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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Baliani, Giovanni Baptista</author> <title>De Motu Naturali Gravium Solidorum et Liquidorum</title> <date>1646</date> <place>Genf</place> <translator></translator> <lang>la</lang> <cvs_file>balia_demot_064_la_1646.xml</cvs_file> <cvs_version></cvs_version> <locator>064.xml</locator> </info> <text> <front></front> <body> <pb xlink:href="064/01/001.jpg"></pb> <chap> <p type="main"> <s id="s.000001">DE MOTV <lb></lb>NATVRALI <lb></lb>GRAVIVM SOLIDORVM <lb></lb>ET LIQUIDORVM <lb></lb>IO: BAPTISTAE BALIANI <lb></lb>PATRITII ENVENSIS. </s> </p> <p type="main"> <s id="s.000002">GENVAE</s> </p> <p type="main"> <s id="s.000003">Ex Typographia IO: Mariæ Farroni 1646 <lb></lb>Superiorum Permiſſu.</s> </p> <pb xlink:href="064/01/002.jpg"></pb> <pb xlink:href="064/01/003.jpg"></pb> <p type="main"> <s id="s.000004">DE MOTV <lb></lb>GRAVIVM <lb></lb>SOLIDORVM <lb></lb>LIBER PRIMVS.<lb></lb></s> </p> <p type="main"> <s id="s.000005">Mihi quoque, sicut & <lb></lb>caeteris hominibus, inest <lb></lb>sciendi cupiditas, nec gra<lb></lb>ve fuit, usque a primis <lb></lb>annis, & aliorum scripta <lb></lb>percurrere, & naturales <lb></lb>effectus observare, qui fa<lb></lb>cile mihi persuaserim, ex hisce fontibus, tum <lb></lb>scientiam, tum sapientiam in animum de<lb></lb>rivare, si tandem ex effectibus diligentius <pb xlink:href="064/01/004.jpg"></pb>perspectis, non modo ad inde consequentes, <lb></lb>sed etiam ad causas, usque ad primam de<lb></lb>veniat intellectus. </s> <s id="s.000006">Statui igitur apud me ip<lb></lb>sum non acquiescere soli relationi pluri<lb></lb>morum, etiam doctiorum; potuisse siquidem <lb></lb>contingere existimavi, ut aliqua laterent, <lb></lb>etiam in plurimis oculatissimos, vel non ple<lb></lb>ne ab eis explicarentur; & ratus sum non <lb></lb>inutilem laborem futurum, si ex accuratiori <lb></lb>naturae rerum investigatione, & ex affection<lb></lb>um inde resultantium deductione, circa <lb></lb>quod omnis demonstrativa scientia versatur, <lb></lb>aut scitis adderem aliqua, aut doctioribus <lb></lb>acuerem desiderium addendi plura: hinc fa<lb></lb>ctum est, ut excitata mens ex praecognitis le<lb></lb>gendo, ad ea, quae se offerebant, secun<lb></lb>dum privatas, aut publicas occupationes per<lb></lb>vestiganda, converteretur studiosus. </s> <s id="s.000007">Inter <lb></lb>alia dum anno millesimo sexcentesimo un<lb></lb>decimo, per paucos menses, ex patriae legis <lb></lb>praescripto, Praefectum Arcis Savonae agerem, <lb></lb>ex militaribus observationibus quae occurre<lb></lb>bant, illud maxime depraehendi, ferreos, <lb></lb>& lapideos tormentorum bellicorum glo<lb></lb>bos, & sic corpora gravia, seu eiusdem, seu <pb xlink:href="064/01/005.jpg"></pb>diversae speciei, in inaequali satis Mole, & <lb></lb>gravitate, per idem spatium, aequali tem<lb></lb>pore, & motu, naturaliter descendere, idque <lb></lb>ita uniformiter, ut repetitis experimentis mihi <lb></lb>plane constiterit, duos ex praedictis globis, <lb></lb>vel ferreos ambos, vel alterum lapideum <lb></lb>alterum plumbeum, eodem plane mo<lb></lb>mento temporis dimissos sibi, per spatium <lb></lb>quinquaginta pedum, etiam si unus es<lb></lb>set librae unius tantum, alter quinquagin<lb></lb>ta, in indivisibili temporis momento, subje<lb></lb>ctum solum ferire, ut unus tantum ambo<lb></lb>rum ictus sensu perciperetur. </s> <s id="s.000008">Repetebam <lb></lb>animo sapientum esse pronunciatum, gravia <lb></lb>moveri naturali motu, secundum gravitatum <lb></lb>proportionem; Processi ulterius, & pericu<lb></lb>lum feci, num forte iuxta eorum sententiam <lb></lb>contingeret, si corpora dimissa, eiusdem fere <lb></lb>essent molis, sed longe diversi ponderis, pu<lb></lb>ta unum plumbeum, cereum alterum; & ex<lb></lb>pertus sum in cereo aliquam longiorem mo<lb></lb>ram in descensu, attamen longe infra propor<lb></lb>tionem gravitatum, globus quippe ille ce<lb></lb>reus, in data distantia quinquaginta pedum <lb></lb>descensus, uno circiter pede distabat a solo,<pb xlink:href="064/01/006.jpg"></pb>quando plumbeus tangebat subjectum pla<lb></lb>num, objecto aere intermedio ni fallor, sen<lb></lb>sibiliter resistente, & impediente motum. <lb></lb></s> <s id="s.000009">Institi adhuc, & globos in gravitate, & in <lb></lb>materia inaequales appendi funiculis aequali<lb></lb>bus, & agitatos animadverti moveri tempo<lb></lb>re aequali, & hoc servare adeo fideliter, ut <lb></lb>globus plumbeus duarum unciarum, alter <lb></lb>librarum duarum, ferreus librarum 34. & la<lb></lb>pideus quadraginta circiter, nec non, & la<lb></lb>pis informis, quorum funiculi comprehen<lb></lb>sis ipsorum semidiametris aequales essent, <lb></lb>uno, & eodem temporis spatio moverentur, <lb></lb>& vibrationes easdem numero darent hinc <lb></lb>inde, sive motus unius globi fieret per aequa<lb></lb>le spatium, sive per inaequale, ita ut qui <lb></lb>maiori impetu jactabatur, & sic majus spa<lb></lb>tium percurrebat, illud tanto velocius per<lb></lb>transiret. </s> <s id="s.000010">In quibus peragendis illud praeter <lb></lb>expectationem sese mihi obtulit, quod quo<lb></lb>tiescunque globi penderent ex funiculis inae<lb></lb>qualibus, ita inaequali motu ferebantur, ut <lb></lb>longitudines funiculorum, durationibus mo<lb></lb>tuum, in duplicata ratione responderent.<lb></lb></s> </p> <p type="main"> <s id="s.000011">Porro cum ex praemissis satis superque li<lb></lb><pb xlink:href="064/01/007.jpg"></pb>queret, in naturali motu gravium, pro<lb></lb>portionem gravitatum communiter credi<lb></lb>tam, non servari; in eam descendi sen<lb></lb>tentiam, ut arbitrater fortasse, gravitatem <lb></lb>se habere ut agens, materiam vero, seu <lb></lb>mavis materiale corpus, ut passum, & <lb></lb>proinde gravia moveri juxta proportionem <lb></lb>gravitatis ad materiam, & ubi sine impedi<lb></lb>mento naturaliter perpendiculari motu fe<lb></lb>rantur, moveri aequaliter, quia ubi plus est <lb></lb>gravitatis, plus pariter sit materiae, seu ma<lb></lb>terialis quantitatis; si vero accedat aliquid <lb></lb>resistentiae, regulari motum secundum ex<lb></lb>cessum virtutis agentis supra resistentiam <lb></lb>passi, seu impedientia motum; qui exces<lb></lb>sus momentum noncupabitur, & quod com<lb></lb>muniter gravitati attributum fuit, momen<lb></lb>to attribui debere, nimirum ut sit momen<lb></lb>tum ad momentum, ut velocitas ad velo<lb></lb>citatem; Et hinc fieri posse, ut cognosca<lb></lb>mus qua mensura, seu proportione corpora <lb></lb>gravia naturali motu ferantur super subje<lb></lb>ctis planis, si super eis quomodolibet in<lb></lb>clinatis, ipsorum gravium momenta ubique <lb></lb>innotescant, quae maiora, aut minora viden<lb></lb><pb xlink:href="064/01/008.jpg"></pb>tur censenda, secundum quod magis, aut <lb></lb>minus super plano quiescunt, & sic secun<lb></lb>dum maiorem, aut minorem inclinationem <lb></lb>plani resistentis; quod demum tali propor<lb></lb>tione facile fieri mihi existimandum vide<lb></lb>tur, juxta quam reciproce momentis pro<lb></lb>portionantur lineae dictorum planorum, si <lb></lb>ambae ductae sint ab eodem puncto ad idem <lb></lb>planum orizontale; de quo Simon Stevi<lb></lb>nus l. p. de Statica prop. 19. & acutissime <lb></lb>Galileus in Mechanica manuscripta, ubi de <lb></lb>Cochlea, & ego æliquali experientia com<lb></lb>pertum habui. </s> <s id="s.000012">Caeterum si per experien<lb></lb>tiam Scientia hominibus efficitur, praedicta <lb></lb>de quibus saepius repetitis actibus expertus <lb></lb>fui, ut principia scientiae habenda fore cen<lb></lb>sui; in quibus occultae conclusiones delites<lb></lb>cant, demonstrationibus duntaxat aperien<lb></lb>dae. </s> <s id="s.000013">Rimari caepi; an deprehenderim alio<lb></lb>rum erit judicium. </s> <s id="s.000014">Subjecta paucula, quae <lb></lb>presens aliquod otium expedire permisit, <lb></lb>de motu naturali solidorum gravium, Ami<lb></lb>ce lector tibi exhibeo, mox de liquidorum, <lb></lb>& deinceps alia plura tam parata daturus, <lb></lb>si haec placuerint. </s> <s id="s.000015">Placuit sane mihi, vel <pb xlink:href="064/01/009.jpg"></pb>paucula tibi dare, qui te eius ingenij esse <lb></lb>confidam, ut non verba, sed res, easque <lb></lb>non mole, sed pondere censeas, felicior si <lb></lb>de eorum genere existimaveris, quae non <lb></lb>mole magna sunt, quod si talia non fue<lb></lb>rint, quo minora minus defatigabunt, sui <lb></lb>exilitate, auctoris partus proprios omnino <lb></lb>esse probatura. </s> <s id="s.000016">Idioma latinum elegi ut <lb></lb>communius. </s> <s id="s.000017">Praemisi aliqua naturalia prin<lb></lb>cipia, sine quibus naturales conclusiones <lb></lb>aliunde duci posse non video. </s> <s id="s.000018">Quae ex prae<lb></lb>dictis experimentis innotuerunt, supposi<lb></lb>tiones appellare, & a reliquis petitionibus <lb></lb>secernere libuit. </s> <s id="s.000019">Petitiones illas, quibus quid <lb></lb>fieri petimus, constructioni deservientes, <lb></lb>tanquam factu, & cognitu faciles, & pro<lb></lb>inde supervacaneas, prudens praetermisi; <lb></lb>ratus siquidem nil inde incredulitatis, aut <lb></lb>difficultatis derivaturum. </s> <s id="s.000020">Septimum po<lb></lb>stulatum ea ratione segregavi, quod il<lb></lb>lud aliquo pacto a 22. prop. pendeat, & <lb></lb>quod in illo etiamsi veritas non deficiat, <lb></lb>evidentiam tamen ut in caeteris non agno<lb></lb>scens, certis dubia quo quo pacto permisce<lb></lb>re noluerim; ut proinde plura eorum, quae <pb xlink:href="064/01/010.jpg"></pb>ex illo deducta sunt, & diversa Methodo & <lb></lb>attingendo potius, quam demonstrando <lb></lb>subjunxerim. </s> <s id="s.000021">Si quae demum minus pro<lb></lb>bata, seu explicata, aut quo quo pacto im<lb></lb>perfecta reperies, velim te tribuere cuidam <lb></lb>naturali meae propensioni, ad nova potius, <lb></lb>qualiacumque ea sint, invenienda, quam <lb></lb>inventa perficienda. </s> <s id="s.000022">Vale.</s> </p> <pb xlink:href="064/01/011.jpg"></pb> <p type="main"> <s id="s.000023">De mandato Reuerendiſſimi Patris Magiſtri <lb></lb>luſtiniani Vagnoni Inquiſitoris Generelis <lb></lb>Genuæ, &c.</s> </p> <p type="main"> <s id="s.000024">Rudi ego infraſcriptus Sancti Officij Conſultor <lb></lb>De Motu Grauium Illuſtriſſimi D. Ioannis <lb></lb>Baptiste Baliani Libros sex.</s> <s id="s.000025">In quibus nil re <lb></lb>peri S. Catholica fidei, bonis moribus, ſacriſ<lb></lb>ue decretis diſſonum; ſed dignam ubique typis, <lb></lb>& publica luce doctrinam, ſi prefato Reue <lb></lb>rendiſſimo Patri ita videbitur.</s> <s id="s.000026">In quorum fi <lb></lb>dem, &c.</s> </p> <p type="main"> <s id="s.000027">Ex Conuentu Sanctiſſime Annunciatæ Veteris <lb></lb>Genue 27. Nouembris 1646.</s> </p> <p type="main"> <s id="s.000028">Magiſt. Fr. Angelicus Riccobonus Aug.</s> </p> <p type="main"> <s id="s.000029">IMPRIMATVR.</s> </p> <p type="main"> <s id="s.000030">F. Iuſtinianus Vagnonus a Calli S. T. M. <lb></lb>Inquiſitor Generalis Genuæ & c.</s> </p> <pb xlink:href="064/01/012.jpg"></pb> <pb xlink:href="064/01/013.jpg"></pb> <subchap1 type="definition"> <p type="head"> <s id="s.000031">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000032">Pendulus dicimus pondus filo <lb></lb>appensum.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000033">Pendula dicuntur aequalia, <lb></lb>seu aequipendula, sive inae<lb></lb>qualia, quae, & longiora, <lb></lb>aut breviora, quatenus <lb></lb>fila, e quibus dependent, sunt <lb></lb>aequalia, longiora, aut breviora.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000034">Vibrationes pendulorum sunt eorum motus hinc <lb></lb>inde </s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000035">Vibrationes aequales dicimus, quae fiunt per spa<lb></lb>tia aequalia, & e contra inaequales.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000036">Vibrationes aeque celeres si fiant per spatia aequa<lb></lb>lia tempore aequali.</s> </p> </subchap2> <pb xlink:href="064/01/014.jpg"></pb> <subchap2 type="definition"> <p type="main"> <s id="s.000037">Vibrationis diuturnitatem dicimus ipsius Dura<lb></lb>tionem, tempus nimirum, quo ipsa vibratio <lb></lb>perficitur.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000038">Vibrationes æquediuturne, sunt, quae fiunt tem<lb></lb>pore aequali, etiamsi per spatia inaequalia, <lb></lb>inde diuturnior est, quae longiori perficitur <lb></lb>tempore.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000039">Vibrationes integras dicimus eas, quae se exten<lb></lb>dunt per integrum semicirculum, se hinc in<lb></lb>de moventes per circuli quadrantem.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000040">Vibrationis portio est pars arcus, quem ipsa vi<lb></lb>bratio disignant.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000041">Vibrationum similes portiones sunt arcus ipsa<lb></lb>rum intercepti inter binas lineas ductas a <lb></lb>centro, a quo concipiuntur pendula pendere.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000042">Vibrationis portionem priorem decimus eam mi<lb></lb>nimam portionem, a qua integra vibratio <lb></lb>initium habet.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000043">Momentum est excessus virtutis moventis supra <lb></lb>motus impedimenta.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/015.jpg"></pb> <subchap1 type="supposition"> <p type="head"> <s id="s.000044">SUPPOSITIONES</s> </p> <subchap2 type="supposition"> <p type="main"> <s id="s.000045">PRIMA. </s> <s id="s.000046">Solidorum aequipendu<lb></lb>lorum cujuscumque gravitatis vibra<lb></lb>tiones aequales sunt aequediu<lb></lb>turnae.</s> </p> </subchap2> <subchap2 type="supposition"> <p type="main"> <s id="s.000047">2 Equipendulorum eorundem vibrationes <lb></lb>sunt aequediuturnae, etiamsi inaequales.</s> </p> </subchap2> <subchap2 type="supposition"> <p type="main"> <s id="s.000048">3 Pendulorum inaequalium longitudines sunt <lb></lb>in duplicata ratione diuturnitatum vi<lb></lb>brationum, seu ut quadrata vibratio<lb></lb>num.</s> </p> </subchap2> <subchap2 type="supposition"> <p type="main"> <s id="s.000049">4 Momentum gravis super plano inclinato <lb></lb>est ad ipsius gravitatem, ut perpendi<pb xlink:href="064/01/016.jpg"></pb>cularis ad inclinatam, si ab eodem <lb></lb>puncto ducta sint ad idem planum <lb></lb>orizontale dicta perpendicularis, & di<lb></lb>ctum planum inclinatum, & proinde <lb></lb>tali casu proportio gravitatis ad mo<lb></lb>mentum est reciproca proportioni li<lb></lb>nearum super quibus grave movetur.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/017.jpg"></pb> <subchap1 type="postulate"> <p type="head"> <s id="s.000050">PETITIONES, SEU POSTULATA</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000051">Pr. </s> <s id="s.000052">Pendulorum inaequalium portiones similes vi<lb></lb>brationum sunt inter se quoad diuturni<lb></lb>tatem, ut vibrationes integrae.<figure id="id.064.01.017.1.jpg" xlink:href="064/01/017/1.jpg"></figure></s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000053">Sint pendula AB, AC; dependentia a puncto A, <lb></lb>& eleventur ad libellam orizontis puncti A, <lb></lb>in E, D, describentia arcus BD, CE, inte<lb></lb>grarum vibrationum, & in arcubus BD, <lb></lb>CE sumantur portiones similes EF, DG, seu <lb></lb>HI, KL ductis EA, FA, seu HA, IA. </s> <s id="s.000054">Peto <lb></lb>mihi concedi, esse pendulorum diuturnitates in <lb></lb>arcubus EC, DB, ut in portionibus EF, DG, <lb></lb>nec non HI, KL, & ita deinceps.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000055">2. Ut est momentum ad momentum solidi <lb></lb>gravis, ita velocitas ad velocitatem.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000056">Huiusmodi passio communiter attribui solet gra<lb></lb>vitati simpliciter, quod eum nimis clare expe<lb></lb>rientijs supra expositis nullo pacto congruere <lb></lb>possit, momentis attribuenda esse visa est, ut <lb></lb>in praefatione explicatum fuit.</s> </p> </subchap2> <pb xlink:href="064/01/018.jpg"></pb> <subchap2 type="postulate"> <p type="main"> <s id="s.000057">3. Portiones minimae peripheriae Circuli con<lb></lb>cipiende sunt, ac si essent lineae rectae.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000058">Quaecumque arcus portio est circularis, atta<lb></lb>men si est minima portio, tam parum aber<lb></lb>rat a linea recta, ut non modo quo ad <lb></lb>sensum, sed quoad quascunque physicas passio<lb></lb>nes, perinde esse videatur, ac si esset linea re<lb></lb>cta, idcirco ut petitionem admittendam cen<lb></lb>seo, quemadmodum in mechanicis admittitur <lb></lb>illa, quod perpendiculares sunt parallelae, etiamsi <lb></lb>in centro concurrant universi, quatenus eis<lb></lb>dem sunt passionibus physicis subjectae, ac si <lb></lb>vere essent parallelae.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000059">4. Data recta linea, possimus concipere cir<lb></lb>culum talis magnitudinis, cujus portio pe<lb></lb>ripheriae aequalis quo ad sensum datae lineae, <lb></lb>concipienda sit, ac si esset linea recta.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000060">Haec petitio videtur concedenda, quia si conci<lb></lb>piamus circulum, eiusque portionem mini<lb></lb>mam, ut in praecedenti, si fiat ut huiusmodi <lb></lb>portio ad datam lineam, ita circulus ad alium, <lb></lb>portio huius, datae lineae aequalis erit, & simi<lb></lb>lis omnino praedicta minimae portioni, & proin<lb></lb>de pariter concipienda ut linea recta.</s> </p> </subchap2> <pb xlink:href="064/01/019.jpg"></pb> <subchap2 type="postulate"> <p type="main"> <s id="s.000061">5. Solida perpendicula libero motu aeque <lb></lb>velociter feruntur, & in tali proportione, <lb></lb>ac si essent pendula, & moverentur in <lb></lb>priori portione vibrationum.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000062">Quoniam prior portio non differt sensibiliter a re<lb></lb>cta, ut in tertia petitione ijsdem physicis passio<lb></lb>nibus subjicitur, & exinde motibus aequalibus.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000063">6. Solida naturaliter mota super plano incli<lb></lb>nato aeque velociter moventur ac si essent <lb></lb>pendula, & moverentur in tali portione vi<lb></lb>brationum, quae quoad sensum esset aequa<lb></lb>lis, & paralella lineae dicti plani super qua <lb></lb>dicta solida moverentur.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000064">Non differt a praecedente, nisi quod in illa mo<lb></lb>tus est perpendicularis, in hac inclinatus, in <lb></lb>reliquis est par ratio.</s> </p> </subchap2> </subchap1> <subchap1 type="enunciation"> <p type="head"> <s id="s.000065">PRONUNCIATA</s> </p> <subchap2 type="enunciation"> <p type="main"> <s id="s.000066">P. </s> <s id="s.000067">Quae sunt aequidiuturna tertio, sunt aequi<lb></lb>diuturna inter se.</s> </p> </subchap2> <subchap2 type="enunciation"> <p type="main"> <s id="s.000068">2. Quadrata datorum temporum, sunt etiam <lb></lb>quadrata aliorum datis aequalium.</s> </p> </subchap2> <subchap2 type="enunciation"> <p type="main"> <s id="s.000069">3. Gravia eadem super planis aequalibus & <lb></lb>pariter inclinatis, pariter moventur.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/020.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000070">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000071">Solidi penduli naturaliter moti vibratio<lb></lb>nes quantumvis semper minores, sunt <lb></lb>aequidiuturnae.<figure id="id.064.01.020.1.jpg" xlink:href="064/01/020/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <s id="s.000072">Sit solidum A pendulum debite applicatum filo <lb></lb>BA, quod ab altera parte elevatum naturaliter, <lb></lb>postea faciat hinc inde vibrationes semper mi<lb></lb>nores, ita ut prior vibratio sit V.G. per spatium <lb></lb>CD maius, posterior vero per spatium EF minus.</s> </p> <p type="main"> <s id="s.000073">Dico quod dicta vibrationes erunt aequidiuturnae, <lb></lb>ita ut vibratio per spatium CD sit eiusdem du<lb></lb>rationis, ac vibratio per spatium EF.</s> </p> <p type="main"> <s id="s.000074">Sit aliud solidum G aequipendulum solido A, de<lb></lb>bite applicatum filo HG, quod elevetur ab una <lb></lb>parte eodem tempore minus quam solidum A <lb></lb>ita ut sint minores vibrationes solidi G, quam, <lb></lb>solidi A, ut sit motus penduli G in initio per <lb></lb>spatium IK aequale spatio EF.</s> </p> <p type="main"> <s id="s.000075">Quoniam spatia EF, & IK, sunt aequalia ex sup<lb></lb>positione, sunt etiam vibrationes EF, & IK, <lb></lb>aequidiuturnae,<arrow.to.target n="marg1"></arrow.to.target>,sed IK, & CD sunt pariter <lb></lb>aequidiuturnae<arrow.to.target n="marg2"></arrow.to.target>, ergo EF, & CD sunt etiam <lb></lb>aequidiuturnae<arrow.to.target n="marg3"></arrow.to.target>. </s> <s id="s.000076">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000077"><margin.target id="marg1"></margin.target>Per primam suppositionem.</s> </p> <p type="margin"> <s id="s.000078"><margin.target id="marg2"></margin.target>Per secundam suppositionem.</s> </p> <p type="margin"> <s id="s.000079"><margin.target id="marg3"></margin.target>Per pr. pron.</s> </p> </subchap2> </subchap1> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000080">PROPOSITIO II. PROB. PRIMUM</s> </p> <pb xlink:href="064/01/021.jpg"></pb> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000081">Pendula constituere, quorum diuturnita<lb></lb>tes vibrationum sint in data ratione.<figure id="id.064.01.021.1.jpg" xlink:href="064/01/021/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <s id="s.000082">Data sit proportio diuturnitatum vibratio<lb></lb>num, quam volumus esse inter solida A,B; <lb></lb>& sit ea, quae est inter C, & D; quae est continuo <lb></lb>eadem,<arrow.to.target n="marg4"></arrow.to.target>,</s> </p> <p type="margin"> <s id="s.000083"><margin.target id="marg4"></margin.target>Per pr. huius.</s> </p> <p type="main"> <s id="s.000084">Venanda est longitudo filorum, quibus applicata <lb></lb>dicta solida producant vibrationes quaesitas.</s> </p> <p type="main"> <s id="s.000085">Fiat L tertia proportionalis ad C, & D,<arrow.to.target n="marg5"></arrow.to.target> & fila <lb></lb>IA, KB fiant inter se ut C ad L,<arrow.to.target n="marg6"></arrow.to.target> & erunt <lb></lb>fila quaesita.</s> </p> <p type="margin"> <s id="s.000086"><margin.target id="marg5"></margin.target>Per 11 sexti.</s> </p> <p type="margin"> <s id="s.000087"><margin.target id="marg6"></margin.target>Per 12 sexti.</s> </p> <p type="main"> <s id="s.000088">Quoniam ita est IA ad KB ut C ad L per constr. <lb></lb>erunt C, & D diuturnitates vibrorum pendu<lb></lb>lorum AB.<arrow.to.target n="marg7"></arrow.to.target> Quod etc</s> </p> <p type="margin"> <s id="s.000089"><margin.target id="marg7"></margin.target>Per 3 Supp.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/022.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000090">PROPOSITIO TERTIA</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000091">Lineae descensus gravium, dum naturali motu <lb></lb>perpendiculariter feruntur, sunt in dupli<lb></lb>cata ratione diuturnitatum.<figure id="id.064.01.022.1.jpg" xlink:href="064/01/022/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="4" type="proof"> <p type="main"> <s id="s.000092">Sint LN, KM linea descensus gravium L, K, <lb></lb>& sint PO ipsorum diuturnitates.</s> </p> <p type="main"> <s id="s.000093">Dico LN, KM esse in duplicata ratione ipsarum P, O.</s> </p> <p type="main"> <s id="s.000094">Sint pendula AH, AI, dependentia a puncto A, & <lb></lb>eleventur ad libellam ipsius A usque ad E, B, <lb></lb>quae in elevatione producant arcus HB, IE, & <lb></lb>sint talis longitudinis, ut ducta ACF, secet ar<lb></lb>cus BC, & EF, tam parvae curvitatis ut pro <lb></lb>rectis habeantur, puta portionis minimae, & <lb></lb>proinde aequales quo ad sensum rectis KM, LN,<arrow.to.target n="marg8"></arrow.to.target> <lb></lb>& fiat V tertia proportionalis ad O, P,<arrow.to.target n="marg9"></arrow.to.target><lb></lb></s> </p> <p type="margin"> <s id="s.000095"><margin.target id="marg8"></margin.target>Per 3 pet.</s> </p> <p type="margin"> <s id="s.000096"><margin.target id="marg9"></margin.target>Per 11 sexti.</s> </p> <p type="main"> <s id="s.000097">Quoniam O, P sunt diuturnitates KM, LN ex <lb></lb>constr., sunt itidem diuturnitates BC, EF, <arrow.to.target n="marg10"></arrow.to.target> & <lb></lb>quia diuturnitates vibrorum AH, AI sunt <lb></lb>etiam ut O ad P <arrow.to.target n="marg11"></arrow.to.target> AH AI sunt ut O, ad V<arrow.to.target n="marg12"></arrow.to.target> <lb></lb>& pariter BC, & EF sunt ut O ad V<arrow.to.target n="marg13"></arrow.to.target> Ergo <lb></lb>KM, LN eis aequales per constr. sunt etiam ut <lb></lb>O ad V, & proinde in duplicata ratione O, P, <lb></lb>temporum seu diuturnitatum earumdem. </s> <s id="s.000098">Quod, etc.</s> </p> <p type="margin"> <s id="s.000099"><margin.target id="marg10"></margin.target>Per 5 pet.</s> </p> <p type="margin"> <s id="s.000100"><margin.target id="marg11"></margin.target>Per p. pet.</s> </p> <p type="margin"> <s id="s.000101"><margin.target id="marg12"></margin.target>Per 3 supp.</s> </p> <p type="margin"> <s id="s.000102"><margin.target id="marg13"></margin.target>Per p. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/023.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000103">PROPOSITIO QUARTA. PROB. II.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000104">Data diuturnitate gravis descendentis a data <lb></lb>altitudine, constituere altitudinem, a qua <lb></lb>idem grave cadat in data alia diuturnitate.<figure id="id.064.01.023.1.jpg" xlink:href="064/01/023/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.000105">Sit A diuturnitas gravis B, dum cadit in C, & <lb></lb>data sit diuturnitas quaecumque D.</s> </p> <p type="main"> <s id="s.000106">Constituenda est alia altitudo, a qua grave de<lb></lb>scendat iuxta diuturnitatem D.</s> </p> <p type="main"> <s id="s.000107">Fiat I, tertia proportionalis ad AD,<arrow.to.target n="marg14"></arrow.to.target> & ut I ad A <lb></lb>fiat altitudo GH ad altitudinem datam BC,<arrow.to.target n="marg15"></arrow.to.target> <lb></lb>Dico GH esse altitudinem quaesitam.</s> </p> <p type="margin"> <s id="s.000108"><margin.target id="marg14"></margin.target>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000109"><margin.target id="marg15"></margin.target>Per 12. sexti.</s> </p> <p type="main"> <s id="s.000110">Quoniam BC, & GH sunt in duplicata ratione <lb></lb>datarum diuturnitatum A, D, per constructio<lb></lb>nem; per ipsas gravia B, & G cadent in diu<lb></lb>turnitatibus A, & D datis<arrow.to.target n="marg16"></arrow.to.target>, unde reperta est <lb></lb>altitudo GH quaesita. </s> <s id="s.000111">Quod fuit faciendum.</s> </p> <p type="margin"> <s id="s.000112"><margin.target id="marg16"></margin.target>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/024.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000113">PROPOSITIO V. PROB. III.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000114">Data altitudine, a qua descendat grave in no<lb></lb>ta diuturnitate; perquirere quanta sit diutur<lb></lb>nitas, qua descendat ab alia altitudine data.<figure id="id.064.01.024.1.jpg" xlink:href="064/01/024/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.000115">Sit A altitudo per quam descendat grave diutur<lb></lb>nitate B nota, & data sit alia altitudo C.</s> </p> <p type="main"> <s id="s.000116">Oportet reperire quanta sit diuturnitas, qua idem <lb></lb>grave descendat per C.</s> </p> <p type="main"> <s id="s.000117">Fiat ut A ad C ita B ad G,<arrow.to.target n="marg17"></arrow.to.target> inter quas media, <lb></lb>proportionalis F<arrow.to.target n="marg18"></arrow.to.target> est diuturnitas quaesita.</s> </p> <p type="margin"> <s id="s.000118"><margin.target id="marg17"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000119"><margin.target id="marg18"></margin.target>Per 13. sexti.</s> </p> <p type="main"> <s id="s.000120">Quoniam A, & C sunt in duplicata ratione diu<lb></lb>turnitatum B, & F per constructionem, per <lb></lb>ipsas gravia descendent in diuturnitatibus B, <lb></lb>F,<arrow.to.target n="marg19"></arrow.to.target> unde F est diuturnitas ipsius C quaesita.</s> </p> <p type="margin"> <s id="s.000121"><margin.target id="marg19"></margin.target>Per 3. huius.</s> </p> <p type="main"> <s id="s.000122">Quod faciendum fuit.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/025.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000123">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000124">Gravia naturali motu descendunt semper velo<lb></lb>cius ea ratione, ut temporibus aequalibus de<lb></lb>scendant per spatia semper maiora, iuxta <lb></lb>proportionem quam habent impares nu<lb></lb>meri ab unitate inter se.<figure id="id.064.01.025.1.jpg" xlink:href="064/01/025/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <s id="s.000125">Sit grave A quod descendat per lineam ABC, <lb></lb>& tempus quo descendit ab A in B sit aequale <lb></lb>tempori, quo descendit a B in C, & a C in D.</s> </p> <p type="main"> <s id="s.000126">Dico quod lineae AB, BC, CD sunt inter se ut 1.<lb></lb>3.5.& sic deinceps.</s> </p> <p type="main"> <s id="s.000127">Sit G linea mensurans tempus, quo A descendit <lb></lb>in B, & H, quo de<lb></lb>scendit a B in C, & I, quo descendit a C in D, quae tempora sunt ex suppositione <lb></lb>aequalia, & sit K latus quadrati ipsius G, & L <lb></lb>quadrati GH, & N quadrati totius GHI.</s> </p> <p type="main"> <s id="s.000128">Quoniam quadrata K, L, N sunt ut AB, AC, A<lb></lb>D<arrow.to.target n="marg20"></arrow.to.target>, quae quadrata sunt ut 1, 4, 9, sunt itidem <lb></lb>AB, AC, AD, ut 1. 4. 9. & dividendo AB, <lb></lb>BC, CD, ut 1. 3. 5. & sic deinceps. </s> <s id="s.000129">Quod <lb></lb>probandum fuit.</s> </p> <p type="margin"> <s id="s.000130"><margin.target id="marg20"></margin.target>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/026.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.000131">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000132">Lineae descensus gravium super plano incli<lb></lb>nato motorum, sunt in duplicata ratione <lb></lb>diuturnitatum.<figure id="id.064.01.026.1.jpg" xlink:href="064/01/026/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.000133">Sint AB, CD plana pariter inclinata, super <lb></lb>quibus moveantur gravia A, C, & sint EF <lb></lb>ipsorum diuturnitates.</s> </p> <p type="main"> <s id="s.000134">Dico AB, CD, esse in duplicata ratione ipsarum E, F.</s> </p> <p type="main"> <s id="s.000135">Secetur AB bifariam in G, & erecta GH, per<lb></lb>pendiculari longissima, fiant pendula HI, HK, <lb></lb>quae sint inter se ut AB, CD, & eleventur in <lb></lb>L, M, describentia arcus LI, KM, secantes <lb></lb>GH in N, O, & ab N hinc inde secentur ar<lb></lb>cus NP, NQ aequales quo ad sensum rectis <lb></lb>GA, GB, & ductis PH, QH, secetur pariter <lb></lb>arcus LI, in R, S, & intelligantur arcus PQ, <lb></lb>RS, tam parvae curvitatis, ob maximam lon<lb></lb>gitudinem pendulorum HI, HK, ut pro re<lb></lb>ctis habeantur, puta portionis minimae, & pro<lb></lb>inde aequales rectis AB, CD.<arrow.to.target n="marg21"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000136"><margin.target id="marg21"></margin.target>Per 3. pet.</s> </p> <p type="main"> <s id="s.000137">Quoniam EF sunt diuturnitates AB, CD per <pb xlink:href="064/01/027.jpg"></pb>construct., sunt etiam diuturnitates portionum <lb></lb>PQ, RS<arrow.to.target n="marg22"></arrow.to.target>, & pariter vibrationum pendulo<lb></lb>rum HK, HI<arrow.to.target n="marg23"></arrow.to.target> sunt autem diuturnitates <lb></lb>praedictae E, F, in subduplicata ratione pendu<lb></lb>lorum HK, HI<arrow.to.target n="marg24"></arrow.to.target> unde pariter portionum PQ, <lb></lb>RS, & proinde plenorum AB, CD, Quod, etc.</s> </p> <p type="margin"> <s id="s.000138"><margin.target id="marg22"></margin.target>Per 6. pet.</s> </p> <p type="margin"> <s id="s.000139"><margin.target id="marg23"></margin.target>Per pr. pet.