Appendix A Vallyupasaṃhāra and continued fractions Ever since the work of Āryabhaṭa on the kuṭṭaka procedure for solving linear indeterminate equations, Indian astronomers and mathematicians have been using this method to solve a variety of problems. The method, also referred to as kuṭṭākāra, basically makes use of a technique called vallyupasaṃhāra which is analogous to the continued fraction expansion of a ratio of integers. The vallī introduced by Āryabhaṭa is nothing but the column composed of the quotients which arise in the mutual division of the integers. The vallyu- pasaṃhāra method of transforming the vallī is essentially the recursive process of calculating the successive convergents of the associated continued fraction. In Karaṇapaddhati Putumana Somayājī displays a very sophisticated un- derstanding of the mathematical properties of the continued fraction expan- sion of a ratio of two integers G, H.1 Usually, G is the guṇa or guṇakāra and H G is the hāra or hāraka, and their ratio ( H ) is the rate of motion of a particular planet or its apogee or node etc. Thus, G being the corrected revolution num- ber and H the total number of civil days, they are indeed very huge numbers. Chapter 2 of Karaṇapaddhati essentially presents the method of approximating H the ratio G by the successive convergents of the associated continued fraction. Karaṇapaddhati also reveals a very sophisticated understanding of the prop- erties of the convergents including a very interesting “remainder theorem”, as we shall explain in this appendix. A.1 Simple continued fraction and its convergents G We start with the ratio of two integers H , where G is the guṇakāra and H is the hāraka. Normally H is larger than G and it is useful to consider the ratio H H G . We now discuss the continued fraction expansion of G . Dividing H by G, we get 1 For an introduction to continued fractions, see Khinchin 1964. © Springer Nature Singapore Pte Ltd. 2018 and Hindustan Book Agency 2018 319 V. Pai et al., Karan. apaddhati of Putumana Somayājī, Sources and Studies in the History of Mathematics and Physical Sciences, https://doi.org/10.1007/978-981-10-6814-0 320 Vallyupasaṃhāra and continued fractions H r = q + 1 , (A.1) G 1 G where q1 is the quotient and r1 < G is the remainder. Now dividing G by r1, we get G r2 = q2 + . (A.2) r1 r1 Continuing in this manner, we obtain a series of quotients and remainders r1 r3 = q3 + , r2 r2 . = . ri−2 ri = qi + . (A.3) ri−1 ri−1 As these quotients (qi) and remainders (ri) are obtained by the mutual division of the numbers H, G, we can write H 1 = q + . (A.4) G 1 1 q + 2 1 q + 3 1 q + ... + 4 ri qi + ri−1 H Since G is a ratio of two integers, the process will terminate for some n, when rn = 0. We thus have H 1 = q + . (A.5) G 1 1 q + 2 1 q + 3 1 q4 + ... + qn This process of mutual division is also the well known process (so called Eu- clidean algorithm) for finding GCD of the numbers H, G which is in fact given by rn−1. H The above equation gives the simple continued fraction expansion of G . If we truncate the above process at any intermediate stage k < n, then we get one of the so called convergents of the continued fraction given by H 1 k = q + . (A.6) G 1 1 k q + 2 1 q + 3 1 q4 + ... + qk A.2 Properties of the convergents 321 In particular, we have H1 q1 = (H1 = q1, G1 = 1), (A.7) G1 1 H2 1 = q1 + G2 q2 q1q2 + 1 = (H2 = q1q2 + 1,G2 = q2), (A.8) q2 and so on. A.2 Properties of the convergents H Consider the simple continued fraction expansion of G : 1 q + . (A.9) 1 1 q + 2 1 q3 + q4 + . .. The successive convergents are H q H q q + 1 H q (q q + 1) + q 1 = 1 , 2 = 2 1 , 3 = 3 2 1 1 ,... (A.10) G1 1 G2 q2 G3 q3q2 + 1 We shall now proceed to explain the properties of convergents. First we shall show that the following recursion relations are satisfied by Hk, Gk for k > 2: Hk = qkHk−1 + Hk−2, (A.11) Gk = qkGk−1 + Gk−2. (A.12) The proof is by induction on k. Clearly from (A.10), we see that (A.11) and (A.12) hold when k = 3. Assuming that these equations hold for k, we shall show that they