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Yet another new version.
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 12:23:20 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Baliani, Giovanni Baptista</author> <title>De Motu Naturali Gravium Solidorum et Liquidorum</title> <date>1646</date> <place>Genf</place> <translator/> <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 id="s.000001">DE MOTV <lb/>NATVRALI <lb/>GRAVIVM SOLIDORVM <lb/>ET LIQUIDORVM <lb/>IO: BAPTISTAE BALIANI <lb/>PATRITII ENVENSIS. </s> </p> <p type="main"> <s id="s.000002">GENVAE</s> </p> <p type="main"> <s id="s.000003">Ex Typographia IO: Mariæ Farroni 1646 <lb/>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 id="s.000004">DE MOTV <lb/>GRAVIVM <lb/>SOLIDORVM <lb/>LIBER PRIMVS.<lb/></s> </p> <p type="main"> <s id="s.000005">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 id="s.000006">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 id="s.000007">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 id="s.000008">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 id="s.000009">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 id="s.000010">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 id="s.000011">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 æliquali experientia com­<lb/>pertum habui. </s> <s id="s.000012">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 id="s.000013">Rimari caepi; an deprehenderim alio­<lb/>rum erit judicium. </s> <s id="s.000014">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 id="s.000015">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 id="s.000016">Idioma latinum elegi ut <lb/>communius. </s> <s id="s.000017">Praemisi aliqua naturalia prin­<lb/>cipia, sine quibus naturales conclusiones <lb/>aliunde duci posse non video. </s> <s id="s.000018">Quae ex prae­<lb/>dictis experimentis innotuerunt, supposi­<lb/>tiones appellare, & a reliquis petitionibus <lb/>secernere libuit. </s> <s id="s.000019">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 id="s.000020">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 id="s.000021">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 id="s.000022">Vale.</s> </p> <pb xlink:href="064/01/011.jpg"/> <p type="main"> <s id="s.000023">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 id="s.000024">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 id="s.000025">In quibus nil re <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 id="s.000026">In quorum fi­ <lb/>dem, &c.</s> </p> <p type="main"> <s id="s.000027">Ex Conuentu Sancti&longs;&longs;ime Annunciatæ Veteris <lb/>Genue 27. Nouembris 1646.</s> </p> <p type="main"> <s id="s.000028">Magi&longs;t. Fr. Angelicus Riccobonus Aug.</s> </p> <p type="main"> <s id="s.000029">IMPRIMATVR.</s> </p> <p type="main"> <s id="s.000030">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 id="s.000031">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000032">Pendulus dicimus pondus filo <lb/>appensum.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000033">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 id="s.000034">Vibrationes pendulorum sunt eorum motus hinc <lb/>inde </s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000035">Vibrationes aequales dicimus, quae fiunt per spa­<lb/>tia aequalia, & e contra inaequales.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000036">Vibrationes aeque celeres si fiant per spatia aequa­<lb/>lia tempore aequali.</s> </p> </subchap2> <pb xlink:href="064/01/014.jpg"/> <subchap2 type="definition"> <p type="main"> <s id="s.000037">Vibrationis diuturnitatem dicimus ipsius Dura­<lb/>tionem, tempus nimirum, quo ipsa vibratio <lb/>perficitur.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000038">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 id="s.000039">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 id="s.000040">Vibrationis portio est pars arcus, quem ipsa vi­<lb/>bratio disignant.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.000041">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 id="s.000042">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 id="s.000043">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 id="s.000044">SUPPOSITIONES</s> </p> <subchap2 type="supposition"> <p type="main"> <s id="s.000045">PRIMA. </s> <s id="s.000046">Solidorum aequipendu­<lb/>lorum cujuscumque gravitatis vibra­<lb/>tiones aequales sunt aequediu­<lb/>turnae.</s> </p> </subchap2> <subchap2 type="supposition"> <p type="main"> <s id="s.000047">2 Equipendulorum eorundem vibrationes <lb/>sunt aequediuturnae, etiamsi inaequales.</s> </p> </subchap2> <subchap2 type="supposition"> <p type="main"> <s id="s.000048">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 id="s.000049">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 id="s.000050">PETITIONES, SEU POSTULATA</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000051">Pr. </s> <s id="s.000052">Pendulorum inaequalium portiones similes vi­<lb/>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 id="s.000053">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 id="s.000054">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 id="s.000055">2. Ut est momentum ad momentum solidi <lb/>gravis, ita velocitas ad velocitatem.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000056">Huiusmodi passio communiter attribui solet gra­<lb/>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 id="s.000057">3. Portiones minimae peripheriae Circuli con­<lb/>cipiende sunt, ac si essent lineae rectae.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000058">Quaecumque arcus portio est circularis, atta­<lb/>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 id="s.000059">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 id="s.000060">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 id="s.000061">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 id="s.000062">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 id="s.000063">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 id="s.000064">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 id="s.000065">PRONUNCIATA</s> </p> <subchap2 type="enunciation"> <p type="main"> <s id="s.000066">P. </s> <s id="s.000067">Quae sunt aequidiuturna tertio, sunt aequi­<lb/>diuturna inter se.</s> </p> </subchap2> <subchap2 type="enunciation"> <p type="main"> <s id="s.000068">2. Quadrata datorum temporum, sunt etiam <lb/>quadrata aliorum datis aequalium.</s> </p> </subchap2> <subchap2 type="enunciation"> <p type="main"> <s id="s.000069">3. Gravia eadem super planis aequalibus & <lb/>pariter inclinatis, pariter moventur.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/020.jpg"/> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000070">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000071">Solidi penduli naturaliter moti vibratio­<lb/>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 id="s.000072">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 id="s.000073">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 id="s.000074">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 id="s.000075">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 id="s.000076">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000077"><margin.target id="marg1"/>Per primam suppositionem.</s> </p> <p type="margin"> <s id="s.000078"><margin.target id="marg2"/>Per secundam suppositionem.</s> </p> <p type="margin"> <s id="s.000079"><margin.target id="marg3"/>Per pr. pron.</s> </p> </subchap2> </subchap1> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000080">PROPOSITIO II. PROB. PRIMUM</s> </p> <pb xlink:href="064/01/021.jpg"/> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000081">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 id="s.000082">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 id="s.000083"><margin.target id="marg4"/>Per pr. huius.</s> </p> <p type="main"> <s id="s.000084">Venanda est longitudo filorum, quibus applicata <lb/>dicta solida producant vibrationes quaesitas.</s> </p> <p type="main"> <s id="s.000085">Fiat L tertia proportionalis ad C, & D,<arrow.to.target n="marg5"/> & 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 id="s.000086"><margin.target id="marg5"/>Per 11 sexti.</s> </p> <p type="margin"> <s id="s.000087"><margin.target id="marg6"/>Per 12 sexti.</s> </p> <p type="main"> <s id="s.000088">Quoniam ita est IA ad KB ut C ad L per constr. <lb/>erunt C, & D diuturnitates vibrorum pendu­<lb/>lorum AB.<arrow.to.target n="marg7"/> Quod etc</s> </p> <p type="margin"> <s id="s.000089"><margin.target id="marg7"/>Per 3 Supp.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/022.jpg"/> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000090">PROPOSITIO TERTIA</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000091">Lineae descensus gravium, dum naturali motu <lb/>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 id="s.000092">Sint LN, KM linea descensus gravium L, K, <lb/>& sint PO ipsorum diuturnitates.</s> </p> <p type="main"> <s id="s.000093">Dico LN, KM esse in duplicata ratione ipsarum P, O.</s> </p> <p type="main"> <s id="s.000094">Sint pendula AH, AI, dependentia a puncto A, & <lb/>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 id="s.000095"><margin.target id="marg8"/>Per 3 pet.</s> </p> <p type="margin"> <s id="s.000096"><margin.target id="marg9"/>Per 11 sexti.</s> </p> <p type="main"> <s id="s.000097">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 id="s.000098">Quod, etc.</s> </p> <p type="margin"> <s id="s.000099"><margin.target id="marg10"/>Per 5 pet.</s> </p> <p type="margin"> <s id="s.000100"><margin.target id="marg11"/>Per p. pet.</s> </p> <p type="margin"> <s id="s.000101"><margin.target id="marg12"/>Per 3 supp.</s> </p> <p type="margin"> <s id="s.000102"><margin.target id="marg13"/>Per p. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/023.jpg"/> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000103">PROPOSITIO QUARTA. PROB. II.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000104">Data diuturnitate gravis descendentis a data <lb/>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 id="s.000105">Sit A diuturnitas gravis B, dum cadit in C, & <lb/>data sit diuturnitas quaecumque D.</s> </p> <p type="main"> <s id="s.000106">Constituenda est alia altitudo, a qua grave de­<lb/>scendat iuxta diuturnitatem D.</s> </p> <p type="main"> <s id="s.000107">Fiat I, tertia proportionalis ad AD,<arrow.to.target n="marg14"/> & 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 id="s.000108"><margin.target id="marg14"/>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000109"><margin.target id="marg15"/>Per 12. sexti.</s> </p> <p type="main"> <s id="s.000110">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 id="s.000111">Quod fuit faciendum.</s> </p> <p type="margin"> <s id="s.000112"><margin.target id="marg16"/>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/024.jpg"/> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000113">PROPOSITIO V. PROB. III.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000114">Data altitudine, a qua descendat grave in no­<lb/>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 id="s.000115">Sit A altitudo per quam descendat grave diutur­<lb/>nitate B nota, & data sit alia altitudo C.</s> </p> <p type="main"> <s id="s.000116">Oportet reperire quanta sit diuturnitas, qua idem <lb/>grave descendat per C.</s> </p> <p type="main"> <s id="s.000117">Fiat ut A ad C ita B ad G,<arrow.to.target n="marg17"/> inter quas media, <lb/>proportionalis F<arrow.to.target n="marg18"/> est diuturnitas quaesita.</s> </p> <p type="margin"> <s id="s.000118"><margin.target id="marg17"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000119"><margin.target id="marg18"/>Per 13. sexti.</s> </p> <p type="main"> <s id="s.000120">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 id="s.000121"><margin.target id="marg19"/>Per 3. huius.</s> </p> <p type="main"> <s id="s.000122">Quod faciendum fuit.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/025.jpg"/> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000123">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000124">Gravia naturali motu descendunt semper velo­<lb/>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 id="s.000125">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 id="s.000126">Dico quod lineae AB, BC, CD sunt inter se ut 1.<lb/>3.5.& sic deinceps.</s> </p> <p type="main"> <s id="s.000127">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 id="s.000128">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 id="s.000129">Quod <lb/>probandum fuit.</s> </p> <p type="margin"> <s id="s.000130"><margin.target id="marg20"/>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/026.jpg"/> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.000131">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000132">Lineae descensus gravium super plano incli­<lb/>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 id="s.000133">Sint AB, CD plana pariter inclinata, super <lb/>quibus moveantur gravia A, C, & sint EF <lb/>ipsorum diuturnitates.</s> </p> <p type="main"> <s id="s.000134">Dico AB, CD, esse in duplicata ratione ipsarum E, F.</s> </p> <p type="main"> <s id="s.000135">Secetur AB bifariam in G, & erecta GH, per­<lb/>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 id="s.000136"><margin.target id="marg21"/>Per 3. pet.</s> </p> <p type="main"> <s id="s.000137">Quoniam EF sunt diuturnitates AB, CD per <pb xlink:href="064/01/027.jpg"/>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 id="s.000138"><margin.target id="marg22"/>Per 6. pet.</s> </p> <p type="margin"> <s id="s.000139"><margin.target id="marg23"/>Per pr. pet.</s> </p> <p type="margin"> <s id="s.000140"><margin.target id="marg24"/>Per 3. supp.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000141">Corollarium</s> </p> <p type="main"> <s id="s.000142">Hinc patet esse longitudines planorum per quae <lb/>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 id="s.000143">PROPOSITIO VIII. PROB. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000144">Dato plano inclinato, super quo per spatium <lb/>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 id="s.000145">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 id="s.000146">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 id="s.000147">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 id="s.000148">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 id="s.000149">Quod <lb/>faciendum erat.</s> </p> <p type="margin"> <s id="s.000150"><margin.target id="marg25"/>Per 6. