Kiddush HaChodesh - Chapter Thirteen

[The following method should be used] if you wish to know the true position1 of the sun on any particular day you desire: First, it is necessary to calculate the mean position of the sun through the methods of calculation we have explained. Then calculate the position of the apogee of the sun.2 Afterwards, subtract the apogee of the sun from the mean position of the sun. The remainder is referred to as the course of the sun.3

[The next step is] to calculate the angular distance of the course of the sun.4 If the angular distance of the course is less than 180 degrees, one should subtract5 the angle [determined by the] course6 from the sun's mean position. If the angular distance of the course is more than 180 degrees, one should add7 the angle [determined by the] course to the sun's mean position. The figure remaining after making this addition or subtraction represents [the sun's] true position.

If the course [of the sun] is an even 180 degrees or an even 360 degrees, there will be no angle [determined by the course to add or to subtract]. Instead, the [sun's] mean position is its true position.8

What is the angle [determined by the] course? If the course is ten degrees, the [resulting] angle will be 20 minutes.

If the course is twenty degrees, the [resulting] angle will be 40 minutes.

If the course is thirty degrees, the [resulting] angle will be 58 minutes.

If the course is forty degrees, the [resulting] angle will be 1 degree and 15 minutes.

If the course is fifty degrees, the [resulting] angle will be 1 degree and 29 minutes.

If the course is sixty degrees, the [resulting] angle will be 1 degree and 41 minutes.

If the course is seventy degrees, the [resulting] angle will be 1 degree and 51 minutes.

If the course is eighty degrees, the [resulting] angle will be 1 degree and 57 minutes.

If the course is ninety degrees, the [resulting] angle will be 1 degree and 59 minutes.

If the course is one hundred degrees, the [resulting] angle will be 1 degree and 58 minutes.9

If the course is one hundred ten degrees, the [resulting] angle will be 1 degree and 53 minutes.

If the course is one hundred twenty degrees, the [resulting] angle will be 1 degree and 45 minutes.

If the course is one hundred thirty degrees, the [resulting] angle will be 1 degree and 33 minutes.

If the course is one hundred forty degrees, the [resulting] angle will be 1 degree and 19 minutes.

If the course is one hundred fifty degrees, the [resulting] angle will be 1 degree and 1 minute.

If the course is one hundred sixty degrees, the [resulting] angle will be 42 minutes.

If the course is one hundred seventy degrees, the [resulting] angle will be 21 minutes.

If the course is an even one hundred eighty degrees, it has no measure. Instead, its mean position is its true position, as we explained.

[The following procedure should be used] if the course [of the sun] is greater than one hundred eighty degrees: One should subtract the course from three hundred sixty degrees and [calculate the resulting] angle accordingly.10

What is implied? If the course is 200 degrees, that figure should be subtracted from 360 degrees, leaving a remainder of 160 degrees. Since you already know that the [resulting] angle of a course of 160 degrees is 42 minutes, that same figure will be the [resulting] angle of a course of 200 degrees.

Similarly, if the course was three hundred degrees, one should subtract that figure from three hundred sixty, leaving a remainder of sixty. Since you already know that the [resulting] angle of a course of 60 degrees is 1 degree and 41 minutes, that same figure will be the [resulting] angle of a course of 300 degrees. Similar procedures should be followed in calculating other figures.

[How is the angle determined by the course calculated] when the course is [an intermediate figure - e.g.,] 65 degrees? You already know that the [resulting] angle of 60 degrees is 1 degree and 41 minutes. And you know that the [resulting] angle of 70 degrees is 1 degree and 51 minutes. Thus, there are ten minutes between these [two] measures. Thus, [an increase of] a degree [of the course] will bring an increase of a minute [in the resulting angle]. Thus, the [resulting] angle of a course of 65 degrees will be 1 degree and 46 minutes.11

Similarly, if the course was 67 degrees, the [resulting] angle would be 1 degree and 48 minutes. A similar procedure should be followed regarding any course that has both units and tens, both for calculations regarding the sun and for calculations regarding the moon.

[To apply these principles]: Should we desire to know the true position of the sun at the beginning of Friday night, the fourteenth of Tammuz for this present year: First, we should calculate the mean position of the sun for this time, which is, as explained,12 105° 37' 25". We should then calculate the apogee of the sun at this time, which is 86° 45' 23". When the apogee is subtracted from the mean position, the remainder, the course [of the sun], will be 18 degrees, 52 minutes and 2 seconds, in symbols 18° 52' 2 ".

With regard to the course [of the sun], the minutes are of no consequence. If they are less than thirty, they should be disregarded entirely. If they are more than thirty, they should be considered an additional degree and added to the sum of the degrees. Accordingly, it should be considered as if there are 19 degrees in this course. The [resulting] angle of such a course can be calculated to be 38 minutes in the manner that we explained.

Since the course is less than 180 degrees, the [resulting] angle [of the course], 38 minutes, should be subtracted from the mean position of the sun, leaving a remainder of 104 degrees, 59 minutes and 25 seconds, in figures 104° 59' 25". Thus, the true position of the sun at the beginning of this night will be fifteen degrees less 35 seconds in the constellation of Cancer.

One need not pay attention to the seconds at all, neither with regard to the position of the sun, nor with regard to the position of the moon, nor in any other calculations regarding the sighting [of the moon]. Instead, if the number of seconds is approximately13 thirty [or more], they should be considered a minute, and added to the sum of the minutes.

Since you are able to calculate the location of the sun on any desired date, you will be able to calculate the true date of the equinox or solstice for any equinox or solstice you desire,14 whether for the equinoxes or solstices that will take place in the future, after the date we established as a starting point, or for the equinoxes or solstices that have taken place in previous years.