Hoernle, August Frederich Rudolf On the Bakshali manuscript ON THK BAKHSHALI MANUSCRIPT 11. IIOERNLE. WITH THKEE I'll OTO Z INCO ti K A 1' VIENNA ALPBED HOLDER E OF THE C U K T ; A N D OF THE UNIVERSITY -I ON THE BAKHSHALI MANUSCRIPT. D a E. HOEKNLE. WITH THREE P H T Z IN C G R AP H S. VIENNA ALFRED H6LDER ED1TOE OF THE COURT AND OF THE UNIVERSITY 1887. 37 Separat- A bd ruck Verhandluugeu des VII. Internationalen Orientalisten-Congresses. Arische Section, S. 127 ff. Druck von Adolf Holzhausen, k. k. Hof- und Uiiiversitiits-Btichdriicker in Wien. Ihe manuscript which I have the honour, this morning, of placing before you, was found, as you will recollect, in May 1881, near a village called Bakhshall, lying in the Yusufzai district of the Peshawer division, at the extreme Northwestern frontier of India. It was dug out by a peasant in a ruined enclosure, where it lay between stones. After the find it was at once forwarded to the Lieutenant Governor of the Panjab who transmitted it to me for examination and eventual publication. The manuscript is written in Sharada character of a rather ancient type, and on leaves of birch-bark which from age have become dry like tinder and extremely fragile. Unfortunately, probably through the careless handling of the finder, it is now in an excessively mutilated condition, both with regard to the size and the number of the leaves. Their present size, as you observe (see Plate I), is about 6 by 3y 2 inches; their original size, however, must have been about 7 by 8Y 4 inches. This might have been presumed from the well-known fact that the old birck-bark manuscripts were always written on leaves of a squarish size. But I was enabled to determine the point by a curious fact. The mutilated leaf which contains a portion of the 27 Vj sutra, shows at top and bottom the remainders of two large square 4 R. Hoernle. figures, such as are used in writing arithmetical notations. These when completed prove that the leaf in its original state must have measured approximately 7 by 8 l / 4 inches. The number of the existing leaves is seventy. This can only be a small portion of the whole manuscript. For neither beginning nor end is pre- served; nor are some leaves forthcoming which are specifically referred to in the existing fragments. 1 ) From all appearances, it must have been a large work, perhaps divided in chapters or sections. The existing leaves include only the middle portion of the work or of a division of it. The earliest sutra that I have found is the 9 th ; the latest is the 57 th . The lateral margins which usually exhibit the numbering of the leaves are broken off. It is thus impossible even to guess what the original number of the leaves may have been. The leaves of the manuscript, when received by me, were found to be in great confusion. Considering that of each leaf the top and bottom (nearly two thirds of the whole leaf) are lost, thus destroying their connection with one another, it may be ima- gined that it was no easy task to read and arrange in order the fragments. After much trouble I have read and transcribed the whole, and have even succeeded in arranging in consecutive order a not inconsiderable portion of the leaves containing eighteen sutras. The latter portion I have also translated in English. v/ The beginning and end of the manuscript being lost, both the name of the work and of its author are unknown. The sub- ject of the work, however, is arithmetic. It contains a great variety of problems relating to daily life. The following are examples. ,In a carriage, instead of 10 horses, there are yoked 5; the distance traversed by the former was one hundred, how much will the other horses be able to accomplish ?' Te following is more complicated: ,A certain person travels 5 yojanas on the ') Thus at the end of the 10 th sutra, instead of the usual explanation, there is the following note: evam sutram | dviflya patre mvaritasti. The leaf referred to is not preserved. On the Bakhshali Manuscript. 5 first day, and 3 more on each succeeding day; another who travels 7 yojanas on each day, has a start of 5 days; in what time will they meet?' The following is still more complicated: 'Of 3 merchants the first possesses 7 horses, the second 9 ponies, the third 10 camels; each of them gives away 3 animals to be equally distributed among themselves; the result is that the value of their respective properties becomes equal; how much was the value of each merchant's original property, and what was the value of , each animal?' The method prescribed in the rules for the solution of these problems is extremely mechanical, and re- duces the labour of thinking to a minimum. For example, the last mentioned problem is solved thus: 'Subtract the gift (3) se- verally from the original quantities (7, 9, 10). Multiply the re- mainders (4, 6, 7) among themselves (168, 168, 168). Divide each of these products by the corresponding remainder (^ ^, ^-p-). The results (42, 28, 24) are the values of the 3 classes of animals. Being multiplied with the numbers of the animals originally pos- sessed by the merchants (42 . 