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55-461: The Bakhshali manuscript is an ancient Indian mathematical text written on birch bark that was found in 1881 in the village of Bakhshali , Mardan (near Peshawar in present-day Pakistan , historical Gandhara ). It is perhaps "the oldest extant manuscript in Indian mathematics ". For some portions a carbon-date was proposed of AD 224–383 while for other portions a carbon-date as late as AD 885–993 in

110-553: A = ( n − 1 ) e + 2 b {\displaystyle (n-1)d+2a=(n-1)e+2b} n = 2 ( b − a ) / ( d − e ) + 1 {\displaystyle n=2(b-a)/(d-e)+1} This formula is contained in Bakshali Manuscript , folio 4v, rule 17 (Kaye III, p. 176) as follows: Ādyor viśeṣa dviguṇam cayasaṃdhiḥ-vibhājitam Rūpādhikaṃ tathā kālaṃ gati sāmyam tadā bhavet. "Twice

165-406: A 2017 study. The open manner and timing of the publication of these test dates was criticised by a group of Indian mathematical historians (Plofker et al. 2017 and Houben 2018 §3). The manuscript contains the earliest known Indian use of a zero symbol. It is written in a form of literary Sanskrit influenced by contemporary dialects. The manuscript was unearthed in a field in 1881, by a peasant in

220-406: A compilation of fragments from different works composed in a number of language varieties. Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical. A colophon to one of the sections states that it was written by a brahmin identified as "the son of Chajaka ", a "king of calculators," for the use of Vasiṣṭha' s son Hasika . The brahmin might have been

275-498: A first communication of these technical and historical matters. The Library thus bypassed standard academic channels that would have permitted serious collegial discussion and peer review prior to public announcements. While the excitement inspired by intriguing discoveries benefits our field and scholarly research in general, the confusion generated by broadcasting over-eager and carelessly inferred conclusions, with their inevitable aftermath of caveats and disputes, does not. Referring to

330-465: A number given in his numeral system by stating ankair api ("in figures this reads"), and then repeating it written with the first nine Brahmi numerals , using a small circle for the zero . Contrary to the word system, however, his numerals were written in descending values from left to right, exactly as we do it today. Therefore, since at least 629, the decimal system was definitely known to Indian scholars. Presumably, Bhāskara did not invent it, but he

385-512: A place-value system, using a dot as a place holder for zero. The dot symbol came to be called the shunya-bindu (literally, the dot of the empty place). References to the concept are found in Subandhu's Vasavadatta , which has been dated between 385 and 465 by the scholar Maan Singh. Prior to the 2017 carbon dating, a 9th-century inscription of zero on the wall of a temple in Gwalior , Madhya Pradesh,

440-754: A proscription on birch to Lugh, warning him; the text of this proscription can be found in the Book of Ballymote . The first letter of Ogham is beith ; beithe means "birch". Buddhist manuscripts written in the Gāndhārī language are likely the oldest extant Indic texts, dating to approximately the 1st century CE. They were written on birch bark and stored in clay jars. The British Library acquired them in 1994. They were written in Kharoṣṭhī and were believed to have originated from Afghanistan , because similar birch bark manuscripts had been discovered in eastern Afghanistan. Since 1994,

495-412: A rule is given, with one or more examples, where each example is followed by a "statement" ( nyāsa / sthāpanā ) of the example's numerical information in tabular form, then a computation that works out the example by following the rule step-by-step while quoting it, and finally a verification to confirm that the solution satisfies the problem. This is a style similar to that of Bhāskara I 's commentary on

550-464: A similar collection of Gāndhārī texts from the same era, called the Senior collection, has also surfaced. The British Library birch bark manuscripts were in the form of scrolls. They were very fragile and had already been damaged. They measured five to nine inches wide, and consisted of twelve- to eighteen-inch long, overlapping rolls that had been glued together to form longer scrolls. A thread sewn through

605-507: A traveller) is 2 and subsequent daily increment is 3. That of another, these are 3 initially and 2 as increment. Find in what time will their distances covered attain equality." The working is lost, but the answer, by the formula in the previous example, n = 2 ( 3 − 2 ) / ( 3 − 2 ) + 1 = 3  days. {\displaystyle n=2(3-2)/(3-2)+1=3{\text{ days.}}} The Bakhshali manuscript uses numerals with

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660-564: Is considered to be a vernacular dialect . The letters are of a personal or business character. A few documents include elaborate obscenities. Very few documents are written in Old Church Slavonic and only one in Old Norse . The school exercises and drawings by a young boy named Onfim have drawn much attention. The document numbered 292 from the Novgorod excavations (unearthed in 1957) is

