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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes> <info> <author>Baliani, Giovanni Baptista</author> <title>De Motu Naturali Gravium Solidorum et Liquidorum</title> <date>1646</date> <place>Genf</place> <translator/> <lang>la</lang> <cvs_file>balia_demot_064_la_1646.xml</cvs_file> <cvs_version/> <locator>064.xml</locator> </info> <text><front/><body><pb xlink:href="064/01/001.jpg"/><chap><p type="main"> <s>DE MOTV <lb/>NATVRALI <lb/>GRAVIVM SOLIDORVM <lb/>ET LIQUIDORVM <lb/>IO: BAPTISTAE BALIANI <lb/>PATRITII ENVENSIS. </s></p><p type="main"> <s>GENVAE</s></p><p type="main"> <s>Ex Typographia IO: Mariæ Farroni 1646 <lb/>Superiorum Permi&longs;&longs;u.</s></p><pb xlink:href="064/01/002.jpg"/><pb xlink:href="064/01/003.jpg"/><p type="main"> <s>DE MOTV<lb/>GRAVIVM<lb/>SOLIDORVM<lb/>LIBER PRIMVS.<lb/></s></p><p type="main"> <s>Mihi quoque, sicut & <lb/>caeteris hominibus, inest <lb/>sciendi cupiditas, nec gra­<lb/>ve fuit, usque a primis <lb/>annis, & aliorum scripta <lb/>percurrere, & naturales <lb/>effectus observare, qui fa­<lb/>cile mihi persuaserim, ex hisce fontibus, tum <lb/>scientiam, tum sapientiam in animum de­<lb/>rivare, si tandem ex effectibus diligentius<pb xlink:href="064/01/004.jpg"/>perspectis, non modo ad inde consequentes, <lb/>sed etiam ad causas, usque ad primam de­<lb/>veniat intellectus. </s> <s>Statui igitur apud me ip­<lb/>sum non acquiescere soli relationi pluri­<lb/>morum, etiam doctiorum; potuisse siquidem <lb/>contingere existimavi, ut aliqua laterent, <lb/>etiam in plurimis oculatissimos, vel non ple­<lb/>ne ab eis explicarentur; & ratus sum non <lb/>inutilem laborem futurum, si ex accuratiori <lb/>naturae rerum investigatione, & ex affection­<lb/>um inde resultantium deductione, circa <lb/>quod omnis demonstrativa scientia versatur, <lb/>aut scitis adderem aliqua, aut doctioribus <lb/>acuerem desiderium addendi plura: hinc fa­<lb/>ctum est, ut excitata mens ex praecognitis le­<lb/>gendo, ad ea, quae se offerebant, secun­<lb/>dum privatas, aut publicas occupationes per­<lb/>vestiganda, converteretur studiosus. </s> <s>Inter <lb/>alia dum anno millesimo sexcentesimo un­<lb/>decimo, per paucos menses, ex patriae legis <lb/>praescripto, Praefectum Arcis Savonae agerem, <lb/>ex militaribus observationibus quae occurre­<lb/>bant, illud maxime depraehendi, ferreos, <lb/>& lapideos tormentorum bellicorum glo­<lb/>bos, & sic corpora gravia, seu eiusdem, seu<pb xlink:href="064/01/005.jpg"/>diversae speciei, in inaequali satis Mole, & <lb/>gravitate, per idem spatium, aequali tem­<lb/>pore, & motu, naturaliter descendere, idque <lb/>ita uniformiter, ut repetitis experimentis mihi <lb/>plane constiterit, duos ex praedictis globis, <lb/>vel ferreos ambos, vel alterum lapideum <lb/>alterum plumbeum, eodem plane mo­<lb/>mento temporis dimissos sibi, per spatium <lb/>quinquaginta pedum, etiam si unus es­<lb/>set librae unius tantum, alter quinquagin­<lb/>ta, in indivisibili temporis momento, subje­<lb/>ctum solum ferire, ut unus tantum ambo­<lb/>rum ictus sensu perciperetur. </s> <s>Repetebam <lb/>animo sapientum esse pronunciatum, gravia <lb/>moveri naturali motu, secundum gravitatum <lb/>proportionem; Processi ulterius, & pericu­<lb/>lum feci, num forte iuxta eorum sententiam <lb/>contingeret, si corpora dimissa, eiusdem fere <lb/>essent molis, sed longe diversi ponderis, pu­<lb/>ta unum plumbeum, cereum alterum; & ex­<lb/>pertus sum in cereo aliquam longiorem mo­<lb/>ram in descensu, attamen longe infra propor­<lb/>tionem gravitatum, globus quippe ille ce­<lb/>reus, in data distantia quinquaginta pedum <lb/>descensus, uno circiter pede distabat a solo,<pb xlink:href="064/01/006.jpg"/>quando plumbeus tangebat subjectum pla­<lb/>num, objecto aere intermedio ni fallor, sen­<lb/>sibiliter resistente, & impediente motum. <lb/></s> <s>Institi adhuc, & globos in gravitate, & in <lb/>materia inaequales appendi funiculis aequali­<lb/>bus, & agitatos animadverti moveri tempo­<lb/>re aequali, & hoc servare adeo fideliter, ut <lb/>globus plumbeus duarum unciarum, alter <lb/>librarum duarum, ferreus librarum 34. & la­<lb/>pideus quadraginta circiter, nec non, & la­<lb/>pis informis, quorum funiculi comprehen­<lb/>sis ipsorum semidiametris aequales essent, <lb/>uno, & eodem temporis spatio moverentur, <lb/>& vibrationes easdem numero darent hinc <lb/>inde, sive motus unius globi fieret per aequa­<lb/>le spatium, sive per inaequale, ita ut qui <lb/>maiori impetu jactabatur, & sic majus spa­<lb/>tium percurrebat, illud tanto velocius per­<lb/>transiret. </s> <s>In quibus peragendis illud praeter <lb/>expectationem sese mihi obtulit, quod quo­<lb/>tiescunque globi penderent ex funiculis inae­<lb/>qualibus, ita inaequali motu ferebantur, ut <lb/>longitudines funiculorum, durationibus mo­<lb/>tuum, in duplicata ratione responderent.<lb/></s></p><p type="main"> <s>Porro cum ex praemissis satis superque li­<lb/><pb xlink:href="064/01/007.jpg"/>queret, in naturali motu gravium, pro­<lb/>portionem gravitatum communiter credi­<lb/>tam, non servari; in eam descendi sen­<lb/>tentiam, ut arbitrater fortasse, gravitatem <lb/>se habere ut agens, materiam vero, seu <lb/>mavis materiale corpus, ut passum, & <lb/>proinde gravia moveri juxta proportionem <lb/>gravitatis ad materiam, & ubi sine impedi­<lb/>mento naturaliter perpendiculari motu fe­<lb/>rantur, moveri aequaliter, quia ubi plus est <lb/>gravitatis, plus pariter sit materiae, seu ma­<lb/>terialis quantitatis; si vero accedat aliquid <lb/>resistentiae, regulari motum secundum ex­<lb/>cessum virtutis agentis supra resistentiam <lb/>passi, seu impedientia motum; qui exces­<lb/>sus momentum noncupabitur, & quod com­<lb/>muniter gravitati attributum fuit, momen­<lb/>to attribui debere, nimirum ut sit momen­<lb/>tum ad momentum, ut velocitas ad velo­<lb/>citatem; Et hinc fieri posse, ut cognosca­<lb/>mus qua mensura, seu proportione corpora <lb/>gravia naturali motu ferantur super subje­<lb/>ctis planis, si super eis quomodolibet in­<lb/>clinatis, ipsorum gravium momenta ubique <lb/>innotescant, quae maiora, aut minora viden­<lb/><pb xlink:href="064/01/008.jpg"/>tur censenda, secundum quod magis, aut <lb/>minus super plano quiescunt, & sic secun­<lb/>dum maiorem, aut minorem inclinationem <lb/>plani resistentis; quod demum tali propor­<lb/>tione facile fieri mihi existimandum vide­<lb/>tur, juxta quam reciproce momentis pro­<lb/>portionantur lineae dictorum planorum, si <lb/>ambae ductae sint ab eodem puncto ad idem <lb/>planum orizontale; de quo Simon Stevi­<lb/>nus l. p. de Statica prop. 19. & acutissime <lb/>Galileus in Mechanica manuscripta, ubi de <lb/>Cochlea, & ego aliquali experientia com­<lb/>pertum habui. </s> <s>Caeterum si per experien­<lb/>tiam Scientia hominibus efficitur, praedicta <lb/>de quibus saepius repetitis actibus expertus <lb/>fui, ut principia scientiae habenda fore cen­<lb/>sui; in quibus occultae conclusiones delites­<lb/>cant, demonstrationibus duntaxat aperien­<lb/>dae. </s> <s>Rimari caepi; an deprehenderim alio­<lb/>rum erit judicium. </s> <s>Subjecta paucula, quae <lb/>presens aliquod otium expedire permisit, <lb/>de motu naturali solidorum gravium, Ami­<lb/>ce lector tibi exhibeo, mox de liquidorum, <lb/>& deinceps alia plura tam parata daturus, <lb/>si haec placuerint. </s> <s>Placuit sane mihi, vel<pb xlink:href="064/01/009.jpg"/>paucula tibi dare, qui te eius ingenij esse <lb/>confidam, ut non verba, sed res, easque <lb/>non mole, sed pondere censeas, felicior si <lb/>de eorum genere existimaveris, quae non <lb/>mole magna sunt, quod si talia non fue­<lb/>rint, quo minora minus defatigabunt, sui <lb/>exilitate, auctoris partus proprios omnino <lb/>esse probatura. </s> <s>Idioma latinum elegi ut <lb/>communius. </s> <s>Praemisi aliqua naturalia prin­<lb/>cipia, sine quibus naturales conclusiones <lb/>aliunde duci posse non video. </s> <s>Quae ex prae­<lb/>dictis experimentis innotuerunt, supposi­<lb/>tiones appellare, & a reliquis petitionibus <lb/>secernere libuit. </s> <s>Petitiones illas, quibus quid <lb/>fieri petimus, constructioni deservientes, <lb/>tanquam factu, & cognitu faciles, & pro­<lb/>inde supervacaneas, prudens praetermisi; <lb/>ratus siquidem nil inde incredulitatis, aut <lb/>difficultatis derivaturum. </s> <s>Septimum po­<lb/>stulatum ea ratione segregavi, quod il­<lb/>lud aliquo pacto a 22. prop. pendeat, & <lb/>quod in illo etiamsi veritas non deficiat, <lb/>evidentiam tamen ut in caeteris non agno­<lb/>scens, certis dubia quo quo pacto permisce­<lb/>re noluerim; ut proinde plura eorum, quae<pb xlink:href="064/01/010.jpg"/>ex illo deducta sunt, & diversa Methodo & <lb/>attingendo potius, quam demonstrando <lb/>subjunxerim. </s> <s>Si quae demum minus pro­<lb/>bata, seu explicata, aut quo quo pacto im­<lb/>perfecta reperies, velim te tribuere cuidam <lb/>naturali meae propensioni, ad nova potius, <lb/>qualiacumque ea sint, invenienda, quam <lb/>inventa perficienda. </s> <s>Vale.</s></p><pb xlink:href="064/01/011.jpg"/><p type="main"> <s>De mandato Reuerendi&longs;&longs;imi Patris Magi&longs;tri <lb/>lu&longs;tiniani Vagnoni Inqui&longs;itoris Generelis <lb/>Genuæ, &c.</s></p><p type="main"> <s>Rudi ego infra&longs;criptus Sancti Officij Con&longs;ultor <lb/>De Motu Grauium Illu&longs;tri&longs;&longs;imi D. Ioannis <lb/>Baptiste Baliani Libros sex.</s> <s>In quibus nilre­ <lb/>peri S. Catholica fidei, bonis moribus, &longs;acri&longs;­ <lb/>ue decretis di&longs;&longs;onum; &longs;ed dignam ubique typis, <lb/>& publica luce doctrinam, &longs;i prefato Reue­ <lb/>rendi&longs;&longs;imo Patri ita videbitur.</s> <s>In quorum fi­ <lb/>dem, &c.</s></p><p type="main"> <s>Ex Conuentu Sancti&longs;&longs;ime Annunciatæ Veteris <lb/>Genue 27. Nouembris 1646.</s></p><p type="main"> <s>Magi&longs;t. Fr. Angelicus Riccobonus Aug.</s></p><p type="main"> <s>IMPRIMATVR.</s></p><p type="main"> <s>F. Iu&longs;tinianus Vagnonus a Calli S. T. M. <lb/>Inqui&longs;itor Generalis Genuæ &c.</s></p><pb xlink:href="064/01/012.jpg"/><pb xlink:href="064/01/013.jpg"/><subchap1 type="definition"><p type="head"> <s>DEFINITIONES</s></p><subchap2 type="definition"><p type="main"> <s>Pendulus dicimus pondus filo <lb/>appensum.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Pendula dicuntur aequalia, <lb/>seu aequipendula, sive inae­<lb/>qualia, quae, & longiora, <lb/>aut breviora, quatenus <lb/>fila, e quibus dependent, sunt <lb/>aequalia, longiora, aut breviora.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationes pendulorum sunt eorum motus hinc <lb/>inde </s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationes aequales dicimus, quae fiunt per spa­<lb/>tia aequalia, & e contra inaequales.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationes aeque celeres si fiant per spatia aequa­<lb/>lia tempore aequali.</s></p></subchap2><pb xlink:href="064/01/014.jpg"/><subchap2 type="definition"><p type="main"> <s>Vibrationis diuturnitatem dicimus ipsius Dura­<lb/>tionem, tempus nimirum, quo ipsa vibratio <lb/>perficitur.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationes æquediuturne, sunt, quae fiunt tem­<lb/>pore aequali, etiamsi per spatia inaequalia, <lb/>inde diuturnior est, quae longiori perficitur <lb/>tempore.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationes integras dicimus eas, quae se exten­<lb/>dunt per integrum semicirculum, se hinc in­<lb/>de moventes per circuli quadrantem.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationis portio est pars arcus, quem ipsa vi­<lb/>bratio disignant.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationum similes portiones sunt arcus ipsa­<lb/>rum intercepti inter binas lineas ductas a <lb/>centro, a quo concipiuntur pendula pendere.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Vibrationis portionem priorem decimus eam mi­<lb/>nimam portionem, a qua integra vibratio <lb/>initium habet.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>Momentum est excessus virtutis moventis supra <lb/>motus impedimenta.</s></p></subchap2></subchap1><pb xlink:href="064/01/015.jpg"/><subchap1 type="supposition"><p type="head"> <s>SUPPOSITIONES</s></p><subchap2 type="supposition"><p type="main"> <s>PRIMA. </s> <s>Solidorum aequipendu­<lb/>lorum cujuscumque gravitatis vibra­<lb/>tiones aequales sunt aequediu­<lb/>turnae.</s></p></subchap2><subchap2 type="supposition"><p type="main"> <s>2 Equipendulorum eorundem vibrationes <lb/>sunt aequediuturnae, etiamsi inaequales.</s></p></subchap2><subchap2 type="supposition"><p type="main"> <s>3 Pendulorum inaequalium longitudines sunt <lb/>in duplicata ratione diuturnitatum vi­<lb/>brationum, seu ut quadrata vibratio­<lb/>num.</s></p></subchap2><subchap2 type="supposition"><p type="main"> <s>4 Momentum gravis super plano inclinato <lb/>est ad ipsius gravitatem, ut perpendi­<pb xlink:href="064/01/016.jpg"/>cularis ad inclinatam, si ab eodem <lb/>puncto ducta sint ad idem planum <lb/>orizontale dicta perpendicularis, & di­<lb/>ctum planum inclinatum, & proinde <lb/>tali casu proportio gravitatis ad mo­<lb/>mentum est reciproca proportioni li­<lb/>nearum super quibus grave movetur.</s></p></subchap2></subchap1><pb xlink:href="064/01/017.jpg"/><subchap1 type="postulate"><p type="head"> <s>PETITIONES, SEU POSTULATA</s></p><subchap2 type="postulate"><p type="main"> <s>Pr. </s> <s>Pendulorum inaequalium portiones similes vi­<lb/>brationum sunt inter se quoad diuturni­<lb/>tatem, ut vibrationes integrae.<figure id="id.064.01.017.1.jpg" xlink:href="064/01/017/1.jpg"/></s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Sint pendula AB, AC; dependentia a puncto A, <lb/>& eleventur ad libellam orizontis puncti A, <lb/>in E, D, describentia arcus BD, CE, inte­<lb/>grarum vibrationum, & in arcubus BD, <lb/>CE sumantur portiones similes EF, DG, seu <lb/>HI, KL ductis EA, FA, seu HA, IA. </s> <s>Peto <lb/>mihi concedi, esse pendulorum diuturnitates in <lb/>arcubus EC, DB, ut in portionibus EF, DG, <lb/>nec non HI, KL, & ita deinceps.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>2. Ut est momentum ad momentum solidi <lb/>gravis, ita velocitas ad velocitatem.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Huiusmodi passio communiter attribui solet gra­<lb/>vitati simpliciter, quod eum nimis clare expe­<lb/>rientijs supra expositis nullo pacto congruere <lb/>possit, momentis attribuenda esse visa est, ut <lb/>in praefatione explicatum fuit.</s></p></subchap2><pb xlink:href="064/01/018.jpg"/><subchap2 type="postulate"><p type="main"> <s>3. Portiones minimae peripheriae Circuli con­<lb/>cipiende sunt, ac si essent lineae rectae.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Quaecumque arcus portio est circularis, atta­<lb/>men si est minima portio, tam parum aber­<lb/>rat a linea recta, ut non modo quo ad <lb/>sensum, sed quoad quascunque physicas passio­<lb/>nes, perinde esse videatur, ac si esset linea re­<lb/>cta, idcirco ut petitionem admittendam cen­<lb/>seo, quemadmodum in mechanicis admittitur­ <lb/>illa, quod perpendiculares sunt parallelae, etiamsi <lb/>in centro concurrant universi, quatenus eis­<lb/>dem sunt passionibus physicis subjectae, ac si <lb/>vere essent parallelae.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>4. Data recta linea, possimus concipere cir­<lb/>culum talis magnitudinis, cujus portio pe­<lb/>ripheriae aequalis quo ad sensum datae lineae, <lb/>concipienda sit, ac si esset linea recta.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Haec petitio videtur concedenda, quia si conci­<lb/>piamus circulum, eiusque portionem mini­<lb/>mam, ut in praecedenti, si fiat ut huiusmodi <lb/>portio ad datam lineam, ita circulus ad alium, <lb/>portio huius, datae lineae aequalis erit, & simi­<lb/>lis omnino praedicta minimae portioni, & proin­<lb/>de pariter concipienda ut linea recta.</s></p></subchap2><pb xlink:href="064/01/019.jpg"/><subchap2 type="postulate"><p type="main"> <s>5. Solida perpendicula libero motu aeque <lb/>velociter feruntur, & in tali proportione, <lb/>ac si essent pendula, & moverentur in <lb/>priori portione vibrationum.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Quoniam prior portio non differt sensibiliter a re­<lb/>cta, ut in tertia petitione ijsdem physicis passio­<lb/>nibus subjicitur, & exinde motibus aequalibus.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>6. Solida naturaliter mota super plano incli­<lb/>nato aeque velociter moventur ac si essent <lb/>pendula, & moverentur in tali portione vi­<lb/>brationum, quae quoad sensum esset aequa­<lb/>lis, & paralella lineae dicti plani super qua <lb/>dicta solida moverentur.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>Non differt a praecedente, nisi quod in illa mo­<lb/>tus est perpendicularis, in hac inclinatus, in <lb/>reliquis est par ratio.</s></p></subchap2></subchap1><subchap1 type="enunciation"><p type="head"> <s>PRONUNCIATA</s></p><subchap2 type="enunciation"><p type="main"> <s>P. </s> <s>Quae sunt aequidiuturna tertio, sunt aequi­<lb/>diuturna inter se.</s></p></subchap2><subchap2 type="enunciation"><p type="main"> <s>2. Quadrata datorum temporum, sunt etiam <lb/>quadrata aliorum datis aequalium.</s></p></subchap2><subchap2 type="enunciation"><p type="main"> <s>3. Gravia eadem super planis aequalibus & <lb/>pariter inclinatis, pariter moventur.</s></p></subchap2></subchap1><pb xlink:href="064/01/020.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA.</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Solidi penduli naturaliter moti vibratio­<lb/>nes quantumvis semper minores, sunt <lb/>aequidiuturnae.<figure id="id.064.01.020.1.jpg" xlink:href="064/01/020/1.jpg"/></s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>Sit solidum A pendulum debite applicatum filo <lb/>BA, quod ab altera parte elevatum naturaliter, <lb/>postea faciat hinc inde vibrationes semper mi­<lb/>nores, ita ut prior vibratio sit V.G. per spatium <lb/>CD maius, posterior vero per spatium EF minus.</s></p><p type="main"> <s>Dico quod dicta vibrationes erunt aequidiuturnae, <lb/>ita ut vibratio per spatium CD sit eiusdem du­<lb/>rationis, ac vibratio per spatium EF.</s></p><p type="main"> <s>Sit aliud solidum G aequipendulum solido A, de­<lb/>bite applicatum filo HG, quod elevetur ab una <lb/>parte eodem tempore minus quam solidum A <lb/>ita ut sint minores vibrationes solidi G, quam, <lb/>solidi A, ut sit motus penduli G in initio per <lb/>spatium IK aequale spatio EF.</s></p><p type="main"> <s>Quoniam spatia EF, & IK, sunt aequalia ex sup­<lb/>positione, sunt etiam vibrationes EF, & IK, <lb/>aequidiuturnae,<arrow.to.target n="marg1"/>,sed IK, & CD sunt pariter <lb/>aequidiuturnae<arrow.to.target n="marg2"/>, ergo EF, & CD sunt etiam <lb/>aequidiuturnae<arrow.to.target n="marg3"/>. </s> <s>Quod fuit probandum.</s></p><p type="margin"> <s><margin.target id="marg1"></margin.target>Per primam suppositionem.</s></p><p type="margin"> <s><margin.target id="marg2"></margin.target>Per secundam suppositionem.</s></p><p type="margin"> <s><margin.target id="marg3"></margin.target>Per pr. pron.</s></p></subchap2></subchap1><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II. PROB. PRIMUM</s></p><pb xlink:href="064/01/021.jpg"/><subchap2 n="2" type="statement"><p type="main"> <s>Pendula constituere, quorum diuturnita­<lb/>tes vibrationum sint in data ratione.<figure id="id.064.01.021.1.jpg" xlink:href="064/01/021/1.jpg"/></s></p></subchap2><subchap2 n="3" type="proof"><p type="main"> <s>Data sit proportio diuturnitatum vibratio­<lb/>num, quam volumus esse inter solida A,B; <lb/>& sit ea, quae est inter C, & D; quae est continuo <lb/>eadem,<arrow.to.target n="marg4"/>,</s></p><p type="margin"> <s><margin.target id="marg4"></margin.target>Per pr. huius.</s></p><p type="main"> <s>Venanda est longitudo filorum, quibus applicata <lb/>dicta solida producant vibrationes quaesitas.</s></p><p type="main"> <s>Fiat L tertia proportionalis ad C, & D,<arrow.to.target n="marg5"/> & fila <lb/>IA, KB fiant inter se ut C ad L,<arrow.to.target n="marg6"/> & erunt <lb/>fila quaesita.</s></p><p type="margin"> <s><margin.target id="marg5"></margin.target>Per 11 sexti.</s></p><p type="margin"> <s><margin.target id="marg6"></margin.target>Per 12 sexti.</s></p><p type="main"> <s>Quoniam ita est IA ad KB ut C ad L per constr. <lb/>erunt C, & D diuturnitates vibrorum pendu­<lb/>lorum AB.<arrow.to.target n="marg7"/> Quod etc</s></p><p type="margin"> <s><margin.target id="marg7"></margin.target>Per 3 Supp.</s></p></subchap2></subchap1><pb xlink:href="064/01/022.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO TERTIA</s></p><subchap2 n="3" type="statement"><p type="main"> <s>Lineae descensus gravium, dum naturali motu <lb/>perpendiculariter feruntur, sunt in dupli­<lb/>cata ratione diuturnitatum.<figure id="id.064.01.022.1.jpg" xlink:href="064/01/022/1.jpg"/></s></p></subchap2><subchap2 n="4" type="proof"><p type="main"> <s>Sint LN, KM linea descensus gravium L, K, <lb/>& sint PO ipsorum diuturnitates.</s></p><p type="main"> <s>Dico LN, KM esse in duplicata ratione ipsarum P, O.</s></p><p type="main"> <s>Sint pendula AH, AI, dependentia a puncto A, & <lb/>eleventur ad libellam ipsius A usque ad E, B, <lb/>quae in elevatione producant arcus HB, IE, & <lb/>sint talis longitudinis, ut ducta ACF, secet ar­<lb/>cus BC, & EF, tam parvae curvitatis ut pro <lb/>rectis habeantur, puta portionis minimae, & <lb/>proinde aequales quo ad sensum rectis KM, LN,<arrow.to.target n="marg8"/> <lb/>& fiat V tertia proportionalis ad O, P,<arrow.to.target n="marg9"/><lb/></s></p><p type="margin"> <s><margin.target id="marg8"></margin.target>Per 3 pet.</s></p><p type="margin"> <s><margin.target id="marg9"></margin.target>Per 11 sexti.</s></p><p type="main"> <s>Quoniam O, P sunt diuturnitates KM, LN ex <lb/>constr., sunt itidem diuturnitates BC, EF, <arrow.to.target n="marg10"/> & <lb/>quia diuturnitates vibrorum AH, AI sunt <lb/>etiam ut O ad P <arrow.to.target n="marg11"/> AH AI sunt ut O, ad V<arrow.to.target n="marg12"/> <lb/>& pariter BC, & EF sunt ut O ad V<arrow.to.target n="marg13"/> Ergo <lb/>KM, LN eis aequales per constr. sunt etiam ut <lb/>O ad V, & proinde in duplicata ratione O, P, <lb/>temporum seu diuturnitatum earumdem. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg10"></margin.target>Per 5 pet.</s></p><p type="margin"> <s><margin.target id="marg11"></margin.target>Per p. pet.</s></p><p type="margin"> <s><margin.target id="marg12"></margin.target>Per 3 supp.</s></p><p type="margin"> <s><margin.target id="marg13"></margin.target>Per p. pet.</s></p></subchap2></subchap1><pb xlink:href="064/01/023.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO QUARTA. PROB. II.</s></p><subchap2 n="4" type="statement"><p type="main"> <s>Data diuturnitate gravis descendentis a data <lb/>altitudine, constituere altitudinem, a qua <lb/>idem grave cadat in data alia diuturnitate.<figure id="id.064.01.023.1.jpg" xlink:href="064/01/023/1.jpg"/></s></p></subchap2><subchap2 n="5" type="proof"><p type="main"> <s>Sit A diuturnitas gravis B, dum cadit in C, & <lb/>data sit diuturnitas quaecumque D.</s></p><p type="main"> <s>Constituenda est alia altitudo, a qua grave de­<lb/>scendat iuxta diuturnitatem D.</s></p><p type="main"> <s>Fiat I, tertia proportionalis ad AD,<arrow.to.target n="marg14"/> & ut I ad A <lb/>fiat altitudo GH ad altitudinem datam BC,<arrow.to.target n="marg15"/> <lb/>Dico GH esse altitudinem quaesitam.</s></p><p type="margin"> <s><margin.target id="marg14"></margin.target>Per 11. sexti.</s></p><p type="margin"> <s><margin.target id="marg15"></margin.target>Per 12. sexti.</s></p><p type="main"> <s>Quoniam BC, & GH sunt in duplicata ratione <lb/>datarum diuturnitatum A, D, per constructio­<lb/>nem; per ipsas gravia B, & G cadent in diu­<lb/>turnitatibus A, & D datis<arrow.to.target n="marg16"/>, unde reperta est <lb/>altitudo GH quaesita. </s> <s>Quod fuit faciendum.</s></p><p type="margin"> <s><margin.target id="marg16"></margin.target>Per 3. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/024.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V. PROB. III.</s></p><subchap2 n="5" type="statement"><p type="main"> <s>Data altitudine, a qua descendat grave in no­<lb/>ta diuturnitate; perquirere quanta sit diutur­<lb/>nitas, qua descendat ab alia altitudine data.<figure id="id.064.01.024.1.jpg" xlink:href="064/01/024/1.jpg"/></s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Sit A altitudo per quam descendat grave diutur­<lb/>nitate B nota, & data sit alia altitudo C.</s></p><p type="main"> <s>Oportet reperire quanta sit diuturnitas, qua idem <lb/>grave descendat per C.</s></p><p type="main"> <s>Fiat ut A ad C ita B ad G,<arrow.to.target n="marg17"/> inter quas media, <lb/>proportionalis F<arrow.to.target n="marg18"/> est diuturnitas quaesita.</s></p><p type="margin"> <s><margin.target id="marg17"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg18"></margin.target>Per 13. sexti.</s></p><p type="main"> <s>Quoniam A, & C sunt in duplicata ratione diu­<lb/>turnitatum B, & F per constructionem, per <lb/>ipsas gravia descendent in diuturnitatibus B, <lb/>F,<arrow.to.target n="marg19"/> unde F est diuturnitas ipsius C quaesita.</s></p><p type="margin"> <s><margin.target id="marg19"></margin.target>Per 3. huius.</s></p><p type="main"> <s>Quod faciendum fuit.</s></p></subchap2></subchap1><pb xlink:href="064/01/025.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>PROPOSITIO VI.</s></p><subchap2 n="6" type="statement"><p type="main"> <s>Gravia naturali motu descendunt semper velo­<lb/>cius ea ratione, ut temporibus aequalibus de­<lb/>scendant per spatia semper maiora, iuxta <lb/>proportionem quam habent impares nu­<lb/>meri ab unitate inter se.<figure id="id.064.01.025.1.jpg" xlink:href="064/01/025/1.jpg"/></s></p></subchap2><subchap2 n="7" type="proof"><p type="main"> <s>Sit grave A quod descendat per lineam ABC, <lb/>& tempus quo descendit ab A in B sit aequale <lb/>tempori, quo descendit a B in C, & a C in D.</s></p><p type="main"> <s>Dico quod lineae AB, BC, CD sunt inter se ut 1.<lb/>3.5.& sic deinceps.</s></p><p type="main"> <s>Sit G linea mensurans tempus, quo A descendit <lb/>in B, & H, quo de­<lb/>scendit a B in C, & I, quo descendit a C in D, quae tempora sunt ex suppositione <lb/>aequalia, & sit K latus quadrati ipsius G, & L <lb/>quadrati GH, & N quadrati totius GHI.</s></p><p type="main"> <s>Quoniam quadrata K, L, N sunt ut AB, AC, A<lb/>D<arrow.to.target n="marg20"/>, quae quadrata sunt ut 1, 4, 9, sunt itidem <lb/>AB, AC, AD, ut 1. 4. 9. & dividendo AB, <lb/>BC, CD, ut 1. 3. 5. & sic deinceps. </s> <s>Quod <lb/>probandum fuit.</s></p><p type="margin"> <s><margin.target id="marg20"></margin.target>Per 3. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/026.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>Lineae descensus gravium super plano incli­<lb/>nato motorum, sunt in duplicata ratione <lb/>diuturnitatum.<figure id="id.064.01.026.1.jpg" xlink:href="064/01/026/1.jpg"/></s></p></subchap2><subchap2 n="8" type="proof"><p type="main"> <s>Sint AB, CD plana pariter inclinata, super <lb/>quibus moveantur gravia A, C, & sint EF <lb/>ipsorum diuturnitates.</s></p><p type="main"> <s>Dico AB, CD, esse in duplicata ratione ipsarum E, F.</s></p><p type="main"> <s>Secetur AB bifariam in G, & erecta GH, per­<lb/>pendiculari longissima, fiant pendula HI, HK, <lb/>quae sint inter se ut AB, CD, & eleventur in <lb/>L, M, describentia arcus LI, KM, secantes <lb/>GH in N, O, & ab N hinc inde secentur ar­<lb/>cus NP, NQ aequales quo ad sensum rectis <lb/>GA, GB, & ductis PH, QH, secetur pariter <lb/>arcus LI, in R, S, & intelligantur arcus PQ, <lb/>RS, tam parvae curvitatis, ob maximam lon­<lb/>gitudinem pendulorum HI, HK, ut pro re­<lb/>ctis habeantur, puta portionis minimae, & pro­<lb/>inde aequales rectis AB, CD.