Melitianus (350 – 500 CE?)
Wrote in Latin a geographical work on Africa, cited by the R C 3.5,
and there said to be an African. The name is otherwise unattested, but cf. Melinianus (PLRE
1 [1971] 594), or Melitius/Meletius, common 350– 500 CE.
(*)
PTK
Melito ̄n (250 BCE – 80 CE)
A, in G CMGen 5.13 (13.843 K.), cites his wound-powder, containing
lime, orpiment, pumice, and realgar, ground fine in water for 30 days, then dried for an
entire day. PIR2 suggests identification with Ti. Claudius Meliton (on whom see Korpela),
physician to some Germanicus, perhaps Tiberius’ adopted heir (s.v.).
PIR2 M-451; Korpela (1987) 167 #69.
PTK
Menaikhmos of Prokonessos (365 – 325 BCE)
A pupil of E and associate of P, who together with his brother D
and A H, also an associate of Plato, “made the whole of
geometry more perfect (or complete)” (P In Eucl. p. 67.8–12 Fr.). Proklos associates
Menaikhmos with metamathematical questions, telling us that he gave an account of the
word “element” ( pp. 72.23–73.14 Fr.), that the mathematicians “around” him considered
everything proved in geometry to be a “problem” rather than a theorem – although he
allowed that some problems seek to determine a feature of some defined thing – ( p. 78.8– 13
Fr.), and that the mathematicians around him and Amphinomos dealt with questions con-
cerning the convertibility of propositions of the form “All A are B.”
But Proklos also reports ( p. 111.20–23 Fr.) that Menaikhmos “conceived” (epinoeisthai,
apparently meaning “discovered”) the conic sections and cites a line of poetry by E-
: “Don’t section the cone with the triads (standardly taken to be the curves we
call parabola, hyperbola, and ellipse) of Menaikhmos.” The line is from an epigram which
Eratosthene ̄s attached to his mechanical solution to the problem of producing a cube
double a given one and urges the reader not to follow the solution of Menaikhmos. The
epigram is quoted by E in his commentary on A’ On the Sphere and
Cylinder ( p. 96.10–27 H.). In the commentary Eutokios describes a number of solutions,
including one which is ascribed to Menaikhmos (78.13–80.24 H.). In it, there is an analysis
and a synthesis. The procedure depends upon the reduction of cube duplication to the
problem of finding two mean proportionals between straight lines a and b, a reduction due
to H K ( p. 88.17–21 H.); for if b = 2a, and a:x :: x:y :: y:b, then, in
algebraic terms, (i) x^2 = ay, (ii) y^2 = 2ax, and (iii) xy = 2a^2 , and any two of these equations
yield that x^3 = 2a^3 , i.e., geometrically, the cube with side x is double the cube with side a.
Eutokios’ presentation of Menaikhmos’ solutions uses the terms “parabola” and “hyper-
bola,” which were introduced by A P to replace his predecessors’
“section of a right-angled cone” and “section of an obtuse-angled cone” (Eutokios, in
Apol. pp. 168.12–170.24 H.; cf. Pappos, Collection 7.30). We cannot determine exactly
how Menaikhmos proceeded and how much he knew about conics. Most scholars assume
that he knew quite a bit, but it has been argued that Menaikhmos only used point-wise
MELITIANUS