Numerations in the Sinhala Language
N umerations in tke Sink ala Language Numerations in the Sinhala Language
N umerations in tke Sinkala Language
by Harsha Wijayawardhana
edited by Aruni Goonetilleke
3 Num erations in the Sinhala Language
Numerations in the Sinhala Language
© Harsha Wijayawardhana 9th Lane, Nawala Road, Rajagiriya, Sri Lanka. [email protected] www.ucsc.cbm.ac.lk/sdu
2009 October
ISBN- 978-955-1199-05-0
Design Sanjaya Epa Senevirathna
Information and Communication Technology Agency of Sri Lanka All rights reserved. No part of this document may be reporoduced or transmitted in any form or by any means without prior written permission from ICTA.
Published Strategic Communications and Media Unit - ICTA 160/24, Kirimandala Mawatha, Colombo 05, Sri Lanka. TP: +94 11 2369099 FAX:+ 94 11 2369091 email: [email protected] web: www.icta.lk Numerations in the Sinhala Language
To my parents and to my daughter Panchali. um a.tlions m llie Sinhala Language
Preface
The research into Sinhala numerals that ICTA initiated has yielded the fact the Sinhala language had several sets of Sinhala numerals, of which two sets had been widely used: one set (Sinhala Illakkam) was in use up to the early part of the nineteenth century, and the other set (Lith Illakkam) was in use well into the 20th century. The latter set clearly includes a zero and a zero place holder. ICTA’s Local Language Working Group, after reviewing the research and after extensive discussions with experts and stakeholders agreed that these two sets should be encoded in (he Sri Lanka Standard Sinhala Character Code for Information Interchange (SLS 1134 : 2004), in the Unicode standard and in ISO/IEC 10646 (the Universal Character set). The research was initiated when ICTA was informed that a proposal had been submitted hy Mr. Michael Everson, - a contributing editor to the Unicode Standard, and also to ISO/IEC 10646 to propose the inclusion of old Sinhala numerals in the Sinhala page of the Unicode standard. ICTA was of the view that there is no agreement of what Sinhala numbers are, and how I hey should be represented. ICTA therefore requested the Unicode Consortium to hold the encoding of Sinhala numerals in abeyance so that there would be time to study the issue and formulate a revised proposal. Thereafter, Sri Lanka was represented at the meeting of the Unicode Technical Committee in February 2008, by H.E Prof. J.B. Disanayaka, an expert in Sinhala. Consequently the Unicode Consortium agreed to hold Mr. Everson’s proposal held in abeyance for a while, giving time for Sri Lanka to conduct research into the subject, and formulate recommendations.
7 Num erations in the Sinhala Language
This is a discovery of which most users if the Sinhala language, up to now have not been aware. For this finding, the work carried out by Mr. Harsha Wijayawardhana of the University of Colombo School*, of Computing (UCSC) together with Prof. k.l). Paranavitana, and the contribution to the research made by all other stakeholders is deeply appreciated. Through information technology it is possible to protect a language and also make certain it is disseminated, made known and spread widely, and it is our mandate to ensure this through the e-Sri Lanka program.
Keshan Dewapura, Chief Operating Officer, ICTA NumniUions m the Sinhala Language
The case for ‘Sinhala Numerals’
The need to count beings and things has been there from the dawn of history. For this purpose ‘numbers’ such as ‘one’ ‘two’ ‘three’ and ‘four’ were created and to symbolize them in graphic form ‘numerals’ such as ‘1’ ‘2’ ‘3’ and ‘4’ were created. Many sets of numerals have come into being, from time to time, in different parts of the world. Of these sets, two have become universally accepted for everyday communication and numeration: ‘Arabic numerals’ such A as 1,2, 3, and 4, and ‘Roman numerals’ such as I, II, III and IV. There is one major difference between the two sets: Arabic numerals contain a ‘zero’ symbolized by ‘0’ but Roman numerals do not. Very few are aware that the Sinhalese have also created some sets of numerals for different purposes. In Sinhala, the words ‘anka’ and ‘illakkam’ denote both numbers and numerals. Mr. Abraham Mendis Gunasekera was one of the few scholars who took pains to record one set of these numerals for posterity. These numerals are given his book titled ‘A Comprehensive Grammar of the Sinhalese Language’ published in 1891. Very few bothered about these numerals because they were, for all intents and purposes, ‘defunct’. However, "a new interest in these numerals was rekindled in the late nineties of the last century when Mr. Michael Everson, a contributing editor to the Unicode Standard, proposed that Sinhala numerals be included in the Sinhala character code. Mr. Everson proposed that the Unicode Sinhala character code must contain not only the Sinhala letters and strokes but also Num erations in the Sinhala Language
the old Sinhala numerals. Since these numerals are not in actual use, Mr. T. Nandasara of the Institute of Computer Technology of the University of Colombo (now designated as the UCSC) and I were able to persuade Mr. Everson to postpone the inclusion of the numerals until a future date. That date, however, was not specifically mentioned and the matter was held in abeyance fpr some time. About a decade later, in February 2008, Mr. Everson took the matter up again at the meeting of the Unicode Technical Committee (UTC) held in Cupertino, California and again proposed that the Sinhala numerals be included in Unicode. 1 was requested by the l( T Agency of Sri Lanka (ICTA) to attend this meeting and persuade the Unicode Consortium and Mr. Everson to withhold the matter again to enable experts at ICTA, SLSI, UCSC and other stakeholders to study the issue more comprehensively. The Unicode Consortium agreed to hold the issue in abeyance because in addition to the set of numerals Mr. Everson proposed should be encoded in the Unicode standard, there is another set of Sinhala numerals that contain even a zero. The discovery of the zero is a major landmark in the history of mathematics. The fact the Sinhalese have created a set of numerals that include a zero augurs well for the mathematical genius of the Sinhalese. It must also be recalled at this point that the Sinhalese .were the first to introduce two distinct symbols [ ep^ ] and [ epi ] , as early as the sixth century, to represent the two vowel sounds that occur at the beginning of English words such as ‘apple’ and ‘ant’. These two symbols also created a problem for the Unicode standard because no other Indian or European alphabet has such a pair of unique symbols, and as such, their order in the alphabet became an issue, which was resolved amicably. One of the experts who continued to work on Sinhala numerals is Mr. Harsha Wijayawardhana of the University of ('olombo School of Computing (UCSC), on ICTA’s request. He has,
10 flKHt' lit dir Sinhala Language
ill.-i many months of intense work, found out that there are, at least, live different kinds of symbols used for numeration by the Sinhalese, in. hiding the one proposed by Mr. Michael Everson on the basis of Mi Abraham Mendis Gunasekera’s findings. Of these, only two sets have been in general circulation: Sinhala illakkam’ as recorded in Gunasekera Grammar, and ‘Lith illakkam’ mostly used by astrologers in casting horoscopes. Mr. Marsha Wijayawardhana and his team must be congratulated for the wonderful task they have performed in order to place Sinhala numerals on the global mathematical map.
