Page:Newton's Principia (1846).djvu/262

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 $\scriptstyle {\frac {tGT}{N}}$ $\scriptstyle {\frac {DR\times Tt}{2N}}$. But a right line, which, if drawn, would touch the hyperbola GTS in G, is parallel to DK, and therefore Tt is $\scriptstyle {\frac {CK\times DR}{DC}}$, and N is $\scriptstyle {\frac {QB\times DC}{CP}}$. And therefore Vr is equal to $\scriptstyle {\frac {DR^{2}\times CK\times CP}{2DC^{2}\times QB}}$, that is, (because DR and DC, DV and DP are proportionals), to $\scriptstyle {\frac {DV^{2}\times CK\times CP}{2DP^{2}\times QB}}$; and the latus rectum $\scriptstyle {\frac {DV^{2}}{Vr}}$ comes out $\scriptstyle {\frac {2DP^{2}\times QB}{CK\times CP}}$, that is (because QB and CK, DA, and AC are proportional), $\scriptstyle {\frac {2DP^{2}\times DA}{AC\times CP}}$, and therefore ist to 2DP as DP $\scriptstyle \times$ DA to CP $\scriptstyle \times$ AC; that is, as the resistance to the gravity.   Q.E.D.

Cor. 4. Hence if a body be projected from any place D with a given velocity, in the direction of a right line DP given by position, and the resistance of the medium, at the beginning of the motion, be given, the curve DraF, which that body will describe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given. Then cutting DC in A, so that CP $\scriptstyle \times$ AC may be to DP $\scriptstyle \times$ DA in the same ratio of the gravity to the resistance, the point A will be given. And hence the curve DraF is also given.

Cor. 5. And, on the contrary, if the curve DraF be given, there will be given both the velocity of the body and the resistance of the medium in each of the places r. For the ratio of CP $\scriptstyle \times$ AC to DP $\scriptstyle \times$ DA being given, there is given both the resistance of the medium at the beginning of the motion, and the latus rectum of the parabola; and thence the velocity at the beginning of the motion is given also. Then from the length of the tangent