Are there ever any simultaneous transits of both Mercury and Venus as seen from the Earth? Transits of Mercury happen fairly frequently due to its short period, but transits of Venus are less frequent.  I've looked over the data available to me and found that there have been transits of both planets in the same calendar-year in 1631 (Nov &amp; Dec) and in 1769 (Jun &amp; Nov).  

What would really be interesting is if the two planets transited the Sun at the same time! With the rule being, that the second transit begins before the first transit completes.  

Can we figure out if they have ever so transited, or if they ever will so transit?
EDIT: As it turns out, I'm not the first or even the second person to run calculations like this:


https://www.fourmilab.ch/documents/canon_transits/
http://www.solexorb.it/SolexOld/Simtrans.pdf


Meeus' work (second link) mentions the 13425 CE event in "Table 1. Simultaneous and near-simultaneous transits of Mercury and Venus, years 1 to 300,000"

Within the limits of DE431 (7 May 13201 BCE to 7 May 17091 CE), there is no 
time at which both Mercury and Venus transit the Sun. 

The closest we get to this: 


On 16 Sep 13425 CE at 11:57pm UTC, Venus starts transiting the Sun. This 
transit ends the next morning (17 Sep 13425 CE) at 7:30am. 
Less than 9 hours later, at 4:27pm, Mercury starts transiting the 
Sun. This transit ends at 10:26pm. 


The program I used to compute this:  

https://github.com/barrycarter/bcapps/blob/master/ASTRO/bc-solve-astro-13227.c 

The list of transits I computed while solving this: 

https://github.com/barrycarter/bcapps/blob/master/ASTRO/mercury-transits.txt.bz2 
https://github.com/barrycarter/bcapps/blob/master/ASTRO/venus-transits.txt.bz2 

Although I believe this answer is correct, Stellarium does not agree with 
me, and HORIZONS doesn't compute positions past 9999 CE, so don't put too 
much faith in this answer, since there's no good way to confirm it. I 
believe that I'm correct and Stellarium is wrong this far in the future, but 
it could be the other way around. 

Even if my calculations are correct, the uncertainty in calculating the 
relevant positions (Sun, Merucry, Venus, Earth) this far in the future is 
high. On their own transit pages, NASA only computes Venus transits from 
2000 BCE to 4000 CE, and Mercury transits from 1601 CE to 2300 CE, even 
though they could've made the same calculations I made from 13201 BCE to 
17091 CE:


http://eclipse.gsfc.nasa.gov/transit/catalog/MercuryCatalog.html
http://eclipse.gsfc.nasa.gov/transit/catalog/VenusCatalog.html 


This suggests NASA isn't confident enough of Mercury/Venus (and Earth/Sun) 
positions to predict that far in the past or future, so my results may be fairly inaccurate.