On the Bakshali manuscript

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

0 E OF THE C 0 U K T ; A N D OF THE UNIVERSITY

-I

ON THE

BAKHSHALI MANUSCRIPT.

Da E. HOEKNLE.

WITH THREE P H 0 T 0 Z IN C 0 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 3y2 inches; their original
size, however, must have been about 7 by 8Y4 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 8l/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 9th; the latest is the 57th. 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 10th 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 25th 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
25th 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 10th and the beginning of the 11th 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 10th 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 8th or 9th 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 3d or 4th 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 5th 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 5th 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
3d 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 3d or 4th 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 1st century B. C., probably extended to about the
end of the 3d 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 2nd 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 50th sutra of
the Bakhshali arithmetic, or rather with the algebraical example
occurring in that sutra, and the 49th 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 position1. 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 place1 (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 dialect1, 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-sadbhi7 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 rhe earns1 (s. 53);
parasm. for atm., vikrindti for vikrmlte 'he sells1 (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
50th, chaupamchdshama 54th, 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«/a£r?/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-bhajay«f 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-bhajayeta2) ta-

tah harasasye krite yutaiii | ^ | samyutaih kanakair bhaktva tada

kanaka 10 anena bhaktaih jatarii | 6603 | 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 || 45f | 56f || ?+ || l+ \\ l+ \\ i90+ | 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 | 3430 | parii-
chadash abhage-sh-chheda kriyate ! phalarii | 7 she 3 | esha ekaika-
mashakaksayaih | pratyaya trairashikena \ 4,5 | 330 1 \ \ phalarii 232 |
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

Pha

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 18th 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
N°II, » 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 4y3 days).

20 B. Hocrnlc.

The 27th 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
wastage1, 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, 0 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 y3 (i. e. 7l/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 53d 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 ig ?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
3d 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 41/3 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.