Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/16

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6
Mathematical Principles
Book II.


Corol. 3. Alſo the differences of the ſpaces, which are deſcribed in equal differences of the times, decreaſe in the ſame geometrical progreſſion.

Corol. 4. The ſpace deſcribed by the body is the difference of two ſpaces, whereof one is as the time taken from the beginning of the deſcent, and the other as the velocity; which [ſpaces] alſo at the beginning of the deſcent are equal among themſelveſ.


Proposition IV. Problem II.

Suppoſing the force of gravity in any ſimilar medium to be uniform, and to tend perpendicular to the plane of the horizon; to define the motion of projectile therein, which ſuffers reſiſŧance proportional to its velocity.

Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. From the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, ſo that DA may be to AC as the reſiſtance of the medium ariſing from its motion upwards at the beginning, to the force of gravity: or (which comes to the ſame) ſo that the rectangle under DA and DP may be to that under AC and CP, as the whole reſiſtance at the beginning of the motion to the force of gravity. With the aſymptotes DC, CP deſcribe any Hyperbola GTBS cutting the perpendiculars DG, AB in G and B; compleat the parallelogram DGKC, and let its ſide GK cut AB in Q. Take a line N in the ſame ratio to QB as DC is in to CP; and from any point R of the right line DC, erect RT perpendicular to it, meeting the Hyperbola in T, and the right: lines EH, GK, DP in I, t and V; in that perpendicular