It seems to me that Otto Toeplitz (1881-1940) has not been mentioned yet. The following paragraph from the Wikipedia article on Toeplitz gives a quick idea of his contributions to the history of mathematics:
In 1929, he cofounded "Quellen und Studien zur Geschichte der Mathematik" with Otto Neugebauer and Julius Stenzel. Beginning in the 1920s, Toeplitz advocated a "genetic method" in teaching of mathematics, which he applied in writing the book Entwicklung der Infinitesimalrechnung ("The Calculus: A Genetic Approach"). The book introduces the subject by giving an idealized historical narrative to motivate the concepts, showing how they developed from classical problems of Greek mathematics. It was left unfinished, edited by Gottfried Köthe and posthumously published in German in 1946 (and translated into English in 1963).
Harold M. Edwards's book on Fermat's Last Theorem was structured having in mind the genetic method. This is what Edwards himself said about the method of his book in the preface to the first edition of it:
«... The basic method of the book is... the genetic method. The dictionary defines the genetic method as "the explanation or evaluation of a thing or event in terms of its origin and development."... It is important to distinguish the genetic method from history. The distinction lies in the fact that the genetic method primarily concerns itself with the subject—its "explanation or evaluation" in the definition above—whereas the primary concern of history is an accurate record of the men, ideas, and events which played a part in the evolution of the subject. In a history there is no place for detailed descriptions of the theory unless it is essential to an understanding of the events. In the genetic method there is no place for a careful study of the events unless it contributes to the appreciation of the subject. This means that the genetic method tends to present the historical record from a false perspective. Questions which were never successfully resolved are ignored. Ideas which led into into blind alleys are not pursued. Months of fruitless effort are passed over in silence and mountains of exploratory calculations are dispensed with. In order to get to the really fruitful ideas, one pretends that human reason moves in straight lines from problems to solutions. I want to emphasize as strongly as I can that this notion that reason moves in straight lines is an outrageous fiction which should not for a moment be taken seriously. Samuel Johnson once said of the writing of biography that "If nothing but the bright side of characters should be shown, we should sit down in despondency, and think it utterly impossible to imitate them in anything. The sacred writers related the vicious as well as the virtuous actions of men; which had this moral effect, that it kept mankind from despair." This book does, for the most part, show only the bright side, only the ideas that work, only the guesses that are correct. You should bear in mind that this is not a history or biography and you should not despair. You may well be interested less in the contrast between history and the genetic method than in the contrast between the genetic method and the more usual method of mathematical exposition. As the mathematician Otto Toeplitz described it, the essence of the genetic method is to look to the historical origins of an idea in order to find the best way to motivate it, to study the context in which the originator of the idea was working in order to find the "burning question" which he was striving to answer... In contrast to this, the more usual method pays no attention to the questions and presents only the answers. From a logical point of view only the answers are needed, but from a psychological point of view, learning the answers without knowing the questions is so difficult that it is almost impossible. That, at any rate, is my own experience...»
It might be noteworthy that Michael Spivak implicitly referred to the genetic approach in the preface to volume one of his A comprehensive introduction to differential geometry:
«... The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary... Of course, I do not think that one should follow all the intricacies of the historical process, with its inevitable duplications and false leads. What is intended, rather, is a presentation of the subject along the lines which its development might have followed; as Bernard Morin said to me, there is no reason, in mathematics any more than in biology, why ontogeny must recapitulate philogeny. When modern terminology is introduced, it should be as an outgrowth of this (mythical) historical development.»