The History of Mathematics: A Brief Course

234 9. MEASUREMENT

circumference of a circle is proportional to its diameter. That is, given two circles

with circumferences C, and diameters Di, i = 1,2, we have C\jCi = D\jO<i. We

shall call the ratio C/Z) the one-dimensional ð. The second formula implies that

if Äé and Ä2 are the areas of disks of radius rj and r-é and S\ and S2 are the

areas of squares of sides r\ and Ã2, then Äé/Ä 2 = S\ISi. We shall call the ratio

Ä/5 the two-dimensional ð, and similarly for volumes. We won't need the ð that

occurs in the formula for the area of a sphere, since everybody seemed to relate

that one to one of the others. Thus, we are dealing with several direct proportions

with different constants of proportionality. It is not obvious that these constants for

different dimensions have any simple relationship to one another. That fact requires

some digging in geometry to discover. Without the abstract concept of a constant

of proportionality, when a mathematician is seeking only numerical approximations

that accord with observation, there is no reason to suspect any connection between

these constants in different dimensions. To be sure, only a small amount of intuition

is required to establish the connection, as shown in Problems 9.13 and 9.20, but in

any discussion of supposed approximations to ð used in different cultures, we need

to keep in mind the dimension of the object being studied: Was it a circle, a disk,

a cone, a cylinder, a sphere, or a ball?

1. Egypt

Foreigners have been interested in the geometry of the Egyptians for a very long

time. In Section 109 of Book 2 of his History, the Greek historian Herodotus writes

that King Sesostris^1 dug a multitude of canals to carry water to the arid parts of

Egypt. He goes on to connect this Egyptian engineering with Greek geometry:

It was also said that this king distributed the land to all the Egyp-

tians, giving an equal quadrilateral farm to each, and that he got

his revenue from this, establishing a tax to be paid for it. If the

river carried off part of someone's farm, that person would come

and let him know what had happened. He would send surveyors

to remeasure and determine the amount by which the land had

decreased, so that the person would pay less tax in proportion to

the loss. It seems likely to me that it was from this source that

geometry was found to have come into Greece. For the Greeks

learned of the sundial and the twelve parts of the day from the

Babylonians.

The main work of Egyptian surveyors was measuring fields. That job cor-
responds well to the Latin word agrimensor, which means surveyor. Our word
surveyor comes through French, but has its origin in the Latin supervideo, meaning
I oversee. The equivalent word in Greek was used by Herodotus in the passage
above. He said that the king would send episkepsomenous kai anametresontas, lit-
erally overseeing and remeasuring men. The process of measuring a field is shown
in a painting from the tomb of an Egyptian noble named Menna at Sheikh Abd
el-Qurna in Thebes (Plate 7). Menna bore the title Scribe of the Fields of the
Lord of the Two Lands during the eighteenth dynasty, probably in the reign of
Amenhotep III or Thutmose IV, around 1400 BCE. His job was probably that of

(^1) There were several pharaohs with this name. Some authorities believe that the one mentioned
by Herodotus was actually Ramses II, who ruled from 1279 to 1212 BCE.