I thought I heard the First say something about starting the calculations (followed by comments from a number of others, culminating in Eccleston's "and for my next trick"). As indicated earlier in the episode, if the First starts the calculations, that then gives centuries until the Thirteenth has the completed set (which he can then broadcast to the others).
That's exactly correct. (Also, note that the whole sonic-screwdriver thing served as exposition/introdution to that concept earlier, so that the audience would get it quickly when applied big-scale to the Doctors).