344 reviel netz
the third millennium bce (if not earlier), surviving, arguably, into our own
time. Th ey persisted almost exclusively as an oral tradition (sometimes,
perhaps, taking a ride for a couple of centuries on the back of written tra-
ditions of the type of text 2, and then proceeding along in the oral mode).
Such texts are called by Høyrup ‘lay algebra’.
Occasionally, lay algebra gets written and systematized (to a certain
extent) in an educational context. It then typically gets transformed into
texts such as text 2: the mere question-and-answer format of text 1 is trans-
formed into a set of indicative and imperative sentences put forward in
the rigid, authoritarian style typical of most written education prior to the
twentieth century. Th is is school algebra which has appeared several times
in Mediterranean cultures. One can mention especially its Babylonian
(early second millennium bce ), Greek (around the year zero) and Italian
(early second millennium ce ) forms. Th e Babylonian layer is important as
the fi rst school algebra of which we are aware; the Greek layer is important,
for our purposes, as providing, possibly, a context for Diophantus’ work; the
Italian layer is important, for our purposes, as providing a context for the
interest in Diophantus in the Renaissance.
Th e historical relationship between various school algebras is not clear
and it may be that they depend on the persistence of lay algebra no less than
on previous school algebras. It should be said that, while essentially based on
the written mode, this is a use of writing fundamentally diff erent from that
of elite literary culture. Writing is understood as a local, ad-hoc aff air. Th e
diff erence between the literacy of school algebra and the oralcy of lay algebra
is huge, in terms of their archaeology : clay tablets, papyri and libri d’abbaco
oft en survive, spoken words never do. But the clay tablets, papyri and libri
d’abbaco of school algebra do not belong to the world of Gilgamesh, Homer
or Dante. Th ey are not faithfully copied and maintained, and the assump-
tions we have for the stability of written culture need not hold for them.
What would happen when such materials become part of elite literate
culture itself? One hypothetical example is text 3: a reworking of the same
material, keeping as closely as possible to the features of elite literate Greek
mathematics (which was developed especially for the treatment of geom-
etry). Th is may be called, then – just so that we have a term – Euclidean
algebra.^16 When transforming the materials of lay and school algebra intothe author, and that as such summaries go it is likely to deviate in some ways from the way in
which Høyrup himself would have summed up his own position.
16 I use the term ‘Euclidean’ to refer to elite, literate mathematical practices. It is true that Euclid –
especially Books i and ii – could have been occasionally part of ancient education (the three
papyrus as fragments P. Mich. 3. 143, P. Berol. Inv 17469 and P. Oxy. 1.29, with defi nitions