Discorsi Propositions |
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Discorsi Proposition2/23-pr-09-schol4 |

Quare admodum rationabile videbitur si, inquirentes quaenam contingant accidentia dum mobile post descensum per aliquod planum inclinatum reflectatur per planum aliquod acclive, accipiamus, gradum illum maximum in descensu acquisitum, idem per se perpetuo in ascendente plano servari; attamen in ascensu ei supervenire naturalem inclinationem deorsum, motum nempe ex {244} quiete acceleratum iuxta semper acceptam proportionem. | It seems altogether reasonable, therefore, if we wish to trace the future history of a body which has descended along some inclined plane and has been deflected along some plane inclined upwards, for us to assume that the maximum speed acquired during descent is permanently maintained during the ascent. In the ascent, however, there supervenes a natural inclination downwards, namely, a motion which, starting from rest, is accelerated at the {244} usual rate. |

Quod si forte haec intelligere fuerit subobscurum, clarius per aliquam delineationem explicabitur. | If perhaps this discussion is a little obscure, the following figure will help to make it clearer. |

Intelligatur itaque, factum esse descensum per planum declive AB, ex quo per aliud acclive BC continuetur motus reflexus, et sint primo, plana aequalia, et ad aequales angulos super horizontem GH elevata: constat iam, quod mobile ex quiete in A descendens per AB, gradus {10} acquirit velocitatis iuxta temporis ipsius incrementum; gradum vero in B esse maximum acquisitorum, et suapte natura immutabiliter impressum, sublatis scilicet causis accelerationis novae aut retardationis: accelerationis, inquam, si adhuc super extenso plano ulterius progrederetur; retardationis vero, dum super planum acclive BC fit reflexio: in horizontali autem GH aequabilis motus, iuxta gradum velocitatis ex A in B acquisitae, in infinitum extenderetur; esset autem talis velocitas, ut in tempore aequali tempori descensus per AB in horizonte conficeret spatium duplum ipsius AB.Modo fingamus, idem mobile eodem celeritatis gradu aequabiliter moveri per planum BC, adeo ut, {20} etiam in hoc, tempore aequali tempori descensus per AB conficeret super BC extenso spatium duplum ipsius AB; verum intelligamus, statim atque ascendere incipit, ei suapte natura supervenire illud idem quod ei contigit ex A super planum AB, nempe descensus quidam ex quiete secundum gradus eosdem accelerationis, vi quorum, ut in AB contigit, tempore eodem tantumdem descendat in plano reflexo, quantum descendit per AB: manifestum est, quod ex eiusmodi mixtione motus aequabilis ascendentis et accelerati descendentis perducetur mobile ad terminum C per planum BC iuxta eosdem velocitatis gradus, qui erunt aequales. | Let us suppose that the descent has been made along the downward sloping plane AB, from which the body is deflected so as to continue its motion along the upward sloping plane BC; and first let these planes be of equal length and placed so as to make equal angles with the horizontal line GH. Now it is well known that a body, starting from rest at A, and descending along AB, acquires a speed which is proportional to the time, which is a maximum at B, and which is maintained by the body so long as all causes of fresh acceleration or retardation are removed; the acceleration to which I refer is that to which the body would be subject if its motion were continued along the plane AB extended, while the retardation is that which the body would encounter if its motion were deflected along the plane BC inclined upwards; but, upon the horizontal plane GH, the body would maintain a uniform velocity equal to that which it had acquired at B after fall from A; moreover this velocity is such that, during an interval of time equal to the time of descent through AB, the body will traverse a horizontal distance equal to twice AB.Now let us imagine this same body to move with the same uniform speed along the plane BC so that here also during a time-interval equal to that of descent along AB, it will traverse along BC extended a distance twice AB; but let us suppose that, at the very instant the body begins its ascent it is subjected, by its very nature, to the same influences which surrounded it during its descent from A along AB, namely, it descends from rest under the same acceleration as that which was effective in AB, and it traverses, during an equal interval of time, the same distance along this second plane as it did along AB; it is clear that, by thus superposing upon the body a uniform motion of ascent and an accelerated motion of descent, it will be carried along the plane BC as far as the point C where these two velocities become equal. |

Discorsi Propositions |
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Discorsi Proposition2/23-pr-09-schol4 |