Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/17

take Vr equal to $\scriptstyle {\frac {tGT}{N}}$ or, which is the ſame thing, take Rr equal to $\scriptstyle {\frac {GTIE}{N}}$; and the projectile in the time DRTG will arrive at the point r, deſcribing the curve line DraF, the locus of the point r; thence it will come to its greateſt height a in the perpendicular AB; and afterwards 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 $\scriptstyle {\frac {DR\times AB-EDGT}{N}}$ Now let the time be expounded by the area RDGR 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. 2. Book 2.) as the line DR, and the height (by Prop. 3. Book 2.) as the area DR x AB - RDGTT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR x 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 length-wiſe 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