I'm looking for any information about the posthumous publication of Gauss' mathematical correspondence and notebooks.
When did these become widely available, and how did it affect progress in mathematics?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign up to join this community
Q1: The mathematical diary that Gauss kept from 1796 to 1814 was rediscovered in 1897 and published in 1903, so almost fifty years after his death. His collected works were published sooner, in 1866.
Q2: According to The Poincaré Conjecture: In Search of the Shape of the Universe (page 124) the posthumous publication of Gauss's correspondence and scientific notebooks made it clear that Gauss had discovered non-Euclidean geometry first, and hastened the acceptance of Bolyai's and Lobachevsky's work.
As an aside: A notable discovery in Gauss' posthumous collected works was the basic algorithm of the fast Fourier transform, which he had already written down in 1805 -- even before Fourier's work from 1822. The FFT was not rediscovered until 1965. Other examples of independent rediscoveries include the Gauss-Seidel method and the quaternion multiplication rule.
The published version of Gauss's collected works, which contains 12 volumes (together with two other volumes that contain treatises by later mathematicians about different aspects of Gauss's work), can be accessed freely on the digital library of Gottingen university - here is a link: https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348. This version contains all of Gauss's publications, together with a lot of unpublished manuscripts, drafts of treatises and etc.
In addition, there are published correspondences of Gauss with different mathematicians/scientists with whom he corresponded regularly. These correspondences (for example: Gauss-Schumacher, Gauss-Bessel,...) fill many more volumes but their quality (in terms of density of mathematical ideas) is much lower than in Gauss's collected works mentioned before.
And finally, and this is something that I'm especially interested in, this is not the whole story - there are many fragments and notebooks of Gauss that have not been digitized yet - here is a link to a website that apparently gives a complete guide to Gauss's work (published as well as unpublished) - https://kalliope-verbund.info/de/findingaid_toc?fa.id=DE-611-BF-61709&fa.enum=1&lastparam=true. Note that some references that you will find in scientific/mathematical articles about Gauss cannot be located within the twelve volumes mentioned before. For example, the famous handnote in which Gauss began to study the topological object known as "braid" - which is the theme of another Mathoverflow question - cannot be found anywhere in his collected works, but according to the guide in the last link, it should be found in Handbuch 7.
Effect on the progress of mathematics
As a source of inspiration
Most of Gauss's unpublished discoveries were independently rediscovered by other mathematicians, albeit in a much later date, before the publication of his collected works. However, even in the case of already known results found in his Nachlass, it is not a specific discovery that affected the progress of mathematics, but rather the organic structure of his body of mathematical work, that served as a source of inspiration for later mathematicians. Once a more complete view of his work was available to the world, it was the interaction between his ideas in different branches of mathematics/science he worked in that fertilized the work of later researchers.
For example, there are deep relations between his work in differential geometry (especially his study of geodesics) and analytical mechanics (a branch of which he contributed his "principle of least constraint") as developed by later mathematicians. However, Gauss himself does not make any remark on such a connection, and Heinrich Herz gave much later an interpretation of Gauss's principle of least constraint using differential geometry. This is just an example to illustrate my point.
Examples of unpublished discoveries that were actually not known
From my experience at reading into Gauss, I gained the impression that it was especially the analytic aspects of his work (elliptic functions, theta functions, modular forms) that did have an impact on the progress of mathematics:
I found a good source of information:
A Critical Survey and Inventory of the Edited Works of Carl Friedrich Gauss