And now let it not be overlooked that in the observing at what time during the next year this extreme limit of the shadow was again reached, and in the inference that the sun had then arrived at the same turning point in his annual course, we have one of the simplest instances of that combined use of equal magnitudes and equal relations, by which all exact science, all quantitative prevision, is reached.  For the relation observed was between the length of the sun’s shadow and his position in the heavens; and the inference drawn was that when, next year, the extremity of his shadow came to the same point, he occupied the same place.  That is, the ideas involved were, the equality of the shadows, and the equality of the relations between shadow and sun in successive years.  As in the case of the scales, the equality of relations here recognised is of the simplest order.  It is not as those habitually dealt with in the higher kinds of scientific reasoning, which answer to the general type—­the relation between two and three equals the relation between six and nine; but it follows the type—­the relation between two and three, equals the relation between two and three; it is a case of not simply equal relations, but coinciding relations.  And here, indeed, we may see beautifully illustrated how the idea of equal relations takes its rise after the same manner that that of equal magnitude does.  As already shown, the idea of equal magnitudes arose from the observed coincidence of two lengths placed together; and in this case we have not only two coincident lengths of shadows, but two coincident relations between sun and shadows.

From the use of the gnomon there naturally grew up the conception of angular measurements; and with the advance of geometrical conceptions there came the hemisphere of Berosus, the equinoctial armil, the solstitial armil, and the quadrant of Ptolemy—­all of them employing shadows as indices of the sun’s position, but in combination with angular divisions.  It is obviously out of the question for us here to trace these details of progress.  It must suffice to remark that in all of them we may see that notion of equality of relations of a more complex kind, which is best illustrated in the astrolabe, an instrument which consisted “of circular rims, movable one within the other, or about poles, and contained circles which were to be brought into the position of the ecliptic, and of a plane passing through the sun and the poles of the ecliptic”—­an instrument, therefore, which represented, as by a model, the relative positions of certain imaginary lines and planes in the heavens; which was adjusted by putting these representative lines and planes into parallelism and coincidence with the celestial ones; and which depended for its use upon the perception that the relations between these representative lines and planes were equal to the relations between those represented.