<arrow.to.target n="marg21"/></s></p><p type="margin"> <s><margin.target id="marg21"></margin.target>Per 3. pet.</s></p><p type="main"> <s>Quoniam EF sunt diuturnitates AB, CD per<pb xlink:href="064/01/027.jpg"/>construct, sunt etiam diuturnitates portionum <lb/>PQ, RS<arrow.to.target n="marg22"/>, & pariter vibrationum pendulo­<lb/>rum HK, HI<arrow.to.target n="marg23"/> sunt autem diuturnitates <lb/>praedictae E, F, in subduplicata ratione pendu­<lb/>lorum HK, HI<arrow.to.target n="marg24"/> unde pariter portionum PQ, <lb/>RS, & proinde plenorum AB, CD, Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg22"></margin.target>Per 6. pet.</s></p><p type="margin"> <s><margin.target id="marg23"></margin.target>Per pr. pet.</s></p><p type="margin"> <s><margin.target id="marg24"></margin.target>Per 3. supp.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Hinc patet esse longitudines planorum per quae <lb/>gravia feruntur ut quadrata temporum, & <lb/>tempora ut radices longitudinum planorum.</s></p></subchap2></subchap1><pb xlink:href="064/01/028.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII. PROB. IV.</s></p><subchap2 n="8" type="statement"><p type="main"> <s>Dato plano inclinato, super quo per spatium <lb/>datum grave moveatur in nota diuturni­<lb/>tate, determinare in eodem plano spatium <lb/>per quod dictum grave moveatur in qua­<lb/>vis alia diuturnitate data.<figure id="id.064.01.028.1.jpg" xlink:href="064/01/028/1.jpg"/></s></p></subchap2><subchap2 n="9" type="proof"><p type="main"> <s>Sit A diuturnitas gravis B, dum descendit in <lb/>C super plano inclinato BC, & data diu­<lb/>turnitas D.</s></p><p type="main"> <s>Praescribendum est aliud spatium in eodem pla­<lb/>no BC, per quod idem grave pertranseat in <lb/>diuturnitate D.</s></p><p type="main"> <s>Fiat H tertia proportionalis ad A & D, & ut <lb/>H ad A fiat BG ad BC, Dico BG esse spa­<lb/>tium quaesitum.</s></p><p type="main"> <s>Quoniam BC, & BG sunt in duplicata ratione <lb/>datorum temporum A, D per constructionem, <lb/>per ipsa cadet grave B diuturnitatibus A, D <lb/>datis<arrow.to.target n="marg25"/>, ergo reperta est BG quaesita. </s> <s>Quod <lb/>faciendum erat.</s></p><p type="margin"> <s><margin.target id="marg25"></margin.target>Per 6. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/029.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX. PROB. V.</s></p><subchap2 n="9" type="statement"><p type="main"> <s>Dato plano inclinato, super quo per spatium <lb/>datum grave moveatur nota diuturnitate; <lb/>& dato alio spatio quocumque; reperire <lb/>diuturnitatem, qua grave per ipsum de­<lb/>scendat.<figure id="id.064.01.029.1.jpg" xlink:href="064/01/029/1.jpg"/></s></p></subchap2><subchap2 n="10" type="proof"><p type="main"> <s>Sit Nota diuturnitas gravis B, dum descendit <lb/>in C super plano inclinato BC, & dato alio <lb/>spatio BG.</s></p><p type="main"> <s>Quaerendum quanta sit diuturnitas gravis in BG.</s></p><p type="main"> <s>Intelligatur BC diuturnitas ipsius BC, & fiat <lb/>BH, media inter BC, & BG, quae erit diu­<lb/>turnitas quaesita.</s></p><p type="main"> <s>Quoniam BC, & BG sunt in duplicata ratio­<lb/>ne diuturnitatum BC, & BH, per constructio­<lb/>nem; per ipsa cadunt gravia diuturnitatibus <lb/>BC, BH,<arrow.to.target n="marg26"/> unde BH est diuturnitas per spa­<lb/>tium BG quaesita. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg26"></margin.target>Per 7. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/030.jpg"/><subchap1 n="10" type="proposition"><p type="head"> <s>PROPOSITIO X.</s></p><subchap2 n="10" type="statement"><p type="main"> <s>Gravia descendunt super planis inclinatis per <lb/>spatia semper maiora, iuxta rationem, quam <lb/>habent impares numeri successive inter se. <figure id="id.064.01.030.1.jpg" xlink:href="064/01/030/1.jpg"/></s></p></subchap2><subchap2 n="11" type="proof"><p type="main"> <s>Sit grave A, quod descendat super plano ABC <lb/>inclinato, & tempus quo descendit ab A in <lb/>B sit aequale tempori, quo descendit a B in C, <lb/>& a C in D.</s></p><p type="main"> <s>Dico quod lineae AB, BC, CD sunt inter se ut <lb/>1. 3. 5. &. sic deinceps.</s></p><p type="main"> <s>Sit E numerus mensurans tempus, quo A descen­<lb/>dit in B, & F quo descendit a B in C, & G <lb/>quo descendit a C in D, quae tempora sunt ex <lb/>suppositione aequalia, & sit H quadratum ip­<lb/>sius E, & I quadratum EF, & K quadra­<lb/>tum totius EFG.</s></p><p type="main"> <s>Quoniam quadrata HIK sunt ut AB, AC, AD<arrow.to.target n="marg27"/>, <lb/>quae quadrata sunt ut 1. 4. 9. sunt pariter <lb/>AB, AC, AD, ut 1. 4. 9. & dividendo AB, <lb/>BC, CD, sunt ut 1. 3. 5. & sic deinceps. <lb/></s> <s>Quod probandum erat.</s></p><p type="margin"> <s><margin.target id="marg27"></margin.target>Per 7. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/031.jpg"/><subchap1 n="11" type="proposition"><p type="head"> <s>PROPOSITIO XI.</s></p><subchap2 n="11" type="statement"><p type="main"> <s>Si Duo gravia descendant alterum super li­<lb/>nea perpendiculari, alterum vero super <lb/>inclinata; proportio velocitatum est reci­<lb/>proca proportioni linearum.<figure id="id.064.01.031.1.jpg" xlink:href="064/01/031/1.jpg"/></s></p></subchap2><subchap2 n="12" type="proof"><p type="main"> <s>Sit ABC planum normaliter erectum super <lb/>lineam orizontalem BC, cuius latus AB sit <lb/>perpendiculare, & AC, inclinatum.</s></p><p type="main"> <s>Dico quod proportio velocitatum solidorum gra­<lb/>vium motorum secundum lineam AB perpen­<lb/>dicularem, & AC inclinatum, est ut propor­<lb/>tio longitudinis inclinatae AC ad longitudinem <lb/>perpendicularis AB; videlicet ita est longitudo <lb/>AB ad longitudinem AC, ut velocitas super <lb/>AC ad velocitatem in AB.</s></p><p type="main"> <s>Quoniam est ut AC ad AB, ita momentum in <lb/>AB, ad momentum in AC<arrow.to.target n="marg28"/>; & ut momentum <lb/>in AB ad momentum in AC, ita velocitas in <lb/>AB ad velocitatem in AC<arrow.to.target n="marg29"/>; ergo est etiam <lb/>ut AC ad AB, ita velocitas in AB ad veloci­<lb/>tatem in AC. </s> <s>Quod fuit probandum.</s></p><p type="margin"> <s><margin.target id="marg28"></margin.target>Per 4. supp.</s></p><p type="margin"> <s><margin.target id="marg29"></margin.target>Per 2. pet.</s></p></subchap2></subchap1><pb xlink:href="064/01/032.jpg"/><subchap1 n="12" type="proposition"><p type="head"> <s>PROPOSITIO XII.</s></p><subchap2 n="12" type="statement"><p type="main"> <s>Gravia descendunt super plana diverse in­<lb/>clinata tali proportione, ut si velocitas ad <lb/>velocitatem reciproca longitudinibus pla­<lb/>norum ductorum ab eodem puncto, ad <lb/>idem planum orizontale.<figure id="id.064.01.032.1.jpg" xlink:href="064/01/032/1.jpg"/></s></p></subchap2><subchap2 n="13" type="proof"><p type="main"> <s>Sint F, D plana inclinata ducta ad idem pla­<lb/>num orizontale.</s></p><p type="main"> <s>Dico esse ut planum D ad planum F, ita veloci­<lb/>tatem gravis ducti super F, ad velocitatem <lb/>eiusdem ducti super D.</s></p><p type="main"> <s>Ducatur perpendicularis E, & sint B, A, C ve­<lb/>locitates gravium latorum super perpendicu­<lb/>lari, & super planis F, D.</s></p><p type="main"> <s>Quoniam est A ad B, ut E ad F, item, & B ad <lb/>C, ut D, ad E<arrow.to.target n="marg30"/>, erit A ad C ut D ad F<arrow.to.target n="marg31"/>, sci­<lb/>licet velocitas gravis super F ad velocitatem <lb/>gravis super D, ut lon­<lb/>gitudo plani D ad longitudinem plani F. </s> <s>Quod fuit probandum.</s></p><p type="margin"> <s><margin.target id="marg30"></margin.target>Per 11. huius.</s></p><p type="margin"> <s><margin.target id="marg31"></margin.target>Per 13. Quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/033.jpg"/><subchap1 n="13" type="proposition"><p type="head"> <s>PROPOSITIO XIII. PROB. VI.</s></p><subchap2 n="13" type="statement"><p type="main"> <s>Reperire inclinationem plani, super quo <lb/>grave moveatur tali velocitate quae cum <lb/>alia super diversa inclinatione sit in ra­<lb/>tione data.<figure id="id.064.01.033.1.jpg" xlink:href="064/01/033/1.jpg"/></s></p></subchap2><subchap2 n="14" type="proof"><p type="main"> <s>Moveatur grave A super recta AB, seu <lb/>perpendiculari, seu inclinata, & data sit <lb/>proportio C ad D.</s></p><p type="main"> <s>Oportet reperire aliud planum inclinatum, ita <lb/>ut velocitas gravis moti super AB ad velo­<lb/>citatem alterius moti super illo reperiendo, <lb/>sit ut D ad C.</s></p><p type="main"> <s>Producatur BA; & fiat ut C ad D ita BA, ad <lb/>AE; & centro A, intervallo AE describatur <lb/>circulus, secans BF in F; ni secet, problema <lb/>insolubile est; si secat, ducatur AF, quam di­<lb/>co esse planum quaesitum.</s></p><p type="main"> <s>Quoniam ut C ad D, ita AB ad AE, seu AF <lb/>per constructionem, erit C velocitas super AF, <lb/>& D super AB<arrow.to.target n="marg32"/>, unde velocitates super ip­<lb/>sis sunt in ratione data. </s> <s>Quod faciendum fuit.</s></p><p type="margin"> <s><margin.target id="marg32"></margin.target>Per 12. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/034.jpg"/><subchap1 n="14" type="proposition"><p type="head"> <s>PROPOSITIO XIV. PROB. VII.</s></p><subchap2 n="14" type="statement"><p type="main"> <s>Data linea perpendiculari, per quam grave <lb/>descendat, cui annectatur linea, seu pla­<lb/>num declinans; in declinante reperire <lb/>punctum, quo grave perveniat eo tempo­<lb/>re, quo pertransiverit perpendicularem.<figure id="id.064.01.034.1.jpg" xlink:href="064/01/034/1.jpg"/></s></p></subchap2><subchap2 n="15" type="proof"><p type="main"> <s>Sit triangulum ABC orthogonaliter erectum <lb/>super plano orizontali BC, cuius latus AB <lb/>intelligatur linea perpendicularis, per quam <lb/>grave descendat, & latus AC planum incli­<lb/>natum.</s></p><p type="main"> <s>Oportet in plano AC reperire punctum quo gra­<lb/>ve perveniat eodem tempore, quo in B.</s></p><p type="main"> <s>Fiat ut AC ad AB, ita AB ad tertiam AD<arrow.to.target n="marg33"/>, <lb/>& D erit punctum quaesitum.</s></p><p type="margin"> <s><margin.target id="marg33"></margin.target>Per 11. Sexti.</s></p><p type="main"> <s>Quoniam velocitas super AD ad velocitatem in <lb/>AB est ut AB ad AC<arrow.to.target n="marg34"/>, & proinde ut AD <lb/>ad AB per const, quae velocitates eadem con­<lb/>tinuo duplicata proportione augentur<arrow.to.target n="marg35"/>, gra­<lb/>via in eis moventur tempore aequali, quia quo­<lb/>tiscunque spatia sunt ut velocitates, aequali <lb/>peraguntur tempore, quod, etc.</s></p><p type="margin"> <s><margin.target id="marg34"></margin.target>Per 11. huius.</s></p><p type="margin"> <s><margin.target id="marg35"></margin.target>Per 3. & 7. huius.</s></p></subchap2><pb xlink:href="064/01/035.jpg"/><subchap2 type="corollary"><p type="head"> <s>Corollarium 1.</s></p><p type="main"> <s>Hinc est quod in D, & B velocitates sunt ut AD, <lb/>AB, & ita in quibuslibet punctis respondenti­<lb/>bus paralellis ad DB cum in AD, & AB ve­<lb/>locitates semper eadem ratione augeantur.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium 2.</s></p><p type="main"> <s>Hinc est etiam quod si esset AE aequalis AB, & <lb/>AF media inter AD, AE, tempus AD, & <lb/>proinde AB ad tempus AE, esset ut AD ad <lb/>AF<arrow.to.target n="marg36"/>.</s></p><p type="margin"> <s><margin.target id="marg36"></margin.target>Per 7. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium 3.</s></p><p type="main"> <s>Si AE est quadrupla AD, AF erit dupla AD, <lb/>unde tempus AE erit duplum tempori AB.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium 4.</s></p><p type="main"> <s>Si AC esset quadrupla AD, grave moveretur <lb/>temporibus aequalibus per AB, AD, DC.</s></p></subchap2></subchap1><pb xlink:href="064/01/036.jpg"/><subchap1 n="15" type="proposition"><p type="head"> <s>PROPOSITIO XV.</s></p><subchap2 n="15" type="statement"><p type="main"> <s>Si duo gravia descendunt alterum quidem <lb/>perpendiculariter, alterum vero super pla­<lb/>no declinante, perveniunt ad idem pla­<lb/>num Orizontale tali ratione, ut sit eadem <lb/>proportio inter diuturnitates eorum, quae <lb/>inter perpendicularem, & declinantem.<figure id="id.064.01.036.1.jpg" xlink:href="064/01/036/1.jpg"/></s></p></subchap2><subchap2 n="16" type="proof"><p type="main"> <s>Sit linea AB perpendiculariter erecta super <lb/>plano Orizontali BC, & AC planum declinans.</s></p><p type="main"> <s>Dico quod diuturnitates gravium descendentium <lb/>per AB, & per AC, sunt ut AB ad AC.</s></p><p type="main"> <s>Fiat AD tertia proportionalis ad AC, & AB<arrow.to.target n="marg37"/>,</s></p><p type="margin"> <s><margin.target id="marg37"></margin.target>Per 11. Sexti.</s></p><p type="main"> <s>Quoniam est ut AD ad AC ita quadratum tem­<lb/>poris AD ad quadratum temporis AC<arrow.to.target n="marg38"/>, & <lb/>tempora AD, & AB sunt aequalia<arrow.to.target n="marg39"/>, & proin­<lb/>de eorum quadrata<arrow.to.target n="marg40"/>, ergo ut AD, ad AC <lb/>ita quadratum temporis AB ad quadratum <lb/>temporis AC, sed ut AD ad AC ita quadra­<lb/>tum AB ad quadratum AC<arrow.to.target n="marg41"/>, ergo ut quadratum temporis AB ad quadratum temporis A<lb/>C, ita quadratum AB ad quadratum AC<arrow.to.target n="marg42"/>, <lb/>sed quia latera sunt inter se ut eorum qua­<lb/>drata<arrow.to.target n="marg43"/>, est ut AB ad AC ita tempus AB ad <lb/>tempus AC. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg38"></margin.target>Per cor. 7. huius.</s></p><p type="margin"> <s><margin.target id="marg39"></margin.target>Per 14. huius.</s></p><p type="margin"> <s><margin.target id="marg40"></margin.target>Per 2. pron.</s></p><p type="margin"> <s><margin.target id="marg41"></margin.target>Per 19. Sexti.</s></p><p type="margin"> <s><margin.target id="marg42"></margin.target>Per 11. Quinti.</s></p><p type="margin"> <s><margin.target id="marg43"></margin.target>Per 22. Sexti.</s></p></subchap2></subchap1><pb xlink:href="064/01/037.jpg"/><subchap1 n="16" type="proposition"><p type="head"> <s>PROPOSITIO XVI. PROBL. VIII.</s></p><subchap2 n="16" type="statement"><p type="main"> <s>Data linea perpendiculari, & plano decli­<lb/>nante; reperire in perpendiculari produ­<lb/>cta punctum, quo perveniat grave eo tem­<lb/>pore, quo pertransit planum inclinatum.<figure id="id.064.01.037.1.jpg" xlink:href="064/01/037/1.jpg"/></s></p></subchap2><subchap2 n="17" type="proof"><p type="main"> <s>Data sit perpendicularis AB, cui connexum <lb/>planum inclinatum AD.</s></p><p type="main"> <s>Oportet in AB producta reperire punctum, quo <lb/>perveniat grave eo tempore, quo pervenit in <lb/>puncto D.</s></p><p type="main"> <s>In puncto D perpendicularis erigatur ad AD, & <lb/>protrahatur usquequo coeat cum AB produ­<lb/>cta in E, & E est punctum quaesitum.</s></p><p type="main"> <s>Quoniam triangula, ADE, AEC sint aequian­<lb/>gula, cum anguli ADE, AEC sint aequales, <lb/>nempe recti, & BAD communis<arrow.to.target n="marg44"/>, sunt etiam <lb/>similia<arrow.to.target n="marg45"/>, ergo ut AC ad AE, ita AE ad AD<arrow.to.target n="marg46"/>, <lb/>unde tempora per AD, & AE sunt aequalia<arrow.to.target n="marg47"/>.</s></p><p type="margin"> <s><margin.target id="marg44"></margin.target>Per 32. prim.</s></p><p type="margin"> <s><margin.target id="marg45"></margin.target>Per 4. sexti.</s></p><p type="margin"> <s><margin.target id="marg46"></margin.target>Per 4. sexti.</s></p><p type="margin"> <s><margin.target id="marg47"></margin.target>Per 14 huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Hinc est quod super plano AC erit AD men­<lb/>sura diuturnitatis motus peracti super AE.</s></p></subchap2></subchap1><pb xlink:href="064/01/038.jpg"/><subchap1 n="17" type="proposition"><p type="head"> <s>PROPOSITIO XVII. PROBL. IX.</s></p><subchap2 n="17" type="statement"><p type="main"> <s>Dato plano declinante, super quo grave de­<lb/>scendat, & dato alio plano minus declinan­<lb/>te, in hoc reperire punctum, quo perveniat <lb/>mobile eo tempore, quo pertransit dictum <lb/>planum magis declinans.<figure id="id.064.01.038.1.jpg" xlink:href="064/01/038/1.jpg"/></s></p></subchap2><subchap2 n="18" type="proof"><p type="main"> <s>Sint plana AB, AC quorum AC minus in­<lb/>clinatum.</s></p><p type="main"> <s>Oportet in AC reperire punctum, quo grave per­<lb/>veniat, quando pervenit in B.</s></p><p type="main"> <s>Fiat ut AC ad AB ita AB ad AD, & dico D <lb/>esse punctum quaesitum.</s></p><p type="main"> <s>Quoniam ut AC ad AD ita est quadratum AC <lb/>ad quadratum AB<arrow.to.target n="marg48"/>, & ut AC ad AD ita <lb/>quadratum temporis AC ad quadratum tem­<lb/>poris AD<arrow.to.target n="marg49"/> ergo ut quadratum AC ad qua­<lb/>dratum AB, ita quadratum temporis AC ad <lb/>quadratum temporis AD Vnde AC ad AB<lb/>ut tempus AC ad tempus AD<arrow.to.target n="marg50"/>, sed ut AC <lb/>ad AB, ita tempus AC ad tempus AB<arrow.to.target n="marg51"/>, ergo <lb/>tempora AB, AD, sunt aequalia. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg48"></margin.target>Per 19. sexti.</s></p><p type="margin"> <s><margin.target id="marg49"></margin.target>Per cot. 7. huius.</s></p><p type="margin"> <s><margin.target id="marg50"></margin.target>Per 22. sexti.</s></p><p type="margin"> <s><margin.target id="marg51"></margin.target>Per 15. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/039.jpg"/><subchap1 n="18" type="proposition"><p type="head"> <s>PROPOSITIO XVIII. PROBL. X.</s></p><subchap2 n="18" type="statement"><p type="main"> <s>Datis planis declinantibus ortis ab eodem <lb/>puncto, reperire in magis declinante pun­<lb/>ctum quo grave perveniat eo tempore, quo <lb/>pertransit planum minus declinans.<figure id="id.064.01.039.1.jpg" xlink:href="064/01/039/1.jpg"/></s></p></subchap2><subchap2 n="19" type="proof"><p type="main"> <s>Datum sit planum minus declinans AC, & <lb/>magis AD, terminantia super plano ori­<lb/>zontali BD.</s></p><p type="main"> <s>Oportet in AD producta reperire punctum, quo <lb/>perveniat grave eo tempore, quo pertransivit <lb/>planum minus declinans AC.</s></p><p type="main"> <s>Fiat ut AD ad AC ita AC ad dictam AD pro­<lb/>ductam in E, quod est punctum quaesitum.</s></p><p type="main"> <s>Quoniam ut AE ad AD ita est quadratum AC <lb/>ad quadratum AD<arrow.to.target n="marg52"/>, sed AE ad AD est ut <lb/>quadratum tempo­<lb/>ris AE, ad quadratum temporis AD<arrow.to.target n="marg53"/>, ergo ut quadra­<lb/>tum AC ad quadratum AD, ita quadratum temporis AE ad qua­<lb/>dratum temporis AD<arrow.to.target n="marg54"/>, unde AC ad AD ut <lb/>tempus AE ad tempus AD<arrow.to.target n="marg55"/>, sed AC ad AD <lb/>est ut tempus AC ad tempus AD<arrow.to.target n="marg56"/>, ergo tem­<lb/>pora AE, AC sunt aequalia. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg52"></margin.target>Per 19. sexti.</s></p><p type="margin"> <s><margin.target id="marg53"></margin.target>Per cor. 7. huius.</s></p><p type="margin"> <s><margin.target id="marg54"></margin.target>Per 11. Quinti.</s></p><p type="margin"> <s><margin.target id="marg55"></margin.target>Per 22. sexti.</s></p><p type="margin"> <s><margin.target id="marg56"></margin.target>Per 15. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/040.jpg"/><subchap1 n="19" type="proposition"><p type="head"> <s>PROPOSITIO XIX. PROBL. XI.</s></p><subchap2 n="19" type="statement"><p type="main"> <s>Dato motus naturali gravis quomodocumque <lb/>ad punctum datum, reperire seu in perpen­<lb/>diculari, seu in plano quomodolibet incli­<lb/>nato punctum, a quo digressum, perveniat <lb/>ad idem punctum quo prius, tempore aequali.<figure id="id.064.01.040.1.jpg" xlink:href="064/01/040/1.jpg"/></s></p></subchap2><subchap2 n="20" type="proof"><p type="main"> <s>Sit AB linea quomodocumque aut perpendicu­<lb/>laris, seu planum inclinatum; super qua <lb/>grave descendat in B, & data sit quaecunque <lb/>linea BC, aut perpendicularis, aut quomodo­<lb/>libet inclinata, quae cum AB, coeat in B.</s></p><p type="main"> <s>Oportet in BC reperire punctum, a quo grave digres­<lb/>sum perveniat in B tempore quo pervenit ab A in idem B.</s></p><p type="main"> <s>Ducatur AC orizontalis, & fiat BD tertia pro­<lb/>portionalis ad CB AB<arrow.to.target n="marg57"/>, & D est punctum <lb/>quaesitum. </s> <s>Quod ut probetur.</s></p><p type="margin"> <s><margin.target id="marg57"></margin.target>Per 11. Sexti.</s></p><p type="main"> <s>Fiat iterum rectae AC paralella, & aequalis BE, & <lb/>ducta EA, secetur recta BF parallela ipsi AD.</s></p><p type="main"> <s>Quoniam AF, BD sunt pariter inclinatae, & <lb/>aequales<arrow.to.target n="marg58"/>, gravia per ipsas aequali tempore mo­<lb/>ventur<arrow.to.target n="marg59"/>, sed per AF, grave movetur tempo­<lb/>re quo per AB<arrow.to.target n="marg60"/>, ergo per BD movetur pari­<lb/>ter tempore quo per AB<arrow.to.target n="marg61"/>, quod, etc.</s></p><p type="margin"> <s><margin.target id="marg58"></margin.target>Per 33. Primi.</s></p><p type="margin"> <s><margin.target id="marg59"></margin.target>Per 3. pronun.</s></p><p type="margin"> <s><margin.target id="marg60"></margin.target>Per 17 huius.</s></p><p type="margin"> <s><margin.target id="marg61"></margin.target>Per 1. pron.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Hinc est quod super plano CB, DB est mensura <lb/>diuturnitatis motus in AB.</s></p></subchap2></subchap1><pb xlink:href="064/01/041.jpg"/><subchap1 n="20" type="proposition"><p type="head"> <s>PROPOSITIO XX. PROBL. XII.</s></p><subchap2 n="20" type="statement"><p type="main"> <s>Datis duobus planis diverse inclinatis lon­<lb/>gitudinis notae; & nota diuturnitate gra­<lb/>vis moti super uno, reperire diuturnita­<lb/>tem si moveatur super alio.<figure id="id.064.01.041.1.jpg" xlink:href="064/01/041/1.jpg"/></s></p></subchap2><subchap2 n="21" type="proof"><p type="main"> <s>Sint plana AB, CD inclinata, & sit data diu­<lb/>turnitas E plani AB.</s></p><p type="main"> <s>Oportet reperire diuturnitatem plani CD.</s></p><p type="main"> <s>Fiat AF, paralella, & aequalis datae CD, in qua <lb/>reperiatur punctum G quo perveniat grave, <lb/>tempore quo in B<arrow.to.target n="marg62"/>, unde E est etiam diuturnitas <lb/>spatij AG, quo dato, & spatio AF perquiratur <lb/>eias diuturnitas, quae sit H<arrow.to.target n="marg63"/>, & dico H esse <lb/>diuturnitatem quae grave descendit in CD.</s></p><p type="margin"> <s><margin.target id="marg62"></margin.target>Per 17. huius.</s></p><p type="margin"> <s><margin.target id="marg63"></margin.target>Per 9. huius.</s></p><p type="main"> <s>Quoniam E, H sunt diuturnitates gravium de­<lb/>scendentium in AG, seu AB, & AF, per con­<lb/>structionem, & AF est aequalis, & paralella <lb/>datae CD per constructionem, sunt etiam E, H <lb/>diuturnitates ipsarum AB, & CD<arrow.to.target n="marg64"/>, unde <lb/>reperta est diuturnitas ipsius CD. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg64"></margin.target>Per 3. pron.</s></p></subchap2></subchap1><pb xlink:href="064/01/042.jpg"/><subchap1 n="21" type="proposition"><p type="head"> <s>PROPOSITIO XXI. PROBL. XIII.</s></p><subchap2 n="21" type="statement"><p type="main"> <s>Datis duabus diuturnitatibus, quarum prior <lb/>sit gravis moti super plano dato longitu­<lb/>dinis notae, & dato alio plano diversimo­<lb/>de declinante; reperiendum est in eo pun­<lb/>ctum, quo grave perveniat in secunda <lb/>diuturnitate data.<figure id="id.064.01.042.1.jpg" xlink:href="064/01/042/1.jpg"/></s></p></subchap2><subchap2 n="22" type="proof"><p type="main"> <s>Dato plano declinante AB, super quo grave <lb/>A moveatur diuturnitate C, & dato alio <lb/>plano D declinationis quae sit dissimilis decli­<lb/>nationi datae AB; data itidem diuturnitate E.</s></p><p type="main"> <s>Oportet reperire in D punctum quo grave per­<lb/>veniat in diuturnitate E.</s></p><p type="main"> <s>Ducatur AF parallela ipsi D, in eaque reperia­<lb/>tur punctum F, quo grave perveniat tempore quo <lb/>in B<arrow.to.target n="marg65"/>, & praescribatur in eadem spatium AG per <lb/>quod moveatur in diuturnitate E<arrow.to.target n="marg66"/>, & fiat DH <lb/>aequalis ipsi AG, & dico H esse punctum quaesitum.</s></p><p type="margin"> <s><margin.target id="marg65"></margin.target>Per 17. huius.</s></p><p type="margin"> <s><margin.target id="marg66"></margin.target>Per 8. huius.</s></p><p type="main"> <s>Quoniam diuturnitates in AB, AF sunt aequales <lb/>per constructionem, & C, E sunt diuturnita­<lb/>tes super planis AF, AG per constructionem, <lb/>sunt etiam diuturnitates super AB, AG, & <lb/>proinde super DH ipsi AG aequali, & para<lb/>lellae, quod, etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/043.jpg"/><subchap1 n="22" type="proposition"><p type="head"> <s>PROPOSITIO XXII.</s></p><subchap2 n="22" type="statement"><p type="main"> <s>Data perpendiculari seu plano quomodoli­<lb/>bet inclinato diuturnitatis notae, & assi­<lb/>gnata ubivis quaecunque eius portione, re­<lb/>perire eius diuturnitatem.<figure id="id.064.01.043.1.jpg" xlink:href="064/01/043/1.jpg"/></s></p></subchap2><subchap2 n="23" type="proof"><p type="main"> <s>Data linea AB perpendiculari aut inclina­<lb/>ta, cuius, diuturnitas sit CD, dataque qua­<lb/>cunque eius portione EF.</s></p><p type="main"> <s>Quaerenda eius diuturnitas.</s></p><p type="main"> <s>Fiat CG diuturnitas AE, & CH diuturnitas <lb/>AF<arrow.to.target n="marg67"/>, GH est diuturnitas quaesita.</s></p><p type="margin"> <s><margin.target id="marg67"></margin.target>Per 5. aut 9. huius.</s></p><p type="main"> <s>Quoniam CH est diuturnitas AF per constr. ab <lb/>ea ablata CG diuturnitate AE per const. resi­<lb/>duum GH est diuturnitas portionis EF quod, <lb/>etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/044.jpg"/><subchap1 n="23" type="proposition"><p type="head"> <s>PROPOSITIO XXIII.</s></p><subchap2 n="23" type="statement"><p type="main"> <s>Duo gravia descendentia super planis diversa <lb/>ratione declinantibus, perveniunt ad idem <lb/>planum orizontale ea ratione, ut sit eadem <lb/>proportio inter diuturnitates, quae inter <lb/>dicta plana si ab eodem puncto ad idem <lb/>planum orizontale producta sint.<figure id="id.064.01.044.1.jpg" xlink:href="064/01/044/1.jpg"/></s></p></subchap2><subchap2 n="24" type="proof"><p type="main"> <s>Datis planis AB, AC declinantibus, ductis <lb/>ab eodem puncto A ad planum orizontale BC. <lb/> </s> <s>Dico quod diuturnitates gravium descendentium <lb/>per AB, AC sint ut AB ad AC.</s></p><p type="main"> <s>Fiat ut AC ad AB ita AB ad AD, ita ut grave <lb/>perveniat in D eodem tempore quo pervenit in B<arrow.to.target n="marg68"/>.</s></p><p type="margin"> <s><margin.target id="marg68"></margin.target>Per 17. huius.</s></p><p type="main"> <s>Quoniam est ut AD ad AC, ita quadratum tem­<lb/>poris AD ad quadratum temporis AC<arrow.to.target n="marg69"/>, & <lb/>tempora AD, AB sunt aequalia<arrow.to.target n="marg70"/>, & proinde <lb/>eorum quadrata; ergo ut AD ad AC ita qua­<lb/>dratum temporis AB, ad quadratum tempo­<lb/>ris AC<arrow.to.target n="marg71"/>, sed ut AD ad AC, ita quadra­<lb/>tum AB ad quadratum AC<arrow.to.target n="marg72"/>, ergo ut quadra­<lb/>tum temporis AB ad quadratum temporis AC, <lb/>ita quadratum AB ad quadratum AC, ergo <lb/>ut tempus AB ad tempus AC, ita AB ad AC<arrow.to.target n="marg73"/>. </s> <s>Quod fuit probandum.</s></p><p type="margin"> <s><margin.target id="marg69"></margin.target>Per Cor. 7. huius.</s></p><p type="margin"> <s><margin.target id="marg70"></margin.target>Per const.</s></p><p type="margin"> <s><margin.target id="marg71"></margin.target>Per 2. pronun.</s></p><p type="margin"> <s><margin.target id="marg72"></margin.target>Per 10. sexti.