# Origin of the “law of quadratic reciprocity”

Today, "reciprocity" is the standard mathematical word used for quadratic reciprocity and its generalizations.

I found that the name dates back to no later than 1832, when a paper of Dirichlet (Crelle's Journal vol. 9 p. 379) made reference to the "theorem so famous under the name of the law of reciprocity" ("théorème si célèbre sous le nom de loi de réciprocité"). I don't actually know French, but it seems Dirichlet is saying that the name pre-dates his work and was already well-known by then.

On the other hand, early writings on the subject do not use the term "reciprocity". Gauss in Disquisitiones Arithmeticae, published in 1801, generally referred to quadratic reciprocity as the "fundamental theorem" (theorema fundamentale). He also called it the "golden theorem" (theorema aureum) in his diary in 1796. From this, I would guess that the name "reciprocity" had not yet been coined then, or at least not yet popularized.

Who, then, introduced the term "reciprocity"? Did Gauss, or his predecessors such as Euler, use the word?

I would be grateful for references attesting to the earliest uses of "reciprocity" in any language.

The term (loi de réciprocité) was introduced by Legendre in his book Essai sur la Théorie des Nombres (1797). It has (p.214) a section titled Théoréme contenant une loi de réciprocité qui existe entre deux nombres premiers quelconques (Theorem containing a law of reciprocity that exists between two arbitrary prime numbers), and he writes in the Preface (p. viii, my translation):

"A memoir that I published in the volume of the Academy Sciences for the year 1785, offers the means of demonstrating the principle in question, and contains, moreover, propositions that seem to advance the science of numbers. I gave 1. A proof of a theorem to determine the possibility or impossibility of solving an indeterminate equation of the second degree reduced to the form $$ax^2+by^2=cz^2$$; 2. A demonstration of a general law that exists between two arbitrary prime numbers, and which may be called the law of reciprocity..."

This is also the book where Legendre introduced what is now called the Legendre symbol, and formulated the law in terms of it. Neither the term nor the symbol appear in the mentioned Recherches d'analyse indéterminée (1785), but instead we find an equivalent formulation split into eight separate cases. A good historical commentary is On Legendre’s Work on the Law of Quadratic Reciprocity by Weintraub, who argues that the changes from 1785 to 1797 were more than cosmetic:

"The first change is the introduction of what we now call the Legendre symbol $$\left(\frac{m}{n}\right)$$. Here $$n$$ is a prime and $$m$$ is an arbitrary integer not divisible by $$n$$. From Theorem A he knows that $$m^{(n-1)/2}$$ leaves a remainder of $$\pm1$$ when divided by $$n$$, and then he sets $$\left(\frac{m}{n}\right)=1$$ or $$-1$$ as this remainder is $$1$$ or $$-1$$. Legendre occasionally uses this symbol when $$n=1$$, in which case he sets $$\left(\frac{m}{n}\right)=1$$ for every nonzero integer $$m$$. In the opinion of the author, this is more than a notational convenience. In introducing this notion, Legendre reifies this concept, and makes it into an object of independent study. This line of thought later led to the Jacobi symbol and the Hilbert symbol.

The second change is the introduction of the term “reciprocity”... Again,in the opinion of the author, this general heading reflects not only an appreciation of the importance of this overall result, but a conception of it as a single result (rather than a collection of eight results), and a conception of it as reflecting a relationship between distinct odd primes.

Euler conjectured an equivalent result in Observationes circa divisionem quadratorum per numeros primos (1772), but did not call it anything in particular. He did introduce "residue", and "quadratic residue". Gauss became aware of Legendre's work late into his thinking on the subject, and was politely unimpressed by it, hence did not adopt the term. Weintraub writes:

"We have mentioned that Euler stated a conjecture equivalent to the law of quadratic reciprocity (though it takes a bit of work to see that), but Euler’s statement seems not to have had any influence on either Legendre or Gauss. Euler coined the terms quadratic residue and nonresidue, which were not used by Legendre but were used by Gauss. Legendre coined the term quadratic reciprocity, but this was never used by Gauss, who always referred to this result as the “fundamental theorem (in the theory of quadratic residues),” nor did Gauss ever use the Legendre symbol in any of his works on the subject.

In the introduction to [3] Gauss writes that his work there had been done without knowledge of prior results in the subject. He also writes there that in the meanwhile, the “excellent” work [11] of the “highly deserving” Legendre appeared, but that he did not rewrite [3] to take it into account, only adding a few additional remarks in the Appendix.

As for Legendre's "demonstration", he repeatedly uses the theorem on primes in arithmetic progressions, which is harder and was only proved by Dirichlet in 1837, and falls into what Cox called "a mess of circular reasoning. [...] At the 3rd edition (1830) of his Essay, there had been enough criticism of Legendre's proof for him to add Gauss's 3rd proof of reciprocity, as well as a statement by Jacobi (all while maintaining that his first proof was valid)" (Cox, Primes of the Form $$x^2 + Ny^2$$).