Discorsi Propositions
3/03-th-02
Discorsi Proposition
3/03-th-02

{281} SALV. Séguita l'Autore per incaminarci a intender quel che accaggia intorno all'impeto d'un mobile mosso pur d'un moto composto di 2, uno cioè orizontale ed equabile, e l'altro perpendicolare ma naturalmente accelerato, de i quali finalmente è composto il moto del proietto e si descrive la linea parabolica, in ciaschedun punto della quale si cerca di determinare quanto sia l'impeto del proietto.Per la cui intelligenza ci dimostra l'Autore il modo, o vogliàn dir metodo, di regolare e misurar cotale impeto sopra l'istessa linea nella quale si fa il moto del grave descendente con moto naturalmente accalerato, {10} partendosi dalla quiete, dicendo:{281} SALV. Our Author next undertakes to explain what happens when a body is urged by a motion compounded of one which is horizontal and uniform and of another which is vertical but naturally accelerated; from these two components results the path of a projectile, which is a parabola. The problem is to determine the speed [impeto] of the projectile at each point. With this purpose in view our Author sets forth as follows the manner, or rather the method, of measuring such speed [impeto] along the path which is taken by a heavy body starting from rest and falling with a naturally accelerated motion.
THEOREMA III, PROPOSITIO III.THEOREM III, PROPOSITION III
Fiat motus per lineam ab ex quiete in a, et accipiatur in ea quodlibet punctum c; et ponatur ipsamet ac esse tempus, seu temporis mensura, casus ipsius per spatium ac, nec non mensura quoque impetus seu momenti in puncto c ex descensu ac acquisiti.Let the motion take place along the line ab, starting from rest at a, and in this line choose any point c. Let ac represent the time, or the measure of the time, required for the body to fall through the space ac; let ac also represent the velocity [impetus seu momentum] at c acquired by a fall through the distance ac.
Modo sumatur in eadem linea ab quodcunque aliud punctum, utputa a, in quo determinandum est de impetu acquisito a mobili per {20} descensum ab, in ratione ad in impetum quem obtinuit in c, cuius mensura posita est ac.Ponatur as media proportionalis inter ba, ac: demonstrabimus, impetum in b ad impetum in c esse ut lineam sa ad ac.Sumantur horizontales cd, dupla ipsius ac, be vero dupla ba: constat, ex demonstratis, cadens per ac, conversum in horizonte cd, atque iuxta impetum {282} in c acquisitum motu aequabili delatum, conficere spatium cd aequali tempore, atque ipsum ac motu accelerato confecit; similiterque, be confici eodem tempore atque ab: sed tempus ipsius descensus ab est as: ergo horizontales be conficitur tempore as.Fiat ut tempus sa ad tempus ac, ita eb {10} ad bl; cumque motus per be sit aequabilis, erit spatium bl peractum tempore ac secundum momentum celeritatis in b: sed tempore eodem ac conficitur spatium cd secundum momentum celeritatis in c; momenta autem celeritatis sunt inter se ut spatia, quae iuxta ipsa momenta codem conficiuntur tempore: ergo momentum celeritatis in c ad momentum celeritatis in b est ut dc ad bl.Quia vero ut dc ad be, ita ipsarum dimidia, nempe ca ad ab; ut autem eb ad bl, ita ba ad as; ergo, ex aequali, ut dc ad bl, ita ca ad as: hoc est, ut momentum celeritatis in c ad momentum celeritatis in b, ita ca ad as, hoc est, tempus per ca ad tempus per ab.In the line ab select any other point b. The problem now is to determine the velocity at b acquired by a body in falling through the distance ab and to express this in terms of the velocity at c, the measure of which is the length ac. Take as a mean proportional between ct and ab. We shall prove that the velocity at b is to that at c as the length as is to the length ac. Draw the horizontal line cd, having twice the length of ac, and be, having twice the length of ba. (Condition 2/23-pr-09-schol1) It then follows, from the preceding theorems, that a body falling through the e distance ac, and turned so as to move along the horizontal cd with a uniform speed equal to that acquired on reaching c {282} will traverse the distance cd in the same interval of time as that required to fall with accelerated motion from a to c. Likewise be will be traversed in the same time as ba. (Condition 202C") But the time of descent through ab is as; hence the horizontal distance be is also traversed in the time as.Take a point I such that the time as is to the time ac as be is to bl; (Condition 1/01-th-01) since the motion along be is uniform, the distance bl, if traversed with the speed [momentum celeritatis] acquired at b, will occupy the time ac; but in this same time-interval, ac, the distance cd is traversed with the speed acquired in c. (Condition 1/02-th-02) Now two speeds are to each other as the distances traversed in equal intervals of time. Hence the speed at c is to the speed at as cd is to bl. But since de is to be as their halves, namely, as ca is to ba, and since be is to bl as be is to sa; it follows that de is to bl as ca is to sa. In other words, the speed at c is to that at b as ca is to sa, that is, as the time of fall through ab.
