ber and geometry, to more concrete phenomena
such as physics. The formula for refraction which Ptolemy helped to
shape, is geometrical in form. With him, as with the discoverer of the
right angle in a semicircle, the mind was working to find a general
ideal statement under which all similar occurrences might be grouped.
Observation, the collection of similar instances, measurement, are all
involved, and the general statement, law or form, when arrived at, is
found to link up other general truths and is then used as a
starting-point in dealing with similar cases in future. Progress in
science consists in extending this mental process to an ever-increasing
area of human experience. We shall see, as we go on, how in the concrete
sciences the growing complexity and change of detail make such
generalizations more and more difficult. The laws of pure geometry seem
to have more inherent necessity and the observations on which they were
originally founded have passed into the very texture of our minds. But
the work of building up, or, perhaps better, of organizing our
experience remains fundamentally the same. Man is throughout both
perceiving and making that structure of truth which is the framework of
progress.

Ptolemy's work brings us to the edge of the great break which occurred
in the growth of science between the Greek and the modern world. In the
interval, the period known as the Middle Ages, the leading minds in the
leading section of the human race were engaged in another part of the
great task of human improvement. For them the most incumbent task was
that of developing the spiritual consciousness of men for which the
Catholic Church provided an incomparable organization. But the interval
was not entirely blank on the scientific side. Our system of
arithmetical notation, including that invaluable item the cipher, took
shape during the Middle Ages at the hands of the Arabs, who appear to
have derived it in the main from India. Its value to science is an
excellent object-lesson on the importance of the details of form. Had
the Greeks possessed it, who can say how far they might have gone in
their applications of mathematics?

Yet in spite of this drawback the most permanent contribution of the
Greeks to science was in the very sphere of exact measurement where they
would have received the most assistance from a better system of
calculation had they possessed it. They founded and largely constructed
both plane and sph