Discorsi Propositions 2/19-pr-06 Discorsi Proposition2/19-pr-06

 {237} PROBLEMA VI, PROPOSITIO XIX. {237} PROBLEM VI, PROPOSITION XIX Dato in perpendiculo spatio quocunque a principio lationis peracto, datoque tempore casus, tempus reperire, quo aliud aequale spatium, ubicunque in eodem perpendiculo acceptum, ab eodem mobili consequenter conficiatur. Given the distance through which a body falls in a vertical line from rest and given also the time of fall, it is required to find the time in which the same body will, later, traverse an equal distance chosen anywhere in the same vertical line. Sit in perpendiculo AB quodcunque spatium AC ex principio lationis in A acceptum, cui aequale sit aliud spatium DB ubicunque acceptum, sitque datum tempus lationis per AC, sitque illud AC: oportet, reperire tempus {10} lationis per DB post casum ex A. Circa totam AB semicirculus describatur AEB, et ex C ad AB perpendicularis sit CE, et iungatur AE, quae maior erit quam EC; secetur EF ipsi EC aequalis: dico, reliquum FA esse tempus lationis per DB. Quia enim AE est media inter BA, AC, estque AC tempus casus per AC, erit AE tempus per totam AB; cumque CE media sit inter DA, AC (est enim DA aequalis ipsi BC), erit CE, hoc est EF, {20} tempus per AD; ergo reliqua AF est tempus per reliquam DB: quod est propositum. On the vertical line AB, lay off AC equal to the distance fallen from rest at A, also locate at random an equal distance DB. Let the time of fall through AC be represented by the length AC. It is required to find the time necessary to traverse DB after fall from rest at A. About the entire length AB describe the semicircle AEB; from C draw CE perpendicular to AB; join the points A and E; the line AE will be longer than EC; lay off EF equal to EC. Then, I say, the difference FA will represent the time required for fall through DB. For since AE is a mean proportional between BA and AC and since AC represents the time of fall through AC, (Condition 2/02-th-02-cor2) it follows that AE will represent the time through the entire distance AB. And since CE is a mean proportional between DA and AC (seeing that DA = BC) (Condition 2/02-th-02-cor2) it follows that CE, that is, EF, will represent the time of fall through AD. (Condition 2/11-th-11) Hence the difference AF will represent the difference DB. Q. E. D.

 Discorsi Propositions 2/19-pr-06 Discorsi Proposition2/19-pr-06