7, 28 . 9, 24 . 10), we obtain the values of their original properties (294, 252, 240). The value of the properts of each merchant after the gift is equal (262, 262, 262).' The rules are expressed in very concise language, but are fully explained by means of examples. Generally there are two examples to each rule (or sutra), but sometimes there are many; the 25 th sutra has no less than 15 examples. The rules and examples are written in verse; the explanations, solutions and all the rest are in prose. The metre used is the shloka. The subject-matter is divided in sutras. In each sutra the matter is arranged as follows. First comes the rule, and then the example, introduced by the word tada. Next, the example is repeated in the form of a notation in figures, which is called sthapana. This is followed by the solution which is called karana. Finally comes the proof, called pratyaya. This arrangement and terminology differ somewhat from those used in the arithmetic of Brahmagupta and Bhaskara. Instead of simply sutra, the latter use the term karana-sutra. The example they call uddeshaka or udaharana. For sthapana they say nyasa. As a rule they give 6 R. Hoernle. no full solution or proof, but the mere answer to the problem. Occasionally a solution is given, but it is not called karana. The system of notation used in the Bakhshali arithmetic is much the same as that employed in the arithmetical works of Brahmagupta and Bhaskara. There is, however, a very impor- tant exception. The sign for the negative quantity is a cross (+). It looks exactly like our modern sign for the positive quantity, but is placed after the number which it qualifies. Thus V l + means 12 7 (i. e. 5). This is a sign which I have not met with in any other Indian arithmetic; nor so far as I have been able to ascertain, is it known in India at all. The sign now used is a dot placed over the number to which it refers. Here, there- fore, there appears to be a mark of great antiquity. As to its origin I am unable to suggest any satisfactory explanation. I have been informed by Dr. Thibaut of Benares, that Diophantus in his Greek arithmetic uses the letter i| (short for Xetyi?) reversed (thus fj>), to indicate the negative quantity. There is undoubtedly a slight resemblance between the two signs; but considering that the Hindus did not get their elements of the arithmetical science from the Greeks, a native origin of the negative sign seems more probable. It is not uncommon in Indian arithmetic to indicate a particular factum by the initial syllable of a word of that import subjoined to the terms which compose it. Thus addition may be indicated by yu (short for yuta\ e. q. f \ w means 5 + 7 (c. e. 12). In the case of substraction or the ne- gative quantity rina would be the indicatory word and ri the indicatory syllable. The difficulty is to explain the connection between the letter ri (^?) and the symbol +. The latter very closely resembles the letter k (^j) in its ancient shape (+) as used in the Ashoka alphabet. The word kana or kaniyas which had once occurred to me, is hardly satisfactory. A whole number, when it occurs in an arithmetical opera- tion, as may be seen from the above given examples, is in- dicated by placing the number 1 under it. This, however, is a practice which is still occasionally observed in India. It may be worth noting that the number one is always designated by On the Bakhslmll Manuscript. 7 the word rupa- 1 ) thus sarupa or rupadhika 'adding one', rupona 'deducting one'. The only other instance of the use of a sym- bolic numeral word is the word rasa for six which occurs once in an example in sutra 53. The following statement, from the first example of the 25 th Sutra, affords a good example of the system of notation employed in the Bakhshali arithmetic: . J | } bha 32 1 3+ 3+ 3+ phalam 108 Here the initial dot is used very much in the same way as we use the letter x to denote the unknown quantity the value of which is sought. The number 1 under the dot is the sign of the whole (in this case, unknown) number. A fraction is denoted by placing one number under the other without any line of separation; thus J is f, i. e. one-third. A mixed number is shown by placing the three numbers under one another; thus 1 is 1 -f- f or If, i. e. one and one-third. Hence i + means 1 f (i. e. f). Multiplication is usually indicated by placing the numbers side by side; thus I g V | phalam 20 means f X 32 = 20. Similarly | + j + ] + means f XIX! or (f) 3 , c. e. . Bha is an abbre- viation of bhaga 'part' and means that the number preceding it is to be divided. Hence ] + i + i + bha means ". The whole statement, therefore, 1 3+ 3+ 3+ bha 32 I phalam 108 means ~ X 32 = 108, and may be thus explained: 'a certain number is found by