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SHORTER NOTICES

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S H O R T E R N O T I C E S

The Bakhshl Manuscript A Study in Mediaeval Mathematics. B y G. R.

K a y e . W i t h p h o t o g r a p h i c fa c si m il es of t h e m a n u s c r i p t a n d t r a n s

l i t e r a t i on of t he t e x t i n r om a n s c r i p t . Ca l c u t t a , Go ve r nm e n t of I nd i a ,

C e n t r a l P u b l i c a t i o n B r a n c h , 1 9 2 7 . P p .

i

+ 1 5 6 + v . P l at es 1 + 4 + 4 2 .

Pr ice Rs 28 or 43s . 6d .

Bakhsh l i s the name of a v i l lage near the c i ty of Peshawar in the

no r th - we s t c o r ne r o f I nd i a . Th e r e i n 1881 , a f a r me r une a r the d a ma n u

sc r i p t of a wor k on ma the m a t i c s . T he g r e a t e r po r t i on of i t h a s be e n de

s t r oye d by r ough ha nd l ing a nd t he r e ma ins c ons i s t o f 70 l e a ve s o f b i r c h -

ba r k bu t some o f t he se a r e me r e s c r a ps . Tha t ma nusc r ip t i s now ge ne r a l l y

know n a s t h e Ba k hsh l M a nu sc r i p t . I t i s p r e se r ve d i n t he Bod le i a n L ib r a r y ,

Oxf o r d . T he wor k of a na ly s ing a nd e d i t i ng t h e M a nu sc r ip t wa s be gun b y

the d i s t i ngu i she d I ndo log i s t , Dr . R . Hoe r n l e , bu t on a c c oun t o f h i s un

t ime ly de a th , i t c a me to t he ha nds o f Mr . G . R . Ka ye who ha s suc c e s s f u l l y

f in ishe d i t . T he e d i t o r ha s a dd e d a n e l a bo r a t e a nd m a s t e r l y i n t r odu c t i on

to t he t e x t c ove r ing 84 pa ge s i n p r i n t wh ic h c on t a in s no t on ly a n a na ly s i s

of t h e c o n t e n t s o f t h e m a nu sc r ip t b u t a l so a s t u dy of i t s r e l a t i on w i th

o the r H indu t r e a t i s e s o f ma the ma t i c s a s f a r a s pos s ib l e .

The Ba khsh l wor k i s a c ompe nd ium o f r u l e s a nd i l l u s t r a t i ve e xa mple s .

I t g ives a l so the so lu t ions of the mos t o f the examples toge ther wi th the i r

ve r i fi c a t ion . I t c on t a in s ma te r i a l r e l a t i ng t o a r i t hm e t i c , a l ge b r a , a n d

ge o me t r y ( i nc lud ing m e ns u r a t i o n ) . Th e t op i c s f or d i s c us s ion a r e found t o

inc lude f r a c t i ons , squa r e - r oo t , a r i t hme t i c a l a nd ge ome t r i c a l p r og r e s s ions ,

i nc ome a nd e xpe nd i tu r e , p ro f it a nd l os s , c om pu t a t i on of go ld , i n t e r e s t ,

r u l e of t h r e e , su m m a t io n of c e r t a in c om ple x s e ri e s , s imp le e qua t i on s ,

s i m u l t a n e o u s l i n e a r e q u a t i o n s , q u a d r a t i c e q u a t i o n s , i n d e t e r m i n a t e e q u a

t i ons o f t he s e c ond de g r e e o f a pa r t i c u l a r t ype , me nsu r a t i on a nd mi sc e l

l a ne ous p r ob l e ms . The t r e a tme n t o f a l l t he se sub j e c t s i s c ommon ly i nc lude d

in o th e r H in du t r e a t i s e s of m a th e m a t i c s . Bu t we mi s s i n i t a n y r e f e re nc e

to t he t r e a tme n t o f t he i nde t e r mina t e e qua t i on o f t he f i r s t de g r e e a nd t he

so-ca l led Pe l l ian equa t ion , bo th of which en te r la rge ly in to la te r Hindu

wor ks a nd i n t he so lu t i on o f wh ic h t he H indus l ong a n t i c ipa t e d t he wor ks

o f Eu l e r a nd La g r a nge .

The mos t no t a b l e ma the ma t i c a l p r i nc ip l e s i n t he Ba khsh l wor k a r e t he

a p p r o x i m a t e s q u a r e - r o o t f o r m u l a ,

t 2_J_ M/ 2 _j_ M

2

* *

2

(a

2

+ r)

1

2

= a

~ --r>

2a 2(a+r/(2a))

the ca lcu la t ion of the e r rors of success ive order

f/ 2fl * x

2

- W . - G 5 >

a nd a c on se qu e n t p r oc e s s of r e c onc i l i a t i on . T he f und a m e n ta l a r i t hm e t i c a l

