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  1. 1. The Mathematics Bdrcrtion SECTION BVo l. V I I I , No.4, D c c . 1 9 74 G L IM P S E S OFA N C IE N T IN D IA N M A T HE M A T I CSNO . 2 1 A lfef$ht and Dlstance Prolrlem From Tlre AnyanrrATiYA 67 R.C. Gupta, Dept. of Mathcnatics Birla InstitutcdTechnolog2 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. Supposethat two equal vcrtical gnomcns (danku) are placed(or the same gnomon is placed succcssively) thc two points P and Q(P being nearer to atthe lamp-post) on the level grourrd such that the horizontal line joining P and Q passesrhro-rrgh O, the foot of the lamppori. Let S and Z be the positionr of the tips (agra) of theshadorvsof the gnomons in the two positions or crses. Thur the tops, M and ,M of thegnonrons will lie on lhe straight lines ZS and LT respectivcly. The points o, P, S, Q, andT n,ill be collinear.z*. t. Let as use the following symbols : First shadow PS:sr Second shadow QT-sz Distance between the tips (or ends) of the slradows ST*d Height of the lamp-post LO:h Lenght or height of each gnomon MP <tr NbC OS:ur - -- r o sq OT:uz FIGUIiE (so that uz-a:d) A rrrle for finding the height of the lampport and distancerof the tips of the shadowsfrom the foot of the lrmp-post is given by qrdCaCqq lryabhata I (born 476 .a.O.| in hisfamols uork cellcd the qrtqqftq Arlebht:ya (abbreviated henceforward as AB;r 1nt *"shall discusr now. Mern b e r . f n t e r n a t i o n a l Co m m issi,r tr o n Hi:to r y o f Il athemati cs
  2. 2. r HE Uf,T II!M ATIC S E D U C A TTON 2. The Rule The author of the AB has called the height of the lamp-post as lfst bhuj-r (arm) andthe distance of the tip or end of a shadow from the foot (mirle) of the larnp-post as 5,ifl koti(upright). These technical terms have been clearly explained in the conrm(.ntaly (on theAB , under t he s ai d ru l e ) b y C (+ s q ( Pa ra me S vara(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 n6p a [ [rjit a k cti / Chayag ur-ritarir ch-yi g r a-vivam-t-l Sairkugur.r-koti s:i chly:i [531t;i bhujl bhavari lll6ll 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.Th e upr ight m ult i p l i e d b y th e (l e n g th o f th e ) gnomon and di vi ded bv the (l ength of therespective)shadow becomesthe, arm ihere the heiglit of the lamppost).That is, upright corresponding to the first shadorv U r:d . s r/(s .:-sr) ...(l) upright corresponding to the second shadow (Jz:d. sz/(ss -sr) ...(2) and then, the height of the lamp-post i - (g n o m o n )x (upri cht)/(shadow ) ...(3)Thus, from either of the above two cases we shall have ft-g .d l (s z - s t) .(4) A pr ac t ic a l m e ri t o f th e fo rm u l a (4 ) i s that i t w i l l gi ve tl re cr)rrect h, i ght even i f u,esubstitute the lengths of the visible shadolvsmeasurecl from tlre edge cr l.rcliptrerlof tlreci rc ular bas eof t h e g n o mo n s .F o r, i f r b e th e radi us of the baseof tl tc (trsrtal crrrri cal ) D on1o,., Bin either case then the first visible sh.adowis tL :s r-r and the second visible shadow is tz :s z -rwhere st and J2 are the lengths of the shadows supposedto be mtasur,d from tlre centres ofthe basesof the gnomons.Thus l z -tr:rz - Tr the lengths of thc visible sltadowsshowing that we shall get the same height even if we Lrsein ( 4) .