dividing with -j^ and multiplying with 32; that number is 108'. The dot is also used for another purpose, namely as one of the ten fundamental figures of the decimal system of notation *) This word was at first read by me upa. The reading rupa was sug- gested to me by Professor A. Weber, and though not so well agreeing with the manuscript characters, is probably the correct one. 2* 8 R. Hoernle. or the zero (0123456789). It is still so used in India for both purposes, to indicate the unknown quantity as well as the naught. With us the dot, or rather its substitute the circle (), has only retained the latter of its two intents, being simply the zero figure, or the 'mark of position' in the decimal system. The Indian usage, however, seems to show, how the zero arose and that it arose in India. The Indian dot, unlike our modern zero, is not properly a numerical figure at all. It is simply a sign to indicate an empty place or a hiatus. This is clearly shown by its name sliunya 'empty". The empty place in an arithmetical statement might or might not be capable of being filled up, according to circumstances. Occurring in a row of figures arranged decimally or according to the Value of position", the empty place could not be filled up, and the dot therefore signified 'naught', or stood in the place of the zero. Thus the two figures 3 and 7, placed in juxtaposition (37) mean 'thirty seven', but with an 'empty space' interposed between them (3 7), they mean 'three hundred and seven'. To prevent misunder- standing the presence of the 'empty space' was indicated by a dot (3*7), or by what in now the zero (307). On the other hand, occurring in the statement of a problem, the 'empty place' could be filled up, and here the dot which marked its presence, signi- fied a 'something" which was to be discovered and to be put in the empty place. In the course of time, and out of India, the latter signification of the dot was discarded; and the dot thus became simply the sign for 'naught' or the zero, and assumed the value of a proper figure of the decimal system of notation, being the 'mark of position'. In its double signification which still survives in India, we can still discern an indication of that country as its birth place. Regarding the age of the manuscript am unable to offer a very definite opinion. The composition of a Hindu work on arithmetic, such as that contained in the Bakhshall MS. seems necessarily to presuppose a country and a period in which Hindu civilisation and Brahmanical learning flourished. Now the country in which Bakhshall lies and which formed part of the Hindu kingdom of Kabul, was early lost to Hindu civilisation through the conquests of the Muhammedan rulers of Ghazni ; and espe- cially through the celebrated expeditions of Mahmiid, towards the end of the 10 th and the beginning of the 11 th centuries A. D. In those troublous times it was a common practice for the learned Hindus to bury their manuscript treasures. Possibly the Bakhshall MS. may be one of these. In any case it cannot well be placed much later than the 10 th century A. D. It is quite possible that it may be somewhat older. The Sharada characters used in it, exhibit in several respects a rather archaic type, and afford some ground for thinking that the manuscript may perhaps go back to the 8 th or 9 th century. But in the present state of our epigraphical knowledge, arguments of this kind are always some- what hazardous. The usual form, in which the numeral figures occur in the manuscript are the following: 1 2 3 4567890 Quite distinct from the question of the age of the manu- script is that of the age of the work contained in it. There is every reason to believe that the Bakhshall arithmetic is of a very considerably earlier date than the manuscript in which it has come down to us. I am disposed to believe that the com- position of the former must be referred to the earliest centuries of our era, and that it may date from the 3 d or 4 th century A. D. The arguments making for this conclusion are briefly the following. In the first place, it appears that the earliest mathematical works of the Hindus were written in the Shloka measure; *) but from about the end of the 5 th century A. D. it became the fashion to use the Arya measure. Aryabhatta c. 500 A. D., Va- raha Mihira c. 550, Brahmagupta c. 630, all wrote in the latter measure. Not only were new works written in it, but also Shloka works were revised and recast in it. Now the Bakhshall arith- See Professor Kern's Introduction to Varaha Mihira. 