</s> </p> <p type="margin"> <s id="s.000140"><margin.target id="marg24"></margin.target>Per 3. supp.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000141">Corollarium</s> </p> <p type="main"> <s id="s.000142">Hinc patet esse longitudines planorum per quae <lb></lb>gravia feruntur ut quadrata temporum, & <lb></lb>tempora ut radices longitudinum planorum.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/028.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.000143">PROPOSITIO VIII. PROB. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000144">Dato plano inclinato, super quo per spatium <lb></lb>datum grave moveatur in nota diuturni<lb></lb>tate, determinare in eodem plano spatium <lb></lb>per quod dictum grave moveatur in qua<lb></lb>vis alia diuturnitate data.<figure id="id.064.01.028.1.jpg" xlink:href="064/01/028/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.000145">Sit A diuturnitas gravis B, dum descendit in <lb></lb>C super plano inclinato BC, & data diu<lb></lb>turnitas D.</s> </p> <p type="main"> <s id="s.000146">Praescribendum est aliud spatium in eodem pla<lb></lb>no BC, per quod idem grave pertranseat in <lb></lb>diuturnitate D.</s> </p> <p type="main"> <s id="s.000147">Fiat H tertia proportionalis ad A & D, & ut <lb></lb>H ad A fiat BG ad BC, Dico BG esse spa<lb></lb>tium quaesitum.</s> </p> <p type="main"> <s id="s.000148">Quoniam BC, & BG sunt in duplicata ratione <lb></lb>datorum temporum A, D per constructionem, <lb></lb>per ipsa cadet grave B diuturnitatibus A, D <lb></lb>datis<arrow.to.target n="marg25"></arrow.to.target>, ergo reperta est BG quaesita. </s> <s id="s.000149">Quod <lb></lb>faciendum erat.</s> </p> <p type="margin"> <s id="s.000150"><margin.target id="marg25"></margin.target>Per 6. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/029.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000151">PROPOSITIO IX. PROB. V.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000152">Dato plano inclinato, super quo per spatium <lb></lb>datum grave moveatur nota diuturnitate; <lb></lb>& dato alio spatio quocumque; reperire <lb></lb>diuturnitatem, qua grave per ipsum de<lb></lb>scendat.<figure id="id.064.01.029.1.jpg" xlink:href="064/01/029/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.000153">Sit Nota diuturnitas gravis B, dum descendit <lb></lb>in C super plano inclinato BC, & dato alio <lb></lb>spatio BG.</s> </p> <p type="main"> <s id="s.000154">Quaerendum quanta sit diuturnitas gravis in BG.</s> </p> <p type="main"> <s id="s.000155">Intelligatur BC diuturnitas ipsius BC, & fiat <lb></lb>BH, media inter BC, & BG, quae erit diu<lb></lb>turnitas quaesita.</s> </p> <p type="main"> <s id="s.000156">Quoniam BC, & BG sunt in duplicata ratio<lb></lb>ne diuturnitatum BC, & BH, per constructio<lb></lb>nem; per ipsa cadunt gravia diuturnitatibus <lb></lb>BC, BH,<arrow.to.target n="marg26"></arrow.to.target> unde BH est diuturnitas per spa<lb></lb>tium BG quaesita. </s> <s id="s.000157">Quod, etc.</s> </p> <p type="margin"> <s id="s.000158"><margin.target id="marg26"></margin.target>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/030.jpg"></pb> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000159">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000160">Gravia descendunt super planis inclinatis per <lb></lb>spatia semper maiora, iuxta rationem, quam <lb></lb>habent impares numeri successive inter se. <figure id="id.064.01.030.1.jpg" xlink:href="064/01/030/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="11" type="proof"> <p type="main"> <s id="s.000161">Sit grave A, quod descendat super plano ABC <lb></lb>inclinato, & tempus quo descendit ab A in <lb></lb>B sit aequale tempori, quo descendit a B in C, <lb></lb>& a C in D.</s> </p> <p type="main"> <s id="s.000162">Dico quod lineae AB, BC, CD sunt inter se ut <lb></lb>1. 3. 5. &. sic deinceps.</s> </p> <p type="main"> <s id="s.000163">Sit E numerus mensurans tempus, quo A descen<lb></lb>dit in B, & F quo descendit a B in C, & G <lb></lb>quo descendit a C in D, quae tempora sunt ex <lb></lb>suppositione aequalia, & sit H quadratum ip<lb></lb>sius E, & I quadratum EF, & K quadra<lb></lb>tum totius EFG.</s> </p> <p type="main"> <s id="s.000164">Quoniam quadrata HIK sunt ut AB, AC, AD<arrow.to.target n="marg27"></arrow.to.target>, <lb></lb>quae quadrata sunt ut 1. 4. 9. sunt pariter <lb></lb>AB, AC, AD, ut 1. 4. 9. & dividendo AB, <lb></lb>BC, CD, sunt ut 1. 3. 5. & sic deinceps. <lb></lb></s> <s id="s.000165">Quod probandum erat.</s> </p> <p type="margin"> <s id="s.000166"><margin.target id="marg27"></margin.target>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/031.jpg"></pb> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000167">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000168">Si Duo gravia descendant alterum super li<lb></lb>nea perpendiculari, alterum vero super <lb></lb>inclinata; proportio velocitatum est reci<lb></lb>proca proportioni linearum.<figure id="id.064.01.031.1.jpg" xlink:href="064/01/031/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="12" type="proof"> <p type="main"> <s id="s.000169">Sit ABC planum normaliter erectum super <lb></lb>lineam orizontalem BC, cuius latus AB sit <lb></lb>perpendiculare, & AC, inclinatum.</s> </p> <p type="main"> <s id="s.000170">Dico quod proportio velocitatum solidorum gra<lb></lb>vium motorum secundum lineam AB perpen<lb></lb>dicularem, & AC inclinatum, est ut propor<lb></lb>tio longitudinis inclinatae AC ad longitudinem <lb></lb>perpendicularis AB; videlicet ita est longitudo <lb></lb>AB ad longitudinem AC, ut velocitas super <lb></lb>AC ad velocitatem in AB.</s> </p> <p type="main"> <s id="s.000171">Quoniam est ut AC ad AB, ita momentum in <lb></lb>AB, ad momentum in AC<arrow.to.target n="marg28"></arrow.to.target>; & ut momentum <lb></lb>in AB ad momentum in AC, ita velocitas in <lb></lb>AB ad velocitatem in AC<arrow.to.target n="marg29"></arrow.to.target>; ergo est etiam <lb></lb>ut AC ad AB, ita velocitas in AB ad veloci<lb></lb>tatem in AC. </s> <s id="s.000172">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000173"><margin.target id="marg28"></margin.target>Per 4. supp.</s> </p> <p type="margin"> <s id="s.000174"><margin.target id="marg29"></margin.target>Per 2. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/032.jpg"></pb> <subchap1 n="12" type="proposition"> <p type="head"> <s id="s.000175">PROPOSITIO XII.</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.000176">Gravia descendunt super plana diverse in<lb></lb>clinata tali proportione, ut si velocitas ad <lb></lb>velocitatem reciproca longitudinibus pla<lb></lb>norum ductorum ab eodem puncto, ad <lb></lb>idem planum orizontale.<figure id="id.064.01.032.1.jpg" xlink:href="064/01/032/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="13" type="proof"> <p type="main"> <s id="s.000177">Sint F, D plana inclinata ducta ad idem pla<lb></lb>num orizontale.</s> </p> <p type="main"> <s id="s.000178">Dico esse ut planum D ad planum F, ita veloci<lb></lb>tatem gravis ducti super F, ad velocitatem <lb></lb>eiusdem ducti super D.</s> </p> <p type="main"> <s id="s.000179">Ducatur perpendicularis E, & sint B, A, C ve<lb></lb>locitates gravium latorum super perpendicu<lb></lb>lari, & super planis F, D.</s> </p> <p type="main"> <s id="s.000180">Quoniam est A ad B, ut E ad F, item, & B ad <lb></lb>C, ut D, ad E<arrow.to.target n="marg30"></arrow.to.target>, erit A ad C ut D ad F<arrow.to.target n="marg31"></arrow.to.target>, sci<lb></lb>licet velocitas gravis super F ad velocitatem <lb></lb>gravis super D, ut lon<lb></lb>gitudo plani D ad longitudinem plani F. </s> <s id="s.000181">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000182"><margin.target id="marg30"></margin.target>Per 11. huius.</s> </p> <p type="margin"> <s id="s.000183"><margin.target id="marg31"></margin.target>Per 13. Quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/033.jpg"></pb> <subchap1 n="13" type="proposition"> <p type="head"> <s id="s.000184">PROPOSITIO XIII. PROB. VI.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.000185">Reperire inclinationem plani, super quo <lb></lb>grave moveatur tali velocitate quae cum <lb></lb>alia super diversa inclinatione sit in ra<lb></lb>tione data.<figure id="id.064.01.033.1.jpg" xlink:href="064/01/033/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="14" type="proof"> <p type="main"> <s id="s.000186">Moveatur grave A super recta AB, seu <lb></lb>perpendiculari, seu inclinata, & data sit <lb></lb>proportio C ad D.</s> </p> <p type="main"> <s id="s.000187">Oportet reperire aliud planum inclinatum, ita <lb></lb>ut velocitas gravis moti super AB ad velo<lb></lb>citatem alterius moti super illo reperiendo, <lb></lb>sit ut D ad C.</s> </p> <p type="main"> <s id="s.000188">Producatur BA; & fiat ut C ad D ita BA, ad <lb></lb>AE; & centro A, intervallo AE describatur <lb></lb>circulus, secans BF in F; ni secet, problema <lb></lb>insolubile est; si secat, ducatur AF, quam di<lb></lb>co esse planum quaesitum.</s> </p> <p type="main"> <s id="s.000189">Quoniam ut C ad D, ita AB ad AE, seu AF <lb></lb>per constructionem, erit C velocitas super AF, <lb></lb>& D super AB<arrow.to.target n="marg32"></arrow.to.target>, unde velocitates super ip<lb></lb>sis sunt in ratione data. </s> <s id="s.000190">Quod faciendum fuit.</s> </p> <p type="margin"> <s id="s.000191"><margin.target id="marg32"></margin.target>Per 12. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/034.jpg"></pb> <subchap1 n="14" type="proposition"> <p type="head"> <s id="s.000192">PROPOSITIO XIV. PROB. VII.</s> </p> <subchap2 n="14" type="statement"> <p type="main"> <s id="s.000193">Data linea perpendiculari, per quam grave <lb></lb>descendat, cui annectatur linea, seu pla<lb></lb>num declinans; in declinante reperire <lb></lb>punctum, quo grave perveniat eo tempo<lb></lb>re, quo pertransiverit perpendicularem.<figure id="id.064.01.034.1.jpg" xlink:href="064/01/034/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="15" type="proof"> <p type="main"> <s id="s.000194">Sit triangulum ABC orthogonaliter erectum <lb></lb>super plano orizontali BC, cuius latus AB <lb></lb>intelligatur linea perpendicularis, per quam <lb></lb>grave descendat, & latus AC planum incli<lb></lb>natum.</s> </p> <p type="main"> <s id="s.000195">Oportet in plano AC reperire punctum quo gra<lb></lb>ve perveniat eodem tempore, quo in B.</s> </p> <p type="main"> <s id="s.000196">Fiat ut AC ad AB, ita AB ad tertiam AD<arrow.to.target n="marg33"></arrow.to.target>, <lb></lb>& D erit punctum quaesitum.</s> </p> <p type="margin"> <s id="s.000197"><margin.target id="marg33"></margin.target>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000198">Quoniam velocitas super AD ad velocitatem in <lb></lb>AB est ut AB ad AC<arrow.to.target n="marg34"></arrow.to.target>, & proinde ut AD <lb></lb>ad AB per const, quae velocitates eadem con<lb></lb>tinuo duplicata proportione augentur<arrow.to.target n="marg35"></arrow.to.target>, gra<lb></lb>via in eis moventur tempore aequali, quia quo<lb></lb>tiscunque spatia sunt ut velocitates, aequali <lb></lb>peraguntur tempore, quod, etc.</s> </p> <p type="margin"> <s id="s.000199"><margin.target id="marg34"></margin.target>Per 11. huius.</s> </p> <p type="margin"> <s id="s.000200"><margin.target id="marg35"></margin.target>Per 3. & 7. huius.</s> </p> </subchap2> <pb xlink:href="064/01/035.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.000201">Corollarium 1.</s> </p> <p type="main"> <s id="s.000202">Hinc est quod in D, & B velocitates sunt ut AD, <lb></lb>AB, & ita in quibuslibet punctis respondenti<lb></lb>bus paralellis ad DB cum in AD, & AB ve<lb></lb>locitates semper eadem ratione augeantur.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000203">Corollarium 2.</s> </p> <p type="main"> <s id="s.000204">Hinc est etiam quod si esset AE aequalis AB, & <lb></lb>AF media inter AD, AE, tempus AD, & <lb></lb>proinde AB ad tempus AE, esset ut AD ad <lb></lb>AF<arrow.to.target n="marg36"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000205"><margin.target id="marg36"></margin.target>Per 7. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000206">Corollarium 3.</s> </p> <p type="main"> <s id="s.000207">Si AE est quadrupla AD, AF erit dupla AD, <lb></lb>unde tempus AE erit duplum tempori AB.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000208">Corollarium 4.</s> </p> <p type="main"> <s id="s.000209">Si AC esset quadrupla AD, grave moveretur <lb></lb>temporibus aequalibus per AB, AD, DC.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/036.jpg"></pb> <subchap1 n="15" type="proposition"> <p type="head"> <s id="s.000210">PROPOSITIO XV.</s> </p> <subchap2 n="15" type="statement"> <p type="main"> <s id="s.000211">Si duo gravia descendunt alterum quidem <lb></lb>perpendiculariter, alterum vero super pla<lb></lb>no declinante, perveniunt ad idem pla<lb></lb>num Orizontale tali ratione, ut sit eadem <lb></lb>proportio inter diuturnitates eorum, quae <lb></lb>inter perpendicularem, & declinantem.<figure id="id.064.01.036.1.jpg" xlink:href="064/01/036/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="16" type="proof"> <p type="main"> <s id="s.000212">Sit linea AB perpendiculariter erecta super <lb></lb>plano Orizontali BC, & AC planum declinans.</s> </p> <p type="main"> <s id="s.000213">Dico quod diuturnitates gravium descendentium <lb></lb>per AB, & per AC, sunt ut AB ad AC.</s> </p> <p type="main"> <s id="s.000214">Fiat AD tertia proportionalis ad AC, & AB<arrow.to.target n="marg37"></arrow.to.target>,</s> </p> <p type="margin"> <s id="s.000215"><margin.target id="marg37"></margin.target>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000216">Quoniam est ut AD ad AC ita quadratum tem<lb></lb>poris AD ad quadratum temporis AC<arrow.to.target n="marg38"></arrow.to.target>, & <lb></lb>tempora AD, & AB sunt aequalia<arrow.to.target n="marg39"></arrow.to.target>, & proin<lb></lb>de eorum quadrata<arrow.to.target n="marg40"></arrow.to.target>, ergo ut AD, ad AC <lb></lb>ita quadratum temporis AB ad quadratum <lb></lb>temporis AC, sed ut AD ad AC ita quadra<lb></lb>tum AB ad quadratum AC<arrow.to.target n="marg41"></arrow.to.target>, ergo ut quadratum temporis AB ad quadratum temporis A<lb></lb>C, ita quadratum AB ad quadratum AC<arrow.to.target n="marg42"></arrow.to.target>, <lb></lb>sed quia latera sunt inter se ut eorum qua<lb></lb>drata<arrow.to.target n="marg43"></arrow.to.target>, est ut AB ad AC ita tempus AB ad <lb></lb>tempus AC. </s> <s id="s.000217">Quod, etc.</s> </p> <p type="margin"> <s id="s.000218"><margin.target id="marg38"></margin.target>Per cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000219"><margin.target id="marg39"></margin.target>Per 14. huius.</s> </p> <p type="margin"> <s id="s.000220"><margin.target id="marg40"></margin.target>Per 2. pron.</s> </p> <p type="margin"> <s id="s.000221"><margin.target id="marg41"></margin.target>Per 19. Sexti.</s> </p> <p type="margin"> <s id="s.000222"><margin.target id="marg42"></margin.target>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000223"><margin.target id="marg43"></margin.target>Per 22. Sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/037.jpg"></pb> <subchap1 n="16" type="proposition"> <p type="head"> <s id="s.000224">PROPOSITIO XVI. PROBL. VIII.</s> </p> <subchap2 n="16" type="statement"> <p type="main"> <s id="s.000225">Data linea perpendiculari, & plano decli<lb></lb>nante; reperire in perpendiculari produ<lb></lb>cta punctum, quo perveniat grave eo tem<lb></lb>pore, quo pertransit planum inclinatum.<figure id="id.064.01.037.1.jpg" xlink:href="064/01/037/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="17" type="proof"> <p type="main"> <s id="s.000226">Data sit perpendicularis AB, cui connexum <lb></lb>planum inclinatum AD.</s> </p> <p type="main"> <s id="s.000227">Oportet in AB producta reperire punctum, quo <lb></lb>perveniat grave eo tempore, quo pervenit in <lb></lb>puncto D.</s> </p> <p type="main"> <s id="s.000228">In puncto D perpendicularis erigatur ad AD, & <lb></lb>protrahatur usquequo coeat cum AB produ<lb></lb>cta in E, & E est punctum quaesitum.</s> </p> <p type="main"> <s id="s.000229">Quoniam triangula, ADE, AEC sint aequian<lb></lb>gula, cum anguli ADE, AEC sint aequales, <lb></lb>nempe recti, & BAD communis<arrow.to.target n="marg44"></arrow.to.target>, sunt etiam <lb></lb>similia<arrow.to.target n="marg45"></arrow.to.target>, ergo ut AC ad AE, ita AE ad AD<arrow.to.target n="marg46"></arrow.to.target>, <lb></lb>unde tempora per AD, & AE sunt aequalia<arrow.to.target n="marg47"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000230"><margin.target id="marg44"></margin.target>Per 32. prim.</s> </p> <p type="margin"> <s id="s.000231"><margin.target id="marg45"></margin.target>Per 4. sexti.</s> </p> <p type="margin"> <s id="s.000232"><margin.target id="marg46"></margin.target>Per 4. sexti.</s> </p> <p type="margin"> <s id="s.000233"><margin.target id="marg47"></margin.target>Per 14 huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000234">Corollarium</s> </p> <p type="main"> <s id="s.000235">Hinc est quod super plano AC erit AD men<lb></lb>sura diuturnitatis motus peracti super AE.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/038.jpg"></pb> <subchap1 n="17" type="proposition"> <p type="head"> <s id="s.000236">PROPOSITIO XVII. PROBL. IX.</s> </p> <subchap2 n="17" type="statement"> <p type="main"> <s id="s.000237">Dato plano declinante, super quo grave de<lb></lb>scendat, & dato alio plano minus declinan<lb></lb>te, in hoc reperire punctum, quo perveniat <lb></lb>mobile eo tempore, quo pertransit dictum <lb></lb>planum magis declinans.<figure id="id.064.01.038.1.jpg" xlink:href="064/01/038/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="18" type="proof"> <p type="main"> <s id="s.000238">Sint plana AB, AC quorum AC minus in<lb></lb>clinatum.</s> </p> <p type="main"> <s id="s.000239">Oportet in AC reperire punctum, quo grave per<lb></lb>veniat, quando pervenit in B.</s> </p> <p type="main"> <s id="s.000240">Fiat ut AC ad AB ita AB ad AD, & dico D <lb></lb>esse punctum quaesitum.</s> </p> <p type="main"> <s id="s.000241">Quoniam ut AC ad AD ita est quadratum AC <lb></lb>ad quadratum AB<arrow.to.target n="marg48"></arrow.to.target>, & ut AC ad AD ita <lb></lb>quadratum temporis AC ad quadratum tem<lb></lb>poris AD<arrow.to.target n="marg49"></arrow.to.target> ergo ut quadratum AC ad qua<lb></lb>dratum AB, ita quadratum temporis AC ad <lb></lb>quadratum temporis AD Vnde AC ad AB<lb></lb>ut tempus AC ad tempus AD<arrow.to.target n="marg50"></arrow.to.target>, sed ut AC <lb></lb>ad AB, ita tempus AC ad tempus AB<arrow.to.target n="marg51"></arrow.to.target>, ergo <lb></lb>tempora AB, AD, sunt aequalia. </s> <s id="s.000242">Quod, etc.</s> </p> <p type="margin"> <s id="s.000243"><margin.target id="marg48"></margin.target>Per 19. sexti.</s> </p> <p type="margin"> <s id="s.000244"><margin.target id="marg49"></margin.target>Per cot. 7. huius.</s> </p> <p type="margin"> <s id="s.000245"><margin.target id="marg50"></margin.target>Per 22. sexti.</s> </p> <p type="margin"> <s id="s.000246"><margin.target id="marg51"></margin.target>Per 15. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/039.jpg"></pb> <subchap1 n="18" type="proposition"> <p type="head"> <s id="s.000247">PROPOSITIO XVIII. PROBL. X.</s> </p> <subchap2 n="18" type="statement"> <p type="main"> <s id="s.000248">Datis planis declinantibus ortis ab eodem <lb></lb>puncto, reperire in magis declinante pun<lb></lb>ctum quo grave perveniat eo tempore, quo <lb></lb>pertransit planum minus declinans.<figure id="id.064.01.039.1.jpg" xlink:href="064/01/039/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="19" type="proof"> <p type="main"> <s id="s.000249">Datum sit planum minus declinans AC, & <lb></lb>magis AD, terminantia super plano ori<lb></lb>zontali BD.</s> </p> <p type="main"> <s id="s.000250">Oportet in AD producta reperire punctum, quo <lb></lb>perveniat grave eo tempore, quo pertransivit <lb></lb>planum minus declinans AC.</s> </p> <p type="main"> <s id="s.000251">Fiat ut AD ad AC ita AC ad dictam AD pro<lb></lb>ductam in E, quod est punctum quaesitum.</s> </p> <p type="main"> <s id="s.000252">Quoniam ut AE ad AD ita est quadratum AC <lb></lb>ad quadratum AD<arrow.to.target n="marg52"></arrow.to.target>, sed AE ad AD est ut <lb></lb>quadratum tempo<lb></lb>ris AE, ad quadratum temporis AD<arrow.to.target n="marg53"></arrow.to.target>, ergo ut quadra<lb></lb>tum AC ad quadratum AD, ita quadratum temporis AE ad qua<lb></lb>dratum temporis AD<arrow.to.target n="marg54"></arrow.to.target>, unde AC ad AD ut <lb></lb>tempus AE ad tempus AD<arrow.to.target n="marg55"></arrow.to.target>, sed AC ad AD <lb></lb>est ut tempus AC ad tempus AD<arrow.to.target n="marg56"></arrow.to.target>, ergo tem<lb></lb>pora AE, AC sunt aequalia. </s> <s id="s.000253">Quod, etc.</s> </p> <p type="margin"> <s id="s.000254"><margin.target id="marg52"></margin.target>Per 19. sexti.</s> </p> <p type="margin"> <s id="s.000255"><margin.target id="marg53"></margin.target>Per cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000256"><margin.target id="marg54"></margin.target>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000257"><margin.target id="marg55"></margin.target>Per 22. sexti.</s> </p> <p type="margin"> <s id="s.000258"><margin.target id="marg56"></margin.target>Per 15. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/040.jpg"></pb> <subchap1 n="19" type="proposition"> <p type="head"> <s id="s.000259">PROPOSITIO XIX. PROBL. XI.</s> </p> <subchap2 n="19" type="statement"> <p type="main"> <s id="s.000260">Dato motus naturali gravis quomodocumque <lb></lb>ad punctum datum, reperire seu in perpen<lb></lb>diculari, seu in plano quomodolibet incli<lb></lb>nato punctum, a quo digressum, perveniat <lb></lb>ad idem punctum quo prius, tempore aequali.<figure id="id.064.01.040.1.jpg" xlink:href="064/01/040/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="20" type="proof"> <p type="main"> <s id="s.000261">Sit AB linea quomodocumque aut perpendicu<lb></lb>laris, seu planum inclinatum; super qua <lb></lb>grave descendat in B, & data sit quaecunque <lb></lb>linea BC, aut perpendicularis, aut quomodo<lb></lb>libet inclinata, quae cum AB, coeat in B.</s> </p> <p type="main"> <s id="s.000262">Oportet in BC reperire punctum, a quo grave digres<lb></lb>sum perveniat in B tempore quo pervenit ab A in idem B.</s> </p> <p type="main"> <s id="s.000263">Ducatur AC orizontalis, & fiat BD tertia pro<lb></lb>portionalis ad CB AB<arrow.to.target n="marg57"></arrow.to.target>, & D est punctum <lb></lb>quaesitum. </s> <s id="s.000264">Quod ut probetur.</s> </p> <p type="margin"> <s id="s.000265"><margin.target id="marg57"></margin.target>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000266">Fiat iterum rectae AC paralella, & aequalis BE, & <lb></lb>ducta EA, secetur recta BF parallela ipsi AD.</s> </p> <p type="main"> <s id="s.000267">Quoniam AF, BD sunt pariter inclinatae, & <lb></lb>aequales<arrow.to.target n="marg58"></arrow.to.target>, gravia per ipsas aequali tempore mo<lb></lb>ventur<arrow.to.target n="marg59"></arrow.to.target>, sed per AF, grave movetur tempo<lb></lb>re quo per AB<arrow.to.target n="marg60"></arrow.to.target>, ergo per BD movetur pari<lb></lb>ter tempore quo per AB<arrow.to.target n="marg61"></arrow.to.target>, quod, etc.</s> </p> <p type="margin"> <s id="s.000268"><margin.target id="marg58"></margin.target>Per 33. Primi.</s> </p> <p type="margin"> <s id="s.000269"><margin.target id="marg59"></margin.target>Per 3. pronun.</s> </p> <p type="margin"> <s id="s.000270"><margin.target id="marg60"></margin.target>Per 17 huius.</s> </p> <p type="margin"> <s id="s.000271"><margin.target id="marg61"></margin.target>Per 1. pron.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000272">Corollarium</s> </p> <p type="main"> <s id="s.000273">Hinc est quod super plano CB, DB est mensura <lb></lb>diuturnitatis motus in AB.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/041.jpg"></pb> <subchap1 n="20" type="proposition"> <p type="head"> <s id="s.000274">PROPOSITIO XX. PROBL. XII.</s> </p> <subchap2 n="20" type="statement"> <p type="main"> <s id="s.000275">Datis duobus planis diverse inclinatis lon<lb></lb>gitudinis notae; & nota diuturnitate gra<lb></lb>vis moti super uno, reperire diuturnita<lb></lb>tem si moveatur super alio.<figure id="id.064.01.041.1.jpg" xlink:href="064/01/041/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="21" type="proof"> <p type="main"> <s id="s.000276">Sint plana AB, CD inclinata, & sit data diu<lb></lb>turnitas E plani AB.</s> </p> <p type="main"> <s id="s.000277">Oportet reperire diuturnitatem plani CD.</s> </p> <p type="main"> <s id="s.000278">Fiat AF, paralella, & aequalis datae CD, in qua <lb></lb>reperiatur punctum G quo perveniat grave, <lb></lb>tempore quo in B<arrow.to.target n="marg62"></arrow.to.target>, unde E est etiam diuturnitas <lb></lb>spatij AG, quo dato, & spatio AF perquiratur <lb></lb>eias diuturnitas, quae sit H<arrow.to.target n="marg63"></arrow.to.target>, & dico H esse <lb></lb>diuturnitatem quae grave descendit in CD.</s> </p> <p type="margin"> <s id="s.000279"><margin.target id="marg62"></margin.target>Per 17. huius.</s> </p> <p type="margin"> <s id="s.000280"><margin.target id="marg63"></margin.target>Per 9. huius.</s> </p> <p type="main"> <s id="s.000281">Quoniam E, H sunt diuturnitates gravium de<lb></lb>scendentium in AG, seu AB, & AF, per con<lb></lb>structionem, & AF est aequalis, & paralella <lb></lb>datae CD per constructionem, sunt etiam E, H <lb></lb>diuturnitates ipsarum AB, & CD<arrow.to.target n="marg64"></arrow.to.target>, unde <lb></lb>reperta est diuturnitas ipsius CD. </s> <s id="s.000282">Quod, etc.</s> </p> <p type="margin"> <s id="s.000283"><margin.target id="marg64"></margin.target>Per 3. pron.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/042.jpg"></pb> <subchap1 n="21" type="proposition"> <p type="head"> <s id="s.000284">PROPOSITIO XXI. PROBL. XIII.</s> </p> <subchap2 n="21" type="statement"> <p type="main"> <s id="s.000285">Datis duabus diuturnitatibus, quarum prior <lb></lb>sit gravis moti super plano dato longitu<lb></lb>dinis notae, & dato alio plano diversimo<lb></lb>de declinante; reperiendum est in eo pun<lb></lb>ctum, quo grave perveniat in secunda <lb></lb>diuturnitate data.<figure id="id.064.01.042.1.jpg" xlink:href="064/01/042/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="22" type="proof"> <p type="main"> <s id="s.000286">Dato plano declinante AB, super quo grave <lb></lb>A moveatur diuturnitate C, & dato alio <lb></lb>plano D declinationis quae sit dissimilis decli<lb></lb>nationi datae AB; data itidem diuturnitate E.</s> </p> <p type="main"> <s id="s.000287">Oportet reperire in D punctum quo grave per<lb></lb>veniat in diuturnitate E.</s> </p> <p type="main"> <s id="s.000288">Ducatur AF parallela ipsi D, in eaque reperia<lb></lb>tur punctum F, quo grave perveniat tempore quo <lb></lb>in B<arrow.to.target n="marg65"></arrow.to.target>, & praescribatur in eadem spatium AG per <lb></lb>quod moveatur in diuturnitate E<arrow.to.target n="marg66"></arrow.to.target>, & fiat DH <lb></lb>aequalis ipsi AG, & dico H esse punctum quaesitum.</s> </p> <p type="margin"> <s id="s.000289"><margin.target id="marg65"></margin.target>Per 17. huius.</s> </p> <p type="margin"> <s id="s.000290"><margin.target id="marg66"></margin.target>Per 8. huius.</s> </p> <p type="main"> <s id="s.000291">Quoniam diuturnitates in AB, AF sunt aequales <lb></lb>per constructionem, & C, E sunt diuturnita<lb></lb>tes super planis AF, AG per constructionem, <lb></lb>sunt etiam diuturnitates super AB, AG, & <lb></lb>proinde super DH ipsi AG aequali, & para<lb></lb>lellae, quod, etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/043.jpg"></pb> <subchap1 n="22" type="proposition"> <p type="head"> <s id="s.000292">PROPOSITIO XXII.</s> </p> <subchap2 n="22" type="statement"> <p type="main"> <s id="s.000293">Data perpendiculari seu plano quomodoli<lb></lb>bet inclinato diuturnitatis notae, & assi<lb></lb>gnata ubivis quaecunque eius portione, re<lb></lb>perire eius diuturnitatem.<figure id="id.064.01.043.1.jpg" xlink:href="064/01/043/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="23" type="proof"> <p type="main"> <s id="s.000294">Data linea AB perpendiculari aut inclina<lb></lb>ta, cuius, diuturnitas sit CD, dataque qua<lb></lb>cunque eius portione EF.</s> </p> <p type="main"> <s id="s.000295">Quaerenda eius diuturnitas.</s> </p> <p type="main"> <s id="s.000296">Fiat CG diuturnitas AE, & CH diuturnitas <lb></lb>AF<arrow.to.target n="marg67"></arrow.to.target>, GH est diuturnitas quaesita.</s> </p> <p type="margin"> <s id="s.000297"><margin.target id="marg67"></margin.target>Per 5. aut 9. huius.</s> </p> <p type="main"> <s id="s.000298">Quoniam CH est diuturnitas AF per constr. ab <lb></lb>ea ablata CG diuturnitate AE per const. resi<lb></lb>duum GH est diuturnitas portionis EF quod, <lb></lb>etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/044.jpg"></pb> <subchap1 n="23" type="proposition"> <p type="head"> <s id="s.000299">PROPOSITIO XXIII.</s> </p> <subchap2 n="23" type="statement"> <p type="main"> <s id="s.000300">Duo gravia descendentia super planis diversa <lb></lb>ratione declinantibus, perveniunt ad idem <lb></lb>planum orizontale ea ratione, ut sit eadem <lb></lb>proportio inter diuturnitates, quae inter <lb></lb>dicta plana si ab eodem puncto ad idem <lb></lb>planum orizontale producta sint.<figure id="id.064.01.044.1.jpg" xlink:href="064/01/044/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="24" type="proof"> <p type="main"> <s id="s.000301">Datis planis AB, AC declinantibus, ductis <lb></lb>ab eodem puncto A ad planum orizontale BC. <lb></lb> </s> <s id="s.000302">Dico quod diuturnitates gravium descendentium <lb></lb>per AB, AC sint ut AB ad AC.</s> </p> <p type="main"> <s id="s.000303">Fiat ut AC ad AB ita AB ad AD, ita ut grave <lb></lb>perveniat in D eodem tempore quo pervenit in B<arrow.to.target n="marg68"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000304"><margin.target id="marg68"></margin.target>Per 17. huius.</s> </p> <p type="main"> <s id="s.000305">Quoniam est ut AD ad AC, ita quadratum tem<lb></lb>poris AD ad quadratum temporis AC<arrow.to.target n="marg69"></arrow.to.target>, & <lb></lb>tempora AD, AB sunt aequalia<arrow.to.target n="marg70"></arrow.to.target>, & proinde <lb></lb>eorum quadrata; ergo ut AD ad AC ita qua<lb></lb>dratum temporis AB, ad quadratum tempo<lb></lb>ris AC<arrow.to.target n="marg71"></arrow.to.target>, sed ut AD ad AC, ita quadra<lb></lb>tum AB ad quadratum AC<arrow.to.target n="marg72"></arrow.to.target>, ergo ut quadra<lb></lb>tum temporis AB ad quadratum temporis AC, <lb></lb>ita quadratum AB ad quadratum AC, ergo <lb></lb>ut tempus AB ad tempus AC, ita AB ad AC<arrow.to.target n="marg73"></arrow.to.target>. </s> <s id="s.000306">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000307"><margin.target id="marg69"></margin.target>Per Cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000308"><margin.target id="marg70"></margin.target>Per const.</s> </p> <p type="margin"> <s id="s.000309"><margin.target id="marg71"></margin.target>Per 2. pronun.</s> </p> <p type="margin"> <s id="s.000310"><margin.target id="marg72"></margin.target>Per 10. sexti.</s> </p> <p type="margin"> <s id="s.000311"><margin.target id="marg73"></margin.target>Per 22. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/045.jpg"></pb> <subchap1 n="24" type="proposition"> <p type="head"> <s id="s.000312">PROPOSITIO XXIV</s> </p> <subchap2 n="24" type="statement"> <p type="main"> <s id="s.000313">Datis planis, & perpendiculari ad eadem li<lb></lb>nea orizontali egressis, quae coeant infra in <lb></lb>eodem puncto, gravia super ipsis mota <lb></lb>procedunt ea ratione, ut sit eadem propor<lb></lb>tion inter diuturnitates, quae inter longitu<lb></lb>dines planorum, & dictam perpendicularem.