If you’d rather not mess with the math, just add 1/2 of the extract when you start the boil, then and 1/2 at the end. There are a lot of posts on here about late extract additions  it has improved the quality of my brews.
What he is doing is accounting for the potential sugars in both the specialty grains and the extract. He calculates the contribution of the grains to the boil SG, and then subtracts that out of the boil SG. Then divides out to see how much extract is needed to make up the difference. Then ratios that out to a 3 gallon size. I don't have my book here at work, so I can't offer a better explanation, but that is what he is doing.
EDIT: Ok, looking at the book over lunch and I'll see if I can clarify. The steeping grains are contributing 24 degrees of gravity per pound, so you multiply the weight you have (1.25 lb in his example) by 24, then divide by the volume of your boil (3 gallons in his example). You subtract that amount (10 in his example) from what the recipe says the Boil SG is (55 in his example, using only the numbers after the decimal point), which leaves you with the degree of gravity that needs to come from extract (45 in his example). LME contributes 36 degrees of gravity per pound, so you divide that by your boil volume (3 gallons in his example) and then divide the needed SG by that number (45/12 in his example = 3.75 pounds of LME).
If you are sticking with 3 gallon boils, all of this works out to: Weight of LME = (Boil SG  8xWeight of Grains)/12 (using the numbers in his example: 3.75 = (558x1.25)/12). You then add the remaining LME at the end of the boil.
Quote:
Originally Posted by BogusOwnz
being able to produce identical end result 5 gallon batches using completely different amounts of lme does not seem possible.

you are using the same amount of LME, just adding it at different times. He added 3.75lb of LME at the start of his boil, then added the remaining 6.5lb of LME at the end of the boil (Step 6 on page 38)
Not sure if I helped or muddied the water.....