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SHORTER NOTICES

[Jul y-Aug.,

ope r a t i ons a r e i nd i c a t e d by pu t t i ng a f t e r , oc c a s iona l l y be f o r e , t he qua n t i t y

a f fe c t ed , t h e a bb r e v i a t i on s of t h e Sa n sk r i t wor ds of r e spe c t i ve im po r t . I n

the c a se o f sub t r a c t i on , t he a bb r e v i a t i on i s f ound t o ha ve de ge ne r a t e d i n to

a s imple c ross -s ign ( -+- ) . The Bakhsh l work i s par t icu la r ly charac te r i sed

by t h e a bse n c e of a n y k ind of a l ge b r a i c sym bo l i sm . Th i s ha s ne c e s s i t a t e d

the preserva t ion of every de ta i l o f the working of the so lu t ions of a lgebra ic

p r ob l e ms ke e p ing up t he i r ge ne r a l i t y t h r o ug ho u t so t h a t t h e f inal s t a t e m e n t

of t h e r e su l t s shou ld c le a r l y p r e se n t t he who le f o r mu la s i nvo lve d . I nde e d

the nume r i c a l qua n t i t i e s i n t hose p r ob l e ms a r e t r e a t e d a lmos t l i ke a lge b r a i c

sym bo l s , r i gh t l y obse r ve s M r . K a y e . M os t of t hose p r ob l e m s ha v e be e n

so lve d by t he me thod o f

false position.

Th e p l a c e of a n unk now n qua n t i t y

in a s t a t em en t of a prob lem is ind i ca te d by a he av y dot ( ) and i s des ign a

t e d a s

nyasthna

o r e m p t y s p a c e , m e a n i n g t h e r e b y t h a t t h e n u m b e r

to be p l a c e d t he r e ha s no t be e n a sc e r t a in e d . O th e r ma t t e r s of i n t e r e s t a r e :

t he l e a s t c ommon mu l t i p l e , t he so lu t i on o f t he qua d r a t i c e qua t i on , t he

e xc lu s ive u se of t h e de c im a l p l a c e - va lue no t a t i on w i th t h e c iphe r , t he a ve r

a ge va lue a nd t he c ha nge r a t i o s o f me a su r e s .

T he t ime of t h e Ba khsh l wor k i s r a t he r unc e r t a in . On pa l e og r a ph i c a l

e v ide nc e Mr . Ka ye ho ld s t ha t i t s hou ld be r e f e r r e d t o a pe r i od a bou t t he

twe l f t h c e n tu r y . Th e r e a r e c l e a r s i gns t o show th a t t he p r e se n t m a nu sc r ip t

i s a n imp e r f e c t c op y of a n o lde r ma n usc r ip t wh ic h a ga in i s of t he c h a r a c t e r

of a pe rpe tua l co m m en ta ry on a s t i ll ea r l ie r wo rk . T he ed i tor has fai led

to de te c t i t . B ut i t d id no t escape th e keen eyes of Dr . Hoe rn l e . H e i s of

o p i n i o n t h a t t h e p r e s e n t c o p y w a s m a d e a b o u t t h e n i n t h c e n t u r y b u t t h e

or ig ina l work must be re fe r red to the ear ly cen tur ies of the Chr i s t ian e ra .

Th e r e a r e m a n y t h ings in t h e wor k t o po in t t o t he s a me pe r iod . M r . K a y e

has drawn a t ten t ion to a few ins tances of fore ign inf luence in the Bakhsh l

w o r k i n c l u d i n g t h e o n e a b o u t t h e a p p r o x i m a t e s q u a r e - r o o t s t a t e d a b o v e

wh ic h he s a ys he ha s no t f ound i n a ny o the r H indu wor ks . Tha t f o r mu la ,

c o r r e c t up t o t he s e c ond a pp r ox ima t ion , wa s known in t he e a r l y J a ina

canonica l works (about 300 B.C. or s t i l l ea r l ie r ) and a l so reappears in la te r

wor ks i nc lud ing t ha t o f r ya bha t a ( 499 A . D . ) a s ha s be e n po in t e d ou t by

Rode t . Ano the r ha s be e n ove r s t a t e d unde r a s imi l a r mi sa pp r e he ns ion a nd

the res t a re minor and of doubt fu l va lue . The ed i tor h imse l f does no t wish

to p r e s s upon t he l a t t e r .

The s i gn i f i c a nc e a nd impor t a nc e o f a pub l i c a t i on l i ke t he p r e se n t one

a r e a p pa r e n t t o a l l l ove r s of t he h i s t o r y of s c ie nc e . An d t he i r t h a n ks a r e

c e r t a in ly due t o Mr . Ka ye f o r t he g r e a t a moun t o f pa in s t ha t he ha s t a ke n

in e xp l a in ing a nd e d i t i ng t h e Ba khsh l ma nu sc r ip t . Th e r e a r e , no doub t ,

severa l inaccurac ies in h i s s tudy of the work , some of which a re no t negl i

g ib le . I t i s no t poss ib le to po in t th em o ut in a sho r t rev iew l ike th i s . A

cr i t ica l examina t ion of them wi l l be found in a paper by the present rev iewer

e n t i t l e d The Bakhshl mathematics in t h e Bu l l e t i n of t h e Ca l c u t t a M a th e

ma t i c a l Soc i e ty ( vo l . 21 , pp .

1-60 .

B ut the y do no t lessen th e labo r of

th e ed i tor . M ore ove r such inaccura c ies a re no t un usu a l in case of a l l

p ioneer s tud ies of d i f f icu l t anc ien t manuscr ip t s .

BlBHUTIBHUSAN DATTA