<figure id="id.064.01.045.1.jpg" xlink:href="064/01/045/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="25" type="proof"> <p type="main"> <s id="s.000314">Data sit linea orizontalis AB, in qua ini<lb></lb>tium sumant plana declinantia AC, DC, <lb></lb>nec non perpendicularis BC coeuntia in puncto C.</s> </p> <p type="main"> <s id="s.000315">Dico quod diuturnitates gravium super ipsis mo<lb></lb>torum, sunt ut AC, DC, BC.</s> </p> <p type="main"> <s id="s.000316">Ducatur CE paralella ipsi AB, & a puncto A du<lb></lb>cantur paralellae ipsis CB, CD, & sint AE, AF.</s> </p> <p type="main"> <s id="s.000317">Quoniam diuturnitates super planis AF, AC, <lb></lb>sunt ut AF, AC<arrow.to.target n="marg74"></arrow.to.target>, & super planis eisdem, & <lb></lb>perpendiculari AE, sunt ut AF, seu AC ad <lb></lb>AE<arrow.to.target n="marg75"></arrow.to.target>, & AE, AF sunt paralellae ipsis CD, <lb></lb>CB, & eisdem aequales,<arrow.to.target n="marg76"></arrow.to.target>, sequitur quod etiam <lb></lb>super AC, DC, BC diuturnitates sunt iuxta <lb></lb>proportiones longitudinum<arrow.to.target n="marg77"></arrow.to.target>, Quod probandum fuit.</s> </p> <p type="margin"> <s id="s.000318"><margin.target id="marg74"></margin.target>Per 23. huius.</s> </p> <p type="margin"> <s id="s.000319"><margin.target id="marg75"></margin.target>Per 15. huius.</s> </p> <p type="margin"> <s id="s.000320"><margin.target id="marg76"></margin.target>Per 33. prim.</s> </p> <p type="margin"> <s id="s.000321"><margin.target id="marg77"></margin.target>Per 3. pron.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/046.jpg"></pb> <subchap1 n="25" type="proposition"> <p type="head"> <s id="s.000322">PROPOSITIO XXV.</s> </p> <subchap2 n="25" type="statement"> <p type="main"> <s id="s.000323">In circulo Orthogonaliter erecto, si a sum<lb></lb>mitate ad puncta peripheriae ducantur pla<lb></lb>na, quo tempore grave perpendiculariter <lb></lb>inde pervenit ad planum orizontale; si de<lb></lb>scendat per dicta plana, eodem perveniet <lb></lb>respective ad quodlibet dictorum puncto<lb></lb>rum peripheriae.<figure id="id.064.01.046.1.jpg" xlink:href="064/01/046/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="26" type="proof"> <p type="main"> <s id="s.000324">Sit circulus cuius centrum B, & diameter AC <lb></lb>erectus super plano orizontali GC, & in eo <lb></lb>ducta sint plana declinantia a puncto A ad <lb></lb>puncta peripheriae DEF, & descendant gravia <lb></lb>super dicta plana, & perpendiculariter.</s> </p> <p type="main"> <s id="s.000325">Dico quod eodem tempore pervenient ad, D, E, F, C.</s> </p> <p type="main"> <s id="s.000326">Ducantur DC, EC, FC.</s> </p> <p type="main"> <s id="s.000327">Quoniam puncta praedicta sunt ea, in quae cadunt <lb></lb>perpendicularia ducta a puncto C in AD, AE, <lb></lb>AF<arrow.to.target n="marg78"></arrow.to.target>, eo perveniunt gravia eodem tempore <lb></lb>quo in C<arrow.to.target n="marg79"></arrow.to.target>. </s> <s id="s.000328">Quod probandum fuit.</s> </p> <p type="margin"> <s id="s.000329"><margin.target id="marg78"></margin.target>Per 30. Tertij.</s> </p> <p type="margin"> <s id="s.000330"><margin.target id="marg79"></margin.target>Per 16. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/047.jpg"></pb> <subchap1 n="26" type="proposition"> <p type="head"> <s id="s.000331">PROPOSITIO XXVI.</s> </p> <subchap2 n="26" type="statement"> <p type="main"> <s id="s.000332">Sit in circulo erecto, a puncto inferiori ducan<lb></lb>tur plana ad puncta peripheriae, & a dictis <lb></lb>punctis descendant gravia super dicta pla<lb></lb>na eodem tempore quo a puncto supremo <lb></lb>descendit aliud grave perpendiculariter; <lb></lb>pervenient omnia eodem instanti ad di<lb></lb>ctum punctum inferius.<figure id="id.064.01.047.1.jpg" xlink:href="064/01/047/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="27" type="proof"> <p type="main"> <s id="s.000333">Sit circulus cuius diameter ABC erectus super <lb></lb>plano orizontali, quod tangat in C, & a C <lb></lb>ducantur plana CD, CE, & a punctis, E, D <lb></lb>gravia descendant super dicta plana, nec non, <lb></lb>& a puncto supremo A perpendiculariter.</s> </p> <p type="main"> <s id="s.000334">Dico quod eodem tempore perveniunt in C.<lb></lb></s> </p> <p type="main"> <s id="s.000335">A puncto A ducantur AF, AG paralellae ipsis <lb></lb>CE, CD, & ducantur AF, FC.</s> </p> <p type="main"> <s id="s.000336">Quoniam in triangulis AEC, AFC anguli al<lb></lb>terni FAC, ACE sint aequales,<arrow.to.target n="marg80"></arrow.to.target>, & anguli<lb></lb> <pb xlink:href="064/01/048.jpg"></pb>AFC, AEC sunt etiam aequales puta re<lb></lb>cti<arrow.to.target n="marg81"></arrow.to.target>, & basis AC communis, Triangula sunt <lb></lb>aequalia<arrow.to.target n="marg82"></arrow.to.target>, & proinde AF est aequalis CE, quod <lb></lb>idem probabitur de reliquis, ergo cum AF, <lb></lb>CE, & reliquae sint paralellae, & aequales, gra<lb></lb>via per CE, CD pervenient in C eodem tem<lb></lb>pore, quo digressa ab A perveniunt ad puncta <lb></lb>FG, sed haec eodem tempore quo perpendicula<lb></lb>riter pervenit in C<arrow.to.target n="marg83"></arrow.to.target>, ergo etiam ea quae per <lb></lb>CE, CD. </s> <s id="s.000337">Quod, etc.</s> </p> <p type="margin"> <s id="s.000338"><margin.target id="marg80"></margin.target>Per 29. primi.</s> </p> <p type="margin"> <s id="s.000339"><margin.target id="marg81"></margin.target>Per 30. Tertij.</s> </p> <p type="margin"> <s id="s.000340"><margin.target id="marg82"></margin.target>Per 16. primi.</s> </p> <p type="margin"> <s id="s.000341"><margin.target id="marg83"></margin.target>Per 25. huius.</s> </p> <pb xlink:href="064/01/049.jpg"></pb> <p type="main"> <s id="s.000342">POSTULATUM VII</s> </p> <p type="main"> <s id="s.000343">Ductis planis inclinatis, & linea perpen<lb></lb>diculari inter binas paralellas orizon<lb></lb>tales, Gravia super illis mota ubi perveni<lb></lb>unt ad paralellam inferiorem habent aequa<lb></lb>les velocitatis gradus; & proinde si ab in<lb></lb>de infra sortiantur parem inclinationem, <lb></lb>aequevelociter moventur.</s> </p> <p type="main"> <s id="s.000344">Videtur probabile. </s> <s id="s.000345">Primo quia si diuturni<lb></lb>tates sunt longitudinibus proportionales, ut <lb></lb>propositione 15. huius probatum fuit, credibile <lb></lb>est motus in fine esse aequales.</s> </p> <p type="main"> <s id="s.000346">Secundo. </s> <s id="s.000347">Argumento ducto ab experientia pen<lb></lb>dulorum, quae quantumvis longiora, aut brevio<lb></lb>ra, & proinde circa finem magis, aut minus in<lb></lb>clinata, pariter ascendunt, si pariter descendant.</s> </p> <p type="main"> <s id="s.000348">Tertio. </s> <s id="s.000349">Quia videmus aquam per siphones rectos, <lb></lb>sive obliquos, seu inclinatos ductam, pariter <lb></lb>ascendere, si pariter descendat. </s> <s id="s.000350">Ceterum fa<lb></lb>teor minorem evidentiam hoc postulatum caete<lb></lb>ris praemissis prae se ferre, quae fuit causa quod <lb></lb>illud, ut in praefatione, segregaverim, & se<lb></lb>quentia, alia methodo, tangendo fere tantum<lb></lb>modo exposuerim, & a pluribus alijs proposi<lb></lb>tionibus, quae hinc deduci facile possent, data <lb></lb>opera abstinuerim.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/050.jpg"></pb> <subchap1 n="27" type="proposition"> <p type="head"> <s id="s.000351">PROPOSITIO XXVII. PROBL. XIV.</s> </p> <subchap2 n="27" type="statement"> <p type="main"> <s id="s.000352">Dato gravi moto perpendiculariter per spa<lb></lb>tium datum diuturnitate data, quod per<lb></lb>ficiat motum super plano inclinato per <lb></lb>spatium itidem datum; perquirere in ipso <lb></lb>diuturnitatem.<figure id="id.064.01.050.1.jpg" xlink:href="064/01/050/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="28" type="proof"> <p type="main"> <s id="s.000353">Moveatur grave A perpendiculariter per <lb></lb>spatium AB diuturnitate C, & perseve<lb></lb>ret in motu super spatio BD in plano incli<lb></lb>nato BD.</s> </p> <p type="main"> <s id="s.000354">Venanda est diuturnitas eius in ipso BD.</s> </p> <p type="main"> <s id="s.000355">Producatur DB donec concurrat cum AE orizon<lb></lb>taliter ducta ab A in E, & fiat ut AB ad EB, <lb></lb>ita diuturnitas C ad diuturnitatem G, quae <lb></lb>idcirco erit diuturnitas ipsius EB<arrow.to.target n="marg84"></arrow.to.target>, & sit H <lb></lb>quadratum diuturnitatis G, & fiat ut EB <lb></lb>ad ED, ita quadratum H ad aliud quod sit I a <lb></lb>cuius latere K, quod est diuturnitas ipsius <lb></lb>ED, ablata KL aequali G, erit LM reli<lb></lb>quum diuturnitas BD quaesita.</s> </p> <p type="margin"> <s id="s.000356"><margin.target id="marg84"></margin.target>* Est quarta tertij.</s> </p> <pb xlink:href="064/01/051.jpg"></pb> <p type="main"> <s id="s.000357">Quoniam notum est triangulum AEB, cum no<lb></lb>tus sit angulus AEB aequalis alterno EDF <lb></lb>inclinationis notae, & EAB rectus ex constru<lb></lb>ctione, & notum latus AB ex hypotesi, notum <lb></lb>erit etiam latus EB, & quia diuturnitas in <lb></lb>plano BD est eadem ac si motus antecedens <lb></lb>esset per EB<arrow.to.target n="marg85"></arrow.to.target>, EB & ED sunt in duplicata <lb></lb>ratione diuturnitatum G, K ex con<lb></lb>structio<lb></lb>ne; unde a K deducta KL aequali G ex constructione, remanet LM diuturnitas BD. </s> <s id="s.000358">Quod, etc.</s> </p> <p type="margin"> <s id="s.000359"><margin.target id="marg85"></margin.target>Per 22 huius.</s> </p> <p type="main"> <s id="s.000360">Inde sequitur quod summa diuturnitatum C, & <lb></lb>LM, est diuturnitas totius ABD.**</s> </p> <p type="main"> <s id="s.000361">Eadem operatione pariter reperietur diuturni<lb></lb>tas BD si BD sit perpendicularis, & AB <lb></lb>inclinata.</s> </p> <p type="main"> <s id="s.000362">Item si ambo sint plana inclinata.</s> </p> <p type="main"> <s id="s.000363">Ducta AD facile reperietur diuturnitas in ipsa <lb></lb>si fiat ut ED ad AD, ita K ad aliud per <lb></lb>21. huius.</s> </p> <pb xlink:href="064/01/052.jpg"></pb> <p type="main"> <s id="s.000364">Ducto alio plano puta DN, reperietur eius <lb></lb>diuturnitas.<figure id="id.064.01.052.1.jpg" xlink:href="064/01/052/1.jpg"></figure></s> </p> <p type="main"> <s id="s.000365">Si fiat ut ED ad OD ita diuturnitas ipsius <lb></lb>ED puta L ad diuturnitatem OD, quae sit <lb></lb>P, deinde ut OD ad ON ita quadratum <lb></lb>diuturnitatis P ad aliud quadratum, cuius <lb></lb>Radix erit diuturnitas ipsius DN.</s> </p> <p type="main"> <s id="s.000366">Ex his patet quod si addantur plura plana ea<lb></lb>dem ratione reperientur eius diuturnitates.</s> </p> <pb xlink:href="064/01/053.jpg"></pb> <figure id="id.064.01.053.1.jpg" xlink:href="064/01/053/1.jpg"></figure> <p type="main"> <s id="s.000367">Ex his itidem patet quod si in circulo dentur <lb></lb>plura, plana v.g. FA, AC, CB, & data sit <lb></lb>diuturnitas super diametro orizonti perpen<lb></lb>diculari, dabitur diuturnitas cuiusvis dicto<lb></lb>rum FA, AC, CT, & omnium simul.7*</s> </p> <p type="main"> <s id="s.000368">In super ex his facile cognosces esse breviorem, <lb></lb>diuturnitatem per AC, CB, simul, quam per <lb></lb>AB;8* nam ducta AE perpendiculari ad BC <lb></lb>productam in D ad orizontalem AD, diutur<lb></lb>nitas motus in AC, super DB mensuratur per <lb></lb>EC<arrow.to.target n="marg86"></arrow.to.target>, ergo addita CB, quae est eiusdem diutur<lb></lb>nitatis, fuerit ne motus per AC an per DC<arrow.to.target n="marg87"></arrow.to.target>, <lb></lb>tota EB erit mensura diuturnitatis in ACB, <lb></lb>sed AB mensurat diuturnitatem ipsius AB <lb></lb>respectu eiusdem DB<arrow.to.target n="marg88"></arrow.to.target>, quae est maior quam <lb></lb>EB<arrow.to.target n="marg89"></arrow.to.target>, maior ergo est diuturnitas in AB quam <lb></lb>in ACB.</s> </p> <p type="margin"> <s id="s.000369"><margin.target id="marg86"></margin.target>Per 7. post.</s> </p> <p type="margin"> <s id="s.000370"><margin.target id="marg87"></margin.target>** Est pars secunda quartae tertij.</s> </p> <p type="margin"> <s id="s.000371"><margin.target id="marg88"></margin.target>*** Est Tertia tertij.</s> </p> <p type="margin"> <s id="s.000372"><margin.target id="marg89"></margin.target>**** Est corol. quartae tertij.</s> </p> <p type="main"> <s id="s.000373">Eadem prorsus ratione probabitur citius grave <lb></lb>descendere per FA, AC, CB, simul, quam per <lb></lb>planum ductum ab F in B.9*</s> </p> <p type="main"> <s id="s.000374">In figura propositionis 27. si facto H quadrato <lb></lb>diuturnitatis G, fiat ML aequalis C, cui ad<pb xlink:href="064/01/054.jpg"></pb>dita LK aequali G, fiat I quadratum MK, <lb></lb>& ut H ad I, ita EB ad ED; MK erit <lb></lb>diuturnitas ED, & ML diuturnitas BD <lb></lb>aequalis C. diuturnitas ipsius AB, unde diu<lb></lb>turnitates in AB, & in BD aequales erunt.10*</s> </p> <p type="main"> <s id="s.000375">Et si BD esset fere Orizontalis, BE fieret longis<lb></lb>sima, & quia EB ad ED est ut G ad tertiam <lb></lb>proportionalem ad G, & MK, haec tertia exce<lb></lb>deret ipsam G fere duplo ipsius ML, seu C, ob <lb></lb>magnam diferentiam inter G, & C, ob quam <lb></lb>G esset fere aequalis ipsi MK, unde itidem E<lb></lb>D excederet EB fere duplo ipsius AB, & quo <lb></lb>BD esset magis orizontalis, eo BD propinquior <lb></lb>esset duplo AB.11*</s> </p> <p type="main"> <s id="s.000376">Ceterum ex hisce plura alia postmodum deduci <lb></lb>facile poterunt, haec vero in praesentia pauca <lb></lb>sufficere mihi visa sunt.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/055.jpg"></pb> <chap> <p type="main"> <s id="s.000377">DE MOTV <lb></lb>GRAVIVM <lb></lb>SOLIDORVM <lb></lb>LIBER SECVNDVS <lb></lb> VBI DE IMPETV.</s> </p> <p type="main"> <s id="s.000378">LIBELLVM edidi octo ab <lb></lb>bine annis anno ſiquidem <lb></lb>1638 de motu ſolidorum, mox de liquidis editurus, quibus nimirum ſolida ſoli <lb></lb>dius ſtruerent fundamen <lb></lb>tum.</s> <s id="s.000379">Hucuſque diſtuli, exi<lb></lb>ſtimans hos itidem duos libros de ſolidis prae <lb></lb>mittendos; faciliorem ſiquidem viſi ſunt ſter <lb></lb>nere viam ad illorum demonſtrationem cla<lb></lb>riorem.</s> <s id="s.000380">Quod eo libentius feci, quoniam ſe<lb></lb>ptimum poſtulatum, quod inter principia, <lb></lb>connumerandum non videbatur, tanquam <lb></lb>minus euidens, decima huius propoſitione <lb></lb>demonſtrare contigit; ex quo inde deducta, <pb xlink:href="064/01/056.jpg"></pb>ſeu potius leuiter tacta, libro ſequenti re <lb></lb>petere, & clarius explica re coactus mihi vi<lb></lb>ſus ſum.</s> <s id="s.000381">Quæ nihilomimus, citius perfici po<lb></lb>tuiſſent, ni pluribus litigijs, alijque negotijs <lb></lb>proprijs, & alienis, tum muneribus publicis <lb></lb>diſtractus, litterarum ſtudia dimittere ſæpius <lb></lb>mihi opus fuiſſet.</s> <s id="s.000382">Non ignoro litteris auide <lb></lb>deditos nuſquam ijs obrui negotijs, quin horas <lb></lb>furtiuas quotidie reperiant, quibus diſcipli<lb></lb>narum ſtudijs vacent: verum ſatis conſtat in<lb></lb>tellectum libentius elaborare in nouis per di<lb></lb>ſcendis, ſeu aliorum partus ingeniorum in<lb></lb>quiras, ſeu (quod delectabilius longe eſt) <lb></lb>noua proprio marte reperias, quam in iam <lb></lb>repertis poſtmodum expoliendis, in quo ni <lb></lb>mirum labor ingens, nulla animi voluptas. <lb></lb></s> <s id="s.000383">Ex quo mirandum non eſt ſiquid otij occupa<lb></lb>tiones permiſſerunt, meum ad noua potius pro <lb></lb>penſum ingenium, ea ſæpius intermiſiſſe, que <lb></lb>ad opus perficiendum neceſſario requireban <lb></lb>tur: quod cauſa fuit non modo procaſtinatio<lb></lb>nis, ſed cur opus prodeat impolitum, poſtre<lb></lb>ma vide licet lima deficiente; vnde, ſi ani<lb></lb>mo meo morem gerere voluiſſem, ad huc ſub <lb></lb>tenebris latitaret.</s> <s id="s.000384">Qualecunque ſit, tibi nunc <lb></lb>exhibere libuit, & priorem librum iterum edi, <lb></lb>allique alligari ad eorundem captum neceſſarium, <lb></lb>tu illud accipias, & excuſes, & corrigas velim.</s> </p> </chap> <pb xlink:href="064/01/057.jpg"></pb> <chap type="bk"> <subchap1 type="definition"> <p type="head"> <s id="s.000385">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000386">1. Motus dicitur aequabilis, si mobile fera<lb></lb>tur per spatia, quae inter se sint ut <lb></lb>tempora, quibus conficiuntur.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000387">2. Impetus est vis, quia mobile est aptum progre<lb></lb>di absque actione gravitatis, aut cuiusvis al<lb></lb>terius rei.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.000388">Petitio</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000389">Impetus sunt ut spatia, quae eius virtute aequis <lb></lb>temporibus permeantur.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.000390">Axiomata</s> </p> <subchap2 type="axiom"> <p type="main"> <s id="s.000391">1. Pares causae producunt pares effectus.</s> </p> </subchap2> <subchap2 type="axiom"> <p type="main"> <s id="s.000392">2. In effectu procedente a duabus causis, ablata eius <lb></lb>portione proveniente ab una, reliquum erit <lb></lb>portio proveniens ab altera.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/058.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000393">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000394">Grave in motu naturali, sive perpendiculari, <lb></lb>sive inclinato, fertur sine ope gravitatis, <lb></lb>aequali tempore, per duplum spatii praece<lb></lb>dentis.</s> </p> </subchap2> <subchap2 n="1" type="proof"> <p type="main"> <figure id="id.064.01.058.1.jpg" xlink:href="064/01/058/1.jpg"></figure> <s id="s.000395">Dato gravi A naturaliter la<lb></lb>to ab A ad B tempore ab, <lb></lb>cuius aequale sit tempus bc, & <lb></lb>spatium BC, sit duplum spati AB. <lb></lb></s> <s id="s.000396">Dico quod tempore bc fertur grave <lb></lb>sine ope gravitatis per spatium <lb></lb>aequale ipsi BC.</s> </p> <p type="main"> <s id="s.000397">Producatur AB, sumaturque portio <lb></lb>BD aequalis, & DE dupla lineae AB, & pro<lb></lb>inde aequalis ipsi BC.</s> </p> <p type="main"> <s id="s.000398">Quoniam ope gravitatis A tempore ab fertur <lb></lb>in B per constructionem, tempore bc eadem <lb></lb>ope prodibit in D per spatium BD aequale A<lb></lb>B<arrow.to.target n="marg90"></arrow.to.target>, at prodit in E<arrow.to.target n="marg91"></arrow.to.target>, ergo fertur per DE du<lb></lb>plum ipsius AB sine ope gravitatis, cui cum <lb></lb>sit aequalis BC per constructionem, constat, <lb></lb>quod sine ope gravitatis tempore bc fertur per <lb></lb>spatium aequale BC, quod etc.</s> </p> <p type="margin"> <s id="s.000399"><margin.target id="marg90"></margin.target>Per axioma primum.</s> </p> <p type="margin"> <s id="s.000400"><margin.target id="marg91"></margin.target>Per 3. primi huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000401">Corollarium Primum</s> </p> <p type="main"> <s id="s.000402">Hinc sequitur quod si spatium AB sectum esset <lb></lb>in quatuor partes aequales, grave perficeret <pb xlink:href="064/01/059.jpg"></pb>primam tempore aequali illi quo conficit tres <lb></lb>reliquas, quia in fine primae acquisivit virtu<lb></lb>tem, seu impetum, quo perficeret duas partes, <lb></lb>tertiam verum conficit eadem virtute qua per<lb></lb>ficit primam. </s> <s id="s.000403">Quod pari ratione sequitur si <lb></lb>AE producatur, & in ea sumantur tres par<lb></lb>tes aequales ipsi AE, quae tres conficientur tem<lb></lb>pore ei aequali quo perficitur AE.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000404">Corollarium II</s> </p> <p type="main"> <s id="s.000405">Impetus autem non sumpsit initium in B, sed <lb></lb>prius, attamen cum mobile est in B ille impe<lb></lb>tus qui simul cum gravitate tempore ab duxit <lb></lb>mobile ab A in B non est sufficiens tempore bc <lb></lb>aequali ab ducere illud ultra D per dictum pri<lb></lb>mum Axioma, unde impetus ducens grave a <lb></lb>D in E eodem tempore bd necessario est is qui <lb></lb>est acquisitus per motum AB in puncto B.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000406">Corollarium III</s> </p> <p type="main"> <s id="s.000407">Quoniam impetus de nouo acquisitus non <lb></lb>operatur seorsim ab impetu qui simul cum <lb></lb>gravitate duxit mobile ab A in B, sed eo<lb></lb>dem prorsus tempore ducitur mobile non modo <lb></lb>ab impetu de novo acquisito in B, sed etiam, & <lb></lb>gravitate, & ab impetu qui continuo produ<pb xlink:href="064/01/060.jpg"></pb><figure id="id.064.01.060.1.jpg" xlink:href="064/01/060/1.jpg"></figure>citur respondens illi qui duxit mobile ab A in <lb></lb>B, idcirco ipsum mobile a B in E fertur perpe<lb></lb>tuo velocius, unde motus est velocior in E quem <lb></lb>fuerit in quolibet puncto superiori, & pro<lb></lb>inde in E sortitum est impetum maiorem quam <lb></lb>habuerit prius, aptum ducere illud aequali tem<lb></lb>pore per spatium duplum ipsius AE.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/061.jpg"></pb> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000408">PROPOSITIO II. PROBL. I.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000409"><figure id="id.064.01.061.1.jpg" xlink:href="064/01/061/1.jpg"></figure>Dato spatio per quod grave naturali<lb></lb>ter ducatur virtute impetus solius sine <lb></lb>ope gravitatis, in dato tempore: repe<lb></lb>rire eius portionem per quam duca<lb></lb>tur eadem virtute in quavis portione <lb></lb>dicti temporis.</s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <s id="s.000410">Ducatur grave A per spatium AE <lb></lb>tempore ae, nec non per spatium <lb></lb>aequale EB duplum AE virtute impetus <lb></lb>acquisiti in E sine ope gravitatis tempore e<lb></lb>h aequale ipsi ae<arrow.to.target n="marg92"></arrow.to.target> cuius temporis eh data sit <lb></lb>portio quaelibet, & sit primo portio immedia<lb></lb>ta tempori ae, & sit eg.</s> </p> <p type="margin"> <s id="s.000411"><margin.target id="marg92"></margin.target>Per pr. huius.</s> </p> <p type="main"> <s id="s.000412">Oportet reperire portionem spatii EB, per quod <lb></lb>grave A ducatur, virtute impetus solius acqui<lb></lb>siti in E, sine ope gravitatis, in dicta portione <lb></lb>temporis eg.</s> </p> <p type="main"> <s id="s.000413">Concipiantur tempora ae, eh, eg tanquam lineae <lb></lb>rectae metientes tempora ae, eh, eg, & fiat <lb></lb>ac tempus aequale tempori eg, & ut ae <lb></lb>ad ac, fiat AE ad AD<arrow.to.target n="marg93"></arrow.to.target> ad quas fiat tertia <lb></lb>AC<arrow.to.target n="marg94"></arrow.to.target>, ex quo AE, AC sunt in duplicata ratio<lb></lb>ne temporum ae, ac,<arrow.to.target n="marg95"></arrow.to.target>. Fiat ut ae ad ag ita <lb></lb>AE ad AF<arrow.to.target n="marg96"></arrow.to.target>, quibus tertia AG<arrow.to.target n="marg97"></arrow.to.target>, ex quo AG, <lb></lb>AE sunt in duplicata ratione temporum ag, ae<arrow.to.target n="marg98"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000414"><margin.target id="marg93"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000415"><margin.target id="marg94"></margin.target>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000416"><margin.target id="marg95"></margin.target>Per 10. def. quinti.</s> </p> <p type="margin"> <s id="s.000417"><margin.target id="marg96"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000418"><margin.target id="marg97"></margin.target>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000419"><margin.target id="marg98"></margin.target>Per 10 def. 5.</s> </p> <p type="main"> <s id="s.000420"><pb xlink:href="064/01/062.jpg"></pb>Fiat EH aequalis AC, et ab AG abla<lb></lb>ta AH, residuo HG fiat aequalis EI.</s> </p> <p type="main"> <s id="s.000421">Dico EI esse portionem quaesitam.</s> </p> <p type="main"> <s id="s.000422">Quoniam AE est casus gravis A tempore ae per <lb></lb>supp. & AE, AC sunt in dupl. ratione tem<lb></lb>porum ae, ac per constr. </s> <s id="s.000423">AC est casus gravis <lb></lb>tempore ac<arrow.to.target n="marg99"></arrow.to.target>, & proinde EH aequalis AC est <lb></lb>casus tempore eg aequali ipsi ab si grave du<lb></lb>ceretur per EH eadem prorsus virtute qua <lb></lb>ductum fuit per AC<arrow.to.target n="marg100"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000424"><margin.target id="marg99"></margin.target>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.000425"><margin.target id="marg100"></margin.target>Per axioma primum.</s> </p> <p type="main"> <s id="s.000426">Item quia AG, AE sunt in duplicata ratione tem<lb></lb>porum ag, ae per constr., AG est casus tempo<lb></lb>re ag<arrow.to.target n="marg101"></arrow.to.target>, & proinde residuum EG est casus re<lb></lb>sidui eg<arrow.to.target n="marg102"></arrow.to.target>, dum tamen motus proveniat tam <lb></lb>e gravitate quam a quolibet impetu superaddi<lb></lb>to, at EH probatum est esse casum itidem, eg <lb></lb>dum tamen grave ducatur ea solum virtute <lb></lb>qua ductum fuit per AC<arrow.to.target n="marg103"></arrow.to.target>, ig, residuum HG <lb></lb>est spatium quod perficitur eodem tempore eg, <lb></lb>a solo impetu acquisito in E<arrow.to.target n="marg104"></arrow.to.target>, quod est aequa<lb></lb>le EI per constr., unde EI est spatium quaesitum.</s> </p> <p type="margin"> <s id="s.000427"><margin.target id="marg101"></margin.target>Per 3. primi huius.</s> </p> <p type="margin"> <s id="s.000428"><margin.target id="marg102"></margin.target>Per 19. Quinti.</s> </p> <p type="margin"> <s id="s.000429"><margin.target id="marg103"></margin.target>Per axioma primum.</s> </p> <p type="margin"> <s id="s.000430"><margin.target id="marg104"></margin.target>Per axioma secundum.</s> </p> <p type="main"> <s id="s.000431">Sit deinde portio temporis eb disiuncta ab ae, puta <lb></lb>gK, & sit rursus reperienda portio spatij EB <lb></lb>per quod grave A ducatur vi solius impetus <lb></lb>in E acquisiti in dicta portione temporis gk: <lb></lb>reperto prius spatio EC respondenti tempori eg <lb></lb>immediato ipsi ae modo quo supra dictum <lb></lb>fuit; fiat ac tempus aequale tempori gK, & ut<pb xlink:href="064/01/063.jpg"></pb><figure id="id.064.01.063.1.jpg" xlink:href="064/01/063/1.jpg"></figure> ag ad ac fiat AG ad AD, ad quas tertia A<lb></lb>C; AG, AC erunt in duplicata ratione tem<lb></lb>porum ag, ac. </s> <s id="s.000432">Item fiat ut ag ad aK ita AG <lb></lb>ad AL, quibus tertia AK: AK, AH erunt in <lb></lb>duplicata ratione temporum aK, ag; fiat GM <lb></lb>aequalis AC, & ab AK auferatur AM, & <lb></lb>residuo MK fiat aequale IN, & eodem ratio<lb></lb>cinio demonstrabitur IN esse spatium quae<lb></lb>situm. </s> <s id="s.000433">Reperta est igitur portio quaesita, <lb></lb>quod etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/064.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000434">PROPOSITIO TERTIA.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000435">In motu naturali gravium, spatia quae conficiun<lb></lb>tur virtute impetus sine ope gravitatis sunt <lb></lb>inter se ut tempora quibus conficiuntur.<figure id="id.064.01.064.1.jpg" xlink:href="064/01/064/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <s id="s.000436">Descendat grave A in E tempore ae, & tem<lb></lb>pore eh aequali ae, ex solo impetu, sine ope <lb></lb>gravitatis, per spatium aequale EB, duplo ipsius <lb></lb>AE,<arrow.to.target n="marg105"></arrow.to.target> & secetur EI portio dicti spatij EB <lb></lb>quae sit aequalis spatio per quod duci debeat gra<lb></lb>ve A tempore eg portione dicti temporis eh so<lb></lb>la vi impetus acquisiti in E<arrow.to.target n="marg106"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000437"><margin.target id="marg105"></margin.target>Per pr. huius.</s> </p> <p type="margin"> <s id="s.000438"><margin.target id="marg106"></margin.target>Per 2. huius.</s> </p> <p type="main"> <s id="s.000439">Dico spatium EI ad spatium EB esse ut <lb></lb>tempus eg ad tempus eh.</s> </p> <p type="main"> <s id="s.000440">Percipiantur tempora ae, eh, eg tanquam rectae me<lb></lb>tientes tempora ae, eh, eg, & reperiantur ut in <lb></lb>praecedenti puncta C, H, G, e, & describantur <lb></lb>quadrata ab, ad, bd, supra ae, ag, eg<arrow.to.target n="marg107"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000441"><margin.target id="marg107"></margin.target>Per 46. primi.</s> </p> <pb xlink:href="064/01/065.jpg"></pb> <p type="main"> <s id="s.000442">Quoniam AG, AE sunt in duplicata ratione <lb></lb>ad ag, ae per constr., & quadrata ad, ab <lb></lb>sunt pariter in duplicata ratione ad ag, ae,<arrow.to.target n="marg108"></arrow.to.target> <lb></lb>erunt AG, AE ut quadrata ad, ab,<arrow.to.target n="marg109"></arrow.to.target> & di<lb></lb>videndo ut EG ad AE ita ad minus ab, hoc est <lb></lb>gnomon edf, ad ab.<arrow.to.target n="marg110"></arrow.to.target> Pari ratione probabimus <lb></lb>ut AE ad EH esse quadrata ab, ad bd, & <lb></lb>proinde EG ad EH est ut gnomon edf ad <lb></lb>quadratum bd<arrow.to.target n="marg111"></arrow.to.target> unde HG, ad EG, ut com<lb></lb>plementa gb, bf ad gnomonem edf,<arrow.to.target n="marg112"></arrow.to.target> at EG <lb></lb>ad AE sunt ut gnomon edf ad quadratum ab, <lb></lb>ut probatum est supra, ergo HG, seu EI <lb></lb>ipsi <lb></lb>aequalis per constr. ad AE est ut dicta comple<lb></lb>menta gb, bf, ad quadratum ab,<arrow.to.target n="marg113"></arrow.to.target> bisk seu <lb></lb>ut gb ad ab,<emph type="sup"></emph>1<emph.end type="sup"></emph.end> seu ut eg ad ae,m seu eh, ei <lb></lb>aequale per constr. </s> <s id="s.000443">Quod, etc.</s> </p> <p type="margin"> <s id="s.000444"><margin.target id="marg108"></margin.target>Per 20. sexti.</s> </p> <p type="margin"> <s id="s.000445"><margin.target id="marg109"></margin.target>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000446"><margin.target id="marg110"></margin.target>Per 17. Quinti.</s> </p> <p type="margin"> <s id="s.000447"><margin.target id="marg111"></margin.target>Per 22. Quinti.</s> </p> <p type="margin"> <s id="s.000448"><margin.target id="marg112"></margin.target>Per 19. Quinti.</s> </p> <p type="margin"> <s id="s.000449"><margin.target id="marg113"></margin.target>Per 22. Quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000450">Corollarium Primum</s> </p> <p type="main"> <s id="s.000451">Si portio temporis eh non sit immediata tempori <lb></lb>ae sed ab ea seiuncta, puta in schemate propo<lb></lb>sitionis secundae gK, reperto in EB spatio IN<pb xlink:href="064/01/066.jpg"></pb><figure id="id.064.01.066.1.jpg" xlink:href="064/01/066/1.jpg"></figure> ipsi gk, respondenten, eodem ratiocinio quo supra <lb></lb>probabitur spatium EB ad eius portionem IN <lb></lb>esse ut tempus eh ad eius portionem gK, quan<lb></lb>doquidem qua ratione EI respondet tempori eg, <lb></lb>eadem EN respondet tempori eK, & proinde <lb></lb>reliquum IN respondet reliquo gK.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000452">Corollarium II</s> </p> <p type="main"> <s id="s.000453">Motus ab impetu proveniens est aequabilis.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/067.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000454">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000455">In motu naturali impetus successive acquisi<lb></lb>ti sunt ut tempora transacta.</s> </p> </subchap2> <figure id="id.064.01.067.1.jpg" xlink:href="064/01/067/1.jpg"></figure> <subchap2 n="4" type="proof"> <p type="main"> <s id="s.000456">Dato gravi moto naturali motu per AC, tem<lb></lb>pore ac, & per AB, tempore ab.</s> </p> <p type="main"> <s id="s.000457">Dico impetum seu velocitatem in B ad impetum <lb></lb>in C esse ut ab ad ac. </s> <s id="s.000458">Concipiantur tempora ab, ac tanquam lineae re<lb></lb>ctae metientes tempora ab, ac. </s> <s id="s.000459">Fiat BD dupla ipsius AB mensura impetus in B <lb></lb>tempore ab, & CE dupla ipsius AC mensura <lb></lb>impetus in C tempore ac<arrow.to.target n="marg114"></arrow.to.target>, & BF media inter <lb></lb>BD, CE<arrow.to.target n="marg115"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000460"><margin.target id="marg114"></margin.target>k Per 25. Quinti.</s> </p> <p type="margin"> <s id="s.000461"><margin.target id="marg115"></margin.target>l Per 22. Quinti & 43. pr.