* Corporis in data Trajectoria Parabolica moventis, invenire locum ad tempus assignatum.
Sit S umbilicus & A vertex principalis Parabolæ, sitq; 4AS × M area Parabolica APS, quæ radio SP, vel post excessum corporis de vertice descripta fuit, vel ante appulsum ejus ad verticem describenda est.
Innotescit area illa ex tempore ipsi proportionali. Biseca AS in G, erigeq; perpendiculum GH æquale 3M, & circulus centro H, intervallo HS descriptus secabit Parabolam in loco quæsito P.
Nam demissa ad axem perpendiculari PO, est HGq. + GSq. (= HSq. = HPq. = GOq. + PO - HGq.) = GOq. + HGq. - 2HG × PO + POq. Et deleto utrinq; HGq. fiet GSq. = GOq. - 2HG × PO + POq. seu 2HG × PO (= GOq. + POq. - GSq. = AOq. - 2GAO + POq.) = AOq. + ¾POq.
Pro AOq. scribe AO × POq. ÷ 4AS, & applicatis terminis omnibus ad 3PO, ductisq; in 2AS, fiet 4/3GH × AS (= 1/6AO × PO + ½AS × PO = {AO + 3AS} ÷ 6 × PO = {4AO - 3SO} ÷ 6 × PO = areæ APO - SPO) = areæ APS.
Sed GH erat 3M, & inde Page 105 4/3HG × AS est 4AS × M. Ergo area APS æqualis est 4AS × M.
Q. E. D.
Corol. 1. Hinc GH est ad AS, ut tempus quo corpus descripsit arcum AP ad tempus quo corpus descripsit arcum inter verticem