From: dbd@urartu.sdpa.org (David Davidian)
Subject: Seventh Century A.D. Armenian Math Problems
Keywords: philosphy, Greece, Persians, math 
Organization: S.D.P.A. Center for Regional Studies
Lines: 366


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 dbd@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