Halachah 1 – Finding the Sun’s Solar Anomaly
To find the sun’s true position, one must first calculate its mean position and the position of the sun’s apogee, as explained in the previous chapter. Then one subtracts the position of the sun’s apogee from the sun’s mean position. The remainder is called the maslul ha-shemesh, the sun’s anomaly or solar path, and it is the basis for correcting the sun’s mean position into its true position.
Halachah 2 – Correcting the Sun’s Mean Position
After finding the sun’s anomaly, one checks whether it is less than or greater than 180 degrees. If the anomaly is less than 180 degrees, one subtracts the correction amount from the sun’s mean position. If it is greater than 180 degrees, one adds the correction amount to the mean position. The resulting position is the sun’s true position.
Halachah 3 – When No Correction Is Needed
If the sun’s anomaly is exactly 180 degrees or exactly 360 degrees, there is no correction at all. In those cases, the sun’s mean position and true position are the same.
Halachah 4 – The Correction Table for the Sun’s Anomaly
The Rambam gives a table of correction values for the sun’s anomaly. For example, an anomaly of 10 degrees has a correction of 20 minutes, an anomaly of 40 degrees has a correction of 1 degree and 15 minutes, and an anomaly of 90 degrees has a correction of 1 degree and 59 minutes. These values are used to adjust the sun’s mean position and find its true position.
Halachah 5 – Finding the Correction for an Anomaly Greater Than 180 Degrees
If the sun’s anomaly is greater than 180 degrees, one subtracts it from 360 degrees and uses the correction value of the remainder. For example, if the anomaly is 200 degrees, one subtracts 200 from 360, leaving 160 degrees. Since the correction for 160 degrees is 42 minutes, the correction for 200 degrees is also 42 minutes.
Halachah 6 – Another Example of a Large Anomaly
If the anomaly is 300 degrees, one subtracts it from 360 degrees, leaving 60 degrees. Since the correction for 60 degrees is 1 degree and 41 minutes, that is also the correction for an anomaly of 300 degrees. This method applies to any anomaly greater than 180 degrees.
Halachah 7 – Estimating Corrections Between Listed Values
When the anomaly falls between two values in the table, one estimates the correction proportionally. For example, if the anomaly is 65 degrees, it lies halfway between 60 and 70 degrees. The correction for 60 degrees is 1 degree and 41 minutes, and the correction for 70 degrees is 1 degree and 51 minutes. Therefore, the correction for 65 degrees is estimated as 1 degree and 46 minutes.
Halachah 8 – Using the Same Method for Other Calculations
The same method of estimating values between table entries applies not only to the sun, but also to the moon’s calculations. Whenever a value falls between two listed numbers, the correction is estimated according to its place between them.
Halachah 9 – Example of Calculating the Sun’s True Position
The Rambam gives an example for the beginning of Shabbat night, the fourteenth of Tammuz in the reference year. First, the sun’s mean position is calculated as 105 degrees, 37 minutes, and 25 seconds. Then the sun’s apogee is calculated as 26 degrees, 45 minutes, and 23 seconds. Subtracting the apogee from the mean position gives an anomaly of 18 degrees, 52 minutes, and 2 seconds.
Halachah 10 – Applying the Correction in the Example
Since the anomaly in the example is less than 180 degrees, the correction must be subtracted from the sun’s mean position. The anomaly is rounded to 19 degrees, giving a correction of 38 minutes. Subtracting this from the mean position produces the sun’s true position: 104 degrees, 59 minutes, and 25 seconds, which places the sun in the sign of Cancer, about 15 degrees into that sign. The Rambam adds that in these visibility calculations, seconds are generally not important unless they are close enough to be rounded into a minute.
Halachah 11 – Using the Sun’s True Position to Find the True Tekufah
Once one knows how to calculate the sun’s true position for any date, one can also determine the true time of any tekufah, whether in future years or past years. This is because the tekufah depends on the sun’s actual position, not merely its average calculated position.
Chapter Theme
Chapter 13 teaches how to move from the sun’s mean position to its true position. The Rambam explains how to calculate the solar anomaly, use correction values, and apply those corrections to determine the sun’s actual place in the zodiac, which is necessary for the later calculations of moon visibility.