I'd consider something by John Stillwell. For example, Numbers and Geometry, which investigates the connections between number theory and plane geometry -- two subjects which your child has probably seen, but not seen related.
Stillwell is a magnificent writer -- he loves to go on digressions, and to talk about the history of the subject. My impression is that his books are a bit rambling for traditional use as textbooks, but perfect for self-motivated reading for exactly the same reason. He makes the subject fun.
(Disclaimer: I haven't read this book in any sort of depth, but I have read another of Stillwell's books cover to cover.)
Concerning your other recommendations: The Princeton Companion to Mathematics is magnificent, but in practice it's something he'd be more likely proudly own and display on his bookshelf than to read; it's quite dense. Spivak's Calculus, from what I've heard, is magnificent. Probably best in the context of a freshman honors class, but I can imagine that someone disciplined could love it for self-study. Don't know Moor and Mertens.
Stillwell is a magnificent writer -- he loves to go on digressions, and to talk about the history of the subject. My impression is that his books are a bit rambling for traditional use as textbooks, but perfect for self-motivated reading for exactly the same reason. He makes the subject fun.
(Disclaimer: I haven't read this book in any sort of depth, but I have read another of Stillwell's books cover to cover.)
Concerning your other recommendations: The Princeton Companion to Mathematics is magnificent, but in practice it's something he'd be more likely proudly own and display on his bookshelf than to read; it's quite dense. Spivak's Calculus, from what I've heard, is magnificent. Probably best in the context of a freshman honors class, but I can imagine that someone disciplined could love it for self-study. Don't know Moor and Mertens.