Discorsi Propositions |
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Discorsi Proposition2/23-pr-09-schol1 |

SCHOLIUM. | SCHOLIUM |

Si diligenter attendatur, manifestum erit, quod quo minus data linea IR deficit a tripla ipsius AC, eo planum inflexum, super quod facienda est secunda latio, puta CO, accedit vicinius ad perpendiculum, in quo tandem in tempore aequali AC conficitur spatium ad {10} AC triplum. | On careful attention, it will be clear that, the nearer the given line IR approaches to three times the length AC, the nearer the inclined plane, CO, along which the second motion takes place, approaches the perpendicular along which the space traversed, during the time AC, will be three times the distance AC. |

Cum enim IR proxima fuerit ad triplicitatem AC, erit IM aequalis fere ipsi MN; cumque ut IM ad MN, in constructione, ita fiat AC ad CE, constat, ipsam CE paulo maiorem reperiri quam CA, et, quod consequens est, punctum E proximum reperiri puncto A, et CO cum CS acutissimum angulum continere, et fere mutuo coincidere.E contra vero, si data IR minimum quid maior fuerit quam dupla eiusdem AC, erit IM {20} brevissima linea; ex quo accidet, minimam quoque futuram esse AC respectu CE, quae longissima erit et quam proxime accedet ad parallelam horizontalem per C productam.Indeque colligere possumus, quod si, in apposita figura, post descensum per planum inclinatum AC fiat reflexio per lineam horizontalem, qualis esset CT, spatium, tempore aequali tempori descensus per AC, per quod mobile consequenter moveretur, esset duplum spatii AC exacte.Videtur autem et hic accommodari consimilis ratiocinatio: apparet enim ex eo, cum OE ad EF sit ut FE ad EC, ipsam FC determinare tempus per CO.Quod si pars horizontalis TC, dupla CA, divisa sit bifariam in V, extensa versus X in infinitum elongata erit, dum occursum cum {30} producta AE quaerit, et ratio infinitae TX ad infinitam VX non erit alia a ratione infinitae VX ad infinitam XC. | For if IR be taken nearly equal to three times AC, then IM will be almost equal to MN; and since, by construction, IM:MN = AC:CE, it follows that CE is but little greater than CA: consequently the point E will lie near the point A, and the lines CO and CS, forming a very acute angle, will almost coincide.But, on the other hand, if the given line, IR, be only the least bit longer than twice AC, the line IM will be very short; from which it follows that AC will be very small in comparison with CE which is now so long that it almost coincides with the horizontal line drawn through C.Hence we can infer that, if, after descent along the inclined plane AC of the adjoining figure, the motion is continued along a horizontal line, such as CT, the distance traversed by a body, during a time equal to the time of fall through AC, will be exactly twice the distance AC.The argument here employed is the same as the preceding. For it is clear, since OE:EF = EF:EC, that FC measures the time of descent along CO.But, if the horizontal line TC which is twice as long as CA, be divided into two equal parts at V then this line must be extended indefinitely in the direction of X before it will intersect the line AE produced; and accordingly the ratio of the infinite length TX to the infinite length VX is the same as the ratio of the infinite distance VX to the infinite distance CX. |

Istud idem alia aggressione concludere poterimus, consimile resumentes ratiocinium ei, quo usi sumus in propositionis primae demonstratione.Resumentes enim triangulum ABC, nobis repraesentans in suis parallelis basi BC {243} velocitatis gradus continue adauctos iuxta temporis incrementa, ex quibus, cum infinitae sint, veluti infinita sunt puncta in linea AC et instantia in quovis tempore, exurget superficies ipsa trianguli; si intelligamus, motum per alterum tantum temporis continuari, sed non amplius motu accelerato, verum aequabili, iuxta maximum gradum velocitatis acquisitae, qui gradus repraesentatur per lineam BC; ex talibus gradibus conflabitur aggregatum consimile parallelogrammo ADBC, quod duplum est trianguli ABC: quare spatium quod cum gradibus {10} consimilibus tempore eodem conficietur, duplum erit spatii peracti cum gradibus velocitatis a triangulo ABC repraesentatis.At in plano horizontali motus est aequabilis, cum nulla ibi sit causa accelerationis aut retardationis; ergo concluditur, spatium CD peractum tempore aequali tempori AC, duplum esse spatii AC: hoc enim motu ex quiete accelerato, iuxta parallelas trianguli, conficitur; illud vero, iuxta parallelas parallelogrammi, quae, dum fuerint infinitae, duplae sunt ad parallelas infinitas trianguli. | The same result may be obtained by another method of approach, namely, by returning to the same line of argument which was employed in the proof of the first proposition.Let us {243} consider the triangle ABC, which, by lines drawn parallel to its base, (Condition 2/00-th-00-dialog1) represents for us a velocity increasing in proportion to the time; if these lines are infinite in number, just as the points in the line AC are infinite or as the number of instants in any interval of time is infinite, they will form the area of the triangle. Let us now suppose that the maximum velocity attained-that represented by the line BC-to be continued, without acceleration and at constant value through another interval of time equal to the first. From these velocities will be built up, in a similar manner, the area of the parallelogram ADBC, which is twice that of the triangle ABC; (Condition Aristot-space-prop) accordingly the distance traversed with these velocities during any given interval of time will be twice that traversed with the velocities represented by the triangle during an equal interval of time.But along a horizontal plane the motion is uniform since here it experiences neither acceleration nor retardation; therefore we conclude that the distance CD traversed during a time-interval equal to AC is twice the distance AC; for the latter is covered by a motion, starting from rest and increasing in speed in proportion to the parallel lines in the triangle, while the former is traversed by a motion represented by the parallel lines of the parallelogram which, being also infinite in number, yield an area twice that of the triangle. |

Discorsi Propositions |
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Discorsi Proposition2/23-pr-09-schol1 |