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.