J.H.Disanayaka I'mlcssor Emeritus, University of Colombo Ambassador of Sri Lanka to Thailand, Cambodia and Laos. Numerations in the Sinhala Language
Authors profile
Mr. Harsha Wijayawardhana is a Biochemist (who is a graduate of University of Miami, USA in Bio chemistry’and Chemistry ) turned Information and Communication Technology consultant with more than fifteen years experience in hardcore ICT. Mr. Wijayawardhana has been involved in Sri Lankan Government ICT projects and designed and implemented government networks such as Ministry of Foreign Affairs, Information Department of Sri Lanka as well as PRIU (Policy Research Information Unit of Presidential Secretariat). He was responsible for setting up of Software Development Unit (SDU) of UCSC, which employs more than 15 software engineers as well as student trainees. He is also responsible for some of the major national level software projects. Some of the recent national level software implementations are Birth, Marriage and Death Certificate issuance system which is setup throughout the country and House Holder List, which holds individual data on every Sri Lankan resident and is set up at every Divisional Secretariat. Mr. Wijayawardhana designed and implemented secure Election Commissioner’s network for dissemination of Election results for all the elections which were held since 1999 under the guidance of the late Prof. V. K. Samaranayake. He and team at SDU implemented the main software which was used for General Election 2001, 2004 and Presidential Election 2005. Mr. Wijayawardhana is involved in UNICODE Sinhala technology Research in Sri Lanka and serves on numerous language committees of Information and Technology Agency (ICTA) which is involved in implementing UNICODE standard in Sri Lanka. He was a member of committee which was involved in Standardization of SINHALA SLS1134. He lead a team of scholars in search of numeration in the Sinhala Language and the results of the research was published in the recently concluded National Symposium in Archaeology. Mr. Wijayawardhana had been involved in several pioneering Telecenter projects in Sri Lanka and has more than six years experience in providing technology to Rural telecenters. He is a member of the steering committee ofTelecentre.org Academy of Sri Lanka and is a member of drafting committee of syllabi for the Global Telecentre.org academy and he also guides Global Telecentre.org Academy in selecting e-Leaming technology platform. "> ilif Sinhala Language
Contents I Introduction 15 ? Researching into Sinhala Numerals 18 1 Universal Encoding or Unicode - Indie and other numerals encoded in the Unicode standard 20 •1 Numerals or numerations found immediately before the fall of the Kandyan Kingdom 24 5 Brahmi Numerals found in Sri Lanka 44 6 Evolution of Sinhala Illakkam and Lith Illakkam 48 7 Conclusion 59 8 Addendum - i (Discovery of Lith lllakkum at Manikdena Vihara, Dambulla) 60 9 Addendum - ii 123 10 References 133 / S o »<3 © 8082502?ocs 65
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01 Introduction
Sinhala belongs to the Indo-European language family with its roots deeply associated with the Indo-Aryan sub family to which the languages such as Persian and Hindi belong [Histroyl]. People in Sul .anka may have spoken a dialect of Prakrit at the time of arrival nl Buddhism in Sri Lanka. But there is enough evidence that Sinhala evolved from a mix of Sanskrit, Magadi (the language which was spoken in Magada Province of India where Lord Buddha was bom) and the local language spoken by the people of Sri Lanka prior to the arrival of Vijaya in Sri Lanka, the founder of Sinhala Kingdom I I)issa06:lll. It is also surmised that Sinhala had evolved from an A undent variant of Apabramsa (middle Indie) which is known as *Elu\ Some scholars believe that ‘Elu’ is a type of Prakrit which was from India but others argue that it was native to Sri Lanka. When tracing the history of Elu, it transpires that Elu was preceded by Hela or Pali Sihala [Histroyl], Sinhala has close relationships with Indo Aryan languages which are spoken primarily in the north, north eastern and central India, and was very much influenced by Dravidian language families of Southern India, especially Tamil. Sinhala has borrowed many Tamil words which have become part of Sinhala vernacular. Though Sinhala is related closely to Indie languages, it also has its own unique characteristics: Sinhala has symbols for two vowels which are not found in any other Indie languages in India: ‘Ae’ ( e p i ) and ‘Ae:’ ((epx). The Sinhala script had evolved from the Southern Brahmi script from which almost all the Southern Indie Scripts such as Telagu and Oriya had evolved. Later Sinhala had been influenced by Grantha writing of Southern India. Since 1250 AD, the Sinhala
15 Num erations in the Sinhala Language
script had basically remained the same with only a few changes. Although some scholars are of the view that the Brahmi Script ai rived with Buddhism, the Mahavamsa (Great Chronicle) speaks of written language even right after the arrival of Vijaya. Archeologists have found pottery fragments in Anuradhapura, Sri Lanka with old Brahmi script inscriptions, which had been carbon dated to the 5th century BC. The earliest examples Brahmi script found in India had been dated to the 6th century BC in Tamil Nadu though most of Brahmi writing found in India had been attributed to emperor Ashoka in the 3rd century BC [Brahmi02], Sinhala letters are round-shaped and are written from left to right and the Sinhala script is the script most circular in shape found among the Indie scripts. The evolution of the script to the present shapes may have taken place due to writing on Ola leaves. Unlike chiseling on a rock, writing on palm leaves necessitates the script to be more round-shaped to avoid the stylus ripping the palm leaf while writing on it. When drawing vertical or horizontal straight lines on Ola leaves the stylus would have ripped the leaf and this also may have been the reason for Sinhala not having a full stop or period. Instead there is a stylistic stop which is known as the ‘Kundaliya’. Period and commas were later introduced into the Sinhala script after the introduction of paper and due to the influence of Western languages. Although various scholars had mentioned numerations in the Sinhala language in their writings, a systematic study had not been conducted up to now on numerals and numerations found in Sinhala immediately prior to the British occupation of Kandy. In modem Sinhala, Arabic numerals, which were introduced by the Portuguese, Dutch and the English, are used for writing numbers and carrying out calculations. Roman numerals sometimes may be used for writing dates and for listing items or words in Sinhala. But at present, Roman Numvmlurns in the Sinhala Language
numerals which had been introduced by western countries which occupied Sri Lanka, are not commonly used. It is accepted that Arabic numerals had evolved from Brahmi numerals. It had also been discovered by Sri Lankan archeologists that Brahmi numerals were used in the ancient Sri Lanka and these may have evolved into I wo sets of numerals; these were known as archaic Sinhala numerals and Lith Illakkam found in the Kandyan period. This paper will mainly cover numerals and numerations in Sri Lanka at the time of British occupation of the Kandyan Kingdom and their evolution to the forms which were found in 1815, the year the British occupied all of Sri Lanka. This paper will also touch upon Brahmi numerals found in Sri Lanka. Numerations in the Sinhala Language
02 Researching into Sinhala numerals
