How did Le Verrier calculate Neptune's position?

Question

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?

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

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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.”

Answer 1

Chapter 2, "The Discovery of Neptune (1845-1846)," in the biography Le Verrier -- Magnificent and Detestable Astronomer by James Lequeux may give you what you want. If you google on the chapter title, you can find a link that'll download the entire chapter as a pdf.

Answer 2

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.