1 The Birth of Science - MSRI

58 3. Other Hellenistic Scientific Theories
veying and topography, were reclassified under the rubric of “geodesy”. 42
Unfortunately there is meager direct documentation about the evolution
of these techniques from the empirical stage, common to many ancient
civilizations, to Hellenistic science-based surveying and cartography. 43
The basic notion of triangulation — the graphical determination of the
distance to an inaccessible feature by comparing the direction of the lines
of sight from two points lying a known distance apart — is very old. It
was present in Hellenic mathematics from the beginning. 44 But the transformation
of this idea into effective surveying techniques had to await the
creation of instruments for viewing from a distance and the development
of trigonometry. The first documented use of trigonometric methods goes
back to the surviving astronomical work of Aristarchus of Samos, in the
first half of the third century B.C.; his calculations of the distance to the sun
and to the moon are clearly bold extensions of topographic triangulation
methods to an astronomical scale. 45
Geminus, in the first century B.C., describes geodesy by listing the tasks
involved in determining distances and differences in height through the
use of instruments such as rulers, plumb lines, squares and diopters for
looking through. 46 Vitruvius mentions only the diopter as an instrument
used in measuring differences in height, and another Greek instrument page 91
having the same purpose: the chorobate (a water-filled level). 47 We will
return later to the diopter described by Heron.
Indirect evidence about the development of surveying techniques can
be inferred from town planning. Greek town planning goes back to Hippodamus
of Miletus (fifth century B.C.), but it was probably in early Hellenistic
times that the establishment of many new and large cities with
their urban infrastructure stimulated the development of effective surveying
instruments. Such instruments would also have been needed for planning
works such as the citadel of Pergamum, which included not only
building construction but also hill terracing.
42
See, for example, Aristotle, Metaphysica, III, 2, 997b, 26–28, where geodesy is distinguished
from geometry by its concrete nature.
43
An important, if late, source on geodesy is the Dioptrics of Heron. See [Dilke] for a synthesis of
the available information.
44
Compare the Proclus passage mentioned in Chapter 2, footnote 9.
45
Aristarchus of Samos, On the sizes and distances of the sun and moon = [Heath: Aristarchus], Appendix.
Aristarchus uses trigonometric methods in the sense that he computes the ratios between
the sides of a triangle whose angles are known. Of course his values are not exact, nor can he use
tables of approximate values of trigonometric functions (which did not exist at the time), but he
determines small intervals which he can show contain the ratios that interest him.
46
This fragment from Geminus appears in [Heron: OO], vol. IV, 100, 4 – 102, 8.
47 Vitruvius, De architectura, VIII, v, 2–3.
Revision: 1.13 Date: 2002/10/16 19:04:00