struck down half of the Persian army. The second time,
pursuing the Persians, he slaughtered one fourth of the soldiers. The third
time, he destroyed one eleventh of the Persian army. The Persians who were
still alive, numbering two hundred eighty, fled to Nakhichevan. And so, from
this remainder, find how many Persian soldiers there were before the massacre.
Problem 8
During the famous Armenian uprising against the Persians, when Zaurak
Kamsarakan killed Suren, one of the Armenian azats[4] sent an envoy to the
Persian king to report the baleful news. The envoy covered fifty miles in a
day. Fifteen days later, when he learned of this, Zaurak Kamsarakan sent
riders in pursuit to bring the envoy back. The riders covered eighty miles in
a day. And so, find how many days it took them to catch the envoy.
Problem 18 mentions vessels made of varying amounts of metal. In the Russian
translation, they are all called "dishes." But in the original Armenian,
according to Orbeli's note, the dishes in the first and second instances are
called mesur, and in the third instance scutel. Scutel is a common Armenian
word, but mesur had not been encountered in Armenian literature before
Anania's Problems and Solutions.
Problem 18
There was a tray in my house. I melted it down and made other vessels from the
metal. From one third I made a mesur; from one fourth, another mesur; from
one fifth, two goblets; from one sixth, two scutels; and from two hundred ten
drams, I made a bowl. And now, find the weight of the tray.
Several of the problems reflect the richness of the Caucasian fauna in
Anania's time -- for instance, problem 7.
Problem 7
Once I was in Marmet, the capital of the Kamsarakans. Strolling along the bank
of the river Akhuryan, I saw a school of fish and ordered that a net be cast.
We caught a half and a quarter of the school, and all the fishes that slipped
out of the net ended up in a creel. When I looked in the creel, I found
forty-five fishes. And now, find how many fishes here were in all.
The temptation is great to present all 24 problems. But I'll restrain myself
and offer you just one more.
Problem 20 provides some interesting information about the wild animals that
inhabited Armenia at one time but now extinct for so long that there is no
mention of them even in zoological reference books. The wild donkey, according
to the generally accepted view, never roamed the Armenian lands. Yet Anania of
Shirak offers evidence to the contrary .
Problem 20
The hunting preserve of Nersekh Kamsarakan, ter[5] of Shirak and Asharunik,
was at the base of the mountain called Artin. One night great herds of wild
donkey entered the preserve. The hunters could not cope with the donkeys and,
running to the village of Talin, told Nersekh about them. When he arrived with
his brothers and azats and entered the preserve, they began killing the wild
beasts. Half of the animals were caught in traps, one fourth were killed by
arrows. The young, which constituted one twelfth of all the animals, were
caught alive, and three hundred sixty wild donkeys were killed by spears. And
so, find how many beasts there were at the start of this massacre.
"Set in type by me, Iosef Orbeli"
His biography could not be squeezed into the framework of a
bibliography.
-- K Uzbashyan, Academician
Iosef Abgarovich Orbeli
Anyone who is lucky enough to hold a copy (1/n of the small printing--n is the
solution to the epigraph in the previous section) of the Russian translation
of Anania of Shirak's Problems and Solutions, a thin book with yellowed pages,
has probably noticed the variety of the fonts, the elegance of the borders,
and the high quality of the design, printing, and binding. Such great
attention to detail is characteristic of works that fulfill a requirement for
a degree in bookmaking. And this problem book was indeed a kind of diploma
attesting to the professional maturity of the man who created it. An
advertisement at the end of the book reads: "This book was typeset in
December 1917 at the printing offices of the Russian Academy of Sciences by
me, Iosef Orbeli; the text was also proofread, laid out, and decorated with
borders by me. Various circumstances prevented me from carrying this project
to the end; the final pages of the book were typeset by M. Strolman."
Typesetting was neither the first nor the only profession of the renowned
orientalist Iosef Orbeli, who later became the director of the Hermitage
Museum in Leningrad. He was also a cabinetmaker and a locksmith. Orbeli had
already become acquainted with the famous academic printing house Typis
Academiae, founded in 1728 and known all over the scientific world for its
rich collection of fonts and its virtuoso typesetters. In preparing to publish
the corpus of ancient inscriptions preserved on the walls of Armenian
churches, Orbeli found it necessary to create a new font that would preserve
the unique signs and ligatures. This complicated work was done by M. G.
Strolman. (Unfortunately the entire set of letters was destroyed during the
blockade of Leningrad in World War II.)
When Orbeli came to the printing offices of the Academy of Sciences, times
were hard. The only way to publish the newly translated Problems of Anania
was for Orbeli to learn typesetting (he had always been attracted to the
printer's craft). In 1922 Orbeli became the director of printing at the
Academy of Sciences. Even after he retired, he remained a tireless champion
of Russian academic typography.
Back to Earth
This book by definition does not exhaust all the most important
works in this domain. The editor hopes that those who are guilty
of this incompleteness will read these lines and, stung by shame,
will work up, if not a collection like this, at least a monograph.
