A History of Mathematics From Mesopotamia to Modernity

62 A History ofMathematics

Proposition 2.The area of any circle is to the square on its diameter as 11 is to 14.

Proposition 3.The circumference of any circle exceeds three times the diameter by a quantity that is

less than one-seventh of the diameter but greater than ten parts in seventy-one.

It is clear that Proposition 2 is both wrongly placed (it depends on Proposition 3) and probably

not as Archimedes stated it (it claims as exact what is recognized in Proposition 3 to be an approxi-

mation). This of course added to the confusion of medieval readers, who tended to go for the more

usable Proposition 2; but at different times, all three parts were found useful.^3 The first states (in

our terms) that the areaAis^12 rC, whereris the radius andCis the circumference. The Greeks

sometimes worried, as we would not, whether this implied the necessary existence of a straight

line whose length was equal to the curved lineC. The second states thatA=^1114 ( 2 r)^2 (=^227 r^2 ).

The third also gives the approximation 3^17 for the ratio ofCto 2rwhich is still used after over 2000
years, and was gladly taken as the ‘right’ answer by calculators who had no use for Archimedes’
more precise formulation:

3

10

71

<

C

2 r

< 3

1

7

When we use the approximation 3^17 forπ, we are therefore indebted to Archimedes, although

we probably know nothing of his methods. These were interesting in themselves, however, as an

example of how he calculated—again, a more down-to-Earth procedure than the Platonic model of

mathematics would suggest. For the upper bound of 3^17 , for example, he starts with a circumscribed

hexagon (Fig. 3).

Archimedes assumes that any circumscribed figure has a greater perimeter than the circle, and

proceeds to find successively smaller ones, by bisecting angles (Fig. 4); he derives the rules for the

lengths of successive sides:

Rule 1: A′:B=A:B+C

Rule 2: A′^2 +B^2 =C′^2

These two rules make it possible to find the perimeter of polygons with 12, 24, 48, and 96 sides As
aids in calculation, he (a) uses a fractional approximation for

√

3, whose origin is unexplained, but

which is needed in the formula for the hexagon (see Exercise 3), (b) by successive applications of

the rule gets a rather complicated fraction for the 96-sided figure, and (c) shows that this fraction

is larger that 3^17. All this is a very interesting mixture of Euclid-style geometry and computation

with ratios of numbers; the way in which the fractions are written and manipulated recalls the

technique of the ancient Egyptians—unit fractions like^15 rather than sexagesimals. There are

repeated approximations to square roots which, while they seem correct, are not explained and so

have been the basis for much speculation. All this is just what we claimed, perhaps prematurely

(in the last chapter), Greek geometry avoided—the detailed engagement with numbers. This broad

statement, true for Euclid, Apollonius, and the ‘major’ works of Archimedes, is, as we will see,

not at all true for a variety of others. Is the ‘Measurement of the Circle’ intended as an aid for

practitioners, or simply as an exercise in technique? We have no indication. And while Archimedes

is always using the numbers (as a good geometer should) asratios, not as absolute measures of

length, the way is open for land-measurers to use them in other ways.

3. See Chapter 6 for Kepler’s attempt to construct an infinitesimal version, around 1600.