hold for k + 1. From (A.6), it is clear that Hk+1 is same Gk+1 Hk 1 as with qk replaced by qk + . Therefore, Gk qk+1 322 Vallyupasaṃhāra and continued fractions 1 (qk + )Hk−1 + Hk−2 Hk+1 qk+1 = 1 Gk+1 (qk + )Gk−1 + Gk−2 qk+1 (q q + 1)H − + q H − = k+1 k k 1 k+1 k 2 (qk+1qk + 1)Gk−1 + qk+1Gk−2 q (q H − + H − ) + H − = k+1 k k 1 k 2 k 1 qk+1(qkGk−1 + Gk−2) + Gk−1 q H + H − = k+1 k k 1 . qk+1Gk + Gk−1 Thus, we have shown that the recurrence relations (A.11) and (A.12) are valid for all k > 2. We shall now show another important property of Hk’s and Gk’s, namely k HkGk+1 − Hk+1Gk = (−1) . (A.13) From the recurrence relations (A.11) and (A.12), we see that HkGk+1 − Hk+1Gk = Hk(qk+1Gk + Gk−1) − (qk+1Hk + Hk−1)Gk = HkGk−1 − Hk−1Gk = −(Hk−1Gk − HkGk−1) . k−1 = (−1) (H1G2 − H2G1) = (−1)k. (A.14) From the above relation we can also derive yet another interesting property of the convergents, namely Hk+1 Hk 1 − = . (A.15) Gk+1 Gk GkGk+1 We can easily see that H H q H + H − H k+1 − k = k+1 k k 1 − k Gk+1 Gk qk+1Gk + Gk−1 Gk (H − G − H G − ) = k 1 k k k 1 Gk+1Gk (−1)k−1 = . (A.16) Gk+1Gk A.3 Remainder theorem of Karaṇapaddhati 323 A.3 Remainder theorem of Karaṇapaddhati The Karaṇapaddhati states and makes extensive use of a ”remainder theorem” which gives the difference between the number H and its convergents Hi in G Gi terms of the remainder ri which is obtained in the mutual division of H, G. Now from the previous discussions, we know that the ratio H r Gq + r = q + 1 = 1 1 . (A.17) G 1 G G The difference between the actual ratio and its first approximation (A.7) can be written as H H1 Gq1 + r1 − = − q1 G G1 G r = 1 . (A.18) G Therefore, GH1 − HG1 = Gq1 − H = −r1. (A.19) Now, from (A.4) H 1 = q + G 1 r2 q2 + r1 r1 = q1 + q2r1 + r2 (q q + 1)r + q r = 1 2 1 1 2 . (A.20) q2r1 + r2 Using (A.8) in the above we have, H H r + q r = 2 1 1 2 . (A.21) G G2r1 + r2 Therefore, G(H2r1 + q1r2) = HG2r1 + Hr2, or (GH2 − HG2)r1 = −(Gq1 − H)r2 = r1r2, (A.22) where we have used (A.19). Hence, GH2 − HG2 = r2. (A.23) 324 Vallyupasaṃhāra and continued fractions Now we present the general version of the above remainder relation which the Karaṇapaddhati states and makes extensive use of. Theorem Hi th H th If is the i convergent of and ri is the i remainder in the mutual Gi G division of H and G, then i−1 HGi − GHi = (−1) ri. (A.24) Proof: We have already seen that (A.24) is valid for i = 1, 2. Now, we shall assume that relation (A.24) is true for some i and then show that it is true for i + 1. To be specific let i be even so that (HGi − GHi) = −ri. (A.25) Using the recursion relations Gi+1 = Giqi+1 + Gi−1, and Hi+1 = Hiqi+1 + Hi−1, we obtain HGi+1 − GHi+1 = (HGi − GHi)qi+1 + HGi−1 − GHi−1 = −riqi+1 + ri−1. (A.26) H In the expression for the continued fraction of G considered earlier, the re- mainder ri+1 is obtained by dividing ri−1 by ri. The corresponding quotient is qi+1 (see section A.1). Hence, ri−1 can be written as ri−1 = riqi+1 + ri+1, or ri+1 = −(riqi+1 − ri−1). (A.27) Thus from (A.26), we have ri+1 = (HGi+1 − GHi+1), (A.28) thereby proving the above theorem. A.4 Some applications of the Remainder theorem 325 A.4 Some applications of the Remainder theorem The above Remainder theorem is used in the computation of dvitīyahāras de- scribed in the second chapter of Karaṇapaddhati. The third chapter of Karaṇa- paddhati also introduces what are called kendraphalas which are nothing but the remainders (rij) which arise when we mutually divide the alpahāras Hi th and alpaguṇakāras Gi. Let rij be the j remainder when we mutually divide th Hi and Gi. This is called the j kendraphala of the hāraka Hi. Applying the above remainder theorem (A.24) to Hi , we clearly obtain Gi j+1 rij = (−1) (HiGj − GiHj). (A.29) This is what is referred to as hāraśeṣa (when j is odd) and guṇaśeṣa (when j is even) in the Section 3.4. Appendix B Epicycle and eccentric models for manda and śīghra corrections Chapter 7 of the text describes the procedures for finding the true geocen- tric longitudes of the planets beginning with the mean longitudes.