1. [The following method should be used] if you wish to know the true position¹As stated in Chapter 11, the true position of the sun refers to the position at which it is seen in the heavenly sphere. The difference between the sun's true position and its mean position stems from the fact that the Earth is not located at the exact center of the sun's orbit. 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.²The method through which both these figures can be calculated is stated in the previous chapter. 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.³I.e., the path the sun has traveled in its orbit from the apogee until it reached its present position.

2. [The next step is] to calculate the angular distance of the course of the sun.⁴The course of the sun is an arc extending from the mean position of the sun to its apogee. The angular distance of the course is derived by drawing straight lines from the mean position and the apogee to the center of the sun's orbit.

If the apogee of the sun were located at O°, the angular distance of the course and the sun's mean position would be the same. Since, however, the apogee also moves within the heavenly sphere, there is a variance between these two figures. If the angular distance of the course is less than 180 degrees, one should subtract⁵Before the sun reaches the perigee, the point in its orbit that is closest to the Earth, its true position will always be less than its mean position. Hence, the angle referred to as the angle [determined by the] course must be subtracted from its mean position to arrive at the true position. the angle [determined by the] course⁶The angle [determined by the] course refers to the extent of the deviation between the position of the sun that can be observed in the sky and its mean position. The manner of determining this figure is described in Halachah 4.

To express these concepts in geometric terms: The sun's true position represents the angle at which it can be found in the sphere of the heavens of which the Earth is the center (c). When the angle that is called the angle [determined by the] course (a) is added to this angle, the sum is equivalent to the angle of the course of the sun (b).

How is this figure derived? Refer to the accompanying diagram: The sum of the angles of the triangle a, c, and d equals 180°, and the angles b and d equal 180°. Hence, b equals c a.

Thus, the true position of the sun is equivalent to its mean position minus the figure referred to as the angle [determined by the] course. Thus, when the course is less than 180 degrees, the sun's true position is always a small amount less than its mean position. from the sun's mean position. If the angular distance of the course is more than 180 degrees, one should add⁷When the sun passes the perigee, its true position will always be greater than its mean position. Hence, the angle referred to as the angle [determined by the] course must be added to its mean position to arrive at the true position.

Why is this so? Refer to the following diagram: d refers to the true position of the sun, b to its mean position and a to the angle referred to as the angle [determined by the] course. E to its true position minus 180° and f refers to its mean position minus 180°. a + f + c equals 180. E + c equals 180. Thus, a + f equals e. Hence, the mean position plus the angle [determined by the] course will be equal to the true position. 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.

3. 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.⁸I.e., when the sun is at the apogee or perigee, there will be one straight line between the Earth (the center of the heavenly sphere), the center of the sun's orbit, and the actual position of the sun.

4. 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.⁹The largest angle determined by the course is when the angular distance of the course itself is 96 degrees - i.e., shortly after the mean position of the sun passes directly above the center of its orbit. After this point is reached, the angle begins to decrease.

Significantly, the rate of the decrease does not correspond exactly to the rate of increase as the angles approach 96 degrees. The reason for this difference is that, as stated above, the position of the Earth is not at the center of the sun's orbit. Hence, at 100 and 80 degrees, although the mean position of the sun has moved an equal distance from the center of its orbit, it has moved different distances from the Earth.

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.

5. [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.¹⁰Thus, our computations will be based on the negative of the angle measured previously. To put the Rambam's statements in layman's terms: The angle formed will be the same regardless of whether the mean position of the sun is measured in an increase from O° or a decrease from 360°.

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.

6. 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.

7. [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.¹¹Although the correspondence between the angular length of the course and the angle [determined by the] course is not uniform over a large span, within a span of ten degrees the difference between the actual figure and the approximation arrived at by the Rambam is not of consequence.

8. 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.

9. [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,¹²Chapter 12, Halachah 2. 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.

10. 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 approximately¹³The commentaries have questioned the Rambam's use of the word "approximately." Our bracketed additions are made in that light. thirty [or more], they should be considered a minute, and added to the sum of the minutes.

11. 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,¹⁴The calculation of the equinoxes and solstices is significant with regard to the determination of the calendar, as explained in Chapters 9 and 10. Since the vernal (spring) equinox takes place when, according to the sun's true motion, it enters the constellation of Aries, the date on which that takes place can be calculated for any particular year. Similarly, the summer solstice takes place when, according to the sun's true motion, it enters the constellation of Cancer, and that date can be calculated. Similar concepts apply regarding the autumnal equinox and the winter solstice. 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.