10 R. Hoernle. metic is written in the Shloka measure; and this circumstance carries its composition back to a time anterior to that change of literary fashion in the 5 th century A. D. In the second place, the Bakhshall arithmetic is written in that peculiar language which used to be called the 'Gatha dialect', but which is rather the literary form of the ancient Northwestern Prakrit (or Pali). It exhibits a strange mixture of what we should now call Sanskrit and Prakrit forms. As shown by the inscription (e. g., of the Indoscythian kings in Mathura) of that period, it appears to have been in general use, in North- western India, for literary purposes till about the end of the 3 d century A. D., when the proper Sanskrit, hitherto the language of the Brahmanic schools, gradually came into general use also for secular compositions. The older literary language may have lingered on some time longer among the Buddhists and Jains, but this would only have been so in the case of religious, not of secular compositions. Its use, therefore, in the Bakhshall arith- metic points to a date not later than the 3 d or 4 th century A. D. for the composition of that work. In the third place, in several examples, the two words dlndra and dramma occur as denominations of money. These words are the Indian forms of the latin denarius and the greek drachme. The former, as current in India, was a gold coin, the latter a silver coin. Golden denarii were first coined at Rome in 207 B. C. The Indian gold pieces, corresponding in weight to the Roman gold denarius, were those coined by the Indoscy- thian kings, whose line beginning with Kadphises, about the middle of the 1 st century B. C., probably extended to about the end of the 3 d century A. D. Roman gold denarii themselves, as shown by the numerous finds, were by no means uncommon in India, in the earliest centuries of our era. The gold dinars most numerously found are those of the Indoscythian kings Kanishka and Huvishka, and of the Roman emperors Trajan, Hadrian and Antonius Pius, all of whom reigned in the 2 nd century A. D. The way in which the two terms are used in the Bakhshall arithmetic seems to indicate that the gold dinara and the silver dramma formed the ordinary currency of the day. This circum- stance again points to some time within the three first . centuries of the Christian era as the date of its composition. A fourth point, also indicative of antiquity which I have already adverted to, is the peculiar use of the cross (-J-) as the sign of the negative quantity. There is another point which may be worth mentioning though I do not know whether it may help in determining the probable date of the work. The year is reckoned in the Bakhshali arithmetic as consisting of 360 days. Thus in one place the fol- lowing calculation is given: Tf in fff of a year 2982 -J-J-J- is spent, how much is spent in one day?' Here it is explained that the lower denomination (adha-ch-chheda) is 360 days, and the result (phala) is given as ^ (i. e. nfft^iii__). In connection with this question of the age of the Bakhshali work, I may note a circumstance which appears to point to a peculiar connection of it with the Brahmasiddhanta of Brahma- gupta. There is a curious resemblance between the 50 th sutra of the Bakhshali arithmetic, or rather with the algebraical example occurring in that sutra, and the 49 th sutra of the chapter on algebra in the Brahmasiddhanta. In that sutra, Brahmagupta first quotes a rule in prose, and then adds another version of it in the Arya measure. Unfortunately the rule is not preserved in the Bakhshali MS., but as in the case of all other rules, it would have been in the form of a shloka and in the North- western Prakrit (or 'Gatha dialect'). Brahmagupta in quoting it, would naturally put it in what he considered correct Sanskrit prose, and would then give his own version of it in his favourite Arya measure. I believe it is generally admitted that Indian arithmetic and algebra, at least, is of entirely native origin. While siddhanta writers, like Brahmagupta and his predecessor Arya- bhatta, might have borrowed their astronomical elements from the Greeks or from books founded themselves on Greek science, they took their arithmetic from native Indian sources. Of the Jains it is well known that they possess astronomical books of a very ancient type, showing no traces of western or Greek 12 R. Hoernle. influence. In India arithmetic and algebra are usually treated as portions of works on astronomy. In any case it is impossible that the Jains should not have possessed their own treatises on arithmetic when they possessed such on astronomy. The early Buddhists, too, are known to have been proficients in mathe- matics. The prevalence of Buddhism in Northwestern India, in the early centuries of our era, is a well known fact. That in those early times there were also large Jain communities in those regions is testified by the remnants of Jain sculpture found near Mathura and elsewhere. From the fact of the general use of the Northwestern Prakrit (or the 'Gatha dialect') for literary purposes among the early Buddhists it may reasonably be concluded that its use prevailed also among the Jains between whom and the Buddhists there was so much similarity of manners and customs. There is also a diffusedness in the mode of