# How to interpret this old degree notation?

I'm trying to follow the calculations in this paper on page 5 (see image below). The paper is describing traditional methods of calculation for a luni-solar calendar.

The math is not hard, but I can't make sense of the notation.

I figured that the ... = 150 : 1780 (as in step 3) means that 150 is the integer result, and 1780 is the remainder.

But what is ... = 2; 19:28 (step 8)? Degree; minute:second? Okay, but then I'm lost at step 10 with 134 * sin(0:32) = 0:01. I haven't managed to get that result.

• I think you're right about step 8 and degree: minute; second. Understanding step 10 may be dependent on knowing where that '134' comes from. It's likely some kind of conversion factor that gives the answer in seconds. At least, when you take 134*sin(0:32), you get 1.2 which rounds to 1, which is the number of seconds in the answer. – Nathan Stock Sep 8 '15 at 22:33
• Is github.com/profound-labs/calculating-the-uposatha-moondays/blob/… yours? If not, it might be helpful. BTW, for anyone who sees the equal signs as minuses, look at the full size version for more detail: i.stack.imgur.com/bMGRW.png – user21 Sep 8 '15 at 22:45
• @barrycarter yes, that's the document I'm working on – Gambhiro Sep 9 '15 at 6:42

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

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

and step 10 is

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

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 ($3'$).

• This was illuminating! Not only in solving the notation puzzle, but also in connecting the (x; y: z) to the zodiac partitions, which are used for what month of the year the calendar is supposed to be at. Also see The Lunar and Solar Zodiac PDF if you are interested. – Gambhiro Sep 9 '15 at 9:15
• There is a follow-up to the puzzle at astronomy.stackexchange.com/q/12052/9153 – Gambhiro Oct 8 '15 at 14:02

This doesn't really answer your question, but I read https://github.com/profound-labs/calculating-the-uposatha-moondays/blob/master/references/Rules%20For%20Interpolation%20in%20the%20Thai%20Calendar.pdf and noted that footnote 13 (which is referenced by equation 8, although it sort of looks like 3^13) reads:

The routine subtraction of 3 arcmins is a causing Ashadha to be month 10; but by this Wat geographical longitude correction for the sun, as is the Tapotharam reckoning Vaisakha is month 7 (Keng subtraction of 40 arcmins for the moon (sec. C13). Tung style), causing Ashadha to be month 9.

which suggests the calculation is being done in arcminutes, not degrees.

Unfortunately, this doesn't help, since the result is 76.5283 arcminutes, which is nowhere near 2 degrees 19 minutes 28 seconds (which is 139.467 arcminutes).

134*sin(0:32) may refer to 32 seconds of arc, and the result is 0.02 which is 1 minute or arc (0:01). Terrible use of notation here if that actually is the case.