<p>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</p>

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

<p>and step 10 is</p>

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

<p>Some passages in the paper that point towards this interpretation:</p>

<blockquote>
  <p>(2) How does one interpret what the numerical values of the <em>rӕk</em> are taken to represent? It is universally agreed that the counting of the <em>rasi</em> (ราศี, signs of the zodiac) begins with Aries (Mesa) = 0; but the "r" at 0. </p>
  
  <p>...</p>
  
  <p>$^{12}$ 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.</p>
</blockquote>

<p>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 <em>implicitly</em> converted to minutes of arc to be compatible with the subtrahend ($3'$).</p>

CC BY-SA 3.0
2015-09-09T09:15:10.633Z
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](https://github.com/profound-labs/calculating-the-uposatha-moondays/blob/master/references/The%20Lunar%20and%20Solar%20Zodiac%20-%20Hasapannyo%20Bhikkhu.pdf) PDF if you are interested.
Gambhiro
CC BY-SA 3.0
2015-10-08T14:02:46.690Z
There is a follow-up to the puzzle at http://astronomy.stackexchange.com/q/12052/9153
Gambhiro
CC BY-SA 3.0
2015-09-09T01:13:30.747Z
2015-09-09T02:29:08.933Z
2015-09-09T02:29:08.933Z
Stan Liou
<p>I'm trying to follow the calculations in <a href="http://www.siamese-heritage.org/jsspdf/1991/JSS_088_0r_Eade_RulesForInterpolationInThaiCalendar.pdf" rel="nofollow noreferrer">this paper</a> on page 5 (see image below). The paper is describing traditional methods of calculation for a luni-solar calendar.</p>

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

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

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

<p><a href="https://i.sstatic.net/bMGRW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bMGRW.png" alt="calculations"></a></p>

CC BY-SA 3.0
2015-09-08T22:33:37.763Z
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
CC BY-SA 3.0
2015-09-09T06:42:47.397Z
@barrycarter yes, that's the document I'm working on
Gambhiro
<notation>
How to interpret this old degree notation?
Gambhiro
# How to interpret this old degree notation?

(Asked by: username_0 on 2015-09-08)

I'm trying to follow the calculations in [this paper](http://www.siamese-heritage.org/jsspdf/1991/JSS_088_0r_Eade_RulesForInterpolationInThaiCalendar.pdf) 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.

[![calculations](https://i.sstatic.net/bMGRW.png)](https://i.sstatic.net/bMGRW.png)

**Tags:** notation

> **Comments:**
> **username_1**: 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.  
> **username_0**: @barrycarter yes, that's the document I'm working on  

---

## 1 answers

### Answer by username_2 on 2015-09-09. Score: 3 | Accepted answer

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 ($3'$).

> **Comments:**
> **username_0**: 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](https://github.com/profound-labs/calculating-the-uposatha-moondays/blob/master/references/The%20Lunar%20and%20Solar%20Zodiac%20-%20Hasapannyo%20Bhikkhu.pdf) PDF if you are interested.  
> **username_0**: There is a follow-up to the puzzle at http://astronomy.stackexchange.com/q/12052/9153  

---

