I wonder what would be a good/early reference for the fact:

rational points on the unit sphere (centered at the origin) are dense.

Stereographic projection (from a rational point in the sphere) provides a bijection between rational points on the sphere and rational points in euclidean space, where the rationals are dense. (This is a special case of a rational line intersecting a quadric in two rational points)

In many places in the literature the above statement is made, but no reference is given. I am looking for (early) references that provide this fact, perhaps only for the circle or the 2-sphere first.

While related, this question does not quite answer my question; I am looking for early references.


The earliest reference is surely Diophantus' Arithmetica. His "method of adequality" can be used to construct rational points on quadrics that approximate real points arbitrarily well (that is, starting from the existence of a rational point).

This is not of course how Diophantus phrases it, but that is what it comes down to. For example, in Book V, Problem 10, he treats the problem of finding rational $x$ and $y$ satisfying $x^2+y^2=9$ and additionally $x^2>2$ and $y^2>6$. Problem 11 asks for rational $x,y,z$ with $x^2+y^2+z^2=10$, with each of $x^2$, $y^2$, and $z^2$ greater than $3$. Similar problems occur a couple of times more in the same book, and it is easy to satisfy oneself that the method works in the generality described above. Quite an accomplishment for a mathematician working in the Hellenistic era!

For more, see pp. 95-98 of the excellent monograph

Thomas L. Heath. Diophantus of Alexandria; a study in the history of Greek algebra.

as well as the paper

Mikhail G. Katz, David M. Schaps, and Steven Shnider. Almost equal: the method of adequality from Diophantus to Fermat and beyond. Perspectives on Science, Vol. 21, No. 3, pp. 283-324.

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    $\begingroup$ I guess it doesn't get much earlier than that.. $\endgroup$ – Moritz Firsching Jan 26 '17 at 17:53

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