50 A History ofMathematics
This complex definition underlies the statement (see Section 3) that circles are in the same ratio
as the squares on their diameters, for example. You can do much more with ratios, and you need
to. For example, if you increase the length of a rectangle by one ratio and its height by a second,
then you increase the area by a third, and so you have a relation of ratios which is similar to
multiplication. (But again, maybe we should avoid confusing them.) In the section ‘ Plato and
theMeno’, we saw that if side and height ratios were both what we would call√
2, the area
would increase by 2. For a full discussion, look at at least some of book V and (if you can face it)
book X, which introduces a whole classification of different kinds of irrationals, up to fourth roots
or ‘medials’. The exact reason for much of this theory is still debated, but the underlying theory is
more or less understood.
A different take on ratios comes from astronomy, since it shows how they tie in with numbers.
Suppose thatYis the length of a year (from spring equinox to spring equinox, say), andDis the
length of a day. We need to know the ratio ofYtoD(in our terms, the number of days in the year) to
have an efficient calendar, which all ancient—and modern—people needed. It complicates matters
that days are not all of the same length, but let us neglect that, as many of the ancients did. Our first
observation is that the ratio is greater than 365; we find this by using 365-day years and observing
that the date of the equinox gets noticeably earlier over a shortish period. We next introduce one
leap year in every four (one extra day), and get a better result, a ratio of( 4 × 365 + 1 )=1461 to 4.
After a few hundred years, we find this is too big, and so on. If we are astronomers, we continue, as
the Babylonians and later Greeks did, to find more or less accurate approximations for the length
of the year as fractions—sexagesimal or other kinds—and construct our calendars accordingly.^12
But if we are philosophers, we may well think that there is areallength of the year, and that this
fiddling with figures is beside the point. This, perhaps, is the meaning of Plato’s statement in the
Republic, which is often ridiculed, that one should study ideal stars and not simply what one sees:
We shall therefore treat astronomy, like geometry, as setting us problems for solution’, I said, ‘and ignore the visible
heavens, if we want to make a genuine study of the subject and use it to put the mind’s native wit to a useful purpose.
(PlatoRepublic, in F-G 2.E.3, p. 72)Socrates even draws on the example of the calendar which we have been looking at for
his argument:
[The astronomer] will think that the sky and the heavenly bodies have been put together by their maker as well as
such things may be; but he will also think it absurd to suppose that there is anything constant or invariable about the
relation of day to night, or of day and night to month, or month to year, or, again, of the periods of the other stars
to them and to each other. They are all visible and material, and it’s absurd to look for exact truth in them. (Plato
Republic, in F-G 2.E.3, p. 72)That the lengths of months were variable had long been known by Plato’s time; that the lengths
of days were too (since the four seasons had different lengths, for example), was perhaps a more
recent discovery. And the reaction to these facts in theRepublicis that they exhibit the failings
of material stars and planets, as opposed to the ideal counterparts which were designed by their
creator. If we put this together with the statement from thePhilebuswhich opens this chapter, we
could conclude that the ratio (e.g. ofYtoD, above) is what the real mathematician uses in his kind
of mathematics; while, in the other kind used by craftsmen, surveyors, and mere star-watchers,- This is, it has been pointed out to me, an oversimplification of the way ancient calendars were constructed. However, perhaps
it can serve as a guide to thought.