</s></p><p type="margin"> <s><margin.target id="marg73"></margin.target>Per 22. sexti.</s></p></subchap2></subchap1><pb xlink:href="064/01/045.jpg"/><subchap1 n="24" type="proposition"><p type="head"> <s>PROPOSITIO XXIV</s></p><subchap2 n="24" type="statement"><p type="main"> <s>Datis planis, & perpendiculari ad eadem li­<lb/>nea orizontali egressis, quae coeant infra in <lb/>eodem puncto, gravia super ipsis mota <lb/>procedunt ea ratione, ut sit eadem propor­<lb/>tion inter diuturnitates, quae inter longitu­<lb/>dines planorum, & dictam perpendicularem.<figure id="id.064.01.045.1.jpg" xlink:href="064/01/045/1.jpg"/></s></p></subchap2><subchap2 n="25" type="proof"><p type="main"> <s>Data sit linea orizontalis AB, in qua ini­<lb/>tium sumant plana declinantia AC, DC, <lb/>nec non perpendicularis BC coeuntia in puncto C.</s></p><p type="main"> <s>Dico quod diuturnitates gravium super ipsis mo­<lb/>torum, sunt ut AC, DC, BC.</s></p><p type="main"> <s>Ducatur CE paralella ipsi AB, & a puncto A du­<lb/>cantur paralellae ipsis CB, CD, & sint AE, AF.</s></p><p type="main"> <s>Quoniam diuturnitates super planis AF, AC, <lb/>sunt ut AF, AC<arrow.to.target n="marg74"/>, & super planis eisdem, & <lb/>perpendiculari AE, sunt ut AF, seu AC ad <lb/>AE<arrow.to.target n="marg75"/>, & AE, AF sunt paralellae ipsis CD, <lb/>CB, & eisdem aequales,<arrow.to.target n="marg76"/>, sequitur quod etiam <lb/>super AC, DC, BC diuturnitates sunt iuxta <lb/>proportiones longitudinum<arrow.to.target n="marg77"/>, Quod probandum fuit.</s></p><p type="margin"> <s><margin.target id="marg74"></margin.target>Per 23. huius.</s></p><p type="margin"> <s><margin.target id="marg75"></margin.target>Per 15. huius.</s></p><p type="margin"> <s><margin.target id="marg76"></margin.target>Per 33. prim.</s></p><p type="margin"> <s><margin.target id="marg77"></margin.target>Per 3. pron.</s></p></subchap2></subchap1><pb xlink:href="064/01/046.jpg"/><subchap1 n="25" type="proposition"><p type="head"> <s>PROPOSITIO XXV.</s></p><subchap2 n="25" type="statement"><p type="main"> <s>In circulo Orthogonaliter erecto, si a sum­<lb/>mitate ad puncta peripheriae ducantur pla­<lb/>na, quo tempore grave perpendiculariter <lb/>inde pervenit ad planum orizontale; si de­<lb/>scendat per dicta plana, eodem perveniet <lb/>respective ad quodlibet dictorum puncto­<lb/>rum peripheriae.<figure id="id.064.01.046.1.jpg" xlink:href="064/01/046/1.jpg"/></s></p></subchap2><subchap2 n="26" type="proof"><p type="main"> <s>Sit circulus cuius centrum B, & diameter AC <lb/>erectus super plano orizontali GC, & in eo <lb/>ducta sint plana declinantia a puncto A ad <lb/>puncta peripheriae DEF, & descendant gravia <lb/>super dicta plana, & perpendiculariter.</s></p><p type="main"> <s>Dico quod eodem tempore pervenient ad, D, E, F, C.</s></p><p type="main"> <s>Ducantur DC, EC, FC.</s></p><p type="main"> <s>Quoniam puncta praedicta sunt ea, in quae cadunt <lb/>perpendicularia ducta a puncto C in AD, AE, <lb/>AF<arrow.to.target n="marg78"/>, eo perveniunt gravia eodem tempore <lb/>quo in C<arrow.to.target n="marg79"/>. </s> <s>Quod probandum fuit.</s></p><p type="margin"> <s><margin.target id="marg78"></margin.target>Per 30. Tertij.</s></p><p type="margin"> <s><margin.target id="marg79"></margin.target>Per 16. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/047.jpg"/><subchap1 n="26" type="proposition"><p type="head"> <s>PROPOSITIO XXVI.</s></p><subchap2 n="26" type="statement"><p type="main"> <s>Sit in circulo erecto, a puncto inferiori ducan­<lb/>tur plana ad puncta peripheriae, & a dictis <lb/>punctis descendant gravia super dicta pla­<lb/>na eodem tempore quo a puncto supremo <lb/>descendit aliud grave perpendiculariter; <lb/>pervenient omnia eodem instanti ad di­<lb/>ctum punctum inferius.<figure id="id.064.01.047.1.jpg" xlink:href="064/01/047/1.jpg"/></s></p></subchap2><subchap2 n="27" type="proof"><p type="main"> <s>Sit circulus cuius diameter ABC erectus super <lb/>plano orizontali, quod tangat in C, & a C <lb/>ducantur plana CD, CE, & a punctis, E, D <lb/>gravia descendant super dicta plana, nec non, <lb/>& a puncto supremo A perpendiculariter.</s></p><p type="main"> <s>Dico quod eodem tempore perveniunt in C.<lb/></s></p><p type="main"> <s>A puncto A ducantur AF, AG paralellae ipsis <lb/>CE, CD, & ducantur AF, FC.</s></p><p type="main"> <s>Quoniam in triangulis AEC, AFC anguli al­<lb/>terni FAC, ACE sint aequales,<arrow.to.target n="marg80"/>, & anguli<lb/> <pb xlink:href="064/01/048.jpg"/>AFC, AEC sunt etiam aequales puta re­<lb/>cti<arrow.to.target n="marg81"/>, & basis AC communis, Triangula sunt <lb/>aequalia<arrow.to.target n="marg82"/>, & proinde AF est aequalis CE, quod <lb/>idem probabitur de reliquis, ergo cum AF, <lb/>CE, & reliquae sint paralellae, & aequales, gra­<lb/>via per CE, CD pervenient in C eodem tem­<lb/>pore, quo digressa ab A perveniunt ad puncta <lb/>FG, sed haec eodem tempore quo perpendicula­<lb/>riter pervenit in C<arrow.to.target n="marg83"/>, ergo etiam ea quae per <lb/>CE, CD. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg80"></margin.target>Per 29. primi.</s></p><p type="margin"> <s><margin.target id="marg81"></margin.target>Per 30. Tertij.</s></p><p type="margin"> <s><margin.target id="marg82"></margin.target>Per 16. primi.</s></p><p type="margin"> <s><margin.target id="marg83"></margin.target>Per 25. huius.</s></p><pb xlink:href="064/01/049.jpg"/><p type="main"> <s>POSTULATUM VII</s></p><p type="main"> <s>Ductis planis inclinatis, & linea perpen­<lb/>diculari inter binas paralellas orizon­<lb/>tales, Gravia super illis mota ubi perveni­<lb/>unt ad paralellam inferiorem habent aequa­<lb/>les velocitatis gradus; & proinde si ab in­<lb/>de infra sortiantur parem inclinationem, <lb/>aequevelociter moventur.</s></p><p type="main"> <s>Videtur probabile. </s> <s>Primo quia si diuturni­<lb/>tates sunt longitudinibus proportionales, ut <lb/>propositione 15. huius probatum fuit, credibile <lb/>est motus in fine esse aequales.</s></p><p type="main"> <s>Secundo. </s> <s>Argumento ducto ab experientia pen­<lb/>dulorum, quae quantumvis longiora, aut brevio­<lb/>ra, & proinde circa finem magis, aut minus in­<lb/>clinata, pariter ascendunt, si pariter descendant.</s></p><p type="main"> <s>Tertio. </s> <s>Quia videmus aquam per siphones rectos, <lb/>sive obliquos, seu inclinatos ductam, pariter <lb/>ascendere, si pariter descendat. </s> <s>Ceterum fa­<lb/>teor minorem evidentiam hoc postulatum caete­<lb/>ris praemissis prae se ferre, quae fuit causa quod <lb/>illud, ut in praefatione, segregaverim, & se­<lb/>quentia, alia methodo, tangendo fere tantum­<lb/>modo exposuerim, & a pluribus alijs proposi­<lb/>tionibus, quae hinc deduci facile possent, data <lb/>opera abstinuerim.</s></p></subchap2></subchap1><pb xlink:href="064/01/050.jpg"/><subchap1 n="27" type="proposition"><p type="head"> <s>PROPOSITIO XXVII. PROBL. XIV.</s></p><subchap2 n="27" type="statement"><p type="main"> <s>Dato gravi moto perpendiculariter per spa­<lb/>tium datum diuturnitate data, quod per­<lb/>ficiat motum super plano inclinato per <lb/>spatium itidem datum; perquirere in ipso <lb/>diuturnitatem.<figure id="id.064.01.050.1.jpg" xlink:href="064/01/050/1.jpg"/></s></p></subchap2><subchap2 n="28" type="proof"><p type="main"> <s>Moveatur grave A perpendiculariter per <lb/>spatium AB diuturnitate C, & perseve­<lb/>ret in motu super spatio BD in plano incli­<lb/>nato BD.</s></p><p type="main"> <s>Venanda est diuturnitas eius in ipso BD.</s></p><p type="main"> <s>Producatur DB donec concurrat cum AE orizon­<lb/>taliter ducta ab A in E, & fiat ut AB ad EB, <lb/>ita diuturnitas C ad diuturnitatem G, quae <lb/>idcirco erit diuturnitas ipsius EB<arrow.to.target n="marg84"/>, & sit H <lb/>quadratum diuturnitatis G, & fiat ut EB <lb/>ad ED, ita quadratum H ad aliud quod sit I a <lb/>cuius latere K, quod est diuturnitas ipsius <lb/>ED, ablata KL aequali G, erit LM reli­<lb/>quum diuturnitas BD quaesita.</s></p><p type="margin"> <s><margin.target id="marg84"></margin.target>* Est quarta tertij.</s></p><pb xlink:href="064/01/051.jpg"/><p type="main"> <s>Quoniam notum est triangulum AEB, cum no­<lb/>tus sit angulus AEB aequalis alterno EDF <lb/>inclinationis notae, & EAB rectus ex constru­<lb/>ctione, & notum latus AB ex hypotesi, notum <lb/>erit etiam latus EB, & quia diuturnitas in <lb/>plano BD est eadem ac si motus antecedens <lb/>esset per EB<arrow.to.target n="marg85"/>, EB & ED sunt in duplicata <lb/>ratione diuturnitatum G, K ex con­<lb/>structio­<lb/>ne; unde a K deducta KL aequali G ex constructione, remanet LM diuturnitas BD. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg85"></margin.target>Per 22 huius.</s></p><p type="main"> <s>Inde sequitur quod summa diuturnitatum C, & <lb/>LM, est diuturnitas totius ABD.**</s></p><p type="main"> <s>Eadem operatione pariter reperietur diuturni­<lb/>tas BD si BD sit perpendicularis, & AB <lb/>inclinata.</s></p><p type="main"> <s>Item si ambo sint plana inclinata.</s></p><p type="main"> <s>Ducta AD facile reperietur diuturnitas in ipsa <lb/>si fiat ut ED ad AD, ita K ad aliud per <lb/>21. huius.</s></p><pb xlink:href="064/01/052.jpg"/><p type="main"> <s>Ducto alio plano puta DN, reperietur eius <lb/>diuturnitas.<figure id="id.064.01.052.1.jpg" xlink:href="064/01/052/1.jpg"/></s></p><p type="main"> <s>Si fiat ut ED ad OD ita diuturnitas ipsius <lb/>ED puta L ad diuturnitatem OD, quae sit <lb/>P, deinde ut OD ad ON ita quadratum <lb/>diuturnitatis P ad aliud quadratum, cuius <lb/>Radix erit diuturnitas ipsius DN.</s></p><p type="main"> <s>Ex his patet quod si addantur plura plana ea­<lb/>dem ratione reperientur eius diuturnitates.</s></p><pb xlink:href="064/01/053.jpg"/><figure id="id.064.01.053.1.jpg" xlink:href="064/01/053/1.jpg"/><p type="main"> <s>Ex his itidem patet quod si in circulo dentur <lb/>plura, plana v.g. FA, AC, CB, & data sit <lb/>diuturnitas super diametro orizonti perpen­<lb/>diculari, dabitur diuturnitas cuiusvis dicto­<lb/>rum FA, AC, CT, & omnium simul.7*</s></p><p type="main"> <s>In super ex his facile cognosces esse breviorem, <lb/>diuturnitatem per AC, CB, simul, quam per <lb/>AB;8* nam ducta AE perpendiculari ad BC <lb/>productam in D ad orizontalem AD, diutur­<lb/>nitas motus in AC, super DB mensuratur per <lb/>EC<arrow.to.target n="marg86"/>, ergo addita CB, quae est eiusdem diutur­<lb/>nitatis, fuerit ne motus per AC an per DC<arrow.to.target n="marg87"/>, <lb/>tota EB erit mensura diuturnitatis in ACB, <lb/>sed AB mensurat diuturnitatem ipsius AB <lb/>respectu eiusdem DB<arrow.to.target n="marg88"/>, quae est maior quam <lb/>EB<arrow.to.target n="marg89"/>, maior ergo est diuturnitas in AB quam <lb/>in ACB.</s></p><p type="margin"> <s><margin.target id="marg86"></margin.target>Per 7. post.</s></p><p type="margin"> <s><margin.target id="marg87"></margin.target>** Est pars secunda quartae tertij.</s></p><p type="margin"> <s><margin.target id="marg88"></margin.target>*** Est Tertia tertij.</s></p><p type="margin"> <s><margin.target id="marg89"></margin.target>**** Est corol. quartae tertij.</s></p><p type="main"> <s>Eadem prorsus ratione probabitur citius grave <lb/>descendere per FA, AC, CB, simul, quam per <lb/>planum ductum ab F in B.9*</s></p><p type="main"> <s>In figura propositionis 27. si facto H quadrato <lb/>diuturnitatis G, fiat ML aequalis C, cui ad­<pb xlink:href="064/01/054.jpg"/>dita LK aequali G, fiat I quadratum MK, <lb/>& ut H ad I, ita EB ad ED; MK erit <lb/>diuturnitas ED, & ML diuturnitas BD <lb/>aequalis C. diuturnitas ipsius AB, unde diu­<lb/>turnitates in AB, & in BD aequales erunt.10*</s></p><p type="main"> <s>Et si BD esset fere Orizontalis, BE fieret longis­<lb/>sima, & quia EB ad ED est ut G ad tertiam <lb/>proportionalem ad G, & MK, haec tertia exce­<lb/>deret ipsam G fere duplo ipsius ML, seu C, ob <lb/>magnam diferentiam inter G, & C, ob quam <lb/>G esset fere aequalis ipsi MK, unde itidem E<lb/>D excederet EB fere duplo ipsius AB, & quo <lb/>BD esset magis orizontalis, eo BD propinquior <lb/>esset duplo AB.11*</s></p><p type="main"> <s>Ceterum ex hisce plura alia postmodum deduci <lb/>facile poterunt, haec vero in praesentia pauca <lb/>sufficere mihi visa sunt.</s></p></subchap2></subchap1></chap><pb xlink:href="064/01/055.jpg"/><chap><p type="main"> <s>DE MOTV <lb/>GRAVIVM <lb/>SOLIDORVM <lb/>LIBER SECVNDUVS <lb/> VBI DE IMPETV.</s></p><p type="main"> <s>LIBELLVM edidi octo ab <lb/>bine annis anno &longs;iquidem <lb/>1638 de motu &longs;olidorum, mox de liquidis editurus, quibus nimirum &longs;olida &longs;oli­ <lb/>dius &longs;truerent fundamen­ <lb/>tum.</s> <s>Hucu&longs;que di&longs;tuli, exi­ <lb/>&longs;timans hos itidem duos libros de &longs;olidis prae­ <lb/>mittendos; faciliorem &longs;iquidem vi&longs;i &longs;unt &longs;ter­ <lb/>nere viam ad illorum demon&longs;trationem cla­ <lb/>riorem.</s> <s>Quod eo libentius feci, quoniam &longs;e­ <lb/>ptimum po&longs;tulatum, quod inter principia, <lb/>connumerandum non videbatur, tanquam <lb/>minus euidens, decima huius propo&longs;itione <lb/>demon&longs;trare contigit; ex quo inde deducta, <pb xlink:href="064/01/056.jpg"/>&longs;eu potius leuiter tacta, libro &longs;equenti re­ <lb/>petere, & clarius explica re coactus mihi vi­ <lb/>&longs;us &longs;um.</s> <s>Quæ nihilomimus, citius perfici po­ <lb/>tui&longs;&longs;ent, ni pluribus litigijs, alijque negotijs <lb/>proprijs, & alienis, tum muneribus publicis <lb/>di&longs;tractus, litterarum &longs;tndia dimittere &longs;æpius <lb/>mihi opus fui&longs;&longs;et.</s> <s>Non ignoro litteris auide <lb/>deditos nu&longs;quam ijs obrui negotijs, quin horas <lb/>furtiuas quotidie reperiant, quibus di&longs;cipli­ <lb/>narum &longs;tudijs vacent: verum &longs;atis con&longs;tat in­ <lb/>tellectum libentius elaborare in nouis per di­ <lb/>&longs;cendis, &longs;eu aliorum partus ingeniorum in­ <lb/>quiras, &longs;eu (quod delectabilius longe e&longs;t) <lb/>noua proprio marte reperias, quam in iam <lb/>repertis po&longs;tmodum expoliendis, in quo ni­ <lb/>mirum labor ingens, nulla animi voluptas. <lb/></s> <s>Ex quo mirandum non e&longs;t &longs;iquid otij occupa­ <lb/>tiones permi&longs;&longs;erunt, meum ad noua potius pro­ <lb/>pen&longs;um ingenium, ea &longs;æpius intermi&longs;i&longs;&longs;e, que <lb/>ad opus perficiendum nece&longs;&longs;ario requireban­ <lb/>tur: quod cau&longs;a fuit non modo proca&longs;tinatio­ <lb/>nis, &longs;ed cur opus prodeat impolitum, po&longs;tre­ <lb/>ma vide licet lima deficiente; vnde, &longs;i ani­ <lb/>mo meo morem gerere volui&longs;&longs;em, ad huc &longs;ub <lb/>tenebris latitaret.</s> <s>Qualecunque &longs;it, tibi nunc <lb/>exhibere libuit, & priorem librum iterum edi, <lb/>allique alligari ad eorundem captum nece&longs;&longs;arium, <lb/>tu illud accipias, & excu&longs;es, & corrigas velim.</s></p></chap><pb xlink:href="064/01/057.jpg"/><chap type="bk"><subchap1 type="definition"><p type="head"> <s>DEFINITIONES</s></p><subchap2 type="definition"><p type="main"> <s>1. Motus dicitur aequabilis, si mobile fera­<lb/>tur per spatia, quae inter se sint ut <lb/>tempora, quibus conficiuntur.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>2. Impetus est vis, quia mobile est aptum progre­<lb/>di absque actione gravitatis, aut cuiusvis al­<lb/>terius rei.</s></p></subchap2></subchap1><subchap1 type="postulate"><p type="head"> <s>Petitio</s></p><subchap2 type="postulate"><p type="main"> <s>Impetus sunt ut spatia, quae eius virtute aequis <lb/>temporibus permeantur.</s></p></subchap2></subchap1><subchap1 type="postulate"><p type="head"> <s>Axiomata</s></p><subchap2 type="axiom"><p type="main"> <s>1. Pares causae producunt pares effectus.</s></p></subchap2><subchap2 type="axiom"><p type="main"> <s>2. In effectu procedente a duabus causis, ablata eius <lb/>portione proveniente ab una, reliquum erit <lb/>portio proveniens ab altera.</s></p></subchap2></subchap1><pb xlink:href="064/01/058.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA.</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Grave in motu naturali, sive perpendiculari, <lb/>sive inclinato, fertur sine ope gravitatis, <lb/>aequali tempore, per duplum spathuius praece­<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"/> <s>Dato gravi A naturaliter la­<lb/>to ab A ad B tempore ab, <lb/>cuius aequale sit tempus bc, & <lb/>spatium BC, sit duplum spathuius AB. <lb/></s> <s>Dico quod tempore bc fertur grave <lb/>sine ope gravitatis per spatium <lb/>aequale ipsi BC.</s></p><p type="main"> <s>Producatur AB, sumaturque portio <lb/>BD aequalis, & DE dupla lineae AB, & pro­<lb/>inde aequalis ipsi BC.</s></p><p type="main"> <s>Quoniam ope gravitatis A tempore ab fertur <lb/>in B per constructionem, tempore bc eadem <lb/>ope prodibit in D per spatium BD aequale A<lb/>B<arrow.to.target n="marg90"/>, at prodit in E<arrow.to.target n="marg91"/>, ergo fertur per DE du­<lb/>plum ipsius AB sine ope gravitatis, cui cum <lb/>sit aequalis BC per constructionem, constat, <lb/>quod sine ope gravitatis tempore bc fertur per <lb/>spatium aequale BC, quod etc.</s></p><p type="margin"> <s><margin.target id="marg90"></margin.target>Per axioma primum.</s></p><p type="margin"> <s><margin.target id="marg91"></margin.target>Per 3. primi huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium Primum</s></p><p type="main"> <s>Hinc sequitur quod si spatium AB sectum esset <lb/>in quatuor partes aequales, grave perficeret<pb xlink:href="064/01/059.jpg"/>primam tempore aequali illi quo conficit tres <lb/>reliquas, quia in fine primae acquisivit virtu­<lb/>tem, seu impetum, quo perficeret duas partes, <lb/>tertiam verum conficit eadem virtute qua per­<lb/>ficit primam. </s> <s>Quod pari ratione sequitur si <lb/>AE producatur, & in ea sumantur tres par­<lb/>tes aequales ipsi AE, quae tres conficientur tem­<lb/>pore ei aequali quo perficitur AE.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II</s></p><p type="main"> <s>Impetus autem non sumpsit initium in B, sed <lb/>prius, attamen cum mobile est in B ille impe­<lb/>tus qui simul cum gravitate tempore ab duxit <lb/>mobile ab A in B non est sufficiens tempore bc <lb/>aequali ab ducere illud ultra D per dictum pri­<lb/>mum Axioma, unde impetus ducens grave a <lb/>D in E eodem tempore bd necessario est is qui <lb/>est acquisitus per motum AB in puncto B.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium III</s></p><p type="main"> <s>Quoniam impetus de nouo acquisitus non <lb/>operatur seorsim ab impetu qui simul cum <lb/>gravitate duxit mobile ab A in B, sed eo­<lb/>dem prorsus tempore ducitur mobile non modo <lb/>ab impetu de novo acquisito in B, sed etiam, & <lb/>gravitate, & ab impetu qui continuo produ­<pb xlink:href="064/01/060.jpg"/><figure id="id.064.01.060.1.jpg" xlink:href="064/01/060/1.jpg"/>citur respondens illi qui duxit mobile ab A in <lb/>B, idcirco ipsum mobile a B in E fertur perpe­<lb/>tuo velocius, unde motus est velocior in E quem <lb/>fuerit in quolibet puncto superiori, & pro­<lb/>inde in E sortitum est impetum maiorem quam <lb/>habuerit prius, aptum ducere illud aequali tem­<lb/>pore per spatium duplum ipsius AE.</s></p></subchap2></subchap1><pb xlink:href="064/01/061.jpg"/><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II. PROBL. I.</s></p><subchap2 n="2" type="statement"><p type="main"> <s><figure id="id.064.01.061.1.jpg" xlink:href="064/01/061/1.jpg"/>Dato spatio per quod grave naturali­<lb/>ter ducatur virtute impetus solius sine <lb/>ope gravitatis, in dato tempore: repe­<lb/>rire eius portionem per quam duca­<lb/>tur eadem virtute in quavis portione <lb/>dicti temporis.</s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>Ducatur grave A per spatium AE <lb/>tempore ae, nec non per spatium <lb/>aequale EB duplum AE virtute impetus <lb/>acquisiti in E sine ope gravitatis tempore e<lb/>h aequale ipsi ae<arrow.to.target n="marg92"/> cuius temporis eh data sit <lb/>portio quaelibet, & sit primo portio immedia­<lb/>ta tempori ae, & sit eg.</s></p><p type="margin"> <s><margin.target id="marg92"></margin.target>Per pr. huius.</s></p><p type="main"> <s>Oportet reperire portionem spathuius EB, per quod <lb/>grave A ducatur, virtute impetus solius acqui­<lb/>siti in E, sine ope gravitatis, in dicta portione <lb/>temporis eg.</s></p><p type="main"> <s>Concipiantur tempora ae, eh, eg tanquam lineae <lb/>rectae metientes tempora ae, eh, eg, & fiat <lb/>ac tempus aequale tempori eg, & ut ae <lb/>ad ac, fiat AE ad AD<arrow.to.target n="marg93"/> ad quas fiat tertia <lb/>AC<arrow.to.target n="marg94"/>, ex quo AE, AC sunt in duplicata ratio­<lb/>ne temporunn ae, ac,<arrow.to.target n="marg95"/>. Fiat ut ae ad ag ita <lb/>AE ad AF<arrow.to.target n="marg96"/>, quibus tertia AG<arrow.to.target n="marg97"/>, ex quo AG, <lb/>AE sunt in duplicata ratione temporum ag, ae<arrow.to.target n="marg98"/>.</s></p><p type="margin"> <s><margin.target id="marg93"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg94"></margin.target>Per 11. sexti.</s></p><p type="margin"> <s><margin.target id="marg95"></margin.target>Per 10. def. quinti.</s></p><p type="margin"> <s><margin.target id="marg96"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg97"></margin.target>Per 11. sexti.</s></p><p type="margin"> <s><margin.target id="marg98"></margin.target>Per 10 def. 5.</s></p><p type="main"> <s><pb xlink:href="064/01/062.jpg"/>Fiat EH aequalis AC, et ab AG abla­<lb/>ta AH, residuo HG fiat aequalis EI.</s></p><p type="main"> <s>Dico EI esse portionem quaesitam.</s></p><p type="main"> <s>Quoniam AE est casus gravis A tempore ae per <lb/>supp. & AE, AC sunt in dupl. ratione tem­<lb/>porum ae, ac per constr. </s> <s>AC est casus gravis <lb/>tempore ac<arrow.to.target n="marg99"/>, & proinde EH aequalis AC est <lb/>casus tempore eg aequali ipsi ab si grave du­<lb/>ceretur per EH eadem prorsus virtute qua <lb/>ductum fuit per AC<arrow.to.target n="marg100"/>.</s></p><p type="margin"> <s><margin.target id="marg99"></margin.target>Per 3. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg100"></margin.target>Per axioma primum.</s></p><p type="main"> <s>Item quia AG, AE sunt in duplicata ratione tem­<lb/>porum ag, ae per constr., AG est casus tempo­<lb/>re ag<arrow.to.target n="marg101"/>, & proinde residuum EG est casus re­<lb/>sidui eg<arrow.to.target n="marg102"/>, dum tamen motus proveniat tam <lb/>e gravitate quam a quolibet impetu superaddi­<lb/>to, at EH probatum est esse casum itidem, eg <lb/>dum tamen grave ducatur ea solum virtute <lb/>qua ductum fuit per AC<arrow.to.target n="marg103"/>, ig, residuum HG <lb/>est spatium quod perficitur eodem tempore eg, <lb/>a solo impetu acquisito in E<arrow.to.target n="marg104"/>, quod est aequa­<lb/>le EI per constr., unde EI est spatium quaesitum.</s></p><p type="margin"> <s><margin.target id="marg101"></margin.target>Per 3. primi huius.</s></p><p type="margin"> <s><margin.target id="marg102"></margin.target>Per 19. Quinti.</s></p><p type="margin"> <s><margin.target id="marg103"></margin.target>Per axioma primum.</s></p><p type="margin"> <s><margin.target id="marg104"></margin.target>Per axioma secundum.</s></p><p type="main"> <s>Sit deinde portio temporis eb disiuncta ab ae, puta <lb/>gK, & sit rursus reperienda portio spathuius EB <lb/>per quod grave A ducatur vi solius impetus <lb/>in E acquisiti in dicta portione temporis gk: <lb/>reperto prius spatio EC respondenti tempori eg <lb/>immediato ipsi ae modo quo supra dictum <lb/>fuit; fiat ac tempus aequale tempori gK, & ut<pb xlink:href="064/01/063.jpg"/><figure id="id.064.01.063.1.jpg" xlink:href="064/01/063/1.jpg"/> ag ad ac fiat AG ad AD, ad quas tertia A<lb/>C; AG, AC erunt in duplicata ratione tem­<lb/>porum ag, ac. </s> <s>Item fiat ut ag ad aK ita AG <lb/>ad AL, quibus tertia AK: AK, AH erunt in <lb/>duplicata ratione temporum aK, ag; fiat GM <lb/>aequalis AC, & ab AK auferatur AM, & <lb/>residuo MK fiat aequale 1N, & eodem ratio­<lb/>cinio demonstrabitur IN esse spatium quae­<lb/>situm. </s> <s>Reperta est igitur portio quaesita, <lb/>quod etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/064.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO TERTIA.</s></p><subchap2 n="3" type="statement"><p type="main"> <s>In motu naturali gravium, spatia quae conficiun­<lb/>tur virtute impetus sine ope gravitatis sunt <lb/>inter se ut tempora quibus conficiuntur.<figure id="id.064.01.064.1.jpg" xlink:href="064/01/064/1.jpg"/></s></p></subchap2><subchap2 n="3" type="proof"><p type="main"> <s>Descendat grave A in E tempore ae, & tem­<lb/>pore eh aequali ae, ex solo impetu, sine ope <lb/>gravitatis, per spatium aequale EB, duplo ipsius <lb/>AE,<arrow.to.target n="marg105"/> & secetur EI portio dicti spathuius EB <lb/>quae sit aequalis spatio per quod duci debeat gra­<lb/>ve A tempore eg portione dicti temporis eh so­<lb/>la vi impetus acquisiti in E<arrow.to.target n="marg106"/>.</s></p><p type="margin"> <s><margin.target id="marg105"></margin.target>Per pr. huius.</s></p><p type="margin"> <s><margin.target id="marg106"></margin.target>Per 2. huius.</s></p><p type="main"> <s>Dico spatium EI ad spatium EB esse ut <lb/>tempus eg ad tempus eh.</s></p><p type="main"> <s>Percipiantur tempora ae, eh, eg tanquam rectae me­<lb/>tientes tempora ae, eh, eg, & reperiantur ut in <lb/>praecedenti puncta C, H, G, e, & describantur <lb/>quadrata ab, ad, bd, supra ae, ag, eg<arrow.to.target n="marg107"/>.</s></p><p type="margin"> <s><margin.target id="marg107"></margin.target>Per 46. primi.</s></p><pb xlink:href="064/01/065.jpg"/><p type="main"> <s>Quoniam AG, AE sunt in duplicata ratione <lb/>ad ag, ae per constr., & quadrata ad, ab <lb/>sunt pariter in duplicata ratione ad ag, ae,<arrow.to.target n="marg108"/> <lb/>erunt AG, AE ut quadrata ad, ab,<arrow.to.target n="marg109"/> & di­<lb/>videndo ut EG ad AE ita ad minus ab, hoc est <lb/>gnomon edf, ad ab.<arrow.to.target n="marg110"/> Pari ratione probabimus <lb/>ut AE ad EH esse quadrata ab, ad bd, & <lb/>proinde EG ad EH est ut gnomon edf ad <lb/>quadratum bd<arrow.to.target n="marg111"/> unde HG, ad EG, ut com­<lb/>plementa gb, bf ad gnomonem edf,<arrow.to.target n="marg112"/> at EG <lb/>ad AE sunt ut gnomon edf ad quadratum ab, <lb/>ut probatum est supra, ergo HG, seu EI <lb/>ipsi <lb/>aequalis per constr. ad AE est ut dicta comple­<lb/>menta gb, bf, ad quadratum ab,<arrow.to.target n="marg113"/> bisk seu <lb/>ut gb ad ab,<emph type="sup"/>1<emph.end type="sup"/> seu ut eg ad ae,m seu eh, ei <lb/>aequale per constr. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg108"></margin.target>Per 20. sexti.</s></p><p type="margin"> <s><margin.target id="marg109"></margin.target>Per 11. Quinti.</s></p><p type="margin"> <s><margin.target id="marg110"></margin.target>Per 17. Quinti.</s></p><p type="margin"> <s><margin.target id="marg111"></margin.target>Per 22. Quinti.</s></p><p type="margin"> <s><margin.target id="marg112"></margin.target>Per 19. Quinti.</s></p><p type="margin"> <s><margin.target id="marg113"></margin.target>Per 22. Quinti.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium Primum</s></p><p type="main"> <s>Si portio temporis eh non sit immediata tempori <lb/>ae sed ab ea seiuncta, puta in schemate propo­<lb/>sitionis secundae gK, reperto in EB spatio IN<pb xlink:href="064/01/066.jpg"/><figure id="id.064.01.066.1.jpg" xlink:href="064/01/066/1.jpg"/> ipsi gk, respondenten, eodem ratiocinio quo supra <lb/>probabitur spatium EB ad eius portionem IN <lb/>esse ut tempus eh ad eius portionem gK, quan­<lb/>doquidem qua ratione EI respondet tempori eg, <lb/>eadem EN respondet tempori eK, & proinde <lb/>reliquum IN respondet reliquo gK.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II</s></p><p type="main"> <s>Motus ab impetu proveniens est aequabilis.