In my first post on History of Numbers, I had a brief overview of how mathematics evolved in India, and the factors leading to it, starting with the Indus Valley Civilization. In this post, I shall look at Satapatha Brahmana, and it’s contribution to mathematics.
Satapatha Brahmana, is considered one of the earliest texts on geometry and mathematics, along with the Sulabha Sutras. It is believed to have been composed around 1000-800 B.C, and as per Seidenberg, the concept that equivalent sacrificial altars should have equal areas, led to some of the basic geometrical concepts. Again like most Indian mathematical texts, the Satapatha Brahmana was not considered for serious study, until Seidenberg did a serious study on it’s contents.
One of the main factors that led to the development of geometrical theories was the Vedic sacrificial rites themselves. Most of these rites took place for one full year, and marked the passage of time. The Satapatha Brahmana majorly deals with the construction of the sacrificial altars for the agnicayana rite, where agni is the year. This Agnicayana rite goes on for 12 days in a large trapezoidal area called the Mahavedi and there is a smaller rectangular area to the west of it called the Pragvamsa. This Mahavedi measures 30 prakarma on the West, 24 prakarma on the East and 36 prakarma long. If we take the sum of these 3 prakarma it comes to 90, which is approximately,a quarter of the year. The brick altar in Mahavedi was in the form of a falcon about to take wing, while in the Pragvamsa, there are fire altars in 3 specified positions-Garhapatya( representing the Earth), Ahavaniya(the Sky) and 8 Dhisnya hearths representing Space. So in a way the Agnicayana alters represented the entire Universe.
Bricks that were to be used in construction of the altar, were classified into two kinds-lokamprna( ordinary) and yajusmati(special). Each brick had it’s own shape and measurements as needed. As per the Satapatha Brahmana, the number of yajusmati bricks needed was 395, which roughly corresponded to 360 days of the year, and an additional 35 days. If we go layer wise layout of yajusmati bricks, it was 98 in first, 41 in second,71 in 3rd, 47 in 4th and around 138 in 5th. Now if we add up the layers it comes as follows
- Sum of bricks in layers 4 and 5 is around 186 that refers to the number of tithis in a half year.
- Sum of bricks in layers 3 and 4 is 118, which is like 1/3 number of days in a lunar year.
- Again 71 in 3rd layer is around 1/5th number of days in a lunar year.
So as we can see, the number of bricks per each layer, had a connection with the year and days.
There are 10,8000 lokamprna bricks , which roughly corresponds to the number of muhurats in a year, assuming that there are like 30 muhurats per day.Most of these lokamprna bricks are used in the Ahavaniya altar while 21 go to the Garhapatya, and around 78 to the 8 Dhisnya hearths.These 5 altars are again surrounded by 360 enclosing stones called parisrita, 21 around the garhapatya, 78 around dhisnya and the rest around the Ahavaniya. The Dhisnya hearths are in one layer in a size of 18 angulas, either in a square or circular form.
The Satapatha Brahmana also defined the measures, which are the same used in the Sualaba Sutras too.
1 pada=10-12 angulas( Here, pada being foot, and angulas being the inches).
The Garhapatya, Ahavinya need to have an area of one square purusa. And here purusa is approximately 4 sq m. If we define by those standards
1 purusa=120 angulas.
The mahadevi altars, were generally made in 5 layers of bricks, and each layer in the falcon altar had around 200 bricks, making it totally around 1000 bricks in 5 layers. Now if we consider the basic falcon shaped altar it’s area was around 7.5 square purusha, roughly around 30 sq m. The body was around 4 sq. purushas, which is around 16 sq m, the wings and the tail were around 1 sq purusha that was around 4 sqm each. The construction of these altars as also the larger altars, required the use of several geometric concepts, including the theory of the diagonal.
Equivalence through Numbers
Most of the early Vedic texts had the concept of equivalence through numbers which is basically, representing a particular phenomena through a number, and seek connection. For eg the circadian biological cycle and the earth’s rotation have the same number, and there is a linkage somewhere to be sought out. The Aitreya Aranyaka for eg, draws parallels between human body and the earth’s rotation cycle, the body has 720 bones, joints and marrows on both sides, which is also equivalent to number of days and nights in a year. So the self is pretty much equivalent to a year. The Satapatha Brahmana also states that if we take the factors of 720(1, 2,3,4,5,6,8,9,10,12, 15,16,18, 20,24) there are exactly 15 which again is parallel to 15 days waxing and waning of the moon in a month.
So in many ways we find that the Satapatha Brahmana, has laid the foundation for some of the basic geometric and number concepts too.
Astronomy of Satapatha Brahmana by Subash Kak- http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b5c_15.pdf