treats these problems in his new, algebraic way. Books 2–3 teach and apply some funda-
mental methods which are then extended to further problems in Books 4–7; for their scope
is, as Diophantos says, “experience and skill” (introductions to Books 4 and 7, pp. 87, 156).
(Thus no really new methods are taught here, perhaps explaining why the lacuna in the
middle of the six Greek books escaped notice.) The last three Greek books contain problems
of a notably higher level. All the problems require that the solution be a rational and
positive number.
The Arithme ̄tika’s historical importance is twofold. First, it is the only surviving testimony
of higher algebra in antiquity. Secondly, the extant Greek books initiated the first modern
studies on number theory, beginning principally with the observations of the French math-
ematician Pierre de Fermat (1601–1665) in his copy of Diophantos. (Fermat’s note on
problem 2.8 is well-known: the equality xn+yn=zn with n any integer larger than 2 is impos-
sible in rational numbers; this assertion was to occupy mathematicians for more than three
centuries until proved in 1992–1995.)
Most of Diophantos’ problems are indeterminate ones of the second degree, that is, with
more unknowns than equations. Now such problems may be soluble or not. Since Diophan-
tos had few general methods at his disposal, only through skillful assumptions were his
problems made determinate and reduced to an already known problem or method; depart-
ing form Diophantos’ assumptions may lead to quite another situation. Furthermore,
Diophantos states, when necessary, that certain numbers cannot be represented as the sum
of two squares, or as the sum of three squares, but without offering any proof. So it is hardly
surprising that later mathematicians found in Diophantos an incentive for further research
to the extent that Diophantos’ name is now associated with various fields of modern
mathematics quite foreign to the content and spirit of his Arithme ̄tika.
Ed.: P. Tannery, Diophanti Alexandrini Opera omnia 2 vv. (1893–1895, repr. 1974); Jacques Sesiano, Books
IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qust.a ̄ ibn Lu ̄qa ̄ (1982).
Th. Heath, Diophantus of Alexandria (1910, repr. 1964); Jacques Sesiano “An early form of Greek
algebra,” Centaurus 40 (1998) 276–302.
Jacques Sesiano
Diophantos of Lukia (40 – 10 BCE)
Physician and surgeon, probably identifiable with C. Iulius Diophantos, son of C. Iulius
He ̄liodo ̄ros of Lude ̄ in Lukia, responsible for inscribing a remedy on the base of an Askl-
e ̄pios statue in the Ludean agora (JHS 10 [1889] 59, #11); the father may have been among
I C’s freedmen (ibid., #8). Diophantos’ colic remedy, admired by A (II),
included centaury sap (either Centaurium nemoralis Jord., native to Spain and Portugal, or
common or lesser centaury, C. erythraea Rafin.: Durling 1993: 199), plus castoreum, squill,
both white and long pepper, myrrh, rue, hyssop, wormwood, Illyrian iris, saffron,
ammo ̄niakon incense, yellow iris root, Pontic nard, ginger, and black hellebore, adminis-
tered with oxumel (G CMLoc 9.4, 13.281 K.). K, in Gale ̄n CMLoc 5.3 (12.845
K.), records his salve for burns and intertrigo, effective also against erusipelas, com-
pounded from litharge, stag marrow, psimuthion, beeswax, terebinth, frankincense
and olive oil, prepared as needed. A records three remedies: Diophantos’
Aphra (sc. “foaming”?) quince-yellow emollient for drawing and squeezing out to heal
joints in Gale ̄n CMGen 2.7 (13.507 K.); and two antidotes for scorpion and spider bites in
Antid. 2.12, 13 (14.175–176, 181 K.), the second of which treats also all serpent bites.
DIOPHANTOS OF LUKIA