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/029.jpg"/> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000151">PROPOSITIO IX. PROB. V.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000152">Dato plano inclinato, super quo per spatium <lb/>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 id="s.000153">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 id="s.000154">Quaerendum quanta sit diuturnitas gravis in BG.</s> </p> <p type="main"> <s id="s.000155">Intelligatur BC diuturnitas ipsius BC, & fiat <lb/>BH, media inter BC, & BG, quae erit diu­<lb/>turnitas quaesita.</s> </p> <p type="main"> <s id="s.000156">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 id="s.000157">Quod, etc.</s> </p> <p type="margin"> <s id="s.000158"><margin.target id="marg26"/>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/030.jpg"/> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000159">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000160">Gravia descendunt super planis inclinatis per <lb/>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 id="s.000161">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 id="s.000162">Dico quod lineae AB, BC, CD sunt inter se ut <lb/>1. 3. 5. &. sic deinceps.</s> </p> <p type="main"> <s id="s.000163">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 id="s.000164">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 id="s.000165">Quod probandum erat.</s> </p> <p type="margin"> <s id="s.000166"><margin.target id="marg27"/>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/031.jpg"/> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000167">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000168">Si Duo gravia descendant alterum super li­<lb/>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 id="s.000169">Sit ABC planum normaliter erectum super <lb/>lineam orizontalem BC, cuius latus AB sit <lb/>perpendiculare, & AC, inclinatum.</s> </p> <p type="main"> <s id="s.000170">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 id="s.000171">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 id="s.000172">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000173"><margin.target id="marg28"/>Per 4. supp.</s> </p> <p type="margin"> <s id="s.000174"><margin.target id="marg29"/>Per 2. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/032.jpg"/> <subchap1 n="12" type="proposition"> <p type="head"> <s id="s.000175">PROPOSITIO XII.</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.000176">Gravia descendunt super plana diverse in­<lb/>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 id="s.000177">Sint F, D plana inclinata ducta ad idem pla­<lb/>num orizontale.</s> </p> <p type="main"> <s id="s.000178">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 id="s.000179">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 id="s.000180">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 id="s.000181">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000182"><margin.target id="marg30"/>Per 11. huius.</s> </p> <p type="margin"> <s id="s.000183"><margin.target id="marg31"/>Per 13. Quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/033.jpg"/> <subchap1 n="13" type="proposition"> <p type="head"> <s id="s.000184">PROPOSITIO XIII. PROB. VI.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.000185">Reperire inclinationem plani, super quo <lb/>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 id="s.000186">Moveatur grave A super recta AB, seu <lb/>perpendiculari, seu inclinata, & data sit <lb/>proportio C ad D.</s> </p> <p type="main"> <s id="s.000187">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 id="s.000188">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 id="s.000189">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 id="s.000190">Quod faciendum fuit.</s> </p> <p type="margin"> <s id="s.000191"><margin.target id="marg32"/>Per 12. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/034.jpg"/> <subchap1 n="14" type="proposition"> <p type="head"> <s id="s.000192">PROPOSITIO XIV. PROB. VII.</s> </p> <subchap2 n="14" type="statement"> <p type="main"> <s id="s.000193">Data linea perpendiculari, per quam grave <lb/>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 id="s.000194">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 id="s.000195">Oportet in plano AC reperire punctum quo gra­<lb/>ve perveniat eodem tempore, quo in B.</s> </p> <p type="main"> <s id="s.000196">Fiat ut AC ad AB, ita AB ad tertiam AD<arrow.to.target n="marg33"/>, <lb/>& D erit punctum quaesitum.</s> </p> <p type="margin"> <s id="s.000197"><margin.target id="marg33"/>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000198">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 id="s.000199"><margin.target id="marg34"/>Per 11. huius.</s> </p> <p type="margin"> <s id="s.000200"><margin.target id="marg35"/>Per 3. & 7. huius.</s> </p> </subchap2> <pb xlink:href="064/01/035.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.000201">Corollarium 1.</s> </p> <p type="main"> <s id="s.000202">Hinc est quod in D, & B velocitates sunt ut AD, <lb/>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 id="s.000203">Corollarium 2.</s> </p> <p type="main"> <s id="s.000204">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 id="s.000205"><margin.target id="marg36"/>Per 7. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000206">Corollarium 3.</s> </p> <p type="main"> <s id="s.000207">Si AE est quadrupla AD, AF erit dupla AD, <lb/>unde tempus AE erit duplum tempori AB.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000208">Corollarium 4.</s> </p> <p type="main"> <s id="s.000209">Si AC esset quadrupla AD, grave moveretur <lb/>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 id="s.000210">PROPOSITIO XV.</s> </p> <subchap2 n="15" type="statement"> <p type="main"> <s id="s.000211">Si duo gravia descendunt alterum quidem <lb/>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 id="s.000212">Sit linea AB perpendiculariter erecta super <lb/>plano Orizontali BC, & AC planum declinans.</s> </p> <p type="main"> <s id="s.000213">Dico quod diuturnitates gravium descendentium <lb/>per AB, & per AC, sunt ut AB ad AC.</s> </p> <p type="main"> <s id="s.000214">Fiat AD tertia proportionalis ad AC, & AB<arrow.to.target n="marg37"/>,</s> </p> <p type="margin"> <s id="s.000215"><margin.target id="marg37"/>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000216">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 id="s.000217">Quod, etc.</s> </p> <p type="margin"> <s id="s.000218"><margin.target id="marg38"/>Per cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000219"><margin.target id="marg39"/>Per 14. huius.</s> </p> <p type="margin"> <s id="s.000220"><margin.target id="marg40"/>Per 2. pron.</s> </p> <p type="margin"> <s id="s.000221"><margin.target id="marg41"/>Per 19. Sexti.</s> </p> <p type="margin"> <s id="s.000222"><margin.target id="marg42"/>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000223"><margin.target id="marg43"/>Per 22. Sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/037.jpg"/> <subchap1 n="16" type="proposition"> <p type="head"> <s id="s.000224">PROPOSITIO XVI. PROBL. VIII.</s> </p> <subchap2 n="16" type="statement"> <p type="main"> <s id="s.000225">Data linea perpendiculari, & plano decli­<lb/>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 id="s.000226">Data sit perpendicularis AB, cui connexum <lb/>planum inclinatum AD.</s> </p> <p type="main"> <s id="s.000227">Oportet in AB producta reperire punctum, quo <lb/>perveniat grave eo tempore, quo pervenit in <lb/>puncto D.</s> </p> <p type="main"> <s id="s.000228">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 id="s.000229">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 id="s.000230"><margin.target id="marg44"/>Per 32. prim.</s> </p> <p type="margin"> <s id="s.000231"><margin.target id="marg45"/>Per 4. sexti.</s> </p> <p type="margin"> <s id="s.000232"><margin.target id="marg46"/>Per 4. sexti.</s> </p> <p type="margin"> <s id="s.000233"><margin.target id="marg47"/>Per 14 huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000234">Corollarium</s> </p> <p type="main"> <s id="s.000235">Hinc est quod super plano AC erit AD men­<lb/>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 id="s.000236">PROPOSITIO XVII. PROBL. IX.</s> </p> <subchap2 n="17" type="statement"> <p type="main"> <s id="s.000237">Dato plano declinante, super quo grave de­<lb/>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 id="s.000238">Sint plana AB, AC quorum AC minus in­<lb/>clinatum.</s> </p> <p type="main"> <s id="s.000239">Oportet in AC reperire punctum, quo grave per­<lb/>veniat, quando pervenit in B.</s> </p> <p type="main"> <s id="s.000240">Fiat ut AC ad AB ita AB ad AD, & dico D <lb/>esse punctum quaesitum.</s> </p> <p type="main"> <s id="s.000241">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 id="s.000242">Quod, etc.</s> </p> <p type="margin"> <s id="s.000243"><margin.target id="marg48"/>Per 19. sexti.</s> </p> <p type="margin"> <s id="s.000244"><margin.target id="marg49"/>Per cot. 7. huius.</s> </p> <p type="margin"> <s id="s.000245"><margin.target id="marg50"/>Per 22. sexti.</s> </p> <p type="margin"> <s id="s.000246"><margin.target id="marg51"/>Per 15. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/039.jpg"/> <subchap1 n="18" type="proposition"> <p type="head"> <s id="s.000247">PROPOSITIO XVIII. PROBL. X.</s> </p> <subchap2 n="18" type="statement"> <p type="main"> <s id="s.000248">Datis planis declinantibus ortis ab eodem <lb/>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 id="s.000249">Datum sit planum minus declinans AC, & <lb/>magis AD, terminantia super plano ori­<lb/>zontali BD.</s> </p> <p type="main"> <s id="s.000250">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 id="s.000251">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 id="s.000252">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 id="s.000253">Quod, etc.</s> </p> <p type="margin"> <s id="s.000254"><margin.target id="marg52"/>Per 19. sexti.</s> </p> <p type="margin"> <s id="s.000255"><margin.target id="marg53"/>Per cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000256"><margin.target id="marg54"/>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000257"><margin.target id="marg55"/>Per 22. sexti.</s> </p> <p type="margin"> <s id="s.000258"><margin.target id="marg56"/>Per 15. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/040.jpg"/> <subchap1 n="19" type="proposition"> <p type="head"> <s id="s.000259">PROPOSITIO XIX. PROBL. XI.</s> </p> <subchap2 n="19" type="statement"> <p type="main"> <s id="s.000260">Dato motus naturali gravis quomodocumque <lb/>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 id="s.000261">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 id="s.000262">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 id="s.000263">Ducatur AC orizontalis, & fiat BD tertia pro­<lb/>portionalis ad CB AB<arrow.to.target n="marg57"/>, & D est punctum <lb/>quaesitum. </s> <s id="s.000264">Quod ut probetur.</s> </p> <p type="margin"> <s id="s.000265"><margin.target id="marg57"/>Per 11. Sexti.</s> </p> <p type="main"> <s id="s.000266">Fiat iterum rectae AC paralella, & aequalis BE, & <lb/>ducta EA, secetur recta BF parallela ipsi AD.</s> </p> <p type="main"> <s id="s.000267">Quoniam AF, BD sunt pariter inclinatae, & <lb/>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 id="s.000268"><margin.target id="marg58"/>Per 33. Primi.</s> </p> <p type="margin"> <s id="s.000269"><margin.target id="marg59"/>Per 3. pronun.</s> </p> <p type="margin"> <s id="s.000270"><margin.target id="marg60"/>Per 17 huius.</s> </p> <p type="margin"> <s id="s.000271"><margin.target id="marg61"/>Per 1. pron.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000272">Corollarium</s> </p> <p type="main"> <s id="s.000273">Hinc est quod super plano CB, DB est mensura <lb/>diuturnitatis motus in AB.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/041.jpg"/> <subchap1 n="20" type="proposition"> <p type="head"> <s id="s.000274">PROPOSITIO XX. PROBL. XII.</s> </p> <subchap2 n="20" type="statement"> <p type="main"> <s id="s.000275">Datis duobus planis diverse inclinatis lon­<lb/>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 id="s.000276">Sint plana AB, CD inclinata, & sit data diu­<lb/>turnitas E plani AB.</s> </p> <p type="main"> <s id="s.000277">Oportet reperire diuturnitatem plani CD.</s> </p> <p type="main"> <s id="s.000278">Fiat AF, paralella, & aequalis datae CD, in qua <lb/>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 id="s.000279"><margin.target id="marg62"/>Per 17. huius.</s> </p> <p type="margin"> <s id="s.000280"><margin.target id="marg63"/>Per 9. huius.</s> </p> <p type="main"> <s id="s.000281">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 id="s.000282">Quod, etc.</s> </p> <p type="margin"> <s id="s.000283"><margin.target id="marg64"/>Per 3. pron.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/042.jpg"/> <subchap1 n="21" type="proposition"> <p type="head"> <s id="s.000284">PROPOSITIO XXI. PROBL. XIII.</s> </p> <subchap2 n="21" type="statement"> <p type="main"> <s id="s.000285">Datis duabus diuturnitatibus, quarum prior <lb/>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 id="s.000286">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 id="s.000287">Oportet reperire in D punctum quo grave per­<lb/>veniat in diuturnitate E.</s> </p> <p type="main"> <s id="s.000288">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 id="s.000289"><margin.target id="marg65"/>Per 17. huius.</s> </p> <p type="margin"> <s id="s.000290"><margin.target id="marg66"/>Per 8. huius.</s> </p> <p type="main"> <s id="s.000291">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 id="s.000292">PROPOSITIO XXII.</s> </p> <subchap2 n="22" type="statement"> <p type="main"> <s id="s.000293">Data perpendiculari seu plano quomodoli­<lb/>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 id="s.000294">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 id="s.000295">Quaerenda eius diuturnitas.</s> </p> <p type="main"> <s id="s.000296">Fiat CG diuturnitas AE, & CH diuturnitas <lb/>AF<arrow.to.target n="marg67"/>, GH est diuturnitas quaesita.</s> </p> <p type="margin"> <s id="s.000297"><margin.target id="marg67"/>Per 5. aut 9. huius.</s> </p> <p type="main"> <s id="s.000298">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 id="s.000299">PROPOSITIO XXIII.</s> </p> <subchap2 n="23" type="statement"> <p type="main"> <s id="s.000300">Duo gravia descendentia super planis diversa <lb/>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 id="s.000301">Datis planis AB, AC declinantibus, ductis <lb/>ab eodem puncto A ad planum orizontale BC. <lb/> </s> <s id="s.000302">Dico quod diuturnitates gravium descendentium <lb/>per AB, AC sint ut AB ad AC.