composition of the Bakhshall work which reminds one of the similar characteristic observed in Buddhist and Jain literature. All these circumstances put together -seem to render it probable that in the Bakhshall MS. we have preserved to us a fragment of an early Buddhist or Jain work on arithmetic (perhaps a portion of a larger work on astronomy) which may have been one of the sources from which the later Indian astronomers took their arithmetical information. These earlier sources, as we know, were written in the shloka measure, and when they belonged to the Buddhist or Jain lite- rature, must have been composed in the ancient Northwestern Prakrit. Both these points are characteristics of the Bakhshall work. I may add that one of the reasons why the earlier works were, as we are told by tradition, revised and rewritten in the Arya measure by later writers such as Brahmagupta, may have been that in their time the literary form (Gatha dialect) of the Northwestern Prakrit had come to be looked upon as a barba- rous and ungrammatical jargon as compared with their own classi- cal Sanskrit. In any case the Buddhist or Jain character of the Bakhshall arithmetic would be a further mark of its high antiquity. Throughout the Bakhshall arithmetic the decimal system of notation is employed. This system rests on the principle of On the Bakhshall Manuscript. 13 the 'value of position' of the numbers. It is certain that this prin- ciple was known in India as early as 500 A. D. There is no good reason why it should not have been discovered there con- siderably earlier. In fact, if the antiquity of the Bakhshall arith- metic be admitted on other grounds, it affords evidence of an earlier date of the discovery of that principle. As regards the zero, in its modern sense of a 'mark of position' and one of the ten fundamental figures of the decimal system (0123456789), its discovery is undoubtedly much later than the discovery of the Value of position 1 . It is quite certain, however, that the appli- cation of the latter principle to numbers in ordinary writing would have been nearly impossible without the employment of some kind of 'mark of position', or some mark to indicate the 'empty place 1 (shunya). Thus the figure 7 may mean either 'seven' or 'seventy' or 'seven hundred' according as it be or be not supposed to be preceded by one (7 or 70) or two (7 or 700) 'empty places'. Unless the presence of these 'empty places' or the 'position' of the figure 7 be indicated, it would be impossible to read its 'value' correctly. Now what the Indians did, and in- deed still do, was simply to use for this purpose the sign which they were in the habit of using for the purpose of indicating any empty place or omission whatsoever in a written composi- tion; that is the dot. It seems obvious from the exigencies of writing that the use of the well known dot as the mark of an empty place must have suggested itself to the Indians as soon as they began to employ their discovery of the principle of 'value position' in ordinary writing. In India the use of the dot as a substitute of the zero must have long preceded the disco- very of the proper zero, and must have been contemporaneous with the discovery of that principle. There is nothing in the Bakhshall arithmetic to show that the dot is used as a proper zero, and that it is any thing more than the ordinary 'mark of an empty place'. The employment, therefore, of the decimal system of notation, such as it is, in the Bakhshall arithmetic is quite consistent with the suggested antiquity of it. 14 R- Hoernle. I have already stated that the Bakhshall arithmetic is written in tho so-called 'Gatha dialect 1 , or in that literary form of the Northwestern Prakrit, which preceded the employment, in secu- lar composition, of the classical Sanskrit. Its literary form con- sisted in what may be called (from the Sanskrit point of view) an imperfect sanskritisation of the vernacular Prakrit. Hence it exhibits at every turn the peculiar characteristics of the under- lying vernacular. The following are some specimens of ortho- graphical peculiarities. Insertion of euphonic consonants: of m, in eka-m-ekatvam, bhritako - m - ekapanditah of r, in tri - r - dshlti, labhate-r- astau. Insertion of s: vibhdktam-s-uttare, Ksiyate-s-traya. This is a peculiarity not elsewhere known to me, either in Pra- krit or in Pali. Doubling of consonants: in compounds, prathama-d-dhdnttt, eka-s-samkhyd; in sentences, yadi-s-sadbhi 7 ete-s-sama- dhand. Peculiar spellings: trinslia or trinsha for trimshat. The spelling with the guttural nasal before sh occurs only in this word; e. q., chatvaliihsha 40. Again ri for ri in tridine, kriyate, vimishritam, krindti; and ri for ri in rinam, dristah. Again katthyatam for kathyatdm. Again the jihvdmullyo and the upadhmdniya are always used before gutturals and palatals respectively. Irregular sandhi: ko so ra for kah sa ra, dvayo kechi for dvayah k, dvayo cha for dvayash cha, dvibhi kri for dvibhih kri, adyo vi* for ddyor vi, vivaritdsti for vivaritam asti. Confusion of the sibilants: sh for s, in shasti 60, mdshako; s for sh, in dashdmsha, visodliayet, sesam; sh for s, in sd- shyam, sdsyatdm; s for sh, in esa ,this^. Confusion of n and n: utpanna. Elision of a final consonant: bhdjaye, kechi for bhdjayet, kechit. On the Bakhshali Manuscript. 