</s></p></subchap2></subchap1><pb xlink:href="064/01/067.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO IV.</s></p><subchap2 n="4" type="statement"><p type="main"> <s>In motu naturali impetus successive acquisi­<lb/>ti sunt ut tempora transacta.</s></p></subchap2><figure id="id.064.01.067.1.jpg" xlink:href="064/01/067/1.jpg"/><subchap2 n="4" type="proof"><p type="main"> <s>Dato gravi moto naturali motu per AC, tem­<lb/>pore ac, & per AB, tempore ab.</s></p><p type="main"> <s>Dico impetum seu velocitatem in B ad impetum <lb/>in C esse ut ab ad ac. </s> <s>Concipiantur tempora ab, ac tanquam lineae re­<lb/>ctae metientes tempora ab, ac. </s> <s>Fiat BD dupla ipsius AB mensura impetus in B <lb/>tempore ab, & CE dupla ipsius AC mensura <lb/>impetus in C tempore ac<arrow.to.target n="marg114"/>, & BF media inter <lb/>BD, CE<arrow.to.target n="marg115"/>.</s></p><p type="margin"> <s><margin.target id="marg114"></margin.target>k Per 25. Quinti.</s></p><p type="margin"> <s><margin.target id="marg115"></margin.target>l Per 22. Quinti & 43. pr.</s></p><p type="main"> <s>Quoniam AB, AC sunt in duplicata ratione <lb/>temporum ab, ac<arrow.to.target n="marg116"/>, BD, CE sunt pariter in <lb/>duplicata ratione eorundem temporum ab, ac<arrow.to.target n="marg117"/>, <lb/>sed BD, CE sunt etiam in duplicitata ratione <lb/>spatiorum BD, BF per constructionem, ergo BD, BF <lb/>sunt ut tempora ab, ac<arrow.to.target n="marg118"/>. Sed BD mensura <lb/>impetus in B tempore ab, est spatium per <lb/>quod percurrit mobile virtute solius impetus <lb/>acquisiti in B tempore ab per constructionem, erit igitur<pb xlink:href="064/01/068.jpg"/>BF spatium per quod percurret idem mobile <lb/>eadem virtute impetus acquisiti in B tempore <lb/>ac<arrow.to.target n="marg119"/>, at CE est spatium quod percurrit mobile <lb/>eodem tempore ac per constr. </s> <s>Igitur eodem tem­<lb/>pore ac mobile in C perficit spatium CE, & in <lb/>B perficit spatium BF; sed impetus sunt ut spa­<lb/>tia quae aequali tempore transiguuntur<emph type="sup"/>g<emph.end type="sup"/><arrow.to.target n="marg120"/>. Ergo <lb/>impetus in C, & B sunt ut CE ad BF spatia, <lb/>quae probatum est esse ut tempora ac, ab, unde <lb/>impetus in C & B sunt ut tempora ac, ab<arrow.to.target n="marg121"/>, <lb/>quod etc.</s></p><p type="margin"> <s><margin.target id="marg116"></margin.target>m Per 36. primi.</s></p><p type="margin"> <s><margin.target id="marg117"></margin.target>n Per 2. huius.</s></p><p type="margin"> <s><margin.target id="marg118"></margin.target>o Per primam defin.</s></p><p type="margin"> <s><margin.target id="marg119"></margin.target>Per primam huius.</s></p><p type="margin"> <s><margin.target id="marg120"></margin.target>Per 13 Sexti.</s></p><p type="margin"> <s><margin.target id="marg121"></margin.target>Per tertiam pr. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/069.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V.</s></p><subchap2 n="5" type="statement"><p type="main"> <s>In motu naturali gravium impetus successive <lb/>acquisiti sunt in subduplicata ratione spa­<lb/>tiorum transactorum.</s></p></subchap2><figure id="id.064.01.069.1.jpg" xlink:href="064/01/069/1.jpg"/><subchap2 n="5" type="proof"><p type="main"> <s>Iisdem positis.</s></p><p type="main"> <s>Dico impetus, seu velocitates in B, & in C <lb/>esse in subduplicata ratione spatiorum <lb/>AB, & AC.</s></p><p type="main"> <s>Quoniam impetus in B, & C sunt ut tempora ab, <lb/>ac transacta<arrow.to.target n="marg122"/>.</s></p><p type="margin"> <s><margin.target id="marg122"></margin.target>Per 11. Quinti.</s></p><p type="main"> <s>Sed tempora ab, ac sunt in subduplicata ra­<lb/>tione spatiorum AB, AC<arrow.to.target n="marg123"/>. </s> <s>Pariter impetus <lb/>in B, & in C sunt in subduplicata ratione <lb/>spatiorum AB, AC, quod etc.</s></p><p type="margin"> <s><margin.target id="marg123"></margin.target>Per 11 Quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/070.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>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"/> <s>Datis in perpendiculari quibuslibet pun­<lb/>ctis reperire impetus singulorum in­<lb/>ter se.</s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Data linea perpendiculari AB, & <lb/>in ea punctis C, D,</s></p><p type="main"> <s>Venandi impetus in C, D dum grave ab <lb/>A dimissum fertur per AB.</s></p><p type="main"> <s>Sit E media inter AC, AD, item fiat AF media <lb/>inter AC, AB.</s></p><p type="main"> <s>Dico impetus in C, D, B esse ut AC, AE, AF.</s></p><p type="main"> <s>Quoniam AE est media inter AC, AD per con­<lb/>structionem, AD, AC sunt in duplicata ratio­<lb/>ne rectarum AE, AC<arrow.to.target n="marg124"/>.</s></p><p type="margin"> <s><margin.target id="marg124"></margin.target>Per 3. huius.</s></p><p type="main"> <s>Ergo AC, AE metiuntur impetus in C & D<arrow.to.target n="marg125"/>.</s></p><p type="margin"> <s><margin.target id="marg125"></margin.target>Per pet. huius.</s></p><p type="main"> <s>Item quoniam AF est media inter AC, AB per <lb/>constructionem, AF, AC sunt in subduplicata <lb/>ratione rectarum AB, AC, igitur AC, AF <lb/>metiuntur impetus in C & B, quod etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/071.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>In quolibet puncto motus reperire spatium, <lb/>per quod mobile sit aptum duci sine ope <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"/><p type="main"> <s>Ducatur grave tempore ab a puncto B per <lb/>spatium aequale rectae BD sine ope gravi­<lb/>tatis ut in praecedenti.</s></p><p type="main"> <s>Oportet reperire in alio puncto ipsius motus, puta <lb/>C, spatium aequale ei, per quod ducetur sine ope <lb/>gravitatis eodem tempore ab.</s></p><p type="main"> <s>Sit ac tempus, per quod ducitur grave naturali­<lb/>ter motum ab A in C, & fiat CE dupla ad AC, & <lb/>secetur CE in F ea ratione, ut partes CF, FE <lb/>sint partibus ab, bc proportionales<arrow.to.target n="marg126"/>.</s></p><p type="margin"> <s><margin.target id="marg126"></margin.target>Per 11. Quinti.</s></p><p type="main"> <s>Dico CF spatium aequari illi, per quod ducetur­<lb/>grave digressum a C tempore ab.</s></p><p type="main"> <s>Qunoniam CF ad FE est ut ab ad bc per constructionem, <lb/>erit ut CE ad CF ita ac ad ab<arrow.to.target n="marg127"/>, & permutando <lb/>ut CE ad ac, ita CF ad ab<arrow.to.target n="marg128"/> at spatium aequa­<lb/>le CE perficitur tempore ac<arrow.to.target n="marg129"/> motu aequabili<arrow.to.target n="marg130"/>.</s></p><p type="margin"> <s><margin.target id="marg127"></margin.target>Per 4. huius.</s></p><p type="margin"> <s><margin.target id="marg128"></margin.target>Per 3. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg129"></margin.target>Per 10. def. Quinti.</s></p><p type="margin"> <s><margin.target id="marg130"></margin.target>Per 5. huius.</s></p><p type="main"> <s>Ergo spatium aequale CF conficitur tempore ab, quod etc.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Huic sequitur quod eodem tempore, puta ab, <lb/>grave ducitur per BD, & per CF.</s></p></subchap2></subchap1><pb xlink:href="064/01/072.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII.</s></p><subchap2 n="8" type="statement"><p type="main"> <s>Si lineae perpendicularis, & inclinata ab eo­<lb/>dem puncto digressae, per quas idem grave <lb/>naturaliter ducatur, secentur a recta norma­<lb/>lis ad inclinatam; impetus in punctis sectionis, <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"/><subchap2 n="8" type="proof"><p type="main"> <s>Sint rectae AB perpendicularis, & AC quomo­<lb/>documque; inclinata per quas grave naturaliter <lb/>ducatur, sectae a BD normali ad AC declinantem.</s></p><p type="main"> <s>Dico impetum in B ad impetum in D esse ut AB <lb/>ad AD.</s></p><p type="main"> <s>Fiat BE dupla AB mensura impetus in B, & DF <lb/>dupla AD mensura impetus in D<arrow.to.target n="marg131"/>.</s></p><p type="margin"> <s><margin.target id="marg131"></margin.target>Per 10. sexti.</s></p><p type="main"> <s>Quoniam grave ducitur per AB AD eodem <lb/>tempore<arrow.to.target n="marg132"/>. Ducitur etiam sine ope gravitatis eo­<lb/>dem tempore per spatia aequalia ipsis BE, DF<arrow.to.target n="marg133"/> <lb/>& proinde BE, DF sunt ut impetus in B & D<arrow.to.target n="marg134"/>.</s></p><p type="margin"> <s><margin.target id="marg132"></margin.target>Per 18. Quinti.</s></p><p type="margin"> <s><margin.target id="marg133"></margin.target>Per 16. Quinti.</s></p><p type="margin"> <s><margin.target id="marg134"></margin.target>Per pr. huius.</s></p><p type="main"> <s>At BE, DF sunt ut AB, AD per constr. quip­<lb/>pe earum duplae. </s> <s>Igitur AB, AD sun t ut im­<lb/>petus in B & D<arrow.to.target n="marg135"/> quod, etc.</s></p><p type="margin"> <s><margin.target id="marg135"></margin.target>Per cor. 3. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Impetus sive velocitas in B ad impetum in D <lb/>est ut AC ad AB.</s></p></subchap2></subchap1><pb xlink:href="064/01/073.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX.</s></p><subchap2 n="9" type="statement"><p type="main"> <s>Ductis a puncto superno perpendiculari, & <lb/>inclinata ad planum Orizontale, & a pun­<lb/>cto inferno perpendicularis ducta normali <lb/>ad inclinatam, impetus inclinatae in pun­<lb/>ctis, in quibus secat normalem, & orizon­<lb/>talem, sunt ut perpendicularis, & inclinata.</s></p></subchap2><figure id="id.064.01.073.1.jpg" xlink:href="064/01/073/1.jpg"/><subchap2 n="9" type="proof"><p type="main"> <s>Sint rectae AB AC ductae a puncto A ad orizon­<lb/>talem CB & a B ducatur normalis BD ad <lb/>AC.</s></p><p type="main"> <s>Dico impetum in D ad impetum in C esse ut AB <lb/>ad AC.</s></p><p type="main"> <s>Quoniam AC AD sunt in duplicata ratione im­<lb/>petus C ad impetum D<arrow.to.target n="marg136"/>.</s></p><p type="margin"> <s><margin.target id="marg136"></margin.target>Per pr. huius.</s></p><p type="main"> <s>Sunt itidem in duplicata ratione AC ad AB<arrow.to.target n="marg137"/>.</s></p><p type="margin"> <s><margin.target id="marg137"></margin.target>Per 14. pr. huius.</s></p><p type="main"> <s>Igitur impetus in C ad impetum in D sunt ut AC <lb/>AB<arrow.to.target n="marg138"/> quod, etc.</s></p><p type="margin"> <s><margin.target id="marg138"></margin.target>Per pr. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/074.jpg"/><subchap1 n="10" type="proposition"><p type="head"> <s>PROPOSITIO X.</s></p><subchap2 n="10" type="statement"><p type="main"> <s>Ductis a puncto superno per pendiculari, & <lb/>inclinata in punctis in quibus secant lineam <lb/>orizontalem sortiuntur impetus aequales.</s></p></subchap2><figure id="id.064.01.074.1.jpg" xlink:href="064/01/074/1.jpg"/><subchap2 n="10" type="proof"><p type="main"> <s>A puncto A superno ducatur AB perpendi­<lb/>cularis, & AC declinans ad BC Orizon­<lb/>talem.</s></p><p type="main"> <s>Dico, quod in B, & C sunt impetus aequales.</s></p><p type="main"> <s>Quoniam impetus in C ad impetum in D est ut <lb/>AC ad AB<arrow.to.target n="marg139"/>.</s></p><p type="margin"> <s><margin.target id="marg139"></margin.target>Per pet. huius.</s></p><p type="main"> <s>Item impetus in B ad impetum in D est pariter <lb/>ut AC ad AB<arrow.to.target n="marg140"/>.</s></p><p type="margin"> <s><margin.target id="marg140"></margin.target>Per 11. Quinti.</s></p><p type="main"> <s>Igitur impetus in C, & B sunt aequales<arrow.to.target n="marg141"/>. </s> <s>Quod <lb/>etc.</s></p><p type="margin"> <s><margin.target id="marg141"></margin.target>Per 5. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/075.jpg"/><subchap1 n="11" type="proposition"><p type="head"> <s>PROPOSITIO XI. PROBL. IV.</s></p><subchap2 n="11" type="statement"><p type="main"> <s>Datis pluribus lineis æqualibus ab eodem <lb/>puncto superno descendentibus, etiam si <lb/>una sit perpendicularis, reperire impetus <lb/>in fine ipsarum inter se.</s></p></subchap2><figure id="id.064.01.075.1.jpg" xlink:href="064/01/075/1.jpg"/><subchap2 n="11" type="proof"><p type="main"> <s>Datis aequalibus AB, AC, AD, inclinatis, <lb/>& AE perpendiculari oportet venari im­<lb/>petus inter se in B, C, D, E.</s></p><p type="main"> <s>Ducantur BF, CG, DH normales ad AE,<arrow.to.target n="marg142"/> & <lb/>proinde orizontales, & fiat AI media inter <lb/>AF, AG, & fiat AK media inter AF, AH, <lb/>item fiat AL media inter AF, AE.</s></p><p type="margin"> <s><margin.target id="marg142"></margin.target>Per 10. definit. quinti.</s></p><p type="main"> <s>Dico impetus in B, C, D, E esse inter se ut AF, <lb/>AI, AK, AL.</s></p><p type="main"> <s>Quoniam impetus in B, & F sunt aequales nec <lb/>non in CL, & in DH<arrow.to.target n="marg143"/>, & impetus in F, G, <lb/>H, E sunt ut AF, AI, AK, AL<arrow.to.target n="marg144"/>,</s></p><p type="margin"> <s><margin.target id="marg143"></margin.target>Per 16. Quinti.</s></p><p type="margin"> <s><margin.target id="marg144"></margin.target>Per 9. huius.</s></p><p type="main"> <s>Igitur impetus in B, C, D, E, sunt ut AF, AI, <lb/>AK, AL, Quod etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/076.jpg"/><subchap1 n="12" type="proposition"><p type="head"> <s>PROPOSITIO XII</s></p><subchap2 n="12" type="statement"><p type="main"> <s>Ductis pluribus lineis diversi mode inclinatis, & <lb/>etiam perpendiculari, quae ab eadem li­<lb/>nea Orizontali terminentur in idem pun­<lb/>ctum inferius; ibi sortiuntur impetus aequales.<figure id="id.064.01.076.1.jpg" xlink:href="064/01/076/1.jpg"/></s></p></subchap2><subchap2 n="12" type="proof"><p type="main"> <s>Sint lineae BD CD diversimode inclinatae, & AD <lb/>perpendicularis, ductae a linea Orizontali AC <lb/>ad punctum inferius D. </s> <s>Dico gravia a punctis <lb/>A B C digressa, & in eis lata, in D sortiri im­<lb/>petus aequales.</s></p><p type="main"> <s>Fiat DEF parallela ad AC<arrow.to.target n="marg145"/>, & proinde ori­<lb/>zontalis, ad quam dimittantur perpendicula­<lb/>res BE CF<arrow.to.target n="marg146"/>.</s></p><p type="margin"> <s><margin.target id="marg145"></margin.target>Per cor. 8. huius.</s></p><p type="margin"> <s><margin.target id="marg146"></margin.target>Per 11. Quinti.</s></p><p type="main"> <s>Quoniam gravia ducta per AD, BE, CF in DEF <lb/>habent impetus aequales, quia omnia paria<arrow.to.target n="marg147"/>, <lb/>& gravia ducta per BD, BE in DE habent im­<lb/>petus aequales, item per CD, CF in DF habent <lb/>impetus aequales<arrow.to.target n="marg148"/> sequitur quod etiam ducta <lb/>per AD, BD, CD sortita sunt in D impetus <lb/>aequales. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg147"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg148"></margin.target>Per 10. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Hinc sequitur, quod si ABC non sit linea, sed planum <lb/>Orizontale, item loco puncti D sint plura puncta, <lb/>dummodo in plano Orizontali; gravia in punctis <lb/>D habebunt impetus aequales.</s></p></subchap2></subchap1><pb xlink:href="064/01/077.jpg"/><subchap1 n="13" type="proposition"><p type="head"> <s>PROPOSITIO XIII. PROBL. V.</s></p><subchap2 n="13" type="statement"><p type="main"> <s>Datis gravibus descendentibus per perpendi­<lb/>cularem, & declinantem reperire rationes im­<lb/>petus in punctis datis.<figure id="id.064.01.077.1.jpg" xlink:href="064/01/077/1.jpg"/></s></p></subchap2><subchap2 n="13" type="proof"><p type="main"> <s>Descendat grave per AC per pendicularem , <lb/>& AB declinantem, & dentur puncta B, C.</s></p><p type="main"> <s>Reperire proportionem impe­<lb/>tus in B ad impetum in C.</s></p><p type="main"> <s>Ducatur BD normalis ad AC<arrow.to.target n="marg149"/>, & fiat AE <lb/>media inter AC, AD<arrow.to.target n="marg150"/>, Dico impetum in C ad <lb/>impetum in B esse ut AE ad AD.</s></p><p type="margin"> <s><margin.target id="marg149"></margin.target>Per 6. huius.</s></p><p type="margin"> <s><margin.target id="marg150"></margin.target>Per 31. primi.</s></p><p type="main"> <s>Quoniam impetus in C ad impetum in D est ut <lb/>AE ad AD<arrow.to.target n="marg151"/>, & impetus in D & B sunt aequa­<lb/>les<arrow.to.target n="marg152"/>, ergo impetus in C ad impetum in B est <lb/>ut AE ad AD, Quod etc.<figure id="id.064.01.077.2.jpg" xlink:href="064/01/077/2.jpg"/></s></p><p type="margin"> <s><margin.target id="marg151"></margin.target>Per 13. primi.</s></p><p type="margin"> <s><margin.target id="marg152"></margin.target>Per axioma primum.</s></p></subchap2><pb xlink:href="064/01/078.jpg" pagenum="78"/><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Eodem pacto reperies impetus in planis ut­<lb/>cumque declinantibus ductis perpendicula­<lb/>ribus ad AC.</s></p></subchap2></subchap1></chap><pb xlink:href="064/01/079.jpg"/><chap type="bk"><p type="main"> <s>DE MOTV<lb/>GRAVIVM<lb/>SOLIDORVM<lb/>LIBER TERTIVS.<lb/>VBI DE MOTV SVPER<lb/>PLVRIBVS PLANIS<lb/>DIVERSIMODE INCLINATIS.</s></p><subchap1 type="preface"><subchap2 type="preface"><p type="main"> <s>Ex libro secundo praecedenti con­<lb/>stat, mobile dum movetur fieri ap­<lb/>tum ex se moveri, quatenus post <lb/>priorem motum ei tribuitur, & im­<lb/>primitur quaedam virtus, seu vis, a qua fit <lb/>aptum duci, sine alicuius ope, ea velocitate qua <lb/>movebatur, dum illa virtus imprimebatur, & <lb/>proinde motu aequabili; quae virtus dicitur Im­<lb/>petus, differens solum fortasse a velocitate, quia <lb/>impetus sit velocitas in actu primo, ita ut ali­<lb/>quo pacto impetus sit causa velocitatis; conve­<lb/>niunt tamen, quatenus velocitates sunt ut spa­<lb/>tia quae mobilia aequali tempore permeant, <lb/>impetus vero ut spatia quae virtute ipsius im­<pb xlink:href="064/01/080.jpg"/>petus sunt apta, & in potentia proxima per­<lb/>meare, & de facto permeant ni impedimen­<lb/>tum aliquod obijciatur, secus enim effectus <lb/>causae non responderet. </s> <s>Porro ex impe­<lb/>tu provenit quod missilia quaelibet, a mo­<lb/>tore velociter ducta, deficiente motoris actio­<lb/>ne, nihilominus a solo impetu ferantur, quod <lb/>in proiectis quotidie experimur. </s> <s>De quibus <lb/>locus postularet ut aliquid agerem, ni via <lb/>quam eorum motu conficiunt, me adhuc late­<lb/>ret; quamvis non ignorem viris oculatissimis <lb/>visam esse parabolicam. </s> <s>Cum illis igitur sup­<lb/>pono proiecta a motore seiuncta, motu du­<lb/>plici moveri, nimirum ab impetu, aequabili <lb/>motu, eadem prorsus directe via qua a motore <lb/>novissime ducta fuerant, & itidem a gravitate <lb/>deorsum, & proinde motu mixto secundum <lb/>quamdam lineam curvam mihi ignotam, <lb/>quamhoc argumento ducti parabolicam ar­<lb/>bitrantur.<figure id="id.064.01.080.1.jpg" xlink:href="064/01/080/1.jpg"/></s></p><p type="main"> <s>Prohuiusciatur missile A versus D motu violento <lb/>quo virtute impetus temporibus aequalibus <lb/>conficiat aequalia spatia AB, BC, CD, & in<pb xlink:href="064/01/081.jpg"/>priori tempore, vi gravitatis descendat per <lb/>spatium aequale AE, quod sit BF, motu mix­<lb/>to describet curvam AF; ducatur mox ab <lb/>impetu eodem quo prius tramite, ab F ver­<lb/>sus G, unde si moveretur eo simplici motu <lb/>violento, in tantundem temporis adiret ip­<lb/>sum G, at quoniam urget etiam gravitas, <lb/>ducitur in H, ita ut GH sit triplum ipsius <lb/>AE, & proinde CH ad BF sit in duplicata <lb/>ratione AC ad AB, describens motu mixto <lb/>curvam FH, & demum eadem ratione du­<lb/>citur in I. </s> <s>Probant puncta AF HI esse in­ <lb/>parabola, per 20 primi A poll. quoniam <lb/>quadrata rectarum AC, AB ordinatim ap­<lb/>plicatarum, seu eis aequalium, sunt ut CH, BF <lb/>ab eis ex diametro praecisae, seu ut eis aequa­<lb/>les. </s> <s>At vero mihi quidem, contra id quod sup­<lb/>ponitur, apparet proiectum descendere mi­<lb/>nori celeritate, quam si a sola ducatur grav­<lb/>itate, & libere dimissum, celerius solum <lb/>attingere, quam orizontaliter latum. </s> <s>Insu­<lb/>per si aequis temporibus proiectum conficit <lb/>curvas AF, FH, HI, successive longiores <lb/>motus est successive velocior, quippe maius <lb/>spatium aequo tempore permeat, unde si vis pro­<lb/>huiuscientis provenit a maiori velocitate, ictus <lb/>eo est validior, quo missile longius a prohuius­<lb/>ciente distat; contra id quod quotidie experi­<pb xlink:href="064/01/082.jpg"/>mur, nec sit tardior ab aeris resistentia, quam <lb/>gravia deorsum mota persentirent, unde <lb/>quo graviora, celerius descenderent; quod <lb/>experientiae repugnat. </s> <s>Sed quia adducere <lb/>inconveniens non est solvere argumentum, <lb/>eius fallaciam pro viribus detegere conabor. <lb/></s> <s>Dum supponitur ab impetu duci perpetuo <lb/>mobile iuxta orizontalem AD, ego equi­<lb/>dem verum esse censeo, ubi mobile unico so­<lb/>lum violento motu ducatur; sed quia fertur <lb/>motu mixto, ab impetu nimirum, & a gravi­<lb/>tate secundum curvam AFH, quemadmodum <lb/>proiectum, a funda circumlatum, sibi dimis­<lb/>sum fertur per tangentem curvae a funda <lb/>descriptae, ita pariter censendum est, quo­<lb/>tiescumque orizontaliter latum pervenit <lb/>in H, non amplius dirigi secundum rectam <lb/>orizontalem HL, sed secundurn contingen­<lb/>tem ipsam curvam FH, fuerit ne ea para­<lb/>bola nec ne, quae contingens sit HK; unde <lb/>proiectum ab H digressum, motu violento, <lb/>remota gravitate, tenderet non in L, sed in <lb/>K; & proinde motu mixto tanto inferius <lb/>puncto L, quanta est recta LK, puta in M, de­<lb/>scribens curvam non HI, sed HM; at M non est <lb/>in parabola, ut facile demonstrari posset ex ea­<lb/>dem 20. primi Apollon. cum DM maior quam DI, <lb/>& BF non sint in duplicata ratione ordina­<pb xlink:href="064/01/083.jpg"/>tim aplicatarum AD, AB. </s> <s>Ex quo satis con­<lb/>stare existimo proiectum suo moto parabo­<lb/>lam non describere, quod probandum pro­<lb/>posueram. </s> <s>De quibus proiectis aliquid in­ <lb/>sequentibus addam fortasse ubi occasio <lb/>tulerit. </s> <s>Reliquum est quod hoc tertio <lb/>libro repetam ea quae in calce libri prio­<lb/>ris dicta fuere, sed parum accurate, quippe <lb/>pendentia ab eo septimo postulato, non satis <lb/>tunc fidem merente, in praesentia vero deci­<lb/>ma secundi huius, ut alibi dixi, ni fallor de­<lb/>monstratum. </s> <s>Interim ibi in notis marginali­<lb/>bus adnotari volui quem locum in hoc ter­<lb/>tio libro sortiantur.</s></p></subchap2></subchap1><pb xlink:href="064/01/084.jpg"/><subchap1 type="postulate"><p type="head"> <s>PETITIONES</s></p><p type="main"> <s>PRIMA</s></p><p type="main"> <s>Peripheria circuli concipiatur tanquam <lb/>constans plurimis, seu mavis infinitis <lb/>lineis rectis.</s></p><p type="main"> <s>SECUNDA</s></p><p type="main"> <s>Mobile naturaliter motum caeteris pari­<lb/>bus, quo longius distat a puncto quie­<lb/>tis sortitur maiorem impetum, & velocius <lb/>movetur.</s></p></subchap1><pb xlink:href="064/01/085.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA.</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Si grave perpendiculariter ductum perse­<lb/>veret in motu super plano declinante; pro­<lb/>dibit eadem velocitate, ac si motus praece­<lb/>dens fuisset cum eadem declinatione, ini­<lb/>tio ducto ab eodem plano Orizontali.<figure id="id.064.01.085.1.jpg" xlink:href="064/01/085/1.jpg"/></s></p></subchap2><subchap2 n="1" type="proof"><p type="main"> <s>Ducatur grave perpendiculariter per AB, & <lb/>perseveret in motu super BE declinante.</s></p><p type="main"> <s>Dico, quod fertur per BE eadem velocitate ac si <lb/>cepisset moveri in D; quod sit ad libellam ipsius A.</s></p><p type="main"> <s>Quoniam in B sortitum est eumdem impetum <lb/>ductum per AB, ac si latum fuisset per DB<arrow.to.target n="marg153"/>.</s></p><p type="margin"> <s><margin.target id="marg153"></margin.target>Per 12. secundi huius.</s></p><p type="main"> <s>Ergo per BE ducitur ab eadem virtute seu vi, <lb/>ac si motus initiuum fuisset in D, quippe ubique <lb/>ducitur a gravitate, & ab impetu in B, & pro­<lb/>inde fertur eadem velocitate. </s> <s>Quod etc.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium primum.</s></p><p type="main"> <s>Si initium motus fuisset per lineam declinantem, <lb/>& demum per perpendicularem, seu declinantem <lb/>diversa inclinatione, idem probabitur eadem ratione.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Hinc sequitur, quod impetus in E est idem si <lb/>motus fuerit per ABE, ac si fuisset per DE.</s></p></subchap2></subchap1><pb xlink:href="064/01/086.jpg"/><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II.</s></p><subchap2 n="2" type="statement"><p type="main"> <s>Grave ductum perpendiculariter per spatium <lb/>datum diuturnitate data, perseveret in <lb/>motu super plano inclinato; perquirere in <lb/>eo motum in data diuturnitate.<figure id="id.064.01.086.1.jpg" xlink:href="064/01/086/1.jpg"/></s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>Ducatur grave A perpendiculariter per AB <lb/>diuturnitate quae sit AB, & perseveret <lb/>in motu super BD plano inclinationis notae.</s></p><p type="main"> <s>Venandus ibi motus in dicta diuturnitate AB.</s></p><p type="main"> <s>Producatur BD in C donec concurrat cum AC <lb/>orizontaliter ducta ab A ad C. </s> <s>Erit BC diu­<lb/>turnitas ipsius BC<arrow.to.target n="marg154"/>.</s></p><p type="margin"> <s><margin.target id="marg154"></margin.target>Per 15. primi huius.</s></p><p type="main"> <s>Fiat BE aequalis AB, & CD tertia ad CB, CE<arrow.to.target n="marg155"/>.</s></p><p type="margin"> <s><margin.target id="marg155"></margin.target>Per 11. sexti.</s></p><p type="main"> <s>Dico BD esse quaesitum, nempe spatium transa­<lb/>ctum diuturnitate AB.</s></p><p type="main"> <s>Quoniam CE est diuturnitas CD<arrow.to.target n="marg156"/>, & CB est diu­<lb/>turnitas motus per eundem CB ut supra pro­<lb/>batum fuit.</s></p><p type="margin"> <s><margin.target id="marg156"></margin.target>Per 7. pr. huius.</s></p><p type="main"> <s>Erit BE diuturnitas BD stante motu praecedenti <lb/>per BC<arrow.to.target n="marg157"/>.</s></p><p type="margin"> <s><margin.target id="marg157"></margin.target>Per 19. quinti.</s></p><p type="main"> <s>Et pariter si fuerit per AB, BE est diuturni­<lb/>tas motus per BD<arrow.to.target n="marg158"/>.</s></p><p type="margin"> <s><margin.target id="marg158"></margin.target>Per pr. huius.</s></p><pb xlink:href="064/01/087.jpg"/><p type="main"> <s>At AB est aequalis ipsi BE per constructionem.</s></p><p type="main"> <s>Ergo motus per BD fit diuturnitate AB. </s> <s>Quod <lb/>etc.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Hinc sequitur, quod in quolibet puncto infra <lb/>B est par impetus, fuerit ne motus per C<lb/>D aut per ABD, cum fuerit par impetus in B<arrow.to.target n="marg159"/>.</s></p><p type="margin"> <s><margin.target id="marg159"></margin.target>Per 12. secundi huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Quotiescunque CE est media inter CB, CD, <lb/>etiamsi motus praecedens fuerit per AB; <lb/>BE est diuturnitas motus per BD.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium III.</s></p><p type="main"> <s>Idem sequitur etiamsi AB noni esset perpendicu­<lb/>laris, nam probatur eodem pacto.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium IV.</s></p><p type="main"> <s>Sequitur etiam, quod si datis AB, & CB, <lb/>fiat AB lineae aequalis BE, & ad CB, CE <lb/>fiat tertia CD; mobile cadens aC, seu ab A, <lb/>movebitur super BD aequali tempore quo per AB.</s></p><p type="main"> <s>Et notandum pr. quod BD semper excedit du­<lb/>plum ipsius AB, quia excedit duplum rectae BE.</s></p><pb xlink:href="064/01/088.jpg"/><p type="main"> <s>Nota secundo quod quo AC est longior, & proinde <lb/>quo BD magis accedit ad orizontalem DE fit <lb/>semper proximior longitudini EB.