1. 1 5 hs 7, tl't 2, tl^&l
EARLY INDIANS ON SECOND OR,DER,
SINE DIFFERENCES
R. C. Guptl
Assistant Professor of llathemat'ics, Birla Institute of Technology,
P.O. Mesra, Ranchi, (Bihar)
(Receiuecl, JuIy 1972)
31
The well known property that the socond ordor diffsrences of sines aro pro-
portional to tho sines themselves was knorvn evon to iryabhata I (born A,
D. 476) whoso Aryabh.tliya is tho earliest extant historicsl work (of the dated
type) containing a sine tablo, The paper describes the various forms of tho
propottionality factor involvod in the mathemabical formula expressing
tho above property,
Relovant references and rules aro givon from the
Indian astronomical works guch as Argabhatiya, Surya-Siddhdnla, Golo,sdra
and Tantra-Saqgraha (A.D. f 500).
The commoncry of Nilakantha Somaydji(born A. D. 1443) on the Aryabhatrya
discusses the property in details and contains an ingenious geometrical
proof of it. The paper gives a brief description of this proof which is merely
bosed on tho similarity of triangles.
Tho Indian mabhomatical method based on the implied differontial process
is founcl, in the words of Delambrc, "neither amongst the Greeks nor
omongst tho Arabs."
1. IxrnonucrroN
Lei (n being a positiveinteger)
So : 'B si:nnh ,.. (t)
Dr:Sr
Dn+ t : Sr * r - Sr . (9
It is easily seen tha,t
D ,-D n * , - F .S , ... (3)
where the propoftionality factor -F (indepenclentof ra) is given by
I :2 (I-c o g h ). ... (4)
Relation (3) represents the fact that in a, set of equidistant tabulated fndian.Sines
defined by (1), the differences of the first Sine-differences(Bn+l-/S,), that is, the
second Sine-differences (Dn-Dn*r) are proportional to the sines B, themselves.
This fact seems to be recognised in India almost since the very begiiLning of fndian
Trigonometry. In Section 2 below we sha,ll describe some of the forms of the
rule (3) alongwith va,riousforms of the factor -F as iound in important Indian lvorks.
In Section3 we shall outline an'Indian proof of the rule'as found in Nilaka4lha
Somayd,ji's AryabhaQiyo-Bhd1ya (: NAB) which was written in the early part of
the sixteenth century of our era,.
YOL .7 , N o. 2.

2. 8i R. C. GUPTA
2. Fonus or rnp Rur,n
It is oasy to see thir,t /
trt : (Dr_D2)lDr ...(5)
tr4ren the norru (r'aclius or ,Sanustotus) R is equal to 3-138minntes and the uniform
tabular interval /r, is equal bo 225 minutes (as is the case with the usual fndian
Sine Tables), we have
Dr : 3438 sin 225' : 224'86 nearl.v'
Dr : 3438 sin 450'-3'138 sin 225' : 213'89nearly,
Dr-D-r: 0'97 : I aPproximatel.v'
Using this value antl (5), rve can put (3) as
D n t.t : D n-S nl D t " ' (6)
A rule rvhich is equivalent to (6) is founcll in tlte Aryabha{iya II,12 of Aryabhala
I (born 476 A.D.) which is the earliest extant historical rvork of the datecl type
containing a Sine table. The rule founde in the Surya-Sidclhd,nta, 15-16 is also
Il,
equivalent to (6) according to thc interpretations of the commcntators Mallikdrjuna
(1I78 A.D.) anci Rimakrsna (I.172 A.D.). The -l/rl.B also acceptsthat t'he Surya-
Sitld,hdntarule is s&me as above and further gives au exact form of the rule (3)
lvhich can be expressedin our notat'ion as follorvs3
D n+t : D n- S o'(DL- Dr)I D t
or,
D n + r: D D r+ ... + D ,).( D r- D 2)D L.
|
" -(D r+
Te Gola.sdro III, l3-1{ gives a rule equir-alent to{
S u -r : S ,-[(2 /n ).{ " t? s in 90" --B si n (90" -D )}' S n* D n+ r]
which implies (3) rvit'h
F :2 @-n cosh)/" rR .
The NAB (part I, p. 53) quotes the Golasdra-rule and further aclds tirat rve
equivalent'ly have
F :2(R versh)/-R .
The actual value of .F (independent of r?) is given by
F : (2 sin 112'5')2: l/233'53 very nearly.
The Tantrasar.ngraha(:78) II,4 givessthe value of the reciprocal of -F as 233.5
and the commenbator therebf even gives it as
233+32160
which is almost equal to the true value.
A rule equivalent to (3) occurs in the ?S II, S-9 (p. 18), v'hich was m'itten in
A.D. 1500, as follorvs :
oFi*qr?iqr<(' taai gq) qt({d ((: r
srr?rsqr4rR?nfr (q'r( (cv€-fcFil(g',rkd: IIq ||
ilr*qr'g gsr{TCrEqT'
fafrcrkfr ficr( r
s{+{c<Eq-Esrr}(t: frrsgqt{il: | |( | |

