1 The Birth of Science - MSRI

260 10. Lost Science
The difference between the abstract meaning that semeion had to the page 324
Greek mathematicians and the concrete notion of signum (a sign on the
paper) that a Roman writer like Vitruvius may substitute for it can be important
in the interpretation of passages of astronomical arguments. The
same Latin term signum can in fact mean a sign of the zodiac (a meaning
expressed in Greek by a different word, ). In all cases the Latin term
maintained a concrete meaning; of course the context would generally allow
one to tell which signum was meant: for example, a sign on paper or
a sign of the zodiac. But we see the problems that could arise in the case
of a Greek scientific work making a statement about a ¨ in astronomical
space. A Roman reader might think that, since in Latin a signum
on the sky is a zodiacal sign, the same is true of a semeion on the sky; and
even if he was aware that this is not so, he would lack the linguistic tools
needed to correctly translate the text into Latin. As to the association between
letter and signum, since to Vitruvius it is at most a matter of spatial
proximity, it cannot be easily extended to the “signs” in the sky, since there
are no letters there. Since in Greek the letters of the alphabet are used also
as ordinal numerals, it would be natural in this case to assume this latter
meaning for the letters and to intrepret expressions such as semeion B
(“point B”) as “the second sign”.
Assuming that the source really did refer to the points in a geometric
construction, can the diagram underlying Vitruvius’ passage be reconstructed?
Note first that Vitruvius mentions triangles with equal sides (paribus lateribus).
This might seem to refer to equilateral triangles, but it can equally
mean an isosceles triangle, especially since the Greek adjective “isosceles”
() means “equal-sided” and Vitruvius may have translated
it componentwise instead of adopting the Greek word itself as became
customary later. The surrounding sentence, “its rays extend out . . . using
lines in the shape of a triangle with equal sides”, suggests triangles
whose equal sides are formed by rays originating in the sun, and so favors
the meaning “isosceles”. More than one such triangle, each sharing
one ray-side with the next, would all have the same side length. Thus the page 325
geometric construction might involve adjacent isosceles triangles fanning
out from the sun; note that the rays of the sun (radii in Latin) are also radii
of a circle, since they are all of the same length.
The use of alter (“alterius trigoni”) indicates that there were two triangles
being considered. Thus we obtain Figure 10.1, left, where H marks the
position of the Sun.
The last position considered for the planet is, in Vitruvius’ words, at the
fifth signum, which is a signum in the second triangle. In our interpretation,
the original meaning of this statement was that the planet, at the end of
Revision: 1.11 Date: 2003/01/06 02:20:46