</s></p><p type="main"> <s>Nota tertio quod si AC sit fere infinita, ex quo <lb/>BD fere Orizontalis, DE insensibiliter differt <lb/>ab EB, & proinde DB erit dupla ipsius AB, <lb/>seu ab eius dupla insensibiliter differens.</s></p><p type="main"> <s>Et quia in BD tali casu gravitas insensibiliter <lb/>agit, quippe cum grave insensibiliter descendat, <lb/>motus erit fere uniformis, & proinde par ve­<lb/>locitas in BED.</s></p><p type="main"> <s>Ex quo, etiam apparet velocitas inquocunque <lb/>puncto descensus, puta in B; nam est talis, ut <lb/>mobile ubi non agat gravitas, sit aptum duci <lb/>per spatium duplum eius, per quod fuerit de­<lb/>scensus, & paulo amplius.</s></p></subchap2></subchap1><pb xlink:href="064/01/089.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO III</s></p><subchap2 n="3" type="statement"><p type="main"> <s>Ducto gravi super plano inclinato, & in­<lb/>de perpendiculariter; perquirere eius mo­<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"/><p type="main"> <s>Ducatur grave super AB incli­<lb/>niationis notae, diuturnitate AB <lb/>data, & inde perpendiculariter, per <lb/>BD; venari motum perpendicularem <lb/>in diuturnitate AB.</s></p><p type="main"> <s>Producatur DB, donec concurrat cum AC <lb/>orizontaliter ducta in C, et sit BC <lb/>diuturnitas motus per BC<arrow.to.target n="marg160"/>. Fiat <lb/>BE aequalis AB, & CD tertia ad CB, CE<arrow.to.target n="marg161"/>.</s></p><p type="margin"> <s><margin.target id="marg160"></margin.target>Per 15. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg161"></margin.target>Per 11. sexti.</s></p><p type="main"> <s>Dico BD esse quaesitum.</s></p><p type="main"> <s>Quoniam CE est diuturnitas CD<arrow.to.target n="marg162"/>, erit BE <lb/>diuturnitas BD, si motus præcedens fuerit per <lb/>CB; at pariter si per AB<arrow.to.target n="marg163"/>. </s> <s>Ergo diuturni­<lb/>tate AB aequali BE pervenit in D. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg162"></margin.target>Per 7. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg163"></margin.target>Per pr. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Hinc sequitur ut in praecedenti, quod impetus <lb/>infra B idem est, fuerit ne motus praecedens <lb/>per CD, ac per ABD.</s></p></subchap2></subchap1><pb xlink:href="064/01/090.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO IV</s></p><subchap2 n="4" type="statement"><p type="main"> <s>Dato gravi moto perpendiculariter per spa­<lb/>tium datum, diuturnitate data, quod per­<lb/>ficiat motum super plano declinante, per <lb/>spatium itidem datum; Perquirenda in ip­<lb/>so diuturnitas.<figure id="id.064.01.090.1.jpg" xlink:href="064/01/090/1.jpg"/></s></p></subchap2><subchap2 n="4" type="proof"><p type="main"> <s>Moveatur grave per AB perpendiculariter <lb/>diuturnitate data, quae sit eadem AB, inde <lb/>super planum inclinatum BD.</s></p><p type="main"> <s>Perquirenda est diuturnitas motus per BD, & per ABD.</s></p><p type="main"> <s>Fiat CE media inter CB, CD, & AF nor­<lb/>malis ad BD productam usquequo concurrat <lb/>cum orizontali AC.</s></p><p type="main"> <s>Dico BE esse diuturnitatem per motus BD, & <lb/>FE esse diuturnitatem motus per ABD.</s></p><p type="main"> <s>Quoniam nota est diuturnitas CB<arrow.to.target n="marg164"/>, & nota est <lb/>EC per constructionem, nota est etiam BE diu­<lb/>turnitas motus per BD, si motus praecedens fue­<lb/>rit per CB; at idem est si fuerit per AB<arrow.to.target n="marg165"/>.</s></p><p type="margin"> <s><margin.target id="marg164"></margin.target>Per 15. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg165"></margin.target>Per pr. huius.</s></p><p type="main"> <s>Ergo EB est diuturnitas motus per BD; At <lb/>FB est diuturnitas motus per AB<arrow.to.target n="marg166"/>. </s> <s>Igitur <lb/>FE est diuturnitas motus per ABD. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg166"></margin.target>Per Co. 19. pr. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Idem sequitur eadem ratione, si AB non sit <lb/>perpendicularis.</s></p></subchap2></subchap1><pb xlink:href="064/01/091.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V</s></p><subchap2 n="5" type="statement"><p type="main"> <s>Data diuturnitate in plano perpendiculari <lb/>motus gravis, quod perseveret moveri super <lb/>plano declinante; & data super eo diutur­<lb/>nitate, reperire longitudinem.<figure id="id.064.01.091.1.jpg" xlink:href="064/01/091/1.jpg"/></s></p></subchap2><subchap2 n="5" type="proof"><p type="main"> <s>Ducatur grave perpendiculariter per AB diu­<lb/>turnitate C, & demum super plano incli­<lb/>nato BD, & data sit diuturnus E.</s></p><p type="main"> <s>Perquirenda sit longitudo super BD quam grave <lb/>conficiat diuturnitate E.</s></p><p type="main"> <s>Fiat ut C ad E ita AB ad BF<arrow.to.target n="marg167"/>, unde si AB <lb/>concipiatur tanquam diuturnitas motus super <lb/>AB, erit BF diuturnitas motus super BD. <lb/></s> <s>Producatur FB donec concurrat cum A G ori­<lb/>zontaliter ducta in G. </s> <s>Et fiat CD tertia pro­<lb/>portionalis ad GB, GF<arrow.to.target n="marg168"/>.</s></p><p type="margin"> <s><margin.target id="marg167"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg168"></margin.target>Per 11. sexti.</s></p><p type="main"> <s>Dico BD esse longitudinem quaesitam.</s></p><p type="main"> <s>Quoniam AB est diuturnitas ipsius AB per sup­<lb/>pos; GB erit diuturnitas ipsius GB<arrow.to.target n="marg169"/>, at GF <lb/>est diuturnitas ipsius GD<arrow.to.target n="marg170"/>, igitur residuum BF <lb/>est diuturnitas BD. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg169"></margin.target>Per 15. primi huius.</s></p><p type="margin"> <s><margin.target id="marg170"></margin.target>Per 3. pr. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium.</s></p><p type="main"> <s>Grave prodibit per AB, BD aequis tempo­<lb/>ribus si diuturnitas E fiat aequalis diu­<lb/>turnitati C.</s></p></subchap2></subchap1><pb xlink:href="064/01/092.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>PROPOSITIO VI.</s></p><subchap2 n="6" type="statement"><p type="main"> <s>Moto gravi super pluribus planis diversimo­<lb/>de inclinatis, venari diuturnitates in quo­<lb/>libet eorurn.<figure id="id.064.01.092.1.jpg" xlink:href="064/01/092/1.jpg"/></s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Ducatur grave per AB diuturnitate data, <lb/>quae sit eadem AB; inde a B in D, & a D <lb/>in H. </s> <s>Venanda est diuturnitam motus per DH.</s></p><p type="main"> <s>Producatur DB in E donec concurrat cum <lb/>AG orizontaliter ducta. </s> <s>Item producatur H<lb/>D donec concurrat cum eadem AG. </s> <s>Fiat <lb/>EC media inter EB, ED<arrow.to.target n="marg171"/>. </s> <s>Fiat itidem GF <lb/>media inter GD, GH.</s></p><p type="margin"> <s><margin.target id="marg171"></margin.target>Per 13. Sexti.</s></p><p type="main"> <s>Dico DF esse diuturnitate motus per DH.<arrow.to.target n="marg172"/></s></p><p type="margin"> <s><margin.target id="marg172"></margin.target>Per 7. pr. huius.</s></p><p type="main"> <s>Quoniam DF est diuturnitas motus per DH <lb/>etiamsi motus praecedens fuerit per ED<arrow.to.target n="marg173"/>. At <lb/>impetus in D est idem si motus praecedens fue­<lb/>rit per GD, an per ED<arrow.to.target n="marg174"/>. </s> <s>Ergo etiam si mo­<lb/>tus fuerit per BD, DF est diuturnitas motus <lb/>per DH. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg173"></margin.target>Per cor. 3.2. huius.</s></p><p type="margin"> <s><margin.target id="marg174"></margin.target>Per 12. secundi huius.</s></p></subchap2><pb xlink:href="064/01/093.jpg"/><subchap2 type="corollary"><figure id="id.064.01.093.1.jpg" xlink:href="064/01/093/1.jpg"/><p type="head"> <s>Corollarium I</s></p><p type="main"> <s>Datis pluribus lineis in quadrante circuli <lb/>puta FA, AB, seu FA, AC, CB, inno­<lb/>tescent diuturnitates in quibuslibet earum, & <lb/>etiam in omnibus simul sumptis.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Impetus infra D est idem fuerit ne motus prae­<lb/>cedens per GD, an per ED, vero per ABD.</s></p></subchap2></subchap1><pb xlink:href="064/01/094.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>Grave naturaliter motum velocius ad idem <lb/>ducitur punctum duabus lineis, quam una <lb/>tantum.</s></p></subchap2><subchap2 n="7" type="proof"><p type="main"> <s>Progrediatur grave per AB in B.</s></p><p type="main"> <s>Dico quod citius perveniet in B motum per <lb/>A CB.</s></p><p type="main"> <s>Protrahatur BC, puta in D; & ab A in BD de­<lb/>mittatur normalis AE.</s></p><p type="main"> <s>Quoniam grave per BC pariter movetur, ductum per <lb/>A CB, ac per DB<arrow.to.target n="marg175"/>, & per eamdem CB ve­<lb/>locius fertur digressum a D quam ab E<arrow.to.target n="marg176"/>, per <lb/>illam itidem velocius fertur motum per ACB, <lb/>quam per EB, sed per A C aeque velociter fer­<lb/>tur ac per CE,<arrow.to.target n="marg177"/> ergo per totum ACB velocius <lb/>fertur quam per EB; sed aequali tempore fer­<lb/>tur per EB ac per AB<arrow.to.target n="marg178"/>; ergo per ACB ve­<lb/>locius fertur quam per AB. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg175"></margin.target>Per pr. huius.</s></p><p type="margin"> <s><margin.target id="marg176"></margin.target>Per 2. peti.</s></p><p type="margin"> <s><margin.target id="marg177"></margin.target>Per 19. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg178"></margin.target>Per 19. pr. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium.</s></p><p type="main"> <s>Hinc est, quod si motus fuerit per ACB, im­<lb/>petus in B est maior ac si fuisset per AB <lb/>secundum proportionem AB ad EB.</s></p></subchap2></subchap1><pb xlink:href="064/01/095.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII</s></p><subchap2 n="8" type="statement"><p type="main"> <s>Grave naturaliter ductum, velocius fertur su­<lb/>per tribus lineis descendentibus, quam su­<lb/>per una tantum.<figure id="id.064.01.095.1.jpg" xlink:href="064/01/095/1.jpg"/></s></p></subchap2><subchap2 n="8" type="proof"><p type="main"> <s>Feratur grave per AB, BC, CD.</s></p><p type="main"> <s>Dico citius duci in D quam per AD.</s></p><p type="main"> <s>Producantur CB, DC ad orizontalem AF in EF.</s></p><p type="main"> <s>Ducantur normales AG, BH, & ducatur AC.</s></p><p type="main"> <s>Quoniam grave pervenit citius in C per ABC, <lb/>quam per AC<arrow.to.target n="marg179"/>. Item citius in D per ACD <lb/>quam per AD<arrow.to.target n="marg180"/>, tanto citius perveniet in D <lb/>per ABCD quam per AD. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg179"></margin.target>Per 7. huius.</s></p><p type="margin"> <s><margin.target id="marg180"></margin.target>Per eamdem.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium. I.</s></p><p type="main"> <s>Eodem pacto facile probabitur quod citius <lb/>perveniet in D, quatenus ducitur pluribus <lb/>inclinationibus.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium. II.</s></p><p type="main"> <s>Impetus in D est maior, si fuerit motus per AB<lb/>CD, quam per AD.</s></p></subchap2></subchap1><pb xlink:href="064/01/096.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX</s></p><subchap2 n="9" type="statement"><p type="main"> <s>In quadrante inferiori circuli grave celerius <lb/>fertur, si moveatur super peripheria, quam <lb/>si una, aut pluribus rectis lineis.<figure id="id.064.01.096.1.jpg" xlink:href="064/01/096/1.jpg"/></s></p></subchap2><subchap2 n="9" type="proof"><p type="main"> <s>Sit ABC quadrans inferius.</s></p><p type="main"> <s>Dico grave B velocius duci si moveatur in <lb/>peripheria, quam si per BC, aut BDC, aut <lb/>BDEFC.</s></p><p type="main"> <s>Quoniam in peripheria ducitur pluribus inclina­<lb/>tionibus<arrow.to.target n="marg181"/>.</s></p><p type="margin"> <s><margin.target id="marg181"></margin.target>Per primam pet.</s></p><p type="main"> <s>Ergo grave super ipsa motum celerius transigit.<arrow.to.target n="marg182"/> Quod etc.</s></p><p type="margin"> <s><margin.target id="marg182"></margin.target>Per cor. primum 8. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Idem sequitur, si digrediatur a quovis puncto <lb/>Peripheriae, puta a D.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>In C impetus est maior, si motus fuerit per <lb/>Peripheriam, quam aliter quomodocunque.</s></p></subchap2></subchap1></chap><pb xlink:href="064/01/097.jpg"/><chap type="bk"><p type="main"> <s>DE MOTV<lb/>GRAVIVM<lb/>LIBER QVARTVS.<lb/>ET LIQVIDORVM PRIMVS.</s></p><subchap1 type="preface"><subchap2 type="preface"><p type="main"> <s>Hactenus<arrow.to.target n="note1"/> mihi videor de <lb/>scientia motus naturalis <lb/>gravium solidorum satis <lb/>pro viribus dixisse, dum <lb/>ex quibusdam proprieta­<lb/>tibus sensui notis, plures <lb/>ignotae deductae, & patefa­<lb/>ctae sunt: in hoc enim so­<lb/>lummodo ex Aristotele omnis scientia ver­<lb/>satur: ut in praxi apud Euclidem, & alios, qui <lb/>veras, & simplices scientias tractant, videre <lb/>est: unde nec agit Geometra de natura quan­<lb/>titatis, nec Musicus de natura soni, nec per­<lb/>spectivus de natura luminis, nec mechanicus <lb/>de natura ponderis.</s> <s>At<arrow.to.target n="note2"/> vero meus intelle­<lb/>ctus non omnino acquiescit, ni causas priores, <lb/>a quibus hi effectus demum proveniunt, si non<pb xlink:href="064/01/098.jpg"/>assequatur, saltem investiget; perquirendo <lb/>quae sit natura mobilium, corporum nimi­<lb/>rum prout mobilia sunt; etiam si hoc non <lb/>ad scientiam de motu, sed ad habitum supe­<lb/>riorem, nimirum sapientiae pertineat; quo <lb/>non effectus, sed rerum naturae, & principia <lb/>nobis innotescunt, ut Aristoteles in Metaphis. <lb/>etiam si in moralibus videatur secus sentire, <lb/>seu quia ex communi potius quam ex propria <lb/>sententia ibi loquutus fuerit, ubi exactam di­<lb/>scussionem locus non postulabat, seu mavis <lb/>culpa transcriptoris; in quo nihilominus plu­<lb/>rimos, & magni nominis habuit sectatores. <lb/></s> <s>Ut<arrow.to.target n="note3"/> ut sit ego quid tale delibavi, dum in prae­<lb/>fatione priori libro praeposita, causam aperire <lb/>conatus sum, cur duo quaelibet gravia, quan­<lb/>tumvis inaequalia, aequalia spatia conficiant; <lb/>videlicet quia natura gravium talis sit, ut <lb/>utrobique gravitas tali pacto sit materiae con­<lb/>nexa, & ita eam perpetuo sequatur, ut quanta <lb/>sit gravitas, seu eius actio; tantumdem sit pa­<lb/>riter materiae, & proinde resistentiae; ex quo <lb/>demum aequales sequantur effectus: quod ta­<lb/>men ad motuum indaginem supervacaneum <lb/>erat.</s> <s>Non tamen ex hoc ego me adhuc gra­<lb/>vium naturam omnino assecutum esse pro <lb/>certo habeo. </s> <s>Non quilibet collimans scopum <lb/>ferit; at quotus quisque propius dirigit, non<pb xlink:href="064/01/099.jpg"/>inutiliter laborasse censendus est. </s> <s>Ut<arrow.to.target n="note4"/> cumque <lb/>sit, quod tum factum est, hic pariter peragere <lb/>libuit, videlicet naturam motus pro viribus <lb/>investigare, causas nimirum, & principia, a <lb/>quibus hae demum motus passiones proveni­<lb/>ant. </s> <s>Iam<arrow.to.target n="note5"/> ante plures annos mihi visus sum <lb/>assequi causam accelerationis motus , dum ad <lb/>huc mobile a motore impellitur; quia nimirum <lb/>mobili moto imprimatur impetus, causa mo­<lb/>tus subsequentis; ex quo in secundo tempore <lb/>adsunt duo motores, unde est velocior, & im­<lb/>petus maior; in tertio tempore sunt duo iti­<lb/>dem motores, at alter puta impetus maioris <lb/>virtutis, unde motus adhuc celerior; & ita de­<lb/>inceps.</s> <s>Non<arrow.to.target n="note6"/> vero ex hoc constabat qua pro­<lb/>portione talis acceleratio fieret. </s> <s>Interim dum <lb/>pendulorum motus, perquirerem, praeter ex­<lb/>pectationem se se mihi obtulit, eorum longi­<lb/>tudines diuturnitatibus in duplicata respon­<lb/>dere ratione; de quo in prioris libri praefatio­<lb/>ne; ex quo demum, nihil minus cogitanti mi­<lb/>hi, in sexta propositione eiusdem deducere con­<lb/>tigit, motum tali pacto accelerari, ut in secun­<lb/>do tempore sit prioris triplum, in tertio quin­<lb/>tuplum, & deinceps iuxta numerorum impa­<lb/>rium progressionem: quod<arrow.to.target n="note7"/> miram mihi exci­<lb/>tavit cupidinem venandi a qua nam virtute, in <lb/>secundo tempore tanta motus fieret accretio,<pb xlink:href="064/01/100.jpg"/>dum nec videbatur esse impetus primum im­<lb/>pressi maior activitas, quam ipsius motoris a <lb/>quo ortum duxerat; nec quid aliud ibi esse de <lb/>novo productum suspicandum videbatur. </s> <s>Non <lb/>tamen deterreri potui, quin ulterius progre­<lb/>diens huius adhuc causam consequi sperarem: <lb/>quamvis se mihi dificillimum obtulerit, & <lb/>pluries me esse assecutum perperam existima­<lb/>verim, meque demum fuisse deceptum com­<lb/>pererim. </s> <s>Contigit<arrow.to.target n="note8"/> interim reperire, quod est <lb/>in Corol. Tertiae Secundi huius, motum or­<lb/>tum ab impetu esse aequabilem; quod a natu­<lb/>ra ipsiusmet mobilis emanere censendum vi­<lb/>sum fuit: ex quo in spem adductus sum ut ip­<lb/>sammet mobilis naturam assequi valerem. <lb/></s> <s>Pluries<arrow.to.target n="note9"/> cogitaveram esse naturae consentane­<lb/>um, ut ex simplicissimis principhuiuss quam plur­<lb/>imi mirabiles effectus educantur. </s> <s>Cuius rei, & <lb/>si plura habeam, unicum tantum in praesentia <lb/>aut alterum adducam exemplum. </s> <s>Perpen­<lb/>das amabo quot qualia, & quanta, ex Solis sub <lb/>Ecliptica circumlatione, in inferioribus gi­<lb/>gnantur; et quot qualia, et quanta hominibus <lb/>deficerent, ni eis necessitas quotidiani cibi <lb/>imposita fuisset: ex<arrow.to.target n="note10"/> quo mihi pariter probabi­<lb/>le visum est, eam fuisse naturam mobilibus tri­<lb/>butam, ut ex eius aliqua simplici immediata <lb/>proprietate emanent caeterae.</s> <s>Cum igitur ut <pb xlink:href="064/01/101.jpg" pagenum="101"/>mox dictum fuit mobile motum aequabiliter <lb/>demum moveatur sine motore; videtur infe­<lb/>rendum, quod motus motum producat, seu <lb/>potius quod motus perseveret, & se ipsum, <lb/>ut ita dicam, extendat, & continuet; quatenus<arrow.to.target n="note11"/> <lb/>dum semel mobile motum est, sit potens, <lb/>seu in potentia proxima se ipsum eadem ra­<lb/>tione movendi: ex<arrow.to.target n="note12"/> quibus in eam incidi sen­<lb/>tentiam, ut existimem, eam esse fortasse na­<lb/>turam mobilium, ut indiferenter se habeant <lb/>tam ad quietem, quam ad quemlibet motum; <lb/>unde, dummodo motus praecedat, a quacumque <lb/>causa proveniens, seu naturali seu violenta, <lb/>similis postmodum subsequatur, seu idem <lb/>perseveret, eadem velocitate quam in quoli­<lb/>bet instanti sortitum fuerit, donec impedia­<lb/>tur; & hanc motus continuationem ab ipsa­<lb/>met immobilis natura immediate emanantem, <lb/>forsitam esse unicam, & simplicem causam, a <lb/>qua fluant omnes illi effectus, & passiones, <lb/>quae in motu demum tum naturali, tum vio­<lb/>lento a nobis percipiuntur.</s> <s>Et<arrow.to.target n="note13"/> quamvis huius­<lb/>modi motus continuatio non sit nova entitas <lb/>superaddita, eam nihilominus intellectus con­<lb/>cipere tanquam quid noviter ortum, nimirum <lb/>posito motu, ex eo oriri virtutem, novum pro­<lb/>ducentem motum, ad faciliorem de motu ra­<lb/>tiocinationem non parum deservientem, quam vir­<pb xlink:href="064/01/102.jpg"/>tutem appellamus impetum; qui<arrow.to.target n="note14"/> re vera nil <lb/>aliud sit, nisi naturalis propensio ad motum, <lb/>seu potentia mobili inexistens continuandi mo­<lb/>tum semel adeptum quae potentia dum mo­<lb/>bile quiescit, sit in actu primo, & mediante, <lb/>motu reducatur in secundum, ea ratione qua <lb/>homini discurrenti non additur nova rationa­<lb/>litas; seu<arrow.to.target n="note15"/> novum principium, & nova poten­<lb/>tia ratiocinandi, sed eademmet, quam intrin­<lb/>secus habet, & est in actu primo, reducitur in <lb/>secundum.</s> <s>Porro<arrow.to.target n="note16"/> quod vere talis fuerit <lb/>natura mobilibus tradita, ut indiferenter se <lb/>habeant ad motum, & quietem, quamvis ex <lb/>dicta uniformis motus continuatione satis pro­<lb/>babile videatur, non ego tamen pro certo af­<lb/>firmare ausim: sumus<arrow.to.target n="note17"/> in physicis, ubi demon­<lb/>strationes rariores: non<arrow.to.target n="note18"/> tamen videri deberet le­<lb/>viter probatum, si ex hoc solummodo prin­<lb/>cipio omnes probarentur sequi passiones, quae <lb/>in motu quolibet percipiuntur absque quo ali­<lb/>quid aliud, vel de novo oriatur, vel ortum de­<lb/>pereat.</s> <s>Ex<arrow.to.target n="note19"/> eo autem sequitur, quod dum mo­<lb/>bile impellitur motus necessario augetur; un­<lb/>de<arrow.to.target n="note20"/> quo per maius spatium impellitur eo cor­<lb/>pus obsistens validius percutit; ex<arrow.to.target n="note21"/> quo tamen <lb/>motus ipse fit debilior, respondens siquidem <lb/>oppositi resistentiae; quae<arrow.to.target n="note22"/> si augeatur, velocitas <lb/>taliter minuitur, ut tandem deficiat, absque<pb xlink:href="064/01/103.jpg"/>quo aliquid oriri, aut deperire supponatur: ex <lb/>quibus vires percussionis metiri licet, de quo <lb/>alibi.</s> <s>Inde<arrow.to.target n="note23"/> est quod si manubrio parietem per­<lb/>cutias, illud intra melleum intruditur, quoniam <lb/>melleo minor obijcitur resistentia; facilius <lb/>siquidem is a manubrio permeatur quam murus <lb/>a manubrio. </s> <s>Si<arrow.to.target n="note24"/> vero mo­bile expellatur, mo­<lb/>veri perseverat, sine cuiusvis ope adiutoris de <lb/>novo orti; cum ex ipsiusmet natura, prout <lb/>mobile est, eiusdem motus continuatio neces­<lb/>sario subsequatur.</s> <s>Si<arrow.to.target n="note25"/> offendit in via quod mo­<lb/>tum urgeat, aut retundat; augetur velocitas, <lb/>aut minuitur; at<arrow.to.target n="note26"/> quaecumque ea sit inde per­<lb/>severat, quia ea motus natura ut continuetur; <lb/>unde<arrow.to.target n="note27"/> si permeet murum quem feriat, ei proin­<lb/>de resistentem, remissius fertur, quatenus est <lb/>maior muri durities, & proinde resistentia; ex <lb/>quo velocitas magis retunditur; quae tamen si <lb/>non omnino perit, qualis tandem remanet <lb/>talis perseverat; idem quippe continuatur mo­<lb/>tus; quousque<arrow.to.target n="note28"/> tamen resistentia perdurat, <lb/>motus paulatim minuitur, & tandem extin­<lb/>guitur.</s> <s>Ceterum<arrow.to.target n="note29"/> cum huiusmodi continuatio <lb/>emanet a propria ipsiusmet mobilis natura, <lb/>subsequi necessario debet quemlibet motum, <lb/>etiamsi per brevem fuerit morulam; quod<arrow.to.target n="note30"/> ap­<lb/>paret in pila lignea, malleo ligneo lusorio lon­<lb/>gioris manubrhuius longe propulsa, quamvis a<pb xlink:href="064/01/104.jpg"/>malleo per parvam morulam, & per minimum <lb/>spatium lata fuerit.</s> <s>Ex<arrow.to.target n="note31"/> quo itidem sequitur, <lb/>quod pila lusoria ad murum illidens, resilit; <lb/>quia pars murum feriens, vi compressa, ictui <lb/>cedens densatur, & ex curva complanatur; & <lb/>si sit talibus praedita viribus, ut deficiente vio­<lb/>lentia propellente, queat ex se in pristinam re­<lb/>duci rotunditatem; pars explanata, quae ite­<lb/>rum incurvatur, retrocedens versus locum cen­<lb/>tri, eo fertur celeri motu; qui quamvis in tali <lb/>reductione brevis fuerit, & proinde per brevem <lb/>morulam, idem continuatur eadem celeritate, <lb/>cum eius naturae competat, motum etiamsi per <lb/>parvum fuerit spatium continuare. </s> <s>Quod idem <lb/>sequitur si non pila, sed murus ipse caedat pri­<lb/>us, & demum se in pristinum reducat; unde <lb/>si neutrum caedat non fit resilitio. </s> <s>Si<arrow.to.target n="note32"/> vero <lb/>non perpendiculariter sed oblique murum <lb/>feriat, resilit ea lege, ut angulus reflexionis sit <lb/>angulo incidentiae proxime aequalis; quoniam <lb/>dum impingit, centrum adhuc fertur antrorsum; <lb/>unde pars pressa dum se in rotunditatem iterum <lb/>reducit, pilam dirigit secundum lineam tran­<lb/>seuntem per centrum iam antrorsum latum; <lb/>qui motus etiamsi per breve spatium, postmodum <lb/>continuatur: quoniam vero ex ea centri pro­<lb/>gressione pilae plures successive partes super <lb/>murum vertuntur, is motus itidem continua­<pb xlink:href="064/01/105.jpg"/>tur unde pila ipsa vertiginem acquirit, eo ce­<lb/>leriorem, quo angulus incidentiae plus acuitur; <lb/>qua vertigine adepta, ex eius continuatione, <lb/>ubi pila in planum iterum incidat, non servat <lb/>praedictam regulam in angulo reflexionis, qui <lb/>erit acutior, si pilae motus sit secundum ver­<lb/>tiginis ordinem, si vero contra obtusior.</s> <s>Quae <lb/>clarius apparent in pila reticulo, aut alio quo­<lb/>libet transversim percussa, in qua maior impri­<lb/>matur vertigo, quae demum eadem continuatur. <lb/></s> <s>Inde<arrow.to.target n="note33"/> item est quod pila eadem dum lusoria <lb/>palmula percussa, libere demum fertur, velo­<lb/>cius prodit ipsam et palmula movente; expul­<lb/>sa siquidem non modo ab ipsius impellentis <lb/>motu, sed etiam quoniam ipsiusmet pilae pars <lb/>percussa, ob modo dictam compressionem ce­<lb/>dens, & exinde densata, & mox in pristinam <lb/>redacta formam, ducitur versus ipsius pilae cen­<lb/>trum maiori velocitate, quam si a sola impel­<lb/>lentis vi ducta fuisset; quae maior velocitas con­<lb/>tinuatur. </s> <s>Imo<arrow.to.target n="note34"/> reticulo expulsa, fertur etiam ve­<lb/>locius, a triplici nempe motore ducta, addito <lb/>tertio, nimirum rete, cedente prius, & mox se <lb/>in pristinum reducente.</s> <s>Hinc<arrow.to.target n="note35"/> est etiam quod <lb/>quandocumque sphaera circumvolvitur, continua­<lb/>tur vertigo: unde<arrow.to.target n="note36"/> contingere potest, ut inde, <lb/>sequatur motus ipsius sphaerae progressivus, ei <lb/>supposito nimirum plano, suo contactu motum<pb xlink:href="064/01/106.jpg"/>partis inferioris impediente, ex quo pars su­<lb/>perior non impedita, & libere mota celerius <lb/>fertur, et quo vergit, vergit item centrum, & <lb/>talis continuatur motus, unde tota sphaera pro­<lb/>dit ulterius, absque quo alius novus motor su­<lb/>peraddatur. Hinc<arrow.to.target n="note37"/> itidem est, quod si sphaeram <lb/>quiescentem ex aliqua sui parte digito com­<lb/>primas contra subiectum planum, ea sortitur <lb/>duplicem motum, progressivum antrorsiim, <lb/>& validiorem in gyrum retrorsum: unde cessan­<lb/>te priori, si circumlatio continuatur, retro­<lb/>cedit, ac si tum ei planum supponeretur, <lb/>absque eo quod aliquid oriatur, aut depereat. <lb/></s> <s>Quod<arrow.to.target n="note38"/> pariter evenit in trochulo puerorum, <lb/>qui dum digitis in gyrum ducitur, circa pro­<lb/>prium axem circumfertur, eius inferiori pro­<lb/>minenti polo innixus; qui ubi demum ob im­<lb/>petum diminutum declinans subiectum plan­<lb/>um latere tangit, super illud circumvolvi­<lb/>tur, fere ad instar asinariae molae, cuius pro­<lb/>inde axis sua circumversione conum efficit, <lb/>cuius vertex est polus inferior, superior vero <lb/>dum rotatur circulum describit ipsius coni basim, <lb/>contra ordinem vertiginis peripheriae, motu tali, <lb/>qui minus diligenter intuentibus, apparet es­<lb/>se prioris, adhuc perseverantis, inversio; pluri­<lb/>bus mirabile visum, non percipientibus esse<pb xlink:href="064/01/107.jpg"/>naturae congruum, ambos ibi continuari mo­<lb/>tus, priorem quidem peripheriae circum, <lb/>axem trochi, postremum vero poli superioris <lb/>contra prioris ordinem; quod quibuslibet <lb/>motibus, ut dictum fuit, commune est, ex <lb/>ipsius mobilis natura proveniens, absque <lb/>quod aliquid aliud oriatur, aut ortum depereat, <lb/>remanente siquidem solummodo cuiuslibet <lb/>velocitatis semel impressae, naturali continua­<lb/>tione, quam quodlibet mobile, quocumque <lb/>pacto, ubivis a quocumque motore sortitum <lb/> fuerit; ex quo non modo praedictae oriuntur mo­<lb/>tus passiones, sed omnes alias passim obvias <lb/>emanare, facile demonstrabitur.