15 Interpolation of r: hrlnam for hlnam. The following are specimens of etymological and syntactical peculiarities. Absence of inflection: nom. sing, masc., esha sd rdshi for ra- shih (s. 50), gavdm vishesa kartavyam for vishesah (s. 51). Nom. plur., sevya santi for sevydh (s. 53). Ace. plur., dlnara dattavdn for dmdrdn (s. 53). Peculiar inflection: gen. sing., gatisya for gateh (s. 15); atm. for parasm., dr jay ate for arjayati r he earns 1 (s. 53); parasm. for atm., vikrindti for vikrmlte 'he sells 1 (s. 54). Change of gender: masc. for neut., mida for mulani (s. 55); neut. for masc., vargam for vargah (s. 50); neut. for fern., yutim cha kartavyd for yutish (s. 50). Exchange of numbers: plur. for sing., (bhavet) Idbhdh for Idbhah (s. 54). Exchange of cases: ace. for nom., dvitlyam pamchadivase rasam ar jay ate for dmtiyah (s. 53); ace. for instr., ksayam samgunya for ksayena (s. 27); ace. for loc., kim kdlam for kasmin kale (s. 52); instr. for loc., anena kdlena for asmin kale (s. 53); instr. for nom., prathamena dattavdn for prathamo (s. 53), or ekena ydti for eko (s. 15); loc. for instr., prathame dattd for prathamena (s. 53), or ma- nave grihltam for mdnavena (s. 57); gen. for dat., dvi- tlyasya dattd for dvitiydya (s. 53). Abnormal concord: incongruent cases, ay am praste for as- min (s. 52); incongruent numbers, esha Idbhdh for Id- bhah (s. 54) rdjaputro kechi for rdjaputrah (s. 53); incon- gruent genders, sd kdlam for tat kdlam (s. 52), vishesa kartavyam for kartavyah (s. 51), sd rdshih for sa (s. 50), kdryam sthitah for sthitam (s. 14). Peculiar forms: nivarita for nivrita, drja for drjana, divad- dha 'one and one-half, chatvdlimsha 40, pamchdshama 50 th , chaupamchdshama 54 th , chaturdshUi 84, tri-r-dshlti 83, etc. The following extracts may serve as specimens of the text. 16 R. Hoernle. Sutram | Adyor vishesadvigunam chayashuddhi vibhajitam Hup&dhikam tathd &alam gatisasyam tada bhavet || tada I Dvayaditrichayash chaiva dvic/zo/ar?/adikottarah | Dvayo cha bhavate pamtha kena kalena sasyatam || sthapanam kriyate | esdm || a j || u I \\ pa \ \\ dvi | a I || u I \\ pa karanam | adyor vishesa . ta dvi 2 . tada a } a \ u J uj pa ; pa { dha ; 1 dha 1 karanam | adi d \ 10 u adyor vishesam vishesa 5 cha- yashuddhi chayam 6 3 shuddhi 3 adishesa 5 dvigunam 10 utta- ravishesa 3 vibhaktam " sarupam esa padaih anena ka^ena sa- madhana bhavanti || pratyayam || ruponakaranena phalam ^ v { 65 || Asthadasashamasutram 18 || -^ I IdanTm suvarnaksayam vaksyami yasyedam sutram | Siitram | Ksayam samgunya kanakas tadyuti-b-bhajayf tatah ] Samyutair eva kanakair ekaikasya ksayo hi sah | tada || Ekadvitrichatussamkhya, suvarna masakai rinai | Ekadvitrichatussamkhi/a/ rahita samabhagatam || sthapanam kriyate | esam J \ + \\ l + \\ l + \\ t + [ karanam | ksayam samgunya kanakadibhi ksayena samgunya jatam 1 | 4 9 16 | tadyuti esa yati 30 kanaka yuti 10 anena bhaktva labdham On the Bakh shall Manuscript. 17 10 ; so ; i i ; i j i pha mase 3 : 10 ; so : 2 ; i ; i I i \ pha mdse i 10 30 1 3 | pha mase i 30 I 3 1 1 1 | pha mase \ 2 tada || Ek&d.viti'ichatussamkhyd suvarna projjhita line | Masaka dvitritarii chaiva chatuhpariichakarariishakaih *) kim ksa- yarii || karanarii | ksayarii saihgunya kanaka esha stha- pyate | | \ \ j \ I \ { | -s-tadyuti-b-bhajayeta 2 ) ta- tah harasasye krite yutaiii | ^ | samyutaih kanakair bhaktva tada kanaka 10 anena bhaktaih jatarii | 6 6 3 | esha ekaikasuvarnasya ksayam || pY&tyayam ^-airashikena kartavya ||' 10 163 ! 1 1 ; 60 : 1 'io"":""i63'"; 2 i ; eo i i 10 I 163 j 3 1 I 60 I 1 Jg pha ' 1 163 j 60 j 4 1 -nV.0 103 i | pna eoo tada shrunushva me ] Kramena dvaya masadi uttare ekahmatam | Suvarnam me tu sammishrya katthyatam ganakottama || sthapanaih || 4 5 f | 5 6 f || ? + || l + \\ l + \\ i 9 + | 2 + || ! + II l + II ksa- yam saiiigunya jatarii 20 30 | 42 | 56 | 72 90 | 2 | 6 1 3) esaiii yuti 330 kana&anaiii yuti 45 I anena bhaktva labdham | 3 4 30 | parii- chadash abhage-sh-chheda kriyate ! phalarii | 7 she 3 | esha ekaika- mashakaksayaih | pratyaya trairashikena \ 4 , 5 | 330 1 \ \ phalarii 2 3 2 | evarii sarvesam pratyaya kartavya \\ Saptavimshatimasufram 27 | ^ || 1 ) Read chatuhpamchamsham kim ksayam, metri causa. 2 ) liead bhajayet. 3 ) Here | 12 | is omitted in the text, by mistake. 