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Mahavira (850) gives twenty-four notational places: 3 eka, dasa, sat a, sahasra, dasa-sahasra, laksa, dasa-laksa, koti, das'a-koti, sata-koti, arbuda, nyarbuda, kharva, mahdkharva, padma, mahd-padma, ksoni, mahd-ksoni, sankha, mahd-sankha, ksiti, mahd-ksiti, ksobha, mahd- ksobha. In Arabia the new notation was introduced in the 8th century A. The exact date of the invention, however, would be nearer to the ist century B. or even earlier, because for a long time after its invention, the system must have been looked upon as a mere curiosity' and used simply for expressing large numbers. In the Veda we do not find the use of names of things to denote numbers, but we do find instances of numbers denoting things. 1 Amongst such works may be mentioned the Mahd-siddhdnta (950), the Siddhdnta-Iekhara (1036), the Siddbdnta-tattva-viveka (1658), etc.

Bhaskara II's (11 50) list agrees with that of Sridhara except for mahdsaroja and saritdpati which are replaced by their synonyms mahdpadma and jaladhi respectively. D., but it came into common use about five or six hundred years later. The Arabs got the complete decimal arith- metic, including the method of performing the various operations, at a period when intellectual activity in Arabia was at its greatest height, but they could not make the decimal system common before about five or six hundred years had elapsed. 397k 2 Heath, History of Greek Mathematics, I, Oxford, 1921, p. 3 The arithmetic written by Al-Kharki in the eleventh century does not use the decimal system, showing that at the time there were two schools amongst the Arab mathematicians, one favouring THE DECIMAL PLACE-VALUE SYSTEM J I and in recording historical dates, the Arabs even now use their old alphabetic notation. A still longer time must have elapsed before the method of perform- ing the operations of addition, subtraction, multiplica- tion, division and the extraction of roots, could be perfected. For instance, in the Rgveda the number 'twelve' has been used to denote a year 2 and in the A.tharvaveda the number 'seven' has been used to denote a group of se Ven things (the seven seas, etc.). 2 It is stated by Bhaskara II that Lalla wrote a separate treatise on patiganita. In India conciseness of composition, especially in scientific matters, was highly prized.

HISTORY OF HINDU MATHEMATICS A SOURCE BOOK Parts I and II BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH ASIA PUBLISHING HOUSK BOMBAY CALCUTTA NEW DELHI MADRAS - LONDON ' N E W YORK © 1935, 1938 AVADHESH NARAYAN SINGH Part I First Published: 1935 Part II First Published: 1938 Single Volume Edition: 1962 All Rights Reserved Part I: pp. 1-308 PRINTED IN INDIA BY SOMESHWAR DAYAL AT THE MUDRAN KALA MANDIH , LUCKNOW AND PUBLISHED BY P. JAYASIKGHE, ASIA PUBLISHING HOUSE, BOMBAY TRANSLITERATION Vowels Short : ^ 5 hundred ayutas niyuta, hundred niyutas hankara, hundred hankaras vivara, hundred vivaras ksobbya, hundred kso- bbyas vlvdha, hundred vivdbas utsanga, hundred utsangas balntla, hundred bahulas ndgabala y hundred ndgabalas tip- 1 xvii. Hundred-hundred- thousand kotis give pakoti S In this manner the further terms are formed. hundred hundred-thousands is koti, hundred-hundred- ' Thus ta//aksana—io 53 . The follow- ing numbers are in the denomination kpti-koti. 12 NUMERAL NOTATION thousand kotis is pakoti, hundred-hundred-thousand pa kotis is koiippakoti, hundred-hundred-thousand koti- ppakotis is nabuta, hundred-hundred-fhousand nahutas is ninnabuta, hundred-hundred-thousand ninnahutas is ak- kbobbini\ similarly we have bindu, abbuda, nir abbuda, cthaha, ababa, atata, sogandbika, uppala, kumuda, pundarika padnwa, katbdna, mabdkaibdna, asankbyeya." 1 In the Anuyogadvdra-sutra 2 (c. C.), a Jaina canonical work written before the commencement of the Christian era; the total number of human beings in the world is given thus: "a number which when expressed in terms of the denominations, koti-koti, etc., occupies twenty-nine places (stbdnd), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square, (i.e., 2 06 ), or it is a number which can be divided (by two) ninety-six times." Another big number that occurs in the Jaina works is the number representing the period of time known as Sirsaprahelikd. 1089) 3 , this number is so large as to occupy 194 notational places (anka-stbdnehi). For instance, according to Aryabhata I (499) the denominations are the names of 'places'. He bore the title of pattopddhyaya, i.e., the teacher (charged with the preparation) of title deeds. Examples are also given to illustrate the use of the rules enunciated.