The research into Sinhala numerals was carried out from both linguistic and mathematical perspectives. The researchers looked specifically for the existence of zero in any form of numeration in the Sinhala language, since the invention of zero had been a major demarcation point in mathematics. Advancement in modem pure mathematics would have not been possible without the concept of zero. Although zero had been discovered and re discovered independently by various civilizations in the world, it is now accepted that zero as an independent number was discovered and used for the first time by the Indian mathematicians and it had been taken to the west by the Arabs with other numerals which were developed in India from Brahmi numerals. E.T. Bell in his book, the Development of Mathematics, describes the development of zero by Indian mathematicians in the following manner:
“The problem of numeration was finally solved by Hindus at some controversial date before A.D. 800. The introduction of zero as a symbol denoting the absence of units or of certain powers of ten in a number represented by the Hindu numerals has been rated as one of the greatest practical inventions of all time” [Bell40:51] Ill ilicir research into Sinhala numerals or numerations, the author Inn I looked into the following:
i Papers or publications on Sinhala numerals, ii. Original documents which had some of form of numerals or numerations, iii. Rock inscriptions, iv. Ola leaf page numbering, v. Any evidence for zero in Sinhala numerals or numerations, vi. Numismatics
Shapes of several numeral sets which belong to Indie languages were compared with of the numeral sets which were identified as numerals or numerations in the Sinhala language.,The Indie numerals sets which were studied extensively were Thai, Lao, Burmese, and Malayalam numerals. The museums in Colombo and Kandy were visited many limes to study Ola leaf pagination by the authors. The Colombo museum library hosts an Ola leaf collection known as the W. A. De Silva Collection and which amounts to 5000. Some of the original and older Ola leaf collections were found to be outside Sri Lanka. A major Sinhala Ola leaf collection known as the Hugh Neville collection is located in Britain. The catalogue of this collection is available in Sri Lanka. Other countries in which museums are reported to host to Sinhala Ola leaf collections are in Arizona, US, Brussels, Belgium and Netherlands. Num erations in the Sinhala Language
03 Universal Encoding or Unicode - Indie and other numerals encoded in the Unicode standard
In the early nineteen eighties, text information in computers was represented as American Standard Code for Information Interchange (ASCII). This 7-bit code was able to represent 128 characters and it was originally designed to represent the English alphabet. Later an 8-bit extension was added to the earlier 128 encoding, which could then represent 256 characters. This allowed representation of other Latin Characters. Developers of Sinhala fonts mapped ASCII characters, replacing English letters with Sinhala letters. Different producers used different encoding for their Sinhala fonts which ensued in lack of cross application usage. The Unicode (Universal Encoding) standard was'introduced to allow non Latin scripts, such as Sinhala, to be encoded. Unicode is based upon the Universal Character Set defined by the standard ISO/ IEC 10646 adopted by the International Standards Organization. The 16—bit encoding in the Unicode standard allows encoding of 65536 unique characters. It allows backward compatibility with ASCII and other legacy encodings. A contributing editor to the Unicode Standard, Mr. Michael Everson from the US and Ireland, submitted a draft proposal on Sinhala to the Unicode Consortium, http://www.egt.ie/standards/ si/si.html. This paper included supposedly Sinhala numerals and in
2 0 Niimri.itums in the Sinhala Language
addition, the two Sinhala letters Ae’ (Cfi) and ‘Ae:’ (CfO had been pushed to the end of Sinhala Unicode page to facilitate transliteration with other Indian languages. Sri Lanka made a counter proposal, which was accepted by the Unicode Consortium and the two letters Ac’ (Cpt) and ‘ Ae:’ (£?;)> thereby were assigned to their proper positions aligned to the alphabet. At present, there are more than thirty unique written scripts encoded in the Unicode standard. Except for Sinhala, all other Indie and South East languages such as Khmer, Thai and Malayalam have included their respective archaic numerals in their Universal Character Sets (UCS). The Khmer UCS has two separate numeral sets, one of which is classified as a non calculation set (http://www.unicode.org/charts/PDF/U1780. pdf). 0 > ()
t9 b e-5 m O on ..■•a.. b d tv**
& d 9 gf too b nrY) g* d> X/ Figure 1: Figure 2: Figure 3: Khmer numerals Thai Numerals Malayalam numerals NiinmmUtum in the Sinhala Language Although ancient Tamil did not have a zero, a zero had been introduced into the Unicode standard when the UCS was prepared (http://www.unicode.org/charts/PDF/U0B80.pdi) for Tamil. Unicode chart of the Tamil language contains three signs for Ten, Hundred and Thousand other than zero. o e _ Hi yjSiUJii) Gjasirru QLpQSTFU jtro n c g , pukkiyam onnj irantu mOnru naranku 0 1 2 3 4 iP © &TT 6T < 9 4 nFTTOSTJg? 6TL_(lj) naranku aintu aru elu ettu 4 5 6 7 8 u ) ITT % 6^657 LJfchl pga GTnF) onpatu pattu nuru en 9 1D 100 1000 Figure 4: Tamil numerals 2 2 MmtifMiit'ii* lit III*- SlttltitlM I angmige Shapes of Tamil numerals look very similar to some of the I uml loiters and it is believed that Tamil numerals had evolved from llulhmi numerals. The following chart shows the evolution of Tamil iiuiiu-ials from Brahmi numerals in different periods from 200AD. NUMERALS IN ANCIENT TAMIL SCRIPTS 200 300 400 800 1000 1100 1200 1500 1800 •2000 A.D. AD. A.D. AD. A.D. AD. A.D. AD. A.D. AD. 1 — "N, \ -r -1- + f- & a S 9L 2_ 2. t SL- ""7C ' = 'i . ft. 1 + % t* T X * 5 h O AA 0 0 © <3. 6 f 5 ■D b "3 *1 * 7 ? 3 1 1 1 In parallel the Evolutions of Brahmi numerals in the form of Tamil letters in South India. Thanks to: S. Ganesan and R. Jayaratnan Jr. Epigrapher, Department of archeology, Thanjavur, Tamil Nadu, INDIA Figure 5: Evolution of Tamil numerals from Brahmi Num erations in the Sinhala Language 04 Numerals or numerations found immediately before the fall of KandyanKingdom to the British Five different types of numerations were used in the Sinhala language at the time of occupation of Kandyan Kingdom by the British. Out of the five types of numerations, two sets of numerations were in use in the twentieth century. These were mainly used for astrological calculations and to express traditional year and dates in ephemerides. The five types or sets of numerals or numerations are listed below. i. Sinhala Numerals or Sinhala Illakkam, ii. Lith Illakkam or ephemeris numerals, iii. Katapayadia, iv. Page numbering of Ola leaves using Sinhala ‘Swara (e38d)’, v. Bhootha Anka or Butha Samkaya i. Sinhala Numerals or Sinhala Illakkam Abraham Mendis Gunasekara, in his book “A Comprehensive Grammar of the Sinhalese Language”, describes a set of archaic numerals which had not been in use even at the time of publication of his book in 1891. According to Mr. Gunasekara, these numerals were used for ordinary calculations and to express simple numbers. These numerals had separate symbols for 10, 40, 50, 100, 1000 [Gunal891:144], These numerals were also regarded as Lith Lakunu or ephemeris numbers by W. A. De Silva in his “Catalogue of Palm leaf manuscripts”, in the library of Colombo Museum. This set of numerals was known as Sinhala Illakkam or Sinhala archaic numerals. 