Tuesday, 29 October 2013

How to interpret this old degree notation?

The notation $(x;;y:z)$ seems to be $(30times 60)x+60y+z$ in minutes of arc. The calculation seems to take just the integer part rather than rounding. Thus, for example, step 8 is



$$begin{eqnarray}((64552 / 292207) times 360) - 3 &=& 79^circ lfloor 31.7rfloor' - 3' \ &=& (2times 30 + 19)^circ 28' \ &=& (2;;19:28)text{,}end{eqnarray}
$$



and step 10 is



$$134,sin(0:32) = lfloor 1.247rfloor = 1 = (0:01)text{.}$$



Some passages in the paper that point towards this interpretation:




(2) How does one interpret what the numerical values of the rӕk are taken to represent? It is universally agreed that the counting of the rasi (ราศี, signs of the zodiac) begins with Aries (Mesa) = 0; but the "r" at 0.



...



$^{12}$ Where desirable, values in arcmins are here converted to signs, degrees, and arcmins in order to make them compatible with following operations. Thus at stage C12, the value 258 arcmins becomes 0; 4, 18 to make it compatible with 2; 19, 28.




Since the zodiac partitions the celestial longitude into twelve $30^circ$ divisions, this makes sense, and seems to work out fairly straightforwardly though a bit cryptically, because the calculation mixes degrees and minutes: e.g., the minuend (first part) of Step 8 is in degrees, which is only implicitly converted to minutes of arc to be compatible with the subtrahend ($3'$).

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