Solar Theory

In his "Laws of the Snctification of the Moon", Chapter 12, Law 1, Rambam states:

The mean rate of movement of the sun in one day (that is, twenty-four hours) is fifty-nine minutes and eight seconds, in symbols - 59′8″. It follows that its movement in ten days is nine degrees fifty-one minutes and twenty-three seconds, in symbols - 9°51′23″. It also follows that its movement in a hundred days is ninety-eight degrees thirty-three minutes and fifty-three seconds, in symbols - 98°33′53″. It also follows that remainder of its movement in a thousand days, after you subtract all [multiples of] 360° (as was explained) is two hundred sixty-five degrees thirty-eight minutes and fifty seconds, in symbols - 265°38′50″. It also follows that the remainder of its movement in ten thousand days is one hundred thirty-six degrees twenty-eight minutes and twenty seconds, in symbols - 136°28′20″.

And in this way you can multiply and calculate its movement for any number [of days] that you want. Similarly, if you want to make known to you values of its movement for two days, three, four and so on to ten - do it. Similarly, if you want for to have known and ready values of its movement for twenty days, thirty, forty and so on to a hundred - do it. This is clear and known once you know its movement in 1 day.

And you should have ready and known to you mean movement of the Sun for twenty-nine days and for three hundred and fifty-four (which is the number of days in the lunar year when its months are "regular" (TODO link), and it is called "regular year"). The reason is: if you have those movement values ready, these calculations of the visibility of the moon will be easy, because there are twenty-nine complete days from the night of observation to the night of observation of the following month, and so it is every month: no less than twenty-nine days and no more. Since our sole desire in all those calculations is exclusively to determine visibility [of the moon]. And between the night of sighting of this month and night of sighting of the same month next year there is either a regular year or a year and 1 day; and the same every year. Mean movement of the Sun in twenty-nine days is twenty-eight degrees thirty-five minutes and one second, in symbols - 28°35′1″. Its movement in a regular year is three hundred forty-eight degrees, fifty-five minutes and fifteen seconds, in symbols - 348°55′15″.

In table format:

days printed
1 0°59′8″
10 9°51′23″
100 98°33′53″
1000 265°38′50″
10000 136°28′20″
29 28°35′1″
354 348°55′15″

Now let's see if the printed values can be recovered by multiplying the value for one day by the number of days (and discarding full circles as appropriate):

days printed reconstructed difference
1 0°59′8″    
10 9°51′23″ 9°51′20″ 0°0′3″
29 28°35′1″ 28°34′52″ 0°0′9″
100 98°33′53″ 98°33′20″ 0°0′33″
354 348°55′15″ 348°53′12″ 0°2′3″
1000 265°38′50″ 265°33′20″ 0°5′30″
10000 136°28′20″ 135°33′20″ 0°55′
days printed reconstructed difference
10 9°51′23″    
100 98°33′53″ 98°33′50″ 0°0′3″
1000 265°38′50″ 265°38′20″ 0°0′30″
10000 136°28′20″ 136°23′20″ 0°5′
days printed reconstructed difference
100 98°33′53″    
1000 265°38′50″ 265°38′50″
10000 136°28′20″ 136°28′20″
days printed reconstructed difference
1000 265°38′50″    
10000 136°28′20″ 136°28′20″