TITLE: Good "history of mathematical ideas" book?
QUESTION [63 upvotes]: All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living subject in itself, which is not enhanced by knowing about the practitioners themselves (unless it can be shown how their lives link to their ideas, which is far too complex, speculative, and rarely as successful for shedding light on the ideas, as would a direct analysis of how their new idea grew from previous ones). Besides, can we really claim to know the details of a person's life enough to be able to draw inferences on why they did something? This is why I'm looking for a good history of maths book that focuses on how the ideas developed through time, also including how (and ideally why) the notation changed, why the new ideas were introduced, and so on. In fact, this isn't too hard, as Lagrange admirably demonstrates in his "lectures on elementary mathematics" with his short and insightful exposition on the development of logarithms, where he ends it by remarking that:

"Since the calculation of logarithms is now a thing of the past,
  except in isolated instances, it may be thought that the details [i.e.
  the history/development of the theory of logs] into which we have here
  entered are devoid of value. We may, however, justly be curious to
  know the trying and tortuous paths which the great inventors have
  trodden, the different steps which they have taken to attain their
  goal, and the extent to which we are indebted to these veritable
  benefactors of the human race. Such knowledge, moreover, is not matter
  of idle curiosity. It can afford us guidance in similar inquiries and
  sheds an increased light on the subjects with which we are employed."

(Lagrange was known to focus on the history of the ideas involved whenever he wrote a large treatise, such as the excellent history of mechanics that he opens off his Mechanique Analytique with.)
I couldn't sum up the reason for my interest in the history of the development of mathematical ideas any better.

REPLY [3 votes]: A Concise History of Mathematics

This compact, well-written history — first published in 1948, and now in its fourth revised edition — describes the main trends in the development of all fields of mathematics from the first available records to the middle of the 20th century. Students, researchers, historians, specialists — in short, everyone with an interest in mathematics — will find it engrossing and stimulating.
  Beginning with the ancient Near East, the author traces the ideas and techniques developed in Egypt, Babylonia, China, and Arabia, looking into such manuscripts as the Egyptian Papyrus Rhind, the Ten Classics of China, and the Siddhantas of India. He considers Greek and Roman developments from their beginnings in Ionian rationalism to the fall of Constantinople; covers medieval European ideas and Renaissance trends; analyzes 17th- and 18th-century contributions; and offers an illuminating exposition of 19th century concepts. Every important figure in mathematical history is dealt with — Euclid, Archimedes, Diophantus, Omar Khayyam, Boethius, Fermat, Pascal, Newton, Leibniz, Fourier, Gauss, Riemann, Cantor, and many others.
  For this latest edition, Dr. Struik has both revised and updated the existing text, and also added a new chapter on the mathematics of the first half of the 20th century. Concise coverage is given to set theory, the influence of relativity and quantum theory, tensor calculus, the Lebesgue integral, the calculus of variations, and other important ideas and concepts. The book concludes with the beginnings of the computer era and the seminal work of von Neumann, Turing, Wiener, and others.
  "The author's ability as a first-class historian as well as an able mathematician has enabled him to produce a work which is unquestionably one of the best." — Nature Magazine.