18 K. Hoornle. Sutram | Ahadravyaharashauta *) tadvishesaih vibhajayet \ Yallabdharii dvigunarii kalarii datta samadhana prati || tada || Tridine arjaye pamcha bhritako-m-ekapanditah | Dvitlyam pamchacfo'vase rasam arjayate budhah || Prathamena dvitiyasya sapta dattani . . tah | Datva samadhana jata kena kalena katthyatam I I! 3 ru II 5 ru II . . . m ....... m ha?'mshauta tadvishesam anena kalena samadhana bhavanti || pratyaya trairashike kriyate pha 50 P ha prathame dvitlyasya-s-sapta datta | 7 she- sam 43 || 43 | 43 ete samadhana jata || tada -Kojaputro dvayo kechi nripati-s-sevya santi vaih | M-ekasyahne dvaya-s-sadbhaga 2 ) dvitiyasya divarddhakaih || Prathamena dvitiyasya dasha dinara dattavan | Kena kalena samatarii ganayitva vadashu me || karanam | aha dravyavishesam cha tatra 2 I dattam pratyayam trairashik tiyasya 10 datta jata jata 1 Sutram tripam ena 55 chas i 1 is i 30 30 1 pha pha 65 45 prathamena dvi- 55 1 samadhana hamah sutram 53 I 3 TRANSLATION. The 18 th Sutra. Let twice the difference of the two initial terms be divided by the difference of the (two) increments. The result augmented by one shall be the time that determines the progression. ') Read haramshauta. 2 ) Read ekasyalme dvi^adbhaga. The error appears to have been no- ticed by the scribe of the manuscript. On the Bakhshali Manuscript. 19 First Example. A person has an initial (speed) of two and an increment of three, another has an increment of two and an initial (speed) of three. Let it now be determined in what time the two persons will meet in their journey. The statement is as follows: N I, init. term 2, increment 3, period x NII, 3, 2, x. Solution: the difference of the two initial terms (2 and 3 is 1 ; the difference of the two increments 3 and 2 is 1 ; twice the diffe- rence of the initial terms 1 is 2, and this, divided by the diffe- rence of the increments 1 is 2 /j , and augmented by 1 is Vj ; this is the period. In this time [3] they meet in their journey which is 15). Second Example. (The problem in words is wanting; it would be something to this effect: A earns 5 on the first and 6 more on every fol- lowing day; B earns 10 on the first and 3 more on every fol- lowing day; when will both have earned an equal amount?) Statement: N 1, init. term 5, increment 6, period x, possession x N 2, 10, 3, x, x. Solution: 'Let twice the difference of the two initial terms', etc. ; the initial terms are 5 and 10, their difference is 5. 'By the difference of the (two) increments'; the increments are 6 and 3; their difference is 3. The difference of the initial terms 5, being doubled, is 10, and divided by the difference of the increments 3, is -^, and augmented by one is . This (i. e. -^ or 4J-) is the period; in that time the two persons become possessed of the some amount of wealth. Proof: by the 'ruponcC method the sum of either progres- sion is found to be 65 (i. e., each of the two persons earns 65 in 4y 3 days). 20 B. Hocrnlc. The 27 th Sutra. Now I shall discuss the wastage (in the working) of gold, the rule about which is the following. Sutra. Multiplying severally the parts of gold with the wastage, let the total wastage be divided by the sum of the parts of gold. The result is the wastage of each part (of the whole mass) of gold. First Example. Suvarnas numbering respectively one, two, three, four are subject to a wastage of masakas numbering respectively one, two, three, four. Irrespective of such wastage they suffer an equal distribution of wastage. (What is the latter?) The statement is as follows: Wastage -- 1, 2, 3, 4 masaka Gold 1, 2, 3, 4 suvarna. Solution: 'Multiplying severally the parts of gold with the wastage 1 , etc. 5 by multiplying with the wastage, the product 1, 4, 9, 16 is obtained; 'let the total wastage', its sum is 30; the sum of the parts of gold is 10; dividing with it, we obtain 3. (This is the wastage of each part, or the average wastage, of the whole mass of gold.) (Proof by the rule of three is the following :) as the sum of gold 10 is to the total wastage of 30 masakas, so the sum of gold 4 is to the wastage of 12 masakas , etc. Second Example. There are suvarnas numbering one, two, three, four. There are thrown out the following masakas: one-half, one-third, one- fourth, one-fifth. What is the (average) wastage (in the whole mass of gold)? Statement : quantities of gold, 1, 2, 3, 4 suvarnas wastage y. 2 , Vs> V*? Vs masakas. Solution : ' Multiplying severally the parts of gold with the wastage', the products may thus be stated: \i^ 2 /3? 3 /j? 4 /&- 'Let On the Bakhshali Manuscript. 