292 10. EUCLIDEAN GEOMETRYbe added (first to first and third to third), that is, if á : b :: c : d and e :&::/: d,
then (a+e) : b :: (c+f) : d. But he did not think of the second and fourth terms in a
proportion as denominators or try to get a common denominator. For multiplication
of ratios, Euclid gives three separate definitions. In Book 5, Definition 9, he defines
the duplicate (which we would call the square) of the ratio a : b to be the ratio á : c
if b is the mean proportional between a and c, that is, á : b :: b : c. Similarly, when
there are four terms in proportion, as in the problem of two mean proportionals,
so that a : b :: b : c :: c : d, he calls the ratio a : d the triplicate of á : b. We would
call it the cube of this ratio. Not until Book 6, Definition 5 is there any kind of
general definition of the product of two ratios. Even that definition is not in all
manuscripts and is believed to be a later interpolation. It goes as follows: A ratio
is said to be the composite of two ratios when the sizes in the two ratios produce
something when multiplied by themselves.^1 * This rather vague definition is made
worse by the fact that the word for composed (sygkeimena) is simply a general word
for combined. It means literally lying together and is the same word used when two
lines are placed end to end to form a longer line. In that context it corresponds
to addition, whereas in the present one it corresponds (but only very loosely) to
multiplication. It can be understood only by seeing the way that Euclid operates
with it. Given four lines a, 6, c, and d, to form the compound ratio a : b.c : d,
Euclid first takes any two lines m and ç such that a : b :: m : n. He then finds a
line ñ such that n:p::c:d and defines the compound ratio á : b.c : d to be m : p.
There is some arbitrariness in this procedure, since m could be any line. A
modern mathematician looking at this proof would note that Euclid could have
shortened the labor by taking m = a and ç = b. The same mathematician would
add that Euclid ought to have shown that the final ratio is the same independently
of the choice of m, which he did not do. But one must remember that the scholarly
community around Euclid was much more intimate than in today's world; he did not
have to write a "self-contained" book. In the present instance a glance at Euclid's
Data shows that he knew what he was doing. The first proposition in that book says
that "if two magnitudes A and Β are given, then their ratio is given." In modern
language, any quantity can be replaced by an equal quantity in a ratio without
changing the ratio. The proof is that if A = Ã and Β = Ä, then A : Ã :: Β : Ä,
and hence by Proposition 16 of Book 5 of the Elements, A : Β :: Ã : Ä. The second
proposition of the Data draws the corollary that if a given magnitude has a given
ratio to a second magnitude, then the second magnitude is also given. That is, if two
quantities have the same ratio to a given quantity, then they are equal. From these
principles, Euclid could see that the final ratio m : ñ is what mathematicians now
call "well-defined," that is, independent of the initial choice of m.^15 The first use
made of this process is in Proposition 23 of Book 6, which asserts that equiangular
parallelograms are in the compound ratio of their (corresponding) sides.
With the departure of Eudoxus for Cnidus, we can bring to a close our discus-
sion of Plato's influence on mathematics. If relations between Plato and Eudoxus
were less than intimate, as Diogenes Laertius implies, Eudoxus may have drawn
off some of Plato's students whose interests were more scientific (in modern terms)
(^14) I am aware that the word "in" here is not a literal translation, since the Greek has the genitive
case—the sizes of the two ratios. But I take of here to mean belonging to, which is one of the
meanings of the genitive case.
(^15) A good exposition of the purpose of Euclid's Data and its relation to the Elements was given
by Il'ina (ZOOS), elaborating a thesis of I.G. Bashmakova.