2 4 Num erations in the Sinhala Language ______ The following is an excerpt from “A Comprehensive Grammar of the Sinhalese Language”, by Abraham Mendis Gunasekara describing Sinhala Archaic numerals: “The Sinhalese had symbols of their own to represent the different numerals which were in use until the beginning of the present century. Arabic Figures are now universally used.” These old numerals are depicted below: Plate m. GS Gv, Gvu CU Gbrx> 0 1 2 3 4 5 6 (jj C2 ©n loo 200 *00 400 g r p aoo eoo < 2 3 * 800 8 00 lOOO W - I t * W&S “ Figure 6: Archaic Sinhala Numerals from Plate III of Abraham Mendis Gunasekara’s book ‘A Comprehensive Grammar of the Sinhalese Language’. These numerals did not have a zero and they also did not have zero concept holder. 2 5 N um erations in the Sinhala Language ( «** ) , Uttk *»wx~U 544 th» numlBtfin* m w rn m n a b f with k» * witti the won) « l m'iio the n uni ® In i'tiii'i 1 *f>jjwart. in iwrtwift mmmIm book* * form of I* aatd u» gbe I lUalpbalwt. 1i.o« ligunw mo known m Lit Lwkuna or Mgm umdtn M*ruk«iull r»M». 11m M kniw » a list 6N S V » v 1 5 a o ( j T t a 6 7 8 9 10 sy (O < $ 30 4 0 5 0 r 8 $ X «o 00 Ail* -sKS 1*-: 1 sNsSSf 100 Figure 7: Archaic numerals from ‘Catalogue of Palm leaf manuscripts in the library of Colombo Museum’, Volume I, compiled by W.A. De Silva, published by the Government Printer in 1938 One of the major findings had been the discovery of Sinhala numerals or Sinhala Illakkam in the Kandyan convention signed between Kandyan chieftains and the British governor in 1815. Eleven clauses had been numbered in Arabic numerals in the English part of the agreement and the parallel Sinhala clauses were numbered in Sinhala Archaic numerals. 2 6 MMMm ii* hi tin- Smhiilii l anguage F figure 8: First page of the Kandyan convention 2 7 Num erations in the Sinhala Language w ^ S > * '-" " " ' y ^ c n ^ <^<5sf7,OJ^ ' r-eW’J i- /!•,/ ’/+*/!/, yl ■— ’ v yrf£ ^■/■Gsv,<5«yi«V2>*' Ar**} y "T/t* rf^y* <4- srffln ^ fieQcsj S 8^ ^ ii )>##/' ^///> /•> * ^ O^h ^ ^aiScofc ,.„ . ^ >y#vV f,/Ss*S/sJ/^jstj’ jtftfe**'«/.^ aTSSv.d^aC0'vO®C ^ ’®'-t? | ,;%£/ jff* ftlr **+■//"-' ' yiy*'rsj4'/im **•* *y /(/' rn * fy ^ s Q jA.^Cv) ^)0 '/•/^*/v*^»' i '/f',/ /?s ,,> ■& $wS?Te^#b;^n^ancfc«*'r.} i &t-/sSjt/*,r s * r , t y STtcisnd^^Dl '^'r^'' -*v,y d f , y , „ . s ^ 2 ,-o,• / • / '<9/ A e 'fV/y/z' J J // /✓ / / ✓V, , /fSy y * fiirmt '. am mr0 .,i* y »»M»i^«w0'at>«555^ ! ./~„/'S A,.!') f r»/iA.rr,.S,r.. , r//r/y /?/■'■?/'/Zftp/ivis, OC^tJs^fe'KO'^^3' 1 *%/*£■<*■*+/*'¥•/ ^/V*^/^v/./ ' ‘ " / / / - 0*7, (5oC a. fO( Figure 9: Second page of Kandyan convention with number one depicted in Sinhala Illakkam. 2 8 ■ i it I It m im in l he Sinhala Language * vv •/.»/ * •■ - v * / ^ r/ - y C ? 3 : £//{/' S.*r r'y — . j /Yff'tcvV s/ft*/ ' .*** Y i ( S r'ffrv-/!* r / rs-nsr* * ■ I . y -*// >/ - n r ~ P , y /&£•/'y .-/*>*& >^u 50 « n * r -^ k * V^ ^3 0*1 fc^^ncov c r ry ..,./, ✓...... ✓ - . 5 r- , .'. tfe «^A -ft:3;!£yrfL S.r/ti?' ;$? -.•‘//ty* Si4"+r *.ts,f ^ ex-i^ostie Ss/r * ■ - ^ " ^ *> A ' J , ; j J „•».'>■.-> /,J s/sf/srtrt*tt \p.''.'*.-cjijC '-^*V, ..1 ».V /’>ry/i-"- ‘.‘S y f t • /* /y ^ 's s ? . / / * i ^ ‘4^1 j j / t . / Z , C>J 5/* r-^53 ^ cX - i ' > / . > . _ ’.y.An ?/y yv>* : /A t'/W 1 (' , “ v ' V ^ s^i '/•■ *C‘ f . - . - -• 1 ' O •■ "• /M’-'Asfrr ts‘-f s>'.*;**. ■ •■■’■1 ..-■ I ' S'VI V'l O C?0(!> A/& v £ , . s l igure 10: Third page of the Kandyan convention with numbers 2 and 3 clearly depicted in Sinhala Illakkam 29 Numerations in the Sinhala Language 2 3 4 5 6 V V 7 8 9 10 11 c / C ^ * r Figure 11: Sinhala numerals in the Kandyan convention of 1815 1 2 3 4 5 6 V % a GS GVv cn cvw .£> 7 8 9 10 11 0 > v i > < / o r v < 3 @1 CM CMGN Figure 12: Sinhala numerals from the Kandyan convention with Sinhala numerals from Mendis Gunesekara’s book. 3 0 The set of numerals found in the Kandyan convention is given above with Mendis Gunasekara’s depiction of the same (minerals. The numbers 2 and 3 of the Sinhala numerals found in ilif Kandyan convention differ from the manner in which Mendis (limaseskara had depicted the same set of numerals. The above a I has been taken as the most accurate set when the shapes of the numerals are considered. The above assumption has been proven by i summing the pocket watch of Sri Wickrama Rajasinghe, the last king of Kandy. The watch gives considerable information as to how time w us expressed in Sinhala. A Sinhala day is made of sixty Sinhala hours. Therefore the watch has thirty Sinhala hours for a half day. Figure 13: the pocket watch of Sri Wickrama Rajasinghe, the last king of Kandy. Numerations in the Sinhala Language ii. Lith Illakkam or ephemeris numerals Although this numeral set was commonly used for casting horoscopes and for carrying out astrological calculations, this set had been used for numbering pages of Ola palm leaf books which covered primarily of non Buddhist topics. Dr. P. De S. Kularatne in his article on Numerals and Numerations in the Sinhala Language published in the Sinhala Encyclopedia claims that this set was used only for astrological calculations [Kula67], r ’' ~ f 7 WT r-o Figure 14: number 1 in Lith Illakkam ""111V 11 wiWBiwwwTnwinn mvnin n w ffn iw ■ »m t 'i m :i iimininnifirwB^fniiwwi V ~ -. * - Vi 03 <*v *1 AlV-Y *5.'£sr) '*51 Figure 15: N um ber 2 3 2 lUftMtMinii'. 111 ilu- Sinhala Language * }<5T?v~rr> . ” V > t '*>»» * • > £ £ » xx i > c e ^ ) ' - X '- 'A - .,<- *o ASo c x s jq ■f‘''r'-) ~-i •'-->^ figure 16: N um ber 6 <3\ Number 1 or kombuwa Number 3 or Naa from 8th Century Number 4 or letter Ja % Number 5 or letter Da Number 9 or letter Ni Number 11 or two kombuwas Number 13 or Naa from 8th Century f igure 17: Lith Illakkam Numbers are depicted in Lith Illakkam by using Sinhala letters and vowel modifiers. There are mainly two versions of these Illakkam according to the way numbers 2, 3 and 9 are depicted. The number liix is known as ‘akma’ in Lith Illakkam. These numerals were in use continuously for writing horoscopes on Ola leaves. This tradition continued until the beginning of the twentieth century. Both versions of Lith Illakkam have a zero. The zero is depicted by the Halantha or Hal lakuna (kodiya) in the Sinhala language. Although it is not understood whether Sinhala mathematicians treated zero as a number, it is possible that they had known the concept of zero. In Lith Illakkam, numbers greater than zero were written in the same way as Arabic numbers, with the value of the number on the left being increased by ten with the zero. In other words, Lith Illakkam had a zero and a zero place holder concept. Lith Illakkam version 1 used the Sinhala letter “Murthaja Na” for numbers 2, 3 and 9 from the 6th to the 8th century. In the second version of Lith Illakkam W. A. De Silva had depicted in his book, the Sinhala letter, ‘Na’ (zn) with vowel modifiers to depict the numbers 2, 3 and 9. 