21 the total wastage be divided' ; the division being directed to be made, the total wastage is -~ ; dividing 'by the sum of the parts of gold'; here the sum of the parts of gold is 10; being divided by this, the result is ||f. This is the wastage of each part of the whole mass of gold. Proof: may be made by the rule of three: as the sum of the parts of gold 10 is to the total wastage of ^ masaka, so the sum of gold 4 is to the wastage of f| masaka, etc. Third Example. (The problem in words is only partially preserved, but from its statement in figures and the subsequent explanation, its purport may be thus restored.) Of gold masakas numbering respectively five, six, seven, eight, nine, ten, quantities numbering respectively four, five, six, seven, eight, nine, are wasted. Of another metal numbering in order two masaka, etc. (i. e., two, three, four) also quantities numbering in order one etc. (i. e. one, two, three) are wasted. Mixing the gold with the alloy, best of arithmeticians, tell me (what is the average wastage of the whole mass of gold)? Statement : wastage: -- 4, - 5, - 6, - 7, - 8, - - 9; - 1, -2, -3, gold: 5, 6, 7, 8, 9, 10; 2, 3, 4. (Solution:) 'Multiplying severally the parts of gold with the wastage', the product is 20, 30, 42, 56, 72, 90, 2, 6, 12; their sum is 330; the sum of the parts of gold is 45; dividing by this we obtain ~^; this is reduced by 15 (i. e. -~); the result is 7 leaving y 3 (i. e. 7 l / 3 ); that is the wastage of each masaka (of mixed gold). Proof: by the rule of three : as the total gold 45 is to the total wastage 330, so 1 masaka of gold is to parts of wastage. In the same way the proof of all (the other) items is to be made (i. e., 45 : 330 = 5 : ^; 45 : 330 = 6 : 44; 45 : 330 ' = 7 : ifi; 45 : 330 = 8 : ip ; 45 : 330 = 9 : 66; 45 : 330 = 10 : ^f). 22 ft. Hoernle. The 53 d sutra. Let the portion given from the daily earnings be divided by the difference of the latter. The quotient, being doubled, is the time (in which), through the gift, their possessions become equal. First Example. Let one serving pandit earn five in three days; another learned man earns six in five days. The first gives seven to the second from his earnings; having given it, their possessions become equal; say, in what time (this takes place)? Statement N 1, | earnings of 1 day, N 2, f earnings of 1 day; gift 7. Solution: 'Let the portion of the daily earnings be divided by the difference of the latter'; (here the daily earnings are f and f; their difference is VIM tne gift i g ?5 divided by the difference of the daily earnings 7 / 15 , the result is 15; being doubled, it is 30; this is the time), in which their possessions become equal. Proof: may be made by the rule of three : 3 : 5 = 30 : 50 and 5 : 6 = 30 : 36 ; 'the first gives seven to the second' 7, remainder 43; hence 43 and 43 are their equal possessions. Second Example. Two Rajputs are the servants of a king. The wages of one per day are two and one-sixth, of the other one and one-half. The first gives to the second ten dinars. Calculate and tell me quickly, in what time there will be equality (in their possessions)? Statement: daily wages -^ and f; gift 10. Solution: 'and the daily earnings'; here (the daily earnings are ^ and f; their difference is f; the gift is 10; divided by the difference of the daily earnings |, the result is 15; being doubled, it is 30. This is the time, in which their possessions become equal). Proof by the rule of three: 1 : - 1 / = 30 : 65; and 1 : f - 30:55. The first gives 10 to the second; hence 55 and 55 are their equal possessions. tin the BakhshSlI Manuscript. Jo NOTES. 1. In the text, the italicised words are conjecturally re- stored portions. The dots signify the number of syllables (aksara) which are wanting in the manuscript. The serpentine lines in- dicate the lines lost at the top and bottom of the leaves of the manuscript. In the translation the bracketed portions supply lost portions of the manuscript. The latter can, to a great extent, be restored by a comparison of the several examples. Occasionally words are added in brackets to facilitate the understanding of the passage. 2. Sutra 18. Problems on progression. Two persons advance from the same point. At starting B has the advantage over A] but afterwards A advances at a quicker rate than B. Question : when will they have made an equal distance? In other words, that period of the two progressions is to be found, where their sums coincide. The first example is taken from the case of two persons travelling. B makes 3 miles on the first day against 2 miles of A] but A makes 3 miles more on each succeding day against 5's 2 miles. The result is that at the end of the 3 d day they meet, after each has travelled 15 miles. For A tra- vels 2 -f- (2 -4- 3) + (2 -f 3 + 3) = 15 miles, and B 3 -f- (3 + 2) -f- (3 + 2 -f- 2) = 15 miles. The second example is taken from the case of two traders. At starting B has the ad- vantage of possessing 10 dinars against the 5 of A ; but in the sequel A gains 6 dinars more on each day against the 3 of B. The result is that after 4 1 / 3 days, they possess an equal amount of dinars f viz. 65. 3. Sutra 27. Problems on averages (samabhagata) . Certain quantities of gold suffer loss at different rates. Question: what is the average loss of the whole? The first problem is very concisely expressed; the question is understood; some words, like kutogatoij must be supplied to samabhagatam. PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY QA 27 I4H6 1887 Hoernle, August Frederich Rudolf On the Bak shall manuscript P&AScl.