nt, 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.

Were there space, we might go on to point out how the conception of the
heavens as a revolving hollow sphere, the discovery of the