Page:Newton's Principia (1846).djvu/261

 ever approach to the asymptote PC. And its velocity in any point r will be as the tangent rL to the curve. Q.E.I.

For N is to QB as DC to CP or DR to RV, and therefore RV is equal to $$\scriptstyle \frac{DR\times QB}{N}$$, and Rr (that is, RV -Vr, or $$\scriptstyle \frac{DR\times QB-tGT}{N}$$) is equal to $$\scriptstyle \frac{DR\times AB-RDGT}{N}$$. Now let the time be expounded by the area RDGT and (by Laws, Cor. 2), distinguish the motion of the body into two others, one of ascent, the other lateral. And since the resistance is as the motion, let that also be distinguished into two parts proportional and contrary to the parts of the motion: and therefore the length described by the lateral motion will be (by Prop. II, Book II) as the line DR, and the height (by Prop. III, Book II) as the area DR $$\scriptstyle \times$$ AB - RDGT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR $$\scriptstyle \times$$ AQ, and therefore that line Rr (or $$\scriptstyle \frac{DR\times AB-DR\times AQ}{N}$$) will then be to DR as AB - AQ or QB to N, that is, as CP to DC; and therefore as the motion upwards to the motion lengthwise at the beginning. Since, therefore, Rr is always as the height, and DR always as the length, and Rr is to DR at the beginning as the height to the length, it 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.

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

. 2. Whence if innumerable lines CR, or, which is the same, innumerable lines ZX, be taken in a geometrical progression, there will be as many lines Xr in an arithmetical progression. And hence the curve DraF is easily delineated by the table of logarithms.

. 3. If a parabola be constructed to the vertex D, and the diameter DG produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the notion 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, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the same place D in the direction of the same right line DP, so as to describe