How to identify the number of members of a smart list on a given date?

Sorry, but there's no way to do this if you're not

1. Completely aware of the Smart List filters on Sept. 9 (that's probably the easy part)
2. Extracting all the relevant fields and/or the Change Data Value activities using the API, so you have a snapshot of how your db looked as of that day

Sandford is right.  but, you might be able to accomplish something close.  It depends on the complexity of your smart list.

For example, smart list is 

state = colorado

you could try this.

(state = colorado

AND

not data value changed (state) on or after the 9th.)

OR 

data value changed (state) since the 9th and the previous value was colorado.

However, the gap here is that you will get people who were say texas on the 9th, and then colorado on the 10th, and then kansas on the 11th.

Again, this depends on the filters you are using.  It might not work out for you.

You could create a tweak of this smart list that solely identifies the people who have changed since the 9th and audit that list to weed out the ones who qualified but shouldn't have.  But that depends also on the size of your list to see if it is worth it.

If you see this need coming often, you might want to create a subscription to your smart list and email it to yourself so you can refer back to what it looked like at a certain moment of time.

Hope this helps!

Oof, yeah, like others have said, there's not a simple way to determine that retroactively. The only thing I can think of for recording the data moving forward is maintaining a spreadsheet with a log of the SmartList member values outside of Marketo. This could be something you just check daily and input, or maybe create an email subscription to send a daily report on the # of members on the list each day. And then maybe if you have a developer write a script and macro to easily extract the necessary tally from the daily reports into a single spreadsheet, you could automate it. But it's not pretty, either way.