But supposition aside, it is clear that the habitual recognition of these claims in their laws implied some prevision of social phenomena.  Even thus early there was a certain amount of social science.  Nay, it may even be shown that there was a vague recognition of that fundamental principle on which all the true social science is based—­the equal rights of all to the free exercise of their faculties.  That same idea of equality which, as we have seen, underlies all other science, underlies also morals and sociology.  The conception of justice, which is the primary one in morals; and the administration of justice, which is the vital condition of social existence; are impossible without the recognition of a certain likeness in men’s claims in virtue of their common humanity. Equity literally means equalness; and if it be admitted that there were even the vaguest ideas of equity in these primitive eras, it must be admitted that there was some appreciation of the equalness of men’s liberties to pursue the objects of life—­some appreciation, therefore, of the essential principle of national equilibrium.

Thus in this initial stage of the positive sciences, before geometry had yet done more than evolve a few empirical rules—­before mechanics had passed beyond its first theorem—­before astronomy had advanced from its merely chronological phase into the geometrical; the most involved of the sciences had reached a certain degree of development—­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.