0 12 3 4 5 6 7 8 9 Figure 18: Lith Illakkam version 1 3 4 mis 111 ihe Sinhala Language • ||i |Am* of the above sign* certain letters of the Sinhalese Alphabet are frequently used ; j ^ l 2 3 4 5 ti 7 8 9 0 I ip.mc 19: Lith Illakkam version 2. W.A. De Silva describes of Lith Illakkam, and lie hud placed a vowel modifier for number 7 which has to be Letter OO 6 60 6 0 ) £ l 2 3 4 5 C <5 ©T 6 7 8 9 io 3 0 ©&) © 60) 6CS it 12 13 14 15 6 ©0 Figure 20: 14 Dr. P. De S. Kularatne on Lith Illakkam (20 is incorrectly depicted) 3 5 Num erations in the Sinhala Language i * 2 3 4 5 t o S C J c f 6 7 8 9 1 0 C2 i 9 C < 5 < h i Figure 21: (g s$ f£ o 23>) or Ephemeris numbers from Hendrick De Silva Hettigoda’s astrology book Figure 21 provides the third version of Lith Illakkam but this version was observed only in this particular book. The difference of this version is that to depict the zero, instead of using the hal lakuna only, this version uses a ‘Pilla’ with the hal lakuna[Hetti87]. Abraham Mendis Gunasekara, in his article on numerals and numerations in the Sinhala language, published in his magazine “Granthanvaya” refers to Lith Illakkam as well as to Sinhala Illakkam. For Sinhala Illakkam, he produces the same shapes which had been given in his English book. Abraham Mendis Gunasekara uses modern Sinhala letters and vowel modifiers which is Version 2 of Lith Illakkam. In this article, he clearly mentions that zero is depicted by the Hal lakuna or ‘Kodiyak In other words, ‘Sunyathana’ filled with a kodiya will multiply by ten, the number which is on the left side of Sunyasthana. Abraham Mendis Gunasekara clearly states that instead of hal lakuna of the Sinhala language, a ‘Shunya binduawa’ (zero place holder) can be used to fill the ‘Sunyasthana’. In other words, Lith Illakkam uses duality of zero to write numbers greater than 9[Guna-Articlel891], This article counters that Lith Illakkam may have been used up to 60 by P. De. S Kularatne[Kula67]. 3 6 h i t i m i util m i 111 ihc Sinhala Language ($}€£$$ g i© S ) 5 f . &&* : — • W«b2r> *>»>. % 3 • 1 so t& co & S z n : 2 . 3 cr 4 /« •* <^C625tf*0 , t‘ 5 qvVS ii . ' 6 c C .v s z s S t t 7 a* v,# IS® $<%35i£fl©9<3d c?3?4^ ®c0££a(3 $©£»•*>&>© qtaiD atfccstf 9 < ® £ . d S h S s r f ©*•* ceag X) % 00^9 20% 2S5J <3»3 S0&5 ^JWjCCO *& © 3 t>}Z$€i& ( 0 ) 4 © 8 < § s 2 tf$ 9 0 ® : O « 0 $> 3 4 03.25 <5d"j 1 0 2 cMQ €»*' *ft©S3j 1547c«0 «^cS*C; 1807 cos£ ©«5^C- Figure 22: Abraham Mendis Gunasekara on Lith Illakkam. In his examples, he had used zero to write 102 instead hal lakuna According to the above article, the following combinations in the figure 22 are possibilities. N um erations in the Sinhala Language A c c o f J ' i h ftt, fr/fain^ Oo r» Gi n aho-oS #r<_ 4 ^CVS.Oit^ (o10 <5Vl o { I 0 z 10 2. 6$GT) ^ 2 0 G o t ) <------elo-n.Jfrcj 101 I O - j. >0-2. tent, in tv. o,f tfl a*«M- 6\f ? 200 j hm Mlk. OunriOe'-f G-. f tj not Lof r. -Teen V j u t/' 1 *" Figure 23: Possible combinations using Lith Illakkam 3 8 l igure 25: Ephemeris in Kandy Museum written with Lith Illakka 3 9 N um erations in the Sinhala Language Figure 26: Fioroscope which was cast in Figure 27: Same horoscope with October 1936 and some of the numerals Lith Illakkam are Lith Illakkam with older ‘Na’ 4 0 Il was found that the tradition of using Lith Illakkam 11m i aiding horoscopes survived into the twentieth century. The lit mi I'n i * | u* depicted in the figures 26 and 27 had planets’ positions fcilfii n wilh version 1 of Lith Illakkam. It was surmised that Lith llltikknni version 1 had been replaced by version 2 after the British mi 111|mil mu of Sri Lanka. On the contrary, the above horoscope Mlivldes ample evidence that Lith Illakkam version I had been 11 mi ii i ion I y used by some of the Sri Lankan astrologers even at the |u'y iniiing of twentieth century. This horoscope provides evidence of in I'M'.lenee of Arabic numerals with Lith Illakkam. iii. Katapayadia Even to this day, years are given in the front page of popular r|ilu ineris in Sri Lanka, ‘Panchanga Lith’ using ‘Katapayadia’. kiiinpayadia is a unique numbering scheme where numbers 1 to 9 tiiul 0 have been depicted by Sinhala consonants. The katapayadia is 11 in inly used for writing dates. 1 2 3 s 6 1 8 9 0 e » s> <53 © © 6 d 0 <± © £3 oC *» £> < © * 3 C3 e © 8 c o . <5 <3 e © a © © I inure 28: Katapayadia (borrowed from Life and Planets, the book written by I Iriulrick De Silva Hettigoda This is numeration is known as Katapayadial since number one is assigned the Sinhala letters ‘Ka’ (as), ‘Ta’ (O), ‘Pa’ (o)and ‘Ya’ (io). In this tradition of writing numbers, the year 2007 can be written Num erations in the Sinhala Language for instance as ‘Ka’ (S) ‘Na’ (zn) ‘Na’ (zn) ‘Sa’ (ea). Traditionally, 2007 will be written from right to left: 7002. Ordinarily, using vowel modifiers, a word in Sanskrit will be created for the year 2007 (7002 right to left) \$ith the allocated letters for 7002. When reading, one has to remove the vowel modifier. Katapayadia was widely used by South Indian astrologers and some of Chola rock inscriptions in Sri Lanka have dates inscribed in Katapyadia. iv. Page numbering of Ola leaves using Sinhala ‘Swara The method of page numbering of Ola using Sinhala Swara with consonants had been the common tradition in the ancient and recent history of Sri Lanka. The author had found that using Sinhala Swara in place of numeration could be traced back to Aryabhata’s (the great Indian Mathematician and Astronomer) numbering system where he used Sanskrit Swaras in place of numerals. Sinhala scribes had developed their own numeration based upon Sinhala characters according to the order of the position of consonants and vowels in the Sinhala Alphabet without the modem two vowels: ‘Ae’ (e^) and ‘Ae:’ (fpj) in the Sinhala Alphabet (the Sinhala alphabet without the above mentioned two vowels is known as ‘Pansal Hodiya’ or the alphabet of the temple). The numeration method which is similar to the use of Sinhala Swara is found in the Burmese Ola collection. The tradition of Swara as numeration in page numbering in Ola had been commonly used for Buddhist manuscripts. The author had the opportunity of examining several Ola palm leaf books which are in the Colombo museum 1 and the catalogues of Hugh Neville5 collection in the London Museum2. Having investigated paging of Ola leaves, the majority of palm leaf manuscripts which are in the ItlliMi iim laid Sinhala consonants with ‘swara’ (ed©d) (combinations Ml mi miiuIs) for numbering. The number of combinations which can lie iimdi out of consonants is 544 and once the first 544 finishes, |ti»yiii[i begins with the second cycle of 544 with the word ‘dwi:’ h i second in English. If the second cycle does not end the palm Uuil hook, it goes into third cycle of 544 which begins with the word ’llin t (!*/>) or Three in English [DeSilva38], v. Bhootha Anka or Butha Samkaya In Sinhala literature, certain words in the language were used to denote numbers. For instance, the sky is associated with zero hi ‘Sonya’, and a number which was denoted by words is known o'. Hhoota Anka. Bhootha Anka was created by ancient Sanskrit M.illicmaticians and Astronomers prior to the invention of a symbol Ini zero. Some of the words which are associated with numbers are Moon = one Eye = two Fire = three If one were to write 130, he or she would place moon, fire, ■Ay together to form the number. Pierre-Sylvain Filliozat in his .nl icle ‘Ancient Sanskrit Mathematics: an oral tradition and a written literature”, he describes Bhootha Anka as object-number metronomic expressions. As it was mentioned previously, knowledge was transferred through memory rather than writing it down. In order to make memorization easier, it is natural that the numbers are placed as words and the words are formulated sequentially so that they would sound rhythmical. The Indian tradition of Bhootha anka was imported to Sri Lanka as it was used in India and the tradition continued with Sinahala words that had same meanings. 