Well that wasn't so bad. I spent only seven hours translating rather than the usual couple of days. The cleaning was hellish though, but it was worth it; I hope you enjoy the resulting higher quality scans!
Download Link: Volume 1 Chapter 3
Translator Notes: You'll notice that Eureka uses more or less the same speech from Chapter 1 for the little memory-transfer ritual. It's about word-for-word identical in the Japanese version, but previously I used "Please dream for a little bit longer" whereas this time I used "Please sleep a while longer" for variation purposes. The closer translation is the latter one, in any case.
[Spoilers Beyond This Point!]
WHY BONES!? Why are you so cruel, and why do I love it!? In my reading of this chapter the first time, I glossed over the last couple of pages, so I had assumed that Renton was upset that Eureka hadn't told him about her physical change, not her mental state. I find it remarkable that he pieced all of that together and went straight for the big question: if she's forgetting all her emotions and feelings, at what point will she forget that she loves him? Somehow I just never thought about it this way; I thought that Eureka retained the memory and "submitted a copy", so to speak, but that the price was a physical change based on the loss of identity metaphorically, not literally. It's like a seven-day accelerated version of Alzheimer's, and that he was able to ask her outright about the obvious truth was heartwrenching. Not only will he lose her physically, but at some point in their last seven days together, she will forget everything about him.
In other less tear-worthy observations, I found it interesting that Eureka chooses to focus on a specific idea whenever she returns a Fragment. The first Fragment was themed on beauty, on perceivable joy based on the world around her, and thus she's lost the ability to judge things as pretty or pleasing. She can't enjoy a sunset, she can't enjoy music, and she can't see the beauty of the world that she painstakingly learned throughout her life as a human. The second Fragment is themed on suffering, on that which causes physical pain and discomfort. Thus she no longer feels thirst or (presumably) hunger, and has moved another step further from humanity. I assume that each of the five remaining fragments will encompass other portions of her identity and humanity, and that once she loses them all, she'll have completely lost herself and be reabsorbed as a part of the Scub.
I really have to say that I appreciate Renton far more in this version. I've always loved him as a character, even if he could verge on whiny in the first quarter or so of the series, but he's so much bolder in this version. It's as if Miyama tried to make Renton as Ao-like as possible while retaining his individualism, and here he comes off as more perceptive and stronger-willed than in his anime counterpart. I think it's a good choice, considering he's still as sweet and clownish as always when he's not realizing horrible truths about the girl he loves.