1 The Birth of Science - MSRI

efraction angle
50 ◦
40 ◦
30 ◦
20 ◦
10 ◦
0 ◦
8 ◦
10 ◦
15.5 ◦
20 ◦
22.5 ◦
30 ◦
3.2 Geodesy and Mathematical Geography 57
29 ◦
40 ◦
35 ◦
50 ◦
45.5 ◦
Ptolemy’s
40.5 ◦
60 ◦
70 ◦
50 ◦
80 ◦
actual
error (expanded scale)
3 ◦
2 ◦
1 ◦
incidence angle
FIGURE 3.1. Angle of refraction versus angle of incidence for the air-water interface.
show constant second differences for each interface (see Figure 3.1 for the
water-air case). His values match reality with remarkable accuracy in the
central range, but are far off at the extremes, especially when the incidence
angle is 80 ◦ . Clearly these values come from two procedures: careful experimentation
on the one hand, and subsequent extrapolation (or “correction”)
on the other, based on the a priori belief that the second differences
should be constant. The two procedures represent such disparate attitudes
toward experimental data that it is plausible to attribute them to different
people, possibly from distinct periods.
In Book V Ptolemy examines refraction between two media separated
by a plane or cylindrical surface. At that point the text stops and the translator
adds that the rest of the work could not be found. What did the missing
part of Book V contain? One might hope to learn this from the statement
of contents and purpose usually present at the beginning of such
works, but unfortunately Book I is missing too.
3.2 Geodesy and Mathematical Geography
Herodotus attributes to the Egyptians the introduction of geometry, in the
original sense of the measuring of the earth, specifying that it arose from page 90
the need to measure, for taxation purposes, by how much plots of land
were eroded by the Nile. 41 When Greek geometry embarked on its spectacular
course of development, its concrete applications, such as to sur-
41 Herodotus, Histories, II, 109.
Revision: 1.13 Date: 2002/10/16 19:04:00