</s> </p> <p type="main"> <s id="s.000303">Fiat ut AC ad AB ita AB ad AD, ita ut grave <lb/>perveniat in D eodem tempore quo pervenit in B<arrow.to.target n="marg68"/>.</s> </p> <p type="margin"> <s id="s.000304"><margin.target id="marg68"/>Per 17. huius.</s> </p> <p type="main"> <s id="s.000305">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 id="s.000306">Quod fuit probandum.</s> </p> <p type="margin"> <s id="s.000307"><margin.target id="marg69"/>Per Cor. 7. huius.</s> </p> <p type="margin"> <s id="s.000308"><margin.target id="marg70"/>Per const.</s> </p> <p type="margin"> <s id="s.000309"><margin.target id="marg71"/>Per 2. pronun.</s> </p> <p type="margin"> <s id="s.000310"><margin.target id="marg72"/>Per 10. sexti.</s> </p> <p type="margin"> <s id="s.000311"><margin.target id="marg73"/>Per 22. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/045.jpg"/> <subchap1 n="24" type="proposition"> <p type="head"> <s id="s.000312">PROPOSITIO XXIV</s> </p> <subchap2 n="24" type="statement"> <p type="main"> <s id="s.000313">Datis planis, & perpendiculari ad eadem li­<lb/>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 id="s.000314">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 id="s.000315">Dico quod diuturnitates gravium super ipsis mo­<lb/>torum, sunt ut AC, DC, BC.</s> </p> <p type="main"> <s id="s.000316">Ducatur CE paralella ipsi AB, & a puncto A du­<lb/>cantur paralellae ipsis CB, CD, & sint AE, AF.</s> </p> <p type="main"> <s id="s.000317">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 id="s.000318"><margin.target id="marg74"/>Per 23. huius.</s> </p> <p type="margin"> <s id="s.000319"><margin.target id="marg75"/>Per 15. huius.</s> </p> <p type="margin"> <s id="s.000320"><margin.target id="marg76"/>Per 33. prim.</s> </p> <p type="margin"> <s id="s.000321"><margin.target id="marg77"/>Per 3. pron.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/046.jpg"/> <subchap1 n="25" type="proposition"> <p type="head"> <s id="s.000322">PROPOSITIO XXV.</s> </p> <subchap2 n="25" type="statement"> <p type="main"> <s id="s.000323">In circulo Orthogonaliter erecto, si a sum­<lb/>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 id="s.000324">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 id="s.000325">Dico quod eodem tempore pervenient ad, D, E, F, C.</s> </p> <p type="main"> <s id="s.000326">Ducantur DC, EC, FC.</s> </p> <p type="main"> <s id="s.000327">Quoniam puncta praedicta sunt ea, in quae cadunt <lb/>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 id="s.000328">Quod probandum fuit.</s> </p> <p type="margin"> <s id="s.000329"><margin.target id="marg78"/>Per 30. Tertij.</s> </p> <p type="margin"> <s id="s.000330"><margin.target id="marg79"/>Per 16. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/047.jpg"/> <subchap1 n="26" type="proposition"> <p type="head"> <s id="s.000331">PROPOSITIO XXVI.</s> </p> <subchap2 n="26" type="statement"> <p type="main"> <s id="s.000332">Sit in circulo erecto, a puncto inferiori ducan­<lb/>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 id="s.000333">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 id="s.000334">Dico quod eodem tempore perveniunt in C.<lb/></s> </p> <p type="main"> <s id="s.000335">A puncto A ducantur AF, AG paralellae ipsis <lb/>CE, CD, & ducantur AF, FC.</s> </p> <p type="main"> <s id="s.000336">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 id="s.000337">Quod, etc.</s> </p> <p type="margin"> <s id="s.000338"><margin.target id="marg80"/>Per 29. primi.</s> </p> <p type="margin"> <s id="s.000339"><margin.target id="marg81"/>Per 30. Tertij.</s> </p> <p type="margin"> <s id="s.000340"><margin.target id="marg82"/>Per 16. primi.</s> </p> <p type="margin"> <s id="s.000341"><margin.target id="marg83"/>Per 25. huius.</s> </p> <pb xlink:href="064/01/049.jpg"/> <p type="main"> <s id="s.000342">POSTULATUM VII</s> </p> <p type="main"> <s id="s.000343">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 id="s.000344">Videtur probabile. </s> <s id="s.000345">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 id="s.000346">Secundo. </s> <s id="s.000347">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 id="s.000348">Tertio. </s> <s id="s.000349">Quia videmus aquam per siphones rectos, <lb/>sive obliquos, seu inclinatos ductam, pariter <lb/>ascendere, si pariter descendat. </s> <s id="s.000350">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 id="s.000351">PROPOSITIO XXVII. PROBL. XIV.</s> </p> <subchap2 n="27" type="statement"> <p type="main"> <s id="s.000352">Dato gravi moto perpendiculariter per spa­<lb/>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 id="s.000353">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 id="s.000354">Venanda est diuturnitas eius in ipso BD.</s> </p> <p type="main"> <s id="s.000355">Producatur DB donec concurrat cum AE orizon­<lb/>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 id="s.000356"><margin.target id="marg84"/>* Est quarta tertij.</s> </p> <pb xlink:href="064/01/051.jpg"/> <p type="main"> <s id="s.000357">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 id="s.000358">Quod, etc.</s> </p> <p type="margin"> <s id="s.000359"><margin.target id="marg85"/>Per 22 huius.</s> </p> <p type="main"> <s id="s.000360">Inde sequitur quod summa diuturnitatum C, & <lb/>LM, est diuturnitas totius ABD.**</s> </p> <p type="main"> <s id="s.000361">Eadem operatione pariter reperietur diuturni­<lb/>tas BD si BD sit perpendicularis, & AB <lb/>inclinata.</s> </p> <p type="main"> <s id="s.000362">Item si ambo sint plana inclinata.</s> </p> <p type="main"> <s id="s.000363">Ducta AD facile reperietur diuturnitas in ipsa <lb/>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 id="s.000364">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 id="s.000365">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 id="s.000366">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 id="s.000367">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 id="s.000368">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 id="s.000369"><margin.target id="marg86"/>Per 7. post.</s> </p> <p type="margin"> <s id="s.000370"><margin.target id="marg87"/>** Est pars secunda quartae tertij.</s> </p> <p type="margin"> <s id="s.000371"><margin.target id="marg88"/>*** Est Tertia tertij.</s> </p> <p type="margin"> <s id="s.000372"><margin.target id="marg89"/>**** Est corol. quartae tertij.</s> </p> <p type="main"> <s id="s.000373">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 id="s.000374">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 id="s.000375">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 id="s.000376">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 id="s.000377">DE MOTV <lb/>GRAVIVM <lb/>SOLIDORVM <lb/>LIBER SECVNDVS <lb/> VBI DE IMPETV.</s> </p> <p type="main"> <s id="s.000378">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 id="s.000379">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 id="s.000380">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 id="s.000381">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;tudia dimittere &longs;æpius <lb/>mihi opus fui&longs;&longs;et.</s> <s id="s.000382">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 id="s.000383">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 id="s.000384">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 id="s.000385">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000386">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 id="s.000387">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 id="s.000388">Petitio</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000389">Impetus sunt ut spatia, quae eius virtute aequis <lb/>temporibus permeantur.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.000390">Axiomata</s> </p> <subchap2 type="axiom"> <p type="main"> <s id="s.000391">1. Pares causae producunt pares effectus.</s> </p> </subchap2> <subchap2 type="axiom"> <p type="main"> <s id="s.000392">2. In effectu procedente a duabus causis, ablata eius <lb/>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 id="s.000393">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000394">Grave in motu naturali, sive perpendiculari, <lb/>sive inclinato, fertur sine ope gravitatis, <lb/>aequali tempore, per duplum spatii 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 id="s.000395">Dato gravi A naturaliter la­<lb/>to ab A ad B tempore ab, <lb/>cuius aequale sit tempus bc, & <lb/>spatium BC, sit duplum spati AB. <lb/></s> <s id="s.000396">Dico quod tempore bc fertur grave <lb/>sine ope gravitatis per spatium <lb/>aequale ipsi BC.</s> </p> <p type="main"> <s id="s.000397">Producatur AB, sumaturque portio <lb/>BD aequalis, & DE dupla lineae AB, & pro­<lb/>inde aequalis ipsi BC.</s> </p> <p type="main"> <s id="s.000398">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 id="s.000399"><margin.target id="marg90"/>Per axioma primum.</s> </p> <p type="margin"> <s id="s.000400"><margin.target id="marg91"/>Per 3. primi huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000401">Corollarium Primum</s> </p> <p type="main"> <s id="s.000402">Hinc sequitur quod si spatium AB sectum esset <lb/>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 id="s.000403">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 id="s.000404">Corollarium II</s> </p> <p type="main"> <s id="s.000405">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 id="s.000406">Corollarium III</s> </p> <p type="main"> <s id="s.000407">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 id="s.000408">PROPOSITIO II. PROBL. I.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000409"><figure id="id.064.01.061.1.jpg" xlink:href="064/01/061/1.jpg"/>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 id="s.000410">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 id="s.000411"><margin.target id="marg92"/>Per pr. huius.</s> </p> <p type="main"> <s id="s.000412">Oportet reperire portionem spatii 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 id="s.000413">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 temporum 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 id="s.000414"><margin.target id="marg93"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000415"><margin.target id="marg94"/>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000416"><margin.target id="marg95"/>Per 10. def. quinti.</s> </p> <p type="margin"> <s id="s.000417"><margin.target id="marg96"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000418"><margin.target id="marg97"/>Per 11. sexti.</s> </p> <p type="margin"> <s id="s.000419"><margin.target id="marg98"/>Per 10 def. 5.</s> </p> <p type="main"> <s id="s.000420"><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 id="s.000421">Dico EI esse portionem quaesitam.</s> </p> <p type="main"> <s id="s.000422">Quoniam AE est casus gravis A tempore ae per <lb/>supp. & AE, AC sunt in dupl. ratione tem­<lb/>porum ae, ac per constr. </s> <s id="s.000423">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 id="s.000424"><margin.target id="marg99"/>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.000425"><margin.target id="marg100"/>Per axioma primum.</s> </p> <p type="main"> <s id="s.000426">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 id="s.000427"><margin.target id="marg101"/>Per 3. primi huius.</s> </p> <p type="margin"> <s id="s.000428"><margin.target id="marg102"/>Per 19. Quinti.</s> </p> <p type="margin"> <s id="s.000429"><margin.target id="marg103"/>Per axioma primum.</s> </p> <p type="margin"> <s id="s.000430"><margin.target id="marg104"/>Per axioma secundum.</s> </p> <p type="main"> <s id="s.000431">Sit deinde portio temporis eb disiuncta ab ae, puta <lb/>gK, & sit rursus reperienda portio spatij 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 id="s.000432">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 IN, & eodem ratio­<lb/>cinio demonstrabitur IN esse spatium quae­<lb/>situm. </s> <s id="s.000433">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 id="s.000434">PROPOSITIO TERTIA.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000435">In motu naturali gravium, spatia quae conficiun­<lb/>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 id="s.000436">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 spatij 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 id="s.000437"><margin.target id="marg105"/>Per pr. huius.</s> </p> <p type="margin"> <s id="s.000438"><margin.target id="marg106"/>Per 2. huius.</s> </p> <p type="main"> <s id="s.000439">Dico spatium EI ad spatium EB esse ut <lb/>tempus eg ad tempus eh.</s> </p> <p type="main"> <s id="s.000440">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 id="s.000441"><margin.target id="marg107"/>Per 46. primi.</s> </p> <pb xlink:href="064/01/065.jpg"/> <p type="main"> <s id="s.000442">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 id="s.000443">Quod, etc.</s> </p> <p type="margin"> <s id="s.000444"><margin.target id="marg108"/>Per 20. sexti.</s> </p> <p type="margin"> <s id="s.000445"><margin.target id="marg109"/>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.000446"><margin.target id="marg110"/>Per 17. Quinti.</s> </p> <p type="margin"> <s id="s.000447"><margin.target id="marg111"/>Per 22. Quinti.</s> </p> <p type="margin"> <s id="s.000448"><margin.target id="marg112"/>Per 19. Quinti.</s> </p> <p type="margin"> <s id="s.000449"><margin.target id="marg113"/>Per 22. Quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000450">Corollarium Primum</s> </p> <p type="main"> <s id="s.000451">Si portio temporis eh non sit immediata tempori <lb/>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 id="s.000452">Corollarium II</s> </p> <p type="main"> <s id="s.000453">Motus ab impetu proveniens est aequabilis.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/067.jpg"/> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.000454">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000455">In motu naturali impetus successive acquisi­<lb/>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 id="s.000456">Dato gravi moto naturali motu per AC, tem­<lb/>pore ac, & per AB, tempore ab.</s> </p> <p type="main"> <s id="s.000457">Dico impetum seu velocitatem in B ad impetum <lb/>in C esse ut ab ad ac. </s> <s id="s.000458">Concipiantur tempora ab, ac tanquam lineae re­<lb/>ctae metientes tempora ab, ac. </s> <s id="s.000459">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 id="s.000460"><margin.target id="marg114"/>k Per 25. Quinti.</s> </p> <p type="margin"> <s id="s.000461"><margin.target id="marg115"/>l Per 22. Quinti & 43. pr.