seventh century a d armenian math problems from quantum magazine march april pages 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 to 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 problems constitute 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 anania of shirak armenian mathematician of the seventh century translated and published by i a orbeli petrograd 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 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 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 the learned monk father kaloust used two manuscripts to publish the problem book supplementing it with an introduction and commentary in 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 when it is necessary to write fractions with numerators other than 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 and relate to the armenian uprising against the persians in a d problem 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 during the famous armenian uprising against the persians when zaurak kamsarakan killed suren one of the armenian azats 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 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 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 problem 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 problems but i ll restrain myself and offer you just one more problem 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 the hunting preserve of nersekh kamsarakan ter 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 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 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 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 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 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 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 pontus or pontus euxinus was an old name for the black sea azats were members one of several strata of freemen in ancient armenia 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 even explain cambridge ma turkish mp march