05 Brahmi numerals found in Sri Lanka Dr. Senarath Paranavitana (the first Sri Lankan Commissioner of Archaeology) and Dr. Abaya Aryasinghe had independently found in their research that Sinhalese had used numerals which closely resembled Brahmi numerals of India in the early days of Sinhala civilization. The evidence for use of Brahmi numerals had been discovered primarily in rock inscriptions which were inscribed in between AD 200 and 400. These numerals were used to record donations given by royals and other people who belonged to the upper echelons of ancient Sinhala society, to Buddhist temples [Abi90:90], Brahmi numerals are ancestors of Arabic numerals which are used presently worldwide. Brahmi numerals had symbols for 10, 100, and 1000. Numbers 1 and 10 in Brahmi have not been found in Sri Lanka up to now. Therefore shapes of these two numerals have been hypothesized without physical evidence [Menninger92:395], In addition, Sri Lanka Brahmi numerals seemed to be void of symbols for 30, 40, 80 and 90. Although further research into these symbols is required, the author came upon what looked like “90” in Anuradhapura. In Gupta numerals in India, “90” was represented by a circle divided perpendicularly into four equal parts. This symbol was very common in Sri Lanka on rock inscriptions, and it had been categorized as a non Brahmi symbol by Dr. Senarath Paranvitana. Figure 31 is the Bavachakra found at the Ranmasu Uyana. This contains symbols which look similar to the Gupta 90 and they occur on this inscription as a group of four bound by a four equally divided square. Four of 90 added up to 360. The number 360 has always had 4 4 Ill mu r ,is degrees for a full circle. Therefore it can be surmised I ll'«' Si I 1 ankan archaeologists may have overlooked the non §*!**' " i' ol '>() in Sri Lanka! Nlnhulu roek inscriptions suddenly become barren of IIIIHmtil*. |roni A.l). 400 onwards. Tradition of writing numbers in MMd hm uinC8 more prevalent from the above period. 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 o 7 7 3 8 ® © 9 1 2 3 4 5 6 7 8 9 10 - - = 20 30 40 50 60 70 80 90 1001000 75- - 3 - >| - - -A J 1 '^urc 29: The first set is found during 3 BC and the second was commonly found 111 * All -) 'n India and the Sri Lanka set without 80 and 90 Figure 30: number 4 in Kandy museum igure 31: Bavachakra at Ranmasu Uyana 01) (- 13 2 0 --- 02 30 03 40 04 9 5 0 l 05 60 06 b 7 0 1 07 80 08 90 2 09 100 as 10 (?)) § 1000 figure 32: Brahmi numerals in Sri Lanka. Numbers 1 and 10 had not been found Num erations in the Sinhala Language 06 Evolution of Sinhala Illakkam and Lith Illakkam i. Sinhala Illakkam Sinhala Illakkam had evolved from Brahmi numerals which were found in Sri Lanka. Since Brahmi numbers 1 and 10 had not been discovered in Sri Lanka, it is not possible to know the shapes of those two numerals in order to compare with Sinhala Illakkam. Shapes of these two numerals were surmised by comparing with the Indian Brahmi numerals of the same period. In Sri Lanka Brahmi number 4 had evolved parallel with the Indian Brahmi, but Sinhala Illakkam had evolved into more complex shapes than those of its Indian counterparts (Arabic numerals had evolved from sets of Indian numerals) due to cursive shapes of Sinhala letters. Shapes of Brahmi number 100 and 1000 found in Sri Lanka are different from of Brahimi 100 and 1000 which are commonly found in India. The following is the probable evolution process of Brahmi into Sinhala Illakkam. a 1 \ Figure 33: Evolution of Sinhala Illakkam ^Hlii'inlm its in the Sinhala Language I igure 34: Evolution of Devanagri Sinhala lllakkum — ■'AGV Figure 35: Evolution of Sinhala numerals Num erations in the Sinhala Language The author modeled the evolution of Sinhala Illakkam using Morphing Software. Modeling enabled the author to see possible stages of evolution not discovered physically. The evolution of Numbers 1, 2 #nd 3 of Brahmi into Sinhala Illakkam can be easily seen, compared with how Indian numerals had evolved from the Indian Brahmi. The author found some evidence for Sinhala Illakkam at the Vatadageya in Polonnaruwa. Floor tiles of Vatadageya in Pollonnaruwa were made from rock slabs which were quarried and were brought from a distance far from the present location. Since the Vatadageya is a circular structure, the builders had numbered the tiles before transporting them to the present location. At the Vatadageya, three different categories of numbering systems were found. At the first level, rock slabs were numbered using Sinhala ‘Swara’ and at the second level, a clearly depicted number 1 of Sinhala Illakkam was found. Figure 36: Sinhala ‘Ku’ which resembles 5 im ilii Sliihdla I.anguage I ii'iire 37: Sinhala Illakkam number 1 The three categories of numbering found at Vatadageya were: 1. Possible Sinhala Illakkam 2. Swara based numbering found in the Ola leaf paging 3. Sanskrit letter (swara) and Indian Numerals. Figure 37 provides evidence that some of Sinhala Illakkam had been used for numbering rock tiles at Vatadageya. Figure U> provides evidence for Swara based numbering. Figure 38 is evidence for the third category of numbering at Vatadageya. When considering the history of Vatadageya, it was originally built by King Parakramabahu the first and it was again restored by Nissanka Malla. At the end of Pollonnaruwa period, the Vatadageya had been destroyed and it was again restored for the last time during the I )ambadeniya period in the ancient times until the 20th century. N um erations in the Sinhala Language Figure 38: Deva nagri ‘tha’, and surmised to be 7 i. Evolution of Lith Illakkam or Ephemeris numbers Unlike Sinhala Illakkam, the author had difficulty in tracing the evolution of Lith Illakkam. The author had looked for any evidence in India for the origins of this set in vain. At present, almost all evidence points to distinctly Sri Lankan origins for Lith Illakkam. Shapes of number 2, 3 and 9 of the first version of Lith Illakkam are self evident when their evolution is considered. They date back to the sixth to eighth century depiction of the Sinhala Letter ‘Murthaja N a’. 5 2 § 9 £ G < 5 cDc»® »C d S H l A z t : t i l / 5 H - - l A ? * 0 LP 6 H ^ ’ - A 9 ^ ^ 8 0 ] » ^ £ ) v> t£ < j in 12 £/» ffv> -<3 - ^ «<> 13is « n ls &> a) t A Ui d I E ^ 1> ,2. + A d £ £.+ 2) o 2) d ® 29 ^~) a ; 29 ^ I f u® JL ^ (TV ^ 0 0 r } 0 c f 0/" B‘'5»C o32J>3®4S igure 39 a: Evolution of Sinhala letters 53 Numerations in the Sinhala Language 0 <5 Q a CD 0 5 . c?* c o d I A o ;> a I 5 G t T h ' L 5 1 c C Co I. I 8 L/ ° 9 % CO b 12 \ V O 3 U (9 O H y A { li 6 < 3 u o A & ^ ) 0 6 o U O 8 t~> <) 6 i 8 o ^ Cn 9 u) U Figure 39 b: Evolution of Sinhala letters The very selection of specific letters for 1 to 9 had been a puzzle to the author from the inception of the research. The author had come up with a possible hypothesis for the selection of the specific Sinhala consonants, vowel and vowel modifiers. 5 4 i. Hypothesis It is assumed that the both Sinhala Illakkam and Lith Illakkam iinst their origins to Brahmi numerals. It had been observed that up In MK) Al) Sinhala had been written with Brahmi letters with slight UliMlilications. From 600 AD, Sinhala letters had begun to be more i in mvc like the other scripts in the region. This may be attributed to iln introduction of writing utensils such as the stylus and Ola palm willing to the region. Sinhala was also influenced primarily by the I'ullava Grantha writing during this period. It is possible to surmise tlini Brahmi numerals had also begun to be more cursive and they gave rise to two variations of Brahmi numerals. The numerals that NIC known as Sinhala Illakkam began their evolution without a zero Imi with the symbols becoming more cursive, elaborate and complex. I Ins set may have taken some time to evolve into the numerals which Inn! been found in the Kandyan period. The other set took a shorter unite and an anonymous inventor had seen the similarities of cursive Bi.thmi numbers to evolved cursive consonants, vowel modifiers in iln then existing Sinhala alphabet. A single horizontal line became Kombuwa as it became more cursive. The numbers two, three and nine took the shapes of ‘Murthaja NA’. Other letters took similar shapes of similar looking letters. For example, seven took the shape id Sinhala vowel ‘0 0 ’. The idea of introducing Sinhala