</s> <s>A<arrow.to.target n="note39"/> nullo au­<lb/>tem experimento praedicta natura mobilium <lb/>tam clare apparere videtur, quam a facilitate, <lb/>qua mobilia quiescentia, a quolibet etiam mi­<lb/>nimo saepius impelluntur motore. </s> <s>Quod ap­<lb/>paret in cymbula in aqua natante, ponderis <lb/>librarum quinquaginta, & ultra; quam non <lb/>modo duces capillo mulieris, sed si illum ex <lb/>alio capite uspiam alligaveris, suo solum pon­<lb/>dere cymbulam trahit, & ad litus, ut ita dicam, <lb/>appellere coarctat, non obstante aqua renu­<lb/>ente, propriae siquidem divisioni saltem ali­<lb/>qualiter obsistente: quod aliunde non vi­<lb/>detur oriri nisi ex eademmet praedicta mo­<pb xlink:href="064/01/108.jpg"/>bilis natura, indiferenter nimirum se haben­<lb/>tis ad motum, & quietem. </s> <s>Vi autem ex eadem <lb/>tandem videamus, qua proportione motus ac­<lb/>celeratio fieri debeat, & an experimentis <lb/>respondeat.<figure id="id.064.01.108.1.jpg" xlink:href="064/01/108/1.jpg"/> Ducatur mobile A, ab <lb/>A versus E a quovis motore, seu <lb/>naturaliter a gravitate deorsum, seu <lb/>violenter ab impellente; et spatium AE con­<lb/>cipiatur sectum in portiones aequales in pun­<lb/>ctis B, C, D tali ratione, ut in B mobile <lb/>ductum virtute motus ab A in B acquirat impe­<lb/>tum, ex quo motus item subsequatur; seu quod <lb/>idem est, cuius virtute potentia mobilis eun­<lb/>dem continuendi motum, reducatur ad actum <lb/>secundum modo superius explicato; si conci­<lb/>piamus in B deficere actionem motoris, idem <lb/>nihilominus eiusdem velocitatis perseverat, & <lb/>continuatur motus; unde per tantundem tem­<lb/>poris fertur per portionem aequalem ipsi AB, <lb/>puta in C. </s> <s>Ni vero motoris actio deficiat, eius <lb/>virtute fertur itidem mobile per portionem <lb/>aequalem ipsi a AB; unde in secundo tempo­<lb/>re conficit duas spathuius portiones, eidem AB <lb/>aequales; & proinde dum prodit in D, movetur <lb/>motu dupliciter velociori, & sortitur dupli­<lb/>cem impetum, seu huius duplicis motus con­<lb/>tinuationem; ex quo in tertio tempore, ducitur <lb/>per duplum eiusdem portionis AB, at per<pb xlink:href="064/01/109.jpg"/>aequale a motore, ergo conficit tres portiones; <lb/>in quarta quatuor, in decima decem, & ita de<lb/>inceps. </s> <s>Obhuiuscies<arrow.to.target n="note40"/> primo, in portione AB iam <lb/>adesse impetum; nec mobile recedere ab A <lb/>quin impetus adsit: cum etenim impetus ema­<lb/>net a motu, & sit eius passio, est ab eo insepa­<lb/>rabilis, & proinde ubi est motus, est pariter im­<lb/>petus, quemadmodum ubi est ignis, est pari­<lb/>ter calor: nec causa est prior effectu tempore, <lb/>sed natura; quod non obstat, quin in eo­<lb/>dem instanti in quo est ignis, seu motus, <lb/>sit pariter calor seu impetus.</s> <s>Responditur<arrow.to.target n="note41"/> conceden­<lb/>dum, quod etiam in eodem instanti in <lb/>quo est motus, fieri possit ut sit pariter im­<lb/>petus, si vice versa mihi concedatur, nil <lb/>esse prius sua causa, & proinde impetum non <lb/>antecedere motum a quo est productus: at <lb/>dum mobile est adhuc in A non movetur, sed <lb/>quiescit: nec potest vere dici quod moveatur, <lb/>quin ab A recedens perveniat in B, unde sicut <lb/>est absurdum dicere ignem producere calorem, <lb/>quin prius sit productus ipsemet ignis, ita pa­<lb/>riter esset obsurdum asserere, motum produ­<lb/>cere impetum, quin sit productus ipsemet mo­<lb/>tus, & proinde prius quam mobile sit in B. </s> <s>Nec <lb/>dicas iam motum adesse priusquam perveniat <lb/>in B; nam quocumque primo perventum <lb/>erit, tum in eo puncto intelligo esse B: non<pb xlink:href="064/01/110.jpg"/>enim quaerimus, portio AD sit ne magna <lb/>aut parva, divisibilis an indivisibilis, & ma­<lb/>thematice vel physice; quod ad praesentem spe­<lb/>culationem non est necessarium; sufficit mi­<lb/>hi namque in praesentia, aliquem motum non <lb/>dici adesse ab impetu dependentem, quin ali­<lb/>us a quocumque impetu independenter prae­<lb/>cedat, productus siquidem a solo motore, cu­<lb/>ius virtute, potentia mobilis in actum secun­<lb/>dum reducatur, per quam demum continuetur <lb/>motus ut supra dictum fuit; secus enim causa <lb/>produceret suam causam in eodem genere <lb/>causae; immo idem esset causa sui ipsius, quippe <lb/>causa suae propriae causae. </s> <s>Obhuiuscies<arrow.to.target n="note42"/> secundo <lb/>motum non augeri iuxta progressionem Arith­<lb/>meticam naturalem, ut hic asseritur, sed secun­<lb/>dum numeros impares, ut in sexta primi <lb/>huius, & ut apud doctiores in praesentia fere <lb/>communiter creditur.</s> <s>Responditur<arrow.to.target n="note43"/> hanc sextam pro­<lb/>positionem inniti experimentis, sensui dece­<lb/>ptioni obnoxhuiuss, quibus insensibilis error de­<lb/>tegi nequit; quod hic evenit ex eo, quia por­<lb/>tiones temporis aequales ei priori, in qua confi­<lb/>citur prima motus portio independens ab im­<lb/>petu, percipi nequeant, utpote insensibiles, <lb/>prout est insensibilis dicta motus prima por­<lb/>tio; quae si perciperentur, videremus augeri <lb/>motum iuxta naturalem progressionem: At<arrow.to.target n="note44"/><pb xlink:href="064/01/111.jpg"/>in temporibus, & motibus sensibilibus res di­<lb/>verse se habet, ubi cognosci nequit motus <lb/>pars aliqua, nec tempus in quo conficiatur, <lb/>quin iam sint plures temporis peractae portio­<lb/>nes, ei aequales, in qua fuit motus ab impetu non <lb/>adiutus; cui tempori si plures aequales subse­<lb/>quantur, motus in eis, seu motus portiones, <lb/>portionibus temporum, iuxta numerorum im­<lb/>parium progressionem fere respondebunt.<figure id="id.064.01.111.1.jpg" xlink:href="064/01/111/1.jpg"/></s></p><p type="foot"> <s><foot.target id="foot.1"></foot.target>1 Actum est de scientia motus naturalis.</s></p><p type="foot"> <s><foot.target id="foot.2"></foot.target>2 Modo perquirendae causae.</s></p><p type="foot"> <s><foot.target id="foot.3"></foot.target>3 Ut supra respectu gravitatis factum fuit.</s></p><p type="foot"> <s><foot.target id="foot.4"></foot.target>4 Natura igitur motus investiganda.</s></p><p type="foot"> <s><foot.target id="foot.5"></foot.target>5 Iam quaesiveram causam accel.</s></p><p type="foot"> <s><foot.target id="foot.6"></foot.target>6 At non proportionem.</s></p><p type="foot"> <s><foot.target id="foot.7"></foot.target>7 Reperta iuxta progressionem numerorum imparium. Quaesivi causam.</s></p><p type="foot"> <s><foot.target id="foot.8"></foot.target>8 Repertus motus ab impetu aequabilis.</s></p><p type="foot"> <s><foot.target id="foot.9"></foot.target>10 Natura utitur principhuiuss simplicibus.</s></p><p type="foot"> <s><foot.target id="foot.10"></foot.target>11 Unde visum ex simplici mobilis proprietate emanandas caeteras.</s></p><p type="foot"> <s><foot.target id="foot.11"></foot.target>12 Quae sit motum ex se continuari.</s></p><p type="foot"> <s><foot.target id="foot.12"></foot.target>13 Quia mobilia indiferenter se habeant, ad motum & quietem.</s></p><p type="foot"> <s><foot.target id="foot.13"></foot.target>14 Huiusmodi continuationem non est nova entitas.</s></p><p type="foot"> <s><foot.target id="foot.14"></foot.target>15 At ut nova concipitur. Dicitur & impetus.</s></p><p type="foot"> <s><foot.target id="foot.15"></foot.target>16 Huiusmodi indiferentiam esse mobili naturalem.</s></p><p type="foot"> <s><foot.target id="foot.16"></foot.target>17 Probatur per dictam naturalem motus continuationem.</s></p><p type="foot"> <s><foot.target id="foot.17"></foot.target>18 Ex quo caeterae motus passiones.</s></p><p type="foot"> <s><foot.target id="foot.18"></foot.target>19 Absque quo quid oriatur aut pereat.</s></p><p type="foot"> <s><foot.target id="foot.19"></foot.target>20 Unde dum mobile impellitur motus augetur.</s></p><p type="foot"> <s><foot.target id="foot.20"></foot.target>21 Et quo longius, ictus validior.</s></p><p type="foot"> <s><foot.target id="foot.21"></foot.target>22 At motus debilior. Si resistentia maior motus tardior.</s></p><p type="foot"> <s><foot.target id="foot.22"></foot.target>23 Et tandem deficit.</s></p><p type="foot"> <s><foot.target id="foot.23"></foot.target>24 Patet experimento mallei.</s></p><p type="foot"> <s><foot.target id="foot.24"></foot.target>25 Expulsum moveri perseverat.</s></p><p type="foot"> <s><foot.target id="foot.25"></foot.target>26 Si quid urgeat aut retundat, variatur velocitas.</s></p><p type="foot"> <s><foot.target id="foot.26"></foot.target>27 Et talis perseverat.</s></p><p type="foot"> <s><foot.target id="foot.27"></foot.target>28 Si murum permeet remittitur.</s></p><p type="foot"> <s><foot.target id="foot.28"></foot.target>29 Si perseveret, velocitas minuitur.</s></p><p type="foot"> <s><foot.target id="foot.29"></foot.target>30 Idem etiam per morulam.</s></p><p type="foot"> <s><foot.target id="foot.30"></foot.target>31 Ut in ludo mallei.</s></p><p type="foot"> <s><foot.target id="foot.31"></foot.target>32 Unde pilae resilitio.</s></p><p type="foot"> <s><foot.target id="foot.32"></foot.target>33 Si oblique feriat, oblique resilit.</s></p><p type="foot"> <s><foot.target id="foot.33"></foot.target>34 Pila celerior instrumento expellente.</s></p><p type="foot"> <s><foot.target id="foot.34"></foot.target>35 Et eo magis reticulo expulsa.</s></p><p type="foot"> <s><foot.target id="foot.35"></foot.target>36 Vertigo durat.</s></p><p type="foot"> <s><foot.target id="foot.36"></foot.target>37 Unde motus localis.</s></p><p type="foot"> <s><foot.target id="foot.37"></foot.target>38 Pila digito compressa acquirit duplicem motum.</s></p><p type="foot"> <s><foot.target id="foot.38"></foot.target>39 Ex quo trochulum retrocedere videtur.</s></p><p type="foot"> <s><foot.target id="foot.39"></foot.target>40 Motus est a minimo motore.</s></p><p type="foot"> <s><foot.target id="foot.40"></foot.target>41 Objectio prima non dari primam motus portionem sine impetu.</s></p><p type="foot"> <s><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><foot.target id="foot.42"></foot.target>43 Objectio 2. motum non augeri iuxta progressionem naturalem.</s></p><p type="foot"> <s><foot.target id="foot.43"></foot.target>44 Responditur quod motus augetur iuxta progressionem naturalem per tempora insensibilia.</s></p><p type="foot"> <s><foot.target id="foot.44"></foot.target>45 At per sensibilia fere iuxta progressionem numerorum imparium.</s></p><p type="main"> <s>Quod ut planius fiat, Moveatur mobile A ab <lb/>A in B sensibiliter, & tempore sensibili ab, <lb/>cui subsequantur aequalia tempora bc, cd, & <lb/>primum tempus ab intelligatur divisum in por­<lb/>tiones minimas aequales, in quarum priori a<lb/>e, latum fuerit mobile ab A in E independen­<lb/>ter ab impetu, qui in puncto E motui con­<lb/>currere incipiat; has portiones patet esse eo <lb/>plures quo minores; sint decem, & mobile fe­<lb/>ratur temporibus ab, bc, cd, per spatia AB, <lb/>BC, CD; erunt portiones aequales portioni <lb/>AE in AB 55, in BC 155, in CD 255, inter <lb/>se ut 11, 31, 51. Si vero portio temporis ae <lb/>sit adhuc minor, cui aequales sint in ab cen­<lb/>tum, portiones spathuius aequales portioni AE<pb xlink:href="064/01/112.jpg"/><figure id="id.064.01.112.1.jpg" xlink:href="064/01/112/1.jpg"/> erunt in AB 5050, in BC 15050, in CD <lb/>25050, inter se ut 101, 301, 501, fere iuxta <lb/>rationem numerorum imperium 1, 3, 5. Ex <lb/>quibus constat, quod eo portiones spatiorum <lb/>magis accedunt ad rationem numerorum impa­<lb/>rium, quo portio temporis, in qua motus est in­<lb/>dependenter ab impetu, minor est. </s> <s>Eadem<arrow.to.target n="note45"/> pror­<lb/>sus ratione probabitur, quo est itidem minor, <lb/>spatia propius esse in duplicata ratione tem­<lb/>porum.</s> <s>Si namque concipiamus impetum incipere <lb/>in b, tempora ab, ac, ad sunt ut 1, 2, 3, spatia <lb/>vero AB, AC, AD, quae in duplicata ratione <lb/>temporum essent ut 1, 4, 9, sunt ut 1, 3, 6, val­<lb/>de ab eis discrepantes: si vero tempora ab, ac, <lb/>ad, intelligantur divisa in portiones, quarum <lb/>ab, contineat decem, erunt temporum in­<lb/>ter se portiones, ut 10, 20, 30, seu ut prius ut <lb/>1, 2, 3, at vero portiones spatiorum, quarum <lb/>prior ut supra sit AE, erunt ut 55, 210, 455 <lb/>seu ut 11, 42, 93; si denique portiones tempo­<lb/>rum sint 100, 200, 300, portiones spatiorum erunt <lb/>5050, 20100, 45150, ut 101, 402, 903, mi­<lb/>nimus, & insensibiliter discrepantes ab 1, 4, 9, & <lb/>proinde fere in duplicata temporum ratione;<pb xlink:href="064/01/113.jpg"/>unde quo plures temporum portiones, spatia <lb/>ad duplicatam rationem magis accedunt. </s> <s>Ut <lb/>autem datis temporibus, facile spatia peracta <lb/>reperiant, qui parum in arithmeticis progres­<lb/>sionibus versati sunt, duc numerum tempo­<lb/>rum, si sit par, in medietatem, & adde medie­ <lb/>tatem, si impar, in portionem maiorem medie­<lb/>tatis, & prodibit summa spatiorum in dato tem­<lb/>pore peractorum. </s> <s>Dentur 4 tempora, duc in <lb/>2 producto 8 adde medietatem 2, sit 10 sum­<lb/>ma spatiorum. </s> <s>Dentur tempora 9, duc in 5, <lb/>productum 45 est summa spatiorum. </s> <s>Auge­<lb/>tur<arrow.to.target n="note46"/> igitur, ni fallor, motus iuxta progressionem <lb/>arithmeticam, non numerorum imparium ab <lb/>unitate huc usque creditam, sed naturalem; at<arrow.to.target n="note47"/> <lb/>nihilominus, cum fere huiusdem effectus subse­<lb/>quantur, ob insensibilem discrepantiam; mi­<lb/>randum non est, creditum fuisse spatia esse in <lb/>duplicata ratione temporum; quandoquidem <lb/>etiam si verum precise fortasse non sit, est <lb/>attamen adeo veritati proximum, ut verita­<lb/>tem in adhibitis experimentis sensus percipe­<lb/>re nequiverit, quamobrem excusandi sunt <lb/>quicunque ita censuerunt. </s> <s>Ego autem modo <lb/>veritatem delitescentem detexisse spero, cau­<lb/>sam nimirum a qua huiusmodi proportio ema­<lb/>nat aperuisse, & insuper quales errores fue­<lb/>rint in suppositionibus, & experimentis huc<pb xlink:href="064/01/114.jpg"/>usque habitis, quod an re vera consecutus fue­<lb/>rim aliorum sit indicium: neque enim is sum <lb/>qui tantum mihi tribuam, ut rerum arcana <lb/>intimius caeteris rimari mihi videar, cui satis <lb/>superque est inter illos connumerari, quo­<lb/>rum disputationi mundus traditus fuit: nec <lb/>inutiliter me laborasse existimavero, si cre­<lb/>dar vitam silentio non pertransisse. </s> <s>Caete­<lb/>rum cum ea, quae de solidis dicenda videban­<lb/>tur, iuxta mei vires ingenhuius, pertractata sint, <lb/>superest, ut ad naturalis motus liquidorum <lb/>passiones inquirendas accedam.</s></p><p type="foot"> <s><foot.target id="foot.45"></foot.target>46 Et fere in duplicata ratione temporum.</s></p><p type="foot"> <s><foot.target id="foot.46"></foot.target>47 Augetur motus iuxta progressionem naturalem.</s></p><p type="foot"> <s><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"/><subchap1 type="definition"><p type="head"> <s>DEFINITIONES</s></p><subchap2 type="definition"><p type="main"> <s>Canale est vas oblongum, per quod aqua de­<lb/>currit; quod in praesentia supponitur habere <lb/>latera erecta, & basi perpendicularia, & pa­<lb/>rallela inter se. </s> <s>Sectio vasis, est parallelogramum quod supponi­<lb/>tur secare canale ad angulos rectos.</s></p></subchap2></subchap1><subchap1 type="postulate"><p type="head"> <s>PETITIONES</s></p><subchap2 type="postulate"><p type="main"> <s>Aqua transiens per eandem sectionem corre­<lb/>spondet tempori.</s></p></subchap2></subchap1><pb xlink:href="064/01/116.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Aqua aequaliter introducta in pluribus cana­<lb/>libus inaequaliter inclinatis correspondet <lb/>diuturnitatibus.<figure id="id.064.01.116.1.jpg" xlink:href="064/01/116/1.jpg"/></s></p></subchap2><subchap2 n="1" type="proof"><p type="main"> <s>Sint Canales AB, CD, in quibus introducatur <lb/>aqua aequalis, & aqua A ducatur in B diu­<lb/>turnitate E, & aqua C perveniat in D diutur­<lb/>nitate F.</s></p><p type="main"> <s>Dico aquam AB ad aquam CD esse ut E ad F.</s></p><p type="main"> <s>Quoniam aqua A B est ea, quae transit per A, diu­<lb/>turnitate E, & aqua CD est ea quae transit <lb/>per C, diuturnitate F per constructionem; sequi­<lb/>tur quod aqua AB est ad aquam CD ut E ad F<arrow.to.target n="marg183"/>.</s></p><p type="margin"> <s><margin.target id="marg183"></margin.target>Per pet. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium.</s></p><p type="main"> <s>Si diuturnitates sint aequales, aquae quantita­<lb/>tes sunt pariter aequales.</s></p></subchap2></subchap1><pb xlink:href="064/01/117.jpg"/><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II.</s></p><subchap2 n="2" type="statement"><p type="main"> <s>In pluribus canalibus ductis ad idem planum <lb/>orizontale, aquae quantitates sunt ut canales.</s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>Sint canalia AB, AC, ducta ad planum Orizon­<lb/>tale CB.</s></p><p type="main"> <s>Dico aquam AB esse ad aquam AC, ut longitudo <lb/>AB ad longitudinem AC.</s></p><p type="main"> <s>Quoniam diuturnitas AB ad diuturnitatem AC <lb/>est ut AB ad AC<arrow.to.target n="marg184"/>, at ut diuturnitas AB ad <lb/>diuturnitatem AC, ita aqua AB ad aquam <lb/>AC<arrow.to.target n="marg185"/>; ergo ut aqua AB ad aquam <lb/>AC, ita <lb/>longitudo AB ad longitudinem AC<arrow.to.target n="marg186"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg184"></margin.target>Per 15. primi. huius.</s></p><p type="margin"> <s><margin.target id="marg185"></margin.target>Per primam huius.</s></p><p type="margin"> <s><margin.target id="marg186"></margin.target>Per 11. Quinti.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Idem sequitur si alterum canale sit perpendi­<lb/>culare.</s></p></subchap2></subchap1><pb xlink:href="064/01/118.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO III. PROBL. I.</s></p><subchap2 n="3" type="statement"><p type="main"> <s>In canali declinante, reperire portionem con­<lb/>tinentem aquam, aequalem eius quae est in <lb/>perpendiculari.<figure id="id.064.01.118.1.jpg" xlink:href="064/01/118/1.jpg"/></s></p></subchap2><subchap2 n="3" type="proof"><p type="main"> <s>Sit AC canale inclinatum, & AB perpendicu­<lb/>lare; oportet reperire in AC portionem con­<lb/>tinentem aquam aequalem aquae AB.</s></p><p type="main"> <s>Ducatur BD normalis ad AC.</s></p><p type="main"> <s>Dico AD esse portionem quaesitam.</s></p><p type="main"> <s>Quoniam aqua ab A ducitur in B eodem tempore, <lb/>quo in D<arrow.to.target n="marg187"/>, erit aqua AB aequalis aqua AD<arrow.to.target n="marg188"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg187"></margin.target>Per 16. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg188"></margin.target>Per Co. primae huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium.</s></p><p type="main"> <s>Eadem ratione Dato canali AD reperietur <lb/>in AB portio continens aquam aequalem <lb/>AD, erecta a puncto D perpendiculari DB.</s></p></subchap2></subchap1><pb xlink:href="064/01/119.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO IV. PROBL. II.</s></p><subchap2 n="4" type="statement"><p type="main"> <s>In quibusvis canalibus quomodolibet inclina­<lb/>tis, reperire portiones continentes aquam <lb/>aequalem cuiusvis dicti canalis.<figure id="id.064.01.119.1.jpg" xlink:href="064/01/119/1.jpg"/></s></p></subchap2><subchap2 n="4" type="proof"><p type="main"> <s>A Canalibus AB, AC, AD, etc. sint secandae <lb/>portiones continentes aquam aequalem aquae <lb/>canalis AE.</s></p><p type="main"> <s>Iungantur omnes praedicti canales, retentis incli­<lb/>nationibus, in puncto superiori A; et si AE est <lb/>perpendicularis ad orizontem, circa ipsum <lb/>tanquam diametrum, describatur circulus AE; <lb/>sin minus a puncto E, erigatur ipsi AE perpen­<lb/>dicularis EF, & ab A demittatur perpendicu­<lb/>laris ad orizontem, donec cum EF coeat in <lb/>F, & circa AF describatur circulus secans <lb/>omnes praedictos canales in G, H, I.</s></p><p type="main"> <s>Dico portiones AG, AH, AI continere aquam <lb/>aequalem aquae canalis AE.</s></p><p type="main"> <s>Quoniam in AG, AE, AH, AI diuturnitates sunt <lb/>aequales<arrow.to.target n="marg189"/>, ergo sunt ibidem quantitates aquae <lb/>aequales<arrow.to.target n="marg190"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg189"></margin.target>Per 25. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg190"></margin.target>Per primam huius.</s></p></subchap2><pb xlink:href="064/01/120.jpg"/><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Si describantur quot vis circuli minores, seu <lb/>maiores, cuiuscumque magnitudinis, se invicem <lb/>tangentes in A, secabunt portiones dictorum <lb/>canalium ea ratione, ut sectiones intra quem­<lb/>vis circulum contineant aquam aequalem.</s></p></subchap2></subchap1><pb xlink:href="064/01/121.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V.</s></p><subchap2 n="5" type="statement"><p type="main"> <s>In canali secto quomodolibet; aquae quantita­<lb/>tes in eius portionibus correspondent diu­<lb/>turnitatibus.<figure id="id.064.01.121.1.jpg" xlink:href="064/01/121/1.jpg"/></s></p></subchap2><subchap2 n="5" type="proof"><p type="main"> <s><figure id="id.064.01.121.2.jpg" xlink:href="064/01/121/2.jpg"/>Sit canale AC sectum in B quomodolibet, & <lb/>sit DE diuturnitas aquae donec perveniat in <lb/>B, & DF diuturnitas donec perveniat <lb/>in C, & proinde EF diuturnitas aquae <lb/>ductae a B in C.</s></p><p type="main"> <s>Dico aquam AB ad aquam BC esse ut diuturni­<lb/>tas DE ad diuturnitatem EF.</s></p><p type="main"> <s>Quoniam aqua AB est ea, quae transit per A diu­<lb/>turnitate DE, & AC ea quae transit per idem <lb/>A diuturnitate DF per constructionem; aqua <lb/>AB ad aquam AC est ut diuturnitas DE ad <lb/>diuturnitatem DF<arrow.to.target n="marg191"/>; igitur dividendo, aqua <lb/>AB ad aquam BC est ut diuturnitas DE ad <lb/>diuturnitatem EF<arrow.to.target n="marg192"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg191"></margin.target>Per pet. huius.</s></p><p type="margin"> <s><margin.target id="marg192"></margin.target>Per 19. quinti.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Si Diuturnitates DE, EF sint aequales, aqua <lb/>AB aequatur aquae BC.</s></p></subchap2></subchap1><pb xlink:href="064/01/122.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>PROPOSITIO VI.</s></p><subchap2 n="6" type="statement"><p type="main"> <s>In canali secto quomodocumque; aqua in <lb/>priori portione ad aquam totius est in sub­<lb/>duplicata ratione longitudinum.<figure id="id.064.01.122.1.jpg" xlink:href="064/01/122/1.jpg"/></s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Sit canale AC sectum quomodocumque in D. </s> <s>Dico, quod aqua AD ad aquam AC est in sub­<lb/>duplicata ratione longitudinum AD, AC.</s></p><p type="main"> <s>Quoniam longitudines AD, AC sunt in duplicata <lb/>ratione diuturnitatum<arrow.to.target n="marg193"/>, at diuturnitates sunt <lb/>ut quantitates aquae<arrow.to.target n="marg194"/>, ergo quantitates aquae <lb/>sunt in subduplicata ratione longitudinum<arrow.to.target n="marg195"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg193"></margin.target>Per 3. & 7. primi huius.</s></p><p type="margin"> <s><margin.target id="marg194"></margin.target>Per 5. huius.</s></p><p type="margin"> <s><margin.target id="marg195"></margin.target>Per 11. quinti.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Unde si fiat AE media proportionalis inter <lb/>AD, AC, aqua AD ad aquam AC erit ut <lb/>AD ad AE.</s></p></subchap2></subchap1><pb xlink:href="064/01/123.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII. PROBL. III.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>Dato canali perpendiculari, & alio inclinato <lb/>eiusdem longitudinis; reperire propor­<lb/>tiones aquarum.<figure id="id.064.01.123.1.jpg" xlink:href="064/01/123/1.jpg"/></s></p></subchap2><subchap2 n="7" type="proof"><p type="main"> <s>Sint canalia AC inclinatum, & AB perpen­<lb/>diculare aequalia, & venanda sit proportio <lb/>inter aquas AB, AC.</s></p><p type="main"> <s>Ducatur BD perpendicularis ad AC, & fiat <lb/>AE media proportionalis inter AD, AC.</s></p><p type="main"> <s>Dico esse aquam AB ad aquam AC ut AD ad <lb/>AE.</s></p><p type="main"> <s>Quoniam aqua AD ad aquam AC est ut AD <lb/>ad AE<arrow.to.target n="marg196"/>, sed aqua AD est aequalis aquae AB<arrow.to.target n="marg197"/>, <lb/>ergo aqua AB ad aquam AC est ut AD ad <lb/>AE<arrow.to.target n="marg198"/>: Quod etc.</s></p><p type="margin"> <s><margin.target id="marg196"></margin.target>Per 6. huius.</s></p><p type="margin"> <s><margin.target id="marg197"></margin.target>Per 3. huius.</s></p><p type="margin"> <s><margin.target id="marg198"></margin.target>Per 11. quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/124.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII. PROBL. IV.</s></p><subchap2 n="8" type="statement"><p type="main"> <s>Datis canalibus aequalis longitudinis maio­<lb/>ris aut minoris inclinationis; venari pro­<lb/>portiones aquarum.<figure id="id.064.01.124.1.jpg" xlink:href="064/01/124/1.jpg"/></s></p></subchap2><subchap2 n="8" type="proof"><p type="main"> <s>Sit canale AC minus, AF magis inclinatum <lb/>ei aequale; & venandae sint proportiones aqua­<lb/>rum ab eis contentorum.</s></p><p type="main"> <s>Ducatur AB perpendicularis ad orizontem eiu­<lb/>sdem longitudinis, & ductis perpendiculari­<lb/>bus BD, BG, fiat AE media inter AD, AC, <lb/>& AH inter AG, AF, & ut AG ad AH, ita <lb/>AD ad AI.</s></p><p type="main"> <s>Dico aquam AC ad aquam AF esse ut AE ad AI.</s></p><p type="main"> <s>Quoniam ut aqua AC ad aquam AB ita AE ad <lb/>AD; & ut aqua AB ad aquam AF, ita AG <lb/>ad AH,<arrow.to.target n="marg199"/> seu ut AD ad AI per constructio­<lb/>nem; erit aqua AC ad aquam AF ut AE ad <lb/>AI<arrow.to.target n="marg200"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg199"></margin.target>Per 7. huius.</s></p><p type="margin"> <s><margin.target id="marg200"></margin.target>Per 22. quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/125.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX.</s></p><subchap2 n="9" type="statement"><p type="main"> <s>In canali secto iuxta proportionem nume­<lb/>rorum imparium, in portionibus ex ea re­<lb/>sultantibus sunt quantitates aquae aequales <lb/>inter se.<figure id="id.064.01.125.1.jpg" xlink:href="064/01/125/1.jpg"/></s></p></subchap2><subchap2 n="9" type="proof"><p type="main"> <s>Sit canale AD sectum in BC, & deinceps, ut <lb/>portiones AB, BC, CD, etc. sint inter se ut <lb/>1, 3, 5, 7.</s></p><p type="main"> <s>Dico quantitates aquae AB, BC, CD, esse <lb/>aequales inter se.</s></p><p type="main"> <s>Quoniam aqua aequali tempore progreditur ab A <lb/>in B, quo a B in C, & deinceps<arrow.to.target n="marg201"/>, erit aqua <lb/>AB aequalis aquae BC<arrow.to.target n="marg202"/>, etc. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg201"></margin.target>Per 10. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg202"></margin.target>Per cor. quintae huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/126.jpg"/><subchap1 n="10" type="proposition"><p type="head"> <s>PROPOSITIO X.</s></p><subchap2 n="10" type="statement"><p type="main"> <s>In quavis priori portione canalis, est aqua <lb/>aequalis portioni sequenti, triplae prioris.<figure id="id.064.01.126.1.jpg" xlink:href="064/01/126/1.jpg"/></s></p></subchap2><subchap2 n="10" type="proof"><p type="main"> <s>Dato canali A C secto in D ita ut AD sit <lb/>1/4 ipsius A C.</s></p><p type="main"> <s>Dico aquam AD aequari aquae DC.