Elementary books by good mathematicians I'm interested in elementary books written by good mathematicians.
For example:


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*Gelfand (Algebra, Trigonometry, Sequences)

*Lang (A first course in calculus, Geometry)


I'm sure there are many other ones. Can you help me to complete this short list?
As for the level, I'm thinking of pupils (can be advanced ones) whose age is less than 18.
But books a bit more advanced could interest me. For example Roger Godement books: Analysis I & II are full of nice little results that could be of interest at an elementary level.
 A: An introduction to theory of numbers by Hardy and Wright
A: A course of Pure Mathematics by G.H. Hardy.
A: By the way, I don't think age is necessarily the optimal criterium. And at 17 -18, some are already in college. So ability and intent are perhaps more relevant.
If you are not constrained to books per se, here is a link to a free download of what can be considered verbatim transcripts of lectures on real analysis by Fields Medal winner Vaughan Jones. 
Here you will find a great mathematician artfully taking the student along assuming no prior knowledge, giving just the right amount of guidance each step of the way. 
I personally feel they are akin (although on a smaller scale) to Feynman's lectures on physics: where a real master knows just how to present challenging material to (at the outset) beginning students.
As well, real analysis can be considered the transitional material going from a somewhat mechanical approach to a conceptual, rigorous study of math.
https://sites.google.com/site/math104sp2011/lecture-notes
Also here are books on geometry from Berkeley Math Circle:


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*Kiselev's Geometry: Book 1, Planimetry
Translated from Russian by Alexander Givental
Published by: Sumizdat
This is a wonderful, easy-going introduction to plane geometry, which was used for decades as a regular textbook in Russian middle schooles. It has been translated from its original Russian to English by one of UC Berkeley's very own math instructors, Professor Alexander Givental.
Price: $25


Highly Recommended for BMC Intermediate and Advanced


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*Kiselev's Geometry: Book 2, Stereometry
Translated from Russian by Alexander Givental
Published by: Sumizdat
This is the second volume of the famous Kiselev's work. A marvelous self-contained exposition on stereometry that proved to be a favorite for generations of students and mathematicians in Russia. Thanks to our UC Berkeley Professor Alexander Givental this book is now available in English.
Price: $15


and a link to their recommended publications:
http://mathcircle.berkeley.edu/index.php?options=bmc|recommendedbooks|Recommended%20Books
A: Concise introduction to pure mathematics by Martin liebeck
A: Introduction to Calculus and Analysis by Courant
A: Hilbert, Geometry and the Imagination
A: Solving Mathematical Problems: A Personal Perspective by Terence Tao
A: Calculus by Michael Spivak.
It's very rigorous, but it starts from ground zero. 
A: Not sure how easy an 18 year old might grasp this, but
Paul R. Halmos' Naive Set Theory is definitely a keeper. (obviously on set theory; relatively non-axiomatic)
This is a little more elementary and I think is definitely a good read especially since most high school students live in the world of pre-rigorous mathematics; I think everyone's first exposure to rigorous math is through proofs:
David C. Velleman's How to Prove it (introductory set theory and proof-writing)
EDIT: Age is not an indicator of ability.
A: Geometry Revisited by H Coxeter
A: Introduction to Geometry By Coxeter
A: Geometry Euclid and Beyond by Hartshorne
A: The Shape of Space by Jeffrey R. Weeks.
To get an idea for some of the topics covered in this book, check this out.
A: V. Arnol'd's book Mathematical Methods of Classical Mechanics is superb. 
A: Walter Rudin Priciples of Mathematical Analysis
A: André Weil's Number theory for beginners is wonderful.
A: Serre's "A Course in Arithmetic".
A: How to Solve it By George Polya.
A: "Mathematics - Form and Function", by Saunders MacLane could be read by a bright 18-year old.
A: Analysis I and II by Terence Tao
A: My suggestion is:


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*A Survey of Modern Algebra by Birkhoff and Mac Lane, especially chapters 1 and 6.


Note that this is not the (much harder) book Algebra by the same authors. I was fortunate in that this book was in my school library when I was doing A-levels (so I was around 17 or 18), and it continues to influence my interest in abstract algebra.
