TITLE: How did Le Verrier calculate Neptune's position?
QUESTION [6 upvotes]: In the Wikipdia article on Neptune the discovery is described as a mathematical achievement:

Subsequent observations revealed substantial deviations from the
  tables, leading Bouvard to hypothesize that an unknown body was
  perturbing the orbit through gravitational interaction. In 1843,
  John Couch Adams began work on the orbit of Uranus using the data he
  had. Via Cambridge Observatory director James Challis, he requested
  extra data from Sir George Airy, the Astronomer Royal, who supplied it
  in February 1844. Adams continued to work in 1845–46 and produced
  several different estimates of a new planet.
  [...]
  In 1845–46, Urbain Le Verrier, independently of Adams, developed his 
  own calculations [...]. Neptune was discovered within 1° of where Le 
  Verrier had predicted it 
  to be, and about 12° from Adams' prediction.

Q1: Which method did Le Verrier employ to calculate Neptune's position with such accuracy? 
Q2: How would it be done with today's tools?

Edit: Springer has the chapter in the answer available for download.

Edit 2: since the work of Le Verrier seems beyond the scope of a stackexchange question, is it possible to explain the general approach?
Quote from the article above:

Jean-Baptiste Biot attempted to explain Le Verrier's methods in six
  papers in *Journal des Savants (October 1846, pp. 577–596; November
  1846, pp. 641–664; December 1846, pp. 750–768; January 1847, pp.
  18–35; February 1847, pp. 65–86; March 1847, pp. 182–187). Arrived at
  the third paper, he writes: “As I progress in the task I have
  undertaken, the difficulty of the subject seems to increase.”

REPLY [3 votes]: There are some very nice illustrations and a detailed explanation on the equations that led to the discovery of Neptune in a thread about solar system dynamics. Here is the link:
http://forum.spaceengine.org/viewtopic.php?t=446&start=60#p22455
Since the thread is long, I cannot transcribe the entirety of it. They discuss the concept of perturbations on the angular momentum of a planet caused by the gravity of another nearby planet. And by some calculations and beautiful charts, pretty much explain how the two mathematicians were able to predict the existence of an unknown planet. I enjoyed reading it very much and highly recommend it to the future visitor of this question.