Mercury's precession due to other planets

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Hi.

I'm interested in how physicists in the mid-19th century such as Urbain Le Verrier were able to compute the gravitational influences of the other planets on the apsidal precession of Mercury's orbit, finding that they can only explain about 531" (per century!), not the observed 574". This is only a relative error of about 7 % between theory and observations, but many astronomers were confident enough in the calculations to search for a hitherto undiscovered planet. Einstein later showed that general relativity can exactly explain the missing 43".

What kinds of methods did they use to make such incredibly accurate computations more than a century before the invention of electronic computers? Is there a book or other publication (preferably modern, not the original papers) that goes into the details of this?
 

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Bandersnatch
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