</s></p><p type="main"> <s>Quoniam eo tempore, quo A ducitur in D, D du­<lb/>citur in C<arrow.to.target n="marg203"/>, ergo aqua AD est aequalis aquae <lb/>DC<arrow.to.target n="marg204"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg203"></margin.target>Per 9. huius.</s></p><p type="margin"> <s><margin.target id="marg204"></margin.target>Per cor. quintae huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/127.jpg"/><subchap1 n="11" type="proposition"><p type="head"> <s>PROPOSITIO XI.</s></p><subchap2 n="11" type="statement"><p type="main"> <s>In canali declinante, duplo perpendicularis <lb/>ductae ad idem planum orizontale sectum <lb/>a linea ad illud normaliter ducta a puncto <lb/>inferiori dictae perpendicularis, portiones <lb/>continent aequales aquae quantitates.<figure id="id.064.01.127.1.jpg" xlink:href="064/01/127/1.jpg"/></s></p></subchap2><subchap2 n="11" type="proof"><p type="main"> <s>Sit canale AC duplum AB, sectum in D a <lb/>perpendiculari BD.</s></p><p type="main"> <s>Dico aquam AD aequari aquae DC.</s></p><p type="main"> <s>Quoniam AB est media inter AD, AC<arrow.to.target n="marg205"/>, <lb/>& AB est medietas ipsius AC per constructio­<lb/>nem, AD est medietas ipsius AB, & proinde <lb/>quarta pars totius AC; igitur aqua in AD <lb/>aequalis aquae in DC<arrow.to.target n="marg206"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg205"></margin.target>Per ea quae ad 16. pri. huius.</s></p><p type="margin"> <s><margin.target id="marg206"></margin.target>Per 10. huius.</s></p></subchap2></subchap1></chap><pb xlink:href="064/01/128.jpg"/><pb xlink:href="064/01/129.jpg"/><chap type="bk"><p type ="main"> <s>DE MOTV<lb/>GRAVIVM<lb/>LIBER QVINTVS<lb/>ET LIBER LIQVIDORVM SECVNDUVS.<lb/>VBI DE CANALIVM SECTIONIBVS.</s></p><subchap1 type="preface"><subchap2 type="preface"><p type="main"> <s>Etiamsi simus in liqui­<lb/>dis, lubet adhuc aliquid <lb/>de solidis praefari, sum­<lb/>pta occasione a Quest. <lb/>19. Mech. </s> <s>Arist. ubi cau­<lb/>sam perquirit cur lignum <lb/>facilius scindat qui secu­<lb/>rim extollens percutit, <lb/>quam qui securim impositam, addito pondere prae­<lb/>mat. </s> <s>Quod perinde est ac si dicas, cur plus scin­<lb/>das leviori securi mota, quam graviori quies­<lb/>cente. </s> <s>Nimirum Quoniam grave, motionem <lb/>gravitatis magis assumit, motum quam quies­<lb/>cens: pro qua gravitatis motione impetus in­<lb/>telligitur, qui primo delitescens, a gravi dein­<pb xlink:href="064/01/130.jpg"/>de per motum assumitur; scilicet qui erat in <lb/>potentia, in actum per motum reductus, mo­<lb/>tum inde auget, ipsum reddens velociorem, <lb/>suplente impetu vicem ponderis. </s> <s>Mihi ta­<lb/>men semper visus est Arist. problema non in­<lb/>tegre solvisse, reticuit siquidem cur huiusmo­<lb/>di motio gravitatis, seu impetus sit talis virtu­<lb/>tis, ut efficacius agat quam pondus additum, ex <lb/>quo demum maior scissio subsequatur. </s> <s>Cuius<arrow.to.target n="note48"/> <lb/>quidem ego causam pro viribus investigare <lb/>mihi proposui, quonam nimirum modo me­<lb/>tiri queat actio percutientis securis, & passio <lb/>ligni resistentis, ut demum percipi possit quan­<lb/>tum sit pondus addendum, ut impetus eius vi­<lb/>ribus respondeat.</s> <s>Quod<arrow.to.target n="note49"/> ut breviter de more <lb/>discutiatur, respectu actionis securis certum <lb/>est, quod si eius potentia non excedit li­<lb/>gni resistentiam, quamvis sit ei aequalis, nulla <lb/>fiet actio; atqui<arrow.to.target n="note50"/> si securis extollatur, quantum­<lb/>vis minimum, actio subsequetur, quoniam mo­<lb/>vens motum plus agit quam dum prius quiescebat, <lb/>quatenus actio gravitatis adhuc perseverat, & <lb/>insuper additur impetus, unde potentia quae <lb/>prius erat aequalis resistentiae, iam eam excedit; <lb/>& eius demum continuatur motus, quousque po­<lb/>tentia minuatur, aut augeatur resistentia: Et<arrow.to.target n="note51"/> <lb/>quo magis securis extollitur, validius scindit; <lb/>acquirit namque impetum maiorem, tali ad<pb xlink:href="064/01/131.jpg"/>priorem proportione, ut sint impetus in sub­<lb/>duplicata ratione spatiorum peractorum; ut <lb/>in quinta secundi huius: Unde<arrow.to.target n="note52"/> data minori <lb/>actione, facile metieris maiorem, percipiens <lb/>quantane ea sit, ex qualibet proveniens altitu­<lb/>dine.</s> <s>Quod<arrow.to.target n="note53"/> item sequitur in quavis percus­<lb/>sione seu a securi, seu a quolibet ad percutien­<lb/>dum idoneo naturaliter moto; trabes siqui­<lb/>dem, seu pali longiores, fortius in terram pan­<lb/>guntur, quo fistuca non modo est ponderosior, <lb/>sed altius effertur, tali ratione, ut altitudines <lb/>in duplicata proportione, percussionum viri­<lb/>bus respondeant. </s> <s>Si vero securis a motore <lb/>impellatur, validius percutit; quoniam motus <lb/>in initio, est celerior ab impulsu, quam a gra­<lb/>vitate; cuius perseverante actione, maior pro­<lb/>ducitur impetus, unde motus celerior, & ictus <lb/>validior, etiam nulla concurrente gravitate, <lb/>ut si motus non deorsum sed ad latera tendat, <lb/>aut sursum. </s> <s>Unde<arrow.to.target n="note54"/> quo malleus a pariete re­<lb/>motior in eum fortius impellitur, clavus ma­<lb/>gis figitur, & longe facilius quam si omnibus <lb/>adhibitis viribus, malleum contra clavum com­<lb/>primas.</s> <s>Unde<arrow.to.target n="note55"/> etiam est, quod mobile vehe­<lb/>mentius impulsum, expulsum demum, in <lb/>quodcumque illidat, validius ferit, & intimius <lb/>intruditur, quod in ictu a funda, arcu, sclopo <lb/>passim videre est. </s> <s>Huius autem vim impulsus pon­<pb xlink:href="064/01/132.jpg"/>dere proxime metiri licebit, si illud adeo con­<lb/>sentanee aptetur, ut illud extollas, eodem pa­<lb/>cto illi innixus, eademque prorsus directio­<lb/>ne, quemadmodum securim, aut quodvis aliud <lb/>impellere lubeat. </s> <s>Quod<arrow.to.target n="note56"/> facile continget, dua­<lb/>bus adhibitis trochleis, unius tantum modo <lb/>rotulae, altera superne appensa, inferne altera; <lb/>quibus ductarius circunductus funis, altero <lb/>extremo pondus, sustineat, alterum vero a po­<lb/>tentia trahatur, modo quo mox dictum fuit, <lb/>sit ne ea totum corpus animalis, seu hominis, <lb/>sive eius ambo brachia, aut ipsorum alterum, <lb/>seu tantum digiti, quorum omnium singilla­<lb/>tim vim, seu potentiam, proxime metietur ma­<lb/>ius aut minus pondus, quod ab uno, quoque eo­<lb/>rum, hac ratione in altum ducatur.</s> <s>Ex qui­<lb/>bus vires percussionis satis aperte apparere ar­<lb/>bitror, nimirum a vi motoris, seu sit gravitas, <lb/>seu impulsus, nec non ab impetu per motum <lb/>acquisito, maiori aut minori, prout motor est <lb/>maioris virtutis. </s> <s>Quo<arrow.to.target n="note57"/> vero ad ligni resisten­<lb/>tis passionem secundo loco propositam, certum <lb/>est, quod si resistentia est maior, aut aequalis <lb/>activitati securis, nulla fiet actio; si vero sit <lb/>resistentia minoris virtutis, unde vires agen­<lb/>tis securis excedant vires ligni resistentis, ali­<lb/>qua fiet scissio; eo<arrow.to.target n="note58"/> maior, quo minor erit resi­<lb/>stentia, quam non vi duntaxat portionis ligni<pb xlink:href="064/01/133.jpg"/>metiemur, quae securi opponitur; sed partium <lb/>itidem ei a latere cohaerentium, & sic porro <lb/>affixarum, ut ab eis difficulter divelli queat. <lb/></s> <s>Quantumvis autem huius resistentiae poten­<lb/>tia minus percipiatur, hoc unum est, quod qualis <lb/>qualis sit, velocitati securis contranititur, eam­<lb/>que tali ratione retundit, ut quantum ei tri­<lb/>buitur, tantundem velocitati detrahatur; un­<lb/>de<arrow.to.target n="note59"/> si resistentia addita sit priori decupla, aut <lb/>centupla, velocitas reducitur ad decimam par­<lb/>tem seu centesimam eius quae prius aderat, <lb/>unde spathuius quod securis per aerem peregit dum <lb/>nil obstaret, addita postmodum ligni obvhuius re­<lb/>sistentia, in aequali tempore, decimam pariter <lb/>aut centesimam conficit portionem. </s> <s>Quandiu<arrow.to.target n="note60"/> <lb/>vero lignum permeat, resistentia success­<lb/>ive augetur; partes quippe ligni ab ipsiusmet <lb/>securis compressione fiunt densiores, praeter <lb/>quam quod saepius, quo ea altius intruditur, <lb/>eo plures sunt partes cohaerentes divellendae. <lb/></s> <s>Utcunque sit, certum est quod dum impetus inci­<lb/>pit minui, & est successive minor proportio ac­<lb/>tionis securis ad ligni resistentiam, velocitas <lb/>non modo successive minuitur, sed paula­<lb/>tim deficit. </s> <s>Quod<arrow.to.target n="note61"/> idem sequitur de impetu, <lb/>qui cum velocitate pari passu procedit; unde<lb/><arrow.to.target n="note62"/> quantum velocitati detrahitur, tantundem <lb/>impetus minuitur; qui proinde cessante mo­<pb xlink:href="064/01/134.jpg"/>tu prorsus deperit.</s> <s>Et<arrow.to.target n="note63"/> quoniam mox adducta <lb/>communia sunt tam motae securi, quam cuili­<lb/>bet mobili, quod nimirum resistentia motum <lb/>retundit, & magis, quo maior proportio resi­<lb/>stentis ad mobilis vires, duae pilae, etiam aequales <lb/>in terram naturaliter cadentes, quae proinde <lb/>in aere aequali feruntur celeritate, etiamsi pon­<lb/>dere inaequales, terram inaequaliter perme­<lb/>ant, resistente nimirum terra magis pilae le­<lb/>viori, quam graviori. </s> <s>Unde est etiam quod si, <lb/>mobili proiecto, aliud addatur quiescens, & <lb/>proinde resistens, impetus minuitur; & quo<arrow.to.target n="note64"/> <lb/>maius mobile superadditur, tardius fertur, & <lb/>minus, aequo tempore conficit spatium, tali ra­<lb/>tione, ut ratio mobilis compositi, ad anterius <lb/>simplex, spathuiuss aequali peractis tempore, reci­<lb/>proce respondeat: unde<arrow.to.target n="note65"/> si mobile composi­<lb/>tum sit prioris quadruplum, velocitas demum <lb/>subsequens sit praecedentis quadrans, & talis <lb/>demum continuetur.</s> <s>Ut<arrow.to.target n="note66"/> autem tandem ad <lb/>propositam quaestionem propius accedamus, <lb/>& innotescat quale pondus addi debeat se­<lb/>curi, ut aequa fiat scissio, ac si ea extollatur, <lb/>hoc, ex dictis visum est erui non posse a viribus <lb/>ligni resistentis, utpote pariter se opponentis, <lb/>& contranitentis viribus securis motae levioris, <lb/>& immotae ponderosioris: Igitur tota quaestio <lb/>pendet ab ipsamet vi securis, seu motae, seu<pb xlink:href="064/01/135.jpg"/>quiescentis. </s> <s>Cum itaque iam visus sit, acti­<lb/>vitatem securis motae a duobus pendere prin­<lb/>ciphuiuss, a vi nimirum impellentis, & imprimen­<lb/>tis motum, quam metiuntur pondera ab eadem <lb/>vi sublata, & itidem a vi impetus, virtute dicti <lb/>motus a securi acquisiti, quam metiuntur <lb/>spatia, quae dum percurruntur, impulsus perse­<lb/>verat eiusdem virtutis; inde sequitur quod <lb/>ratio potentiae, seu momenti, seu virium se­<lb/>curis motae, ad potentiam eiusdem sensibili­<lb/>ter immotae, componitur ex ratione ponderum <lb/>inter se, nimirum eius quod aequipolet vi se­<lb/>curis impulsae, additi ad percutientis securis <lb/>pondus, ad pondus eiusdem quiescentis; nec <lb/>non ex ratione spatiorum peractorum maio­<lb/>ris securis in altum elatae, ad minus, fortasse <lb/>insensibile, eiusdem sensibiliter immotae, adeo <lb/>ut si vires tali pacto mensuratae utriusque se­<lb/>curis motae, & immotae, sint v.g. in ratione de­<lb/>cupla, & spatia peracta sint in centupla, ratio <lb/>porro virium securis motae, ad vires quiescen­<lb/>tis, sit in millecupla; unde si quiescens sit mil­<lb/>lies gravior, aequa fiet scissio. </s> <s>Nec dicas inter <lb/>spatia motae, & immotae nullam dari propor­<lb/>tionem, quia agitur hic de sensibiliter immo­<lb/>ta, & non praecise, seu mathematice, sed phy­<lb/>sice, nec videtur dari posse casum quin securis <lb/>imposita tantulum moveatur, etiamsi insen­<pb xlink:href="064/01/136.jpg"/>sibiliter; quod eo facilius existimandum vide­<lb/>tur, cum in hypotesi suppositum fuerit, secu­<lb/>ris vires esse viribus resistentiae prorsus aequa­<lb/>les: ex hoc tamen insensibili motu oritur, non <lb/>modo ut videamus, quantum vires percussionis <lb/>excedant vires ponderis, ex quo adeo facile li­<lb/>gnum scinditur; sed ex illo itidem oritur difficul­<lb/>tas percipiendi, qua precise proportione per­<lb/>cussio, vi prementi respondeat. </s> <s>Caeterum haec <lb/>sunt quae mihi in mentem venerunt de vi per­<lb/>cussionis sapientioribus proponenda, ut ad­<lb/>dant meliora.</s></p><p type="foot"> <s><foot.target id="foot.48"></foot.target>1 De vi percussionis.</s></p><p type="foot"> <s><foot.target id="foot.49"></foot.target>2 De activitate securis seu percutientis.</s></p><p type="foot"> <s><foot.target id="foot.50"></foot.target>3 Quia motum plus agit ob impetum.</s></p><p type="foot"> <s><foot.target id="foot.51"></foot.target>4 Et quo per longius spatium impetus est maior.</s></p><p type="foot"> <s><foot.target id="foot.52"></foot.target>5 Proportio inter impetus et spatia.</s></p><p type="foot"> <s><foot.target id="foot.53"></foot.target>6 In quavis percussione.</s></p><p type="foot"> <s><foot.target id="foot.54"></foot.target>7 Etiamsi motus non sit deorsum.</s></p><p type="foot"> <s><foot.target id="foot.55"></foot.target>8 Unde vis percussionis.</s></p><p type="foot"> <s><foot.target id="foot.56"></foot.target>9 Vim impulsus pondus metitur.</s></p><p type="foot"> <s><foot.target id="foot.57"></foot.target>10 De ligni resistentia.</s></p><p type="foot"> <s><foot.target id="foot.58"></foot.target>11 Quae pendet etiam a partibus cohaerentibus.</s></p><p type="foot"> <s><foot.target id="foot.59"></foot.target>12 Quo resistentia est maior minor est motus.</s></p><p type="foot"> <s><foot.target id="foot.60"></foot.target>13 Et inde resistentia augetur.</s></p><p type="foot"> <s><foot.target id="foot.61"></foot.target>14 Et velocitas minuitur. Et deficit.</s></p><p type="foot"> <s><foot.target id="foot.62"></foot.target>15 Et pariter impetus.</s></p><p type="foot"> <s><foot.target id="foot.63"></foot.target>16 Quod est commune cuivis mobili.</s></p><p type="foot"> <s><foot.target id="foot.64"></foot.target>17 Cui addito immoto minuitur impetus.</s></p><p type="foot"> <s><foot.target id="foot.65"></foot.target>18 Qua proportione.</s></p><p type="foot"> <s><foot.target id="foot.66"></foot.target>19 Quod pondus percussionis aequivaleat.</s></p></subchap2></subchap1><pb xlink:href="064/01/137.jpg"/><subchap1 type="postulate"><p type="head"> <s>PETITIONAE</s></p><subchap2 type="postulate"><p type="main"> <s>1. In sectionibus aequalibus quantitates aquae <lb/>sunt ut velocitates.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>2. Si velocitates sint aequales, sectiones sunt ut <lb/>quantitates aquae.</s></p></subchap2><subchap2 type="postulate"><p type="main"> <s>3.In canalium sectionibus Impetus, & veloci­<lb/>tates pro eodem sumuntur.</s></p></subchap2></subchap1><pb xlink:href="064/01/138.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA.</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Si sectiones sint aequales; aquarum transeun­<lb/>tium quantitates sunt, ut velocitates.<figure id="id.064.01.138.1.jpg" xlink:href="064/01/138/1.jpg"/></s></p></subchap2><subchap2 n="1" type="proof"><p type="main"> <s>Transeat aqua A per sectionem A, ab A ad <lb/>B; & aqua C per sectionem C aequalem <lb/>sectioni A, a C ad D aequali tempore.</s></p><p type="main"> <s>Dico aquam AB ad aquam CD esse ut velocitas <lb/>aquae A ad velocitatem aquae C.</s></p><p type="main"> <s>Quoniam velocitas in A ad velocitatem in C, est <lb/>ut AB ad CD,<arrow.to.target n="marg207"/> & aqua AB ad aquam CD <lb/>est itidem ut AB ad CD<arrow.to.target n="marg208"/>, sequitur quod velo­<lb/>citas in A ad velocitatem in C, est ut aqua <lb/>AB ad aquam CD<arrow.to.target n="marg209"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg207"></margin.target>Per 32. undec.</s></p><p type="margin"> <s><margin.target id="marg208"></margin.target>Per 11. Quinti.</s></p><p type="margin"> <s><margin.target id="marg209"></margin.target>Per primam huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/139.jpg"/><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II.</s></p><subchap2 n="2" type="statement"><p type="main"> <s>Velocitas aquae in pluribus eiusdem canalis <lb/>sectionibus, est reciproca sectionibus ipsis.<figure id="id.064.01.139.1.jpg" xlink:href="064/01/139/1.jpg"/></s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>Sint A, C, canalis sectiones, diversae magnitu­<lb/>dinis.</s></p><p type="main"> <s>Dico esse, ut magnitudo sectionis A ad magnitu­<lb/>dinem sectionis C, ita velocitatem in C, ad ve­<lb/>locitatem in A.</s></p><p type="main"> <s>Fiat sectio B aequalis ipsi A, per quam intelliga­<lb/>tur transire aquam aequaliter velocem ut in <lb/>sectione C.</s></p><p type="main"> <s>Quoniam ut quantitas aquae A seu C, ad quan­<lb/>titatem aquae B, ita est velocitas aquae in A, ad <lb/>velocitatem aquae in B seu C<arrow.to.target n="marg210"/>; sed ut magni­<lb/>tudo sectionis C ad magnitudinem sectionis B, <lb/>seu A, ita quantitas aquae C seu A, ad quanti­<lb/>tatem aquae B<arrow.to.target n="marg211"/>.</s></p><p type="margin"> <s><margin.target id="marg210"></margin.target>Per 2. pet. huius.</s></p><p type="margin"> <s><margin.target id="marg211"></margin.target>Per 2. huius.</s></p><p type="main"> <s>Ergo ut magnitudo sectionis C ad magnitudi­<lb/>nem sectionis A, ita velo­<lb/>citas aquae A ad velocitatem aquae C. </s> <s>Quod etc.</s></p></subchap2><pb xlink:href="064/01/140.jpg"/><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Idem sequitur, si sectiones sint canalium diversorum, dummodo ducant aquae quantitates aequales.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Impetus sunt ibidem ut sectiones reciproce.</s></p></subchap2></subchap1><pb xlink:href="064/01/141.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO III.</s></p><subchap2 n="3" type="statement"><p type="main"> <s>Sectiones canalis sunt reciproce in subduplicata ratione longitudinum.</s></p></subchap2><subchap2 n="3" type="proof"><p type="main"> <s>Sit canale AB sectum in C.</s></p><p type="main"> <s>Dico sectiones CB esse in subduplicata ratione AB, AC.</s></p><p type="main"> <s>Quoniam sectiones CB sunt ut velocitates in B, & in C<arrow.to.target n="marg212"/>, at velocitas in B ad velocitatem in C est in subduplicata ratione AB ad AC<arrow.to.target n="marg213"/>, Ergo sectio C ad sectionem B est in subduplicata ratione AB ad AC<arrow.to.target n="marg214"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg212"></margin.target>Per 5. secundi huius.</s></p><p type="margin"> <s><margin.target id="marg213"></margin.target>Per 11. quinti.</s></p><p type="margin"> <s><margin.target id="marg214"></margin.target>Per 33. primi.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Igitur si canalis latera sint parallela, altitudines sectionem sunt in subduplicata ratione longitudinum.</s></p><p type="main"> <s>Nam si latera perpendicularia canalis intelligantur bases, & ea ratione latitudines canalis ut altitudines, quae proinde sunt aequales<arrow.to.target n="marg215"/>, sectiones sunt ut dicta latera perpendicularia<arrow.to.target n="marg216"/>,<pb xlink:href="064/01/142.jpg"/>quae cum sint altitudines sectionum, sequitur <lb/>quod propositum fuit.</s></p><p type="margin"> <s><margin.target id="marg215"></margin.target>Per pri. sexti.</s></p><p type="margin"> <s><margin.target id="marg216"></margin.target>Per 3. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Si sectiones sint reciprocae in subduplicata ra­<lb/>tione longitudinum, exit aqua aequalis.</s></p></subchap2></subchap1><pb xlink:href="064/01/143.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO IV.</s></p><subchap2 n="4" type="statement"><p type="main"> <s>Impetus sectionum canalis, sunt in subdupli­<lb/>cata ratione longitudinum ipsarum a pun­<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"/><p type="main"> <s>In canali ACB.</s></p><p type="main"> <s>Dico impetum sectionis B ad impe­<lb/>tum sectionis C esse in subduplicata <lb/>ratione longitudinum AB ad AC.</s></p><p type="main"> <s>Quoniam sectio C ad sectionem B est in <lb/>subduplicata ratione AB ad AC<arrow.to.target n="marg217"/>. <lb/>Impetus in B ad impetum in C est in eadem sub­<lb/>duplicata ratione AB ad AC<arrow.to.target n="marg218"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg217"></margin.target>Per 2. huius.</s></p><p type="margin"> <s><margin.target id="marg218"></margin.target>Per 13. sexti.</s></p></subchap2></subchap1><pb xlink:href="064/01/144.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V. PROBL. I.</s></p><subchap2 n="5" type="statement"><p type="main"> <s>Data canalis sectione, reperire sectionem in <lb/>quolibet allo dato puncto.<figure id="id.064.01.144.1.jpg" xlink:href="064/01/144/1.jpg"/></s></p></subchap2><subchap2 n="5" type="proof"><p type="main"> <s>Data sectione C, & puncto B in canali AB, <lb/>Venanda est sectio puncti B.</s></p><p type="main"> <s>Fiat AD media inter AC, AB<arrow.to.target n="marg219"/>, & sectio B ad <lb/>sectionem C ut AC ad AD.</s></p><p type="margin"> <s><margin.target id="marg219"></margin.target>Per 20. sexti.</s></p><p type="main"> <s>Dico B esse sectionem quaesitam.</s></p><p type="main"> <s>Quoniam sectio B ad sectionem C est ut AC ad <lb/>AD per constructionem; erit sectio B ad sectio­<lb/>nem C in subduplicata ratione AC ad AB<arrow.to.target n="marg220"/>, <lb/>unde sectio B est sectio puncti B<arrow.to.target n="marg221"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg220"></margin.target>Per 3. huius.</s></p><p type="margin"> <s><margin.target id="marg221"></margin.target>Defini. pr. quarti huius.</s></p><p type="main"> <s>Fiet sectio B ad sectionem C ut AC ad AD, si fiat <lb/>altitudo laterum sectionis B ad altitudinem <lb/>laterum sectionis C ut AC ad AD<arrow.to.target n="marg222"/>.</s></p><p type="margin"> <s><margin.target id="marg222"></margin.target>Per 2. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/145.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>PROPOSITIO VI.</s></p><subchap2 n="6" type="statement"><p type="main"> <s>Datis pluribus sectionibus; ratio primae ad ter­<lb/>tiam, est composita ex rationibus velocitatis <lb/>secundae ad velocitatem primae, & velo­<lb/>citatis tertiae ad velocitatem secundae.<figure id="id.064.01.145.1.jpg" xlink:href="064/01/145/1.jpg"/></s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Dentur in canali AB sectiones B, C, D. <lb/></s> <s>Dico proportionem sectionis B ad sectionem <lb/>D, esse compositam ex rationibus velocitatis C <lb/>ad veloci­<lb/>tatem B, & velocitatis D ad veloci­<lb/>tatem C.</s></p><p type="main"> <s>Quoniam sectio B ad sectionem C est ut velocitas <lb/>C ad velocitatem B, item sectio D ad veloci­<lb/>tatem C ut velocitas C ad velocitatem D<arrow.to.target n="marg223"/>.</s></p><p type="margin"> <s><margin.target id="marg223"></margin.target>Per 5. def. sexti.</s></p><p type="main"> <s>Sed ratio velocitatis D ad velocitatem B est com­<lb/>posita ex rationibus velocitatis C ad velocita­<lb/>tem B, & velocitatis D ad velocitatem C<arrow.to.target n="marg224"/>.</s></p><p type="margin"> <s><margin.target id="marg224"></margin.target>Per 8. secundi huius.</s></p><p type="main"> <s>Ergo pariter ratio sectionis B ad sectionem D <lb/>est composita ex rationibus velocitatis C ad <lb/>velocitatem B, & velocitatis D ad velocita­<lb/>tem C. </s> <s>Quod etc.</s></p></subchap2><pb xlink:href="064/01/146.jpg"/><subchap2 type="corollary"><p type="head"> <s>Corollarium</s></p><p type="main"> <s>Si sint plures sectiones puta B, C, D, E, F, <lb/>pariter ratio sectionis B ad sectionem F com­<lb/>ponitur ex velocitatibus C ad B, D ad C, E ad <lb/>D, F ad E.</s></p></subchap2></subchap1><pb xlink:href="064/01/147.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>Si canales perpendicularis, & inclinatus ter­<lb/>minentur a recta normali ad inclinatum, <lb/>sectio perpendicularis ad sectionem in­<lb/>clinati est, ut inclinatus ad perpendicu­<lb/>larem.<figure id="id.064.01.147.1.jpg" xlink:href="064/01/147/1.jpg"/></s></p></subchap2><subchap2 n="7" type="proof"><p type="main"> <s>Dentur canales AB perpendicularis, & A<lb/>D inclinatus, terminati a recta BD, ut an­<lb/>gulus ADB sit rectus. </s> <s>Dico sectionem B ad se­<lb/>ctionem D esse ut AD, ad AB.</s></p><p type="main"> <s>Quoniam velocitas in B ad velocitatem in D est <lb/>ut AB ad AD<arrow.to.target n="marg225"/>.</s></p><p type="margin"> <s><margin.target id="marg225"></margin.target>Per 2. huius.</s></p><p type="main"> <s>Erit sectio B ad sectionem D ut AD ad AB<arrow.to.target n="marg226"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg226"></margin.target>Per cor. 8. sexti.</s></p></subchap2></subchap1><pb xlink:href="064/01/148.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII.</s></p><subchap2 n="8" type="statement"><p type="main"> <s>In canalibus perpendiculari, & inclinato; se­<lb/>ctiones terminatae a linea orizontali sunt <lb/>aequales.<figure id="id.064.01.148.1.jpg" xlink:href="064/01/148/1.jpg"/></s></p></subchap2><subchap2 n="8" type="proof"><p type="main"> <s>Dentur canales AB perpendicularis, & AC <lb/>inclinatus, quorum sectiones CB sint ori­<lb/>zontales.</s></p><p type="main"> <s>Dico eas esse aequales inter se.</s></p><p type="main"> <s>Ducatur normalis BD ad AC.</s></p><p type="main"> <s>Quoniam AB est media inter AD, AC<arrow.to.target n="marg227"/>, AD ad <lb/>AC habet duplicatam rationem AD ad AB<arrow.to.target n="marg228"/>. <lb/>Unde sectio D ad sectionem C est ut AB ad AD<arrow.to.target n="marg229"/>. </s> <s>Et eadem sectio D ad sectionem B est pariter <lb/>ut AB ad AD<arrow.to.target n="marg230"/>. Ergo sectiones C, B ha<lb/>bentes eamdem rationem ad sectionem D, sunt <lb/>aequales inter se<arrow.to.target n="marg231"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg227"></margin.target>Per 10. def. quin.</s></p><p type="margin"> <s><margin.target id="marg228"></margin.target>Per 3. huius.</s></p><p type="margin"> <s><margin.target id="marg229"></margin.target>Per 7. huius.</s></p><p type="margin"> <s><margin.target id="marg230"></margin.target>Per 9. quinti.</s></p><p type="margin"> <s><margin.target id="marg231"></margin.target>Per 3. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/149.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX.</s></p><subchap2 n="9" type="statement"><p type="main"> <s>Ductis pluribus canalibus a puncto superno <lb/>quomodocunque; reperire rationes data­<lb/>rum sectionum inter se.<figure id="id.064.01.149.1.jpg" xlink:href="064/01/149/1.jpg"/></s></p></subchap2><subchap2 n="9" type="proof"><p type="main"> <s>Dati sint quilibet canales AB, AC, AD, in <lb/>quibus assignentur puncta B, C, D.</s></p><p type="main"> <s>Oportet reperire rationes dictarum sectionum inter se. <lb/></s> <s>Ducatur perpendicularis AE, & ad eam per­<lb/>pendiculares BF, CG, DE, & sint F, G, E sectio­<lb/>nes canalis AE.</s></p><p type="main"> <s>Quoniam est nota ratio sectionum F, G, E<arrow.to.target n="marg232"/>, & B, C, D <lb/>sectiones aequantur sectionibus F, G, E respective<arrow.to.target n="marg233"/>, <lb/>sequitur notas esse ipsarum rationes. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg232"></margin.target>Per 8. huius.</s></p><p type="margin"> <s><margin.target id="marg233"></margin.target>Per 8. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Si sectiones B, C, D terminentur in <lb/>perpendiculari BD, erit pariter <lb/>ratio inter ipsas nota.</s></p></subchap2></subchap1><pb xlink:href="064/01/150.jpg"/><subchap1 n="10" type="proposition"><p type="head"> <s>PROPOSITIO X</s></p><subchap2 n="10" type="statement"><p type="main"> <s>In canalibus inter binas orizontales, sectiones <lb/>inferiores sunt aequales.