</s> </p> <p type="main"> <s id="s.000462">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 id="s.000463">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 transignuntur <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 id="s.000464"><margin.target id="marg116"/>m Per 36. primi.</s> </p> <p type="margin"> <s id="s.000465"><margin.target id="marg117"/>n Per 2. huius.</s> </p> <p type="margin"> <s id="s.000466"><margin.target id="marg118"/>o Per primam defin.</s> </p> <p type="margin"> <s id="s.000467"><margin.target id="marg119"/>Per primam huius.</s> </p> <p type="margin"> <s id="s.000468"><margin.target id="marg120"/>Per 13 Sexti.</s> </p> <p type="margin"> <s id="s.000469"><margin.target id="marg121"/>Per tertiam pr. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/069.jpg"/> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000470">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000471">In motu naturali gravium impetus successive <lb/>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 id="s.000472">Iisdem positis.</s> </p> <p type="main"> <s id="s.000473">Dico impetus, seu velocitates in B, & in C <lb/>esse in subduplicata ratione spatiorum <lb/>AB, & AC.</s> </p> <p type="main"> <s id="s.000474">Quoniam impetus in B, & C sunt ut tempora ab, <lb/>ac transacta<arrow.to.target n="marg122"/>.</s> </p> <p type="margin"> <s id="s.000475"><margin.target id="marg122"/>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000476">Sed tempora ab, ac sunt in subduplicata ra­<lb/>tione spatiorum AB, AC<arrow.to.target n="marg123"/>. </s> <s id="s.000477">Pariter impetus <lb/>in B, & in C sunt in subduplicata ratione <lb/>spatiorum AB, AC, quod etc.</s> </p> <p type="margin"> <s id="s.000478"><margin.target id="marg123"/>Per 11 Quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/070.jpg"/> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000479">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <figure id="id.064.01.070.1.jpg" xlink:href="064/01/070/1.jpg"/> <s id="s.000480">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 id="s.000481">Data linea perpendiculari AB, & <lb/>in ea punctis C, D,</s> </p> <p type="main"> <s id="s.000482">Venandi impetus in C, D dum grave ab <lb/>A dimissum fertur per AB.</s> </p> <p type="main"> <s id="s.000483">Sit E media inter AC, AD, item fiat AF media <lb/>inter AC, AB.</s> </p> <p type="main"> <s id="s.000484">Dico impetus in C, D, B esse ut AC, AE, AF.</s> </p> <p type="main"> <s id="s.000485">Quoniam AE est media inter AC, AD per con­<lb/>structionem, AD, AC sunt in duplicata ratio­<lb/>ne rectarum AE, AC<arrow.to.target n="marg124"/>.</s> </p> <p type="margin"> <s id="s.000486"><margin.target id="marg124"/>Per 3. huius.</s> </p> <p type="main"> <s id="s.000487">Ergo AC, AE metiuntur impetus in C & D<arrow.to.target n="marg125"/>.</s> </p> <p type="margin"> <s id="s.000488"><margin.target id="marg125"/>Per pet. huius.</s> </p> <p type="main"> <s id="s.000489">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 id="s.000490">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000491">In quolibet puncto motus reperire spatium, <lb/>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 id="s.000492">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 id="s.000493">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 id="s.000494">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 id="s.000495"><margin.target id="marg126"/>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000496">Dico CF spatium aequari illi, per quod ducetur­ <lb/>grave digressum a C tempore ab.</s> </p> <p type="main"> <s id="s.000497">Quonniam CF ad FE est ut ab ad bc per constructionem, <lb/>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 id="s.000498"><margin.target id="marg127"/>Per 4. huius.</s> </p> <p type="margin"> <s id="s.000499"><margin.target id="marg128"/>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.000500"><margin.target id="marg129"/>Per 10. def. Quinti.</s> </p> <p type="margin"> <s id="s.000501"><margin.target id="marg130"/>Per 5. huius.</s> </p> <p type="main"> <s id="s.000502">Ergo spatium aequale CF conficitur tempore ab, quod etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000503">Corollarium</s> </p> <p type="main"> <s id="s.000504">Huic sequitur quod eodem tempore, puta ab, <lb/>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 id="s.000505">PROPOSITIO VIII.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000506">Si lineae perpendicularis, & inclinata ab eo­<lb/>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 id="s.000507">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 id="s.000508">Dico impetum in B ad impetum in D esse ut AB <lb/>ad AD.</s> </p> <p type="main"> <s id="s.000509">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 id="s.000510"><margin.target id="marg131"/>Per 10. sexti.</s> </p> <p type="main"> <s id="s.000511">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 id="s.000512"><margin.target id="marg132"/>Per 18. Quinti.</s> </p> <p type="margin"> <s id="s.000513"><margin.target id="marg133"/>Per 16. Quinti.</s> </p> <p type="margin"> <s id="s.000514"><margin.target id="marg134"/>Per pr. huius.</s> </p> <p type="main"> <s id="s.000515">At BE, DF sunt ut AB, AD per constr. quip­<lb/>pe earum duplae. </s> <s id="s.000516">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 id="s.000517"><margin.target id="marg135"/>Per cor. 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000518">Corollarium</s> </p> <p type="main"> <s id="s.000519">Impetus sive velocitas in B ad impetum in D <lb/>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 id="s.000520">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000521">Ductis a puncto superno perpendiculari, & <lb/>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 id="s.000522">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 id="s.000523">Dico impetum in D ad impetum in C esse ut AB <lb/>ad AC.</s> </p> <p type="main"> <s id="s.000524">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 id="s.000525"><margin.target id="marg136"/>Per pr. huius.</s> </p> <p type="main"> <s id="s.000526">Sunt itidem in duplicata ratione AC ad AB<arrow.to.target n="marg137"/>.</s> </p> <p type="margin"> <s id="s.000527"><margin.target id="marg137"/>Per 14. pr. huius.</s> </p> <p type="main"> <s id="s.000528">Igitur impetus in C ad impetum in D sunt ut AC <lb/>AB<arrow.to.target n="marg138"/> quod, etc.</s> </p> <p type="margin"> <s id="s.000529"><margin.target id="marg138"/>Per pr. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/074.jpg"/> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000530">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000531">Ductis a puncto superno perpendiculari, & <lb/>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 id="s.000532">A puncto A superno ducatur AB perpendi­<lb/>cularis, & AC declinans ad BC Orizon­<lb/>talem.</s> </p> <p type="main"> <s id="s.000533">Dico, quod in B, & C sunt impetus aequales.</s> </p> <p type="main"> <s id="s.000534">Quoniam impetus in C ad impetum in D est ut <lb/>AC ad AB<arrow.to.target n="marg139"/>.</s> </p> <p type="margin"> <s id="s.000535"><margin.target id="marg139"/>Per pet. huius.</s> </p> <p type="main"> <s id="s.000536">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 id="s.000537"><margin.target id="marg140"/>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000538">Igitur impetus in C, & B sunt aequales<arrow.to.target n="marg141"/>. </s> <s id="s.000539">Quod <lb/>etc.</s> </p> <p type="margin"> <s id="s.000540"><margin.target id="marg141"/>Per 5. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/075.jpg"/> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000541">PROPOSITIO XI. PROBL. IV.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000542">Datis pluribus lineis æqualibus ab eodem <lb/>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 id="s.000543">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 id="s.000544">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 id="s.000545"><margin.target id="marg142"/>Per 10. definit. quinti.</s> </p> <p type="main"> <s id="s.000546">Dico impetus in B, C, D, E esse inter se ut AF, <lb/>AI, AK, AL.</s> </p> <p type="main"> <s id="s.000547">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 id="s.000548"><margin.target id="marg143"/>Per 16. Quinti.</s> </p> <p type="margin"> <s id="s.000549"><margin.target id="marg144"/>Per 9. huius.</s> </p> <p type="main"> <s id="s.000550">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 id="s.000551">PROPOSITIO XII</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.000552">Ductis pluribus lineis diversi mode inclinatis, & <lb/>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 id="s.000553">Sint lineae BD CD diversimode inclinatae, & AD <lb/>perpendicularis, ductae a linea Orizontali AC <lb/>ad punctum inferius D. </s> <s id="s.000554">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 id="s.000555">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 id="s.000556"><margin.target id="marg145"/>Per cor. 8. huius.</s> </p> <p type="margin"> <s id="s.000557"><margin.target id="marg146"/>Per 11. Quinti.</s> </p> <p type="main"> <s id="s.000558">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 id="s.000559">Quod etc.</s> </p> <p type="margin"> <s id="s.000560"><margin.target id="marg147"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000561"><margin.target id="marg148"/>Per 10. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000562">Corollarium</s> </p> <p type="main"> <s id="s.000563">Hinc sequitur, quod si ABC non sit linea, sed planum <lb/>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 id="s.000564">PROPOSITIO XIII. PROBL. V.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.000565">Datis gravibus descendentibus per perpendi­<lb/>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 id="s.000566">Descendat grave per AC perpendicularem , <lb/>& AB declinantem, & dentur puncta B, C.</s> </p> <p type="main"> <s id="s.000567">Reperire proportionem impe­<lb/>tus in B ad impetum in C.</s> </p> <p type="main"> <s id="s.000568">Ducatur BD normalis ad AC<arrow.to.target n="marg149"/>, & 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 id="s.000569"><margin.target id="marg149"/>Per 6. huius.</s> </p> <p type="margin"> <s id="s.000570"><margin.target id="marg150"/>Per 31. primi.</s> </p> <p type="main"> <s id="s.000571">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.</s> </p> <p type="margin"> <s id="s.000572"><margin.target id="marg151"/>Per 13. primi.</s> </p> <p type="margin"> <s id="s.000573"><margin.target id="marg152"/>Per axioma primum.</s> </p> </subchap2> <pb pagenum="78" xlink:href="064/01/078.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.000574">Corollarium</s> </p> <p type="main"> <s id="s.000575">Eodem pacto reperies impetus in planis ut­<lb/>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 id="s.000576">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 id="s.000577">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 id="s.000578">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 id="s.000579">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 id="s.000580">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 id="s.000581">Proijciatur 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 id="s.000582">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 id="s.000583">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 id="s.000584">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/>ijcientis provenit a maiori velocitate, ictus <lb/>eo est validior, quo missile longius a proij<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 id="s.000585">Sed quia adducere <lb/>inconveniens non est solvere argumentum, <lb/>eius fallaciam pro viribus detegere conabor. <lb/></s> <s id="s.000586">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 id="s.000587">Ex quo satis con­<lb/>stare existimo proiectum suo moto parabo­<lb/>lam non describere, quod probandum pro­<lb/>posueram. </s> <s id="s.000588">De quibus proiectis aliquid in­ <lb/>sequentibus addam fortasse ubi occasio <lb/>tulerit. </s> <s id="s.000589">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 id="s.000590">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 id="s.000591">PETITIONES</s> </p> <p type="main"> <s id="s.000592">PRIMA</s> </p> <p type="main"> <s id="s.000593">Peripheria circuli concipiatur tanquam <lb/>constans plurimis, seu mavis infinitis <lb/>lineis rectis.</s> </p> <p type="main"> <s id="s.000594">SECUNDA</s> </p> <p type="main"> <s id="s.000595">Mobile naturaliter motum caeteris pari­<lb/>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 id="s.000596">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000597">Si grave perpendiculariter ductum perse­<lb/>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 id="s.000598">Ducatur grave perpendiculariter per AB, & <lb/>perseveret in motu super BE declinante.</s> </p> <p type="main"> <s id="s.000599">Dico, quod fertur per BE eadem velocitate ac si <lb/>cepisset moveri in D; quod sit ad libellam ipsius A.</s> </p> <p type="main"> <s id="s.000600">Quoniam in B sortitum est eundem impetum <lb/>ductum per AB, ac si latum fuisset per DB<arrow.to.target n="marg153"/>.</s> </p> <p type="margin"> <s id="s.000601"><margin.target id="marg153"/>Per 12. secundi huius.</s> </p> <p type="main"> <s id="s.000602">Ergo per BE ducitur ab eadem virtute seu vi, <lb/>ac si motus initium fuisset in D, quippe ubique <lb/>ducitur a gravitate, & ab impetu in B, & pro­<lb/>inde fertur eadem velocitate. </s> <s id="s.000603">Quod etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000604">Corollarium primum.</s> </p> <p type="main"> <s id="s.000605">Si initium motus fuisset per lineam declinantem, <lb/>& demum per perpendicularem, seu declinantem <lb/>diversa inclinatione, idem probabitur eadem ratione.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000606">Corollarium II.</s> </p> <p type="main"> <s id="s.000607">Hinc sequitur, quod impetus in E est idem si <lb/>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 id="s.000608">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000609">Grave ductum perpendiculariter per spatium <lb/>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 id="s.000610">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 id="s.000611">Venandus ibi motus in dicta diuturnitate AB.</s> </p> <p type="main"> <s id="s.000612">Producatur BD in C donec concurrat cum AC <lb/>orizontaliter ducta ab A ad C. </s> <s id="s.000613">Erit BC diu­<lb/>turnitas ipsius BC<arrow.to.target n="marg154"/>.