The Mathematics Bdrcrtion

Vol. VI I I , No.4, Dcc. 1974

SECTION B

GLIMPSES OFANCIENT INDIAN MATHEMATICS NO. 12A lfef$ht and Dlstance Prolrlem From Tlre

AnyanrrATiYA67 R.C. Gupta, Dept. of Mathcnatics Birla Institutc dTechnolog2 P,O. Mesra, RANCHI.

(Recoived l7 October (1974)

l. Introduction

Let LO be a lamp-post standing vertically on a level ground at the point O and carry-ing the lamp at its top point.L. Suppose that two equal vcrtical gnomcns (danku) are placed(or the same gnomon is placed succcssively) at thc two points P and Q(P being nearer tothe lamp-post) on the level grourrd such that the horizontal l ine joining P and Q passes rhro-rrgh O, the foot of the lamp'pori. Let S and Z be the positionr of the tips (agra) of theshadorvs of the gnomons in the tw'o positions or crses. Thur the tops, M and ,M of thegnonrons wil l l ie on lhe straight l ines ZS and LT respectivcly. The points o, P, S, Q, andT n,i l l be coll inear.

z*.

t\. Let as use the following symbols :

First shadow PS:srSecond shadow QT-szDistance between the tips (or ends) of theslradows ST*d

Height of the lamp-post LO:h

Lenght or height of each gnomonMP <tr NbC

OS:ur

OT:uz

(so that uz-a:d)

o - sq -- rFIGUIiE

A rrrle for f inding the height of the lamp'port and distancer of the tips of the shadows

from the foot of the lrmp-post is given by qrdCaCqq l\ryabhata I (born 476 .a.O.| in his

famols u'ork cellcd the qrtq'qftq Arl 'ebht:ya (abbreviated henceforward as AB;r' 1n't *"

shall discusr now.

Mernber . fnternat ional Commissi , r t r on Hi : tory of I \ lathematics '

r HE Uf,TI I !MATICS EDUCATTON

2. The Rule

The author of the AB has called the height of the lamp-post as lfst bhuj-r ( 'arm') and

the distance of the tip or end of a shadow from the foot (mirle) of the larnp-post as 5,if l koti

( 'upright'). These technical terms have been clearly explained in the conrm(.ntaly (on theAB, under the said rule) by C(+sq( ParameSvara (about 1360-1455).

Now we give the original rule. The AB, II, 16 (p.34) states :

qrqrgfqrd qTqrqfbE(Tl4 qlfqdl qlal r

srsiggqfi +]fl sr qIIIT rTffI gsTI cs[il lttqll

Chayag ur-ritarir ch-y'i g r a-vivam-t-l n6p a [ [rjit a k cti /Sairkugur.r- koti s:i chly:i [531t;i bhujl bhavari l l l6l l

'The distance between the tips of the shadows Inultiplied by (the length of ) a sha-

dow and Jivided by the difference (ofthe lengths ofthe shadorvs) is the (repective) upright.

Theupr ight mult ip l ied by the ( lengthof the) gnomon and div ided bv the ( length of the

respective) shadow becomes the, arm ihere the heiglit of the lamp'post).

That is,upright corresponding to the first shadorv

Ur:d. sr / (s. : -sr)

upright corresponding to the second shadow(Jz:d. sz/(ss -sr)

and then, the height of the lamp-post

i - (gnomon) x (upr icht) / (shadow)

Thus, from either of the above two cases' we shall have

ft-g.dl(sz- st)

. . . ( l )

. . . (2)

. . . (3 )

. (4)

A pract ical meri t of the formula (4) is that i t wi l l g ive t l re cr)rrect h, ' ight even i f u,e

substitute the lengths of the visible shadolvs measurecl from tlre edge c'r l.rcliptrerl ' of t lre

circular base of the gnomons. For, i f r be the radius of the base of t l tc ( t rsr ta l crrrr ical) BDon1o,. ,

in either case' then

the first visible sh.adow is

tL:sr-r

and the second visible shadow is

tz:sz-r

where st and J2 are the lengths of the shadows supposed to be mt'asur,'d from tlre centres of

the bases of the gnomons.

Thuslz- t r : rz- Tr

showing that we shall get the same height even if we Lrse the lengths of thc visible sltadows

in (4).

R.C- GUP'T

3. Exernples on the Rule

The following example on the rule is given by Bhlskara I in his commentary (629

A.D.) on the r\83.

Example ( i ) : ' (The lengths of) thc: .hadorvs of the t ' * 'o equal gnomons are seen to be l0 and

l6 respectively. The distance betrveen the shadow-ends is seen to be 30. Give out the

upright and the arrn for the each (gnomon).'

It is understood that the length of the gnomon is l2 (airgulas or f inger-breadths)

which is the usual length of t lre standard gnomon used in ancient Indian astronomy.

Now bv using ( l ) and (z) . the iwo upr ights wi l l be found to be 50 and B0 uni ts.

Putting either of these in (3), or using (4) di 'ectly, the reqtrired height wil l be found to be

60 uni ts.

This example is reporteda to leappear in the comntentaries by S[tryadeva (Yajvan)

(born I l9 l A.D.) , Yal lava (aborr t 1470), and Raghrrnr i tha Raja (1597) on the AB.

Another -. imple example given b,v Blrlskara I in the same comlnentary is as followss.

Exarnple ( i i ) : ' (1 'hc lengt l rs of) thc shadorvs of two eqtral gnomons are stated to be 5 and 7

lespect ively. The distarrcc betrveen the shadcru-cnds is observed to be B. Give out the arm

ancl the upr ight . '

I Iere the upr ights r 'v i l l be fotrnd to be 2C and 28, and the hcightof the lamp-post to be48 uni ts.

