To transfer a line from the right-hand side of the room, around the fireplace, and over to the left-hand side, I used a laser level. I set it up in the room on a base which I leveled front-to-back and side-to-side with shims. I directed the laser to the top of the lowest shelf on the existing case, making sure that the laser sat perfectly level as I did so. The point at which I was aiming was the same one highlighted in red in the photo above.
Next, I pivoted the laser until it was directed to my target point on the left-hand side of the fireplace. I had made a base about 1/2" shorter than I would need and used shims at its four corners to raise it to its final height. That height is represented by the line onto which the laser beam is shining in the above photo. Once this height was correct all the way across the front of the base, I leveled the front-to-back dimension with shims and a regular spirit level.
The photo above shows the almost-complete base. At this point, two steps remained. The first was attaching permanent feet which would hold the base in its final, level position.
The last step was establishing a plumb line from the new mantel shelf above to the floor below. Once I had the base positioned relative to this plumb line right where I wanted it, I attached it to the floor with screws. At this point I could be sure that when I plopped the case onto its base, it would sit in the proper relation to the existing case on the right-hand side, with all the shelves reading as level with one another all the way across.
Here's the installed case. With minimal fuss it slid onto its permanent base just as I'd hoped it would. Next week, I'll trim the case to match the existing one, and I'll get started on the upper cases with their adjustable shelves.
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