</s> </p> <p type="margin"> <s id="s.000614"><margin.target id="marg154"/>Per 15. primi huius.</s> </p> <p type="main"> <s id="s.000615">Fiat BE aequalis AB, & CD tertia ad CB, CE<arrow.to.target n="marg155"/>.</s> </p> <p type="margin"> <s id="s.000616"><margin.target id="marg155"/>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000617">Dico BD esse quaesitum, nempe spatium transa­<lb/>ctum diuturnitate AB.</s> </p> <p type="main"> <s id="s.000618">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 id="s.000619"><margin.target id="marg156"/>Per 7. pr. huius.</s> </p> <p type="main"> <s id="s.000620">Erit BE diuturnitas BD stante motu praecedenti <lb/>per BC<arrow.to.target n="marg157"/>.</s> </p> <p type="margin"> <s id="s.000621"><margin.target id="marg157"/>Per 19. quinti.</s> </p> <p type="main"> <s id="s.000622">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 id="s.000623"><margin.target id="marg158"/>Per pr. huius.</s> </p> <pb xlink:href="064/01/087.jpg"/> <p type="main"> <s id="s.000624">At AB est aequalis ipsi BE per constructionem.</s> </p> <p type="main"> <s id="s.000625">Ergo motus per BD fit diuturnitate AB. </s> <s id="s.000626">Quod <lb/>etc.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000627">Corollarium I.</s> </p> <p type="main"> <s id="s.000628">Hinc sequitur, quod in quolibet puncto infra <lb/>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 id="s.000629"><margin.target id="marg159"/>Per 12. secundi huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000630">Corollarium II.</s> </p> <p type="main"> <s id="s.000631">Quotiescunque CE est media inter CB, CD, <lb/>etiamsi motus praecedens fuerit per AB; <lb/>BE est diuturnitas motus per BD.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000632">Corollarium III.</s> </p> <p type="main"> <s id="s.000633">Idem sequitur etiamsi AB noni esset perpendicu­<lb/>laris, nam probatur eodem pacto.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000634">Corollarium IV.</s> </p> <p type="main"> <s id="s.000635">Sequitur etiam, quod si datis AB, & CB, <lb/>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 id="s.000636">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 id="s.000637">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 id="s.000638">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 id="s.000639">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 id="s.000640">Ex quo, etiam apparet velocitas in quocunque <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 id="s.000641">PROPOSITIO III</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000642">Ducto gravi super plano inclinato, & in­<lb/>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 id="s.000643">Ducatur grave super AB incli­<lb/>nationis notae, diuturnitate AB <lb/>data, & inde perpendiculariter, per <lb/>BD; venari motum perpendicularem <lb/>in diuturnitate AB.</s> </p> <p type="main"> <s id="s.000644">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 id="s.000645"><margin.target id="marg160"/>Per 15. pr. huius.</s> </p> <p type="margin"> <s id="s.000646"><margin.target id="marg161"/>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000647">Dico BD esse quaesitum.</s> </p> <p type="main"> <s id="s.000648">Quoniam CE est diuturnitas CD<arrow.to.target n="marg162"/>, erit BE <lb/>diuturnitas BD, si motus præcedens fuerit per <lb/>CB; at pariter si per AB<arrow.to.target n="marg163"/>. </s> <s id="s.000649">Ergo diuturni­<lb/>tate AB aequali BE pervenit in D. </s> <s id="s.000650">Quod etc.</s> </p> <p type="margin"> <s id="s.000651"><margin.target id="marg162"/>Per 7. pr. huius.</s> </p> <p type="margin"> <s id="s.000652"><margin.target id="marg163"/>Per pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000653">Corollarium</s> </p> <p type="main"> <s id="s.000654">Hinc sequitur ut in praecedenti, quod impetus <lb/>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 id="s.000655">PROPOSITIO IV</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000656">Dato gravi moto perpendiculariter per spa­<lb/>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 id="s.000657">Moveatur grave per AB perpendiculariter <lb/>diuturnitate data, quae sit eadem AB, inde <lb/>super planum inclinatum BD.</s> </p> <p type="main"> <s id="s.000658">Perquirenda est diuturnitas motus per BD, & per ABD.</s> </p> <p type="main"> <s id="s.000659">Fiat CE media inter CB, CD, & AF nor­<lb/>malis ad BD productam usquequo concurrat <lb/>cum orizontali AC.</s> </p> <p type="main"> <s id="s.000660">Dico BE esse diuturnitatem per motus BD, & <lb/>FE esse diuturnitatem motus per ABD.</s> </p> <p type="main"> <s id="s.000661">Quoniam nota est diuturnitas CB<arrow.to.target n="marg164"/>, & 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 id="s.000662"><margin.target id="marg164"/>Per 15. pr. huius.</s> </p> <p type="margin"> <s id="s.000663"><margin.target id="marg165"/>Per pr. huius.</s> </p> <p type="main"> <s id="s.000664">Ergo EB est diuturnitas motus per BD; At <lb/>FB est diuturnitas motus per AB<arrow.to.target n="marg166"/>. </s> <s id="s.000665">Igitur <lb/>FE est diuturnitas motus per ABD. </s> <s id="s.000666">Quod etc.</s> </p> <p type="margin"> <s id="s.000667"><margin.target id="marg166"/>Per Co. 19. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000668">Corollarium</s> </p> <p type="main"> <s id="s.000669">Idem sequitur eadem ratione, si AB non sit <lb/>perpendicularis.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/091.jpg"/> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.000670">PROPOSITIO V</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000671">Data diuturnitate in plano perpendiculari <lb/>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 id="s.000672">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 id="s.000673">Perquirenda sit longitudo super BD quam grave <lb/>conficiat diuturnitate E.</s> </p> <p type="main"> <s id="s.000674">Fiat ut C ad E ita AB ad BF<arrow.to.target n="marg167"/>, unde si AB <lb/>concipiatur tanquam diuturnitas motus super <lb/>AB, erit BF diuturnitas motus super BD. <lb/></s> <s id="s.000675">Producatur FB donec concurrat cum A G ori­<lb/>zontaliter ducta in G. </s> <s id="s.000676">Et fiat CD tertia pro­<lb/>portionalis ad GB, GF<arrow.to.target n="marg168"/>.</s> </p> <p type="margin"> <s id="s.000677"><margin.target id="marg167"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.000678"><margin.target id="marg168"/>Per 11. sexti.</s> </p> <p type="main"> <s id="s.000679">Dico BD esse longitudinem quaesitam.</s> </p> <p type="main"> <s id="s.000680">Quoniam AB est diuturnitas ipsius AB per sup­<lb/>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 id="s.000681">Quod etc.</s> </p> <p type="margin"> <s id="s.000682"><margin.target id="marg169"/>Per 15. primi huius.</s> </p> <p type="margin"> <s id="s.000683"><margin.target id="marg170"/>Per 3. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000684">Corollarium.</s> </p> <p type="main"> <s id="s.000685">Grave prodibit per AB, BD aequis tempo­<lb/>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 id="s.000686">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000687">Moto gravi super pluribus planis diversimo­<lb/>de inclinatis, venari diuturnitates in quo­<lb/>libet eorum.<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 id="s.000688">Ducatur grave per AB diuturnitate data, <lb/>quae sit eadem AB; inde a B in D, & a D <lb/>in H. </s> <s id="s.000689">Venanda est diuturnitam motus per DH.</s> </p> <p type="main"> <s id="s.000690">Producatur DB in E donec concurrat cum <lb/>AG orizontaliter ducta. </s> <s id="s.000691">Item producatur H<lb/>D donec concurrat cum eadem AG. </s> <s id="s.000692">Fiat <lb/>EC media inter EB, ED<arrow.to.target n="marg171"/>. </s> <s id="s.000693">Fiat itidem GF <lb/>media inter GD, GH.</s> </p> <p type="margin"> <s id="s.000694"><margin.target id="marg171"/>Per 13. Sexti.</s> </p> <p type="main"> <s id="s.000695">Dico DF esse diuturnitate motus per DH.<arrow.to.target n="marg172"/></s> </p> <p type="margin"> <s id="s.000696"><margin.target id="marg172"/>Per 7. pr. huius.</s> </p> <p type="main"> <s id="s.000697">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 id="s.000698">Ergo etiam si mo­<lb/>tus fuerit per BD, DF est diuturnitas motus <lb/>per DH. </s> <s id="s.000699">Quod etc.</s> </p> <p type="margin"> <s id="s.000700"><margin.target id="marg173"/>Per cor. 3.2. huius.</s> </p> <p type="margin"> <s id="s.000701"><margin.target id="marg174"/>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 id="s.000702">Corollarium I</s> </p> <p type="main"> <s id="s.000703">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 id="s.000704">Corollarium II.</s> </p> <p type="main"> <s id="s.000705">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 id="s.000706">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000707">Grave naturaliter motum velocius ad idem <lb/>ducitur punctum duabus lineis, quam una <lb/>tantum.</s> </p> </subchap2> <subchap2 n="7" type="proof"> <p type="main"> <figure id="id.064.01.094.1.jpg" xlink:href="064/01/094/1.jpg"/> <s id="s.000708">Progrediatur grave per AB in B.</s> </p> <p type="main"> <s id="s.000709">Dico quod citius perveniet in B motum per <lb/>A CB.</s> </p> <p type="main"> <s id="s.000710">Protrahatur BC, puta in D; & ab A in BD de­<lb/>mittatur normalis AE.</s> </p> <p type="main"> <s id="s.000711">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 id="s.000712">Quod etc.</s> </p> <p type="margin"> <s id="s.000713"><margin.target id="marg175"/>Per pr. huius.</s> </p> <p type="margin"> <s id="s.000714"><margin.target id="marg176"/>Per 2. peti.</s> </p> <p type="margin"> <s id="s.000715"><margin.target id="marg177"/>Per 19. pr. huius.</s> </p> <p type="margin"> <s id="s.000716"><margin.target id="marg178"/>Per 19. pr. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000717">Corollarium.</s> </p> <p type="main"> <s id="s.000718">Hinc est, quod si motus fuerit per ACB, im­<lb/>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 id="s.000719">PROPOSITIO VIII</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000720">Grave naturaliter ductum, velocius fertur su­<lb/>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 id="s.000721">Feratur grave per AB, BC, CD.</s> </p> <p type="main"> <s id="s.000722">Dico citius duci in D quam per AD.</s> </p> <p type="main"> <s id="s.000723">Producantur CB, DC ad orizontalem AF in EF.</s> </p> <p type="main"> <s id="s.000724">Ducantur normales AG, BH, & ducatur AC.</s> </p> <p type="main"> <s id="s.000725">Quoniam grave pervenit citius in C per ABC, <lb/>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 id="s.000726">Quod etc.</s> </p> <p type="margin"> <s id="s.000727"><margin.target id="marg179"/>Per 7. huius.</s> </p> <p type="margin"> <s id="s.000728"><margin.target id="marg180"/>Per eamdem.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000729">Corollarium. I.</s> </p> <p type="main"> <s id="s.000730">Eodem pacto facile probabitur quod citius <lb/>perveniet in D, quatenus ducitur pluribus <lb/>inclinationibus.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000731">Corollarium. II.</s> </p> <p type="main"> <s id="s.000732">Impetus in D est maior, si fuerit motus per AB<lb/>CD, quam per AD.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/096.jpg"/> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000733">PROPOSITIO IX</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000734">In quadrante inferiori circuli grave celerius <lb/>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 id="s.000735">Sit ABC quadrans inferius.</s> </p> <p type="main"> <s id="s.000736">Dico grave B velocius duci si moveatur in <lb/>peripheria, quam si per BC, aut BDC, aut <lb/>BDEFC.</s> </p> <p type="main"> <s id="s.000737">Quoniam in peripheria ducitur pluribus inclina­<lb/>tionibus<arrow.to.target n="marg181"/>.</s> </p> <p type="margin"> <s id="s.000738"><margin.target id="marg181"/>Per primam pet.</s> </p> <p type="main"> <s id="s.000739">Ergo grave super ipsa motum celerius transigit.<arrow.to.target n="marg182"/> Quod etc.</s> </p> <p type="margin"> <s id="s.000740"><margin.target id="marg182"/>Per cor. primum 8. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000741">Corollarium I.</s> </p> <p type="main"> <s id="s.000742">Idem sequitur, si digrediatur a quovis puncto <lb/>Peripheriae, puta a D.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000743">Corollarium II.</s> </p> <p type="main"> <s id="s.000744">In C impetus est maior, si motus fuerit per <lb/>Peripheriam, quam aliter quomodocunque.</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/097.jpg"/> <chap type="bk"> <p type="main"> <s id="s.000745">DE MOTV <lb/>GRAVIVM <lb/>LIBER QVARTVS.<lb/>ET LIQVIDORVM PRIMVS.</s> </p> <subchap1 type="preface"> <subchap2 type="preface"> <p type="main"> <s id="s.000746">Hactenus<arrow.to.target n="note1"/> 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 id="s.000747">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 id="s.000748">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 id="s.000749">Non tamen ex hoc ego me adhuc gra­<lb/>vium naturam omnino assecutum esse pro <lb/>certo habeo. </s> <s id="s.000750">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 id="s.000751">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 id="s.000752">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 id="s.000753">Non<arrow.to.target n="note6"/> vero ex hoc constabat qua pro­<lb/>portione talis acceleratio fieret. </s> <s id="s.000754">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 id="s.000755">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 id="s.000756">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 id="s.000757">Pluries<arrow.to.target n="note9"/> cogitaveram esse naturae consentane­<lb/>um, ut ex simplicissimis principijs quam plur­<lb/>imi mirabiles effectus educantur. </s> <s id="s.000758">Cuius rei, & <lb/>si plura habeam, unicum tantum in praesentia <lb/>aut alterum adducam exemplum. </s> <s id="s.000759">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 id="s.000760">Cum igitur ut <pb pagenum="101" xlink:href="064/01/101.jpg"/>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 id="s.000761">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 id="s.000762">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 id="s.000763">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 id="s.000764">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 id="s.000765">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 id="s.000766">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 id="s.000767">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 manubrij longe propulsa, quamvis a <pb xlink:href="064/01/104.jpg"/>malleo per parvam morulam, & per minimum <lb/>spatium lata fuerit.