To briefly illustrate their respective powers;--arithmetic can express in one formula the value of a _particular_ tangent to a _particular_ curve; algebra can express in one formula the values of _all_ tangents to a _particular_ curve; transcendental analysis can express in one formula the values of _all_ tangents to _all_ curves. Just as arithmetic deals with the common properties of lines, areas, bulks, forces, periods; so does algebra deal with the common properties of the numbers which arithmetic presents; so does transcendental analysis deal with the common properties of the equations exhibited by algebra. Thus, the generality of the higher branches of the calculus, when compared with the lower, is the same kind of generality as that of the lower branches when compared with geometry or mechanics. And on examination it will be found that the like relation exists in the various other cases above given. Having shown that M. Comte's alleged law of progression does not hold among the several parts of the same science, let us see how it agrees with the facts when applied to separate sciences. "Astronomy," says M. Comte, at the opening of Book III., "was a positive science, in its geometrical aspect, from the earliest days of the school of Alexandria; but Physics, which we are now to consider, had no positive character at all till Galileo made his great discoveries on the fall of heavy bodies." On this, our comment is simply that it is a misrepresentation based upon an arbitrary misuse of words--a mere verbal artifice. By choosing to exclude from terrestrial physics those laws of magnitude, motion, and position, which he includes in celestial physics, M. Comte makes it appear that the one owes nothing to the other. Not only is this altogether unwarrantable, but it is radically inconsistent with his own scheme of divisions. At the outset he says--and as the point is important we quote from the original--"Pour la _physique inorganique_ nous voyons d'abord, en nous conformant toujours a l'ordre de generalite et de dependance des phenomenes, qu'elle doit etre partagee en deux sections distinctes, suivant qu'elle considere les phenomenes generaux de l'univers, ou, en particulier, ceux que presentent les corps terrestres. D'ou la physique celeste, ou l'astronomie, soit geometrique, soit mechanique; et la physique terrestre."