10; the list is che same with the exception that niyuta and prayuta change places. This and the following show that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which "are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." Cf. He says: "Eka (unit) dasa (ten), sata (hundred), sabasra (thousand), ayuta (ten thousand), niyuta (hundred thousand), prayuta (million), 1 Thus asankhyeya is (io) 14n =(io,ooo,ooo) 20 . 3 The figures within brackets after the names of authors or works denote dates after Christ. Gwaliot inscription of the reign of Bhojadeva IX, p. The date Vallabhi Sam- vat 5 74 is given in decimal figures. The existence of manuals such as the Lekhapancasika, the Lek/japraka Ja, which give rules for drafting letters, land grants, treaties, and various kinds of bonds and bills of exchange, show beyond doubt that the writing of grant plates was a specialised art and that the style of writing those documents must always have been centuries behind the times, just as it is even to-day with respect to legal and state documents. Besides these there are a number of astronomical works, known as Siddhdnta, each of which contains a section dealing with mathematics.

KHAROSTHI NUMERALS 23 It is accepted by all that the Kharosthi is a foreign script brought into India from the west. There should, therefore, be a gap of about eight centuries between the time of invention and its coming into popular use, just as was the case with the Greek alphabetic notation. PERSISTENCE OF THE OLD SYSTEM The occurrence of the old system of writing numbers, with no place-value, is found generally in inscriptions upto the seventh century A. C., in the Satapatba Brdbmana* and Taittirtya Brdhmana? This compactness is more pronounced in the older works; for instance, the exposition in the Aryabbatiya is more compact than in the later works.

The exact period at -which it -was imported is unknown. 1 The numerals given above undoubtedly belong to this script as they proceed from right to left. There- fore, on epigraphic evidence alone, the invention of the place -value system must be assigned to the begin- ning of the Christian era, very probably the ist century B. This conclusion is supported by literary and other evidences which will be given hereafter. D., after which it was gradually given up in favour of the new system with place-value. 24 is expressed by gdyatri, jina, arbat, siddha, etc. The 1 Generally used for 5 ; also for 7 by Mahavira. This hankering after brevity, in early times, was due chiefly to the dearth of writing material, the fashion of the time and the method of instruction fol- lowed.

The ancient Kharosthl numerals are given in Table I. The forms of these symbols are: 4 6 50 200 64 6.0 4 v, V • -a if Ki ? Ail other nations of the world have given up their indigenous numerical symbols which, they had used without place-value and have adopted the zero and a new set of symbols, which were never in use in those countries previously. 7 is expressed by naga, aga, bhiibhrt, parvata, saila, acala, adri, giri, rsi, muni, yati, atri, vara, svara, 1 Method of comprehending things from particular stand- points — dravydrthika and parydydrthik.a. 14 is expressed by manu, vidyd, indra, sakra, /oka, 10 etc. Thus it would appear that instruction in mathematics, upto a certain minimum standard, was available almost every- where in India. In abnormal times when there were 128 ARITHMETIC foreign invasions, internal warfares or bad government and consequent insecurity, the study of mathematics and, in fact, of all sciences and arts languished.

