# Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus in his Zur Geschichte der Mathematik in Alterthum und Mittelalter (p. 163):

At this point, there is a mistake to be corrected, a mistake that is reinforced by false nomenclature and therefore, or so I fear, impossible to weed out. In education, one designates linear equations of the form $ax+by=c$, that are to be solved in integers $x,y$, as Diophantine. Now, not only did Diophantus not know the solution method of these equations, that in the West was first obtained by his commentator Bachet, but the very problem would have been utterly alien to him, since he never fixes the condition that his solutions should be integral, but is completely satisfied with rational solutions.

(Italics are mine.) Now, in connection with this passage, I have the following questions:

1. When did people start talking/writing about "Diophantine equations"?

And also:

1. Were Diophantine equations originally considered as "polynomial equations to be solved in rational numbers", in accordance with Diophantus' own preference, or was the mistake that Hankel aims to correct made from the beginning?

There is a website, Earliest Known Uses of Some of the Words of Mathematics. Some entries are,

DIOPHANTINE ANALYSIS (named for Diophantus of Alexandria) occurs in French in a letter of March 1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière et surpassait même les règles connues de l'analyse de Diophante” (“this problem appeared to me to be of a singular nature and surpassed the known rules of Diophantine analysis”).

Lagrange used “analyse de Diophante” in a letter to D’Alembert in June 1771.

“Diophantine Analysis” occurs in English in the chapter title “Demonstration of a Theorem in the Diophantine Analysis. By Mr. P. Barlow, of the Royal Military Academy, Woolwich.” in The Mathematical Repository, New Series, Volume III (1809) page 70.

[This entry was contributed by James A. Landau.]

DIOPHANTINE EQUATION. Felix Klein used Diophantische Gleichungen in “Die Eindeutigen automorphen Formen vom Geschlechte Null” in the 1892 issue of Nachrichten (page 286): “Die Relationen kann man in Diophantische Gleichungen umsetzen, welche dann leicht übersehen lassen, unter welchen Umständen Multiplicatorsysteme möglich sind, und in welcher Anzahl.” [James A. Landau]

Diophantine equation appears in English in 1893 in Eliakim Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple Groups," Bulletin of the New York Mathematical Society, vol. III, pp. 73-78, October 13, 1893 [Julio González Cabillón].

Henry B. Fine writes in The Number System of Algebra (1902):

The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.

DIOPHANTINE PROBLEM. The phrase "Diophantus Problemes" appears in 1670 [James A. Landau].

The OED2 has the citation in 1700: Gregory, Collect. (Oxf. Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical or Diophantine problems."

• This is really great. Thanks! (I edited your answer slightly to make the last entry more conspicuous.) – RP_ May 26 '15 at 10:46
• Too bad though that the 1670 citation for "Diophantus Problemes" (French?) is so unclear. I think the year coincides with the re-edition of Bachet's Diophantus, containing Fermat's marginal notes. I will try to find out if that is indeed the intended source. – RP_ May 26 '15 at 12:51
• Google Books seems to turn up several earlier instances of "Diophantine equation." – Timothy Chow May 26 '15 at 14:13
• Ah, you mean earlier than 1893 (for a moment there I assumed you meant pre-1670, which would have been awesome). Yes, the website does not seem to be perfect (in part user-contributed, so you wouldn't expect it to be). – RP_ May 26 '15 at 14:43