</s> </p> <p type="main"> <s id="s.000462">Quoniam AB, AC sunt in duplicata ratione <lb></lb>temporum ab, ac<arrow.to.target n="marg116"></arrow.to.target>, BD, CE sunt pariter in <lb></lb>duplicata ratione eorundem temporum ab, ac<arrow.to.target n="marg117"></arrow.to.target>, <lb></lb>sed BD, CE sunt etiam in duplicitata ratione <lb></lb>spatiorum BD, BF per constructionem, ergo BD, BF <lb></lb>sunt ut tempora ab, ac<arrow.to.target n="marg118"></arrow.to.target>. Sed BD mensura <lb></lb>impetus in B tempore ab, est spatium per <lb></lb>quod percurrit mobile virtute solius impetus <lb></lb>acquisiti in B tempore ab per constructionem, erit igitur <pb xlink:href="064/01/068.jpg"></pb>BF spatium per quod percurret idem mobile <lb></lb>eadem virtute impetus acquisiti in B tempore <lb></lb>ac<arrow.to.target n="marg119"></arrow.to.target>, at CE est spatium quod percurrit mobile <lb></lb>eodem tempore ac per constr. </s> <s id="s.000463">Igitur eodem tem<lb></lb>pore ac mobile in C perficit spatium CE, & in <lb></lb>B perficit spatium BF; sed impetus sunt ut spa<lb></lb>tia quae aequali tempore transignuntur <emph type="sup"></emph>g<emph.end type="sup"></emph.end><arrow.to.target n="marg120"></arrow.to.target>. Ergo <lb></lb>impetus in C, & B sunt ut CE ad BF spatia, <lb></lb>quae probatum est esse ut tempora ac, ab, unde <lb></lb>impetus in C & B sunt ut tempora ac, ab<arrow.to.target n="marg121"></arrow.to.target>, <lb></lb>quod etc.</s> </p> <p type="margin"> <s id="s.000464"><margin.target id="marg116"></margin.target>m Per 36. primi.</s> </p> <p type="margin"> <s id="s.000465"><margin.target id="marg117"></margin.target>n Per 2. huius.</s> </p> <p type="margin"> <s id="s.000466"><margin.target id="marg118"></margin.target>o Per primam defin.</s> </p> <p type="margin"> <s id="s.000467"><margin.target id="marg119"></margin.target>Per primam huius.</s> </p> <p type="margin"> <s id="s.000468"><margin.target id="marg120"></margin.target>Per 13 Sexti.</s> </p> <p type="margin"> <s id="s.000469"><margin.target id="marg121"></margin.target>Per tertiam pr. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/069.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000470">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000471">In motu naturali gravium impetus successive <lb></lb>acquisiti sunt in subduplicata ratione spa<lb></lb>tiorum transactorum.</s> </p> </subchap2> <figure id="id.064.01.069.1.jpg" xlink:href="064/01/069/1.jpg"></figure> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.000472">Iisdem positis.</s> </p> <p type="main"> <s id="s.000473">Dico impetus, seu velocitates in B, & in C <lb></lb>esse in subduplicata ratione spatiorum <lb></lb>AB, & AC.</s> </p> <p type="main"> <s id="s.000474">Quoniam impetus in B, & C sunt ut tempora ab, <lb></lb>ac transacta<arrow.to.target n="marg122"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000475"><margin.target id="marg122"></margin.target>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000476">Sed tempora ab, ac sunt in subduplicata ra<lb></lb>tione spatiorum AB, AC<arrow.to.target n="marg123"></arrow.to.target>. </s> <s id="s.000477">Pariter impetus <lb></lb>in B, & in C sunt in subduplicata ratione <lb></lb>spatiorum AB, AC, quod etc.</s> </p> <p type="margin"> <s id="s.000478"><margin.target id="marg123"></margin.target>Per 11 Quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/070.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000479">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <figure id="id.064.01.070.1.jpg" xlink:href="064/01/070/1.jpg"></figure> <s id="s.000480">Datis in perpendiculari quibuslibet pun<lb></lb>ctis reperire impetus singulorum in<lb></lb>ter se.</s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.000481">Data linea perpendiculari AB, & <lb></lb>in ea punctis C, D,</s> </p> <p type="main"> <s id="s.000482">Venandi impetus in C, D dum grave ab <lb></lb>A dimissum fertur per AB.</s> </p> <p type="main"> <s id="s.000483">Sit E media inter AC, AD, item fiat AF media <lb></lb>inter AC, AB.</s> </p> <p type="main"> <s id="s.000484">Dico impetus in C, D, B esse ut AC, AE, AF.</s> </p> <p type="main"> <s id="s.000485">Quoniam AE est media inter AC, AD per con<lb></lb>structionem, AD, AC sunt in duplicata ratio<lb></lb>ne rectarum AE, AC<arrow.to.target n="marg124"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000486"><margin.target id="marg124"></margin.target>Per 3. huius.</s> </p> <p type="main"> <s id="s.000487">Ergo AC, AE metiuntur impetus in C & D<arrow.to.target n="marg125"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000488"><margin.target id="marg125"></margin.target>Per pet. huius.</s> </p> <p type="main"> <s id="s.000489">Item quoniam AF est media inter AC, AB per <lb></lb>constructionem, AF, AC sunt in subduplicata <lb></lb>ratione rectarum AB, AC, igitur AC, AF <lb></lb>metiuntur impetus in C & B, quod etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/071.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.000490">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000491">In quolibet puncto motus reperire spatium, <lb></lb>per quod mobile sit aptum duci sine ope <lb></lb>gravitatis in dato tempore.</s> </p> </subchap2> <subchap2 n="7" type="proof"> <figure id="id.064.01.071.1.jpg" xlink:href="064/01/071/1.jpg"></figure> <p type="main"> <s id="s.000492">Ducatur grave tempore ab a puncto B per <lb></lb>spatium aequale rectae BD sine ope gravi<lb></lb>tatis ut in praecedenti.</s> </p> <p type="main"> <s id="s.000493">Oportet reperire in alio puncto ipsius motus, puta <lb></lb>C, spatium aequale ei, per quod ducetur sine ope <lb></lb>gravitatis eodem tempore ab.</s> </p> <p type="main"> <s id="s.000494">Sit ac tempus, per quod ducitur grave naturali<lb></lb>ter motum ab A in C, & fiat CE dupla ad AC, & <lb></lb>secetur CE in F ea ratione, ut partes CF, FE <lb></lb>sint partibus ab, bc proportionales<arrow.to.target n="marg126"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000495"><margin.target id="marg126"></margin.target>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000496">Dico CF spatium aequari illi, per quod ducetur <lb></lb>grave digressum a C tempore ab.</s> </p> <p type="main"> <s id="s.000497">Quonniam CF ad FE est ut ab ad bc per constructionem, <lb></lb>erit ut CE ad CF ita ac ad ab<arrow.to.target n="marg127"></arrow.to.target>, & permutando <lb></lb>ut CE ad ac, ita CF ad ab<arrow.to.target n="marg128"></arrow.to.target> at spatium aequa<lb></lb>le CE perficitur tempore ac<arrow.to.target n="marg129"></arrow.to.target> motu aequabili<arrow.to.target n="marg130"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000498"><margin.target id="marg127"></margin.target>Per 4. huius.</s> </p> <p type="margin"> <s id="s.000499"><margin.target id="marg128"></margin.target>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.000500"><margin.target id="marg129"></margin.target>Per 10. def. Quinti.</s> </p> <p type="margin"> <s id="s.000501"><margin.target id="marg130"></margin.target>Per 5. huius.</s> </p> <p type="main"> <s id="s.000502">Ergo spatium aequale CF conficitur tempore ab, quod etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000503">Corollarium</s> </p> <p type="main"> <s id="s.000504">Huic sequitur quod eodem tempore, puta ab, <lb></lb>grave ducitur per BD, & per CF.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/072.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.000505">PROPOSITIO VIII.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000506">Si lineae perpendicularis, & inclinata ab eo<lb></lb>dem puncto digressae, per quas idem grave <lb></lb>naturaliter ducatur, secentur a recta norma<lb></lb>lis ad inclinatam; impetus in punctis sectionis, <lb></lb>sunt ut portiones linearum intra sectiones.</s> </p> </subchap2> <figure id="id.064.01.072.1.jpg" xlink:href="064/01/072/1.jpg"></figure> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.000507">Sint rectae AB perpendicularis, & AC quomo<lb></lb>documque; inclinata per quas grave naturaliter <lb></lb>ducatur, sectae a BD normali ad AC declinantem.</s> </p> <p type="main"> <s id="s.000508">Dico impetum in B ad impetum in D esse ut AB <lb></lb>ad AD.</s> </p> <p type="main"> <s id="s.000509">Fiat BE dupla AB mensura impetus in B, & DF <lb></lb>dupla AD mensura impetus in D<arrow.to.target n="marg131"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000510"><margin.target id="marg131"></margin.target>Per 10. sexti.</s> </p> <p type="main"> <s id="s.000511">Quoniam grave ducitur per AB AD eodem <lb></lb>tempore<arrow.to.target n="marg132"></arrow.to.target>. Ducitur etiam sine ope gravitatis eo<lb></lb>dem tempore per spatia aequalia ipsis BE, DF<arrow.to.target n="marg133"></arrow.to.target> <lb></lb>& proinde BE, DF sunt ut impetus in B & D<arrow.to.target n="marg134"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000512"><margin.target id="marg132"></margin.target>Per 18. Quinti.</s> </p> <p type="margin"> <s id="s.000513"><margin.target id="marg133"></margin.target>Per 16. Quinti.</s> </p> <p type="margin"> <s id="s.000514"><margin.target id="marg134"></margin.target>Per pr. huius.</s> </p> <p type="main"> <s id="s.000515">At BE, DF sunt ut AB, AD per constr. quip<lb></lb>pe earum duplae. </s> <s id="s.000516">Igitur AB, AD sun t ut im<lb></lb>petus in B & D<arrow.to.target n="marg135"></arrow.to.target> quod, etc.</s> </p> <p type="margin"> <s id="s.000517"><margin.target id="marg135"></margin.target>Per cor. 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000518">Corollarium</s> </p> <p type="main"> <s id="s.000519">Impetus sive velocitas in B ad impetum in D <lb></lb>est ut AC ad AB.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/073.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000520">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000521">Ductis a puncto superno perpendiculari, & <lb></lb>inclinata ad planum Orizontale, & a pun<lb></lb>cto inferno perpendicularis ducta normali <lb></lb>ad inclinatam, impetus inclinatae in pun<lb></lb>ctis, in quibus secat normalem, & orizon<lb></lb>talem, sunt ut perpendicularis, & inclinata.</s> </p> </subchap2> <figure id="id.064.01.073.1.jpg" xlink:href="064/01/073/1.jpg"></figure> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.000522">Sint rectae AB AC ductae a puncto A ad orizon<lb></lb>talem CB & a B ducatur normalis BD ad <lb></lb>AC.</s> </p> <p type="main"> <s id="s.000523">Dico impetum in D ad impetum in C esse ut AB <lb></lb>ad AC.</s> </p> <p type="main"> <s id="s.000524">Quoniam AC AD sunt in duplicata ratione im<lb></lb>petus C ad impetum D<arrow.to.target n="marg136"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000525"><margin.target id="marg136"></margin.target>Per pr. huius.</s> </p> <p type="main"> <s id="s.000526">Sunt itidem in duplicata ratione AC ad AB<arrow.to.target n="marg137"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000527"><margin.target id="marg137"></margin.target>Per 14. pr. huius.</s> </p> <p type="main"> <s id="s.000528">Igitur impetus in C ad impetum in D sunt ut AC <lb></lb>AB<arrow.to.target n="marg138"></arrow.to.target> quod, etc.</s> </p> <p type="margin"> <s id="s.000529"><margin.target id="marg138"></margin.target>Per pr. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/074.jpg"></pb> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000530">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000531">Ductis a puncto superno perpendiculari, & <lb></lb>inclinata in punctis in quibus secant lineam <lb></lb>orizontalem sortiuntur impetus aequales.</s> </p> </subchap2> <figure id="id.064.01.074.1.jpg" xlink:href="064/01/074/1.jpg"></figure> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.000532">A puncto A superno ducatur AB perpendi<lb></lb>cularis, & AC declinans ad BC Orizon<lb></lb>talem.</s> </p> <p type="main"> <s id="s.000533">Dico, quod in B, & C sunt impetus aequales.</s> </p> <p type="main"> <s id="s.000534">Quoniam impetus in C ad impetum in D est ut <lb></lb>AC ad AB<arrow.to.target n="marg139"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000535"><margin.target id="marg139"></margin.target>Per pet. huius.</s> </p> <p type="main"> <s id="s.000536">Item impetus in B ad impetum in D est pariter <lb></lb>ut AC ad AB<arrow.to.target n="marg140"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000537"><margin.target id="marg140"></margin.target>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000538">Igitur impetus in C, & B sunt aequales<arrow.to.target n="marg141"></arrow.to.target>. </s> <s id="s.000539">Quod <lb></lb>etc.</s> </p> <p type="margin"> <s id="s.000540"><margin.target id="marg141"></margin.target>Per 5. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/075.jpg"></pb> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000541">PROPOSITIO XI. PROBL. IV.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000542">Datis pluribus lineis æqualibus ab eodem <lb></lb>puncto superno descendentibus, etiam si <lb></lb>una sit perpendicularis, reperire impetus <lb></lb>in fine ipsarum inter se.</s> </p> </subchap2> <figure id="id.064.01.075.1.jpg" xlink:href="064/01/075/1.jpg"></figure> <subchap2 n="11" type="proof"> <p type="main"> <s id="s.000543">Datis aequalibus AB, AC, AD, inclinatis, <lb></lb>& AE perpendiculari oportet venari im<lb></lb>petus inter se in B, C, D, E.</s> </p> <p type="main"> <s id="s.000544">Ducantur BF, CG, DH normales ad AE,<arrow.to.target n="marg142"></arrow.to.target> & <lb></lb>proinde orizontales, & fiat AI media inter <lb></lb>AF, AG, & fiat AK media inter AF, AH, <lb></lb>item fiat AL media inter AF, AE.</s> </p> <p type="margin"> <s id="s.000545"><margin.target id="marg142"></margin.target>Per 10. definit. quinti.</s> </p> <p type="main"> <s id="s.000546">Dico impetus in B, C, D, E esse inter se ut AF, <lb></lb>AI, AK, AL.</s> </p> <p type="main"> <s id="s.000547">Quoniam impetus in B, & F sunt aequales nec <lb></lb>non in CL, & in DH<arrow.to.target n="marg143"></arrow.to.target>, & impetus in F, G, <lb></lb>H, E sunt ut AF, AI, AK, AL<arrow.to.target n="marg144"></arrow.to.target>,</s> </p> <p type="margin"> <s id="s.000548"><margin.target id="marg143"></margin.target>Per 16. Quinti.</s> </p> <p type="margin"> <s id="s.000549"><margin.target id="marg144"></margin.target>Per 9. huius.</s> </p> <p type="main"> <s id="s.000550">Igitur impetus in B, C, D, E, sunt ut AF, AI, <lb></lb>AK, AL, Quod etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/076.jpg"></pb> <subchap1 n="12" type="proposition"> <p type="head"> <s id="s.000551">PROPOSITIO XII</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.000552">Ductis pluribus lineis diversi mode inclinatis, & <lb></lb>etiam perpendiculari, quae ab eadem li<lb></lb>nea Orizontali terminentur in idem pun<lb></lb>ctum inferius; ibi sortiuntur impetus aequales.<figure id="id.064.01.076.1.jpg" xlink:href="064/01/076/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="12" type="proof"> <p type="main"> <s id="s.000553">Sint lineae BD CD diversimode inclinatae, & AD <lb></lb>perpendicularis, ductae a linea Orizontali AC <lb></lb>ad punctum inferius D. </s> <s id="s.000554">Dico gravia a punctis <lb></lb>A B C digressa, & in eis lata, in D sortiri im<lb></lb>petus aequales.</s> </p> <p type="main"> <s id="s.000555">Fiat DEF parallela ad AC<arrow.to.target n="marg145"></arrow.to.target>, & proinde ori<lb></lb>zontalis, ad quam dimittantur perpendicula<lb></lb>res BE CF<arrow.to.target n="marg146"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000556"><margin.target id="marg145"></margin.target>Per cor. 8. huius.</s> </p> <p type="margin"> <s id="s.000557"><margin.target id="marg146"></margin.target>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000558">Quoniam gravia ducta per AD, BE, CF in DEF <lb></lb>habent impetus aequales, quia omnia paria<arrow.to.target n="marg147"></arrow.to.target>, <lb></lb>& gravia ducta per BD, BE in DE habent im<lb></lb>petus aequales, item per CD, CF in DF habent <lb></lb>impetus aequales<arrow.to.target n="marg148"></arrow.to.target> sequitur quod etiam ducta <lb></lb>per AD, BD, CD sortita sunt in D impetus <lb></lb>aequales. </s> <s id="s.000559">Quod etc.</s> </p> <p type="margin"> <s id="s.000560"><margin.target id="marg147"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000561"><margin.target id="marg148"></margin.target>Per 10. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000562">Corollarium</s> </p> <p type="main"> <s id="s.000563">Hinc sequitur, quod si ABC non sit linea, sed planum <lb></lb>Orizontale, item loco puncti D sint plura puncta, <lb></lb>dummodo in plano Orizontali; gravia in punctis <lb></lb>D habebunt impetus aequales.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/077.jpg"></pb> <subchap1 n="13" type="proposition"> <p type="head"> <s id="s.000564">PROPOSITIO XIII. PROBL. V.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.000565">Datis gravibus descendentibus per perpendi<lb></lb>cularem, & declinantem reperire rationes im<lb></lb>petus in punctis datis.<figure id="id.064.01.077.1.jpg" xlink:href="064/01/077/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="13" type="proof"> <p type="main"> <s id="s.000566">Descendat grave per AC perpendicularem , <lb></lb>& AB declinantem, & dentur puncta B, C.</s> </p> <p type="main"> <s id="s.000567">Reperire proportionem impe<lb></lb>tus in B ad impetum in C.</s> </p> <p type="main"> <s id="s.000568">Ducatur BD normalis ad AC<arrow.to.target n="marg149"></arrow.to.target>, & fiat AE <lb></lb>media inter AC, AD<arrow.to.target n="marg150"></arrow.to.target>, Dico impetum in C ad <lb></lb>impetum in B esse ut AE ad AD.</s> </p> <p type="margin"> <s id="s.000569"><margin.target id="marg149"></margin.target>Per 6. huius.</s> </p> <p type="margin"> <s id="s.000570"><margin.target id="marg150"></margin.target>Per 31. primi.</s> </p> <p type="main"> <s id="s.000571">Quoniam impetus in C ad impetum in D est ut <lb></lb>AE ad AD<arrow.to.target n="marg151"></arrow.to.target>, & impetus in D & B sunt aequa<lb></lb>les<arrow.to.target n="marg152"></arrow.to.target>, ergo impetus in C ad impetum in B est <lb></lb>ut AE ad AD, Quod etc.</s> </p> <p type="margin"> <s id="s.000572"><margin.target id="marg151"></margin.target>Per 13. primi.</s> </p> <p type="margin"> <s id="s.000573"><margin.target id="marg152"></margin.target>Per axioma primum.</s> </p> </subchap2> <pb pagenum="78" xlink:href="064/01/078.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.000574">Corollarium</s> </p> <p type="main"> <s id="s.000575">Eodem pacto reperies impetus in planis ut<lb></lb>cumque declinantibus ductis perpendicula<lb></lb>ribus ad AC.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/079.jpg"></pb> <chap type="bk"> <p type="main"> <s id="s.000576">DE MOTV <lb></lb>GRAVIVM <lb></lb>SOLIDORVM <lb></lb>LIBER TERTIVS.<lb></lb>VBI DE MOTV SVPER<lb></lb>PLVRIBVS PLANIS <lb></lb>DIVERSIMODE INCLINATIS.</s> </p> <subchap1 type="preface"> <subchap2 type="preface"> <p type="main"> <s id="s.000577">Ex libro secundo praecedenti con<lb></lb>stat, mobile dum movetur fieri ap<lb></lb>tum ex se moveri, quatenus post <lb></lb>priorem motum ei tribuitur, & im<lb></lb>primitur quaedam virtus, seu vis, a qua fit <lb></lb>aptum duci, sine alicuius ope, ea velocitate qua <lb></lb>movebatur, dum illa virtus imprimebatur, & <lb></lb>proinde motu aequabili; quae virtus dicitur Im<lb></lb>petus, differens solum fortasse a velocitate, quia <lb></lb>impetus sit velocitas in actu primo, ita ut ali<lb></lb>quo pacto impetus sit causa velocitatis; conve<lb></lb>niunt tamen, quatenus velocitates sunt ut spa<lb></lb>tia quae mobilia aequali tempore permeant, <lb></lb>impetus vero ut spatia quae virtute ipsius im<pb xlink:href="064/01/080.jpg"></pb>petus sunt apta, & in potentia proxima per<lb></lb>meare, & de facto permeant ni impedimen<lb></lb>tum aliquod obijciatur, secus enim effectus <lb></lb>causae non responderet. </s> <s id="s.000578">Porro ex impe<lb></lb>tu provenit quod missilia quaelibet, a mo<lb></lb>tore velociter ducta, deficiente motoris actio<lb></lb>ne, nihilominus a solo impetu ferantur, quod <lb></lb>in proiectis quotidie experimur. </s> <s id="s.000579">De quibus <lb></lb>locus postularet ut aliquid agerem, ni via <lb></lb>quam eorum motu conficiunt, me adhuc late<lb></lb>ret; quamvis non ignorem viris oculatissimis <lb></lb>visam esse parabolicam. </s> <s id="s.000580">Cum illis igitur sup<lb></lb>pono proiecta a motore seiuncta, motu du<lb></lb>plici moveri, nimirum ab impetu, aequabili <lb></lb>motu, eadem prorsus directe via qua a motore <lb></lb>novissime ducta fuerant, & itidem a gravitate <lb></lb>deorsum, & proinde motu mixto secundum <lb></lb>quamdam lineam curvam mihi ignotam, <lb></lb>quamhoc argumento ducti parabolicam ar<lb></lb>bitrantur.<figure id="id.064.01.080.1.jpg" xlink:href="064/01/080/1.jpg"></figure></s> </p> <p type="main"> <s id="s.000581">Proijciatur missile A versus D motu violento <lb></lb>quo virtute impetus temporibus aequalibus <lb></lb>conficiat aequalia spatia AB, BC, CD, & in<pb xlink:href="064/01/081.jpg"></pb>priori tempore, vi gravitatis descendat per <lb></lb>spatium aequale AE, quod sit BF, motu mix<lb></lb>to describet curvam AF; ducatur mox ab <lb></lb>impetu eodem quo prius tramite, ab F ver<lb></lb>sus G, unde si moveretur eo simplici motu <lb></lb>violento, in tantundem temporis adiret ip<lb></lb>sum G, at quoniam urget etiam gravitas, <lb></lb>ducitur in H, ita ut GH sit triplum ipsius <lb></lb>AE, & proinde CH ad BF sit in duplicata <lb></lb>ratione AC ad AB, describens motu mixto <lb></lb>curvam FH, & demum eadem ratione du<lb></lb>citur in I. </s> <s id="s.000582">Probant puncta AF HI esse in <lb></lb>parabola, per 20 primi A poll. quoniam <lb></lb>quadrata rectarum AC, AB ordinatim ap<lb></lb>plicatarum, seu eis aequalium, sunt ut CH, BF <lb></lb>ab eis ex diametro praecisae, seu ut eis aequa<lb></lb>les. </s> <s id="s.000583">At vero mihi quidem, contra id quod sup<lb></lb>ponitur, apparet proiectum descendere mi<lb></lb>nori celeritate, quam si a sola ducatur grav<lb></lb>itate, & libere dimissum, celerius solum <lb></lb>attingere, quam orizontaliter latum. </s> <s id="s.000584">Insu<lb></lb>per si aequis temporibus proiectum conficit <lb></lb>curvas AF, FH, HI, successive longiores <lb></lb>motus est successive velocior, quippe maius <lb></lb>spatium aequo tempore permeat, unde si vis pro<lb></lb>ijcientis provenit a maiori velocitate, ictus <lb></lb>eo est validior, quo missile longius a proij<lb></lb>ciente distat; contra id quod quotidie experi<pb xlink:href="064/01/082.jpg"></pb>mur, nec sit tardior ab aeris resistentia, quam <lb></lb>gravia deorsum mota persentirent, unde <lb></lb>quo graviora, celerius descenderent; quod <lb></lb>experientiae repugnat. </s> <s id="s.000585">Sed quia adducere <lb></lb>inconveniens non est solvere argumentum, <lb></lb>eius fallaciam pro viribus detegere conabor. <lb></lb></s> <s id="s.000586">Dum supponitur ab impetu duci perpetuo <lb></lb>mobile iuxta orizontalem AD, ego equi<lb></lb>dem verum esse censeo, ubi mobile unico so<lb></lb>lum violento motu ducatur; sed quia fertur <lb></lb>motu mixto, ab impetu nimirum, & a gravi<lb></lb>tate secundum curvam AFH, quemadmodum <lb></lb>proiectum, a funda circumlatum, sibi dimis<lb></lb>sum fertur per tangentem curvae a funda <lb></lb>descriptae, ita pariter censendum est, quo<lb></lb>tiescumque orizontaliter latum pervenit <lb></lb>in H, non amplius dirigi secundum rectam <lb></lb>orizontalem HL, sed secundurn contingen<lb></lb>tem ipsam curvam FH, fuerit ne ea para<lb></lb>bola nec ne, quae contingens sit HK; unde <lb></lb>proiectum ab H digressum, motu violento, <lb></lb>remota gravitate, tenderet non in L, sed in <lb></lb>K; & proinde motu mixto tanto inferius <lb></lb>puncto L, quanta est recta LK, puta in M, de<lb></lb>scribens curvam non HI, sed HM; at M non est <lb></lb>in parabola, ut facile demonstrari posset ex ea<lb></lb>dem 20. primi Apollon. cum DM maior quam DI, <lb></lb>& BF non sint in duplicata ratione ordina<pb xlink:href="064/01/083.jpg"></pb>tim aplicatarum AD, AB. </s> <s id="s.000587">Ex quo satis con<lb></lb>stare existimo proiectum suo moto parabo<lb></lb>lam non describere, quod probandum pro<lb></lb>posueram. </s> <s id="s.000588">De quibus proiectis aliquid in <lb></lb>sequentibus addam fortasse ubi occasio <lb></lb>tulerit. </s> <s id="s.000589">Reliquum est quod hoc tertio <lb></lb>libro repetam ea quae in calce libri prio<lb></lb>ris dicta fuere, sed parum accurate, quippe <lb></lb>pendentia ab eo septimo postulato, non satis <lb></lb>tunc fidem merente, in praesentia vero deci<lb></lb>ma secundi huius, ut alibi dixi, ni fallor de<lb></lb>monstratum. </s> <s id="s.000590">Interim ibi in notis marginali<lb></lb>bus adnotari volui quem locum in hoc ter<lb></lb>tio libro sortiantur.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/084.jpg"></pb> <subchap1 type="postulate"> <p type="head"> <s id="s.000591">PETITIONES</s> </p> <p type="main"> <s id="s.000592">PRIMA</s> </p> <p type="main"> <s id="s.000593">Peripheria circuli concipiatur tanquam <lb></lb>constans plurimis, seu mavis infinitis <lb></lb>lineis rectis.</s> </p> <p type="main"> <s id="s.000594">SECUNDA</s> </p> <p type="main"> <s id="s.000595">Mobile naturaliter motum caeteris pari<lb></lb>bus, quo longius distat a puncto quie<lb></lb>tis sortitur maiorem impetum, & velocius <lb></lb>movetur.</s> </p> </subchap1> <pb xlink:href="064/01/085.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000596">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000597">Si grave perpendiculariter ductum perse<lb></lb>veret in motu super plano declinante; pro<lb></lb>dibit eadem velocitate, ac si motus praece<lb></lb>dens fuisset cum eadem declinatione, ini<lb></lb>tio ducto ab eodem plano Orizontali.<figure id="id.064.01.085.1.jpg" xlink:href="064/01/085/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="1" type="proof"> <p type="main"> <s id="s.000598">Ducatur grave perpendiculariter per AB, & <lb></lb>perseveret in motu super BE declinante.</s> </p> <p type="main"> <s id="s.000599">Dico, quod fertur per BE eadem velocitate ac si <lb></lb>cepisset moveri in D; quod sit ad libellam ipsius A.</s> </p> <p type="main"> <s id="s.000600">Quoniam in B sortitum est eundem impetum <lb></lb>ductum per AB, ac si latum fuisset per DB<arrow.to.target n="marg153"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000601"><margin.target id="marg153"></margin.target>Per 12. secundi huius.</s> </p> <p type="main"> <s id="s.000602">Ergo per BE ducitur ab eadem virtute seu vi, <lb></lb>ac si motus initium fuisset in D, quippe ubique <lb></lb>ducitur a gravitate, & ab impetu in B, & pro<lb></lb>inde fertur eadem velocitate. </s> <s id="s.000603">Quod etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000604">Corollarium primum.</s> </p> <p type="main"> <s id="s.000605">Si initium motus fuisset per lineam declinantem, <lb></lb>& demum per perpendicularem, seu declinantem <lb></lb>diversa inclinatione, idem probabitur eadem ratione.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000606">Corollarium II.</s> </p> <p type="main"> <s id="s.000607">Hinc sequitur, quod impetus in E est idem si <lb></lb>motus fuerit per ABE, ac si fuisset per DE.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/086.jpg"></pb> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000608">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000609">Grave ductum perpendiculariter per spatium <lb></lb>datum diuturnitate data, perseveret in <lb></lb>motu super plano inclinato; perquirere in <lb></lb>eo motum in data diuturnitate.<figure id="id.064.01.086.1.jpg" xlink:href="064/01/086/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <s id="s.000610">Ducatur grave A perpendiculariter per AB <lb></lb>diuturnitate quae sit AB, & perseveret <lb></lb>in motu super BD plano inclinationis notae.</s> </p> <p type="main"> <s id="s.000611">Venandus ibi motus in dicta diuturnitate AB.</s> </p> <p type="main"> <s id="s.000612">Producatur BD in C donec concurrat cum AC <lb></lb>orizontaliter ducta ab A ad C. </s> <s id="s.000613">Erit BC diu<lb></lb>turnitas ipsius BC<arrow.to.target n="marg154"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000614"><margin.target id="marg154"></margin.target>Per 15. primi huius.</s> </p> <p type="main"> <s id="s.000615">Fiat BE aequalis AB, & CD tertia ad CB, CE<arrow.to.target n="marg155"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000616"><margin.target id="marg155"></margin.target>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000617">Dico BD esse quaesitum, nempe spatium transa<lb></lb>ctum diuturnitate AB.</s> </p> <p type="main"> <s id="s.000618">Quoniam CE est diuturnitas CD<arrow.to.target n="marg156"></arrow.to.target>, & CB est diu<lb></lb>turnitas motus per eundem CB ut supra pro<lb></lb>batum fuit.</s> </p> <p type="margin"> <s id="s.000619"><margin.target id="marg156"></margin.target>Per 7. pr. huius.</s> </p> <p type="main"> <s id="s.000620">Erit BE diuturnitas BD stante motu praecedenti <lb></lb>per BC<arrow.to.target n="marg157"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000621"><margin.target id="marg157"></margin.target>Per 19. quinti.</s> </p> <p type="main"> <s id="s.000622">Et pariter si fuerit per AB, BE est diuturni<lb></lb>tas motus per BD<arrow.to.target n="marg158"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000623"><margin.target id="marg158"></margin.target>Per pr. huius.</s> </p> <pb xlink:href="064/01/087.jpg"></pb> <p type="main"> <s id="s.000624">At AB est aequalis ipsi BE per constructionem.</s> </p> <p type="main"> <s id="s.000625">Ergo motus per BD fit diuturnitate AB. </s> <s id="s.000626">Quod <lb></lb>etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000627">Corollarium I.</s> </p> <p type="main"> <s id="s.000628">Hinc sequitur, quod in quolibet puncto infra <lb></lb>B est par impetus, fuerit ne motus per C<lb></lb>D aut per ABD, cum fuerit par impetus in B<arrow.to.target n="marg159"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000629"><margin.target id="marg159"></margin.target>Per 12. secundi huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000630">Corollarium II.</s> </p> <p type="main"> <s id="s.000631">Quotiescunque CE est media inter CB, CD, <lb></lb>etiamsi motus praecedens fuerit per AB; <lb></lb>BE est diuturnitas motus per BD.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000632">Corollarium III.</s> </p> <p type="main"> <s id="s.000633">Idem sequitur etiamsi AB noni esset perpendicu<lb></lb>laris, nam probatur eodem pacto.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000634">Corollarium IV.</s> </p> <p type="main"> <s id="s.000635">Sequitur etiam, quod si datis AB, & CB, <lb></lb>fiat AB lineae aequalis BE, & ad CB, CE <lb></lb>fiat tertia CD; mobile cadens aC, seu ab A, <lb></lb>movebitur super BD aequali tempore quo per AB.</s> </p> <p type="main"> <s id="s.000636">Et notandum pr. quod BD semper excedit du<lb></lb>plum ipsius AB, quia excedit duplum rectae BE.</s> </p> <pb xlink:href="064/01/088.jpg"></pb> <p type="main"> <s id="s.000637">Nota secundo quod quo AC est longior, & proinde <lb></lb>quo BD magis accedit ad orizontalem DE fit <lb></lb>semper proximior longitudini EB.</s> </p> <p type="main"> <s id="s.000638">Nota tertio quod si AC sit fere infinita, ex quo <lb></lb>BD fere Orizontalis, DE insensibiliter differt <lb></lb>ab EB, & proinde DB erit dupla ipsius AB, <lb></lb>seu ab eius dupla insensibiliter differens.</s> </p> <p type="main"> <s id="s.000639">Et quia in BD tali casu gravitas insensibiliter <lb></lb>agit, quippe cum grave insensibiliter descendat, <lb></lb>motus erit fere uniformis, & proinde par ve<lb></lb>locitas in BED.</s> </p> <p type="main"> <s id="s.000640">Ex quo, etiam apparet velocitas in quocunque <lb></lb>puncto descensus, puta in B; nam est talis, ut <lb></lb>mobile ubi non agat gravitas, sit aptum duci <lb></lb>per spatium duplum eius, per quod fuerit de<lb></lb>scensus, & paulo amplius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/089.