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The author, L. E. Dickson, was a distinguished algebraist and number theorist in the first half of the 20th century. With collaborators, he compiled this "history", in three volumes, to get himself acquainted with the subject and to launch research on this topic (considered as important in other countries) in the US.
The book is a kind of "Mathematical Reviews" of number theory for the period until 1920: short, technical, notices on hundreds of articles and books, classified by topic, according to the classification valid in 1900. Dickson prefered "just the (mathematical) facts" and thus there is no biographical information, nor sweeping conceptual or sociological description. A useful, slightly more entertaining, introduction summarizes the development of the results in each volume.
Thus, if you are looking 1) for a popular, general presentation of the history of number theory 2) or for an up-to date synthesis of number theory or its history, forget it. It is rather technical and was written in the 1920s. But it is very interesting if your interest in history of maths is (semi-)professional, specially on the 19th century, or if you are looking for mathematical fun in old, mostly elementary, but certainly still fruitful, problems. Better to check the state-of-the-art, though, in both cases, if you have any hope for publication ( many new documents and of course many new theorems, even new research areas, have been found since the 1920s).
And then, it is such a classic for US (history of) maths that, especially in paperback, you may want it for your library in any case. If you want to know more on this book, there is a nice chapter devoted to it, written by D. Fenster (the best specialist of Dickson) in Landmark Writings in Western Mathematics, ed. I. Grattan-Guinness.
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If you are looking for a quick concise history of any technical number theoretical topic (300BC--1900) this set of volumes is a masterpiece and stands alone--very few fields have such a fine authoritative concise summary of their history. The author clearly spent years combing libraries and trying to catalog all of what had been done in number theory. For this reason, these volumes are truly excellent reference books that all large libraries should own (but few individuals).
They are reference books, do not buy them to sit down and learn about the history of number theory. They are not narratives. No quaint stories. No pictures. Very few definitions. Just very long lists of very terse concise references arranged by topic and then listed chronologically. For example, if you want to know who created what length of prime number tables before 1900, no extra words, just a list of dates and references, these volumes are ideal! To fault these volumes for being boring is to entirely miss their purpose, they are not written to entertain you. They are not novels or text books. Just references.
I love my copies, and enjoy seeing who did what on areas of number theory that are no longer explored, but I think others should look in a library and know what they are buying before they do.
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I hate writing negative reviews, and I am not arguing about the completeness or accuracy of the mathematics of the book, but this is the most boring, useless, dreary, tedious, uninteresting, monotonous and dull book on mathematics that I have ever possessed. The whole book is a catalog of mathematical facts and theorem, without proof, and with extremely vague corelations between them, proved by this or that author, each from 1 to 10 lines long. Reading a phone directory can often be more entertaining.