Discorsi Propositions 2/11-th-11 Discorsi Proposition2/11-th-11

 THEOREMA XI, PROPOSITIO XI. THEOREM XI, PROPOSITION XI Si planum, in quo fit motus ex quiete, dividatur utcunque, tempus lationis per priorem partem ad tempus lationis per sequentem est ut {20} ipsamet prima pars ad excessum quo eadem pars superatur a media proportionali inter totum planum et primam eamdem partem. If a plane be divided into any two parts and if motion along it starts from rest, then the time of descent along the first part is to the time of descent along the remainder as the length of this first part is to the excess of a mean proportional between this first part and the entire length over this first part. Fiat latio per totam ab ex quiete in A, quae in C divisa sit utcumque; totius autem BA et prioris partis AC media sit proportionalis AF; erit CF excessus mediae FA super partem AC: dico, tempus lationis per AC ad tempus sequentis lationis per CB esse ut AC ad CF. Quod patet: nam tempus per AC ad tempus per totam AB est ut AC ad mediam AF; ergo, dividendo, tempus per AC ad tempus per reliquam CB erit ut AC ad CF. Si itaque intelligatur, tempus per AC esse ipsamet AC, tempus per CB erit CF: quod est {30} propositum. {230} Quod si motus non fiat per continuatam ACB, sed per inflexas ACD usque ad horizontem BD cui ex F parallela ducta sit FE, demonstrabitur pariter, tempus per AC ad tempus per reflexam CD esse ut AC ad CE. Nam tempus per AC ad tempus per CB est ut AC ad CF; tempus vero per CB post AC ad tempus per CD post eumdem descensum per AC demonstratum est esse ut CB ad CD, hoc est ut CF ad CE; ergo, ex aequali, tempus per AC ad tempus per CD erit ut AC linea ad CE. {10} Let the fall take place, from rest at A, through the entire distance AB which is divided at any point C; also let AF be a mean proportional between the entire length BA and the first part AC; then CF will denote the excess of the mean proportional FA over the first part AC. Now, I say, the time of descent along AC will be to the time of subsequent fall through CB as the length AC is to CF. This is evident, because (Condition 2/02-th-02-cor2) the time along AC is to the time along the entire distance AB as AC is to the mean proportional AF. Therefore, dividendo, the time along AC will be to the time along the remainder CB as AC is to CF. If we agree to represent the time along AC by the length AC then the time along CB will be represented by CF. Q. E. D. {230} In case the motion is not along the straight line ACB but along the broken line ACD to the horizontal line BD, and if from F we draw the horizontal line FE, it may in like manner be proved that the time along AC is to the time along the inclined line CD as AC is to CE. (Condition 211a) For the time along AC is to the time along CB as AC is to CF; (Condition )210 but it has already been shown that the time along CB, after the fall through the distance AC, is to the time along CD, after descent through the same distance AC, as CB is to CD, or, as CF is to CE; therefore, ex aequali, the time along AC will be to the time along CD as the length AC is to the length CE.

 Discorsi Propositions 2/11-th-11 Discorsi Proposition2/11-th-11