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000641">PROPOSITIO III</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000642">Ducto gravi super plano inclinato, & in<lb></lb>de perpendiculariter; perquirere eius mo<lb></lb>tum in pari diuturnitate.</s> </p> </subchap2> <subchap2 n="3" type="proof"> <figure id="id.064.01.089.1.jpg" xlink:href="064/01/089/1.jpg"></figure> <p type="main"> <s id="s.000643">Ducatur grave super AB incli<lb></lb>nationis notae, diuturnitate AB <lb></lb>data, & inde perpendiculariter, per <lb></lb>BD; venari motum perpendicularem <lb></lb>in diuturnitate AB.</s> </p> <p type="main"> <s id="s.000644">Producatur DB, donec concurrat cum AC <lb></lb>orizontaliter ducta in C, et sit BC <lb></lb>diuturnitas motus per BC<arrow.to.target n="marg160"></arrow.to.target>. Fiat <lb></lb>BE aequalis AB, & CD tertia ad CB, CE<arrow.to.target n="marg161"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000645"><margin.target id="marg160"></margin.target>Per 15. pr. huius.</s> </p> <p type="margin"> <s id="s.000646"><margin.target id="marg161"></margin.target>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000647">Dico BD esse quaesitum.</s> </p> <p type="main"> <s id="s.000648">Quoniam CE est diuturnitas CD<arrow.to.target n="marg162"></arrow.to.target>, erit BE <lb></lb>diuturnitas BD, si motus præcedens fuerit per <lb></lb>CB; at pariter si per AB<arrow.to.target n="marg163"></arrow.to.target>. </s> <s id="s.000649">Ergo diuturni<lb></lb>tate AB aequali BE pervenit in D. </s> <s id="s.000650">Quod etc.</s> </p> <p type="margin"> <s id="s.000651"><margin.target id="marg162"></margin.target>Per 7. pr. huius.</s> </p> <p type="margin"> <s id="s.000652"><margin.target id="marg163"></margin.target>Per pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000653">Corollarium</s> </p> <p type="main"> <s id="s.000654">Hinc sequitur ut in praecedenti, quod impetus <lb></lb>infra B idem est, fuerit ne motus praecedens <lb></lb>per CD, ac per ABD.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/090.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000655">PROPOSITIO IV</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000656">Dato gravi moto perpendiculariter per spa<lb></lb>tium datum, diuturnitate data, quod per<lb></lb>ficiat motum super plano declinante, per <lb></lb>spatium itidem datum; Perquirenda in ip<lb></lb>so diuturnitas.<figure id="id.064.01.090.1.jpg" xlink:href="064/01/090/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="4" type="proof"> <p type="main"> <s id="s.000657">Moveatur grave per AB perpendiculariter <lb></lb>diuturnitate data, quae sit eadem AB, inde <lb></lb>super planum inclinatum BD.</s> </p> <p type="main"> <s id="s.000658">Perquirenda est diuturnitas motus per BD, & per ABD.</s> </p> <p type="main"> <s id="s.000659">Fiat CE media inter CB, CD, & AF nor<lb></lb>malis ad BD productam usquequo concurrat <lb></lb>cum orizontali AC.</s> </p> <p type="main"> <s id="s.000660">Dico BE esse diuturnitatem per motus BD, & <lb></lb>FE esse diuturnitatem motus per ABD.</s> </p> <p type="main"> <s id="s.000661">Quoniam nota est diuturnitas CB<arrow.to.target n="marg164"></arrow.to.target>, & nota est <lb></lb>EC per constructionem, nota est etiam BE diu<lb></lb>turnitas motus per BD, si motus praecedens fue<lb></lb>rit per CB; at idem est si fuerit per AB<arrow.to.target n="marg165"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000662"><margin.target id="marg164"></margin.target>Per 15. pr. huius.</s> </p> <p type="margin"> <s id="s.000663"><margin.target id="marg165"></margin.target>Per pr. huius.</s> </p> <p type="main"> <s id="s.000664">Ergo EB est diuturnitas motus per BD; At <lb></lb>FB est diuturnitas motus per AB<arrow.to.target n="marg166"></arrow.to.target>. </s> <s id="s.000665">Igitur <lb></lb>FE est diuturnitas motus per ABD. </s> <s id="s.000666">Quod etc.</s> </p> <p type="margin"> <s id="s.000667"><margin.target id="marg166"></margin.target>Per Co. 19. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000668">Corollarium</s> </p> <p type="main"> <s id="s.000669">Idem sequitur eadem ratione, si AB non sit <lb></lb>perpendicularis.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/091.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000670">PROPOSITIO V</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000671">Data diuturnitate in plano perpendiculari <lb></lb>motus gravis, quod perseveret moveri super <lb></lb>plano declinante; & data super eo diutur<lb></lb>nitate, reperire longitudinem.<figure id="id.064.01.091.1.jpg" xlink:href="064/01/091/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.000672">Ducatur grave perpendiculariter per AB diu<lb></lb>turnitate C, & demum super plano incli<lb></lb>nato BD, & data sit diuturnus E.</s> </p> <p type="main"> <s id="s.000673">Perquirenda sit longitudo super BD quam grave <lb></lb>conficiat diuturnitate E.</s> </p> <p type="main"> <s id="s.000674">Fiat ut C ad E ita AB ad BF<arrow.to.target n="marg167"></arrow.to.target>, unde si AB <lb></lb>concipiatur tanquam diuturnitas motus super <lb></lb>AB, erit BF diuturnitas motus super BD. <lb></lb></s> <s id="s.000675">Producatur FB donec concurrat cum A G ori<lb></lb>zontaliter ducta in G. </s> <s id="s.000676">Et fiat CD tertia pro<lb></lb>portionalis ad GB, GF<arrow.to.target n="marg168"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000677"><margin.target id="marg167"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000678"><margin.target id="marg168"></margin.target>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000679">Dico BD esse longitudinem quaesitam.</s> </p> <p type="main"> <s id="s.000680">Quoniam AB est diuturnitas ipsius AB per sup<lb></lb>pos; GB erit diuturnitas ipsius GB<arrow.to.target n="marg169"></arrow.to.target>, at GF <lb></lb>est diuturnitas ipsius GD<arrow.to.target n="marg170"></arrow.to.target>, igitur residuum BF <lb></lb>est diuturnitas BD. </s> <s id="s.000681">Quod etc.</s> </p> <p type="margin"> <s id="s.000682"><margin.target id="marg169"></margin.target>Per 15. primi huius.</s> </p> <p type="margin"> <s id="s.000683"><margin.target id="marg170"></margin.target>Per 3. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000684">Corollarium.</s> </p> <p type="main"> <s id="s.000685">Grave prodibit per AB, BD aequis tempo<lb></lb>ribus si diuturnitas E fiat aequalis diu<lb></lb>turnitati C.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/092.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000686">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000687">Moto gravi super pluribus planis diversimo<lb></lb>de inclinatis, venari diuturnitates in quo<lb></lb>libet eorum.<figure id="id.064.01.092.1.jpg" xlink:href="064/01/092/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.000688">Ducatur grave per AB diuturnitate data, <lb></lb>quae sit eadem AB; inde a B in D, & a D <lb></lb>in H. </s> <s id="s.000689">Venanda est diuturnitam motus per DH.</s> </p> <p type="main"> <s id="s.000690">Producatur DB in E donec concurrat cum <lb></lb>AG orizontaliter ducta. </s> <s id="s.000691">Item producatur H<lb></lb>D donec concurrat cum eadem AG. </s> <s id="s.000692">Fiat <lb></lb>EC media inter EB, ED<arrow.to.target n="marg171"></arrow.to.target>. </s> <s id="s.000693">Fiat itidem GF <lb></lb>media inter GD, GH.</s> </p> <p type="margin"> <s id="s.000694"><margin.target id="marg171"></margin.target>Per 13. Sexti.</s> </p> <p type="main"> <s id="s.000695">Dico DF esse diuturnitate motus per DH.<arrow.to.target n="marg172"></arrow.to.target></s> </p> <p type="margin"> <s id="s.000696"><margin.target id="marg172"></margin.target>Per 7. pr. huius.</s> </p> <p type="main"> <s id="s.000697">Quoniam DF est diuturnitas motus per DH <lb></lb>etiamsi motus praecedens fuerit per ED<arrow.to.target n="marg173"></arrow.to.target>. At <lb></lb>impetus in D est idem si motus praecedens fue<lb></lb>rit per GD, an per ED<arrow.to.target n="marg174"></arrow.to.target>. </s> <s id="s.000698">Ergo etiam si mo<lb></lb>tus fuerit per BD, DF est diuturnitas motus <lb></lb>per DH. </s> <s id="s.000699">Quod etc.</s> </p> <p type="margin"> <s id="s.000700"><margin.target id="marg173"></margin.target>Per cor. 3.2. huius.</s> </p> <p type="margin"> <s id="s.000701"><margin.target id="marg174"></margin.target>Per 12. secundi huius.</s> </p> </subchap2> <pb xlink:href="064/01/093.jpg"></pb> <subchap2 type="corollary"> <figure id="id.064.01.093.1.jpg" xlink:href="064/01/093/1.jpg"></figure> <p type="head"> <s id="s.000702">Corollarium I</s> </p> <p type="main"> <s id="s.000703">Datis pluribus lineis in quadrante circuli <lb></lb>puta FA, AB, seu FA, AC, CB, inno<lb></lb>tescent diuturnitates in quibuslibet earum, & <lb></lb>etiam in omnibus simul sumptis.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000704">Corollarium II.</s> </p> <p type="main"> <s id="s.000705">Impetus infra D est idem fuerit ne motus prae<lb></lb>cedens per GD, an per ED, vero per ABD.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/094.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.000706">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000707">Grave naturaliter motum velocius ad idem <lb></lb>ducitur punctum duabus lineis, quam una <lb></lb>tantum.</s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <figure id="id.064.01.094.1.jpg" xlink:href="064/01/094/1.jpg"></figure> <s id="s.000708">Progrediatur grave per AB in B.</s> </p> <p type="main"> <s id="s.000709">Dico quod citius perveniet in B motum per <lb></lb>A CB.</s> </p> <p type="main"> <s id="s.000710">Protrahatur BC, puta in D; & ab A in BD de<lb></lb>mittatur normalis AE.</s> </p> <p type="main"> <s id="s.000711">Quoniam grave per BC pariter movetur, ductum per <lb></lb>A CB, ac per DB<arrow.to.target n="marg175"></arrow.to.target>, & per eamdem CB ve<lb></lb>locius fertur digressum a D quam ab E<arrow.to.target n="marg176"></arrow.to.target>, per <lb></lb>illam itidem velocius fertur motum per ACB, <lb></lb>quam per EB, sed per A C aeque velociter fer<lb></lb>tur ac per CE,<arrow.to.target n="marg177"></arrow.to.target> ergo per totum ACB velocius <lb></lb>fertur quam per EB; sed aequali tempore fer<lb></lb>tur per EB ac per AB<arrow.to.target n="marg178"></arrow.to.target>; ergo per ACB ve<lb></lb>locius fertur quam per AB. </s> <s id="s.000712">Quod etc.</s> </p> <p type="margin"> <s id="s.000713"><margin.target id="marg175"></margin.target>Per pr. huius.</s> </p> <p type="margin"> <s id="s.000714"><margin.target id="marg176"></margin.target>Per 2. peti.</s> </p> <p type="margin"> <s id="s.000715"><margin.target id="marg177"></margin.target>Per 19. pr. huius.</s> </p> <p type="margin"> <s id="s.000716"><margin.target id="marg178"></margin.target>Per 19. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000717">Corollarium.</s> </p> <p type="main"> <s id="s.000718">Hinc est, quod si motus fuerit per ACB, im<lb></lb>petus in B est maior ac si fuisset per AB <lb></lb>secundum proportionem AB ad EB.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/095.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.000719">PROPOSITIO VIII</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000720">Grave naturaliter ductum, velocius fertur su<lb></lb>per tribus lineis descendentibus, quam su<lb></lb>per una tantum.<figure id="id.064.01.095.1.jpg" xlink:href="064/01/095/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.000721">Feratur grave per AB, BC, CD.</s> </p> <p type="main"> <s id="s.000722">Dico citius duci in D quam per AD.</s> </p> <p type="main"> <s id="s.000723">Producantur CB, DC ad orizontalem AF in EF.</s> </p> <p type="main"> <s id="s.000724">Ducantur normales AG, BH, & ducatur AC.</s> </p> <p type="main"> <s id="s.000725">Quoniam grave pervenit citius in C per ABC, <lb></lb>quam per AC<arrow.to.target n="marg179"></arrow.to.target>. Item citius in D per ACD <lb></lb>quam per AD<arrow.to.target n="marg180"></arrow.to.target>, tanto citius perveniet in D <lb></lb>per ABCD quam per AD. </s> <s id="s.000726">Quod etc.</s> </p> <p type="margin"> <s id="s.000727"><margin.target id="marg179"></margin.target>Per 7. huius.</s> </p> <p type="margin"> <s id="s.000728"><margin.target id="marg180"></margin.target>Per eamdem.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000729">Corollarium. I.</s> </p> <p type="main"> <s id="s.000730">Eodem pacto facile probabitur quod citius <lb></lb>perveniet in D, quatenus ducitur pluribus <lb></lb>inclinationibus.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000731">Corollarium. II.</s> </p> <p type="main"> <s id="s.000732">Impetus in D est maior, si fuerit motus per AB<lb></lb>CD, quam per AD.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/096.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000733">PROPOSITIO IX</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000734">In quadrante inferiori circuli grave celerius <lb></lb>fertur, si moveatur super peripheria, quam <lb></lb>si una, aut pluribus rectis lineis.<figure id="id.064.01.096.1.jpg" xlink:href="064/01/096/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.000735">Sit ABC quadrans inferius.</s> </p> <p type="main"> <s id="s.000736">Dico grave B velocius duci si moveatur in <lb></lb>peripheria, quam si per BC, aut BDC, aut <lb></lb>BDEFC.</s> </p> <p type="main"> <s id="s.000737">Quoniam in peripheria ducitur pluribus inclina<lb></lb>tionibus<arrow.to.target n="marg181"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000738"><margin.target id="marg181"></margin.target>Per primam pet.</s> </p> <p type="main"> <s id="s.000739">Ergo grave super ipsa motum celerius transigit.<arrow.to.target n="marg182"></arrow.to.target> Quod etc.</s> </p> <p type="margin"> <s id="s.000740"><margin.target id="marg182"></margin.target>Per cor. primum 8. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000741">Corollarium I.</s> </p> <p type="main"> <s id="s.000742">Idem sequitur, si digrediatur a quovis puncto <lb></lb>Peripheriae, puta a D.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000743">Corollarium II.</s> </p> <p type="main"> <s id="s.000744">In C impetus est maior, si motus fuerit per <lb></lb>Peripheriam, quam aliter quomodocunque.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/097.jpg"></pb> <chap type="bk"> <p type="main"> <s id="s.000745">DE MOTV <lb></lb>GRAVIVM <lb></lb>LIBER QVARTVS.<lb></lb>ET LIQVIDORVM PRIMVS.</s> </p> <subchap1 type="preface"> <subchap2 type="preface"> <p type="main"> <s id="s.000746">Hactenus<arrow.to.target n="note1"></arrow.to.target> mihi videor de <lb></lb>scientia motus naturalis <lb></lb>gravium solidorum satis <lb></lb>pro viribus dixisse, dum <lb></lb>ex quibusdam proprieta<lb></lb>tibus sensui notis, plures <lb></lb>ignotae deductae, & patefa<lb></lb>ctae sunt: in hoc enim so<lb></lb>lummodo ex Aristotele omnis scientia ver<lb></lb>satur: ut in praxi apud Euclidem, & alios, qui <lb></lb>veras, & simplices scientias tractant, videre <lb></lb>est: unde nec agit Geometra de natura quan<lb></lb>titatis, nec Musicus de natura soni, nec per<lb></lb>spectivus de natura luminis, nec mechanicus <lb></lb>de natura ponderis.</s> <s id="s.000747">At<arrow.to.target n="note2"></arrow.to.target> vero meus intelle<lb></lb>ctus non omnino acquiescit, ni causas priores, <lb></lb>a quibus hi effectus demum proveniunt, si non <pb xlink:href="064/01/098.jpg"></pb>assequatur, saltem investiget; perquirendo <lb></lb>quae sit natura mobilium, corporum nimi<lb></lb>rum prout mobilia sunt; etiam si hoc non <lb></lb>ad scientiam de motu, sed ad habitum supe<lb></lb>riorem, nimirum sapientiae pertineat; quo <lb></lb>non effectus, sed rerum naturae, & principia <lb></lb>nobis innotescunt, ut Aristoteles in Metaphis. <lb></lb>etiam si in moralibus videatur secus sentire, <lb></lb>seu quia ex communi potius quam ex propria <lb></lb>sententia ibi loquutus fuerit, ubi exactam di<lb></lb>scussionem locus non postulabat, seu mavis <lb></lb>culpa transcriptoris; in quo nihilominus plu<lb></lb>rimos, & magni nominis habuit sectatores. <lb></lb></s> <s id="s.000748">Ut<arrow.to.target n="note3"></arrow.to.target> ut sit ego quid tale delibavi, dum in prae<lb></lb>fatione priori libro praeposita, causam aperire <lb></lb>conatus sum, cur duo quaelibet gravia, quan<lb></lb>tumvis inaequalia, aequalia spatia conficiant; <lb></lb>videlicet quia natura gravium talis sit, ut <lb></lb>utrobique gravitas tali pacto sit materiae con<lb></lb>nexa, & ita eam perpetuo sequatur, ut quanta <lb></lb>sit gravitas, seu eius actio; tantumdem sit pa<lb></lb>riter materiae, & proinde resistentiae; ex quo <lb></lb>demum aequales sequantur effectus: quod ta<lb></lb>men ad motuum indaginem supervacaneum <lb></lb>erat.</s> <s id="s.000749">Non tamen ex hoc ego me adhuc gra<lb></lb>vium naturam omnino assecutum esse pro <lb></lb>certo habeo. </s> <s id="s.000750">Non quilibet collimans scopum <lb></lb>ferit; at quotus quisque propius dirigit, non <pb xlink:href="064/01/099.jpg"></pb>inutiliter laborasse censendus est. </s> <s id="s.000751">Ut<arrow.to.target n="note4"></arrow.to.target> cumque <lb></lb>sit, quod tum factum est, hic pariter peragere <lb></lb>libuit, videlicet naturam motus pro viribus <lb></lb>investigare, causas nimirum, & principia, a <lb></lb>quibus hae demum motus passiones proveni<lb></lb>ant. </s> <s id="s.000752">Iam<arrow.to.target n="note5"></arrow.to.target> ante plures annos mihi visus sum <lb></lb>assequi causam accelerationis motus , dum ad <lb></lb>huc mobile a motore impellitur; quia nimirum <lb></lb>mobili moto imprimatur impetus, causa mo<lb></lb>tus subsequentis; ex quo in secundo tempore <lb></lb>adsunt duo motores, unde est velocior, & im<lb></lb>petus maior; in tertio tempore sunt duo iti<lb></lb>dem motores, at alter puta impetus maioris <lb></lb>virtutis, unde motus adhuc celerior; & ita de<lb></lb>inceps.</s> <s id="s.000753">Non<arrow.to.target n="note6"></arrow.to.target> vero ex hoc constabat qua pro<lb></lb>portione talis acceleratio fieret. </s> <s id="s.000754">Interim dum <lb></lb>pendulorum motus, perquirerem, praeter ex<lb></lb>pectationem se se mihi obtulit, eorum longi<lb></lb>tudines diuturnitatibus in duplicata respon<lb></lb>dere ratione; de quo in prioris libri praefatio<lb></lb>ne; ex quo demum, nihil minus cogitanti mi<lb></lb>hi, in sexta propositione eiusdem deducere con<lb></lb>tigit, motum tali pacto accelerari, ut in secun<lb></lb>do tempore sit prioris triplum, in tertio quin<lb></lb>tuplum, & deinceps iuxta numerorum impa<lb></lb>rium progressionem: quod<arrow.to.target n="note7"></arrow.to.target> miram mihi exci<lb></lb>tavit cupidinem venandi a qua nam virtute, in <lb></lb>secundo tempore tanta motus fieret accretio,<pb xlink:href="064/01/100.jpg"></pb>dum nec videbatur esse impetus primum im<lb></lb>pressi maior activitas, quam ipsius motoris a <lb></lb>quo ortum duxerat; nec quid aliud ibi esse de <lb></lb>novo productum suspicandum videbatur. </s> <s id="s.000755">Non <lb></lb>tamen deterreri potui, quin ulterius progre<lb></lb>diens huius adhuc causam consequi sperarem: <lb></lb>quamvis se mihi dificillimum obtulerit, & <lb></lb>pluries me esse assecutum perperam existima<lb></lb>verim, meque demum fuisse deceptum com<lb></lb>pererim. </s> <s id="s.000756">Contigit<arrow.to.target n="note8"></arrow.to.target> interim reperire, quod est <lb></lb>in Corol. Tertiae Secundi huius, motum or<lb></lb>tum ab impetu esse aequabilem; quod a natu<lb></lb>ra ipsiusmet mobilis emanere censendum vi<lb></lb>sum fuit: ex quo in spem adductus sum ut ip<lb></lb>sammet mobilis naturam assequi valerem. <lb></lb></s> <s id="s.000757">Pluries<arrow.to.target n="note9"></arrow.to.target> cogitaveram esse naturae consentane<lb></lb>um, ut ex simplicissimis principijs quam plur<lb></lb>imi mirabiles effectus educantur. </s> <s id="s.000758">Cuius rei, & <lb></lb>si plura habeam, unicum tantum in praesentia <lb></lb>aut alterum adducam exemplum. </s> <s id="s.000759">Perpen<lb></lb>das amabo quot qualia, & quanta, ex Solis sub <lb></lb>Ecliptica circumlatione, in inferioribus gi<lb></lb>gnantur; et quot qualia, et quanta hominibus <lb></lb>deficerent, ni eis necessitas quotidiani cibi <lb></lb>imposita fuisset: ex<arrow.to.target n="note10"></arrow.to.target> quo mihi pariter probabi<lb></lb>le visum est, eam fuisse naturam mobilibus tri<lb></lb>butam, ut ex eius aliqua simplici immediata <lb></lb>proprietate emanent caeterae.</s> <s id="s.000760">Cum igitur ut <pb pagenum="101" xlink:href="064/01/101.jpg"></pb>mox dictum fuit mobile motum aequabiliter <lb></lb>demum moveatur sine motore; videtur infe<lb></lb>rendum, quod motus motum producat, seu <lb></lb>potius quod motus perseveret, & se ipsum, <lb></lb>ut ita dicam, extendat, & continuet; quatenus<arrow.to.target n="note11"></arrow.to.target> <lb></lb>dum semel mobile motum est, sit potens, <lb></lb>seu in potentia proxima se ipsum eadem ra<lb></lb>tione movendi: ex<arrow.to.target n="note12"></arrow.to.target> quibus in eam incidi sen<lb></lb>tentiam, ut existimem, eam esse fortasse na<lb></lb>turam mobilium, ut indiferenter se habeant <lb></lb>tam ad quietem, quam ad quemlibet motum; <lb></lb>unde, dummodo motus praecedat, a quacumque <lb></lb>causa proveniens, seu naturali seu violenta, <lb></lb>similis postmodum subsequatur, seu idem <lb></lb>perseveret, eadem velocitate quam in quoli<lb></lb>bet instanti sortitum fuerit, donec impedia<lb></lb>tur; & hanc motus continuationem ab ipsa<lb></lb>met immobilis natura immediate emanantem, <lb></lb>forsitam esse unicam, & simplicem causam, a <lb></lb>qua fluant omnes illi effectus, & passiones, <lb></lb>quae in motu demum tum naturali, tum vio<lb></lb>lento a nobis percipiuntur.</s> <s id="s.000761">Et<arrow.to.target n="note13"></arrow.to.target> quamvis huius<lb></lb>modi motus continuatio non sit nova entitas <lb></lb>superaddita, eam nihilominus intellectus con<lb></lb>cipere tanquam quid noviter ortum, nimirum <lb></lb>posito motu, ex eo oriri virtutem, novum pro<lb></lb>ducentem motum, ad faciliorem de motu ra<lb></lb>tiocinationem non parum deservientem, quam vir<pb xlink:href="064/01/102.jpg"></pb>tutem appellamus impetum; qui<arrow.to.target n="note14"></arrow.to.target> re vera nil <lb></lb>aliud sit, nisi naturalis propensio ad motum, <lb></lb>seu potentia mobili inexistens continuandi mo<lb></lb>tum semel adeptum quae potentia dum mo<lb></lb>bile quiescit, sit in actu primo, & mediante, <lb></lb>motu reducatur in secundum, ea ratione qua <lb></lb>homini discurrenti non additur nova rationa<lb></lb>litas; seu<arrow.to.target n="note15"></arrow.to.target> novum principium, & nova poten<lb></lb>tia ratiocinandi, sed eademmet, quam intrin<lb></lb>secus habet, & est in actu primo, reducitur in <lb></lb>secundum.</s> <s id="s.000762">Porro<arrow.to.target n="note16"></arrow.to.target> quod vere talis fuerit <lb></lb>natura mobilibus tradita, ut indiferenter se <lb></lb>habeant ad motum, & quietem, quamvis ex <lb></lb>dicta uniformis motus continuatione satis pro<lb></lb>babile videatur, non ego tamen pro certo af<lb></lb>firmare ausim: sumus<arrow.to.target n="note17"></arrow.to.target> in physicis, ubi demon<lb></lb>strationes rariores: non<arrow.to.target n="note18"></arrow.to.target> tamen videri deberet le<lb></lb>viter probatum, si ex hoc solummodo prin<lb></lb>cipio omnes probarentur sequi passiones, quae <lb></lb>in motu quolibet percipiuntur absque quo ali<lb></lb>quid aliud, vel de novo oriatur, vel ortum de<lb></lb>pereat.</s> <s id="s.000763">Ex<arrow.to.target n="note19"></arrow.to.target> eo autem sequitur, quod dum mo<lb></lb>bile impellitur motus necessario augetur; un<lb></lb>de<arrow.to.target n="note20"></arrow.to.target> quo per maius spatium impellitur eo cor<lb></lb>pus obsistens validius percutit; ex<arrow.to.target n="note21"></arrow.to.target> quo tamen <lb></lb>motus ipse fit debilior, respondens siquidem <lb></lb>oppositi resistentiae; quae<arrow.to.target n="note22"></arrow.to.target> si augeatur, velocitas <lb></lb>taliter minuitur, ut tandem deficiat, absque<pb xlink:href="064/01/103.jpg"></pb>quo aliquid oriri, aut deperire supponatur: ex <lb></lb>quibus vires percussionis metiri licet, de quo <lb></lb>alibi.</s> <s id="s.000764">Inde<arrow.to.target n="note23"></arrow.to.target> est quod si manubrio parietem per<lb></lb>cutias, illud intra melleum intruditur, quoniam <lb></lb>melleo minor obijcitur resistentia; facilius <lb></lb>siquidem is a manubrio permeatur quam murus <lb></lb>a manubrio. </s> <s id="s.000765">Si<arrow.to.target n="note24"></arrow.to.target> vero mobile expellatur, mo<lb></lb>veri perseverat, sine cuiusvis ope adiutoris de <lb></lb>novo orti; cum ex ipsiusmet natura, prout <lb></lb>mobile est, eiusdem motus continuatio neces<lb></lb>sario subsequatur.</s> <s id="s.000766">Si<arrow.to.target n="note25"></arrow.to.target> offendit in via quod mo<lb></lb>tum urgeat, aut retundat; augetur velocitas, <lb></lb>aut minuitur; at<arrow.to.target n="note26"></arrow.to.target> quaecumque ea sit inde per<lb></lb>severat, quia ea motus natura ut continuetur; <lb></lb>unde<arrow.to.target n="note27"></arrow.to.target> si permeet murum quem feriat, ei proin<lb></lb>de resistentem, remissius fertur, quatenus est <lb></lb>maior muri durities, & proinde resistentia; ex <lb></lb>quo velocitas magis retunditur; quae tamen si <lb></lb>non omnino perit, qualis tandem remanet <lb></lb>talis perseverat; idem quippe continuatur mo<lb></lb>tus; quousque<arrow.to.target n="note28"></arrow.to.target> tamen resistentia perdurat, <lb></lb>motus paulatim minuitur, & tandem extin<lb></lb>guitur.</s> <s id="s.000767">Ceterum<arrow.to.target n="note29"></arrow.to.target> cum huiusmodi continuatio <lb></lb>emanet a propria ipsiusmet mobilis natura, <lb></lb>subsequi necessario debet quemlibet motum, <lb></lb>etiamsi per brevem fuerit morulam; quod<arrow.to.target n="note30"></arrow.to.target> ap<lb></lb>paret in pila lignea, malleo ligneo lusorio lon<lb></lb>gioris manubrij longe propulsa, quamvis a <pb xlink:href="064/01/104.jpg"></pb>malleo per parvam morulam, & per minimum <lb></lb>spatium lata fuerit.</s> <s id="s.000768">Ex<arrow.to.target n="note31"></arrow.to.target> quo itidem sequitur, <lb></lb>quod pila lusoria ad murum illidens, resilit; <lb></lb>quia pars murum feriens, vi compressa, ictui <lb></lb>cedens densatur, & ex curva complanatur; & <lb></lb>si sit talibus praedita viribus, ut deficiente vio<lb></lb>lentia propellente, queat ex se in pristinam re<lb></lb>duci rotunditatem; pars explanata, quae ite<lb></lb>rum incurvatur, retrocedens versus locum cen<lb></lb>tri, eo fertur celeri motu; qui quamvis in tali <lb></lb>reductione brevis fuerit, & proinde per brevem <lb></lb>morulam, idem continuatur eadem celeritate, <lb></lb>cum eius naturae competat, motum etiamsi per <lb></lb>parvum fuerit spatium continuare. </s> <s id="s.000769">Quod idem <lb></lb>sequitur si non pila, sed murus ipse caedat pri<lb></lb>us, & demum se in pristinum reducat; unde <lb></lb>si neutrum caedat non fit resilitio. </s> <s id="s.000770">Si<arrow.to.target n="note32"></arrow.to.target> vero <lb></lb>non perpendiculariter sed oblique murum <lb></lb>feriat, resilit ea lege, ut angulus reflexionis sit <lb></lb>angulo incidentiae proxime aequalis; quoniam <lb></lb>dum impingit, centrum adhuc fertur antrorsum; <lb></lb>unde pars pressa dum se in rotunditatem iterum <lb></lb>reducit, pilam dirigit secundum lineam tran<lb></lb>seuntem per centrum iam antrorsum latum; <lb></lb>qui motus etiamsi per breve spatium, postmodum <lb></lb>continuatur: quoniam vero ex ea centri pro<lb></lb>gressione pilae plures successive partes super <lb></lb>murum vertuntur, is motus itidem continua<pb xlink:href="064/01/105.jpg"></pb>tur unde pila ipsa vertiginem acquirit, eo ce<lb></lb>leriorem, quo angulus incidentiae plus acuitur; <lb></lb>qua vertigine adepta, ex eius continuatione, <lb></lb>ubi pila in planum iterum incidat, non servat <lb></lb>praedictam regulam in angulo reflexionis, qui <lb></lb>erit acutior, si pilae motus sit secundum ver<lb></lb>tiginis ordinem, si vero contra obtusior.</s> <s id="s.000771">Quae <lb></lb>clarius apparent in pila reticulo, aut alio quo<lb></lb>libet transversim percussa, in qua maior impri<lb></lb>matur vertigo, quae demum eadem continuatur. <lb></lb></s> <s id="s.000772">Inde<arrow.to.target n="note33"></arrow.to.target> item est quod pila eadem dum lusoria <lb></lb>palmula percussa, libere demum fertur, velo<lb></lb>cius prodit ipsam et palmula movente; expul<lb></lb>sa siquidem non modo ab ipsius impellentis <lb></lb>motu, sed etiam quoniam ipsiusmet pilae pars <lb></lb>percussa, ob modo dictam compressionem ce<lb></lb>dens, & exinde densata, & mox in pristinam <lb></lb>redacta formam, ducitur versus ipsius pilae cen<lb></lb>trum maiori velocitate, quam si a sola impel<lb></lb>lentis vi ducta fuisset; quae maior velocitas con<lb></lb>tinuatur. </s> <s id="s.000773">Imo<arrow.to.target n="note34"></arrow.to.target> reticulo expulsa, fertur etiam ve<lb></lb>locius, a triplici nempe motore ducta, addito <lb></lb>tertio, nimirum rete, cedente prius, & mox se <lb></lb>in pristinum reducente.</s> <s id="s.000774">Hinc<arrow.to.target n="note35"></arrow.to.target> est etiam quod <lb></lb>quandocumque sphaera circumvolvitur, continua<lb></lb>tur vertigo: unde<arrow.to.target n="note36"></arrow.to.target> contingere potest, ut inde, <lb></lb>sequatur motus ipsius sphaerae progressivus, ei <lb></lb>supposito nimirum plano, suo contactu motum <pb xlink:href="064/01/106.jpg"></pb>partis inferioris impediente, ex quo pars su<lb></lb>perior non impedita, & libere mota celerius <lb></lb>fertur, et quo vergit, vergit item centrum, & <lb></lb>talis continuatur motus, unde tota sphaera pro<lb></lb>dit ulterius, absque quo alius novus motor su<lb></lb>peraddatur. Hinc<arrow.to.target n="note37"></arrow.to.target> itidem est, quod si sphaeram <lb></lb>quiescentem ex aliqua sui parte digito com<lb></lb>primas contra subiectum planum, ea sortitur <lb></lb>duplicem motum, progressivum antrorsum, <lb></lb>& validiorem in gyrum retrorsum: unde cessan<lb></lb>te priori, si circumlatio continuatur, retro<lb></lb>cedit, ac si tum ei planum supponeretur, <lb></lb>absque eo quod aliquid oriatur, aut depereat. <lb></lb></s> <s id="s.000775">Quod<arrow.to.target n="note38"></arrow.to.target> pariter evenit in trochulo puerorum, <lb></lb>qui dum digitis in gyrum ducitur, circa pro<lb></lb>prium axem circumfertur, eius inferiori pro<lb></lb>minenti polo innixus; qui ubi demum ob im<lb></lb>petum diminutum declinans subiectum plan<lb></lb>um latere tangit, super illud circumvolvi<lb></lb>tur, fere ad instar asinariae molae, cuius pro<lb></lb>inde axis sua circumversione conum efficit, <lb></lb>cuius vertex est polus inferior, superior vero <lb></lb>dum rotatur circulum describit ipsius coni basim, <lb></lb>contra ordinem vertiginis peripheriae, motu tali, <lb></lb>qui minus diligenter intuentibus, apparet es<lb></lb>se prioris, adhuc perseverantis, inversio; pluri<lb></lb>bus mirabile visum, non percipientibus esse <pb xlink:href="064/01/107.jpg"></pb>naturae congruum, ambos ibi continuari mo<lb></lb>tus, priorem quidem peripheriae circum, <lb></lb>axem trochi, postremum vero poli superioris <lb></lb>contra prioris ordinem; quod quibuslibet <lb></lb>motibus, ut dictum fuit, commune est, ex <lb></lb>ipsius mobilis natura proveniens, absque <lb></lb>quod aliquid aliud oriatur, aut ortum depereat, <lb></lb>remanente siquidem solummodo cuiuslibet <lb></lb>velocitatis semel impressae, naturali continua<lb></lb>tione, quam quodlibet mobile, quocumque <lb></lb>pacto, ubivis a quocumque motore sortitum <lb></lb> fuerit; ex quo non modo praedictae oriuntur mo<lb></lb>tus passiones, sed omnes alias passim obvias <lb></lb>emanare, facile demonstrabitur.