The consequence of this principle is, that all bodies of similar figure, and equal density, at equal distances from the earth, fall with equal velocity; and if a body describes a space of 16ft. in the first second of time, it will, in the next second of time, fall _three_ times 16, or 48 feet, and thus will have fallen, from the time it first dropped, four times 16 feet, or 64 feet, because 4 is the square of 2, the time the body was falling. In the third second, it will fall 5 times 16 feet, or 80 feet, and these sums collectively, viz., 16 + 48 + 80 = 144 feet, the whole distance described by the falling body in three seconds of time. From this it is evident, that instead of moving in a straight line A. B., (plate 21, fig. 5.), the bullet will be drawn from that course. ~Parabolic theory.~ From the point C., draw C. F., equal to the space that the bullet may be supposed to fall in one second of time, then at the end of the first second of time the bullet will be at F., instead of at C., and will have moved in the direction A. F., instead of A. C.; at the end of the next second it will have fallen a total distance D. G., equal to four times C. F., thus the bullet will have fallen at the end of the third second a distance E. H., equal to nine times C. F., and it will have moved in the line A. F. G. H. instead of the straight line A. B., in which it would have moved, had it not been affected by the force of gravity. The curve A. H., is of the form called a Parabola, and hence the theory is called the “Parabolic Theory.” It is founded on the principle that the velocity given to the bullet by the explosion of the gunpowder is continued throughout its course, but this would only be true in vacuo, and is therefore of little value in calculating the real course of the bullet in the air. ON THE TIME TAKEN TO DRAW A BALL TO THE GROUND BY THE FORCE OF GRAVITY. ~If fired with axis parallel to the ground.~ 1st Case. Supposing a ball to be fired when the axis of the piece is parallel to the ground and 16 feet above it, then the projectile will strike the earth in the same length of time that it would have done, had it been rolled out of the muzzle, quite irrespective of the velocity with which it may have been propelled, or the consequent extent of range; that is to say the ball will have reached the point B., (plate 22, fig. 1.), in the same length of time that it would require to fall from the muzzle A., to the earth C.; _i. e._, in one second. 2nd Case. Were three guns to be fired at the same instant, with their three axes parallel to the horizon as before, and loaded respectively with ¹⁄₂ drm., 1 drm., and 1¹⁄₂ drm. of powder of the same strength, then, although the three initial velocities and three ranges would consequently all be different, yet the three balls would strike the ground at the same time, _i. e._ at the points B. B. B. in one second. (Plate 22, fig. 2.) ~If axis at an angle to the ground.~