CHAPTER XXI. THEORY OF APPROXIMATION.

according to which a projectile, such as a cannon ball, moves through the atmosphere is a problem very imperfectly solved at the present day, but in which many successive advances have been made. So little was known concerning the subject three or four centuries ago that a cannon ball was supposed to move at first in a straight line, and after a time to be deflected into a curve. Tartaglia ventured to maintain that the path was curved throughout, as by the principle of continuity it should be; but the ingenuity of Galileo was required to prove this opinion, and to show that the curve was approximately a parabola. It is only, however, under forced hypotheses that we can assert the path of a projectile to be truly a parabola: the path must be through a perfect vacuum, where there is no resisting medium of any kind; the force of gravity must be uniform and act in parallel lines; or else the moving body must be either a mere point, or a perfect centrobaric body, that is a body possessing a definite centre of gravity. These conditions cannot be really fulfilled in practice. The next great step in the problem was made by Newton and Huyghens, the latter of whom asserted that the atmosphere would offer a resistance proportional to the velocity of the moving body, and concluded that the path would have in consequence a logarithmic character. Newton investigated in a general manner the subject of resisting media, and came to the conclusion that the resistance is more nearly proportional to the square of the velocity. The subject then fell into the hands of Daniel Bernoulli, who pointed out the enormous resistance of the air in cases of rapid movement, and calculated that a cannon ball, if fired vertically in a vacuum, would rise eight times as high as in the atmosphere. In recent times an immense amount both of theoretical and experimental investigation has been spent upon the subject, since it is one of importance in the art of war. Successive approximations to the true law have been made, but nothing like a complete and final solution has been achieved or even hoped for.‍⁸