</s> <s id="s.000768">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 id="s.000769">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 id="s.000770">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 id="s.000771">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 id="s.000772">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 id="s.000773">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 id="s.000774">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 antrorsum, <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 id="s.000775">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 id="s.000776">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 id="s.000777">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 id="s.000778">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 id="s.000779">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 spatij 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 id="s.000780">Obijcies<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 id="s.000781">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 id="s.000782">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 id="s.000783">Obijcies<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 id="s.000784">Responditur<arrow.to.target n="note43"/> hanc sextam pro­<lb/>positionem inniti experimentis, sensui dece­<lb/>ptioni obnoxijs, 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 id="s.000785"><foot.target id="foot.1"/>1 Actum est de scientia motus naturalis.</s> </p> <p type="foot"> <s id="s.000786"><foot.target id="foot.2"/>2 Modo perquirendae causae.</s> </p> <p type="foot"> <s id="s.000787"><foot.target id="foot.3"/>3 Ut supra respectu gravitatis factum fuit.</s> </p> <p type="foot"> <s id="s.000788"><foot.target id="foot.4"/>4 Natura igitur motus investiganda.</s> </p> <p type="foot"> <s id="s.000789"><foot.target id="foot.5"/>5 Iam quaesiveram causam accel.</s> </p> <p type="foot"> <s id="s.000790"><foot.target id="foot.6"/>6 At non proportionem.</s> </p> <p type="foot"> <s id="s.000791"><foot.target id="foot.7"/>7 Reperta iuxta progressionem numerorum imparium. Quaesivi causam.</s> </p> <p type="foot"> <s id="s.000792"><foot.target id="foot.8"/>8 Repertus motus ab impetu aequabilis.</s> </p> <p type="foot"> <s id="s.000793"><foot.target id="foot.9"/>10 Natura utitur principijs simplicibus.</s> </p> <p type="foot"> <s id="s.000794"><foot.target id="foot.10"/>11 Unde visum ex simplici mobilis proprietate emanandas caeteras.</s> </p> <p type="foot"> <s id="s.000795"><foot.target id="foot.11"/>12 Quae sit motum ex se continuari.</s> </p> <p type="foot"> <s id="s.000796"><foot.target id="foot.12"/>13 Quia mobilia indiferenter se habeant, ad motum & quietem.</s> </p> <p type="foot"> <s id="s.000797"><foot.target id="foot.13"/>14 Huiusmodi continuationem non est nova entitas.</s> </p> <p type="foot"> <s id="s.000798"><foot.target id="foot.14"/>15 At ut nova concipitur. Dicitur & impetus.</s> </p> <p type="foot"> <s id="s.000799"><foot.target id="foot.15"/>16 Huiusmodi indiferentiam esse mobili naturalem.</s> </p> <p type="foot"> <s id="s.000800"><foot.target id="foot.16"/>17 Probatur per dictam naturalem motus continuationem.</s> </p> <p type="foot"> <s id="s.000801"><foot.target id="foot.17"/>18 Ex quo caeterae motus passiones.</s> </p> <p type="foot"> <s id="s.000802"><foot.target id="foot.18"/>19 Absque quo quid oriatur aut pereat.</s> </p> <p type="foot"> <s id="s.000803"><foot.target id="foot.19"/>20 Unde dum mobile impellitur motus augetur.</s> </p> <p type="foot"> <s id="s.000804"><foot.target id="foot.20"/>21 Et quo longius, ictus validior.</s> </p> <p type="foot"> <s id="s.000805"><foot.target id="foot.21"/>22 At motus debilior. Si resistentia maior motus tardior.</s> </p> <p type="foot"> <s id="s.000806"><foot.target id="foot.22"/>23 Et tandem deficit.</s> </p> <p type="foot"> <s id="s.000807"><foot.target id="foot.23"/>24 Patet experimento mallei.</s> </p> <p type="foot"> <s id="s.000808"><foot.target id="foot.24"/>25 Expulsum moveri perseverat.</s> </p> <p type="foot"> <s id="s.000809"><foot.target id="foot.25"/>26 Si quid urgeat aut retundat, variatur velocitas.</s> </p> <p type="foot"> <s id="s.000810"><foot.target id="foot.26"/>27 Et talis perseverat.</s> </p> <p type="foot"> <s id="s.000811"><foot.target id="foot.27"/>28 Si murum permeet remittitur.</s> </p> <p type="foot"> <s id="s.000812"><foot.target id="foot.28"/>29 Si perseveret, velocitas minuitur.</s> </p> <p type="foot"> <s id="s.000813"><foot.target id="foot.29"/>30 Idem etiam per morulam.</s> </p> <p type="foot"> <s id="s.000814"><foot.target id="foot.30"/>31 Ut in ludo mallei.</s> </p> <p type="foot"> <s id="s.000815"><foot.target id="foot.31"/>32 Unde pilae resilitio.</s> </p> <p type="foot"> <s id="s.000816"><foot.target id="foot.32"/>33 Si oblique feriat, oblique resilit.</s> </p> <p type="foot"> <s id="s.000817"><foot.target id="foot.33"/>34 Pila celerior instrumento expellente.</s> </p> <p type="foot"> <s id="s.000818"><foot.target id="foot.34"/>35 Et eo magis reticulo expulsa.</s> </p> <p type="foot"> <s id="s.000819"><foot.target id="foot.35"/>36 Vertigo durat.</s> </p> <p type="foot"> <s id="s.000820"><foot.target id="foot.36"/>37 Unde motus localis.</s> </p> <p type="foot"> <s id="s.000821"><foot.target id="foot.37"/>38 Pila digito compressa acquirit duplicem motum.</s> </p> <p type="foot"> <s id="s.000822"><foot.target id="foot.38"/>39 Ex quo trochulum retrocedere videtur.</s> </p> <p type="foot"> <s id="s.000823"><foot.target id="foot.39"/>40 Motus est a minimo motore.</s> </p> <p type="foot"> <s id="s.000824"><foot.target id="foot.40"/>41 Objectio prima non dari primam motus portionem sine impetu.</s> </p> <p type="foot"> <s id="s.000825"><foot.target id="foot.41"/>42 Responditur etiam si adsit impetus prima motus portio est ab eo independens.</s> </p> <p type="foot"> <s id="s.000826"><foot.target id="foot.42"/>43 Objectio 2. motum non augeri iuxta progressionem naturalem.</s> </p> <p type="foot"> <s id="s.000827"><foot.target id="foot.43"/>44 Responditur quod motus augetur iuxta progressionem naturalem per tempora insensibilia.</s> </p> <p type="foot"> <s id="s.000828"><foot.target id="foot.44"/>45 At per sensibilia fere iuxta progressionem numerorum imparium.</s> </p> <p type="main"> <s id="s.000829">Quod ut planius fiat, Moveatur mobile A ab <lb/>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 spatij 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 id="s.000830">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 id="s.000831">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 id="s.000832">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 id="s.000833">Dentur 4 tempora, duc in <lb/>2 producto 8 adde medietatem 2, sit 10 sum­<lb/>ma spatiorum. </s> <s id="s.000834">Dentur tempora 9, duc in 5, <lb/>productum 45 est summa spatiorum. </s> <s id="s.000835">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 ijdem 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 id="s.000836">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 id="s.000837">Caete­<lb/>rum cum ea, quae de solidis dicenda videban­<lb/>tur, iuxta mei vires ingenij, pertractata sint, <lb/>superest, ut ad naturalis motus liquidorum <lb/>passiones inquirendas accedam.</s> </p> <p type="foot"> <s id="s.000838"><foot.target id="foot.45"/>46 Et fere in duplicata ratione temporum.</s> </p> <p type="foot"> <s id="s.000839"><foot.target id="foot.46"/>47 Augetur motus iuxta progressionem naturalem.</s> </p> <p type="foot"> <s id="s.000840"><foot.target id="foot.47"/>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 id="s.000841">DEFINITIONES</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.000842">Canale est vas oblongum, per quod aqua de­<lb/>currit; quod in praesentia supponitur habere <lb/>latera erecta, & basi perpendicularia, & pa­<lb/>rallela inter se. </s> <s id="s.000843">Sectio vasis, est parallelogramum quod supponi­<lb/>tur secare canale ad angulos rectos.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.000844">PETITIONES</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000845">Aqua transiens per eandem sectionem corre­<lb/>spondet tempori.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/116.jpg"/> <subchap1 n="1" type="proposition"> <p type="head"> <s id="s.000846">PROPOSITIO PRIMA</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.000847">Aqua aequaliter introducta in pluribus cana­<lb/>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 id="s.000848">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 id="s.000849">Dico aquam AB ad aquam CD esse ut E ad F.</s> </p> <p type="main"> <s id="s.000850">Quoniam aqua A B est ea, quae transit per A, diu­<lb/>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 id="s.000851"><margin.target id="marg183"/>Per pet. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000852">Corollarium.</s> </p> <p type="main"> <s id="s.000853">Si diuturnitates sint aequales, aquae quantita­<lb/>tes sunt pariter aequales.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/117.jpg"/> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.000854">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.000855">In pluribus canalibus ductis ad idem planum <lb/>orizontale, aquae quantitates sunt ut canales.</s> </p> </subchap2> <subchap2 n="2" type="proof"> <p type="main"> <figure id="id.064.01.117.1.jpg" xlink:href="064/01/117/1.jpg"/> <s id="s.000856">Sint canalia AB, AC, ducta ad planum Orizon­<lb/>tale CB.</s> </p> <p type="main"> <s id="s.000857">Dico aquam AB esse ad aquam AC, ut longitudo <lb/>AB ad longitudinem AC.</s> </p> <p type="main"> <s id="s.000858">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 id="s.000859">Quod etc.</s> </p> <p type="margin"> <s id="s.000860"><margin.target id="marg184"/>Per 15. primi. huius.</s> </p> <p type="margin"> <s id="s.000861"><margin.target id="marg185"/>Per primam huius.</s> </p> <p type="margin"> <s id="s.000862"><margin.target id="marg186"/>Per 11. Quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000863">Corollarium</s> </p> <p type="main"> <s id="s.000864">Idem sequitur si alterum canale sit perpendi­<lb/>culare.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/118.jpg"/> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.000865">PROPOSITIO III. PROBL. I.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.000866">In canali declinante, reperire portionem con­<lb/>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 id="s.000867">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 id="s.000868">Ducatur BD normalis ad AC.</s> </p> <p type="main"> <s id="s.000869">Dico AD esse portionem quaesitam.</s> </p> <p type="main"> <s id="s.000870">Quoniam aqua ab A ducitur in B eodem tempore, <lb/>quo in D<arrow.to.target n="marg187"/>, erit aqua AB aequalis aqua AD<arrow.to.target n="marg188"/>. </s> <s id="s.000871">Quod etc.</s> </p> <p type="margin"> <s id="s.000872"><margin.target id="marg187"/>Per 16. pr. huius.</s> </p> <p type="margin"> <s id="s.000873"><margin.target id="marg188"/>Per Co. primae huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000874">Corollarium.</s> </p> <p type="main"> <s id="s.000875">Eadem ratione Dato canali AD reperietur <lb/>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 id="s.000876">PROPOSITIO IV. PROBL. II.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.000877">In quibusvis canalibus quomodolibet inclina­<lb/>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 id="s.000878">A Canalibus AB, AC, AD, etc. sint secandae <lb/>portiones continentes aquam aequalem aquae <lb/>canalis AE.</s> </p> <p type="main"> <s id="s.000879">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 id="s.000880">Dico portiones AG, AH, AI continere aquam <lb/>aequalem aquae canalis AE.</s> </p> <p type="main"> <s id="s.000881">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 id="s.000882">Quod etc.</s> </p> <p type="margin"> <s id="s.000883"><margin.target id="marg189"/>Per 25. pr. huius.</s> </p> <p type="margin"> <s id="s.000884"><margin.target id="marg190"/>Per primam huius.</s> </p> </subchap2> <pb xlink:href="064/01/120.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.000885">Corollarium</s> </p> <p type="main"> <s id="s.000886">Si describantur quot vis circuli minores, seu <lb/>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 id="s.000887">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.000888">In canali secto quomodolibet; aquae quantita­<lb/>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 id="s.000889"><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 id="s.000890">Dico aquam AB ad aquam BC esse ut diuturni­<lb/>tas DE ad diuturnitatem EF.</s> </p> <p type="main"> <s id="s.000891">Quoniam aqua AB est ea, quae transit per A diu­<lb/>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 id="s.000892">Quod etc.</s> </p> <p type="margin"> <s id="s.000893"><margin.target id="marg191"/>Per pet. huius.</s> </p> <p type="margin"> <s id="s.000894"><margin.target id="marg192"/>Per 19. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000895">Corollarium</s> </p> <p type="main"> <s id="s.000896">Si Diuturnitates DE, EF sint aequales, aqua <lb/>AB aequatur aquae BC.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/122.jpg"/> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.000897">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.000898">In canali secto quomodocumque; aqua in <lb/>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 id="s.000899">Sit canale AC sectum quomodocumque in D. </s> <s id="s.000900">Dico, quod aqua AD ad aquam AC est in sub­<lb/>duplicata ratione longitudinum AD, AC.</s> </p> <p type="main"> <s id="s.000901">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 id="s.000902">Quod etc.</s> </p> <p type="margin"> <s id="s.000903"><margin.target id="marg193"/>Per 3. & 7. primi huius.</s> </p> <p type="margin"> <s id="s.000904"><margin.target id="marg194"/>Per 5. huius.</s> </p> <p type="margin"> <s id="s.000905"><margin.target id="marg195"/>Per 11. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.000906">Corollarium</s> </p> <p type="main"> <s id="s.000907">Unde si fiat AE media proportionalis inter <lb/>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 id="s.000908">PROPOSITIO VII. PROBL. III.