Misquoted: Gauss on the Queen of Mathematics

The following two quotes from Carl Friedrich Gauss are frequently met in books on Mathematics and Science (sometimes with slight differences in formulation as they were translated from German):

"Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics." 
"Mathematics is the Queen of the Sciences, and Number Theory the Queen of Mathematics."

The earliest occurrence of the quotes I met first in Eric T Bell's "Mathematics: Queen & Servant of Science" (1951), which is probably one of the most quoted sources for the quotes:

"Mathematics is Queen of the Sciences and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all  circumstances the first place is her due."

However, doing a little research I found an earlier reference in John T Merz' "European Thought in the Nineteenth Century" Vol II (1903), who cites Sartorius von Waltershausen's "Gauss zum Gedächtniss" (1856) as source:

"Mathematics is the Queen of the Sciences, and arithmetic the Queen of Mathematics. She frequently condescends to do service for astronomy and other natural sciences, but to her belongs, under all circumstances, the foremost place."

Arithmetic, at least the way introduced in schools, is the study of numbers and their properties in report with the basic operations (addition, subtraction, multiplication, division, exponentiation and extraction of roots). In exchange, Number Theory appeared while studying the intrinsic properties of integers and integer-valued functions, what is known also as Higher Arithmetic. Even if the earliest discoveries date from early antiquity, the basis were put by Fermat, Euler, Gauss and others mathematicians, incorporating over time domains like Complex Analysis, Group theory or Galois theory.

According to Merriam-Webster, "Number Theory" was used for the first time around 1864 and it took some time until it was incorporated in mathematical texts. Arithmetic and Number Theory become synonymous in the 20th century (which might be strange for some). Thus, in some occurrence of the quote "Arithmetic" was replaced by "Number Theory" to reflect nowadays’ interpretation of Gauss' words. The two quotes seem to mean the same thing, though they probably need to be interpreted upon case through the historical context.  

'Queen' is used here metaphorically as an analogy between the meanings associated with a queen (ruler, fertility, authority), respectively the roles taken by Mathematics and Arithmetic in science. Therefore, a science considered as a queen could be seen as having a high fecundity of ideas and authority in another field. As "Queen of Science" were considered also Metaphysics (Immanuel Kant), Philosophy and even Theology. Also Physics and Chemistry have a statute of royalty.

The importance of Higher Arithmetic in respect to Mathematics, is stressed by Gauss in a letter to Dirichlet from 1838: 

"I place this part of Mathematics above all others (and have always done so)."

The idea is reiterated by David Hilbert half of century later:

"We thus see how arithmetic, the queen of mathematical science, has conquered large domains and has assumed the leadership. That this was not done earlier and more completely, seems to me to depend on the fact that the theory of numbers has only in quite recent times arrived at maturity." (David Hilbert, "Theorie der Algebraischen Zahlkörper", Bericht der Mathematiker-Vereinigung,' vol. IV, 1897)

Paraphrasing Godfrey H Hardy, even if Number Theory began as an experimental science, its role in the further development of Mathematics is important, worthily of the status of a 'queen'.

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