Discorsi Propositions
2/21-th-14
Discorsi Proposition
2/21-th-14

THEOREMA IV, PROPOSITIO XXI.THEOREM XIV, PROPOSITION XXI
Si in perpendiculo fiat casus ex quiete, in quo a principio lationis sumatur pars, quovis tempore peracta, post quam sequatur motus inflexus per aliquod planum utcunque inclinatum, spatium quod in {20} tali plano conficitur in tempore aequali tempori casus iam peracti in perpendiculo, ad spatium iam peractum in perpendiculo, maius erit quam duplum minus vero quam triplum. If, on the path of a body falling vertically from rest, one lays off a portion which is traversed in any time you please and whose upper terminus coincides with the point where the motion begins, and if this fall is followed by a motion deflected along any inclined plane, then the space traversed along the inclined plane, during a time-interval equal to that occupied in the previous vertical fall, will be greater than twice, and less than three times, the length of the vertical fall.
Infra horizontem AE sit perpendiculum AB, in quo ex principio A fiat casus, cuius sumatur quaelibet pars AC; inde ex C inclinetur {30} utcunque planum CG, super quo post casum in AC continuetur motus: dico, quod {239} spatium tali motu peractum per CG in tempore aequali tempori casus per AC, est plus quam duplum, minus vero quam triplum, eiusdem spatii AC. Ponatur enim CF aequalis AC, et extenso plano GC usque ad horizontem in E, fiat ut CE ad EF, ita FE ad EG. Si itaque ponatur, tempus casus per AC esse ut linea AC, erit CE tempus per EC, et CF, seu CA, tempus motus per CG: ostendendum itaque est, spatium CG ipsius CA maius esse quam duplum, minus vero quam triplum. Cum enim sit ut CE ad EF, ita FE ad EG, erit etiam ita CF ad FG; minor autem est EC quam EF; quare et CF minor erit quam FG, et GC maior quam dupla ad FC, seu AC. Cumque {10} rursus FE minor sit quam dupla ad EC (est enim EC maior CA, seu CF), erit quoque GF minor quam dupla ad FC, et GC minor quam tripla ad CF, seu CA: quod erat demonstrandum. Poterat autem universalius idem proponi: quod enim accidit in perpendiculari et plano inclinato, contingit etiam si post motum in plano quodam inclinato inflectatur per magis inclinatum, ut videtur in altera figura; eademque est demonstratio.Let AB be a vertical line drawn downwards from the horizontal line AE, and let it represent the path of a body falling from rest at A; choose any portion AC of this path.Through C draw any inclined plane, CG, along which the motion is continued after fall through AC. Then, I say, that the distance {239} traversed along this plane CG, during the time-interval equal to that of the fall through AC, is more than twice, but less than three times, this same distance AC.Let us layoff CF equal to AC, and extend the plane GC until it meets the horizontal in E; choose G such that CE:EF = EF:EG. If now we assume that the time of fall along AC is represented by the length AC, (Condition 2/03-th-03-cor) then CE will represent the time of descent along CE, (Condition 2/11-th-11) while CF, or CA, will represent the time of descent along CG. It now remains to be shown that the distance CG is more than twice, and less than three times, the distance CA itself. Since CE:EF = EF:EG, it follows that CE:EF = CF:FG; but EC

Discorsi Propositions
2/21-th-14
Discorsi Proposition
2/21-th-14