’Progression from the simple to the composite, says M. Petin, is the universal law.  In the works of nature, the action of this law is everywhere visible; and man, in his works, follows the path thus consecrated by the footsteps of the Creator.  Thus we find, he continues, that the point multiplied by itself produces the line; the line, in like manner, produces the plane; and the plane, the cube; an ascending series, which he conceives to have its exact analogy in that furnished by the earth, the water, and the air, considered as media of locomotion.  In other words, the point, or primary germ of extension, corresponds, according to the theory of M. Petin, with the fulcrum, or primary condition of locomotion; the line, first and simplest form of extension, corresponds with locomotion on the surface of the earth, where, owing to topographic inequalities, and other obstacles, locomotion can take place only in its first and simplest mode—­namely, in a linear direction; the plane, produced by the movement of the line, and constituting a higher term of superficial development, corresponds with locomotion upon the water, whose unencumbered surface, which can be traversed in every direction, presents a locomotive medium, the facilities of which, compared with those offered by the surface of the earth, increase in the ratio of the difference of extension between the line and the plane.

’The cube, product of the plane multiplied by itself, corresponds with locomotion in the air, where the aeronaut, being surrounded on every side by fulcra furnished by the various strata of the atmosphere, moves at will in every direction; pressing on the higher strata in ascending, on the lower in descending, on the lateral in turning to the right or to the left, and thus commanding a sphere of locomotion whose extent and facilities, compared with those afforded by the water, are as the cube to the plane.

’Aerial navigation being thus, according to his theory, the highest form of locomotion, M. Petin considers himself as justified in assuming, a priori, that this mode of transportation will offer facilities superior to those of every other in point of safety, speed, power, and cheapness; but on condition of its being carried into effect upon a scale commensurate with the vastness of its field and the importance of its results.

’To convince ourselves that such is really the intention of Providence, and that balloons are destined to transport the heaviest loads, we have only, continues M. Petin, to examine the law which presides over the development of spheric bodies; the surface of a sphere being represented by the square of the radius, while its contenance, or containing power, is represented by the cube of the radius.  In other words, if we increase the diameter of a sphere three times, although we increase its surface only nine times, we increase its containing power twenty-seven times.  Therefore, by constructing balloons on a very large