715-549: Is divided into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x : which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation 16 5 π − 1 ≈ 1.859 % {\displaystyle {\frac {16}{5\pi }}-1\approx 1.859\%} at x = 0 {\displaystyle x=0} ). Additionally, he gives relations between sine and cosine, as well as relations between

770-422: Is explained by a deep culture layer in Novgorod (up to eight meters, or 25 feet) and heavy waterlogged clay soil which prevents the access of oxygen . Serious excavations in Novgorod started only in 1932, although some attempts had been made in the 19th century. Although their existence was mentioned in some old East Slavic manuscripts (along with a mention of Slavs writing upon "white wood" by Ibn al-Nadim ),

825-574: Is still used today in India and Nepal for writing sacred mantras. Russian texts discovered in Veliky Novgorod have been dated to approximately the 9th to 15th century CE. Most of those documents are letters written by various people in the Old Novgorod dialect . The Irish language's native writing system Ogham , sometimes called the "tree alphabet", was traditionally attributed to the god Ogma who wrote

880-782: The Bamiyan Caves . The approximately 3,000 scroll fragments are in Sanskrit or Buddhist Sanskrit, in the Brāhmī script, and date to a period from the 2nd to 8th century CE. The Bower Manuscript is one of the oldest Sanskrit texts on birch bark in the Brāhmī script. It includes several texts covering subjects including a medical treatise and proverbs. It was discovered in Kucha (currently in Aksu Prefecture in Xinjiang , China), an ancient Buddhist kingdom on

935-810: The Bodleian Library at the University of Oxford (MS. Sansk. d. 14), though folio are periodically loaned to museums. The manuscript is a compendium of rules and illustrative examples. Each example is stated as a problem, the solution is described, and it is verified that the problem has been solved. The sample problems are in verse and the commentary is in prose associated with calculations. The problems involve arithmetic , algebra and geometry , including mensuration . The topics covered include fractions, square roots, arithmetic and geometric progressions , solutions of simple equations, simultaneous linear equations , quadratic equations and indeterminate equations of

990-547: The UNESCO " Memory of the World " Register. During World War II , propaganda newspapers and leaflets published by guerilla fighters were sometimes printed on birch bark due to shortage of paper. Bh%C4%81skara I Bhāskara ( c.  600  – c.  680 ) (commonly called Bhāskara I to avoid confusion with the 12th-century mathematician Bhāskara II ) was a 7th-century Indian mathematician and astronomer who

1045-433: The gaṇita (mathematics) chapter of the Āryabhaṭīya , including the emphasis on verification that became obsolete in later works. The rules are algorithms and techniques for a variety of problems, such as systems of linear equations, quadratic equations, arithmetic progressions and arithmetico-geometric series, computing square roots approximately, dealing with negative numbers (profit and loss), measurement such as of

1100-665: The 16th century. A fragment of a birch bark scroll in Sanskrit, in the Brāhmī script , was part of the British Library Gandhara scroll collection. It is presumed to be from North India, dating to sometime during the first few centuries CE. Birch bark manuscripts in Brāhmī script were discovered in an ancient Buddhist monastery in Jaulian , near Taxila in the Punjab in Pakistan, and dated to

1155-584: The 20th century, most notably by victims of the repressions of the Soviet Stalinist regime. People in Soviet forced settlements and GULAG camps in Siberia used strips of birch bark to write letters to their loved ones back home, due to inaccessibility of paper. In 2023 birch bark letters from Siberia Estonia, Latvia, Lithuania, Poland and Ukraine, applied to include birch bark letters from Siberia (1945-1965) in

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1210-468: The 3rd or 4th century. Indian scholars assigned it an earlier date. Datta assigned it to the "early centuries of the Christian era". Channabasappa dated it to AD 200–400, on the grounds that it uses mathematical terminology different from that of Aryabhata . Hayashi noted some similarities between the manuscript and Bhaskara I 's work (AD 629), and said that it was "not much later than Bhaskara I". To settle

1265-494: The 5th century CE. The Bakhshali manuscript consists of seventy birch bark fragments written in Sanskrit and Prakrit, in the Śāradā script . Based on the language and content, it is estimated to be from the 2nd to 3rd century CE. The text discusses various mathematical techniques. A large collection of birch bark scrolls were discovered in Afghanistan during the civil war in the late 20th and early 21st centuries, possibly in

1320-555: The Devīkavaca text, a hymn praising the goddess Durga , were thought to protect the person who carries them from evil influences like an amulet or charm. An example of one of these texts in Devanagari script from Nepal is held at Cambridge University Library (MS Add. 1578). Birch bark is still used in some parts of India and Nepal for writing sacred mantras . This practice was first mentioned c.  8th or 9th century CE, in

