I realize this is an old thread but in case anybody else looks here I will share my thoughts. This book is not out of date, if any math graduate student can find the time to read it, they definitely should. Had it been written 20 years earlier then it would be too old to read today, but happily the notation has been pretty well locked in since the 1920's. He even gives nice histories of the development of the terminology and notation, usually in footnotes.
The real problem I see with this book is the first couple chapters where he lays down the foundations. He talks at extreme length almost philosophically about the construction of the real numbers in an almost Shakespearean style. It could be cut down by 2/3 and be much more readable. Also, he uses "nests" instead of Cauchy sequences to complete $\Bbb Q$. In modern treatments it's almost always Cauchy sequences. But once he gets past this, the rest of the book read just like modern math, the meat and potatoes of it is great and wouldn't need to be modified for a current student of the 21st century.
So recently I undertook the task of rewriting the first couple chapters to modern exposition, and then I plan to just transcribe the rest of the book nearly verbatim. I assume this book is no longer under copyright protection, if it is then my version will just have to wait until it becomes public domain. But I would think 90 years would be enough. In any case this book deserves to be read by future generations for a long time to come.