Reasoning and symbolism in Diophantus 343
Text 3:
Given two numbers, the fi rst greater than the second, and given the ratio of a third
number to unity, to fi nd a fourth number so that, added to the second and removed
from the fi rst, it makes the ratio of the second to the fi rst equal to the given ratio of
the third number to unity.
Let the fourth have been found. Since the second number together with the
fourth has to the fi rst lacking the fourth the ratio of the third to unity, make a fi ft h
number which is the third multiplied by the fi rst lacking the third multiplied by
the fourth. Th is fi fth number is equal to the second together with the fourth. So
the third multiplied by the fi rst lacking the third multiplied by the fourth is equal
to the second with the fourth.So the third multiplied by the fi rst is equal to the
second with the fourth with the third multiplied by the fourth, or to the second with
the fourth taken the third and one times. Th at is, the third multiplied by the fi rst,
lacking the second, is equal to the fourth taken the third and one times. Multiply
all by the third and one fraction. Th us the third multiplied by the fi rst, multiplied
by the third and one fraction, lacking the second multiplied by the third and one
fraction, is equal to the fourth taken the third and one times, multiplied by the third
and one fraction, which is the fourth. So the third multiplied by the fi rst, multiplied
by the third and one fraction, lacking the second multiplied by the third and one
fraction, is equal to the fourth.
So it shall be constructed as follows. Let one be added to the third to make
the sixth. Let the seventh be made to be the fraction of the sixth. Let the third be
multiplied by the fi rst and by the seventh to make the eighth.
Again, let the second be multiplied by the seventh to make the ninth.
Now let the ninth be taken away from the eighth, to make the fourth. I say that
the fourth produces the task.
[Here it is straightforward to add an explicit synthesis, showing that the ratio
obtains; for brevity’s sake, I omit this part.]
I suggest that we see Diophantus’ text with reference to texts 1 and 2 – of
which it must have been aware – and with reference to text 3 – which it
deliberately avoided. 14 Based on Høyrup’s work, 15 I assume that texts such
as text 1 were widespread in Mediterranean cultures from as far back as
14 Text 3 is my invention; perhaps not the most elegant one possible. All I did was to try to
write, in an idiom as close as possible to that of Diophantus, a general analysis of the problem,
following a line of reasoning hewing closely to the steps of the solution in Diophantus’
own solution. (Th is is not a mechanical translation: obviously, a particular solution such
as Diophantus’ underdetermines the general analysis from which it may be derived, since
any particular term may be understood as the result of more than one kind of general
confi guration.)
15 See, for instance, H2002: 362–7. It is fair to say that my summary is based not so much on this
reference from the book, as on numerous discussions, conference papers and preprints from