1375-642: The Dharmaguptaka sect and probably "represent a random but reasonably representative fraction of what was probably a much larger set of texts preserved in the library of a monastery of the Dharmaguptaka sect in Nagarāhāra ", according to Richard Salomon. The collection includes a variety of known commentaries and sutras, including a Dhammapada , discourses of Shakyamuni Buddha that include the Rhinoceros Sutra , avadānas , and abhidharma texts. The condition of

1430-783: The Lakshmi Tantra. In Indian sculpture, a birch bark manuscript is easily identified by the droop. A palm leaf manuscript is stiff. On July 26, 1951, during excavations in Novgorod , an expedition led by Artemiy Artsikhovsky discovered the first birch bark manuscripts in Russia in a stratigraphic layer dated to around the year 1400. Since then, more than 1,000 similar documents were discovered in Staraya Russa , Smolensk , Polatsk , Vitebsk , Mstsislaw , Torzhok , Pskov , Tver , Moscow , Ryazan , and Vologda , although Novgorod remains by far

1485-515: The Pell equation 8 x 2 + 1 = y 2 {\displaystyle 8x^{2}+1=y^{2}} (or y 2 − 8 x 2 = 1 {\displaystyle y^{2}-8x^{2}=1} relative to pell's equation). This equation has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, such as (x,y) = (6,17). Bhāskara clearly believed that π

1540-479: The author of the commentary as well as the scribe of the manuscript. Near the colophon appears a broken word rtikāvati , which has been interpreted as the place Mārtikāvata mentioned by Varāhamihira as being in northwestern India (along with Takṣaśilā , Gandhāra etc.), the supposed place where the manuscript might have been written. The manuscript is a compilation of mathematical rules and examples (in verse), and prose commentaries on these verses. Typically,

1595-517: The date of the Bakhshali manuscript, language use and especially palaeography are other major parameters to be taken into account. In this context Houben observed: "it is difficult to derive a linear chronological difference from the observed linguistic variation," and therefore it is necessary to "take quite seriously the judgement of palaeographists such as Richard Salomon who observed that, what he teleologically called “Proto-Śāradā,” “first emerged around

1650-498: The detailed reconsideration of the evidence by Plofker et al., Sanskrit scholar, Jan Houben remarked: "If the finding that samples of the same manuscript would be centuries apart is not based on mistakes ... there are still some factors that have evidently been overlooked by the Bodleian research team: the well-known divergence in exposure to cosmic radiation at different altitudes and the possible variation in background radiation due to

1705-458: The difference of the initial terms divided by the difference of the common differences is increased by one. That will be time (represented by n {\displaystyle n} , cf. kāla iha padasyopalakṣaṇam) when the distances moved (by the two travellers) will be same." Dvayāditricayaś caiva dvicayatryādikottaraḥ Dvayo ca bhavate paṃthā kena kālena sāsyatāṃ kriyate? The accompanying example reads: "The initial speed (of

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1760-613: The discovery of birch bark documents ( Russian : берестяна́я гра́мота , berestyanáya grámota , and also grámota in those documents) significantly changed the understanding of the cultural level and language spoken by the East Slavs between the 11th and 15th centuries. Over two hundred styluses have also been found, mostly made of iron , some of bone or bronze . According to historians Valentin Yanin and Andrey Zaliznyak , most documents are ordinary letters by various people written in what

1815-506: The eastern Punjab, which was ruled by Harsha . Therefore, a reasonable guess would be that Bhāskara was born in Saurastra and later moved to Aśmaka . Bhāskara I is considered the most important scholar of Aryabhata 's astronomical school. He and Brahmagupta are two of the most renowned Indian mathematicians; both made considerable contributions to the study of fractions. The most important mathematical contribution of Bhāskara I concerns

1870-472: The edges helped to hold them together. The script was written in black ink. The manuscripts were written on both sides of the scrolls, beginning at the top on one side, continuing with the scroll turned over and upside down, so that the text concluded at the top and back of the scroll. The longest intact scroll from the British Library collection is eighty-four inches long. The texts were likely compiled by

1925-694: The fineness of gold, etc. Equality of Two Uniformly Accelerated Growths Let, S 1 = a + ( a + d ) + ( a + 2 d ) + …  to  n  terms, {\displaystyle S_{1}=a+(a+d)+(a+2d)+\ldots {\text{ to }}n{\text{ terms,}}} S 2 = b + ( b + e ) + ( b + 2 e ) + …  to  n  terms, {\displaystyle S_{2}=b+(b+e)+(b+2e)+\ldots {\text{ to }}n{\text{ terms,}}} If these two are equal, we must have ( n − 1 ) d + 2