</s> <s id="s.000776">A<arrow.to.target n="note39"></arrow.to.target> nullo au<lb></lb>tem experimento praedicta natura mobilium <lb></lb>tam clare apparere videtur, quam a facilitate, <lb></lb>qua mobilia quiescentia, a quolibet etiam mi<lb></lb>nimo saepius impelluntur motore. </s> <s id="s.000777">Quod ap<lb></lb>paret in cymbula in aqua natante, ponderis <lb></lb>librarum quinquaginta, & ultra; quam non <lb></lb>modo duces capillo mulieris, sed si illum ex <lb></lb>alio capite uspiam alligaveris, suo solum pon<lb></lb>dere cymbulam trahit, & ad litus, ut ita dicam, <lb></lb>appellere coarctat, non obstante aqua renu<lb></lb>ente, propriae siquidem divisioni saltem ali<lb></lb>qualiter obsistente: quod aliunde non vi<lb></lb>detur oriri nisi ex eademmet praedicta mo<pb xlink:href="064/01/108.jpg"></pb>bilis natura, indiferenter nimirum se haben<lb></lb>tis ad motum, & quietem. </s> <s id="s.000778">Vi autem ex eadem <lb></lb>tandem videamus, qua proportione motus ac<lb></lb>celeratio fieri debeat, & an experimentis <lb></lb>respondeat.<figure id="id.064.01.108.1.jpg" xlink:href="064/01/108/1.jpg"></figure> Ducatur mobile A, ab <lb></lb>A versus E a quovis motore, seu <lb></lb>naturaliter a gravitate deorsum, seu <lb></lb>violenter ab impellente; et spatium AE con<lb></lb>cipiatur sectum in portiones aequales in pun<lb></lb>ctis B, C, D tali ratione, ut in B mobile <lb></lb>ductum virtute motus ab A in B acquirat impe<lb></lb>tum, ex quo motus item subsequatur; seu quod <lb></lb>idem est, cuius virtute potentia mobilis eun<lb></lb>dem continuendi motum, reducatur ad actum <lb></lb>secundum modo superius explicato; si conci<lb></lb>piamus in B deficere actionem motoris, idem <lb></lb>nihilominus eiusdem velocitatis perseverat, & <lb></lb>continuatur motus; unde per tantundem tem<lb></lb>poris fertur per portionem aequalem ipsi AB, <lb></lb>puta in C. </s> <s id="s.000779">Ni vero motoris actio deficiat, eius <lb></lb>virtute fertur itidem mobile per portionem <lb></lb>aequalem ipsi a AB; unde in secundo tempo<lb></lb>re conficit duas spatij portiones, eidem AB <lb></lb>aequales; & proinde dum prodit in D, movetur <lb></lb>motu dupliciter velociori, & sortitur dupli<lb></lb>cem impetum, seu huius duplicis motus con<lb></lb>tinuationem; ex quo in tertio tempore, ducitur <lb></lb>per duplum eiusdem portionis AB, at per<pb xlink:href="064/01/109.jpg"></pb>aequale a motore, ergo conficit tres portiones; <lb></lb>in quarta quatuor, in decima decem, & ita de<lb></lb>inceps. </s> <s id="s.000780">Obijcies<arrow.to.target n="note40"></arrow.to.target> primo, in portione AB iam <lb></lb>adesse impetum; nec mobile recedere ab A <lb></lb>quin impetus adsit: cum etenim impetus ema<lb></lb>net a motu, & sit eius passio, est ab eo insepa<lb></lb>rabilis, & proinde ubi est motus, est pariter im<lb></lb>petus, quemadmodum ubi est ignis, est pari<lb></lb>ter calor: nec causa est prior effectu tempore, <lb></lb>sed natura; quod non obstat, quin in eo<lb></lb>dem instanti in quo est ignis, seu motus, <lb></lb>sit pariter calor seu impetus.</s> <s id="s.000781">Responditur<arrow.to.target n="note41"></arrow.to.target> conceden<lb></lb>dum, quod etiam in eodem instanti in <lb></lb>quo est motus, fieri possit ut sit pariter im<lb></lb>petus, si vice versa mihi concedatur, nil <lb></lb>esse prius sua causa, & proinde impetum non <lb></lb>antecedere motum a quo est productus: at <lb></lb>dum mobile est adhuc in A non movetur, sed <lb></lb>quiescit: nec potest vere dici quod moveatur, <lb></lb>quin ab A recedens perveniat in B, unde sicut <lb></lb>est absurdum dicere ignem producere calorem, <lb></lb>quin prius sit productus ipsemet ignis, ita pa<lb></lb>riter esset obsurdum asserere, motum produ<lb></lb>cere impetum, quin sit productus ipsemet mo<lb></lb>tus, & proinde prius quam mobile sit in B. </s> <s id="s.000782">Nec <lb></lb>dicas iam motum adesse priusquam perveniat <lb></lb>in B; nam quocumque primo perventum <lb></lb>erit, tum in eo puncto intelligo esse B: non <pb xlink:href="064/01/110.jpg"></pb>enim quaerimus, portio AD sit ne magna <lb></lb>aut parva, divisibilis an indivisibilis, & ma<lb></lb>thematice vel physice; quod ad praesentem spe<lb></lb>culationem non est necessarium; sufficit mi<lb></lb>hi namque in praesentia, aliquem motum non <lb></lb>dici adesse ab impetu dependentem, quin ali<lb></lb>us a quocumque impetu independenter prae<lb></lb>cedat, productus siquidem a solo motore, cu<lb></lb>ius virtute, potentia mobilis in actum secun<lb></lb>dum reducatur, per quam demum continuetur <lb></lb>motus ut supra dictum fuit; secus enim causa <lb></lb>produceret suam causam in eodem genere <lb></lb>causae; immo idem esset causa sui ipsius, quippe <lb></lb>causa suae propriae causae. </s> <s id="s.000783">Obijcies<arrow.to.target n="note42"></arrow.to.target> secundo <lb></lb>motum non augeri iuxta progressionem Arith<lb></lb>meticam naturalem, ut hic asseritur, sed secun<lb></lb>dum numeros impares, ut in sexta primi <lb></lb>huius, & ut apud doctiores in praesentia fere <lb></lb>communiter creditur.</s> <s id="s.000784">Responditur<arrow.to.target n="note43"></arrow.to.target> hanc sextam pro<lb></lb>positionem inniti experimentis, sensui dece<lb></lb>ptioni obnoxijs, quibus insensibilis error de<lb></lb>tegi nequit; quod hic evenit ex eo, quia por<lb></lb>tiones temporis aequales ei priori, in qua confi<lb></lb>citur prima motus portio independens ab im<lb></lb>petu, percipi nequeant, utpote insensibiles, <lb></lb>prout est insensibilis dicta motus prima por<lb></lb>tio; quae si perciperentur, videremus augeri <lb></lb>motum iuxta naturalem progressionem: At<arrow.to.target n="note44"></arrow.to.target><pb xlink:href="064/01/111.jpg"></pb>in temporibus, & motibus sensibilibus res di<lb></lb>verse se habet, ubi cognosci nequit motus <lb></lb>pars aliqua, nec tempus in quo conficiatur, <lb></lb>quin iam sint plures temporis peractae portio<lb></lb>nes, ei aequales, in qua fuit motus ab impetu non <lb></lb>adiutus; cui tempori si plures aequales subse<lb></lb>quantur, motus in eis, seu motus portiones, <lb></lb>portionibus temporum, iuxta numerorum im<lb></lb>parium progressionem fere respondebunt.<figure id="id.064.01.111.1.jpg" xlink:href="064/01/111/1.jpg"></figure></s> </p> <p type="foot"> <s id="s.000785"><foot.target id="foot.1"></foot.target>1 Actum est de scientia motus naturalis.</s> </p> <p type="foot"> <s id="s.000786"><foot.target id="foot.2"></foot.target>2 Modo perquirendae causae.</s> </p> <p type="foot"> <s id="s.000787"><foot.target id="foot.3"></foot.target>3 Ut supra respectu gravitatis factum fuit.</s> </p> <p type="foot"> <s id="s.000788"><foot.target id="foot.4"></foot.target>4 Natura igitur motus investiganda.</s> </p> <p type="foot"> <s id="s.000789"><foot.target id="foot.5"></foot.target>5 Iam quaesiveram causam accel.</s> </p> <p type="foot"> <s id="s.000790"><foot.target id="foot.6"></foot.target>6 At non proportionem.</s> </p> <p type="foot"> <s id="s.000791"><foot.target id="foot.7"></foot.target>7 Reperta iuxta progressionem numerorum imparium. Quaesivi causam.</s> </p> <p type="foot"> <s id="s.000792"><foot.target id="foot.8"></foot.target>8 Repertus motus ab impetu aequabilis.</s> </p> <p type="foot"> <s id="s.000793"><foot.target id="foot.9"></foot.target>10 Natura utitur principijs simplicibus.</s> </p> <p type="foot"> <s id="s.000794"><foot.target id="foot.10"></foot.target>11 Unde visum ex simplici mobilis proprietate emanandas caeteras.</s> </p> <p type="foot"> <s id="s.000795"><foot.target id="foot.11"></foot.target>12 Quae sit motum ex se continuari.</s> </p> <p type="foot"> <s id="s.000796"><foot.target id="foot.12"></foot.target>13 Quia mobilia indiferenter se habeant, ad motum & quietem.</s> </p> <p type="foot"> <s id="s.000797"><foot.target id="foot.13"></foot.target>14 Huiusmodi continuationem non est nova entitas.</s> </p> <p type="foot"> <s id="s.000798"><foot.target id="foot.14"></foot.target>15 At ut nova concipitur. Dicitur & impetus.</s> </p> <p type="foot"> <s id="s.000799"><foot.target id="foot.15"></foot.target>16 Huiusmodi indiferentiam esse mobili naturalem.</s> </p> <p type="foot"> <s id="s.000800"><foot.target id="foot.16"></foot.target>17 Probatur per dictam naturalem motus continuationem.</s> </p> <p type="foot"> <s id="s.000801"><foot.target id="foot.17"></foot.target>18 Ex quo caeterae motus passiones.</s> </p> <p type="foot"> <s id="s.000802"><foot.target id="foot.18"></foot.target>19 Absque quo quid oriatur aut pereat.</s> </p> <p type="foot"> <s id="s.000803"><foot.target id="foot.19"></foot.target>20 Unde dum mobile impellitur motus augetur.</s> </p> <p type="foot"> <s id="s.000804"><foot.target id="foot.20"></foot.target>21 Et quo longius, ictus validior.</s> </p> <p type="foot"> <s id="s.000805"><foot.target id="foot.21"></foot.target>22 At motus debilior. Si resistentia maior motus tardior.</s> </p> <p type="foot"> <s id="s.000806"><foot.target id="foot.22"></foot.target>23 Et tandem deficit.</s> </p> <p type="foot"> <s id="s.000807"><foot.target id="foot.23"></foot.target>24 Patet experimento mallei.</s> </p> <p type="foot"> <s id="s.000808"><foot.target id="foot.24"></foot.target>25 Expulsum moveri perseverat.</s> </p> <p type="foot"> <s id="s.000809"><foot.target id="foot.25"></foot.target>26 Si quid urgeat aut retundat, variatur velocitas.</s> </p> <p type="foot"> <s id="s.000810"><foot.target id="foot.26"></foot.target>27 Et talis perseverat.</s> </p> <p type="foot"> <s id="s.000811"><foot.target id="foot.27"></foot.target>28 Si murum permeet remittitur.</s> </p> <p type="foot"> <s id="s.000812"><foot.target id="foot.28"></foot.target>29 Si perseveret, velocitas minuitur.</s> </p> <p type="foot"> <s id="s.000813"><foot.target id="foot.29"></foot.target>30 Idem etiam per morulam.</s> </p> <p type="foot"> <s id="s.000814"><foot.target id="foot.30"></foot.target>31 Ut in ludo mallei.</s> </p> <p type="foot"> <s id="s.000815"><foot.target id="foot.31"></foot.target>32 Unde pilae resilitio.</s> </p> <p type="foot"> <s id="s.000816"><foot.target id="foot.32"></foot.target>33 Si oblique feriat, oblique resilit.</s> </p> <p type="foot"> <s id="s.000817"><foot.target id="foot.33"></foot.target>34 Pila celerior instrumento expellente.</s> </p> <p type="foot"> <s id="s.000818"><foot.target id="foot.34"></foot.target>35 Et eo magis reticulo expulsa.</s> </p> <p type="foot"> <s id="s.000819"><foot.target id="foot.35"></foot.target>36 Vertigo durat.</s> </p> <p type="foot"> <s id="s.000820"><foot.target id="foot.36"></foot.target>37 Unde motus localis.</s> </p> <p type="foot"> <s id="s.000821"><foot.target id="foot.37"></foot.target>38 Pila digito compressa acquirit duplicem motum.</s> </p> <p type="foot"> <s id="s.000822"><foot.target id="foot.38"></foot.target>39 Ex quo trochulum retrocedere videtur.</s> </p> <p type="foot"> <s id="s.000823"><foot.target id="foot.39"></foot.target>40 Motus est a minimo motore.</s> </p> <p type="foot"> <s id="s.000824"><foot.target id="foot.40"></foot.target>41 Objectio prima non dari primam motus portionem sine impetu.</s> </p> <p type="foot"> <s id="s.000825"><foot.target id="foot.41"></foot.target>42 Responditur etiam si adsit impetus prima motus portio est ab eo independens.</s> </p> <p type="foot"> <s id="s.000826"><foot.target id="foot.42"></foot.target>43 Objectio 2. motum non augeri iuxta progressionem naturalem.</s> </p> <p type="foot"> <s id="s.000827"><foot.target id="foot.43"></foot.target>44 Responditur quod motus augetur iuxta progressionem naturalem per tempora insensibilia.</s> </p> <p type="foot"> <s id="s.000828"><foot.target id="foot.44"></foot.target>45 At per sensibilia fere iuxta progressionem numerorum imparium.</s> </p> <p type="main"> <s id="s.000829">Quod ut planius fiat, Moveatur mobile A ab <lb></lb>A in B sensibiliter, & tempore sensibili ab, <lb></lb>cui subsequantur aequalia tempora bc, cd, & <lb></lb>primum tempus ab intelligatur divisum in por<lb></lb>tiones minimas aequales, in quarum priori a<lb></lb>e, latum fuerit mobile ab A in E independen<lb></lb>ter ab impetu, qui in puncto E motui con<lb></lb>currere incipiat; has portiones patet esse eo <lb></lb>plures quo minores; sint decem, & mobile fe<lb></lb>ratur temporibus ab, bc, cd, per spatia AB, <lb></lb>BC, CD; erunt portiones aequales portioni <lb></lb>AE in AB 55, in BC 155, in CD 255, inter <lb></lb>se ut 11, 31, 51. Si vero portio temporis ae <lb></lb>sit adhuc minor, cui aequales sint in ab cen<lb></lb>tum, portiones spatij aequales portioni AE<pb xlink:href="064/01/112.jpg"></pb><figure id="id.064.01.112.1.jpg" xlink:href="064/01/112/1.jpg"></figure> erunt in AB 5050, in BC 15050, in CD <lb></lb>25050, inter se ut 101, 301, 501, fere iuxta <lb></lb>rationem numerorum imperium 1, 3, 5. Ex <lb></lb>quibus constat, quod eo portiones spatiorum <lb></lb>magis accedunt ad rationem numerorum impa<lb></lb>rium, quo portio temporis, in qua motus est in<lb></lb>dependenter ab impetu, minor est. </s> <s id="s.000830">Eadem<arrow.to.target n="note45"></arrow.to.target> pror<lb></lb>sus ratione probabitur, quo est itidem minor, <lb></lb>spatia propius esse in duplicata ratione tem<lb></lb>porum.</s> <s id="s.000831">Si namque concipiamus impetum incipere <lb></lb>in b, tempora ab, ac, ad sunt ut 1, 2, 3, spatia <lb></lb>vero AB, AC, AD, quae in duplicata ratione <lb></lb>temporum essent ut 1, 4, 9, sunt ut 1, 3, 6, val<lb></lb>de ab eis discrepantes: si vero tempora ab, ac, <lb></lb>ad, intelligantur divisa in portiones, quarum <lb></lb>ab, contineat decem, erunt temporum in<lb></lb>ter se portiones, ut 10, 20, 30, seu ut prius ut <lb></lb>1, 2, 3, at vero portiones spatiorum, quarum <lb></lb>prior ut supra sit AE, erunt ut 55, 210, 455 <lb></lb>seu ut 11, 42, 93; si denique portiones tempo<lb></lb>rum sint 100, 200, 300, portiones spatiorum erunt <lb></lb>5050, 20100, 45150, ut 101, 402, 903, mi<lb></lb>nimus, & insensibiliter discrepantes ab 1, 4, 9, & <lb></lb>proinde fere in duplicata temporum ratione;<pb xlink:href="064/01/113.jpg"></pb>unde quo plures temporum portiones, spatia <lb></lb>ad duplicatam rationem magis accedunt. </s> <s id="s.000832">Ut <lb></lb>autem datis temporibus, facile spatia peracta <lb></lb>reperiant, qui parum in arithmeticis progres<lb></lb>sionibus versati sunt, duc numerum tempo<lb></lb>rum, si sit par, in medietatem, & adde medie <lb></lb>tatem, si impar, in portionem maiorem medie<lb></lb>tatis, & prodibit summa spatiorum in dato tem<lb></lb>pore peractorum. </s> <s id="s.000833">Dentur 4 tempora, duc in <lb></lb>2 producto 8 adde medietatem 2, sit 10 sum<lb></lb>ma spatiorum. </s> <s id="s.000834">Dentur tempora 9, duc in 5, <lb></lb>productum 45 est summa spatiorum. </s> <s id="s.000835">Auge<lb></lb>tur<arrow.to.target n="note46"></arrow.to.target> igitur, ni fallor, motus iuxta progressionem <lb></lb>arithmeticam, non numerorum imparium ab <lb></lb>unitate huc usque creditam, sed naturalem; at<arrow.to.target n="note47"></arrow.to.target> <lb></lb>nihilominus, cum fere ijdem effectus subse<lb></lb>quantur, ob insensibilem discrepantiam; mi<lb></lb>randum non est, creditum fuisse spatia esse in <lb></lb>duplicata ratione temporum; quandoquidem <lb></lb>etiam si verum precise fortasse non sit, est <lb></lb>attamen adeo veritati proximum, ut verita<lb></lb>tem in adhibitis experimentis sensus percipe<lb></lb>re nequiverit, quamobrem excusandi sunt <lb></lb>quicunque ita censuerunt. </s> <s id="s.000836">Ego autem modo <lb></lb>veritatem delitescentem detexisse spero, cau<lb></lb>sam nimirum a qua huiusmodi proportio ema<lb></lb>nat aperuisse, & insuper quales errores fue<lb></lb>rint in suppositionibus, & experimentis huc<pb xlink:href="064/01/114.jpg"></pb>usque habitis, quod an re vera consecutus fue<lb></lb>rim aliorum sit indicium: neque enim is sum <lb></lb>qui tantum mihi tribuam, ut rerum arcana <lb></lb>intimius caeteris rimari mihi videar, cui satis <lb></lb>superque est inter illos connumerari, quo<lb></lb>rum disputationi mundus traditus fuit: nec <lb></lb>inutiliter me laborasse existimavero, si cre<lb></lb>dar vitam silentio non pertransisse. </s> <s id="s.000837">Caete<lb></lb>rum cum ea, quae de solidis dicenda videban<lb></lb>tur, iuxta mei vires ingenij, pertractata sint, <lb></lb>superest, ut ad naturalis motus liquidorum <lb></lb>passiones inquirendas accedam.</s> </p> <p type="foot"> <s id="s.000838"><foot.target id="foot.45"></foot.target>46 Et fere in duplicata ratione temporum.</s> </p> <p type="foot"> <s id="s.000839"><foot.target id="foot.46"></foot.target>47 Augetur motus iuxta progressionem naturalem.</s> </p> <p type="foot"> <s id="s.000840"><foot.target id="foot.47"></foot.target>48 Et apparet esse in duplicata ratione temporum.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/115.jpg"></pb> <subchap1 type="definition"> <p type="head"> <s id="s.000841">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000842">Canale est vas oblongum, per quod aqua de<lb></lb>currit; quod in praesentia supponitur habere <lb></lb>latera erecta, & basi perpendicularia, & pa<lb></lb>rallela inter se. </s> <s id="s.000843">Sectio vasis, est parallelogramum quod supponi<lb></lb>tur secare canale ad angulos rectos.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.000844">PETITIONES</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000845">Aqua transiens per eandem sectionem corre<lb></lb>spondet tempori.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/116.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000846">PROPOSITIO PRIMA</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000847">Aqua aequaliter introducta in pluribus cana<lb></lb>libus inaequaliter inclinatis correspondet <lb></lb>diuturnitatibus.<figure id="id.064.01.116.1.jpg" xlink:href="064/01/116/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="1" type="proof"> <p type="main"> <s id="s.000848">Sint Canales AB, CD, in quibus introducatur <lb></lb>aqua aequalis, & aqua A ducatur in B diu<lb></lb>turnitate E, & aqua C perveniat in D diutur<lb></lb>nitate F.</s> </p> <p type="main"> <s id="s.000849">Dico aquam AB ad aquam CD esse ut E ad F.</s> </p> <p type="main"> <s id="s.000850">Quoniam aqua A B est ea, quae transit per A, diu<lb></lb>turnitate E, & aqua CD est ea quae transit <lb></lb>per C, diuturnitate F per constructionem; sequi<lb></lb>tur quod aqua AB est ad aquam CD ut E ad F<arrow.to.target n="marg183"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.000851"><margin.target id="marg183"></margin.target>Per pet. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000852">Corollarium.</s> </p> <p type="main"> <s id="s.000853">Si diuturnitates sint aequales, aquae quantita<lb></lb>tes sunt pariter aequales.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/117.jpg"></pb> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000854">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000855">In pluribus canalibus ductis ad idem planum <lb></lb>orizontale, aquae quantitates sunt ut canales.</s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <figure id="id.064.01.117.1.jpg" xlink:href="064/01/117/1.jpg"></figure> <s id="s.000856">Sint canalia AB, AC, ducta ad planum Orizon<lb></lb>tale CB.</s> </p> <p type="main"> <s id="s.000857">Dico aquam AB esse ad aquam AC, ut longitudo <lb></lb>AB ad longitudinem AC.</s> </p> <p type="main"> <s id="s.000858">Quoniam diuturnitas AB ad diuturnitatem AC <lb></lb>est ut AB ad AC<arrow.to.target n="marg184"></arrow.to.target>, at ut diuturnitas AB ad <lb></lb>diuturnitatem AC, ita aqua AB ad aquam <lb></lb>AC<arrow.to.target n="marg185"></arrow.to.target>; ergo ut aqua AB ad aquam <lb></lb>AC, ita <lb></lb>longitudo AB ad longitudinem AC<arrow.to.target n="marg186"></arrow.to.target>. </s> <s id="s.000859">Quod etc.</s> </p> <p type="margin"> <s id="s.000860"><margin.target id="marg184"></margin.target>Per 15. primi. huius.</s> </p> <p type="margin"> <s id="s.000861"><margin.target id="marg185"></margin.target>Per primam huius.</s> </p> <p type="margin"> <s id="s.000862"><margin.target id="marg186"></margin.target>Per 11. Quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000863">Corollarium</s> </p> <p type="main"> <s id="s.000864">Idem sequitur si alterum canale sit perpendi<lb></lb>culare.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/118.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000865">PROPOSITIO III. PROBL. I.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000866">In canali declinante, reperire portionem con<lb></lb>tinentem aquam, aequalem eius quae est in <lb></lb>perpendiculari.<figure id="id.064.01.118.1.jpg" xlink:href="064/01/118/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <s id="s.000867">Sit AC canale inclinatum, & AB perpendicu<lb></lb>lare; oportet reperire in AC portionem con<lb></lb>tinentem aquam aequalem aquae AB.</s> </p> <p type="main"> <s id="s.000868">Ducatur BD normalis ad AC.</s> </p> <p type="main"> <s id="s.000869">Dico AD esse portionem quaesitam.</s> </p> <p type="main"> <s id="s.000870">Quoniam aqua ab A ducitur in B eodem tempore, <lb></lb>quo in D<arrow.to.target n="marg187"></arrow.to.target>, erit aqua AB aequalis aqua AD<arrow.to.target n="marg188"></arrow.to.target>. </s> <s id="s.000871">Quod etc.</s> </p> <p type="margin"> <s id="s.000872"><margin.target id="marg187"></margin.target>Per 16. pr. huius.</s> </p> <p type="margin"> <s id="s.000873"><margin.target id="marg188"></margin.target>Per Co. primae huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000874">Corollarium.</s> </p> <p type="main"> <s id="s.000875">Eadem ratione Dato canali AD reperietur <lb></lb>in AB portio continens aquam aequalem <lb></lb>AD, erecta a puncto D perpendiculari DB.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/119.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000876">PROPOSITIO IV. PROBL. II.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000877">In quibusvis canalibus quomodolibet inclina<lb></lb>tis, reperire portiones continentes aquam <lb></lb>aequalem cuiusvis dicti canalis.<figure id="id.064.01.119.1.jpg" xlink:href="064/01/119/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="4" type="proof"> <p type="main"> <s id="s.000878">A Canalibus AB, AC, AD, etc. sint secandae <lb></lb>portiones continentes aquam aequalem aquae <lb></lb>canalis AE.</s> </p> <p type="main"> <s id="s.000879">Iungantur omnes praedicti canales, retentis incli<lb></lb>nationibus, in puncto superiori A; et si AE est <lb></lb>perpendicularis ad orizontem, circa ipsum <lb></lb>tanquam diametrum, describatur circulus AE; <lb></lb>sin minus a puncto E, erigatur ipsi AE perpen<lb></lb>dicularis EF, & ab A demittatur perpendicu<lb></lb>laris ad orizontem, donec cum EF coeat in <lb></lb>F, & circa AF describatur circulus secans <lb></lb>omnes praedictos canales in G, H, I.</s> </p> <p type="main"> <s id="s.000880">Dico portiones AG, AH, AI continere aquam <lb></lb>aequalem aquae canalis AE.</s> </p> <p type="main"> <s id="s.000881">Quoniam in AG, AE, AH, AI diuturnitates sunt <lb></lb>aequales<arrow.to.target n="marg189"></arrow.to.target>, ergo sunt ibidem quantitates aquae <lb></lb>aequales<arrow.to.target n="marg190"></arrow.to.target>. </s> <s id="s.000882">Quod etc.</s> </p> <p type="margin"> <s id="s.000883"><margin.target id="marg189"></margin.target>Per 25. pr. huius.</s> </p> <p type="margin"> <s id="s.000884"><margin.target id="marg190"></margin.target>Per primam huius.</s> </p> </subchap2> <pb xlink:href="064/01/120.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.000885">Corollarium</s> </p> <p type="main"> <s id="s.000886">Si describantur quot vis circuli minores, seu <lb></lb>maiores, cuiuscumque magnitudinis, se invicem <lb></lb>tangentes in A, secabunt portiones dictorum <lb></lb>canalium ea ratione, ut sectiones intra quem<lb></lb>vis circulum contineant aquam aequalem.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/121.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000887">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000888">In canali secto quomodolibet; aquae quantita<lb></lb>tes in eius portionibus correspondent diu<lb></lb>turnitatibus.<figure id="id.064.01.121.1.jpg" xlink:href="064/01/121/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.000889"><figure id="id.064.01.121.2.jpg" xlink:href="064/01/121/2.jpg"></figure>Sit canale AC sectum in B quomodolibet, & <lb></lb>sit DE diuturnitas aquae donec perveniat in <lb></lb>B, & DF diuturnitas donec perveniat <lb></lb>in C, & proinde EF diuturnitas aquae <lb></lb>ductae a B in C.</s> </p> <p type="main"> <s id="s.000890">Dico aquam AB ad aquam BC esse ut diuturni<lb></lb>tas DE ad diuturnitatem EF.</s> </p> <p type="main"> <s id="s.000891">Quoniam aqua AB est ea, quae transit per A diu<lb></lb>turnitate DE, & AC ea quae transit per idem <lb></lb>A diuturnitate DF per constructionem; aqua <lb></lb>AB ad aquam AC est ut diuturnitas DE ad <lb></lb>diuturnitatem DF<arrow.to.target n="marg191"></arrow.to.target>; igitur dividendo, aqua <lb></lb>AB ad aquam BC est ut diuturnitas DE ad <lb></lb>diuturnitatem EF<arrow.to.target n="marg192"></arrow.to.target>. </s> <s id="s.000892">Quod etc.</s> </p> <p type="margin"> <s id="s.000893"><margin.target id="marg191"></margin.target>Per pet. huius.</s> </p> <p type="margin"> <s id="s.000894"><margin.target id="marg192"></margin.target>Per 19. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000895">Corollarium</s> </p> <p type="main"> <s id="s.000896">Si Diuturnitates DE, EF sint aequales, aqua <lb></lb>AB aequatur aquae BC.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/122.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000897">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000898">In canali secto quomodocumque; aqua in <lb></lb>priori portione ad aquam totius est in sub<lb></lb>duplicata ratione longitudinum.<figure id="id.064.01.122.1.jpg" xlink:href="064/01/122/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.000899">Sit canale AC sectum quomodocumque in D. </s> <s id="s.000900">Dico, quod aqua AD ad aquam AC est in sub<lb></lb>duplicata ratione longitudinum AD, AC.</s> </p> <p type="main"> <s id="s.000901">Quoniam longitudines AD, AC sunt in duplicata <lb></lb>ratione diuturnitatum<arrow.to.target n="marg193"></arrow.to.target>, at diuturnitates sunt <lb></lb>ut quantitates aquae<arrow.to.target n="marg194"></arrow.to.target>, ergo quantitates aquae <lb></lb>sunt in subduplicata ratione longitudinum<arrow.to.target n="marg195"></arrow.to.target>. </s> <s id="s.000902">Quod etc.</s> </p> <p type="margin"> <s id="s.000903"><margin.target id="marg193"></margin.target>Per 3. & 7. primi huius.</s> </p> <p type="margin"> <s id="s.000904"><margin.target id="marg194"></margin.target>Per 5. huius.</s> </p> <p type="margin"> <s id="s.000905"><margin.target id="marg195"></margin.target>Per 11. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000906">Corollarium</s> </p> <p type="main"> <s id="s.000907">Unde si fiat AE media proportionalis inter <lb></lb>AD, AC, aqua AD ad aquam AC erit ut <lb></lb>AD ad AE.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/123.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.000908">PROPOSITIO VII. PROBL. III.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000909">Dato canali perpendiculari, & alio inclinato <lb></lb>eiusdem longitudinis; reperire propor<lb></lb>tiones aquarum.<figure id="id.064.01.123.1.jpg" xlink:href="064/01/123/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <s id="s.000910">Sint canalia AC inclinatum, & AB perpen<lb></lb>diculare aequalia, & venanda sit proportio <lb></lb>inter aquas AB, AC.</s> </p> <p type="main"> <s id="s.000911">Ducatur BD perpendicularis ad AC, & fiat <lb></lb>AE media proportionalis inter AD, AC.</s> </p> <p type="main"> <s id="s.000912">Dico esse aquam AB ad aquam AC ut AD ad <lb></lb>AE.</s> </p> <p type="main"> <s id="s.000913">Quoniam aqua AD ad aquam AC est ut AD <lb></lb>ad AE<arrow.to.target n="marg196"></arrow.to.target>, sed aqua AD est aequalis aquae AB<arrow.to.target n="marg197"></arrow.to.target>, <lb></lb>ergo aqua AB ad aquam AC est ut AD ad <lb></lb>AE<arrow.to.target n="marg198"></arrow.to.target>: Quod etc.</s> </p> <p type="margin"> <s id="s.000914"><margin.target id="marg196"></margin.target>Per 6. huius.</s> </p> <p type="margin"> <s id="s.000915"><margin.target id="marg197"></margin.target>Per 3. huius.</s> </p> <p type="margin"> <s id="s.000916"><margin.target id="marg198"></margin.target>Per 11. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/124.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.000917">PROPOSITIO VIII. PROBL. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000918">Datis canalibus aequalis longitudinis maio<lb></lb>ris aut minoris inclinationis; venari pro<lb></lb>portiones aquarum.<figure id="id.064.01.124.1.jpg" xlink:href="064/01/124/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.000919">Sit canale AC minus, AF magis inclinatum <lb></lb>ei aequale; & venandae sint proportiones aqua<lb></lb>rum ab eis contentorum.</s> </p> <p type="main"> <s id="s.000920">Ducatur AB perpendicularis ad orizontem eiu<lb></lb>sdem longitudinis, & ductis perpendiculari<lb></lb>bus BD, BG, fiat AE media inter AD, AC, <lb></lb>& AH inter AG, AF, & ut AG ad AH, ita <lb></lb>AD ad AI.</s> </p> <p type="main"> <s id="s.000921">Dico aquam AC ad aquam AF esse ut AE ad AI.</s> </p> <p type="main"> <s id="s.000922">Quoniam ut aqua AC ad aquam AB ita AE ad <lb></lb>AD; & ut aqua AB ad aquam AF, ita AG <lb></lb>ad AH,<arrow.to.target n="marg199"></arrow.to.target> seu ut AD ad AI per constructio<lb></lb>nem; erit aqua AC ad aquam AF ut AE ad <lb></lb>AI<arrow.to.target n="marg200"></arrow.to.target>. </s> <s id="s.000923">Quod etc.</s> </p> <p type="margin"> <s id="s.000924"><margin.target id="marg199"></margin.target>Per 7. huius.</s> </p> <p type="margin"> <s id="s.000925"><margin.target id="marg200"></margin.target>Per 22. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/125.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000926">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000927">In canali secto iuxta proportionem nume<lb></lb>rorum imparium, in portionibus ex ea re<lb></lb>sultantibus sunt quantitates aquae aequales <lb></lb>inter se.<figure id="id.064.01.125.1.jpg" xlink:href="064/01/125/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.000928">Sit canale AD sectum in BC, & deinceps, ut <lb></lb>portiones AB, BC, CD, etc. sint inter se ut <lb></lb>1, 3, 5, 7.</s> </p> <p type="main"> <s id="s.000929">Dico quantitates aquae AB, BC, CD, esse <lb></lb>aequales inter se.</s> </p> <p type="main"> <s id="s.000930">Quoniam aqua aequali tempore progreditur ab A <lb></lb>in B, quo a B in C, & deinceps<arrow.to.target n="marg201"></arrow.to.target>, erit aqua <lb></lb>AB aequalis aquae BC<arrow.to.target n="marg202"></arrow.to.target>, etc. </s> <s id="s.000931">Quod etc.</s> </p> <p type="margin"> <s id="s.000932"><margin.target id="marg201"></margin.target>Per 10. pr. huius.</s> </p> <p type="margin"> <s id="s.000933"><margin.target id="marg202"></margin.target>Per cor. quintae huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/126.jpg"></pb> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000934">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000935">In quavis priori portione canalis, est aqua <lb></lb>aequalis portioni sequenti, triplae prioris.<figure id="id.064.01.126.1.jpg" xlink:href="064/01/126/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.000936">Dato canali A C secto in D ita ut AD sit <lb></lb>1/4 ipsius A C.</s> </p> <p type="main"> <s id="s.000937">Dico aquam AD aequari aquae DC.</s> </p> <p type="main"> <s id="s.000938">Quoniam eo tempore, quo A ducitur in D, D du<lb></lb>citur in C<arrow.to.target n="marg203"></arrow.to.target>, ergo aqua AD est aequalis aquae <lb></lb>DC<arrow.to.target n="marg204"></arrow.to.target>. </s> <s id="s.000939">Quod etc.</s> </p> <p type="margin"> <s id="s.000940"><margin.target id="marg203"></margin.target>Per 9. huius.