  3. 3. R.C- GUP T 73 3. Exernples on the Rule The following example on the rule is given by Bhlskara I in his commentary (629A.D.) on the r83.Exa mple ( i) : ( T he l e n g th so f) th c :.h a d o rv so f th e t * o equal gnomons are seento be l 0 andl6 respectively. The distance betrveenthe shadow-ends is seen to be 30. Give out theupright and the arrn for the each (gnomon). It is understood that the length of the gnomon is l2 (airgulas or finger-breadths)which is the usual length of tlre standard gnomon used in ancient Indian astronomy. Now bv us ing (l ) a n d (z ). th e i w o u p ri g h ts w i l l be found to be 50 and B 0 uni ts.Putting either of thesein (3), or using (4) diectly, the reqtrired height will be found to be6 0 u n it s . This example is reporteda to leappear in the comntentaries by S[tryadeva (Yajvan)(b o rn I l9l A . D. ) , Y a l l a v a (a b o rrt 1 4 7 0 ),a n d R a g hrrnri tha aj a (1597) on the A B . R Another -.imple example given b,v Blrlskara I in the same comlnentary is as followss.Exa rn ple ( ii) : ( 1 hc l e n g tl rso f) th c s h a d o rv s f t w o eqtralgnomons are statedto be 5 and 7 ol e sp e c t iv ely . T he d i s ta rrc cb e trv e e nth e s h a d c ru-cndss observedto be B . Gi ve out the arm ia n cl the upr ight . I I er e t he upr ig h ts r v i l lb e fo trn d to b e 2 C a n d 28, and the hci ghtof the l amp-postto be4 8 u n it s . In botli the :rbovc exarnJrles the tip o{the first shadorvwill be found to fall betweenth e tw o pos it ior , s th e g rto n o l n s . of Arr examprlcin rvhich the tip of thr::first shadow lies (theoretically) beyond the secondsn o n )or ris giv c n br . P a ra me i v a ta i rr I i s c o rtme ntary (p.35) en the A B i n the fol l ow i ngsta n za fEftcrqigqrflrwcq] qrt stqF(( dq]: I cdseq ftvgwr der]ft s Fnqsai rr D i -g h b h i s -s o ri a s a b h i s trrlc hi ye c?i gri ntararhtayoh / yo A rk a tu l y a ri r d l p a b h rrj a ta tkoti ca ni gadyatem l lExa m ple ( iii) : f he s h :rd o w sa re e i tu a l (i n l e n g t hs) to l 0 and 16 and the di stance betw eenth e i r tips is equal t o 1 2 . T e l l th t: h ti g l rt o f th e l a mp-postand the upri ghts. T he c om m en ta to r tl ;e n c o n e c tl y g i v e s th e upri ghts as 20 and 32, and the hei ght ofth e l a r lp- pos t as 114u n i ts . Rationale of the Rule If s be the shadorv corresponding to any positiorr of a gnomon and z be the corres-p o n d ingupr ight , t ha ti s ,tl i e d i s ta n c e o fth c ti p of theshadow fromthefootofthel amp post,we e a s ilyget , f r om s i mi l a r tri a n g l e s ,
  4. 4. !EE UIIIEDU T!IOT E D U OA TIOII7+ slp:ul h ...( s) or 7a-:(hlg.) s ...(6) Therefore shall have we y 1 :(h lg). st ..(7) a 2 :(h l g). sz ...(B) wlsr:uzlsz: hlg ..(e) : (uz- ur)I $r -sr) - d/(52 -sr; ...(10)a nd t he r es ult s (l ), (2 ) a n d (3 ) fo l l o w . The commentator Parame5vara, however, gives (p.35) a different approach to derivethese results. His arguments are based on considering the changes in the length of the shadow andthe ,upright as the position of the gnomon is changed. In modern, notation these changcs,as we can seefrom the relation (6), are connected by the formula tru-(rtlg).trs ...(l l ) That is, the change in the upright is (linearly) proportional to the change in the shadow and thus the Rule of Three (*<rFnt) can be used. With this preliminary explanation, the rationale given by Pararneivara may be outli- ned as follows : He starts by saying that €tq{qwr$T {rqslft qIqT? Tlut[d When the gnomon is at the foot of the lamp-post, there is no shadow.G Then he considersthe changes (here decrements)in the lerrghsof 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;TqTql6s4 sr qFIIcEfqgolr;d(reTfcq : I qFuRFo; Gq tlf corresponding to a diminution equal to the difference of shadows there is obtained a (in the tvvo bhlmi (here a horizontal distance) equal to the distance between the sltadow-tips equal to a position), then how much bhi,mi will be obtained corre:pondir)g to tlie dirninution (the restrlt is that ) we given shudorv (when the gnomon moves to the foot of the lamp-post); get the horizontal distance between the foot lamp-post and the tip olthe isaid) shadow. That is (s z -s r) 3 (u z -u1):s:11 giv ing s1: 51 (uz- ur)/ (sz-rr) )/ (rz- sr) a nd uz: sz (uz-ur)/(sl -sr) :dsz/ .sz- rr) rvhich are the re q u i re d re l a ti c n s (l ) a n d (2). (5) Finally, he gives another proportionality rule which is eqtivalerrt to tbe relation
  5. 5. R.C. GUP N f, /Jand which leads to the desired rule expressed by (3). Sengupta? has partially derived the results from the relations h l u :g l s t hluz:glszwhich firllow from similar triangles. The d,erivationsgiven by Baladeva MishraE are unnecessarilylong. Refercncer and Notesl. Fo r abr ief not eon h i s w o rk s , s e e R .C . Gu p ta , " A ryrbhata I s V al ue of P i . " (Gl i mpses of Ancient Indian Math. No. 5), The MathentaticsEducation, Vol. VII, No. I (March 1 9 73) ,s ec .B , p. 1 7 .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, Brill, Leiden, 1874.3. K.S. Shukla, "Hindu Mathemrrics ir thc Seventh Century as Found in BhFskara ls C o mm ent ar y on t h e AB " (II), C a q i 1 3 ,V o l .2 2 , N o.2 (D ecemberl 9Tl ) p.67.4. Ibid. (foot-note)s. Ib i d . ( 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 Senguptas expla- naticn is not draun properly becausethe 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<hSccitty, Parna (about 1966).