Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/18

8 follows, that Rr is always to DR as the height to the length; and therefore that the body will move in the line DraF, which is the locus of the point r. Q. E. D.

Therefore Rr is equal to $$\scriptstyle \frac {DR \times AB}{N} - \frac {RDGT}{N}$$; and therefore if RT be produced to X, o that RX may be equal to $$\scriptstyle \frac {DR \times AB}{N}$$, that is, if the parallelogram ACPT be compleated, and DT cutting CP in Z be drawn, and RT be produced till it meets DT in X; Xr will be equal $$\scriptstyle \frac {RDGT}{N}$$, and therefore proportional to the time.

Whence if innumerable lines CR, or, which is the ame, innumerable lines ZX, be taken in a geometrical progreion; there will be as many lines Xr in an arithmetical progreion. And hence the curve DraF is eaily delineated by the Table of Logarithms.

If a Parabola be contructed to the vertex D, and the diameter DG, produced downwards, and its latus rectum is to 2DP as the whole reitance at the beginning of the motion to the gravitating force: the velocity with which the body ought to go from the place D, in the direction of the right line DP, o as in an uniform reiting medium to decribe the curve DraF, will be the ame as that with which it ought to go from the ame place D, in the direction of the ame right line DP, o as to decribe a Parabola in a non-resisting medium. For the latus rectum of this Parabola, at the very beginning of the motion, is $$\scriptstyle \frac {DV^2}{Vr}$$; and Vr is to $$\scriptstyle \frac {tGT}{N}$$ or $$\scriptstyle \frac {DR \times Tt}{2N}$$. But a right