What day of the week was the Battle of the Eclipse 585 BC?

This answer is rather longer than necessary, since it is an opportunity to educate about calendars and weekdays.

I believe that the seven day week originated in the Middle East. Thus if that battle was in the Middle East the people who fought in it might have know what day of the week it was.

And quite possibly the ancient accounts which mentioned the battle didn't date it any more closely than by the year and the weekday was unrecorded.

But it seems quite certain that with millions of people using a seven day week since long before that battle in 585 BC, it would be impossible for the rhythm of weekdays to be disrupted and for people to start using a different sequence of weekdays.

Modern historians tend to date events by the Julian calendar since AD 1 and the Gregorian calendar since AD 1582.

The Julian calendar was invented about 46 BC. The year 46 BC was lengthened to 445 days to realignthe calendar with the seasons, and 45 BC was the first year with the new calendar operating normally.

The only thing needed to keep the new calendar properly aligned was to add a leap day to February every fourth year. Unfortunately, there are different methods of counting, and the Roman priests interpreted "every fouth year" to mean what we would call "every third year". So they inserted a few extra leap years until the error was discovered in the reign of the first emperor, Augusutus. Augustus then discontinued or reduced adding leap days until the calendar was back in proper alignment.

So every complete date with year, month, and day number in the Julian calendar since about AD 8 is belieed to be correct, making it easy to calculate the day of the week it happened on.

Events between 45 BC and AD 8 which have a complete date, year, month, and day number, in ancient Roman sources are not so easy. Historians are uncetain which years were leap years and which were not in that period, thus making dates possibly off by a few days, which makes the weekday harder to calculate in that era. Of course it is much harder to calculate the weekday for Roman dates before the Julian calendar was adopted.

If someone uses a calendar to date an event before that calendar was used they are using a "proleptic" version of that calendar. Since AD and BC are mostly used with the Julian and Gregorian calendars, the date of 28 May, 585 BC should be in either the proleptic Julian calendar or the proleptic Gregorian calendar, whose dates in 585 BC would be several days different.

So you need to make certain whether Asimov gave the date as 28th May 585 BC in the proleptic Julian calendar or the proleptic Gregorian calendar and then finding the weekday should be as simple as using a weekday calculator for either the proleptic Julian calendar or the proleptic Gregorian calendar.

This calculator for Julian Day Number gives the Julian Day numbers as:

• May 28, 585 B.C. Julian, as 1,507,899

• Dec. 12, 2020 A.D. Gregorian, as 2,459,195

Subtracting these numbers gives an elapsed day count as 951,296 = 7 * 135,899 + 3.

Thus since Dec. 12, 2020 A.D. Gregorian, is a Saturday, the Thales eclipse occurred exactly 135,899 weeks, and 3 days, ago (from today as I write); meaning the Thales Eclipse occurred on a Wednesday.

Note that the fractional part of the Julian Day Numbering, representing time-of-day UTC, cancels out when all one is interested in is a day-number-difference at a single point on the Earth's surface. They only matter when one needs to compare different location on the Earth's surfaces, as astronomers are want to do when calculating eclipses and the like.

The Julian Day Numbering system is precisely the one used by astronomers, particularly to calculate the may 28, 585 B.C. Julian, date for the eclipse, and thus is a valid means to count back to determine Day of the Week. The extent to which Day of the Week is meaningful, at that time and in that culture, is for the reader to determine.

According to the system of numbering days called Julian day numbers, used by astronomers and calendricists (those who study calendars, unfortunately not for a living), the temporal sequence of days is mapped onto the sequence of integers, -2, -1, 0, 1, 2, 3, etc. This makes it easy to determine the number of days between two dates (just subtract one Julian day number from the other).

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Following Herschel's lead astronomers adopted this system and took noon GMT -4712-01-01 JC (January 1st, 4713 B.C.) as their zero point. (Note that 4713 B.C. is the year -4712 according to the astronomical year numbering.) For astronomers a "day" begins at noon (GMT) and runs until the next noon (so that the nighttime falls conveniently within one "day", unless they are making their observations in a place such as Australia). Thus they defined the Julian day number of a day as the number of days elapsed since January 1st, 4713 B.C. in the proleptic Julian Calendar.