He ̄rakleitos (Math.) (450 – 150 BCE?)
In commenting on A P’s
lost Neuseis, P (Coll. 7.128 Jones) quotes
He ̄rakleitos’ neusis-construction for the
square (see figure), which depends on the previ-
ous lemma (7.127 Jones), probably Pappos’ own
contribution but perhaps also his. Since, as
mentioned by Pappos ( p. 203.18–19 Jones), the
problem solves a particular case of Apollo ̄nios’
neusis in a rhombus ( problems 8 and 9 in his
Neuseis I, evoked by Pappos at 7.27 and again
7.126 Jones) and since Pappos criticizes
Apollo ̄nios’ (alleged) disdain for his predecessors’ efforts (7.34–35 Jones), He ̄rakleitos is most
probably earlier than Apollo ̄nios. E ( p. 228 H.) cites a He ̄rakleide ̄s who wrote a life
of A, ( probably the same as the He ̄rakleios mentioned in the first lines of
Eutokios’ commentary on the Conics) whom Heiberg and Knorr unconvincingly identified
with Archime ̄de ̄s’ companion mentioned in his On Spirals and with He ̄rakleitos, although
Eutokios’ He ̄rakleide ̄s is most probably a later biographer (Decorps 2000: n.5 p.10), perhaps
the same as the biographer and doxographer H K.
Jones (1986) 436; Knorr (1986) 294–302.
Alain Bernard
He ̄rakleitos of Ephesos (ca 510 – 490 BCE)
Known for his dark sayings and caustic invectives against his predecessors, He ̄rakleitos has
an anomalous position in the history of science. On the one hand, he seems to continue the
cosmological theories of the philosophers of Mile ̄tos; on the other, he calls into question
many of their assumptions and turns attention to human concerns. He ̄rakleitos criticized
famous wise men for pursuing information without a unifying theory. He seemingly
admired the natural philosophers for trying to explain the world in a systematic way. Stress-
ing the unity underlying the natural phenomena of the world, he criticized people for not
recognizing that unity.
He ̄rakleitos posited fire as the source of all things. The main forms of matter were fire,
water and earth. There is a constant interchange of matter such that portions of fire turn
into water, while portions of water return into fire and other portions transform into earth.
A given amount of one element becomes a proportionate amount of the other. Overall, the
relative amounts of the three kinds of element remain fixed in the world, so that a balance is
maintained. He ̄rakleitos first envisaged a “law of conservation of matter.” Although a given
portion of matter can change its form, it remains some kind of matter, and changes in a
law-like way.
Earlier cosmologists saw cosmic unity arising from a single stable substance such as water
or air, but He ̄rakleitos seems to have chosen fire for its instability. Fire is matter and also a
symbol of the changeability of matter. What makes the world possible is balanced cycles in
the interchange of elements: a hidden harmony. He ̄rakleitos seems to have identified this
harmony with the Logos, which is at once a message about the world and its underlying
structure. His own sayings are often cast in the form of riddles whose solution manifests the
unity of different meanings, as though in imitation of the world’s structure.
He ̄rakleitos’ neusis in a given square
A∆: the given length EZ should verge toward
point B. © Bernard
HE ̄RAKLEITOS (MATH.)