a development
without which no progress in other sciences was possible.

Only noting as we pass, how, thus early, we may see that the progress of
exact science was not only towards an increasing number of previsions,
but towards previsions more accurately quantitative--how, in astronomy,
the recurring period of the moon's motions was by and by more correctly
ascertained to be nineteen years, or two hundred and thirty-five
lunations; how Callipus further corrected this Metonic cycle, by leaving
out a day at the end of every seventy-six years; and how these
successive advances implied a longer continued registry of observations,
and the co-ordination of a greater number of facts--let us go on to
inquire how geometrical astronomy took its rise.

The first astronomical instrument was the gnomon. This was not only
early in use in the East, but it was found also among the Mexicans; the
sole astronomical observations of the Peruvians were made by it; and we
read that 1100 B.C., the Chinese found that, at a certain place, the
length of the sun's shadow, at the summer solstice, was to the height of
the gnomon as one and a half to eight. Here again it is observable, not
only that the instrument is found ready made, but that Nature is
perpetually performing the process of measurement. Any fixed, erect
object--a column, a dead palm, a pole, the angle of a building--serves
for a gnomon; and it needs but to notice the changing position of the
shadow it daily throws to make the first step in geometrical astronomy.
How small this first step was, may be seen in the fact that the only
things ascertained at the outset were the periods of the summer and
winter solstices, which corresponded with the least and greatest lengths
of the mid-shadow; and to fix which, it was needful merely to mark the
point to which each day's shadow reached.

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 poi