In botl i the :rbovc exarnJrles the tip o{'the first shadorv wil l be found to fall between

the two posi t ior ,s of the grtonolns.

Arr examprlc in rvhich the tip of thr:: f ir 'st shadow lies (theoretically) beyond the second

snon)orr is g ivcn br.Parameivata i r r I ' is cortmentary (p.35) en the AB in the fo l lowing

stanza

fEftcrq'igqrflrwcq] qrt stqF(( dq]: I

cdseq ftvgwr der]ft s Fnqsai rr

Di-ghbhis-sor iasabhistrr lyo chiye c?igr intararh tayoh /Arkatulyar i r d lpabhrr ja tatkot i ca nigadyatem l l

Example ( i i i ) : ' fhe sh:rdows are ei tual ( in lengths) to l0 and 16 and the distance between

their t ips is equal to 12. Tel l tht : ht ig l r t of the lamp-post and the upr ights. '

The commentator t l ;en conect ly gives the upr ights as 20 and 32, and the height of

the lar lp-post as 114 uni ts.

Rationale of the Rule

If s be the shadorv corresponding to any positiorr of a gnomon and z be the corres-

pondingupr ight, that is, t l iedistanceofthct ip of theshadowfromthefootof thelamp'post,

we easi ly get, f rom simi lar t r iangles,

73

_.-

7+ !EE UII IEDUT!IOT EDUOATIOII

slp:ul hor 7a- :(hlg.) s

Therefore we shall havey1:(hlg). st

a2:(hlg). sz

wlsr:uzlsz: hlg: (uz- ur) I $r -sr) - d/(52 -sr;

and the resul ts ( l ) , (2) and (3) fo l low.

The commentator Parame5vara, however, gives (p.35) a different approach to derive

these results.

His arguments are based on considering the changes in the length of the shadow and

the ,upright' as the position of the gnomon is changed. In modern, notation these changcs,

as we can see from the relation (6), are connected by the formula

tru-(rtlg). trsThat is, the change in the upright is (l inearly) proportional to the

shadow and thus the Rule of Three (*<rFnt) can be used.

With this preliminary explanation, the rationale given by Pararneivara

ned as follows :

He starts by saying that

€tq{qwr$T {rqslft qIqT ? Tlut[d

'When the gnomon is at the foot of the lamp-post, there is no shadow'.G

Then he considers the changes (here decrements) in the lerrghs of shadows and upri-

ghts as the gnomon supposedly moves from the outer-most position to the nearer positiorr and

afterwards, to the foot of the lamp-post; and applies the trairaiika fAf{fo (Rule of l hree)

in the following words6

qf< qrqr;e<gda qr46tta qrar(n)ra< g"qr rlfrdr+t, a?raqrqrld;T qTql6s4 sr

qFuRFo; qFIIc Efqgolr;d(reTfcq Gq : It l f corresponding to a diminution equal to the difference of shadows there is obtained a

bhlmi (here a horizontal distance) equal to the distance between the sltadow-tips (in the tvvo

position), then how much bhi,mi wil l be obtained corre:pondir)g to tl i 'e dirninution equal to a

given shudorv (when the gnomon moves to the foot of the lamp-post); (the restrlt is that ) we

get the horizontal distance between the foot lamp-post and the tip ol 'the isaid) shadow.'

(sz-sr) 3(uz-u1):s:11

s1: 5 1 (uz- ur) / (sz-rr) :d.sr )/ (rz - sr)

uz: sz (uz-ur)/(sl -sr) :d'sz/ .sz- rr)

required relat icns ( l ) and (2).

he gives another proportionality rule which is eqtivalerrt to tbe relation (5)

. . . (s)

. . . (6)

. . (7). . . (B). . ' (e). . . (10)

. . . ( l l )

change in the

may be outl i-

That is

giv ingand

rvhich are the

Finally,

R.C. GUP Nf,

and which leads to the desired rule expressed by (3).

Sengupta? has partially derived the results from the relations

hlu:glst

hluz:glsz

which fir l low from similar triangles.

The d,erivations given by Baladeva MishraE are unnecessarily long.

Refercncer and Notes

l . For abr ief noteon his works, see R.C. Gupta, "Aryrbhata I 's Value of Pi . " (Gl impsesof Ancient Indian Math. No. 5), The Mathentatics Education, Vol. VII, No. I (March

1973), sec. B, p. 17.

2. Several cditions and translation of th AB hava been published. The page-references in

the present article are according to the edition (wirh the commentary of Parametivara)

by Dr. H. Kern, Bril l , Leiden, 1874.

3. K.S. Shukla, "Hindu Mathemrrics ir thc Seventh Century as Found in BhFskara l 's

Commentary on the AB" ( I I ) , Caqi13, Vol .22, No.2 (Decemberl9Tl) p.67.

4. Ibid. (foot-note)

s. Ib id. (p. 67)

6. We have emended the text at one place as indicated by the brackets.

7. P.C. Sengupta (translator) : "'fhe Ar1abhotri1am." Journal oJ the Departmcnt of Letters(Calcutta University), Vol. XVI, 1927, atticle No. 6, p. 20.

However, it may be pointed out tbat the diagram accompanying Sengupta's expla-naticn is not drau'n properly because the tip of the first shadow is assumed, quite unne-cessarily, to fall on the (entre of the) base of the second gnomon.

B See his edition of the Aryabhatiyam uith Scnskrit commentary and Hindi translatien,pp 35-36; Bihar Resear<h Sccitty, Parna (about 1966).

/J