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These forms clearly dis- tinguish the Brahmi notation from the Kharosthi and the Semitic systems. We, however, know of at least one very distinguished mathematician, Sripati, who probably lived in Kashmir at that time.2 The formation of other numbers may be illustrated by the number 274 which is written with the help of the symbols for 2, 100, 20, 10 and 4 arranged as nvfii in the right to left order. Due to the lack of early documents, we are not in a position to say what exactly were the original forms of the Brahmi symbols. 9 n , A ncient Na gari Numeration; from an inscription at Nf.naghat /o»w. -^ 2 E- Senart, "The inscriptions in the caves at Nasik," El, Vol. 39 - 9 6; "The inscriptions in the cave at Karle " EI, Vol. Thus the gradual development of these forms can be easily traced. 3 is expressed by rdma, guna, triguna, loka, trijagat, bhitvana, kdla, trikdla, trigata, trinetra, haramtra, sahodardh, agni, ana la , vafmi, pdraka, vaisvdnara, dahana, tapana, hutdsana, jvalana, sikhin, krsdnu, botr, pur a, ratna? 5 is expressed by banc, sara, sastra, sdyaka, isu, bhuta, parva, prdna^pavana* pdndava,artha, visaya, tuahdbhiita, tatva, bhdva, indriya, ratna, karamya* vrata, etc. ii is expressed by rudra, isvara, uirda, hara, isa, bhava, bharga, sulirt, mahadeva, aksaubini, etc, iz is expressed by ravi, stirya, ina, arka, mdrtanda, dyumani, bbdnu, dditya, divdkara, wdsa, rdsi, vyaya, etc. Moreover, there have always been, from very early times, a class of people known as ganaka whose profession was fortune-telling.The 2 on the right of 100 multiplies 100, whilst the numbers written to the left are added, thus giving 274. Our knowledge of these symbols goes back to the time of King Asoka (c. C.) whose vast dominions included the whole of India and extended in the north upto Central Asia. o//fe Bewfc*,- Branch of the Royal Asiatic Society, 1876, Vol. This gradual change from the old system without place-value to the new system with the zero and the place-value is to be met with in India alone. The date Vikrama Sarhvat 894 is given in decimal figures. 6 is expressed by rasa, anga, kdya, rtu, mdsdrdha, dar- s'ana, rdga, art, sdstra, tarka, kdraka, lekhya, dravya, 6 ik/jara, kumdravadana, sanmukha, etc. 13 is expressed by visvedevdh, visva, kdma, atijagati, agbosa, etc. These people were astrologers, and in order to impress their clients with their learning, they used to have some knowledge of mathematics and astronomy.The difference in writing the symbols 1 to 3, seems to be due to the inherent difference between the two systems of writing. 28 NUMERAL NOTATION there are separate signs for each of the numbers i, 4 to 9 and 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 1000, 2000, etc., while in the oldest Kharosth I and in the earliest Semitic writings, the Hieroglyphic and the Phoenician, the only symbols are those for 1, 10, zo and 100. Aryabhata I gives the rules for finding the square- and cube-roots only, whilst Brahma- gupta gives the cube-root rule only.

The principles upon which numeri- cal signs are formed in the two systems are quite different. In the Brahmi 3 It has been jncorfectly stated by Smith and Karpinski that the Nanaghat forms were vertical. The Hieratic and the Demotic numerals, however, resemble the Brahmi in having nineteen symbols for the numbers from 1 to 100, but the principle of forma- tion of the numbers 200, 300, 400, 2,000, 3,000 and 4,000 are different, as will appear from Table 11(f). In the works on arithmetic (pdtiganita), the methods of addition and subtraction have not been mentioned at all or men- tioned very briefly.2 The Nabatean numerals resemble the Kharosthi also in the use of the scale of twenty and in the method of formation of the hundreds. D., in which the dates, although written in the old notation, are incorrectly inscribed, showing thereby that people had already forgotten the old system. In this the sign of 8 is written for 80 and that of 30 for 3. 1 The document records gifts made on several occasions ranging over thirty-seven years, the last entry corresponding to 905 A. In this inscription the old notation is used in the first six lines whilst in the follow- ing lines it has been discarded and the new place-value notation appears T It is evident from the forms that the writer did not know the old system. The ancient Hindu mathema- ticians and astronomers wrote their works in verse. Along with each step in the process of calculation the sutra (rule) was repeated by the student, the teacher supervising and helping the student where he made mistakes.