This paper is one in a series of papers in which du Bois Reymond studied functions on the positive real line ordered by the "order of infinity" (order of growth at infinity), what he later called infinitary pantachy. This was motivated by attempts to find "ideal boundary" between converging and diverging series in terms of the growth of their terms by analogy to filling in Dedekind cuts to get real numbers. Du Bois Reymond more or less established that such completion would not work. Later Hausdorff developed modern theory in Cantorian terms which explains why, the "gaps" in the orders of growth are more severe, beyond what countable limits can fill in. These are now called Hausdorff gaps, but du Bois Reymond did not work in cardinality terms.
Fisher gives a detailed account in Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception, here is the reference with direct link:
Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre
Auflösung yon Gleichungen, Mathematische Annalen 8 (1875) 363-414 (Nachtrage zur Abhandlung: Ueber asymptotische Werthe etc., 574-576).
Du Bois-Reymond defines one order of infinity to be larger than another if the limit of the quotient of their representing functions is infinite. He is not very specific about the class of functions considered, and apparently overlooks the possibility of incomparable orders, an issue later investigated by Borel and Hausdorff. He then analogizes, and contrasts, this continuum of orders to the linear continuum of real numbers:
"Between the two domains there are many analogies... Instead of numbers as fixed signs in the domain of numbers, one has in the infinitary domain of quantities an unlimited number of simple functions: the exponential functions, the powers, the logarithmic functions, that likewise form fixed points of comparisons, and between whose arbitrarily close infinities a limitless number of infinities different from each other can still be inserted.
These functions serving as numbers stretch, in accordance with the present
state of analysis, from exponential functions stacked arbitrarily high... down to logarithms repeated arbitrarily, often to infinity... The question which arises here, whether one actually includes in this interval of infinity the entire domain of infinity, in such a way that one can enclose any given infinity between two infinities of functions of that interval, as is possible for any number in the number series: This question I have already answered in substance and in fact negatively..."
He refers to 1873 paper where only a particular case was considered. This time he gives a more general version, which Borel highly praised and termed
du Bois-Reymond's theorem. A consequence of it, and the motivation, is the non-existence of the "ideal boundary" that can be specified by a sequence of converging/diverging series, as Bertrand earlier proposed (he was thinking of reciprocals to products of powers and powers of iterated logarithms as terms). The du Bois-Reymond's theorem would be the "diagonal argument":
"...if an unlimited family of more and more slowly increasing functions $\lambda_1(x), \lambda_2(x), \lambda_3(x)...$ is given which for each $r$ satisfies the condition $\lim\lambda_r(X)/\lambda_{r+1}(X) = \infty$, one can
always specify a function $\psi(x)$ which becomes infinite with $x$, but more slowly than any function of that family".
The construction of $\psi(x)$ is in a footnote on p. 365, and does show some "diagonality" if one looks hard. There is however no cardinality involved in du Bois-Reymond's setting (Hausdorff will relate gaps to cardinalities only later), so making it into the diagonal argument takes some reading in. In his 1882 book Die allgemeine Functionentheorie (The General Theory of Functions) du Bois-Reymond touches base with Cantorian set theory, and mentions that Cantor showed "continuum of the idealists" to be uncountable. He does not however point out any affinity between the diagonal argument by which it was shown and his earlier construction, let alone lay any claim to it. So whatever the relation between the two it was not apparent to du Bois-Reymond.