1980-645: The line of Aryabhata's school: the Mahābhāskarīya ("Great Book of Bhāskara") and the Laghubhāskarīya ("Small Book of Bhāskara"). On 7 June 1979, the Indian Space Research Organisation launched the Bhāskara I satellite , named in honour of the mathematician. Little is known about Bhāskara's life, except for what can be deduced from his writings. He was born in India in the 7th century, and

2035-555: The middle of the seventh century” (Salomon 1998: 40). This excludes the earlier dates attributed to manuscript folios on which a fully developed form of Śāradā appears." Birch bark document Sanskrit birch bark manuscripts written with Brahmi script have been dated to the first few centuries CE. Several early Sanskrit writers, such as Kālidāsa ( c.  4th century CE ), Sushruta ( c.  3rd century CE ), and Varāhamihira (6th century CE) mention its use for manuscripts. The bark of Betula utilis (Himalayan Birch)

2090-536: The most prolific source of them. In Ukraine, birch bark documents were found in Zvenyhorod , Volynia ; while those from Belarus was unearthed in Vitebsk . The contents of the birch bark writings included not only religious writings but also document death of princes, conclusions of peace, dignitary arrivals, folk verses and local proverbs, even casual doodles. While legal related matters include accusations, witnesses and

2145-585: The northern Silk Road , and is estimated to be from around 450 CE. The Gilgit Manuscripts were Buddhist texts discovered in the Gilgit area of Pakistan in 1931 and include various sutras, including the Lotus Sutra , along with folk tales, medicine, and philosophy. They are dated to approximately the 5th to 6th centuries AD, and were written in Buddhist Sanskrit in the Śāradā script . Manuscripts containing

2200-459: The number 5 was given by the (5) senses . Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were to the right of the lower ones. Bhāskara's numeral system was truly positional, in contrast to word representations, where the same word could represent multiple values (such as 40 or 400). He often explained

2255-1562: The oldest known document in any Finnic language. It is dated to the beginning of the 13th century. The language used in the document is thought to be an archaic form of the language spoken in Olonets Karelia , a dialect of the Karelian language . For details and full text, see Birch bark letter no. 292 . Novgorod birch-bark letter №366, about 1360-1380 A.D. Case of trampled wheat, release. Original text (with added word division): сь урѧдѣсѧ ѧковь съ гюргьмо и съ харѣтономъ по бьсудьнои грамотѣ цто былъ возѧлъ гюргѣ грамоту в ызьѣжьнои пьшьнѣцѣ а харѣтоно во проторѣхо своѣхъ и возѧ гюрьгѣ за вьсь то рубьль и трѣ грѣвоны и коробью пьшьнѣцѣ а харѣтонъ возѧ дьсѧть локотъ сукона и грѣвону а боль не надобѣ гюрьгю нѣ харѣтону до ѧкова нѣ ѧкову до гюргѧ нѣ до харитона а на то рѧдьцѣ и послусѣ давыдъ лукѣнъ сынъ и сьтьпанъ таишѣнъ Translation (with explanations in square brackets): Here, Yakov has settled with Gyurgiy and with Khariton by courtless deed Gyurgiy has gotten [ at court ] concerning trampled [ by horses ] wheat and Khariton concerning his loss. Gyurgiy got one rouble [ money ], three grivnas [ money ], and basket [ measure ] of wheat for all that, and Khariton got ten cubits of cloth and one grivna. And Gyurgiy and Khariton have no more concern to Yakov, nor Yakov to Gyurgiy and Khariton. And arrangers and perceivers to that are Davyd, son of Luka, and Stepan Taishin. There are birch bark letters written in

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2310-401: The presence of certain minerals in exposed, mountainous rock have nowhere been taken into account." Prior to the proposed radiocarbon dates of the 2017 study, most scholars agreed that the physical manuscript was a copy of a more ancient text, whose date had to be estimated partly on the basis of its content. Hoernlé thought that the manuscript was from the 9th century, but the original was from

2365-482: The procedure of evidence, payments and fines, theft, fraud as well as wife-beating. One mundane personal writing reads "Sell the house and come to Smolensk or Kiev; bread is cheap; if you cannot come, write to me about your health. The document №752 stratigraphically dated as 1080–1100 AD is a passionate letter of an abandoned young woman torn in two and thrown away (by her addressee?). The late discovery of birch documents, as well as their amazing state of preservation,