3. EARLY INDIANS ON SECOND ORDER O'""''ENCES 83
"*U
'Trvice the difference between the last and the last-but-one (Sines)is the multi.
plier; the semi-diameter is the divisor. The first Sine then (that is, when operated
by the multiplier and divisor defined above) becomes the difference of the initial
Sine-differences. lVith those very multiplier ancl divisor (operated upon) the
tabular Sines starting from the second, (rve get) the successivedifferences of
Sine-differencesrespectively.' That is,
2[.Rsin 90"-.R sin (90'-h)] : Multiplier, .L1;
Semidiametet ot radius .B : Divisor D.
Then
(tll lD)S, : Dt-Dz
(MlD)Sn: Dn-Dn+t, n:2,3, ,,.,
So that we have
D,-Dn+r: 2(l-cos D)S,
whichr is equivalent to (3).
Finally, we also ha,Ye
I : (c rd h )2 1 B 2 ... (A )
where crd b denotes the full chord of the arc h in a circle of radius ,8. lviih (A)
as the yalue of the proportionality factor, the N;lB (part I, p. 52) gives the verbal
statement of-the rule (3) as follorvs (NAB rvas composed after ?B)
ilqgfrq.rilgqrqr: gcwsq-rqli girqK!,
ffif g1lrq.K:I s'af (Eq-siqr+Tqq
I
'X'or the Sine at any arc-junction (that is, at any point rvhere two adjacent
elemental arcs meet,) the square of tho full chord is the multiplier; the square of
the radius is the divisor. The result (of operating the Sine by multiplier and
divisor) is the difference of the (trvo adjacent) Sine-differences.'
That is,
D ,- D n * , :' S' ' (c rd h )2R 2 .
| ...(7)
From this, the .lf.llB rightly concludes that
'Uilsrrgtrlf$iq eqrcqq-dni1l€3 1'
'The (numerical) increases of the Sine.differences is proportional to the very
Sines.'
3. Pnoon oF TEE RrrLE
An Indian proof of the rule (7) as found in the NAB (pafi I, pp. 48-52) may
be briefly outlined in the modern language as follows :
Make tho reference circle on a lovel gtound and draw the reference lines XOX'
and YOY' (seethe accompanfng figure where only a quadrant is shown). Mark
the parts of the arc on the circumference (by points, such as L, LItNt which are
at tho &roual interval l,).

4. 84 R. C. GUFIA
!.
I
t
r
t.'
l
I
o X
Take a rcd OQ equal in length to the radius -& and fix firmly and crossly (ancl
symmetrically) another rod jllrY whose length is equal to tho full chorcl of the
(elenrental) arc h at the point P rvhich is at a distance equal to the Versed Sine of
half the elemental arc D from the end Q of the first rod.
The sides of the similar triangles NKII and OAQ are proportional. There-
fore. bv the Rule of Three we have
NK : OA.LINIOQ
x IK :
QA .MN IOQ
fn other worcls rve have*
Lemma, I : The difference of Sines, corresponding to the end-points of an5r
elemental arc, is proportional to the Cosine at the middle of the arc;
Lemma II '. The difference of Cosines, corresponding to the enrl-points of
any element arc, is proportional to the Sine at the middle of the arc;
the proporbionality factor in both casesbeing
: (chord of the arc)/Radius : (crd ft.)/.B
* The Sanslsit text (q-+qTT€qRGTr . qr-e-slizfil as quoted in the lLlB,
Cs:),
statos the Lemmas as two Rules of Three. ,Seb Gupta, R.C., Some fmportant IndianMathematical
llethods as Conceived in Sanskrit Lenguage, peper presonted at the International Sanskrit
Conferenco, New Delhi, llarch 1972, p. 3. For a nice stetemont of tho Lemmas, oee Gupta,
R,. C., Second Order Interpolatioo in Indisn lVletherqatics etc., .f, J.E.5., Vol. 4 (196g),
p. 95, verses 7-8.