--V. Bonch-Bruyevich introduction to the
Russian translation of Solid-Body
Symmetry by R. Knox and A. Gold
Let's imagine a time when space flight is an everyday thing, and high
schoolers will spend their breaks as astronauts-in training in the Perelman
crater on the far side of the Moon. Maybe one of the space travelers will take
this very copy of Quantum, and another, looking over her shoulder, will read
this article and say to himself: "This Anania from Shirak seems like a pretty
interesting guy. When I get home I'll try to find his problems."
Good luck, my young friend! Anania is sure to entertain you. Perhaps by then
there will be more than n copies of his timeless Problems and Solutions. And
we can hope they will be as lovingly printed as the masterpieces created by
Iosef Orbeli.
[1] Vardapet (or vartabed) means teacher or learned man in Armenian. (The
Armenian language suffers in English from a dual transliteration scheme.
Thus, Mesrop is often rendered as "Mesrob", Komitas as "Gomidas," and so
on).
[2] Fourth Armenia was one of fifteen provinces into which, according to
Armenian Geography in the Seventh Century A.D., so-called Great Armenia
was divided.
[3] "Pontus" (or "Pontus Euxinus") was an old name for the Black Sea.
[4] "Azats" were members one of several strata of freemen in ancient Armenia.
[5] "Ter" was the title of the heads of sovereign royal families in ancient
Armenia.
--
David Davidian d...@urartu.sdpa.org | "How do we explain Turkish troops on
S.D.P.A. Center for Regional Studies | the Armenian border, when we can't
P.O. Box 382761 | even explain 1915?"
Cambridge, MA 02238 | Turkish MP, March 1992
Cetin Kaya Koc
>
> My father told me the following story. During the famous wars between the
> Armenians and the Persians, prince Zaurak Kamsarakan performed extraordinary
> heroic deeds. Three times in a single month he attacked the Persian troops.
> The first time, he struck down half of the Persian army. The second time,
> pursuing the Persians, he slaughtered one fourth of the soldiers. The third
> time, he destroyed one eleventh of the Persian army. The Persians who were
> still alive, numbering two hundred eighty, fled to Nakhichevan. And so, from
> this remainder, find how many Persian soldiers there were before the
massacre.
>
Answer: a(1-1/2-1/4-1/11)=280 -> a = 1760
Corollary: Armenians strike, slaughter, destroy, and massacre. After all,
they are not as innocent as the asala network claims.
Adam L. Schwartz
>> Problem 1
>>
>> My father told me the following story. During the famous wars between the
>> Armenians and the Persians, prince Zaurak Kamsarakan performed extraordinary
>> heroic deeds. Three times in a single month he attacked the Persian troops.
>> The first time, he struck down half of the Persian army. The second time,
>> pursuing the Persians, he slaughtered one fourth of the soldiers. The third
>> time, he destroyed one eleventh of the Persian army. The Persians who were
>> still alive, numbering two hundred eighty, fled to Nakhichevan. And so, from
>> this remainder, find how many Persian soldiers there were before the
>massacre.
>>
>
>Answer: a(1-1/2-1/4-1/11)=280 -> a = 1760
>
I thought the implication was that the prince destroyed one fourth of the
remaining Persian troops on the second round, and then 1/11 of those remaining
on the third round. This would mean
Answer: a*(1 - 1/2)*(1 - 1/4)*(1 - 1/11) = 280 --> a = 821.333
Mark W Kramer
A delightful message, interesting, and so kindly written. Thanks.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Prof. M. Kramer, Boston University
David Davidian
sdpa.org (David Davidian) who wrote:
[DD] Problem 1
[DD]
[DD] My father told me the following story. During the famous wars between the
[DD] Armenians and the Persians, prince Zaurak Kamsarakan performed
[DD] extraordinary heroic deeds. Three times in a single month he attacked the
[DD] Persian troops. The first time, he struck down half of the Persian army.
[DD] The second time, pursuing the Persians, he slaughtered one fourth of the
[DD] soldiers. The third time, he destroyed one eleventh of the Persian army.
[DD] The Persians who were still alive, numbering two hundred eighty, fled to
[DD] Nakhichevan. And so, from this remainder, find how many Persian soldiers
[DD] there were before the massacre.
[Koc] Answer: a(1-1/2-1/4-1/11)=280 -> a = 1760
Good for you! You win the prize -- a free trip to Karabakh as an Azeri
soldier! Now, calculate the odds of you coming back after trying to de-populate
the area of Armenians!
[Koc] Corollary: Armenians strike, slaughter, destroy, and massacre. After
[Koc] all, they are not as innocent as the asala network claims.
Fact: I didn't notice any mention of Turks in Shirak, Van, or Trebizon in
this seventh century story!
Fact: These places were filled with Armenians as of 1915.
Fact: By the end of 1916, after the Turkish genocide of the Armenians, there
were no Armenians left in Shirak, Van, or Trebizon -- only Turks and
Kurds! In fact, there were no Pontus Greeks left alive in Trebizon
either!
Conclusion: Numbers don't lie in either case!
Hovig Heghinian
Hmm ... Turks sure know how to keep track of deaths, but they seem to
lose count around 1.5 million.
Hovig