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 number 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?