5. EARLY INDIAS ON SECOD ORDEIi, SI)iE DIFFEREiiCES do
Thus, in our s1'rnbolsrve ]tave (rvhen arc 'lIX: nh)
D ,+ r: @ rc lh ).OA l R
and, similarly
D,, : (crd h).OBiR.
Therefore,
D, - Dn * , - (c rc lh .(OB-OA )l n . ...(8)
Norv tlre seconilhaf (T,V) of the first. (los'er) arc LTM ancl the first haff QIQ)
of the second (upper) arc illQli together fonn the arc T,IIQ *'hose longth is equal
to that of an elemental arc 1.. Thus rve can place the above frame of tu'o rocls
such that the raclial rod coincicles rvith O-il1 ancl the cross radial rod (therefore)
coinciclesrvit'h the full chorcl of the arc ?Q. ancl consiclerthe proportionality of sides
as before.
frr ot'her rvorcls rve use Lentnta /1 for thc arc 7Q. This will mean that the
difference of the Cosines, OB ancl O-{, corresl>ondingto the encl-points T and.Q,
rvill be proportional to the Sine, ,]1C, at thr. micldle point M of the arc TQ. That is,
rve have
OB-OA : (crd h).:llClR
: (crcli,).(-Esin nh)lR.
Hence by (8)
D n -D ,,., : (c rd tr):.(J s i rt n h ) l R 2
?
rvhich is equivalenb to (7).
4. CoNcr,uorNc Rnrnnrs
An Indian mcthoclof conrputingtabular Sinesbv using a process given basicallS'
by the tule expressedby (7) has been regarclecl curioug b;' Delambre whom Dat'ta6
quotes as remarliing thus :
"This clifferential procoss has not upto norv beeu enrployed except b5r Briggs
(c. f6l5 A.D.) rvho lrirnself dicl not larorv t.hat the const'antfactor rvas the squarc
of the chorcl or the intcrvrr,l (taking unit ratlius). ancl rvho coulcl not obtain it, except
by comparing the second differences obtainecl in a different rnanner. The fndians
also hacl probably clone tlr.e same; they obtair-rthe methocl of differences only from
a table calculated previousl.v bv a geonretrical process. Here then is a methorl
which the fndians possessecl and which is founcl neither arnongst the Greeks nor
amongst the Arabs".
Like Delambre, BurgessTalso thinks that the property, that the second differ-
ences of Sines are proportional to Sines themselvcs, 'rvas knour to the Hindus
only by observation. Had their trigonometry sufficed to demonstra.te it, they
might easily have constructecl much more complete and accurate table of Sines'.
Datta (op. cil.), borvever, sees no reason to suspect that fndians obtained the
above formula (6) by inspection after having calculatecl t'he table by a different
method; "there is no doubt that the early Hindus lvert in possession necessary of
resourcesto deriye the formula". he adds.

6. 86 GIII{TA : EAII,LY I){DIAS ON SECOIfD ON,DER,SINE DIFFXR,ENCES
Finall5' it nr.ay be statecl that various geometrical proofs of the rule have been
givens by moclern schola'rs lilie Nervton, I(rishnas'rvami Ayvanger, Naraharawa
and Srini'r'asiengar. Hori'over, it nray be pointecl out that the rule given by (T)
is exact, ancl not approximato as assumecl b_v some of the above scholars. The
exposition ancl the limiting forms of th.e nrles and results from the NAB and
Yu,kti-Bhdqd. (lTth century A.D.) as given b1' Sarast'atie shoulcl also be noted.
Many other moclern proofs hilvrr been givcn,l0
R,nlonpxcns aND Xott,
I The Aryabhaliyo (rvith the commentary of Paramedvara) edited by H.Kern, Leiclen 1874; p.30.
For a fi esh moclern esposition of tho rulc see Sen, S. N. : Aryabhato's llathem atics, Bulletht
National Institute oJ Scicnces of Itvlia No. 21 (196:)), p. 213.
2 Tho Silrya Siddhdnta (rrith the comment&ry of Paramesivara) editecl by K. S. Shukla, Lucknorv
1957:'p.27. For references to the commentators }lallikdrjuna ancl Ramakrsr.ra see Lucknorv
University transcripts No. 45747 ancl No. 457{9 respectively.
3 The Aryabhaliyct with Lhe Bhagya (gloss) of l{ilakar.rtha Part I (Ganita) edited by S. Sambasiva
Sastri, Trivanclrum, 1930; p. i16.
a Golasdra of Nilakaltha SorntryEji editcd by K. V. Sarma, Hoshiarpur 1970; p. 19.
6 "fhe Tantrasamlyaha of Niltlktrnthtr, Somasutvan (rvith commentary of Sankara, Variar) eclitecl
b y S. K. Pilla i, T r iva n cln r m l9 5E ; p. 17.
6 Drrtta, B. B. : Hinclrr Contribution to llathematics. Bulletht Allahabad Uniu. Math. Assoc..
Vo ls. I & 2 ( 1 9 2 7 - 2 9 ) ; p . 6 :1 .
7 Burgess, E. (translator) Silryo, Siddhdnta. Calcutta reprint lgl)5; p. 62.
6 (i) Bu r g e ss, E., o p . cit., p . s:ji r vhe.rc I{. A . N ervton's prcof i s quoted.
( i i) Ayya n g a r ' , A. A. K.: l' h c Hinr-l u S i ne.Tabl es. ,J. Irtrl i an fuIath. S oc., V ol . 15 (1921),
fir ' st p a r t, p . l:.1 3 .
( i i i ) Na r a h a r a yya , S. N.: No te s o n the H i ncl u Tabl es of S i nes, J.Incl i a,n Math. S oa., V ol . l 5
( 1 9 2 - { ) , No te s a n d Qu e stio n s, p p. l 0S -110.
(iv) Srinivasiengar, C. N.. ?/rc History oJ Ancient Indian Mathematics, Celcutta, lg67; p. 52.
e Sarusn'athi, T. A., The Devclopmr.nt of llathcmatical Series in Inclia after Bhaskara II.
Bulletin oJ the National Irt.st. of Scierrcesof Inrlia No. 2l (106:]), pp. 335-339.
ro See Bina Chatterjee (editor ancl tlanslator): The KhantlakhidyaLa of Brahmagupta. New
De lh i a n d Ca lcu tta , 1 0 7 0 . Vo l. I, pp. f98-205.