> Dear List members,
> I have a student doing a project about the argument as to
> whether
> Euclid was really doing geometrical algebra, and she would like to ask
> any
> Greek scholars amongst you about the following (fine?) point of
> translation.
>
> ============
> ==================================================================
> I have just finished a translation of Book II of Euclid's Elements for
> a
> research project at Canterbury university. I have a couple of queries
>
> regarding the translation for anybody who is familiar with the book:
>
> (1) How do you handle the phrase
>
> to upo twn A, BC periexomenon orthogwnion
>
> Is it 'the rectangle contained by A, BC'
> or is 'the rectangle contained by A and BC' okay?
>
> I just don't understand whether Euclid means us to think of the two
> line
> segments as being 'one thing' (I have here rectangle formation in
> mind) or
> whether they are two seperate things (and hence the latter translation
> is
> better. ( I have here in mind the arguments for and against
> geometrical
> algebra.....)
>
> (2) Also what is the general feeling about Heath's translation?
>
> ===============================================================
> ===============
>
> Thanks for your help.
> John Hannah
Dear John,
The point, I think, is that the rectangle is ONE thing contained by
TWO lines. In this, it is, perhaps, useful to compare the phrase
to upo twn A,BC periexomenwn orthogwnion
with a phrase such as this (from Apollonius's CONICA):
ti shmeion...to D entos ths upo twn asumptwtwn
(a point D in the angle contained by the asymptotes)
So, like the rectangle, there is ONE angle contained by TWO asymptotes.
While it is by no means a sin to write 'the rectangle contained by A
and BC', I generally translate 'to upo twn A,BC...' as 'the rectangle
contained by A,BC' to stay as close to the Greek text as possible. The
problem with the latter translation, especially in connection with the
issue of 'geometrical algebra', is that it feeds the temptation (to
which Heath quite willingly succumbed) of thinking not of the rectangle
contained by A and BC but of the product of the lengths of A and BC.
You probably know that even Heiberg will write a phrase such as
kai to upo twn KZ,ZH ara ison esti tw apo ths ZQ
as
quare etiam KZ x ZH=ZQ2
About Heath's translations, his Euclid, on the whole, is very good and
faithful to the Greek text. In his Apollonius and Archimedes, on the
other hand, he took far to many liberties -- to the extent that it is
moot even calling his translations translations.
Yours,
Michael Fried