<figure id="id.064.01.150.1.jpg" xlink:href="064/01/150/1.jpg"/></s></p></subchap2><subchap2 n="10" type="proof"><p type="main"> <s>Sint canales AB, CD inter orizontales AC, BD. <lb/></s> <s>Dico sectiones B, D esse aequales.</s></p><p type="main"> <s>Fiat canale CE.</s></p><p type="main"> <s>Sectio E aequatur sectioni D<arrow.to.target n="marg234"/>. </s> <s>Aequatur pariter <lb/>sectioni B, quia est par ratio. </s> <s>Ergo sectiones B, <lb/>D sunt aequales. </s> <s>Quod etc.<figure id="id.064.01.150.2.jpg" xlink:href="064/01/150/2.jpg"/></s></p><p type="margin"> <s><margin.target id="marg234"></margin.target>Per 3. huius.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Si canales AB, CB ducti ab orizontali A C ter­<lb/>minantur in B, sectio in B erit aequaliter de­<lb/>serviens utrique canali.</s></p></subchap2></subchap1><pb xlink:href="064/01/151.jpg"/><subchap1 n="11" type="proposition"><p type="head"> <s>PROPOSITIO XI.</s></p><subchap2 n="11" type="statement"><p type="main"> <s>Dato canali inflexo quomodolibet, venari quan­<lb/>titatem datae sectionis.<figure id="id.064.01.151.1.jpg" xlink:href="064/01/151/1.jpg"/></s></p></subchap2><subchap2 n="11" type="proof"><p type="main"> <s>Canalis AB inflectatur in B quovis angulo <lb/>ABC, in quo data sectione C venanda sit <lb/>eius quantitas.</s></p><p type="main"> <s>Protrahatur CB ad orizontalem AD, & fiat DE <lb/>media inter DB, DC, & sectionis C altitudo <lb/>ad altitudinem sectionis B fiat ut DB ad DE.</s></p><p type="main"> <s>Dico C esse sectionem in C.</s></p><p type="main"> <s>Quoniam si canale sit DC, sectio C ad sectionem B <lb/>est ut DB ad DE<arrow.to.target n="marg235"/>. At sectio B est eadem <lb/>etiam, respectu canalis AB<arrow.to.target n="marg236"/>. </s> <s>Ergo sectio <lb/>C ad sectionem B est ut DB ad DE.</s></p><p type="margin"> <s><margin.target id="marg235"></margin.target>Per co. decimae huius.</s></p><p type="margin"> <s><margin.target id="marg236"></margin.target></s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"><figure id="id.064.01.151.2.jpg" xlink:href="064/01/151/2.jpg"/> <s>Eadem via reperietur quantitas se­<lb/>ctionis C, si canalis sit declinans, <lb/>& demum perpendicularis ut A, B, C.</s></p></subchap2></subchap1></chap><pb xlink:href="064/01/152.jpg"/><pb xlink:href="064/01/153.jpg"/><chap type="bk"><p type="main"> <s>DE MOTV<lb/>GRAVIVM<lb/>LIBER SEXTVS<lb/>ET LIQUIDORVM TERTIVS<lb/>VBI DE FORAMINIBVS VASIS.</s></p><subchap1 type="preface"><p type="main"> <s>Non alienum ab instituto <lb/>arbitratus sum adhuc ali­<lb/>quid huic postremo prae­<lb/>fari libro, ubi nodum sol­<lb/>vere conabor ab eruditis­<lb/>simo Mersenno proposi­<lb/>tum prop. 15. Ballist. <lb/>quod quidem, explican­<lb/>do, quantum ingenhuius fert imbecilitas, qua diu­<lb/>turnitate pendulum, tam prius descendendo, <lb/>quam inde ascendendo feratur, suppositis ex­<lb/>perimentis cum ipso primo habitis, postmo­<lb/>dum a me repetitis, quibus percipere mihi vi­<lb/>sus sum diuturnitatem penduli in integra <lb/>vibratione aequari diuturnitati gravis moti per<pb xlink:href="064/01/154.jpg"/>spatium eius quadruplum, & in descensu, <lb/>aequari diuturnitati gravis moti per eiusdem <lb/>penduli duplum: quod non omnino congruit <lb/>cum eo quod prop. 9. Terthuius huius proba­<lb/>tum fuit, quoniam experimenta veritatem <lb/>proxime, at non praecise patefaciunt. </s><figure id="id.064.01.154.1.jpg" xlink:href="064/01/154/1.jpg"/> <s>Sit pen­<lb/>dulum AB, quod in C translatum sua integra <lb/>vibratione describat circulum CBD: ex dictis <lb/>experimentis compertum est diuturnitatem il­<lb/>lius percurrentis per quadrantem CB, aequari <lb/>diuturnitati gravis descendentis per FB dia­<lb/>metrum, ipsius penduli duplam; diuturnita­<lb/>tem vero eiusdem conficientis integram vibra­<lb/>tionem CBD, aequari diuturnitati eiusdem gravis <lb/>descendentis per duplum ipsius FB, puta per FG. <lb/></s> <s>Quibus positis, mihi assequi visus sum, qua pro­<pb xlink:href="064/01/155.jpg"/>portione sibi respondeant diuturnitates pen­<lb/>duli moti in descensu a C in B, & in ascensu <lb/>a B in D, secta CD in E tali ratione, ut E tan­<lb/>tundem destet a C, quantum B; existimans diu­<lb/>turnitates motuum per CB, & BD quadrantes, <lb/>esse inter se ut CE ad ED. </s> <s>Quoniam ratio diu­<lb/>turnitatum per FB, & FG est eadem ac per <lb/>AB, & FB, cum utrobique sit subdupla pro­<lb/>portio, quae ratio est pariter inter CB, & <lb/>FB<arrow.to.target n="marg237"/>, cum CB sit media inter AB, FB,<arrow.to.target n="marg238"/> erit <lb/>ratio diuturnitatum per FB, & FG, & itidem <lb/>per quadrantem CB, & per semic. CBD eis <lb/>aequalium<arrow.to.target n="marg239"/> ut CB ad FB, seu ut CE ad CD eis <lb/>aequales: & dividendo, ratio diuturnitatum <lb/>per CB, & BD quadrantes erit ut CE ad ED<arrow.to.target n="marg240"/>. <lb/></s> <s>Quod etc. </s> <s>Unde si ex Mersenno, grave ab A in <lb/>B pedum 3 regiorum, qui quatuor palmis nostra­<lb/>tibus proxime respondent, descendit in 30 ter­<lb/>thuiuss, a C in B fertur non in 30 sed in 42, unde <lb/>a B in D ascendit in 17 sibi respondentes ut <lb/>99 ad 41. Caeterum ex dictis facile demonstrabi­<lb/>tur quod si vibrationes sint minores, v.g. ab <lb/>H in I, pariter diuturnitates per HB, & per <lb/>BI erunt ut CE ad ED, cum iam probatum <lb/>fuerit, & experientia constet vibrationes CB, HB <lb/>nec non CD, HI esse aequediuturnas. </s> <s>Ex his <lb/>etiam constat esse aequales diuturnitates per <lb/>BG, & BD, etiamsi per BD fiat ascensus, &<pb xlink:href="064/01/156.jpg"/>proinde motus successive tardior, & per BG <lb/>descensus, & proinde motus successive velo­<lb/>cior. </s> <s>Quem nodum, de quo in praesentia <lb/>nil addam, alhuiuss enodandum relinquo.</s></p><p type="margin"> <s><margin.target id="marg237"></margin.target>Per 3. pr. huius.</s></p><p type="margin"> <s><margin.target id="marg238"></margin.target>Per cor. 8. sexti.</s></p><p type="margin"> <s><margin.target id="marg239"></margin.target>Per Observat.</s></p><p type="margin"> <s><margin.target id="marg240"></margin.target>Per 17. quinti.</s></p></subchap1><pb xlink:href="064/01/157.jpg"/><subchap1 type="definition"><p type="head"> <s>DEFINITIONES.</s></p><subchap2 type="definition"><p type="main"> <s>1 Vas aquae intelligitur, cuius latera sint <lb/>retangula, & basis orizontalis.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>2. Foramen intelligitur rectangulum cuius basis <lb/>orizontalis.</s></p></subchap2><subchap2 type="definition"><p type="main"> <s>3. Foramina inaequalia eiusdem altitudinis, quo­<lb/>rum inaequalitas pendet a sola latitudine.</s></p></subchap2></subchap1><pb xlink:href="064/01/158.jpg"/><subchap1 type="axiom"><p type="head"> <s>DIGNITATES</s></p><subchap2 type="postulate"><p type="main"> <s>Ubi omnia sint paria, effectus sunt aequa­<lb/>les.</s></p></subchap2></subchap1><subchap1 type="postulate"><p type="head"> <s>PETITIONES</s></p><subchap2 type="axiom"><p type="main"> <s>1 Quantitates eiusdem generis sunt omnes <lb/>commensurabiles.</s></p></subchap2><subchap2 type="axiom"><p type="main"> <s>2. Aqua transiens per vasis foramen, decurrit a <lb/>summo vasis ad foramen tanquam per cana­<lb/>lem perpendicularem.</s></p></subchap2><subchap2 type="axiom"><p type="main"> <s>Quod experieris, si vas aqua plenum, in cuius <lb/>imo sit foramen, sit perspicuum; videbis etenim <lb/>in eo formari canale, per quod aqua supe­<lb/>rior exeat.</s></p></subchap2></subchap1><pb xlink:href="064/01/159.jpg"/><subchap1 n="1" type="proposition"><p type="head"> <s>PROPOSITIO PRIMA</s></p><subchap2 n="1" type="statement"><p type="main"> <s>Aquarum quantitates exeuntium per forami­<lb/>na aequalia, aeque distantia a summo vasis, <lb/>aequali tempore; sunt aequales.<figure id="id.064.01.159.1.jpg" xlink:href="064/01/159/1.jpg"/></s></p></subchap2><subchap2 n="1" type="proof"><p type="main"> <s>In vase AB, sint foramina C, D aequalia, & <lb/>orizontalia, per quae aqua aequali tempore de­<lb/>currat.</s></p><p type="main"> <s>Dico aquas decursas esse aequales inter se.</s></p><p type="main"> <s>Quoniam ubi omnia sunt paria, effectus sunt <lb/>aequales<arrow.to.target n="marg241"/>.</s></p><p type="margin"> <s><margin.target id="marg241"></margin.target>Per ax. huius.</s></p><p type="main"> <s>Sed hic sunt omnia paria ex constructione.</s></p><p type="main"> <s>Ergo habent effectus aequales.</s></p><p type="main"> <s>Sed aquae decursa sunt effectus, & proinde aequa­<lb/>les. </s> <s>Quod etc.</s></p><p type="main"> <s>Seu mavis.</s></p><p type="main"> <s>Ubi omnia paria effectus sunt aequales, & <lb/>proinde si effectus sunt aquae decursae, ipsae <lb/>sunt aequales.</s></p><p type="main"> <s>Sed hic sunt omnia paria, & effectus sunt aquae <lb/>decursae, ex constructione. </s> <s>Ergo aquae decursae sunt aequales. </s> <s>Quod etc.</s></p></subchap2></subchap1><pb xlink:href="064/01/160.jpg"/><subchap1 n="2" type="proposition"><p type="head"> <s>PROPOSITIO II.</s></p><subchap2 n="2" type="statement"><p type="main"> <s>Si foramina sint orizontalia, eiusdem altitudi­<lb/>nis, quantitates aquarum decursarum sunt <lb/>inter se ut foramina.<figure id="id.064.01.160.1.jpg" xlink:href="064/01/160/1.jpg"/></s></p></subchap2><subchap2 n="2" type="proof"><p type="main"> <s>In vase AB dentur foramina orizontalia aeque <lb/>alta C minus, D vero maius.</s></p><p type="main"> <s>Dico aquam decursam per C, quae sit E, se habere ad aquam <lb/>decursam per D, quae sit F, ut foramen C ad foramen D.</s></p><p type="main"> <s>Longitudinum C, & D commensurabilium,<arrow.to.target n="marg242"/> <lb/>sit G communis mensura, & secentur lon­<lb/>gitudines C, D in partes, quae sint aequales ipsi <lb/>G, quibus divisis a perpendicularibus, producan­<lb/>tur tot foramina, quot sunt dictae partes.</s></p><p type="margin"> <s><margin.target id="marg242"></margin.target>Per pr. pet.</s></p><p type="main"> <s>Quoniam huiusmodi foramina erunt inter se <lb/>aequalia<arrow.to.target n="marg243"/>. Ex eis effluent quantitates aquae <lb/>aequales<arrow.to.target n="marg244"/>. </s> <s>Quot igitur sunt foramina in C, D, <lb/>tot sunt quantitates aquarum in E, F. </s> <s>Igitur <lb/>sunt quatuor quantitates C, D, E, F, quarum <lb/>prima, C, est ad E, 2., ut D, 3., ad F, 4.; & per­<lb/>mutando erit C ad D ut E ad F<arrow.to.target n="marg245"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg243"></margin.target>Per 36. primi.</s></p><p type="margin"> <s><margin.target id="marg244"></margin.target>Per primum huius.</s></p><p type="margin"> <s><margin.target id="marg245"></margin.target>Per 16. quinti.</s></p><p type="main"> <s>Dices, quod fieri potest quod longitudines C, D, <lb/>non sint commensurabiles, nec proinde G sit eo­<lb/>rum communis mensura: sed hic non sumus in <lb/>Mathematicis, sed in physicis, ubi non habetur <lb/>ratio insensibilium.</s></p></subchap2></subchap1><pb xlink:href="064/01/161.jpg"/><subchap1 n="3" type="proposition"><p type="head"> <s>PROPOSITIO III.</s></p><subchap2 n="3" type="statement"><p type="main"> <s>Foramina vasis perinde se habent ac sectio­<lb/>nes canalis, respectu impetus.<figure id="id.064.01.161.1.jpg" xlink:href="064/01/161/1.jpg"/></s></p></subchap2><subchap2 n="3" type="proof"><p type="main"> <s>Sit vas CD in quo foramen D, & sit AB ca­<lb/>nalis perpendicularis in quo sectio B, & <lb/>AB, CD, altitudines sint aequales.</s></p><p type="main"> <s>Dico in B, & D esse impetus aequales.</s></p><p type="main"> <s>Quoniam aqua fluens a foramine D decurrit per <lb/>spatium CD, ac si decurreret per canalem AB <lb/>perpendicularem, eiusdem longitudinis<arrow.to.target n="marg246"/>, in <lb/>D, & B sortitur impetus aequales. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg246"></margin.target>Per 2. pet.</s></p></subchap2></subchap1><pb xlink:href="064/01/162.jpg"/><subchap1 n="4" type="proposition"><p type="head"> <s>PROPOSITIO IV.</s></p><subchap2 n="4" type="statement"><p type="main"> <s>Impetus foraminum aequalium vasis, sunt in <lb/>duplicata ratione distantiae a summo va­<lb/>sis.<figure id="id.064.01.162.1.jpg" xlink:href="064/01/162/1.jpg"/></s></p></subchap2><subchap2 n="4" type="proof"><p type="main"> <s>In vase AC, distantiae foraminum aequalium <lb/>B, C a summo vasis AB, AC; media sit AD.</s></p><p type="main"> <s>Dico impetus in C ad impetum in B esse ut AD <lb/>ad AB.</s></p><p type="main"> <s>Quoniam foramina B, C, sunt ac si essent sectio­<lb/>nes canalis AC respectu impetus<arrow.to.target n="marg247"/>, impetus in <lb/>B & C sunt ut AB ad AD<arrow.to.target n="marg248"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg247"></margin.target>Per 3. huius.</s></p><p type="margin"> <s><margin.target id="marg248"></margin.target>Per 4. quinti huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/163.jpg"/><subchap1 n="5" type="proposition"><p type="head"> <s>PROPOSITIO V.</s></p><subchap2 n="5" type="statement"><p type="main"> <s>Altitudines a foraminibus aequalibus ad sum­<lb/>mum vasis, sunt in duplicata ratione aqua­<lb/>rum per ea decurrentium.<figure id="id.064.01.163.1.jpg" xlink:href="064/01/163/1.jpg"/></s></p></subchap2><subchap2 n="5" type="proof"><p type="main"> <s>In vase AC altitudines a foraminibus aequa­<lb/>libus B, C, ad summum vasis A sint AB, <lb/>AC, quarum media sit AD.</s></p><p type="main"> <s>Dico AD ad AB esse ut aqua fluens per C ad <lb/>aquam fluentem per B.</s></p><p type="main"> <s>Quoniam ut AD ad AB ita est impetus in C ad <lb/>impetum in B<arrow.to.target n="marg249"/>, & impetus sunt ut velocita­<lb/>tes<arrow.to.target n="marg250"/>; impetus in C ad impetum B est ut aqua <lb/>fluens per C ad aquam effluentem per B. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg249"></margin.target>Per quartam huius.</s></p><p type="margin"> <s><margin.target id="marg250"></margin.target>Per 3. petit.</s></p></subchap2></subchap1><pb xlink:href="064/01/164.jpg"/><subchap1 n="6" type="proposition"><p type="head"> <s>PROPOSITIO VI. PROBL. II.</s></p><subchap2 n="6" type="statement"><p type="main"> <s>Secto foramine in partes aliquotas a rectis <lb/>orizontalibus, venari rationes aquarum ex <lb/>eis fluentium.<figure id="id.064.01.164.1.jpg" xlink:href="064/01/164/1.jpg"/></s></p></subchap2><subchap2 n="6" type="proof"><p type="main"> <s>Secetur foramen AB in partes AC, CD, DB <lb/>aequales, quorum altitudines sint notae, & <lb/>ab AC fluat aqua E, a CD aqua F, a DB <lb/>aqua G, tempore aequali.</s></p><p type="main"> <s>Venanda proportio aquarum E, F, G.</s></p><p type="main"> <s>Fiant HI, KL, MN, altitudines foraminum A<lb/>C, CD, DB a summo vasis; & inter ipsas <lb/>mediae OP, QR<arrow.to.target n="marg251"/>.</s></p><p type="margin"> <s><margin.target id="marg251"></margin.target>Per 13. sexti.</s></p><p type="main"> <s>Quoniam aqua E ad aquam F, est ut HI ad OP<arrow.to.target n="marg252"/>, <lb/>Nota est ratio aquae E ad aquam F. Item quoniam <lb/>aqua F ad aquam G est ut KL, ad QR<arrow.to.target n="marg253"/>, <lb/>nota est pariter ratio aquae F ad aquam G. <lb/>at ratio aquae E ad aquam G, composita ra­<lb/>tionum inter EF & FG notarum, est pariter <lb/>nota. </s> <s>Reperta est igitur ratio aquarum E, F, G. </s> <s>Quod, etc.</s></p><p type="margin"> <s><margin.target id="marg252"></margin.target>Per 5. huius.</s></p><p type="margin"> <s><margin.target id="marg253"></margin.target>Per 5. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/165.jpg"/><subchap1 n="7" type="proposition"><p type="head"> <s>PROPOSITIO VII. PROBL. III.</s></p><subchap2 n="7" type="statement"><p type="main"> <s>Secto foramine vasis in partes a recta orizon­<lb/>tali, reperire rationes aquarum effluen­<lb/>tium ab ipsis.<figure id="id.064.01.165.1.jpg" xlink:href="064/01/165/1.jpg"/></s></p></subchap2><subchap2 n="7" type="proof"><p type="main"> <s>Foramen CD vasis AB secetur a recta E in <lb/>partes CE, CD, & effluat a parte superio­<lb/>ri CE aqua F, & ab inferiori ED aqua G eo­<lb/>dem tempore.</s></p><p type="main"> <s>Quaeritur proportio F ad G.</s></p><p type="main"> <s>Si ED foramen minus non mensurat CE, repe­<lb/>riatur eorum maxima communis mensura<arrow.to.target n="marg254"/>, <lb/>quae sit H, & iuxta eam secetur CE in partes <lb/>CQ, QK, KE, item ED in partes EI, ID.</s></p><p type="margin"> <s><margin.target id="marg254"></margin.target>Per 3. decimi.</s></p><p type="main"> <s>Quoniam foramen CD sectum est in partes CQ, <lb/>QK, KE, EI, ID aequales per constructionem; <lb/>venabitur ratio aquarum per eos fluentium<arrow.to.target n="marg255"/>, & <lb/>proinde aquarum per CE, & ED. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg255"></margin.target>Per 6. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/166.jpg"/><subchap1 n="8" type="proposition"><p type="head"> <s>PROPOSITIO VIII. PROBL. IV.</s></p><subchap2 n="8" type="statement"><p type="main"> <s>Datis foraminibus inaequalibus super eadem <lb/>orizontali, venari rationes aquarum.<figure id="id.064.01.166.1.jpg" xlink:href="064/01/166/1.jpg"/></s></p></subchap2><subchap2 n="8" type="proof"><p type="main"> <s>Sint foramina AB, & CD super orizontali <lb/>BD.</s></p><p type="main"> <s>Quaerenda proportio aquarum ex eis eodem tem­<lb/>pore fluentium.</s></p><p type="main"> <s>Producatur CE FG parallela DB.</s></p><p type="main"> <s>Quoniam nota est ratio aquarum fluentium ex <lb/>CD, & FB<arrow.to.target n="marg256"/>, item per FB, & AG<arrow.to.target n="marg257"/>, Nota est <lb/>pariter ratio ex eis composita inter aquas flu­<lb/>entes per CD, & AG. </s> <s>Cum igitur sit nota ra­<lb/>tio aquae fluentis per CD, ad fluentem per <lb/>FB, & per AG partes, nota erit ratio eiusdem <lb/>ad totam fluentem per AB. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg256"></margin.target>Per 2. huius.</s></p><p type="margin"> <s><margin.target id="marg257"></margin.target>Per 7. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/167.jpg"/><subchap1 n="9" type="proposition"><p type="head"> <s>PROPOSITIO IX. PROBL. V.</s></p><subchap2 n="9" type="statement"><p type="main"> <s>Datis foraminibus, quorum unum superius, <lb/>alterum inferius inter easdem parallelas <lb/>perpendiculares: Reperire rationes aqua­<lb/>rum.<figure id="id.064.01.167.1.jpg" xlink:href="064/01/167/1.jpg"/></s></p></subchap2><subchap2 n="9" type="proof"><p type="main"> <s>Dentur foramina AB, CD inter parallelas <lb/>AC, & DB.</s></p><p type="main"> <s>Venanda ratio aquarum ex eis, aequo tempore, <lb/>fluentium.</s></p><p type="main"> <s>Concipiatur BC tanquam foramen.</s></p><p type="main"> <s>Quoniam nota est ratio aquarum fluentium ex CD, <lb/>& ex CB, item ex CB, & ex AB<arrow.to.target n="marg258"/>, nota est <lb/>pariter ratio ex eis composita aquarum fluen­<lb/>tium per CD, & per AB. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg258"></margin.target>Per 7. huius.</s></p></subchap2></subchap1><pb xlink:href="064/01/168.jpg"/><subchap1 n="10" type="proposition"><p type="head"> <s>PROPOSITIO X. PROBL. VI.</s></p><subchap2 n="10" type="statement"><p type="main"> <s>Datis foraminibus venari aquas.<figure id="id.064.01.168.1.jpg" xlink:href="064/01/168/1.jpg"/></s></p></subchap2><subchap2 n="10" type="proof"><p type="main"> <s>Data sint foramina AD, EH.</s></p><p type="main"> <s>Oportet reperire rationem aquarum per <lb/>illa aequo tempore fluentium.</s></p><p type="main"> <s>Duc orizontales HI, FK, & producta DB in L, con­<lb/>cipiatur IL tanquam foramen; & quaeratur <lb/>ratio aquarum per AD, IL<arrow.to.target n="marg259"/>, & sit ut M ad N. <lb/></s> <s>Item quaeratur ratio IL ad EH,<arrow.to.target n="marg260"/>, & sit ut N ad O.</s></p><p type="margin"> <s><margin.target id="marg259"></margin.target>Per 9. huius.</s></p><p type="margin"> <s><margin.target id="marg260"></margin.target>Per 2. huius.</s></p><p type="main"> <s>Dico M ad O esse rationem aquarum per AD, HE.</s></p><p type="main"> <s>Quoniam ut M ad N ita est AD ad IL, & ut <lb/>N ad O, ita IL ad EH per constr. </s> <s>Erit ex <lb/>aequo ut M ad O, ita AD ad EH<arrow.to.target n="marg261"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg261"></margin.target>Per 22. quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/169.jpg"/><subchap1 n="11" type="proposition"><p type="head"> <s>PROPOSITIO XI. PROBL. VII</s></p><subchap2 n="11" type="statement"><p type="main"> <s>Dato foramine, & linea orizontali intermi­<lb/>nata; constituere super illa foramen, a quo <lb/>aequalis aqua fluat.<figure id="id.064.01.169.1.jpg" xlink:href="064/01/169/1.jpg"/></s></p></subchap2><subchap2 n="11" type="proof"><p type="main"> <s>Dato foramine AB, & orizontali CD.</s></p><p type="main"> <s>Describendum sit foramen super CD, a <lb/>quo effluat aqua ut per AB.</s></p><p type="main"> <s>Erigantur perpendiculares AE, BC, & produca­<lb/>tur DC in E, & super EC fait foramen aequale <lb/>AB, & sit FC, & ducta FG parallela CD, fiat <lb/>HI media inter K summum vasis B, & KE, <lb/>& ut HI ad KE, ita DL ad EC.</s></p><p type="main"> <s>Dico per LG foramen fluere aquam ut per AB.</s></p><p type="main"> <s>Quoniam aqua LG ad aquam FC est ut HI ad <lb/>KE<arrow.to.target n="marg262"/>, & aqua AB ad aquam CF est ut HI ad <lb/>KE<arrow.to.target n="marg263"/>, erit ut aqua LG ad CF, ita aqua AB <lb/>ad CF<arrow.to.target n="marg264"/>, & proinde aqua AB aequalis aquae <lb/>LG<arrow.to.target n="marg265"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg262"></margin.target>Per 2. huius.</s></p><p type="margin"> <s><margin.target id="marg263"></margin.target>Per 5. huius.</s></p><p type="margin"> <s><margin.target id="marg264"></margin.target>Per 11. quinti.</s></p><p type="margin"> <s><margin.target id="marg265"></margin.target>Per nonam quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/170.jpg"/><subchap1 n="12" type="proposition"><p type="head"> <s>PROPOSITIO XII. PROBL. VIII.</s></p><subchap2 n="12" type="statement"><p type="main"> <s>Dato foramine, & latere alterius, reperire fo­<lb/>ramen, e quo aequalis aqua effluat.<figure id="id.064.01.170.1.jpg" xlink:href="064/01/170/1.jpg"/></s></p></subchap2><subchap2 n="12" type="proof"><p type="main"> <s>Datum sit foramen AB, & latere DC. </s> <s>Oportet describere foramen, a quo effluat <lb/>aqua ut ab AB, cuius latus sit CD.</s></p><p type="main"> <s>Ductis CE, & DF, orizontalibus; protrahatur B<lb/>E, & FE intelligatur foramen, & reperiatur ra­<lb/>tio aquarum fluentium ab AB, & ab FE<arrow.to.target n="marg266"/>, <lb/>quae sit ut C ad H; & fiat ut H ad G, ita <lb/>FI ad FK, & a K erigitur perpendicularis KL, <lb/>& fiat foramen cuius latus DC aequale, & <lb/>simile ipsi FL, et sit DM.</s></p><p type="margin"> <s><margin.target id="marg266"></margin.target>Per 9. huius.</s></p><p type="main"> <s>Dico a foramine DM fluere aquam, ut ab AB.</s></p><p type="main"> <s>Quoniam aqua fluens per AB ad fluentem per FE <lb/>est ut G ad H per const. item aqua fluens per FL <lb/>seu ei aequale DM ad fluentem per eandem F<lb/>E est itidem ut G ad H<arrow.to.target n="marg267"/>, aquae fluentes per A<lb/>B & per DM sunt inter se aequales<arrow.to.target n="marg268"/>, DM ig. </s> <s>Est foramen quaesitum. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg267"></margin.target>Per secundum huius.</s></p><p type="margin"> <s><margin.target id="marg268"></margin.target>Per 9. quinti.</s></p></subchap2></subchap1><pb xlink:href="064/01/171.jpg"/><subchap1 n="13" type="proposition"><p type="head"> <s>PROPOSITIO XIII. PROBL. IX.</s></p><subchap2 n="13" type="statement"><p type="main"> <s>Dato foramine, reperire aliud aequale, a quo <lb/>fluat aqua in ratione data.<figure id="id.064.01.171.1.jpg" xlink:href="064/01/171/1.jpg"/></s></p></subchap2><subchap2 n="13" type="proof"><p type="main"> <s>Detur in vase AB foramen C, & data sit <lb/>ratio aquarum D, E, quarum D fluat in <lb/>dato tempore per foramen C.</s></p><p type="main"> <s>Reperiendum ubi fiat aequale foramen, a quo fluat <lb/>in aequali tempore aqua E.</s></p><p type="main"> <s>Fiat ad D, E, AC quarta preportionalis AF<arrow.to.target n="marg269"/>, <lb/>& ad AC, AF tertia proportionalis AG<arrow.to.target n="marg270"/>, & <lb/>in G fiat foramen: quod si fieri nequit proble­<lb/>ma est insolubile. </s> <s>Dico G esse locum forami­<lb/>nis quaesitum.</s></p><p type="margin"> <s><margin.target id="marg269"></margin.target>Per 12. sexti.</s></p><p type="margin"> <s><margin.target id="marg270"></margin.target>Per 11. sexti.</s></p><p type="main"> <s>Quoniam aquae fluentes per dicta foramina sunt <lb/>in subduplicata ratione altitudinum AC, AG<arrow.to.target n="marg271"/>, <lb/>& aquae D, E, sunt pariter in subduplicata ra­<lb/>tione eorumdem altitudinum AC, AG<arrow.to.target n="marg272"/>, aquae <lb/>fluentes per dicta foramina sunt ut aquae D, <lb/>& E<arrow.to.target n="marg273"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg271"></margin.target>Per 5. huius.</s></p><p type="margin"> <s><margin.target id="marg272"></margin.target>Per eamdem.</s></p><p type="margin"> <s><margin.target id="marg273"></margin.target>Per 9. quinti.</s></p></subchap2><pb xlink:href="064/01/172.jpg"/><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Parum refert sint foramina quadrata nec ne.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Idem sequitur si ambo foramina sint rotunda.</s></p></subchap2></subchap1><pb xlink:href="064/01/173.jpg"/><subchap1 n="14" type="proposition"><p type="head"> <s>PROPOSITIO XIV.</s></p><subchap2 n="14" type="statement"><p type="main"> <s>Dato foramine, aptandum sit aliud datum <lb/>simile, magnitudinis diversae, a quo aqua <lb/>fluens cum fluente a primo, habeat ratio­<lb/>nem datam.<figure id="id.064.01.173.1.jpg" xlink:href="064/01/173/1.jpg"/></s></p></subchap2><subchap2 n="14" type="proof"><p type="main"> <s>In vase AB, dato foramine C, & alio D ut <lb/>supra dictum est; & data sit ratio aquarum E, F.</s></p><p type="main"> <s>Aptandum est foramen D ea lege, ut aqua per il­<lb/>lud fluens, cum aqua fluente a C, sit ut F ad E.</s></p><p type="main"> <s>Super orizontali ducta CG fiat foramen G, <lb/>aequale foramini D; & perquiratur ratio <lb/>aquarum fluentium per C, & G<arrow.to.target n="marg274"/>, & sit ut E <lb/>ad H: quae si est eadem quae est inter E, & F, <lb/>habemus intentum; ni sit, fiat aliud foramen <lb/>infra seu supra G ei simile, & aequale a quo <lb/>fluat aqua quae cum fluente ab ipso G habeat <lb/>rationem ut H ad F<arrow.to.target n="marg275"/>, & sit I. </s> <s>Quod si fieri <lb/>nequit problema est insolubile. </s> <s>Dico I esse <lb/>foramen quaesitum.</s></p><p type="margin"> <s><margin.target id="marg274"></margin.target>Per 8. huius.</s></p><p type="margin"> <s><margin.target id="marg275"></margin.target>Per 13. huius.</s></p><pb xlink:href="064/01/174.jpg"/><p type="main"> <s>Quoniam probatum fuit aquam C ad aquam <lb/>G esse ut E ad H, & aquam G ad aquam I <lb/>esse ut H ad F, constat aquam C ad aquam I <lb/>esse ut E ad F<arrow.to.target n="marg276"/>. </s> <s>Quod etc.</s></p><p type="margin"> <s><margin.target id="marg276"></margin.target>Per 22. quinti.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium I.</s></p><p type="main"> <s>Parum refert sint ne foramina quadrata, <lb/>nec ne.</s></p></subchap2><subchap2 type="corollary"><p type="head"> <s>Corollarium II.</s></p><p type="main"> <s>Idem sequeretur si essent ambo rotunda<arrow.to.target n="marg277"/>.</s></p><p type="margin"> <s><margin.target id="marg277"></margin.target>Per 3. pet.</s></p><p type="main"> <s>FINIS</s></p></subchap2></subchap1></chap> <pb xlink:href="064/01/175.jpg"/><pb xlink:href="064/01/176.jpg"/><pb xlink:href="064/01/177.jpg"/><pb xlink:href="064/01/178.jpg"/><pb xlink:href="064/01/179.jpg"/><pb xlink:href="064/01/180.jpg"/><pb xlink:href="064/01/181.jpg"/></body><back/></text></archimedes>