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.
Best Answer
John Stillwell's Mathematics and Its History is a terrific book for precisely this purpose. See my Amazon review.
I have also written some free history of mathematics course materials myself in a similar spirit.