</s> </p> <p type="margin"> <s id="s.000941"><margin.target id="marg204"></margin.target>Per cor. quintae huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/127.jpg"></pb> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000942">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000943">In canali declinante, duplo perpendicularis <lb></lb>ductae ad idem planum orizontale sectum <lb></lb>a linea ad illud normaliter ducta a puncto <lb></lb>inferiori dictae perpendicularis, portiones <lb></lb>continent aequales aquae quantitates.<figure id="id.064.01.127.1.jpg" xlink:href="064/01/127/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="11" type="proof"> <p type="main"> <s id="s.000944">Sit canale AC duplum AB, sectum in D a <lb></lb>perpendiculari BD.</s> </p> <p type="main"> <s id="s.000945">Dico aquam AD aequari aquae DC.</s> </p> <p type="main"> <s id="s.000946">Quoniam AB est media inter AD, AC<arrow.to.target n="marg205"></arrow.to.target>, <lb></lb>& AB est medietas ipsius AC per constructio<lb></lb>nem, AD est medietas ipsius AB, & proinde <lb></lb>quarta pars totius AC; igitur aqua in AD <lb></lb>aequalis aquae in DC<arrow.to.target n="marg206"></arrow.to.target>. </s> <s id="s.000947">Quod etc.</s> </p> <p type="margin"> <s id="s.000948"><margin.target id="marg205"></margin.target>Per ea quae ad 16. pri. huius.</s> </p> <p type="margin"> <s id="s.000949"><margin.target id="marg206"></margin.target>Per 10. huius.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/128.jpg"></pb> <pb xlink:href="064/01/129.jpg"></pb> <chap type="bk"> <p type="main"> <s id="s.000950">DE MOTV <lb></lb>GRAVIVM <lb></lb>LIBER QVINTVS <lb></lb>ET LIBER LIQVIDORVM SECVNDVS.<lb></lb>VBI DE CANALIVM SECTIONIBVS.</s> </p> <subchap1 type="preface"> <subchap2 type="preface"> <p type="main"> <s id="s.000951">Etiamsi simus in liqui<lb></lb>dis, lubet adhuc aliquid <lb></lb>de solidis praefari, sum<lb></lb>pta occasione a Quest. <lb></lb>19. Mech. </s> <s id="s.000952">Arist. ubi cau<lb></lb>sam perquirit cur lignum <lb></lb>facilius scindat qui secu<lb></lb>rim extollens percutit, <lb></lb>quam qui securim impositam, addito pondere prae<lb></lb>mat. </s> <s id="s.000953">Quod perinde est ac si dicas, cur plus scin<lb></lb>das leviori securi mota, quam graviori quies<lb></lb>cente. </s> <s id="s.000954">Nimirum Quoniam grave, motionem <lb></lb>gravitatis magis assumit, motum quam quies<lb></lb>cens: pro qua gravitatis motione impetus in<lb></lb>telligitur, qui primo delitescens, a gravi dein<pb xlink:href="064/01/130.jpg"></pb>de per motum assumitur; scilicet qui erat in <lb></lb>potentia, in actum per motum reductus, mo<lb></lb>tum inde auget, ipsum reddens velociorem, <lb></lb>suplente impetu vicem ponderis. </s> <s id="s.000955">Mihi ta<lb></lb>men semper visus est Arist. problema non in<lb></lb>tegre solvisse, reticuit siquidem cur huiusmo<lb></lb>di motio gravitatis, seu impetus sit talis virtu<lb></lb>tis, ut efficacius agat quam pondus additum, ex <lb></lb>quo demum maior scissio subsequatur. </s> <s id="s.000956">Cuius<arrow.to.target n="note48"></arrow.to.target> <lb></lb>quidem ego causam pro viribus investigare <lb></lb>mihi proposui, quonam nimirum modo me<lb></lb>tiri queat actio percutientis securis, & passio <lb></lb>ligni resistentis, ut demum percipi possit quan<lb></lb>tum sit pondus addendum, ut impetus eius vi<lb></lb>ribus respondeat.</s> <s id="s.000957">Quod<arrow.to.target n="note49"></arrow.to.target> ut breviter de more <lb></lb>discutiatur, respectu actionis securis certum <lb></lb>est, quod si eius potentia non excedit li<lb></lb>gni resistentiam, quamvis sit ei aequalis, nulla <lb></lb>fiet actio; atqui<arrow.to.target n="note50"></arrow.to.target> si securis extollatur, quantum<lb></lb>vis minimum, actio subsequetur, quoniam mo<lb></lb>vens motum plus agit quam dum prius quiescebat, <lb></lb>quatenus actio gravitatis adhuc perseverat, & <lb></lb>insuper additur impetus, unde potentia quae <lb></lb>prius erat aequalis resistentiae, iam eam excedit; <lb></lb>& eius demum continuatur motus, quousque po<lb></lb>tentia minuatur, aut augeatur resistentia: Et<arrow.to.target n="note51"></arrow.to.target> <lb></lb>quo magis securis extollitur, validius scindit; <lb></lb>acquirit namque impetum maiorem, tali ad <pb xlink:href="064/01/131.jpg"></pb>priorem proportione, ut sint impetus in sub<lb></lb>duplicata ratione spatiorum peractorum; ut <lb></lb>in quinta secundi huius: Unde<arrow.to.target n="note52"></arrow.to.target> data minori <lb></lb>actione, facile metieris maiorem, percipiens <lb></lb>quantane ea sit, ex qualibet proveniens altitu<lb></lb>dine.</s> <s id="s.000958">Quod<arrow.to.target n="note53"></arrow.to.target> item sequitur in quavis percus<lb></lb>sione seu a securi, seu a quolibet ad percutien<lb></lb>dum idoneo naturaliter moto; trabes siqui<lb></lb>dem, seu pali longiores, fortius in terram pan<lb></lb>guntur, quo fistuca non modo est ponderosior, <lb></lb>sed altius effertur, tali ratione, ut altitudines <lb></lb>in duplicata proportione, percussionum viri<lb></lb>bus respondeant. </s> <s id="s.000959">Si vero securis a motore <lb></lb>impellatur, validius percutit; quoniam motus <lb></lb>in initio, est celerior ab impulsu, quam a gra<lb></lb>vitate; cuius perseverante actione, maior pro<lb></lb>ducitur impetus, unde motus celerior, & ictus <lb></lb>validior, etiam nulla concurrente gravitate, <lb></lb>ut si motus non deorsum sed ad latera tendat, <lb></lb>aut sursum. </s> <s id="s.000960">Unde<arrow.to.target n="note54"></arrow.to.target> quo malleus a pariete re<lb></lb>motior in eum fortius impellitur, clavus ma<lb></lb>gis figitur, & longe facilius quam si omnibus <lb></lb>adhibitis viribus, malleum contra clavum com<lb></lb>primas.</s> <s id="s.000961">Unde<arrow.to.target n="note55"></arrow.to.target> etiam est, quod mobile vehe<lb></lb>mentius impulsum, expulsum demum, in <lb></lb>quodcumque illidat, validius ferit, & intimius <lb></lb>intruditur, quod in ictu a funda, arcu, sclopo <lb></lb>passim videre est. </s> <s id="s.000962">Huius autem vim impulsus pon<pb xlink:href="064/01/132.jpg"></pb>dere proxime metiri licebit, si illud adeo con<lb></lb>sentanee aptetur, ut illud extollas, eodem pa<lb></lb>cto illi innixus, eademque prorsus directio<lb></lb>ne, quemadmodum securim, aut quodvis aliud <lb></lb>impellere lubeat. </s> <s id="s.000963">Quod<arrow.to.target n="note56"></arrow.to.target> facile continget, dua<lb></lb>bus adhibitis trochleis, unius tantum modo <lb></lb>rotulae, altera superne appensa, inferne altera; <lb></lb>quibus ductarius circunductus funis, altero <lb></lb>extremo pondus, sustineat, alterum vero a po<lb></lb>tentia trahatur, modo quo mox dictum fuit, <lb></lb>sit ne ea totum corpus animalis, seu hominis, <lb></lb>sive eius ambo brachia, aut ipsorum alterum, <lb></lb>seu tantum digiti, quorum omnium singilla<lb></lb>tim vim, seu potentiam, proxime metietur ma<lb></lb>ius aut minus pondus, quod ab uno, quoque eo<lb></lb>rum, hac ratione in altum ducatur.</s> <s id="s.000964">Ex qui<lb></lb>bus vires percussionis satis aperte apparere ar<lb></lb>bitror, nimirum a vi motoris, seu sit gravitas, <lb></lb>seu impulsus, nec non ab impetu per motum <lb></lb>acquisito, maiori aut minori, prout motor est <lb></lb>maioris virtutis. </s> <s id="s.000965">Quo<arrow.to.target n="note57"></arrow.to.target> vero ad ligni resisten<lb></lb>tis passionem secundo loco propositam, certum <lb></lb>est, quod si resistentia est maior, aut aequalis <lb></lb>activitati securis, nulla fiet actio; si vero sit <lb></lb>resistentia minoris virtutis, unde vires agen<lb></lb>tis securis excedant vires ligni resistentis, ali<lb></lb>qua fiet scissio; eo<arrow.to.target n="note58"></arrow.to.target> maior, quo minor erit resi<lb></lb>stentia, quam non vi duntaxat portionis ligni <pb xlink:href="064/01/133.jpg"></pb>metiemur, quae securi opponitur; sed partium <lb></lb>itidem ei a latere cohaerentium, & sic porro <lb></lb>affixarum, ut ab eis difficulter divelli queat. <lb></lb></s> <s id="s.000966">Quantumvis autem huius resistentiae poten<lb></lb>tia minus percipiatur, hoc unum est, quod qualis <lb></lb>qualis sit, velocitati securis contranititur, eam<lb></lb>que tali ratione retundit, ut quantum ei tri<lb></lb>buitur, tantundem velocitati detrahatur; un<lb></lb>de<arrow.to.target n="note59"></arrow.to.target> si resistentia addita sit priori decupla, aut <lb></lb>centupla, velocitas reducitur ad decimam par<lb></lb>tem seu centesimam eius quae prius aderat, <lb></lb>unde spatij quod securis per aerem peregit dum <lb></lb>nil obstaret, addita postmodum ligni obvij re<lb></lb>sistentia, in aequali tempore, decimam pariter <lb></lb>aut centesimam conficit portionem. </s> <s id="s.000967">Quandiu<arrow.to.target n="note60"></arrow.to.target> <lb></lb>vero lignum permeat, resistentia success<lb></lb>ive augetur; partes quippe ligni ab ipsiusmet <lb></lb>securis compressione fiunt densiores, praeter <lb></lb>quam quod saepius, quo ea altius intruditur, <lb></lb>eo plures sunt partes cohaerentes divellendae. <lb></lb></s> <s id="s.000968">Utcunque sit, certum est quod dum impetus inci<lb></lb>pit minui, & est successive minor proportio ac<lb></lb>tionis securis ad ligni resistentiam, velocitas <lb></lb>non modo successive minuitur, sed paula<lb></lb>tim deficit. </s> <s id="s.000969">Quod<arrow.to.target n="note61"></arrow.to.target> idem sequitur de impetu, <lb></lb>qui cum velocitate pari passu procedit; unde<lb></lb><arrow.to.target n="note62"></arrow.to.target> quantum velocitati detrahitur, tantundem <lb></lb>impetus minuitur; qui proinde cessante mo<pb xlink:href="064/01/134.jpg"></pb>tu prorsus deperit.</s> <s id="s.000970">Et<arrow.to.target n="note63"></arrow.to.target> quoniam mox adducta <lb></lb>communia sunt tam motae securi, quam cuili<lb></lb>bet mobili, quod nimirum resistentia motum <lb></lb>retundit, & magis, quo maior proportio resi<lb></lb>stentis ad mobilis vires, duae pilae, etiam aequales <lb></lb>in terram naturaliter cadentes, quae proinde <lb></lb>in aere aequali feruntur celeritate, etiamsi pon<lb></lb>dere inaequales, terram inaequaliter perme<lb></lb>ant, resistente nimirum terra magis pilae le<lb></lb>viori, quam graviori. </s> <s id="s.000971">Unde est etiam quod si, <lb></lb>mobili proiecto, aliud addatur quiescens, & <lb></lb>proinde resistens, impetus minuitur; & quo<arrow.to.target n="note64"></arrow.to.target> <lb></lb>maius mobile superadditur, tardius fertur, & <lb></lb>minus, aequo tempore conficit spatium, tali ra<lb></lb>tione, ut ratio mobilis compositi, ad anterius <lb></lb>simplex, spatijs aequali peractis tempore, reci<lb></lb>proce respondeat: unde<arrow.to.target n="note65"></arrow.to.target> si mobile composi<lb></lb>tum sit prioris quadruplum, velocitas demum <lb></lb>subsequens sit praecedentis quadrans, & talis <lb></lb>demum continuetur.</s> <s id="s.000972">Ut<arrow.to.target n="note66"></arrow.to.target> autem tandem ad <lb></lb>propositam quaestionem propius accedamus, <lb></lb>& innotescat quale pondus addi debeat se<lb></lb>curi, ut aequa fiat scissio, ac si ea extollatur, <lb></lb>hoc, ex dictis visum est erui non posse a viribus <lb></lb>ligni resistentis, utpote pariter se opponentis, <lb></lb>& contranitentis viribus securis motae levioris, <lb></lb>& immotae ponderosioris: Igitur tota quaestio <lb></lb>pendet ab ipsamet vi securis, seu motae, seu <pb xlink:href="064/01/135.jpg"></pb>quiescentis. </s> <s id="s.000973">Cum itaque iam visus sit, acti<lb></lb>vitatem securis motae a duobus pendere prin<lb></lb>cipijs, a vi nimirum impellentis, & imprimen<lb></lb>tis motum, quam metiuntur pondera ab eadem <lb></lb>vi sublata, & itidem a vi impetus, virtute dicti <lb></lb>motus a securi acquisiti, quam metiuntur <lb></lb>spatia, quae dum percurruntur, impulsus perse<lb></lb>verat eiusdem virtutis; inde sequitur quod <lb></lb>ratio potentiae, seu momenti, seu virium se<lb></lb>curis motae, ad potentiam eiusdem sensibili<lb></lb>ter immotae, componitur ex ratione ponderum <lb></lb>inter se, nimirum eius quod aequipolet vi se<lb></lb>curis impulsae, additi ad percutientis securis <lb></lb>pondus, ad pondus eiusdem quiescentis; nec <lb></lb>non ex ratione spatiorum peractorum maio<lb></lb>ris securis in altum elatae, ad minus, fortasse <lb></lb>insensibile, eiusdem sensibiliter immotae, adeo <lb></lb>ut si vires tali pacto mensuratae utriusque se<lb></lb>curis motae, & immotae, sint v.g. in ratione de<lb></lb>cupla, & spatia peracta sint in centupla, ratio <lb></lb>porro virium securis motae, ad vires quiescen<lb></lb>tis, sit in millecupla; unde si quiescens sit mil<lb></lb>lies gravior, aequa fiet scissio. </s> <s id="s.000974">Nec dicas inter <lb></lb>spatia motae, & immotae nullam dari propor<lb></lb>tionem, quia agitur hic de sensibiliter immo<lb></lb>ta, & non praecise, seu mathematice, sed phy<lb></lb>sice, nec videtur dari posse casum quin securis <lb></lb>imposita tantulum moveatur, etiamsi insen<pb xlink:href="064/01/136.jpg"></pb>sibiliter; quod eo facilius existimandum vide<lb></lb>tur, cum in hypotesi suppositum fuerit, secu<lb></lb>ris vires esse viribus resistentiae prorsus aequa<lb></lb>les: ex hoc tamen insensibili motu oritur, non <lb></lb>modo ut videamus, quantum vires percussionis <lb></lb>excedant vires ponderis, ex quo adeo facile li<lb></lb>gnum scinditur; sed ex illo itidem oritur difficul<lb></lb>tas percipiendi, qua precise proportione per<lb></lb>cussio, vi prementi respondeat. </s> <s id="s.000975">Caeterum haec <lb></lb>sunt quae mihi in mentem venerunt de vi per<lb></lb>cussionis sapientioribus proponenda, ut ad<lb></lb>dant meliora.</s> </p> <p type="foot"> <s id="s.000976"><foot.target id="foot.48"></foot.target>1 De vi percussionis.</s> </p> <p type="foot"> <s id="s.000977"><foot.target id="foot.49"></foot.target>2 De activitate securis seu percutientis.</s> </p> <p type="foot"> <s id="s.000978"><foot.target id="foot.50"></foot.target>3 Quia motum plus agit ob impetum.</s> </p> <p type="foot"> <s id="s.000979"><foot.target id="foot.51"></foot.target>4 Et quo per longius spatium impetus est maior.</s> </p> <p type="foot"> <s id="s.000980"><foot.target id="foot.52"></foot.target>5 Proportio inter impetus et spatia.</s> </p> <p type="foot"> <s id="s.000981"><foot.target id="foot.53"></foot.target>6 In quavis percussione.</s> </p> <p type="foot"> <s id="s.000982"><foot.target id="foot.54"></foot.target>7 Etiamsi motus non sit deorsum.</s> </p> <p type="foot"> <s id="s.000983"><foot.target id="foot.55"></foot.target>8 Unde vis percussionis.</s> </p> <p type="foot"> <s id="s.000984"><foot.target id="foot.56"></foot.target>9 Vim impulsus pondus metitur.</s> </p> <p type="foot"> <s id="s.000985"><foot.target id="foot.57"></foot.target>10 De ligni resistentia.</s> </p> <p type="foot"> <s id="s.000986"><foot.target id="foot.58"></foot.target>11 Quae pendet etiam a partibus cohaerentibus.</s> </p> <p type="foot"> <s id="s.000987"><foot.target id="foot.59"></foot.target>12 Quo resistentia est maior minor est motus.</s> </p> <p type="foot"> <s id="s.000988"><foot.target id="foot.60"></foot.target>13 Et inde resistentia augetur.</s> </p> <p type="foot"> <s id="s.000989"><foot.target id="foot.61"></foot.target>14 Et velocitas minuitur. Et deficit.</s> </p> <p type="foot"> <s id="s.000990"><foot.target id="foot.62"></foot.target>15 Et pariter impetus.</s> </p> <p type="foot"> <s id="s.000991"><foot.target id="foot.63"></foot.target>16 Quod est commune cuivis mobili.</s> </p> <p type="foot"> <s id="s.000992"><foot.target id="foot.64"></foot.target>17 Cui addito immoto minuitur impetus.</s> </p> <p type="foot"> <s id="s.000993"><foot.target id="foot.65"></foot.target>18 Qua proportione.</s> </p> <p type="foot"> <s id="s.000994"><foot.target id="foot.66"></foot.target>19 Quod pondus percussionis aequivaleat.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/137.jpg"></pb> <subchap1 type="postulate"> <p type="head"> <s id="s.000995">PETITIONAE</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000996">1. In sectionibus aequalibus quantitates aquae <lb></lb>sunt ut velocitates.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000997">2. Si velocitates sint aequales, sectiones sunt ut <lb></lb>quantitates aquae.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000998">3.In canalium sectionibus Impetus, & veloci<lb></lb>tates pro eodem sumuntur.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/138.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000999">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.001000">Si sectiones sint aequales; aquarum transeun<lb></lb>tium quantitates sunt, ut velocitates.<figure id="id.064.01.138.1.jpg" xlink:href="064/01/138/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="1" type="proof"> <p type="main"> <s id="s.001001">Transeat aqua A per sectionem A, ab A ad <lb></lb>B; & aqua C per sectionem C aequalem <lb></lb>sectioni A, a C ad D aequali tempore.</s> </p> <p type="main"> <s id="s.001002">Dico aquam AB ad aquam CD esse ut velocitas <lb></lb>aquae A ad velocitatem aquae C.</s> </p> <p type="main"> <s id="s.001003">Quoniam velocitas in A ad velocitatem in C, est <lb></lb>ut AB ad CD,<arrow.to.target n="marg207"></arrow.to.target> & aqua AB ad aquam CD <lb></lb>est itidem ut AB ad CD<arrow.to.target n="marg208"></arrow.to.target>, sequitur quod velo<lb></lb>citas in A ad velocitatem in C, est ut aqua <lb></lb>AB ad aquam CD<arrow.to.target n="marg209"></arrow.to.target>. </s> <s id="s.001004">Quod etc.</s> </p> <p type="margin"> <s id="s.001005"><margin.target id="marg207"></margin.target>Per 32. undec.</s> </p> <p type="margin"> <s id="s.001006"><margin.target id="marg208"></margin.target>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.001007"><margin.target id="marg209"></margin.target>Per primam huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/139.jpg"></pb> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.001008">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.001009">Velocitas aquae in pluribus eiusdem canalis <lb></lb>sectionibus, est reciproca sectionibus ipsis.<figure id="id.064.01.139.1.jpg" xlink:href="064/01/139/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <s id="s.001010">Sint A, C, canalis sectiones, diversae magnitu<lb></lb>dinis.</s> </p> <p type="main"> <s id="s.001011">Dico esse, ut magnitudo sectionis A ad magnitu<lb></lb>dinem sectionis C, ita velocitatem in C, ad ve<lb></lb>locitatem in A.</s> </p> <p type="main"> <s id="s.001012">Fiat sectio B aequalis ipsi A, per quam intelliga<lb></lb>tur transire aquam aequaliter velocem ut in <lb></lb>sectione C.</s> </p> <p type="main"> <s id="s.001013">Quoniam ut quantitas aquae A seu C, ad quan<lb></lb>titatem aquae B, ita est velocitas aquae in A, ad <lb></lb>velocitatem aquae in B seu C<arrow.to.target n="marg210"></arrow.to.target>; sed ut magni<lb></lb>tudo sectionis C ad magnitudinem sectionis B, <lb></lb>seu A, ita quantitas aquae C seu A, ad quanti<lb></lb>tatem aquae B<arrow.to.target n="marg211"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001014"><margin.target id="marg210"></margin.target>Per 2. pet. huius.</s> </p> <p type="margin"> <s id="s.001015"><margin.target id="marg211"></margin.target>Per 2. huius.</s> </p> <p type="main"> <s id="s.001016">Ergo ut magnitudo sectionis C ad magnitudi<lb></lb>nem sectionis A, ita velo<lb></lb>citas aquae A ad velocitatem aquae C. </s> <s id="s.001017">Quod etc.</s> </p> </subchap2> <pb xlink:href="064/01/140.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.001018">Corollarium I.</s> </p> <p type="main"> <s id="s.001019">Idem sequitur, si sectiones sint canalium diversorum, dummodo ducant aquae quantitates aequales.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001020">Corollarium II.</s> </p> <p type="main"> <s id="s.001021">Impetus sunt ibidem ut sectiones reciproce.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/141.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.001022">PROPOSITIO III.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.001023">Sectiones canalis sunt reciproce in subduplicata ratione longitudinum.</s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <figure id="id.064.01.141.1.jpg" xlink:href="064/01/141/1.jpg"></figure> <s id="s.001024">Sit canale AB sectum in C.</s> </p> <p type="main"> <s id="s.001025">Dico sectiones CB esse in subduplicata ratione AB, AC.</s> </p> <p type="main"> <s id="s.001026">Quoniam sectiones CB sunt ut velocitates in B, & in C<arrow.to.target n="marg212"></arrow.to.target>, at velocitas in B ad velocitatem in C est in subduplicata ratione AB ad AC<arrow.to.target n="marg213"></arrow.to.target>, Ergo sectio C ad sectionem B est in subduplicata ratione AB ad AC<arrow.to.target n="marg214"></arrow.to.target>. </s> <s id="s.001027">Quod etc.</s> </p> <p type="margin"> <s id="s.001028"><margin.target id="marg212"></margin.target>Per 5. secundi huius.</s> </p> <p type="margin"> <s id="s.001029"><margin.target id="marg213"></margin.target>Per 11. quinti.</s> </p> <p type="margin"> <s id="s.001030"><margin.target id="marg214"></margin.target>Per 33. primi.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001031">Corollarium I.</s> </p> <p type="main"> <s id="s.001032">Igitur si canalis latera sint parallela, altitudines sectionem sunt in subduplicata ratione longitudinum.</s> </p> <p type="main"> <s id="s.001033">Nam si latera perpendicularia canalis intelligantur bases, & ea ratione latitudines canalis ut altitudines, quae proinde sunt aequales<arrow.to.target n="marg215"></arrow.to.target>, sectiones sunt ut dicta latera perpendicularia<arrow.to.target n="marg216"></arrow.to.target>,<pb xlink:href="064/01/142.jpg"></pb>quae cum sint altitudines sectionum, sequitur <lb></lb>quod propositum fuit.</s> </p> <p type="margin"> <s id="s.001034"><margin.target id="marg215"></margin.target>Per pri. sexti.</s> </p> <p type="margin"> <s id="s.001035"><margin.target id="marg216"></margin.target>Per 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001036">Corollarium II.</s> </p> <p type="main"> <s id="s.001037">Si sectiones sint reciprocae in subduplicata ra<lb></lb>tione longitudinum, exit aqua aequalis.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/143.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.001038">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.001039">Impetus sectionum canalis, sunt in subdupli<lb></lb>cata ratione longitudinum ipsarum a pun<lb></lb>cto superno.</s> </p> </subchap2> <subchap2 n="4" type="proof"> <figure id="id.064.01.143.1.jpg" xlink:href="064/01/143/1.jpg"></figure> <p type="main"> <s id="s.001040">In canali ACB.</s> </p> <p type="main"> <s id="s.001041">Dico impetum sectionis B ad impe<lb></lb>tum sectionis C esse in subduplicata <lb></lb>ratione longitudinum AB ad AC.</s> </p> <p type="main"> <s id="s.001042">Quoniam sectio C ad sectionem B est in <lb></lb>subduplicata ratione AB ad AC<arrow.to.target n="marg217"></arrow.to.target>. <lb></lb>Impetus in B ad impetum in C est in eadem sub<lb></lb>duplicata ratione AB ad AC<arrow.to.target n="marg218"></arrow.to.target>. </s> <s id="s.001043">Quod etc.</s> </p> <p type="margin"> <s id="s.001044"><margin.target id="marg217"></margin.target>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001045"><margin.target id="marg218"></margin.target>Per 13. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/144.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.001046">PROPOSITIO V. PROBL. I.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.001047">Data canalis sectione, reperire sectionem in <lb></lb>quolibet allo dato puncto.<figure id="id.064.01.144.1.jpg" xlink:href="064/01/144/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.001048">Data sectione C, & puncto B in canali AB, <lb></lb>Venanda est sectio puncti B.</s> </p> <p type="main"> <s id="s.001049">Fiat AD media inter AC, AB<arrow.to.target n="marg219"></arrow.to.target>, & sectio B ad <lb></lb>sectionem C ut AC ad AD.</s> </p> <p type="margin"> <s id="s.001050"><margin.target id="marg219"></margin.target>Per 20. sexti.</s> </p> <p type="main"> <s id="s.001051">Dico B esse sectionem quaesitam.</s> </p> <p type="main"> <s id="s.001052">Quoniam sectio B ad sectionem C est ut AC ad <lb></lb>AD per constructionem; erit sectio B ad sectio<lb></lb>nem C in subduplicata ratione AC ad AB<arrow.to.target n="marg220"></arrow.to.target>, <lb></lb>unde sectio B est sectio puncti B<arrow.to.target n="marg221"></arrow.to.target>. </s> <s id="s.001053">Quod etc.</s> </p> <p type="margin"> <s id="s.001054"><margin.target id="marg220"></margin.target>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001055"><margin.target id="marg221"></margin.target>Defini. pr. quarti huius.</s> </p> <p type="main"> <s id="s.001056">Fiet sectio B ad sectionem C ut AC ad AD, si fiat <lb></lb>altitudo laterum sectionis B ad altitudinem <lb></lb>laterum sectionis C ut AC ad AD<arrow.to.target n="marg222"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001057"><margin.target id="marg222"></margin.target>Per 2. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/145.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.001058">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.001059">Datis pluribus sectionibus; ratio primae ad ter<lb></lb>tiam, est composita ex rationibus velocitatis <lb></lb>secundae ad velocitatem primae, & velo<lb></lb>citatis tertiae ad velocitatem secundae.<figure id="id.064.01.145.1.jpg" xlink:href="064/01/145/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.001060">Dentur in canali AB sectiones B, C, D. <lb></lb></s> <s id="s.001061">Dico proportionem sectionis B ad sectionem <lb></lb>D, esse compositam ex rationibus velocitatis C <lb></lb>ad veloci<lb></lb>tatem B, & velocitatis D ad veloci<lb></lb>tatem C.</s> </p> <p type="main"> <s id="s.001062">Quoniam sectio B ad sectionem C est ut velocitas <lb></lb>C ad velocitatem B, item sectio D ad veloci<lb></lb>tatem C ut velocitas C ad velocitatem D<arrow.to.target n="marg223"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001063"><margin.target id="marg223"></margin.target>Per 5. def. sexti.</s> </p> <p type="main"> <s id="s.001064">Sed ratio velocitatis D ad velocitatem B est com<lb></lb>posita ex rationibus velocitatis C ad velocita<lb></lb>tem B, & velocitatis D ad velocitatem C<arrow.to.target n="marg224"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001065"><margin.target id="marg224"></margin.target>Per 8. secundi huius.</s> </p> <p type="main"> <s id="s.001066">Ergo pariter ratio sectionis B ad sectionem D <lb></lb>est composita ex rationibus velocitatis C ad <lb></lb>velocitatem B, & velocitatis D ad velocita<lb></lb>tem C. </s> <s id="s.001067">Quod etc.</s> </p> </subchap2> <pb xlink:href="064/01/146.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.001068">Corollarium</s> </p> <p type="main"> <s id="s.001069">Si sint plures sectiones puta B, C, D, E, F, <lb></lb>pariter ratio sectionis B ad sectionem F com<lb></lb>ponitur ex velocitatibus C ad B, D ad C, E ad <lb></lb>D, F ad E.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/147.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.001070">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.001071">Si canales perpendicularis, & inclinatus ter<lb></lb>minentur a recta normali ad inclinatum, <lb></lb>sectio perpendicularis ad sectionem in<lb></lb>clinati est, ut inclinatus ad perpendicu<lb></lb>larem.<figure id="id.064.01.147.1.jpg" xlink:href="064/01/147/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <s id="s.001072">Dentur canales AB perpendicularis, & A<lb></lb>D inclinatus, terminati a recta BD, ut an<lb></lb>gulus ADB sit rectus. </s> <s id="s.001073">Dico sectionem B ad se<lb></lb>ctionem D esse ut AD, ad AB.</s> </p> <p type="main"> <s id="s.001074">Quoniam velocitas in B ad velocitatem in D est <lb></lb>ut AB ad AD<arrow.to.target n="marg225"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001075"><margin.target id="marg225"></margin.target>Per 2. huius.</s> </p> <p type="main"> <s id="s.001076">Erit sectio B ad sectionem D ut AD ad AB<arrow.to.target n="marg226"></arrow.to.target>. </s> <s id="s.001077">Quod etc.</s> </p> <p type="margin"> <s id="s.001078"><margin.target id="marg226"></margin.target>Per cor. 8. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/148.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.001079">PROPOSITIO VIII.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.001080">In canalibus perpendiculari, & inclinato; se<lb></lb>ctiones terminatae a linea orizontali sunt <lb></lb>aequales.<figure id="id.064.01.148.1.jpg" xlink:href="064/01/148/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.001081">Dentur canales AB perpendicularis, & AC <lb></lb>inclinatus, quorum sectiones CB sint ori<lb></lb>zontales.</s> </p> <p type="main"> <s id="s.001082">Dico eas esse aequales inter se.</s> </p> <p type="main"> <s id="s.001083">Ducatur normalis BD ad AC.</s> </p> <p type="main"> <s id="s.001084">Quoniam AB est media inter AD, AC<arrow.to.target n="marg227"></arrow.to.target>, AD ad <lb></lb>AC habet duplicatam rationem AD ad AB<arrow.to.target n="marg228"></arrow.to.target>. <lb></lb>Unde sectio D ad sectionem C est ut AB ad AD<arrow.to.target n="marg229"></arrow.to.target>. </s> <s id="s.001085">Et eadem sectio D ad sectionem B est pariter <lb></lb>ut AB ad AD<arrow.to.target n="marg230"></arrow.to.target>. Ergo sectiones C, B ha<lb></lb>bentes eamdem rationem ad sectionem D, sunt <lb></lb>aequales inter se<arrow.to.target n="marg231"></arrow.to.target>. </s> <s id="s.001086">Quod etc.</s> </p> <p type="margin"> <s id="s.001087"><margin.target id="marg227"></margin.target>Per 10. def. quin.</s> </p> <p type="margin"> <s id="s.001088"><margin.target id="marg228"></margin.target>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001089"><margin.target id="marg229"></margin.target>Per 7. huius.</s> </p> <p type="margin"> <s id="s.001090"><margin.target id="marg230"></margin.target>Per 9. quinti.</s> </p> <p type="margin"> <s id="s.001091"><margin.target id="marg231"></margin.target>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/149.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.001092">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.001093">Ductis pluribus canalibus a puncto superno <lb></lb>quomodocunque; reperire rationes data<lb></lb>rum sectionum inter se.<figure id="id.064.01.149.1.jpg" xlink:href="064/01/149/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.001094">Dati sint quilibet canales AB, AC, AD, in <lb></lb>quibus assignentur puncta B, C, D.</s> </p> <p type="main"> <s id="s.001095">Oportet reperire rationes dictarum sectionum inter se. <lb></lb></s> <s id="s.001096">Ducatur perpendicularis AE, & ad eam per<lb></lb>pendiculares BF, CG, DE, & sint F, G, E sectio<lb></lb>nes canalis AE.</s> </p> <p type="main"> <s id="s.001097">Quoniam est nota ratio sectionum F, G, E<arrow.to.target n="marg232"></arrow.to.target>, & B, C, D <lb></lb>sectiones aequantur sectionibus F, G, E respective<arrow.to.target n="marg233"></arrow.to.target>, <lb></lb>sequitur notas esse ipsarum rationes. </s> <s id="s.001098">Quod etc.</s> </p> <p type="margin"> <s id="s.001099"><margin.target id="marg232"></margin.target>Per 8. huius.</s> </p> <p type="margin"> <s id="s.001100"><margin.target id="marg233"></margin.target>Per 8. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001101">Corollarium I.</s> </p> <p type="main"> <s id="s.001102">Si sectiones B, C, D terminentur in <lb></lb>perpendiculari BD, erit pariter <lb></lb>ratio inter ipsas nota.<figure id="id.064.01.149.2.jpg" xlink:href="064/01/149/2.jpg"></figure></s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/150.jpg"></pb> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.001103">PROPOSITIO X</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.001104">In canalibus inter binas orizontales, sectiones <lb></lb>inferiores sunt aequales.<figure id="id.064.01.150.1.jpg" xlink:href="064/01/150/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.001105">Sint canales AB, CD inter orizontales AC, BD. <lb></lb></s> <s id="s.001106">Dico sectiones B, D esse aequales.</s> </p> <p type="main"> <s id="s.001107">Fiat canale CE.