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.000909">Dato canali perpendiculari, & alio inclinato <lb/>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 id="s.000910">Sint canalia AC inclinatum, & AB perpen­<lb/>diculare aequalia, & venanda sit proportio <lb/>inter aquas AB, AC.</s> </p> <p type="main"> <s id="s.000911">Ducatur BD perpendicularis ad AC, & fiat <lb/>AE media proportionalis inter AD, AC.</s> </p> <p type="main"> <s id="s.000912">Dico esse aquam AB ad aquam AC ut AD ad <lb/>AE.</s> </p> <p type="main"> <s id="s.000913">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 id="s.000914"><margin.target id="marg196"/>Per 6. huius.</s> </p> <p type="margin"> <s id="s.000915"><margin.target id="marg197"/>Per 3. huius.</s> </p> <p type="margin"> <s id="s.000916"><margin.target id="marg198"/>Per 11. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/124.jpg"/> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.000917">PROPOSITIO VIII. PROBL. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.000918">Datis canalibus aequalis longitudinis maio­<lb/>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 id="s.000919">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 id="s.000920">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 id="s.000921">Dico aquam AC ad aquam AF esse ut AE ad AI.</s> </p> <p type="main"> <s id="s.000922">Quoniam ut aqua AC ad aquam AB ita AE ad <lb/>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 id="s.000923">Quod etc.</s> </p> <p type="margin"> <s id="s.000924"><margin.target id="marg199"/>Per 7. huius.</s> </p> <p type="margin"> <s id="s.000925"><margin.target id="marg200"/>Per 22. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/125.jpg"/> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.000926">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.000927">In canali secto iuxta proportionem nume­<lb/>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 id="s.000928">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 id="s.000929">Dico quantitates aquae AB, BC, CD, esse <lb/>aequales inter se.</s> </p> <p type="main"> <s id="s.000930">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 id="s.000931">Quod etc.</s> </p> <p type="margin"> <s id="s.000932"><margin.target id="marg201"/>Per 10. pr. huius.</s> </p> <p type="margin"> <s id="s.000933"><margin.target id="marg202"/>Per cor. quintae huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/126.jpg"/> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.000934">PROPOSITIO X.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.000935">In quavis priori portione canalis, est aqua <lb/>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 id="s.000936">Dato canali A C secto in D ita ut AD sit <lb/>1/4 ipsius A C.</s> </p> <p type="main"> <s id="s.000937">Dico aquam AD aequari aquae DC.</s> </p> <p type="main"> <s id="s.000938">Quoniam eo tempore, quo A ducitur in D, D du­<lb/>citur in C<arrow.to.target n="marg203"/>, ergo aqua AD est aequalis aquae <lb/>DC<arrow.to.target n="marg204"/>. </s> <s id="s.000939">Quod etc.</s> </p> <p type="margin"> <s id="s.000940"><margin.target id="marg203"/>Per 9. huius.</s> </p> <p type="margin"> <s id="s.000941"><margin.target id="marg204"/>Per cor. quintae huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/127.jpg"/> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.000942">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.000943">In canali declinante, duplo perpendicularis <lb/>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 id="s.000944">Sit canale AC duplum AB, sectum in D a <lb/>perpendiculari BD.</s> </p> <p type="main"> <s id="s.000945">Dico aquam AD aequari aquae DC.</s> </p> <p type="main"> <s id="s.000946">Quoniam AB est media inter AD, AC<arrow.to.target n="marg205"/>, <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 id="s.000947">Quod etc.</s> </p> <p type="margin"> <s id="s.000948"><margin.target id="marg205"/>Per ea quae ad 16. pri. huius.</s> </p> <p type="margin"> <s id="s.000949"><margin.target id="marg206"/>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 id="s.000950">DE MOTV <lb/>GRAVIVM <lb/>LIBER QVINTVS <lb/>ET LIBER LIQVIDORVM SECVNDVS.<lb/>VBI DE CANALIVM SECTIONIBVS.</s> </p> <subchap1 type="preface"> <subchap2 type="preface"> <p type="main"> <s id="s.000951">Etiamsi simus in liqui­<lb/>dis, lubet adhuc aliquid <lb/>de solidis praefari, sum­<lb/>pta occasione a Quest. <lb/>19. Mech. </s> <s id="s.000952">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 id="s.000953">Quod perinde est ac si dicas, cur plus scin­<lb/>das leviori securi mota, quam graviori quies­<lb/>cente. </s> <s id="s.000954">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 id="s.000955">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 id="s.000956">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 id="s.000957">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 id="s.000958">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 id="s.000959">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 id="s.000960">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 id="s.000961">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 id="s.000962">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 id="s.000963">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 id="s.000964">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 id="s.000965">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 id="s.000966">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 spatij quod securis per aerem peregit dum <lb/>nil obstaret, addita postmodum ligni obvij re­<lb/>sistentia, in aequali tempore, decimam pariter <lb/>aut centesimam conficit portionem. </s> <s id="s.000967">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 id="s.000968">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 id="s.000969">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 id="s.000970">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 id="s.000971">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, spatijs 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 id="s.000972">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 id="s.000973">Cum itaque iam visus sit, acti­<lb/>vitatem securis motae a duobus pendere prin­<lb/>cipijs, 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 id="s.000974">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 id="s.000975">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 id="s.000976"><foot.target id="foot.48"/>1 De vi percussionis.</s> </p> <p type="foot"> <s id="s.000977"><foot.target id="foot.49"/>2 De activitate securis seu percutientis.</s> </p> <p type="foot"> <s id="s.000978"><foot.target id="foot.50"/>3 Quia motum plus agit ob impetum.</s> </p> <p type="foot"> <s id="s.000979"><foot.target id="foot.51"/>4 Et quo per longius spatium impetus est maior.</s> </p> <p type="foot"> <s id="s.000980"><foot.target id="foot.52"/>5 Proportio inter impetus et spatia.</s> </p> <p type="foot"> <s id="s.000981"><foot.target id="foot.53"/>6 In quavis percussione.</s> </p> <p type="foot"> <s id="s.000982"><foot.target id="foot.54"/>7 Etiamsi motus non sit deorsum.</s> </p> <p type="foot"> <s id="s.000983"><foot.target id="foot.55"/>8 Unde vis percussionis.</s> </p> <p type="foot"> <s id="s.000984"><foot.target id="foot.56"/>9 Vim impulsus pondus metitur.</s> </p> <p type="foot"> <s id="s.000985"><foot.target id="foot.57"/>10 De ligni resistentia.</s> </p> <p type="foot"> <s id="s.000986"><foot.target id="foot.58"/>11 Quae pendet etiam a partibus cohaerentibus.</s> </p> <p type="foot"> <s id="s.000987"><foot.target id="foot.59"/>12 Quo resistentia est maior minor est motus.</s> </p> <p type="foot"> <s id="s.000988"><foot.target id="foot.60"/>13 Et inde resistentia augetur.</s> </p> <p type="foot"> <s id="s.000989"><foot.target id="foot.61"/>14 Et velocitas minuitur. Et deficit.</s> </p> <p type="foot"> <s id="s.000990"><foot.target id="foot.62"/>15 Et pariter impetus.</s> </p> <p type="foot"> <s id="s.000991"><foot.target id="foot.63"/>16 Quod est commune cuivis mobili.</s> </p> <p type="foot"> <s id="s.000992"><foot.target id="foot.64"/>17 Cui addito immoto minuitur impetus.</s> </p> <p type="foot"> <s id="s.000993"><foot.target id="foot.65"/>18 Qua proportione.</s> </p> <p type="foot"> <s id="s.000994"><foot.target id="foot.66"/>19 Quod pondus percussionis aequivaleat.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/137.jpg"/> <subchap1 type="postulate"> <p type="head"> <s id="s.000995">PETITIONAE</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.000996">1. In sectionibus aequalibus quantitates aquae <lb/>sunt ut velocitates.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000997">2. Si velocitates sint aequales, sectiones sunt ut <lb/>quantitates aquae.</s> </p> </subchap2> <subchap2 type="postulate"> <p type="main"> <s id="s.000998">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 id="s.000999">PROPOSITIO PRIMA.</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.001000">Si sectiones sint aequales; aquarum transeun­<lb/>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 id="s.001001">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 id="s.001002">Dico aquam AB ad aquam CD esse ut velocitas <lb/>aquae A ad velocitatem aquae C.</s> </p> <p type="main"> <s id="s.001003">Quoniam velocitas in A ad velocitatem in C, est <lb/>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 id="s.001004">Quod etc.</s> </p> <p type="margin"> <s id="s.001005"><margin.target id="marg207"/>Per 32. undec.</s> </p> <p type="margin"> <s id="s.001006"><margin.target id="marg208"/>Per 11. Quinti.</s> </p> <p type="margin"> <s id="s.001007"><margin.target id="marg209"/>Per primam huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/139.jpg"/> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.001008">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.001009">Velocitas aquae in pluribus eiusdem canalis <lb/>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 id="s.001010">Sint A, C, canalis sectiones, diversae magnitu­<lb/>dinis.</s> </p> <p type="main"> <s id="s.001011">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 id="s.001012">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 id="s.001013">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 id="s.001014"><margin.target id="marg210"/>Per 2. pet. huius.</s> </p> <p type="margin"> <s id="s.001015"><margin.target id="marg211"/>Per 2. huius.</s> </p> <p type="main"> <s id="s.001016">Ergo ut magnitudo sectionis C ad magnitudi­<lb/>nem sectionis A, ita velo­<lb/>citas aquae A ad velocitatem aquae C. </s> <s id="s.001017">Quod etc.</s> </p> </subchap2> <pb xlink:href="064/01/140.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.001018">Corollarium I.</s> </p> <p type="main"> <s id="s.001019">Idem sequitur, si sectiones sint canalium diversorum, dummodo ducant aquae quantitates aequales.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001020">Corollarium II.</s> </p> <p type="main"> <s id="s.001021">Impetus sunt ibidem ut sectiones reciproce.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/141.jpg"/> <subchap1 n="3" type="proposition"> <p type="head"> <s id="s.001022">PROPOSITIO III.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.001023">Sectiones canalis sunt reciproce in subduplicata ratione longitudinum.</s> </p> </subchap2> <subchap2 n="3" type="proof"> <p type="main"> <figure id="id.064.01.141.1.jpg" xlink:href="064/01/141/1.jpg"/> <s id="s.001024">Sit canale AB sectum in C.</s> </p> <p type="main"> <s id="s.001025">Dico sectiones CB esse in subduplicata ratione AB, AC.</s> </p> <p type="main"> <s id="s.001026">Quoniam sectiones CB sunt ut velocitates in B, & in C<arrow.to.target n="marg212"/>, 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 id="s.001027">Quod etc.</s> </p> <p type="margin"> <s id="s.001028"><margin.target id="marg212"/>Per 5. secundi huius.</s> </p> <p type="margin"> <s id="s.001029"><margin.target id="marg213"/>Per 11. quinti.</s> </p> <p type="margin"> <s id="s.001030"><margin.target id="marg214"/>Per 33. primi.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001031">Corollarium I.</s> </p> <p type="main"> <s id="s.001032">Igitur si canalis latera sint parallela, altitudines sectionem sunt in subduplicata ratione longitudinum.</s> </p> <p type="main"> <s id="s.001033">Nam si latera perpendicularia canalis intelligantur bases, & ea ratione latitudines canalis ut altitudines, quae proinde sunt aequales<arrow.to.target n="marg215"/>, 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 id="s.001034"><margin.target id="marg215"/>Per pri. sexti.</s> </p> <p type="margin"> <s id="s.001035"><margin.target id="marg216"/>Per 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001036">Corollarium II.</s> </p> <p type="main"> <s id="s.001037">Si sectiones sint reciprocae in subduplicata ra­<lb/>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 id="s.001038">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.001039">Impetus sectionum canalis, sunt in subdupli­<lb/>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 id="s.001040">In canali ACB.</s> </p> <p type="main"> <s id="s.001041">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 id="s.001042">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 id="s.001043">Quod etc.</s> </p> <p type="margin"> <s id="s.001044"><margin.target id="marg217"/>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001045"><margin.target id="marg218"/>Per 13. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/144.jpg"/> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.001046">PROPOSITIO V. PROBL. I.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.001047">Data canalis sectione, reperire sectionem in <lb/>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 id="s.001048">Data sectione C, & puncto B in canali AB, <lb/>Venanda est sectio puncti B.</s> </p> <p type="main"> <s id="s.001049">Fiat AD media inter AC, AB<arrow.to.target n="marg219"/>, & sectio B ad <lb/>sectionem C ut AC ad AD.</s> </p> <p type="margin"> <s id="s.001050"><margin.target id="marg219"/>Per 20. sexti.</s> </p> <p type="main"> <s id="s.001051">Dico B esse sectionem quaesitam.</s> </p> <p type="main"> <s id="s.001052">Quoniam sectio B ad sectionem C est ut AC ad <lb/>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 id="s.001053">Quod etc.</s> </p> <p type="margin"> <s id="s.001054"><margin.target id="marg220"/>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001055"><margin.target id="marg221"/>Defini. pr. quarti huius.</s> </p> <p type="main"> <s id="s.001056">Fiet sectio B ad sectionem C ut AC ad AD, si fiat <lb/>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 id="s.001057"><margin.target id="marg222"/>Per 2. