Computing the number of perfumes that constitute the gandhārṇava and kacchapuṭa of Varāhamihira

Abstract

The sixth-century Indian scholar Varāhamihira elaborated on the processes for the preparation of perfumes in the ‘gandhayukti’ chapter of the Bṛhatsaṁhitā. He adapted the combinatorial analysis and magic square structure for arriving at the possible number of perfumes from the set of specified substances. This attempt is an excellent example of the multidisciplinary thought process of the Indian scholars in general and Varāhamihira in particular. This paper focuses on the verses under the captions ‘gandhārṇava’ and ‘kacchapuṭa’. In the first case, four substances are selected from sixteen, and the chosen four are permuted with the pre-defined proportions. In the second case, selected substances are placed in a 4 × 4 magic square. The claims by Varāhamihira and the same amended by the commentator Bhaṭṭotpala for the number of perfumes are verified here. The constraints imposed by Varāhamihira in the first case on the proportions of the two substances are considered. The correct number for the possible perfumes is determined mathematically in the first case and by the specially designed computer program in the second case. Valid lists of perfumes with ingredients obtained as the output are attached in the form of web links in the first case, and the same is placed in the Appendix for the second case.

Appendix

List of the 172 perfumes, where proportions of four ingredients add to 18

No.	Substances				No.	Substances				No.	Substances				No.	Substances
1	S1	S2	S3	S4	46	S5	S9	S15	S3	91	S2	S9	S13	S15	136	S4	S8	S14	S16
2	S5	S6	S7	S8	47	S9	S13	S3	S7	92	S2	S3	S8	S11	137	S4	S7	S13	S16
3	S9	S10	S11	S12	48	S13	S1	S7	S11	93	S2	S9	S11	S15	138	S4	S7	S9	S15
4	S13	S14	S15	S16	49	S2	S6	S12	S16	94	S2	S4	S8	S9	139	S4	S5	S15	S16
5	S1	S5	S9	S13	50	S6	S10	S16	S4	95	S2	S4	S8	S15	140	S5	S7	S9	S11
6	S2	S6	S10	S14	51	S10	S14	S4	S8	96	S2	S6	S8	S9	141	S5	S8	S12	S15
7	S3	S7	S11	S15	52	S14	S2	S8	S12	97	S2	S6	S8	S15	142	S5	S8	S9	S14
8	S4	S8	S12	S16	53	S1	S3	S13	S15	98	S2	S5	S9	S12	143	S5	S6	S9	S16
9	S1	S6	S11	S16	54	S1	S6	S7	S12	99	S2	S5	S14	S15	144	S5	S6	S10	S15
10	S4	S7	S10	S13	55	S1	S3	S6	S8	100	S2	S7	S11	S12	145	S5	S7	S11	S15
11	S2	S5	S12	S15	56	S1	S8	S12	S13	101	S2	S5	S6	S7	146	S5	S7	S9	S13
12	S3	S8	S9	S14	57	S1	S6	S10	S13	102	S2	S3	S12	S16	147	S5	S10	S12	S14
13	S1	S8	S11	S14	58	S1	S8	S13	S14	103	S2	S4	S4	S7	148	S5	S12	S14	S16
14	S4	S5	S10	S15	59	S1	S2	S12	S13	104	S2	S4	S10	S14	149	S6	S8	S10	S12
15	S2	S7	S12	S13	60	S1	S8	S11	S12	105	S2	S6	S10	S12	150	S6	S7	S9	S15
16	S3	S6	S9	S16	61	S1	S2	S11	S14	106	S2	S6	S14	S16	151	S6	S7	S9	S13
17	S1	S2	S5	S6	62	S1	S9	S12	S14	107	S2	S11	S13	S16	152	S6	S8	S12	S16
18	S2	S3	S6	S7	63	S1	S12	S14	S15	108	S2	S10	S11	S13	153	S6	S8	S10	S14
19	S3	S4	S7	S8	109	S2	S3	S12	S15	64	S1	S6	S9	S15	154	S6	S7	S11	S16
20	S5	S6	S9	S10	65	S1	S4	S10	S13	110	S2	S3	S9	S14	155	S7	S9	S12	S14
21	S6	S7	S10	S11	66	S1	S3	S11	S15	111	S3	S8	S12	S15	156	S7	S8	S12	S13
22	S7	S8	S11	S12	67	S1	S5	S6	S8	112	S3	S6	S10	S15	157	S7	S12	S14	S15
23	S9	S10	S13	S14	113	S3	S5	S7	S14	68	S1	S2	S3	S6	158	S7	S8	S11	S14
24	S10	S11	S14	S15	69	S1	S3	S4	S8	114	S3	S5	S13	S16	159	S8	S10	S11	S13
25	S11	S12	S15	S16	70	S1	S4	S11	S16	115	S3	S5	S10	S13	160	S8	S9	S13	S15
26	S1	S3	S9	S11	71	S1	S5	S13	S15	116	S3	S5	S10	S11	161	S8	S11	S13	S16
27	S2	S4	S10	S12	72	S1	S5	S9	S11	117	S3	S8	S9	S12	162	S8	S9	S11	S15
28	S5	S7	S13	S15	73	S1	S2	S4	S5	118	S3	S8	S14	S15	163	S9	S11	S14	S16
29	S6	S8	S14	S16	74	S1	S3	S9	S13	119	S3	S4	S9	S16	164	S9	S13	S14	S16
30	S1	S4	S13	S16	75	S1	S6	S10	S11	120	S3	S4	S10	S15	165	S9	S10	S11	S14
31	S1	S2	S13	S14	76	S1	S6	S13	S16	121	S3	S6	S9	S10	166	S9	S11	S12	S16
32	S2	S3	S14	S15	77	S1	S4	S7	S12	122	S3	S6	S7	S8	167	S9	S10	S12	S13
33	S3	S4	S15	S16	78	S1	S4	S7	S14	123	S3	S7	S9	S11	168	S10	S12	S13	S15
34	S1	S4	S5	S8	79	S1	S6	S7	S14	124	S3	S7	S13	S15	169	S10	S13	S14	S15
35	S5	S8	S9	S12	80	S1	S3	S5	S12	125	S3	S5	S7	S12	170	S10	S11	S12	S15
36	S9	S12	S13	S16	81	S1	S3	S5	S14	126	S3	S5	S11	S16	171	S11	S14	S15	S16
37	S1	S2	S11	S12	82	S1	S4	S9	S15	127	S3	S12	S14	S16	172	S12	S13	S15	S16
38	S2	S3	S12	S9	83	S2	S4	S14	S16	128	S3	S10	S12	S14
39	S3	S4	S9	S10	84	S2	S4	S5	S7	129	S3	S6	S15	S16
40	S4	S1	S10	S11	85	S2	S5	S8	S11	130	S4	S7	S11	S16
41	S5	S6	S15	S16	86	S2	S7	S11	S14	131	S4	S5	S9	S16
42	S6	S7	S16	S13	87	S2	S5	S9	S14	132	S4	S5	S7	S8
43	S7	S8	S13	S14	88	S2	S7	S13	S14	133	S4	S5	S9	S10
44	S8	S5	S14	S15	89	S2	S3	S8	S13	134	S4	S8	S10	S12
45	S1	S5	S11	S15	90	S2	S5	S8	S13	135	S4	S7	S10	S11