Patet itaque ratio mensurandi impetum seu celeritatis momentum super linea in qua fit motus descensus; qui quidem impetus ponitur augeri pro ratione temporis.The method of measuring the speed of a body along the direction of its fall is thus clear; the speed is assumed to increase directly as the time.
Hic autem, antequam ulterius progrediamur, praemonendum est, quod cum de motu composito ex aequabili horizontali et ex naturaliter accelerato deorsum futurus sit sermo (ex tali enim mixtione conflatur ac designatur linea proiecti, nempe parabola), necesse habemus definire aliquam {20} communem mensuram, iuxta quam utriusque motus velocitatem, impetum, seu momentum, dimetiri valeamus; cumque lationis aequabilis innumeri sint velocitatis gradus, quorum non quilibet fortuito, sed unus ex illis innumeris, cum gradu celeritatis per motum naturaliter acceleratum acquisito sit conferendus et coniungendus, nullam faciliorem viam excogitare potui pro eo eligendo atque determinando, quam alium eiusdem generis assumendo.Ut autem clarius me explicem, intelligatur perpendicularis ac ad {30} horizontalem cb; ac vero esse altitudinem, cb autem amplitudinem semiparabolae ab, quale describitur a compositione duarum lationum, quarum una est {283} mobilis descendentis per ac motu naturaliter accellarato ex quiete in a, altera est motus transversalis aequabilis iuxta horizontalem ad.Impetus acquisitus in c per descensum ac determinatur a quantitate eiusdem altitudinis ac; unus enim atque idem est semper impetus mobilis ex eadem altitudine cadentis: verum in horizontali non unus, sed innumeri assignari possunt gradus velocitatis motum aequabilium.Ex quorum multitudine ut illum quem elegero a reliquis segregare et quasi digito monstrare possim, altitudinem ca in sublimi extendam, in qua, prout opus fuerit) sublimitatum ae firmabo: ex qua si cadens ex quiete in e mente concipiam, patet, impetum {10} eius in termino a acquisitum, unum esse cum quo idem mobile, per horizontalem ad conversum, ferri concepero; eiusque gradum celeritatis esse illum, quo, in tempore descensus per ea, spatium in horizontali duplum ipsius ea conficiet.Haec praemonere necessarium visum est.But before we proceed further, since this discussion is to deal with the motion compounded of a uniform horizontal one and one accelerated vertically downwards - the path of a projectile, namely, a parabola - it is necessary that we define some common standard by which we may estimate the velocity, or momentum [velocitatem, impetum seu momentum] of both motions; and since from the innumerable uniform velocities one only, and that not selected at random, is to be compounded with a velocity acquired by naturally accelerated motion, I can think of no simpler way of selecting and measuring this than to assume another of the same kind. For the sake of clearness, draw the vertical line ac to meet the horizontal line bc. Ac is the height and be the amplitude of the semi-parabola ab, which is the resultant of the two motions, one that of a body falling {283} from rest at a, through the distance ac, with naturally accelerated motion, the other a uniform motion along the horizontal ad. The speed acquired at c by a fall through the distance ac is determined by the height ac; for the speed of a body falling from the same elevation is always one and the same; but along the horizontal one may give body an infinite number of uniform speeds. However, in order that I may select one out of this multitude and separate it from the rest in a perfectly definite manner, I will extend the height ca upwards to e just as far as is necessary and will call this distance ae the "sublimity." Imagine a body to fall from rest at e; it is clear that e may make its terminal speed at a the same as that with which the same body travels along the horizontal line ad; this speed will be such that, in the time of descent along ea, it will describe a horizontal distance twice the length of ea. This preliminary remark seems necessary.
Advertatur insuper, semiparabolae ab «amplitudinem» a me vocari horizontalem cb; «altudinem», ac nempe, eiusdem parabolae axem: lineam vero ea, ex cuius descensu determinatur impetus horizontalis. «sublimitatum» appello.The reader is reminded that above I have called the horizontal line cb the "amplitude" of the semi-parabola ab; the axis ac of this parabola, I have called its "altitude"; but the line ea the fall along which determines the horizontal speed I have called the "sublimity."
His declaratis ac definitis, ad demonstrandum me confero. These matters having been explained, I proceed with the demonstration.

Discorsi Propositions
3/03-th-02
Discorsi Proposition
3/03-th-02