Discorsi Propositions 2/17-pr-04 Discorsi Proposition2/17-pr-04

 PROBLEMA IV, PROPOSITIO XVII. PROBLEM IV, PROPOSITION XVII Dato perpendiculo et plano ad ipsum inflexo, in dato plano partem signare, in qua post casum in perpendiculo fiat motus tempore aequali {10} ei, quo mobile datum perpendiculum ex quiete confecit. Given a vertical line and an inclined plane, it is required to lay off a distance along the given plane which will be traversed by a body, after fall along the perpendicular, in the same time-interval which is needed for this body to fall from rest through the given perpendicular. Sit perpendiculum AB, et ad ipsum planum inflexum BE: oportet, in BE spatium signare, per quod mobile post casum in ab moveatur tempore aequali ei, quo ipsum perpendiculum ab ex quiete confecit. Sit horizontalis linea AD, cui occurrat in D planum extensum, et accipiatur FB aequalis BA, et fiat ut BD ad DF, ita FD ad DE: dico, tempus per BE post casum in AB aequari tempori {20} per AB ex quiete in A. Si enim intelligatur, AB esse tempus per AB, erit DB tempus per DB; cumque sit ut BD ad DF, ita FD ad DE, erit DF tempus per totum planum DE, et BF per partem BE ex D: sed tempus per BE post DB est idem ac post AB: ergo tempus per BE post AB erit BF, aequale scilicet tempori AB ex quiete in A: quod erat propositum. Let AB be the vertical line and BE the inclined plane. The problem is to determine on BE a distance such that a body, after falling through AB, will traverse it in a time equal to that required to traverse the perpendicular AB itself, starting from rest. Draw the horizontal AD and extend the plane until it meets this line in D. Lay off FB equal to BA; and choose the point E such that BD:FD = DF:DE. Then, I say, the time of descent along BE, after fall through AB, is equal to the time of fall, from rest at A, through AB. For, if we assume that the length AB represents the time of fall through AB, (Condition 2/03-th-03-cor) then the time of fall through DB will be represented by the time DB; and since BD:FD = DF:DE, (Condition 202C) it follows that DF will represent the time of descent along the entire plane DE (Condition 2/11-th-11) while BF represents the time through the portion BE starting from rest at D; but the time of descent along BE after the preliminary descent along DB is the same as that after a preliminary fall through AB. Hence the time of descent along BE after AB will be BF which of course is equal to the time of fall through AB from rest at A.

 Discorsi Propositions 2/17-pr-04 Discorsi Proposition2/17-pr-04