2420-532: The radio carbon dates, initially via non-academic media, led Kim Plofker , Agathe Keller, Takao Hayashi , Clemency Montelle and Dominik Wujastyk to publicly object to the library making the dates globally available, usurping academic precedence: We express regret that the Bodleian Library kept their carbon-dating findings embargoed for many months, and then chose a newspaper press-release and YouTube as media for

2475-463: The representation of numbers in a positional numeral system . The first positional representations had been known to Indian astronomers approximately 500 years before Bhāskara's work. However, these numbers were written not in figures, but in words or allegories and were organized in verses. For instance, the number 1 was given as moon , since it exists only once; the number 2 was represented by wings , twins , or eyes since they always occur in pairs;

2530-421: The resultant linguistic peculiarities of the text are shared with Buddhist Hybrid Sanskrit . The overlying dialects, though sharing affinities with Apabhraṃśa and with Old Kashmiri , have not been identified precisely. It is probable that most of the rules and examples had been originally composed in Sanskrit, while one of the sections was written entirely in a dialect. It is possible that the manuscript might be

2585-749: The scrolls indicates that they were already in poor condition and fragments by the time they were stored in the clay jars. Scholars concluded that the fragmented scrolls were given a ritual interment, much like Jewish texts stored in a genizah . The bark of Betula utilis (Himalayan Birch) has been used for centuries in India for writing scriptures and texts in various scripts. Its use was especially prevalent in historical Kashmir . Use of bark as paper has been mentioned by early Sanskrit writers such as Kalidasa ( c.  4th century CE ), Sushruta ( c.  3rd century CE ), and Varahamihira (6th century CE). In Kashmir, early scholars recounted that all of their books were written on Himalayan birch bark until

2640-435: The second degree. The manuscript is written in an earlier form of Sharada script , a script which is known for having been in use mainly from the 8th to the 12th century in the northwestern part of South Asia, such as Kashmir and neighbouring regions. The language of the manuscript, though intended to be Sanskrit , was significantly influenced in its phonetics and morphology by a local artist dialect or dialects, and some of

2695-405: The sine of an angle less than 90° and the sines of angles 90°–180°, 180°–270°, and greater than 270°. Moreover, Bhāskara stated theorems about the solutions to equations now known as Pell's equations . For instance, he posed the problem: " Tell me, O mathematician, what is that square which multiplied by 8 becomes – together with unity – a square? " In modern notation, he asked for the solutions of

2750-403: The village of Bakhshali , which is near Mardan , in present-day Khyber Pakhtunkhwa , Pakistan. The first research on the manuscript was done by A. F. R. Hoernlé . After his death, it was examined by G.R.Kaye, who edited the work and published it as a book in 1927. The extant manuscript is incomplete, consisting of seventy leaves of birch bark , whose intended order is not known. It is kept at

2805-446: Was irrational. In support of Aryabhata 's approximation of π , he criticized its approximation to 10 {\displaystyle {\sqrt {10}}} , a practice common among Jain mathematicians. He was the first mathematician to openly discuss quadrilaterals with four unequal, nonparallel sides. The Mahābhāskarīya consists of eight chapters dealing with mathematical astronomy. The book deals with topics such as

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2860-400: Was once thought to be the oldest Indian use of a zero symbol. In 2017, samples from 3 folios of the corpus were radiocarbon dated to three different centuries and empires, from AD 224–383 ( Indo-Scythian ), 680–779 ( Turk Shahis ), and 885–993 ( Saffarid dynasty ). If the dates are valid, it is not known how folios from different centuries came to be collected and buried. The publication of

2915-519: Was probably an astronomer . Bhāskara I received his astronomical education from his father. There are references to places in India in Bhāskara's writings, such as Vallabhi (the capital of the Maitraka dynasty in the 7th century) and Sivarajapura, both of which are in the Saurastra region of the present-day state of Gujarat in India. Also mentioned are Bharuch in southern Gujarat, and Thanesar in

2970-630: Was the first to openly use the Brahmi numerals in a scientific contribution in Sanskrit . Bhāskara I wrote three astronomical contributions. In 629, he annotated the Āryabhaṭīya , an astronomical treatise by Aryabhata written in verses. Bhāskara's comments referred exactly to the 33 verses dealing with mathematics, in which he considered variable equations and trigonometric formulae. In general, he emphasized proving mathematical rules instead of simply relying on tradition or expediency. His work Mahābhāskarīya

3025-460: Was the first to write numbers in the Hindu–Arabic decimal system with a circle for the zero , and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata 's work. This commentary, Āryabhaṭīyabhāṣya , written in 629, is among the oldest known prose works in Sanskrit on mathematics and astronomy . He also wrote two astronomical works in

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