letters may nol have been an accident since Sinhala grammarians in the past had k nown of what halant did to a consonant when placed with it. Indian linguist Pannini who lived around 3rd BC had introduced Shunya ,iiul nullifier concepts to Sanskrit. His language rules shaped almost .ill languages including non Indo Aryan languages in the region. When the halant is placed with a consonant, the consonant loses its inherent vowel, thus neutralizing the consonant. The halant has been identified as a nullifier. In Sinhala the kodiya is used for the halant. N um erations in the Sinhala 1 unguiuM The Sinhala inventor who had known of the Shunya or zero may have realized that by having the Sinhala halant with these nine letters of Sinhala alphabet which denoted numbers would increase by ten. It is interesting*to note Number 2, 3 and 9 remained the same for many centuries into Kandyan period even after ‘Na’ evolved into a entirely different shape. In Tamil, halant is the pulliya or dot, placed on top of a consonant. It is also very interesting to note that the present numerals which are used in the Middle East use a dot for zero. A Figure 40: Numerals used in the Middle East: these are known as Indian Numerals — 9 0 = a j ) Is til V K -Iff*— & _5®L n Fll _25L_ V\\J / li f t . SrfJ <§\ Figure 41: Lith Illakkam would Figure 42: Khmer have been in 6-8AD 56 ■ p i tflnlull.1 I .lll|'ll,lf« ii. Zero in the Indie numerals I ndinn mathematicians had been credited with the introduction " f | 0lii In I he world. It is possible to assume that the concept of zero tMil lim'ii in India with its religions and philosphies for more than Ittu thousand years since Buddhism and Jainism both include the (hiltivn or the zero concept. As mentioned previously, Pannini had H ill given credit for introducing shunya or nullifiers to language. I In* piesent English word zero owes its orgins to the Sanskrit word •liuoyii Shunya traversed through the Middle East as ‘safira’ or iiuplv' in Arabic, into Venice. The great Indian Mathematician Aivuhliuta has been credited with the use of zero in Mathematics. Although there were several claims in India for the first known •ihunyn binduwa or zero, it is now accepted that the first verifiable *i*io was found in a temple in North India, the location of which is known as Gwalior; hence the Gwalio zero. It is dated to 876 AD. I i^ure 43: Gwalior zero In the ancient Khmer kingdom which owed much of its orgins to the Indian civilization, one of the world’s first zeros on a rock inscription is recorded. N um erations in the Sinhala Languayl Q ' f 6 0 5 Figure 44: Khmer zero Shaka year 605 had inscribed with the Khmer numerals and this translates to 683 AD. This is found in a Sanskrit verse. iii. Importance of Sinhala Zero Hitherto many scholars believed that the zero was not used in the Sinhala language. Apparent use of Lith Illakkam provides a clue that the zero had been used in astrological calculations in Sri Lanka. There are no clues as to whether the zero was used as a number in its own right in ancient Sri Lanka with its own operations. But one can surmise that they had known about attributes of zero. Since division and multiplications are required for complex astrological calculations, it is possible to surmise that astrologers of ancient Sri Lanka may have known of the operations on zero. Lith Illakkam zero provides insight into as to how zero came into being. The Sinhala zero provides a clue that the zero in numerals would have owed to nullifier concepts in language. 58 Ml llir Sinhala Language 07 Conclusion Although there is a vacuum and silence where Sinhala mathematics is concerned, it can be said without doubt that the Sinhala language had several well developed numeral sets. Although numbers are depicted in words in many rock inscriptions after 400 A I), the evolution of numerals may have occurred in parallel with the Sinhala letters from Brahmi numerals. Sinhala Illakkam is the older brother or sister of Arabic, and Devangri numerals. But unlike Arabic and Devanagri, they never evolved into a set with a zero. Instead Sinhala astrological mathematicians, ‘Ganithas’, developed I heir own numeral set from Brahmi numerals which is known as I ith Illakkam. This set is unique to Sri Lanka and it was based upon Sinhala consonants and vowel modifiers. This set possessed a clear zero and a zero place holder unlike Sinhala Illakkam. This numeral set was used to carry out tedious calculations such as finding the exact timing of the end and the beginning of the year based upon the traversing of the sun in the zodiac or movement of earth in the solar system. Further research is required to be carried out to establish whether Sinhala mathematicians had treated zero as a number. Numerations in the Sinhala Languor Addendum - i Discovery of Lith Illakkum at Manikdena Vihara, Dambulla The authors surmised that Lith Illakkum would have been invented between 8AD to 10AD. This surmise was done taking into consideration of the shapes of number 2, 3 and 9 of Lith Illakkum. The shape of number 2 in the Lith Illakkum is ‘Murthada Na’ which is found in writings between 8AD to 12 AD. This letter was used for number 2 in the Kandyan period for Astrological calculations as well as for page numbering of ola leaves using lith illakkum. The numbers 3 and 9 were Murthda Na in 8AD with vowel modifiers. The authors further surmised that Lith Illakkum would have evolved from Brahmi numerals. An unknown inventor of Lith Illakkum had chosen similar looking Sinhala letters to Brahmi numerals as they started to be more cursive. Although the time period for origins of lith illakkum was only surmised, the authors lacked any physical evidence to prove the above hypothesis until recently. It was by accident that one of the authors visited newly inaugurated Museum at Sigiriya. The new museum at Sigiriya had been renovated with the Japanese financial assistance and it has artifacts which were discovered from locations, which are closer to Sigiriya. In one of the galleries, the author observed that a base of a small Stupa had been reconstructed with the bricks (with circular shapes to fit a stupa base) which were found from excavations at an ancient temple which is known as Manikdena Vihara at Dambulla. The period, which was given for the bricks, were 9AD to 10 AD. The author observed that Sinhala letters had been inscribed on some of the bricks. It was further observed that one or two bricks had more than (»,. letters but less than six. The majority of bricks had either one or |*vo Sinhala letters. To his surprise, Kombuva played a major role in in Min* of writings on these bricks. In ordinary writings, Kombouva is |ilm ed before a Sinhala consonant not after. Some of the bricks with (wo letters inscribed had kombuva placed after a consonant. One or Iwo bricks had Kombuva alone with a line crossing diagonally. The iiuilior also further observed that Murthada NA from 8AD placed In lore a Kombuva. He surmised that these were Lith Illakkum. The staff at the Sigiriya museum had been very helpful to llu author where he was introduced to the curator, Mr. Kusumsiri I oddlhuwakku. Since it was a late Saturday afternoon (after 5 PM), ■ tin- author was asked to meet Mr. Kusumsiri on Sunday morning ill 7:30 AM. The curator promised the author that he would bring '(lumped pages of the bricks (eye copies) to the meeting. On Sunday morning, Mr. Kusumsiri and the author met. The author exarriined I lie eye copies and he surmised that they might be as 1,4,5,7, possible 10,21, and possible 81 in Lith Illakkum. Kombuwa was clearly marked as well as the number 2, murthada NA. In this period, the shape of Ja was different and present shape was yet to be evolved into. Hence it was difficult to identify the number 4 and the same was true with eight since ‘Ra’, which is number 8 in the Kandyan period had a different shape in 8AD. I 2. I I 81’) 1 a t t s' f ? 