</s> </p> <p type="main"> <s id="s.001108">Sectio E aequatur sectioni D<arrow.to.target n="marg234"></arrow.to.target>. </s> <s id="s.001109">Aequatur pariter <lb></lb>sectioni B, quia est par ratio. </s> <s id="s.001110">Ergo sectiones B, <lb></lb>D sunt aequales. </s> <s id="s.001111">Quod etc.<figure id="id.064.01.150.2.jpg" xlink:href="064/01/150/2.jpg"></figure></s> </p> <p type="margin"> <s id="s.001112"><margin.target id="marg234"></margin.target>Per 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001113">Corollarium I.</s> </p> <p type="main"> <s id="s.001114">Si canales AB, CB ducti ab orizontali A C ter<lb></lb>minantur in B, sectio in B erit aequaliter de<lb></lb>serviens utrique canali.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/151.jpg"></pb> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.001115">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.001116">Dato canali inflexo quomodolibet, venari quan<lb></lb>titatem datae sectionis.<figure id="id.064.01.151.1.jpg" xlink:href="064/01/151/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="11" type="proof"> <p type="main"> <s id="s.001117">Canalis AB inflectatur in B quovis angulo <lb></lb>ABC, in quo data sectione C venanda sit <lb></lb>eius quantitas.</s> </p> <p type="main"> <s id="s.001118">Protrahatur CB ad orizontalem AD, & fiat DE <lb></lb>media inter DB, DC, & sectionis C altitudo <lb></lb>ad altitudinem sectionis B fiat ut DB ad DE.</s> </p> <p type="main"> <s id="s.001119">Dico C esse sectionem in C.</s> </p> <p type="main"> <s id="s.001120">Quoniam si canale sit DC, sectio C ad sectionem B <lb></lb>est ut DB ad DE<arrow.to.target n="marg235"></arrow.to.target>. At sectio B est eadem <lb></lb>etiam, respectu canalis AB<arrow.to.target n="marg236"></arrow.to.target>. </s> <s id="s.001121">Ergo sectio <lb></lb>C ad sectionem B est ut DB ad DE.</s> </p> <p type="margin"> <s id="s.001122"><margin.target id="marg235"></margin.target>Per co. decimae huius.</s> </p> <p type="margin"> <s id="s.001123"><margin.target id="marg236"></margin.target></s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001124">Corollarium I.</s> </p> <p type="main"> <figure id="id.064.01.151.2.jpg" xlink:href="064/01/151/2.jpg"></figure> <s id="s.001125">Eadem via reperietur quantitas se<lb></lb>ctionis C, si canalis sit declinans, <lb></lb>& demum perpendicularis ut A, B, C.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/152.jpg"></pb> <pb xlink:href="064/01/153.jpg"></pb> <chap type="bk"> <p type="main"> <s id="s.001126">DE MOTV <lb></lb>GRAVIVM <lb></lb>LIBER SEXTVS <lb></lb>ET LIQUIDORVM TERTIVS <lb></lb>VBI DE FORAMINIBVS VASIS.</s> </p> <subchap1 type="preface"> <p type="main"> <s id="s.001127">Non alienum ab instituto <lb></lb>arbitratus sum adhuc ali<lb></lb>quid huic postremo prae<lb></lb>fari libro, ubi nodum sol<lb></lb>vere conabor ab eruditis<lb></lb>simo Mersenno proposi<lb></lb>tum prop. 15. Ballist. <lb></lb>quod quidem, explican<lb></lb>do, quantum ingenij fert imbecilitas, qua diu<lb></lb>turnitate pendulum, tam prius descendendo, <lb></lb>quam inde ascendendo feratur, suppositis ex<lb></lb>perimentis cum ipso primo habitis, postmo<lb></lb>dum a me repetitis, quibus percipere mihi vi<lb></lb>sus sum diuturnitatem penduli in integra <lb></lb>vibratione aequari diuturnitati gravis moti per <pb xlink:href="064/01/154.jpg"></pb>spatium eius quadruplum, & in descensu, <lb></lb>aequari diuturnitati gravis moti per eiusdem <lb></lb>penduli duplum: quod non omnino congruit <lb></lb>cum eo quod prop. 9. Terthuius huius proba<lb></lb>tum fuit, quoniam experimenta veritatem <lb></lb>proxime, at non praecise patefaciunt. </s> <figure id="id.064.01.154.1.jpg" xlink:href="064/01/154/1.jpg"></figure> <s id="s.001128">Sit pen<lb></lb>dulum AB, quod in C translatum sua integra <lb></lb>vibratione describat circulum CBD: ex dictis <lb></lb>experimentis compertum est diuturnitatem il<lb></lb>lius percurrentis per quadrantem CB, aequari <lb></lb>diuturnitati gravis descendentis per FB dia<lb></lb>metrum, ipsius penduli duplam; diuturnita<lb></lb>tem vero eiusdem conficientis integram vibra<lb></lb>tionem CBD, aequari diuturnitati eiusdem gravis <lb></lb>descendentis per duplum ipsius FB, puta per FG. <lb></lb></s> <s id="s.001129">Quibus positis, mihi assequi visus sum, qua pro<pb xlink:href="064/01/155.jpg"></pb>portione sibi respondeant diuturnitates pen<lb></lb>duli moti in descensu a C in B, & in ascensu <lb></lb>a B in D, secta CD in E tali ratione, ut E tan<lb></lb>tundem destet a C, quantum B; existimans diu<lb></lb>turnitates motuum per CB, & BD quadrantes, <lb></lb>esse inter se ut CE ad ED. </s> <s id="s.001130">Quoniam ratio diu<lb></lb>turnitatum per FB, & FG est eadem ac per <lb></lb>AB, & FB, cum utrobique sit subdupla pro<lb></lb>portio, quae ratio est pariter inter CB, & <lb></lb>FB<arrow.to.target n="marg237"></arrow.to.target>, cum CB sit media inter AB, FB,<arrow.to.target n="marg238"></arrow.to.target> erit <lb></lb>ratio diuturnitatum per FB, & FG, & itidem <lb></lb>per quadrantem CB, & per semic. CBD eis <lb></lb>aequalium<arrow.to.target n="marg239"></arrow.to.target> ut CB ad FB, seu ut CE ad CD eis <lb></lb>aequales: & dividendo, ratio diuturnitatum <lb></lb>per CB, & BD quadrantes erit ut CE ad ED<arrow.to.target n="marg240"></arrow.to.target>. <lb></lb></s> <s id="s.001131">Quod etc. </s> <s id="s.001132">Unde si ex Mersenno, grave ab A in <lb></lb>B pedum 3 regiorum, qui quatuor palmis nostra<lb></lb>tibus proxime respondent, descendit in 30 ter<lb></lb>tijs, a C in B fertur non in 30 sed in 42, unde <lb></lb>a B in D ascendit in 17 sibi respondentes ut <lb></lb>99 ad 41. Caeterum ex dictis facile demonstrabi<lb></lb>tur quod si vibrationes sint minores, v.g. ab <lb></lb>H in I, pariter diuturnitates per HB, & per <lb></lb>BI erunt ut CE ad ED, cum iam probatum <lb></lb>fuerit, & experientia constet vibrationes CB, HB <lb></lb>nec non CD, HI esse aequediuturnas. </s> <s id="s.001133">Ex his <lb></lb>etiam constat esse aequales diuturnitates per <lb></lb>BG, & BD, etiamsi per BD fiat ascensus, &<pb xlink:href="064/01/156.jpg"></pb>proinde motus successive tardior, & per BG <lb></lb>descensus, & proinde motus successive velo<lb></lb>cior. </s> <s id="s.001134">Quem nodum, de quo in praesentia <lb></lb>nil addam, alijs enodandum relinquo.</s> </p> <p type="margin"> <s id="s.001135"><margin.target id="marg237"></margin.target>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.001136"><margin.target id="marg238"></margin.target>Per cor. 8. sexti.</s> </p> <p type="margin"> <s id="s.001137"><margin.target id="marg239"></margin.target>Per Observat.</s> </p> <p type="margin"> <s id="s.001138"><margin.target id="marg240"></margin.target>Per 17. quinti.</s> </p> </subchap1> <pb xlink:href="064/01/157.jpg"></pb> <subchap1 type="definition"> <p type="head"> <s id="s.001139">DEFINITIONES.</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.001140">1 Vas aquae intelligitur, cuius latera sint <lb></lb>rectangula, & basis orizontalis.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.001141">2. Foramen intelligitur rectangulum cuius basis <lb></lb>orizontalis.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.001142">3. Foramina inaequalia eiusdem altitudinis, quo<lb></lb>rum inaequalitas pendet a sola latitudine.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/158.jpg"></pb> <subchap1 type="axiom"> <p type="head"> <s id="s.001143">DIGNITATES</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.001144">Ubi omnia sint paria, effectus sunt aequa<lb></lb>les.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.001145">PETITIONES</s> </p> <subchap2 type="axiom"> <p type="main"> <s id="s.001146">1 Quantitates eiusdem generis sunt omnes <lb></lb>commensurabiles.</s> </p> </subchap2> <subchap2 type="axiom"> <p type="main"> <s id="s.001147">2. Aqua transiens per vasis foramen, decurrit a <lb></lb>summo vasis ad foramen tanquam per cana<lb></lb>lem perpendicularem.</s> </p> </subchap2> <subchap2 type="axiom"> <p type="main"> <s id="s.001148">Quod experieris, si vas aqua plenum, in cuius <lb></lb>imo sit foramen, sit perspicuum; videbis etenim <lb></lb>in eo formari canale, per quod aqua supe<lb></lb>rior exeat.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/159.jpg"></pb> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.001149">PROPOSITIO PRIMA</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.001150">Aquarum quantitates exeuntium per forami<lb></lb>na aequalia, aeque distantia a summo vasis, <lb></lb>aequali tempore; sunt aequales.<figure id="id.064.01.159.1.jpg" xlink:href="064/01/159/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="1" type="proof"> <p type="main"> <s id="s.001151">In vase AB, sint foramina C, D aequalia, & <lb></lb>orizontalia, per quae aqua aequali tempore de<lb></lb>currat.</s> </p> <p type="main"> <s id="s.001152">Dico aquas decursas esse aequales inter se.</s> </p> <p type="main"> <s id="s.001153">Quoniam ubi omnia sunt paria, effectus sunt <lb></lb>aequales<arrow.to.target n="marg241"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001154"><margin.target id="marg241"></margin.target>Per ax. huius.</s> </p> <p type="main"> <s id="s.001155">Sed hic sunt omnia paria ex constructione.</s> </p> <p type="main"> <s id="s.001156">Ergo habent effectus aequales.</s> </p> <p type="main"> <s id="s.001157">Sed aquae decursa sunt effectus, & proinde aequa<lb></lb>les. </s> <s id="s.001158">Quod etc.</s> </p> <p type="main"> <s id="s.001159">Seu mavis.</s> </p> <p type="main"> <s id="s.001160">Ubi omnia paria effectus sunt aequales, & <lb></lb>proinde si effectus sunt aquae decursae, ipsae <lb></lb>sunt aequales.</s> </p> <p type="main"> <s id="s.001161">Sed hic sunt omnia paria, & effectus sunt aquae <lb></lb>decursae, ex constructione. </s> <s id="s.001162">Ergo aquae decursae sunt aequales. </s> <s id="s.001163">Quod etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/160.jpg"></pb> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.001164">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.001165">Si foramina sint orizontalia, eiusdem altitudi<lb></lb>nis, quantitates aquarum decursarum sunt <lb></lb>inter se ut foramina.<figure id="id.064.01.160.1.jpg" xlink:href="064/01/160/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <s id="s.001166">In vase AB dentur foramina orizontalia aeque <lb></lb>alta C minus, D vero maius.</s> </p> <p type="main"> <s id="s.001167">Dico aquam decursam per C, quae sit E, se habere ad aquam <lb></lb>decursam per D, quae sit F, ut foramen C ad foramen D.</s> </p> <p type="main"> <s id="s.001168">Longitudinum C, & D commensurabilium,<arrow.to.target n="marg242"></arrow.to.target> <lb></lb>sit G communis mensura, & secentur lon<lb></lb>gitudines C, D in partes, quae sint aequales ipsi <lb></lb>G, quibus divisis a perpendicularibus, producan<lb></lb>tur tot foramina, quot sunt dictae partes.</s> </p> <p type="margin"> <s id="s.001169"><margin.target id="marg242"></margin.target>Per pr. pet.</s> </p> <p type="main"> <s id="s.001170">Quoniam huiusmodi foramina erunt inter se <lb></lb>aequalia<arrow.to.target n="marg243"></arrow.to.target>. Ex eis effluent quantitates aquae <lb></lb>aequales<arrow.to.target n="marg244"></arrow.to.target>. </s> <s id="s.001171">Quot igitur sunt foramina in C, D, <lb></lb>tot sunt quantitates aquarum in E, F. </s> <s id="s.001172">Igitur <lb></lb>sunt quatuor quantitates C, D, E, F, quarum <lb></lb>prima, C, est ad E, 2., ut D, 3., ad F, 4.; & per<lb></lb>mutando erit C ad D ut E ad F<arrow.to.target n="marg245"></arrow.to.target>. </s> <s id="s.001173">Quod etc.</s> </p> <p type="margin"> <s id="s.001174"><margin.target id="marg243"></margin.target>Per 36. primi.</s> </p> <p type="margin"> <s id="s.001175"><margin.target id="marg244"></margin.target>Per primum huius.</s> </p> <p type="margin"> <s id="s.001176"><margin.target id="marg245"></margin.target>Per 16. quinti.</s> </p> <p type="main"> <s id="s.001177">Dices, quod fieri potest quod longitudines C, D, <lb></lb>non sint commensurabiles, nec proinde G sit eo<lb></lb>rum communis mensura: sed hic non sumus in <lb></lb>Mathematicis, sed in physicis, ubi non habetur <lb></lb>ratio insensibilium.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/161.jpg"></pb> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.001178">PROPOSITIO III.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.001179">Foramina vasis perinde se habent ac sectio<lb></lb>nes canalis, respectu impetus.<figure id="id.064.01.161.1.jpg" xlink:href="064/01/161/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <s id="s.001180">Sit vas CD in quo foramen D, & sit AB ca<lb></lb>nalis perpendicularis in quo sectio B, & <lb></lb>AB, CD, altitudines sint aequales.</s> </p> <p type="main"> <s id="s.001181">Dico in B, & D esse impetus aequales.</s> </p> <p type="main"> <s id="s.001182">Quoniam aqua fluens a foramine D decurrit per <lb></lb>spatium CD, ac si decurreret per canalem AB <lb></lb>perpendicularem, eiusdem longitudinis<arrow.to.target n="marg246"></arrow.to.target>, in <lb></lb>D, & B sortitur impetus aequales. </s> <s id="s.001183">Quod, etc.</s> </p> <p type="margin"> <s id="s.001184"><margin.target id="marg246"></margin.target>Per 2. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/162.jpg"></pb> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.001185">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.001186">Impetus foraminum aequalium vasis, sunt in <lb></lb>duplicata ratione distantiae a summo va<lb></lb>sis.<figure id="id.064.01.162.1.jpg" xlink:href="064/01/162/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="4" type="proof"> <p type="main"> <s id="s.001187">In vase AC, distantiae foraminum aequalium <lb></lb>B, C a summo vasis AB, AC; media sit AD.</s> </p> <p type="main"> <s id="s.001188">Dico impetus in C ad impetum in B esse ut AD <lb></lb>ad AB.</s> </p> <p type="main"> <s id="s.001189">Quoniam foramina B, C, sunt ac si essent sectio<lb></lb>nes canalis AC respectu impetus<arrow.to.target n="marg247"></arrow.to.target>, impetus in <lb></lb>B & C sunt ut AB ad AD<arrow.to.target n="marg248"></arrow.to.target>. </s> <s id="s.001190">Quod etc.</s> </p> <p type="margin"> <s id="s.001191"><margin.target id="marg247"></margin.target>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001192"><margin.target id="marg248"></margin.target>Per 4. quinti huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/163.jpg"></pb> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.001193">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.001194">Altitudines a foraminibus aequalibus ad sum<lb></lb>mum vasis, sunt in duplicata ratione aqua<lb></lb>rum per ea decurrentium.<figure id="id.064.01.163.1.jpg" xlink:href="064/01/163/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="5" type="proof"> <p type="main"> <s id="s.001195">In vase AC altitudines a foraminibus aequa<lb></lb>libus B, C, ad summum vasis A sint AB, <lb></lb>AC, quarum media sit AD.</s> </p> <p type="main"> <s id="s.001196">Dico AD ad AB esse ut aqua fluens per C ad <lb></lb>aquam fluentem per B.</s> </p> <p type="main"> <s id="s.001197">Quoniam ut AD ad AB ita est impetus in C ad <lb></lb>impetum in B<arrow.to.target n="marg249"></arrow.to.target>, & impetus sunt ut velocita<lb></lb>tes<arrow.to.target n="marg250"></arrow.to.target>; impetus in C ad impetum B est ut aqua <lb></lb>fluens per C ad aquam effluentem per B. </s> <s id="s.001198">Quod etc.</s> </p> <p type="margin"> <s id="s.001199"><margin.target id="marg249"></margin.target>Per quartam huius.</s> </p> <p type="margin"> <s id="s.001200"><margin.target id="marg250"></margin.target>Per 3. petit.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/164.jpg"></pb> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.001201">PROPOSITIO VI. PROBL. II.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.001202">Secto foramine in partes aliquotas a rectis <lb></lb>orizontalibus, venari rationes aquarum ex <lb></lb>eis fluentium.<figure id="id.064.01.164.1.jpg" xlink:href="064/01/164/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="6" type="proof"> <p type="main"> <s id="s.001203">Secetur foramen AB in partes AC, CD, DB <lb></lb>aequales, quorum altitudines sint notae, & <lb></lb>ab AC fluat aqua E, a CD aqua F, a DB <lb></lb>aqua G, tempore aequali.</s> </p> <p type="main"> <s id="s.001204">Venanda proportio aquarum E, F, G.</s> </p> <p type="main"> <s id="s.001205">Fiant HI, KL, MN, altitudines foraminum A<lb></lb>C, CD, DB a summo vasis; & inter ipsas <lb></lb>mediae OP, QR<arrow.to.target n="marg251"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001206"><margin.target id="marg251"></margin.target>Per 13. sexti.</s> </p> <p type="main"> <s id="s.001207">Quoniam aqua E ad aquam F, est ut HI ad OP<arrow.to.target n="marg252"></arrow.to.target>, <lb></lb>Nota est ratio aquae E ad aquam F. Item quoniam <lb></lb>aqua F ad aquam G est ut KL, ad QR<arrow.to.target n="marg253"></arrow.to.target>, <lb></lb>nota est pariter ratio aquae F ad aquam G. <lb></lb>at ratio aquae E ad aquam G, composita ra<lb></lb>tionum inter EF & FG notarum, est pariter <lb></lb>nota. </s> <s id="s.001208">Reperta est igitur ratio aquarum E, F, G. </s> <s id="s.001209">Quod, etc.</s> </p> <p type="margin"> <s id="s.001210"><margin.target id="marg252"></margin.target>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001211"><margin.target id="marg253"></margin.target>Per 5. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/165.jpg"></pb> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.001212">PROPOSITIO VII. PROBL. III.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.001213">Secto foramine vasis in partes a recta orizon<lb></lb>tali, reperire rationes aquarum effluen<lb></lb>tium ab ipsis.<figure id="id.064.01.165.1.jpg" xlink:href="064/01/165/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <s id="s.001214">Foramen CD vasis AB secetur a recta E in <lb></lb>partes CE, CD, & effluat a parte superio<lb></lb>ri CE aqua F, & ab inferiori ED aqua G eo<lb></lb>dem tempore.</s> </p> <p type="main"> <s id="s.001215">Quaeritur proportio F ad G.</s> </p> <p type="main"> <s id="s.001216">Si ED foramen minus non mensurat CE, repe<lb></lb>riatur eorum maxima communis mensura<arrow.to.target n="marg254"></arrow.to.target>, <lb></lb>quae sit H, & iuxta eam secetur CE in partes <lb></lb>CQ, QK, KE, item ED in partes EI, ID.</s> </p> <p type="margin"> <s id="s.001217"><margin.target id="marg254"></margin.target>Per 3. decimi.</s> </p> <p type="main"> <s id="s.001218">Quoniam foramen CD sectum est in partes CQ, <lb></lb>QK, KE, EI, ID aequales per constructionem; <lb></lb>venabitur ratio aquarum per eos fluentium<arrow.to.target n="marg255"></arrow.to.target>, & <lb></lb>proinde aquarum per CE, & ED. </s> <s id="s.001219">Quod etc.</s> </p> <p type="margin"> <s id="s.001220"><margin.target id="marg255"></margin.target>Per 6. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/166.jpg"></pb> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.001221">PROPOSITIO VIII. PROBL. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.001222">Datis foraminibus inaequalibus super eadem <lb></lb>orizontali, venari rationes aquarum.<figure id="id.064.01.166.1.jpg" xlink:href="064/01/166/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="8" type="proof"> <p type="main"> <s id="s.001223">Sint foramina AB, & CD super orizontali <lb></lb>BD.</s> </p> <p type="main"> <s id="s.001224">Quaerenda proportio aquarum ex eis eodem tem<lb></lb>pore fluentium.</s> </p> <p type="main"> <s id="s.001225">Producatur CE FG parallela DB.</s> </p> <p type="main"> <s id="s.001226">Quoniam nota est ratio aquarum fluentium ex <lb></lb>CD, & FB<arrow.to.target n="marg256"></arrow.to.target>, item per FB, & AG<arrow.to.target n="marg257"></arrow.to.target>, Nota est <lb></lb>pariter ratio ex eis composita inter aquas flu<lb></lb>entes per CD, & AG. </s> <s id="s.001227">Cum igitur sit nota ra<lb></lb>tio aquae fluentis per CD, ad fluentem per <lb></lb>FB, & per AG partes, nota erit ratio eiusdem <lb></lb>ad totam fluentem per AB. </s> <s id="s.001228">Quod etc.</s> </p> <p type="margin"> <s id="s.001229"><margin.target id="marg256"></margin.target>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001230"><margin.target id="marg257"></margin.target>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/167.jpg"></pb> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.001231">PROPOSITIO IX. PROBL. V.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.001232">Datis foraminibus, quorum unum superius, <lb></lb>alterum inferius inter easdem parallelas <lb></lb>perpendiculares: Reperire rationes aqua<lb></lb>rum.<figure id="id.064.01.167.1.jpg" xlink:href="064/01/167/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="9" type="proof"> <p type="main"> <s id="s.001233">Dentur foramina AB, CD inter parallelas <lb></lb>AC, & DB.</s> </p> <p type="main"> <s id="s.001234">Venanda ratio aquarum ex eis, aequo tempore, <lb></lb>fluentium.</s> </p> <p type="main"> <s id="s.001235">Concipiatur BC tanquam foramen.</s> </p> <p type="main"> <s id="s.001236">Quoniam nota est ratio aquarum fluentium ex CD, <lb></lb>& ex CB, item ex CB, & ex AB<arrow.to.target n="marg258"></arrow.to.target>, nota est <lb></lb>pariter ratio ex eis composita aquarum fluen<lb></lb>tium per CD, & per AB. </s> <s id="s.001237">Quod etc.</s> </p> <p type="margin"> <s id="s.001238"><margin.target id="marg258"></margin.target>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/168.jpg"></pb> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.001239">PROPOSITIO X. PROBL. VI.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.001240">Datis foraminibus venari aquas.<figure id="id.064.01.168.1.jpg" xlink:href="064/01/168/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.001241">Data sint foramina AD, EH.</s> </p> <p type="main"> <s id="s.001242">Oportet reperire rationem aquarum per <lb></lb>illa aequo tempore fluentium.</s> </p> <p type="main"> <s id="s.001243">Duc orizontales HI, FK, & producta DB in L, con<lb></lb>cipiatur IL tanquam foramen; & quaeratur <lb></lb>ratio aquarum per AD, IL<arrow.to.target n="marg259"></arrow.to.target>, & sit ut M ad N. <lb></lb></s> <s id="s.001244">Item quaeratur ratio IL ad EH,<arrow.to.target n="marg260"></arrow.to.target>, & sit ut N ad O.</s> </p> <p type="margin"> <s id="s.001245"><margin.target id="marg259"></margin.target>Per 9. huius.</s> </p> <p type="margin"> <s id="s.001246"><margin.target id="marg260"></margin.target>Per 2. huius.</s> </p> <p type="main"> <s id="s.001247">Dico M ad O esse rationem aquarum per AD, HE.</s> </p> <p type="main"> <s id="s.001248">Quoniam ut M ad N ita est AD ad IL, & ut <lb></lb>N ad O, ita IL ad EH per constr. </s> <s id="s.001249">Erit ex <lb></lb>aequo ut M ad O, ita AD ad EH<arrow.to.target n="marg261"></arrow.to.target>. </s> <s id="s.001250">Quod etc.</s> </p> <p type="margin"> <s id="s.001251"><margin.target id="marg261"></margin.target>Per 22. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/169.jpg"></pb> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.001252">PROPOSITIO XI. PROBL. VII</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.001253">Dato foramine, & linea orizontali intermi<lb></lb>nata; constituere super illa foramen, a quo <lb></lb>aequalis aqua fluat.<figure id="id.064.01.169.1.jpg" xlink:href="064/01/169/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="11" type="proof"> <p type="main"> <s id="s.001254">Dato foramine AB, & orizontali CD.</s> </p> <p type="main"> <s id="s.001255">Describendum sit foramen super CD, a <lb></lb>quo effluat aqua ut per AB.</s> </p> <p type="main"> <s id="s.001256">Erigantur perpendiculares AE, BC, & produca<lb></lb>tur DC in E, & super EC fait foramen aequale <lb></lb>AB, & sit FC, & ducta FG parallela CD, fiat <lb></lb>HI media inter K summum vasis B, & KE, <lb></lb>& ut HI ad KE, ita DL ad EC.</s> </p> <p type="main"> <s id="s.001257">Dico per LG foramen fluere aquam ut per AB.</s> </p> <p type="main"> <s id="s.001258">Quoniam aqua LG ad aquam FC est ut HI ad <lb></lb>KE<arrow.to.target n="marg262"></arrow.to.target>, & aqua AB ad aquam CF est ut HI ad <lb></lb>KE<arrow.to.target n="marg263"></arrow.to.target>, erit ut aqua LG ad CF, ita aqua AB <lb></lb>ad CF<arrow.to.target n="marg264"></arrow.to.target>, & proinde aqua AB aequalis aquae <lb></lb>LG<arrow.to.target n="marg265"></arrow.to.target>. </s> <s id="s.001259">Quod etc.</s> </p> <p type="margin"> <s id="s.001260"><margin.target id="marg262"></margin.target>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001261"><margin.target id="marg263"></margin.target>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001262"><margin.target id="marg264"></margin.target>Per 11. quinti.</s> </p> <p type="margin"> <s id="s.001263"><margin.target id="marg265"></margin.target>Per nonam quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/170.jpg"></pb> <subchap1 n="12" type="proposition"> <p type="head"> <s id="s.001264">PROPOSITIO XII. PROBL. VIII.</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.001265">Dato foramine, & latere alterius, reperire fo<lb></lb>ramen, e quo aequalis aqua effluat.<figure id="id.064.01.170.1.jpg" xlink:href="064/01/170/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="12" type="proof"> <p type="main"> <s id="s.001266">Datum sit foramen AB, & latere DC. </s> <s id="s.001267">Oportet describere foramen, a quo effluat <lb></lb>aqua ut ab AB, cuius latus sit CD.</s> </p> <p type="main"> <s id="s.001268">Ductis CE, & DF, orizontalibus; protrahatur B<lb></lb>E, & FE intelligatur foramen, & reperiatur ra<lb></lb>tio aquarum fluentium ab AB, & ab FE<arrow.to.target n="marg266"></arrow.to.target>, <lb></lb>quae sit ut C ad H; & fiat ut H ad G, ita <lb></lb>FI ad FK, & a K erigitur perpendicularis KL, <lb></lb>& fiat foramen cuius latus DC aequale, & <lb></lb>simile ipsi FL, et sit DM.</s> </p> <p type="margin"> <s id="s.001269"><margin.target id="marg266"></margin.target>Per 9. huius.</s> </p> <p type="main"> <s id="s.001270">Dico a foramine DM fluere aquam, ut ab AB.</s> </p> <p type="main"> <s id="s.001271">Quoniam aqua fluens per AB ad fluentem per FE <lb></lb>est ut G ad H per const. item aqua fluens per FL <lb></lb>seu ei aequale DM ad fluentem per eandem F<lb></lb>E est itidem ut G ad H<arrow.to.target n="marg267"></arrow.to.target>, aquae fluentes per A<lb></lb>B & per DM sunt inter se aequales<arrow.to.target n="marg268"></arrow.to.target>, DM ig. </s> <s id="s.001272">Est foramen quaesitum. </s> <s id="s.001273">Quod etc.</s> </p> <p type="margin"> <s id="s.001274"><margin.target id="marg267"></margin.target>Per secundum huius.</s> </p> <p type="margin"> <s id="s.001275"><margin.target id="marg268"></margin.target>Per 9. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/171.jpg"></pb> <subchap1 n="13" type="proposition"> <p type="head"> <s id="s.001276">PROPOSITIO XIII. PROBL. IX.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.001277">Dato foramine, reperire aliud aequale, a quo <lb></lb>fluat aqua in ratione data.<figure id="id.064.01.171.1.jpg" xlink:href="064/01/171/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="13" type="proof"> <p type="main"> <s id="s.001278">Detur in vase AB foramen C, & data sit <lb></lb>ratio aquarum D, E, quarum D fluat in <lb></lb>dato tempore per foramen C.</s> </p> <p type="main"> <s id="s.001279">Reperiendum ubi fiat aequale foramen, a quo fluat <lb></lb>in aequali tempore aqua E.</s> </p> <p type="main"> <s id="s.001280">Fiat ad D, E, AC quarta preportionalis AF<arrow.to.target n="marg269"></arrow.to.target>, <lb></lb>& ad AC, AF tertia proportionalis AG<arrow.to.target n="marg270"></arrow.to.target>, & <lb></lb>in G fiat foramen: quod si fieri nequit proble<lb></lb>ma est insolubile. </s> <s id="s.001281">Dico G esse locum forami<lb></lb>nis quaesitum.</s> </p> <p type="margin"> <s id="s.001282"><margin.target id="marg269"></margin.target>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.001283"><margin.target id="marg270"></margin.target>Per 11. sexti.</s> </p> <p type="main"> <s id="s.001284">Quoniam aquae fluentes per dicta foramina sunt <lb></lb>in subduplicata ratione altitudinum AC, AG<arrow.to.target n="marg271"></arrow.to.target>, <lb></lb>& aquae D, E, sunt pariter in subduplicata ra<lb></lb>tione eorumdem altitudinum AC, AG<arrow.to.target n="marg272"></arrow.to.target>, aquae <lb></lb>fluentes per dicta foramina sunt ut aquae D, <lb></lb>& E<arrow.to.target n="marg273"></arrow.to.target>. </s> <s id="s.001285">Quod etc.</s> </p> <p type="margin"> <s id="s.001286"><margin.target id="marg271"></margin.target>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001287"><margin.target id="marg272"></margin.target>Per eamdem.</s> </p> <p type="margin"> <s id="s.001288"><margin.target id="marg273"></margin.target>Per 9. quinti.</s> </p> </subchap2> <pb xlink:href="064/01/172.jpg"></pb> <subchap2 type="corollary"> <p type="head"> <s id="s.001289">Corollarium I.</s> </p> <p type="main"> <s id="s.001290">Parum refert sint foramina quadrata nec ne.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001291">Corollarium II.</s> </p> <p type="main"> <s id="s.001292">Idem sequitur si ambo foramina sint rotunda.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/173.jpg"></pb> <subchap1 n="14" type="proposition"> <p type="head"> <s id="s.001293">PROPOSITIO XIV.</s> </p> <subchap2 n="14" type="statement"> <p type="main"> <s id="s.001294">Dato foramine, aptandum sit aliud datum <lb></lb>simile, magnitudinis diversae, a quo aqua <lb></lb>fluens cum fluente a primo, habeat ratio<lb></lb>nem datam.<figure id="id.064.01.173.1.jpg" xlink:href="064/01/173/1.jpg"></figure></s> </p> </subchap2> <subchap2 n="14" type="proof"> <p type="main"> <s id="s.001295">In vase AB, dato foramine C, & alio D ut <lb></lb>supra dictum est; & data sit ratio aquarum E, F.</s> </p> <p type="main"> <s id="s.001296">Aptandum est foramen D ea lege, ut aqua per il<lb></lb>lud fluens, cum aqua fluente a C, sit ut F ad E.</s> </p> <p type="main"> <s id="s.001297">Super orizontali ducta CG fiat foramen G, <lb></lb>aequale foramini D; & perquiratur ratio <lb></lb>aquarum fluentium per C, & G<arrow.to.target n="marg274"></arrow.to.target>, & sit ut E <lb></lb>ad H: quae si est eadem quae est inter E, & F, <lb></lb>habemus intentum; ni sit, fiat aliud foramen <lb></lb>infra seu supra G ei simile, & aequale a quo <lb></lb>fluat aqua quae cum fluente ab ipso G habeat <lb></lb>rationem ut H ad F<arrow.to.target n="marg275"></arrow.to.target>, & sit I. </s> <s id="s.001298">Quod si fieri <lb></lb>nequit problema est insolubile. </s> <s id="s.001299">Dico I esse <lb></lb>foramen quaesitum.</s> </p> <p type="margin"> <s id="s.001300"><margin.target id="marg274"></margin.target>Per 8. huius.</s> </p> <p type="margin"> <s id="s.001301"><margin.target id="marg275"></margin.target>Per 13. huius.</s> </p> <pb xlink:href="064/01/174.jpg"></pb> <p type="main"> <s id="s.001302">Quoniam probatum fuit aquam C ad aquam <lb></lb>G esse ut E ad H, & aquam G ad aquam I <lb></lb>esse ut H ad F, constat aquam C ad aquam I <lb></lb>esse ut E ad F<arrow.to.target n="marg276"></arrow.to.target>. </s> <s id="s.001303">Quod etc.</s> </p> <p type="margin"> <s id="s.001304"><margin.target id="marg276"></margin.target>Per 22. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001305">Corollarium I.</s> </p> <p type="main"> <s id="s.001306">Parum refert sint ne foramina quadrata, <lb></lb>nec ne.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001307">Corollarium II.</s> </p> <p type="main"> <s id="s.001308">Idem sequeretur si essent ambo rotunda<arrow.to.target n="marg277"></arrow.to.target>.</s> </p> <p type="margin"> <s id="s.001309"><margin.target id="marg277"></margin.target>Per 3. pet.</s> </p> <p type="main"> <s id="s.001310">FINIS</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/175.jpg"></pb> <pb xlink:href="064/01/176.jpg"></pb> <pb xlink:href="064/01/177.jpg"></pb> </body> <back></back> </text> </archimedes>