Hi, Pete. There are a few observations related to this, not widely known although basic, and that includes your colleague. First, Conway gives a quick proof on page 142 of The Sensual Quadratic Form, including over the rationals.
Next, also Conway, the form (five variables) that he and Schneeberger found that represents all the numbers from 1 to 289, fails to represent 290, then represents 291 and on forever, he initially called Methusaleh. It is just a binary added to a ternary that represents the numbers from 1 to 28 consecutive, discriminant 29. However, for ternaries that is not the record. The form he called Little Methusaleh, discriminant 31, represents 1 to 30 consecutive. The theorem is in this material, as the conditions for a positive ternary to represent, say, 1,2,3,5, places strong restrictions on a partly reduced form. Kap wrote this sort of argument up several times, including a repeat in the unpublished 1996 Classification. It is quite easy. OK, Little Methusaleh and your result over the integers are proved on page 81 of The Sensual Quadratic Form
Finally, a positive form is anisotropic at the "prime" infinity. In Cassels Rational Quadratic Forms he shows global relations on the Hilbert Norm Residue symbol that show that any ternary is anisotropic at an even number of primes. So a positive ternary is anisotropic at an odd number of finite primes. Taken with the observation above that at least one number below 31 is missed, and a positive ternary fails to integrally represent an infinite number of positive integers.
I will look up some of my tables and fill things in. Note that some of this is discussed in an early article by William Duke, 1997 Notices, but he mistyped the form with discriminant 29.
Let's see, Conway and Schneeberger probably had an acceptable proof of the 15 Theorem scattered about, but it never got put together. Bhargava was looking for diversions from his own dissertation, Conway mentioned this in passing. Bhargava showed the fundamental result that one of these forms must have a regular ternary as a sub-form, thus the project became a careful inspection of my paper with Kap on all possible regular ternaries. Also, correspondence between Kap and Bhargava first revealed some important errors in Magma relating to calculating the spinor genus, and hilarity ensued.
EDIT: thinking about the history question, it is quite possible that this result was never written down as a separate proposition, by Gauss, Legendre, etc. The reason I suggest this is the great weight placed on positive ternary forms missing certain "progressions," in the language of Jones, Dickson, other early books. So, in Jones, chapter 8, we read "Thus there will be a finite number of arithmetical progressions of this type" of numbers not represented by any form in the genus under consideration. Not much motivation for proving that a form misses at least one number if you are going to quickly show that it misses an entire arithmetic progression.
EDIT TOOO: note that Conway replaces the prime usually called $\infty$ by the prime $-1.$
No definite ternary form is universal
However, a simple argument shows that
any definite ternary form must fail to
represent infinitely many integers,
even over the rationals. For if a
ternary form $f$ of determinant $d$
represents anything in the $p$-adic
squareclass of $-d$ over $\mathbf
> Q_p,$ then it must be $p$-adically
equivalent to $[ -d,a,b]$ where the
"quotient form" $[a,b]$ has
determinant $-1,$ and so $p$-adically,
$f$ must be the isotropic form $[
> -d,1,-1].$
But a positive definite form fails to
represent $-1,$ and so it is not
$p$-adically isotropic for $p=-1.$ By
the global relation, there must be
another $p$ for which it is not
$p$-adically isotropic, and so it
also fails to represent all numbers in
the $p$-adic square-class of $-d$ for
this $p$ too!
The Three Squares Theorem illustrates
this nicely--the form $[ 1,1,1]$ fails
to represent $-1$ both $-1$-adically
and $2$-adically. In the Third
Lecture, we showed that The Little
Methusaleh Form $$ x^2 + 2 y^2 + y z
> + 4 z^2 $$ fails to represent 31. We now see that since it fails to
represent the $-1$-adic class of its
determinant $-31/4$ (i.e., the
negative numbers), it must also fail
to represent the infinitely many
positive integers in the $31$-adic
squareclass of $-31/4.$
Best Answer
I found the following excerpt on this web site: http://jeff560.tripod.com/d.html. It gives references to specific papers of both Lagrange and Gauss. The inverse square law for the electric force is usually associated with Coulomb, but was apparently first inferred by Priestley on the basis of Franklin's observation that there is no electric field inside a hollow conductor and analogy to the known analogous property for gravity.
The history of the theorem is bewildering with many re-discoveries.
O. D. Kellogg Foundations of Potential Theory (1929, p. 38) has the following note on the result “known as the Divergence Theorem, or as Gauss’ Theorem or Green’s Theorem”:
A similar reduction of triple integrals to double integrals was employed by Lagrange: Nouvelles recherches sur la nature et la propagation du son Miscellanea Taurinensis, t. II, 1760-61; Oeuvres t. I, p. 263. The double integrals are given in more definite form by Gauss Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate, Commentationes societas scientiarum Gottingensis recentiores, Vol III, 1813, Werke Bd. V pp. 5-7. A systematic use of integral identities equivalent to the divergence theorem was made by George Green in his Essay on the Mathematical Theory of Electricity and Magnetism; Nottingham, 1828 [Green Papers, pp. 1-115]. Kline (pp. 789-90) writes that Mikhail Ostrogradski obtained the theorem when solving the partial differential equation of heat. He published the result in 1831 in Mem. Ac. Sci. St. Peters., 6, (1831) p. 39. J. C. Maxwell had made the same attribution in the 2nd edition of the Treatise on Electricity and Magnetism (1881). See also the Encyclopaedia of Mathematics entry Ostrogradski formula
For Gauss’s theorem Hermann Rothe “Systeme Geometrischen Analyse, Erster Teil” Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen Volume: 3, T.1, H.2 p. 1345 refers to Gauss’s Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossung-Kräfte 1839 in Werke Bd. V (especially pp. 226-8.)