d
until Descartes and the mathematicians recommenced the work in the
seventeenth century. The second greatest contribution of the Greeks was
the statics and the conics of which Archimedes was the chief creator in
the third century B.C. In his work he gave the first sketch of an
infinitesimal calculus and in his own way performed an integration. The
third invaluable construction was the trigonometry by which Hipparchus
for the first time made a scientific astronomy possible. The fourth, the
optics of Ptolemy based on much true observation and containing an
approximation to the general law.

These are a few outstanding landmarks, peaks in the highlands of Greek
science, and nothing has been said of their zoology or medicine. In all
these cases it will be seen that the advance consisted in bringing
varying instances under the same rule, in seeing unity in difference, in
discovering the true link which held together the various elements in
the complex of phenomena. That the Greek mind was apt in doing this is
cognate to their idealizing turn in art. In their statues they show us
the universal elements in human beauty; in their science, the true
relations that are common to all triangles and all cones.

Ptolemy's work in optics is a good example of the scientific mind at
work.[81] The problem is the general relation which holds between the
angles of incidence and of refraction when a ray passes from air into
water or from air into glass. He groups a series of the angles with a
close approximation to the truth, but just misses the perception which
would have turned his excellent raw material into the finished product
of science. His brick does not quite fit its place in the building. His
formula _i_ (the angle of incidence) = _nr_ (the angle of refraction)
only fits the case of very small angles for which the sine is
negligible, though it had the deceptive advantage of including reflexion
as one case of refraction. He did not pursue the argument and make his
form completely general. Sin _i_ = _n_ sin _r_ escaped him, though he
had all the trigonometry of Hipparchus behind him, and it was left for
Snell and Descartes to take the simple but crucial step at the beginning
of the seventeenth century.

The case is interesting for more than one reason. It shows us what is a
general form, or law of nature in mathematical shape, and it also
illustrates the progress of science as it advances from the most
abstract conceptions of num