7 5 10 * SY1 8 (?) % Figure 1 Manikdena inscription the author’s notes 61 N um erations in the Sinhaln I mt^j § -] tn £—■* C° 0 ' ' I ^ \ ^ ">O 9» »ftft » 1j in» « Btv tn P ft f\ 'G’ /s* xm V I c gI Figure 2 Manikdena inscription the author’s notes In the latter Brahimi period, Brahmi 4 looked very much like an English letter y. On one of the bricks, a letter has been inscribed very clearly which looked like a Sinhala ‘pu’ but bottom pilla or vowel modifier crossing he' zontally cutting across the consonant ‘Pa’. This shape o f‘pu’ looked somewhat similar to the English letter y. Mr. Koddithuwakku pointed that some of the Sinhala letters had been cut across. The author observed that except for one Kombuwa, the other kombuwas which were after consonant were not cut across. 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H{i5t ts»);fl0i>9 . y Soaxg po&ojotBgxa Stodxs cp.xb)q5 t3o6)ox>g)a)o W *9 *(dS>an«o4 © » 6 c qnge S ato& tsl *f»*5 oaca e^SsjswJ 12 aft»0 ijsSQj jSs.Si® 8>3w fi 8 : «><#, <>0, »> Z.S 1 3 0 131 SogKO SJygDQS 12 0 Cf-<5>3 & «• ©cgJ © co— «sD$C$&8*D© «. 0 3 C353 * 8 s S S3} 25) 2 0 9 e ^ o a e a s «syft © s a ©CS5^ sS^s a 8 ci § ^51 Sa» ete 5^3 * 9 4 f « 9 « 9 j a . i9s ^ k, <31 * u i9 (S »9 «? 9»a <»:♦» •213 ©Si^» «s . 25i1 40>S5Cr? ♦ ifll © S--1”-.. c a» 3 2 $ a ©? 2S<«1 4B9 «*« References - epD§z2>c32?f [Bell40] Bell E.T. Development o f Mathematics, Second Edition, , Me- Grew Hill Book Company, Published 1940, 1945, page 51 [DeSilva38] De Silva W.A.,from 'Catalogue of Palm leaf manuscripts ' in the library o f Colombo Museum, Volume I, Government Printer, 1938 [Dissa06] Disanayaka J. B , Sinhala Graphology, Sumathi Publications, 2006 [Epa07] Epa Panchanga (Epa Ephemeris), Epa Printers, 2007 [Ever07] Everson Michael, 1S0/IEC JTC1/SC2/WG2 N3I95R, Proposal to add archaic numbers for Sinhala to the BMP of UCS, 2007 [FilHozat04] Filliozat Pierre-Sylvain, Ancient Sanskrit Mathematics: an oral tradition and a written literature, this article is in the book 147, History o f Science, History Text by Karine Chemla, 2004 [Gunal891] Gunasekara, Abraham Mendis A Comprehensive Grammar of the Sinhalese Language, Sri Lanka Sahitya Mandalaya (Academy o f Letters, 144-150, 1891). [Hetti87] Hettigoda, Hendrick De Silva. Life and Planets, Vishwa Lekha, Sarvodaya, 34 -36, 1987 (Firstprint in 1967) [Indian98] Indian Epigraphy: A Guide to the Study o f Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages, by Richard Salomon, published by Oxford University Press US, 1998 [Jouv95] Jouveau-Dubreuil G., The Pallavas, V. S. Swaminadha Dikshita,Published by Asian Educational Sendees, 1995 [Kula67] Kularatne P. De. S, Article on Numeration, Sinhala Encyclopedia, 1967 8 ocd® PoOsaooeDQ [Masica93] Masica, Colin P., Indo Aryan Languages, Cambridge language surveys, 1993 [Menninger92JMenninger Karl, Paul Broneer Number words and number symkols: a cultural history o f Translated by Paul Broneer Published by Courier Dover Publications, 1992. [SamOOJ Samaranayake V K., Nandasara S. T., Disanayaka J.B., Weerasinghe A. R., Wijayawardhana H., An Introduction to Unicode for Sinhala characters, 2000 [Mahanama03] Mahanama Thera, The Mahavamsa, The Great Chronicle o f Sri Lanka, Vijith Yapa publication 2003. [Abi90]qNq&>®Q$S)&) es®6i ®C33®di esgg, @Q€>® gOaz?) £3oefe)Dc5zs> - £3-sg>Se> epQobcs S@ds@efe>(5 @<^e33te3@®25jq30 - 1990 [ Guna-Article] 1891] [History 1] History o f Sinhala Language, accessed 29 th o f May 2009 http://en. wikipedia. org/wiki/Sinhaleselanguage/ [Brahmi2] Wikipaedia on Accessed on 26th o f May 2009, Brahmi Script: http://en.wikipedia.org/wiki/Br%C4%81hm%C4%AB_script/' [Code3]Code charts, accessed 29th o f May 2009, www.unicode.org [Divehi4] Divehi Akuru, accesed 29th o f May 2009 http://www. omniglot. com/writing/dhivesakuru.htm [5]http://www. Unicode, org/cldr/data/charts/supplemental/ languages_and_scripts.html S o P f e cd j9 y» 8 Oocpgjoc&gxa Acknowledgments This paper would not have been possible without the help of many. Special thank must go to Bhikkhu Mettavihari, for providing some of the documents, which I used for this paper. The help given by the Director of Colombo Museum, Dr. Nanda Wickramasinghe who authorized access to the library of the Museum, and assistance rendered by the librarian Ms. Padma Akarawita of the Colombo Museum who tirelessly found three Ola palm leaves with the Sinhala numerals out of the five thousand manuscripts is specially acknowledged. The assistance rendered by Mr. Senarath Wickramasinghe of the Colombo Museum is gratefully acknowledged. The author was given special access to Royal Asiatic Society library. The president of the society is specially acknowledged. The late Prof. V. K. Samaranayake is remembered for inspiring me to carry out research into Sinhala computerization. The facilitation rendered by the Information and Communication Technology Agency of Sri Lanka (ICTA) and its Board, is hereby acknowledged. Prof P.W. Epasinghe and Mr. Reshan Dewapura are thankfully remembered for giving recognition to this work officially and encouraging me when I was carrying out the research. Mr. Jayantha Fernando is remembered for his sincere encouragement and believing in my. work. Ms. Aruni Goonetillekc is thanked for spending many hours discussing with me finer points on numerals. Dr. Shahani Weerawarana is acknowledged for all the encouragement she gave. The direction given by ICTA’s Local Language Working Group (LLWG) and its Chair, Dr. R.B. Ekanayake is greatly appreciated. Appreciation is given to Mr. Athula Pushpakumara and the ICTA Communications team for the assistance rendered in publication of this work. 1 3 5 8og)(3 Oo6)g)oCDg>C3 Prof. Gihan Dias and Dr. Ruvan Weerasinghe are gratefully acknowledged for their assistance in the preparation of this paper. Prof. J. B. Disanayaka is specially acknowledged for sharing his insight into Sinhala Numerals. Heartfelt thanks are given to the late Mr. Rohan Wijeratne for his invaluable comments which convinced me that the original report had to be re-drafted. , Prof. K. D. Paranavithana is acknowledged for his invaluable assistance, and participation in the project. Special thank must go to Prof. Nimal Silva, Director, Postgraduate Institute of Archaeology, for sharing his thoughts on the subject with me. Acknowledgement is also given to late Prof. Abaya Aryasinghe, who provided us information on the basic concepts of Sinhala numerals at the very early stages. Special acknowledgment is given to the librarian of University of Kelaniya for their help in providing reference materials. I also acknowledge my staff at Software Development Unit of University of Colombo School of Computing who provided immense help for the preparation of this paper, especially Mr. Ashant Hannan, Senior Software Engineer and Mr. Samira Kodithuwakku, Software Engineer who accompanied me to numerous organizations in search of documents with numerals. This work would have not been possible without assistance from my father, Dr. G. Wijayawardhana who helped greatly in the initial preparation of the paper. Finally, but not the least, special thanks are given to my wife, Shalika, for the patience she evinced during this work. Qooxa epixtoaS ooatepooaiQ @5)3@©if @iq@2a2^@erf -£3©@c3d©c3 ©25330x@S)02530 @®@ 0 S 0 Q C 3 O g @23>302g CptZS. @®@ [email protected] £3 @@>® £)Q)25$© O Q 0 @ O 0 8 8 @ © ^ ^l25?g £32030 00^253 ®203©355o Scoszrf © o 8 , C3£0 cfsSsbca (5x©25? 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