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/145.jpg"/> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.001058">PROPOSITIO VI.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.001059">Datis pluribus sectionibus; ratio primae ad ter­<lb/>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 id="s.001060">Dentur in canali AB sectiones B, C, D. <lb/></s> <s id="s.001061">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 id="s.001062">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 id="s.001063"><margin.target id="marg223"/>Per 5. def. sexti.</s> </p> <p type="main"> <s id="s.001064">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 id="s.001065"><margin.target id="marg224"/>Per 8. secundi huius.</s> </p> <p type="main"> <s id="s.001066">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 id="s.001067">Quod etc.</s> </p> </subchap2> <pb xlink:href="064/01/146.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.001068">Corollarium</s> </p> <p type="main"> <s id="s.001069">Si sint plures sectiones puta B, C, D, E, F, <lb/>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 id="s.001070">PROPOSITIO VII.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.001071">Si canales perpendicularis, & inclinatus ter­<lb/>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 id="s.001072">Dentur canales AB perpendicularis, & A<lb/>D inclinatus, terminati a recta BD, ut an­<lb/>gulus ADB sit rectus. </s> <s id="s.001073">Dico sectionem B ad se­<lb/>ctionem D esse ut AD, ad AB.</s> </p> <p type="main"> <s id="s.001074">Quoniam velocitas in B ad velocitatem in D est <lb/>ut AB ad AD<arrow.to.target n="marg225"/>.</s> </p> <p type="margin"> <s id="s.001075"><margin.target id="marg225"/>Per 2. huius.</s> </p> <p type="main"> <s id="s.001076">Erit sectio B ad sectionem D ut AD ad AB<arrow.to.target n="marg226"/>. </s> <s id="s.001077">Quod etc.</s> </p> <p type="margin"> <s id="s.001078"><margin.target id="marg226"/>Per cor. 8. sexti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/148.jpg"/> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.001079">PROPOSITIO VIII.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.001080">In canalibus perpendiculari, & inclinato; se­<lb/>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 id="s.001081">Dentur canales AB perpendicularis, & AC <lb/>inclinatus, quorum sectiones CB sint ori­<lb/>zontales.</s> </p> <p type="main"> <s id="s.001082">Dico eas esse aequales inter se.</s> </p> <p type="main"> <s id="s.001083">Ducatur normalis BD ad AC.</s> </p> <p type="main"> <s id="s.001084">Quoniam AB est media inter AD, AC<arrow.to.target n="marg227"/>, 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 id="s.001085">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 id="s.001086">Quod etc.</s> </p> <p type="margin"> <s id="s.001087"><margin.target id="marg227"/>Per 10. def. quin.</s> </p> <p type="margin"> <s id="s.001088"><margin.target id="marg228"/>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001089"><margin.target id="marg229"/>Per 7. huius.</s> </p> <p type="margin"> <s id="s.001090"><margin.target id="marg230"/>Per 9. quinti.</s> </p> <p type="margin"> <s id="s.001091"><margin.target id="marg231"/>Per 3. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/149.jpg"/> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.001092">PROPOSITIO IX.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.001093">Ductis pluribus canalibus a puncto superno <lb/>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 id="s.001094">Dati sint quilibet canales AB, AC, AD, in <lb/>quibus assignentur puncta B, C, D.</s> </p> <p type="main"> <s id="s.001095">Oportet reperire rationes dictarum sectionum inter se. <lb/></s> <s id="s.001096">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 id="s.001097">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 id="s.001098">Quod etc.</s> </p> <p type="margin"> <s id="s.001099"><margin.target id="marg232"/>Per 8. huius.</s> </p> <p type="margin"> <s id="s.001100"><margin.target id="marg233"/>Per 8. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001101">Corollarium I.</s> </p> <p type="main"> <s id="s.001102">Si sectiones B, C, D terminentur in <lb/>perpendiculari BD, erit pariter <lb/>ratio inter ipsas nota.<figure id="id.064.01.149.2.jpg" xlink:href="064/01/149/2.jpg"/></s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/150.jpg"/> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.001103">PROPOSITIO X</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.001104">In canalibus inter binas orizontales, sectiones <lb/>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 id="s.001105">Sint canales AB, CD inter orizontales AC, BD. <lb/></s> <s id="s.001106">Dico sectiones B, D esse aequales.</s> </p> <p type="main"> <s id="s.001107">Fiat canale CE.</s> </p> <p type="main"> <s id="s.001108">Sectio E aequatur sectioni D<arrow.to.target n="marg234"/>. </s> <s id="s.001109">Aequatur pariter <lb/>sectioni B, quia est par ratio. </s> <s id="s.001110">Ergo sectiones B, <lb/>D sunt aequales. </s> <s id="s.001111">Quod etc.<figure id="id.064.01.150.2.jpg" xlink:href="064/01/150/2.jpg"/></s> </p> <p type="margin"> <s id="s.001112"><margin.target id="marg234"/>Per 3. huius.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001113">Corollarium I.</s> </p> <p type="main"> <s id="s.001114">Si canales AB, CB ducti ab orizontali A C ter­<lb/>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 id="s.001115">PROPOSITIO XI.</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.001116">Dato canali inflexo quomodolibet, venari quan­<lb/>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 id="s.001117">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 id="s.001118">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 id="s.001119">Dico C esse sectionem in C.</s> </p> <p type="main"> <s id="s.001120">Quoniam si canale sit DC, sectio C ad sectionem B <lb/>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 id="s.001121">Ergo sectio <lb/>C ad sectionem B est ut DB ad DE.</s> </p> <p type="margin"> <s id="s.001122"><margin.target id="marg235"/>Per co. decimae huius.</s> </p> <p type="margin"> <s id="s.001123"><margin.target id="marg236"/></s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001124">Corollarium I.</s> </p> <p type="main"> <figure id="id.064.01.151.2.jpg" xlink:href="064/01/151/2.jpg"/> <s id="s.001125">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 id="s.001126">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 id="s.001127">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 ingenij 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 id="s.001128">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 id="s.001129">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 id="s.001130">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 id="s.001131">Quod etc. </s> <s id="s.001132">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/>tijs, 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 id="s.001133">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 id="s.001134">Quem nodum, de quo in praesentia <lb/>nil addam, alijs enodandum relinquo.</s> </p> <p type="margin"> <s id="s.001135"><margin.target id="marg237"/>Per 3. pr. huius.</s> </p> <p type="margin"> <s id="s.001136"><margin.target id="marg238"/>Per cor. 8. sexti.</s> </p> <p type="margin"> <s id="s.001137"><margin.target id="marg239"/>Per Observat.</s> </p> <p type="margin"> <s id="s.001138"><margin.target id="marg240"/>Per 17. quinti.</s> </p> </subchap1> <pb xlink:href="064/01/157.jpg"/> <subchap1 type="definition"> <p type="head"> <s id="s.001139">DEFINITIONES.</s> </p> <subchap2 type="definition"> <p type="main"> <s id="s.001140">1 Vas aquae intelligitur, cuius latera sint <lb/>rectangula, & basis orizontalis.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.001141">2. Foramen intelligitur rectangulum cuius basis <lb/>orizontalis.</s> </p> </subchap2> <subchap2 type="definition"> <p type="main"> <s id="s.001142">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 id="s.001143">DIGNITATES</s> </p> <subchap2 type="postulate"> <p type="main"> <s id="s.001144">Ubi omnia sint paria, effectus sunt aequa­<lb/>les.</s> </p> </subchap2> </subchap1> <subchap1 type="postulate"> <p type="head"> <s id="s.001145">PETITIONES</s> </p> <subchap2 type="axiom"> <p type="main"> <s id="s.001146">1 Quantitates eiusdem generis sunt omnes <lb/>commensurabiles.</s> </p> </subchap2> <subchap2 type="axiom"> <p type="main"> <s id="s.001147">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 id="s.001148">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 id="s.001149">PROPOSITIO PRIMA</s> </p> <subchap2 n="1" type="statement"> <p type="main"> <s id="s.001150">Aquarum quantitates exeuntium per forami­<lb/>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 id="s.001151">In vase AB, sint foramina C, D aequalia, & <lb/>orizontalia, per quae aqua aequali tempore de­<lb/>currat.</s> </p> <p type="main"> <s id="s.001152">Dico aquas decursas esse aequales inter se.</s> </p> <p type="main"> <s id="s.001153">Quoniam ubi omnia sunt paria, effectus sunt <lb/>aequales<arrow.to.target n="marg241"/>.</s> </p> <p type="margin"> <s id="s.001154"><margin.target id="marg241"/>Per ax. huius.</s> </p> <p type="main"> <s id="s.001155">Sed hic sunt omnia paria ex constructione.</s> </p> <p type="main"> <s id="s.001156">Ergo habent effectus aequales.</s> </p> <p type="main"> <s id="s.001157">Sed aquae decursa sunt effectus, & proinde aequa­<lb/>les. </s> <s id="s.001158">Quod etc.</s> </p> <p type="main"> <s id="s.001159">Seu mavis.</s> </p> <p type="main"> <s id="s.001160">Ubi omnia paria effectus sunt aequales, & <lb/>proinde si effectus sunt aquae decursae, ipsae <lb/>sunt aequales.</s> </p> <p type="main"> <s id="s.001161">Sed hic sunt omnia paria, & effectus sunt aquae <lb/>decursae, ex constructione. </s> <s id="s.001162">Ergo aquae decursae sunt aequales. </s> <s id="s.001163">Quod etc.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/160.jpg"/> <subchap1 n="2" type="proposition"> <p type="head"> <s id="s.001164">PROPOSITIO II.</s> </p> <subchap2 n="2" type="statement"> <p type="main"> <s id="s.001165">Si foramina sint orizontalia, eiusdem altitudi­<lb/>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 id="s.001166">In vase AB dentur foramina orizontalia aeque <lb/>alta C minus, D vero maius.</s> </p> <p type="main"> <s id="s.001167">Dico aquam decursam per C, quae sit E, se habere ad aquam <lb/>decursam per D, quae sit F, ut foramen C ad foramen D.</s> </p> <p type="main"> <s id="s.001168">Longitudinum C, & D commensurabilium,<arrow.to.target n="marg242"/> <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 id="s.001169"><margin.target id="marg242"/>Per pr. pet.</s> </p> <p type="main"> <s id="s.001170">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 id="s.001171">Quot igitur sunt foramina in C, D, <lb/>tot sunt quantitates aquarum in E, F. </s> <s id="s.001172">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 id="s.001173">Quod etc.</s> </p> <p type="margin"> <s id="s.001174"><margin.target id="marg243"/>Per 36. primi.</s> </p> <p type="margin"> <s id="s.001175"><margin.target id="marg244"/>Per primum huius.</s> </p> <p type="margin"> <s id="s.001176"><margin.target id="marg245"/>Per 16. quinti.</s> </p> <p type="main"> <s id="s.001177">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 id="s.001178">PROPOSITIO III.</s> </p> <subchap2 n="3" type="statement"> <p type="main"> <s id="s.001179">Foramina vasis perinde se habent ac sectio­<lb/>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 id="s.001180">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 id="s.001181">Dico in B, & D esse impetus aequales.</s> </p> <p type="main"> <s id="s.001182">Quoniam aqua fluens a foramine D decurrit per <lb/>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 id="s.001183">Quod, etc.</s> </p> <p type="margin"> <s id="s.001184"><margin.target id="marg246"/>Per 2. pet.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/162.jpg"/> <subchap1 n="4" type="proposition"> <p type="head"> <s id="s.001185">PROPOSITIO IV.</s> </p> <subchap2 n="4" type="statement"> <p type="main"> <s id="s.001186">Impetus foraminum aequalium vasis, sunt in <lb/>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 id="s.001187">In vase AC, distantiae foraminum aequalium <lb/>B, C a summo vasis AB, AC; media sit AD.</s> </p> <p type="main"> <s id="s.001188">Dico impetus in C ad impetum in B esse ut AD <lb/>ad AB.</s> </p> <p type="main"> <s id="s.001189">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 id="s.001190">Quod etc.</s> </p> <p type="margin"> <s id="s.001191"><margin.target id="marg247"/>Per 3. huius.</s> </p> <p type="margin"> <s id="s.001192"><margin.target id="marg248"/>Per 4. quinti huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/163.jpg"/> <subchap1 n="5" type="proposition"> <p type="head"> <s id="s.001193">PROPOSITIO V.</s> </p> <subchap2 n="5" type="statement"> <p type="main"> <s id="s.001194">Altitudines a foraminibus aequalibus ad sum­<lb/>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 id="s.001195">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 id="s.001196">Dico AD ad AB esse ut aqua fluens per C ad <lb/>aquam fluentem per B.</s> </p> <p type="main"> <s id="s.001197">Quoniam ut AD ad AB ita est impetus in C ad <lb/>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 id="s.001198">Quod etc.</s> </p> <p type="margin"> <s id="s.001199"><margin.target id="marg249"/>Per quartam huius.</s> </p> <p type="margin"> <s id="s.001200"><margin.target id="marg250"/>Per 3. petit.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/164.jpg"/> <subchap1 n="6" type="proposition"> <p type="head"> <s id="s.001201">PROPOSITIO VI. PROBL. II.</s> </p> <subchap2 n="6" type="statement"> <p type="main"> <s id="s.001202">Secto foramine in partes aliquotas a rectis <lb/>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 id="s.001203">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 id="s.001204">Venanda proportio aquarum E, F, G.</s> </p> <p type="main"> <s id="s.001205">Fiant HI, KL, MN, altitudines foraminum A<lb/>C, CD, DB a summo vasis; & inter ipsas <lb/>mediae OP, QR<arrow.to.target n="marg251"/>.</s> </p> <p type="margin"> <s id="s.001206"><margin.target id="marg251"/>Per 13. sexti.</s> </p> <p type="main"> <s id="s.001207">Quoniam aqua E ad aquam F, est ut HI ad OP<arrow.to.target n="marg252"/>, <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 id="s.001208">Reperta est igitur ratio aquarum E, F, G. </s> <s id="s.001209">Quod, etc.</s> </p> <p type="margin"> <s id="s.001210"><margin.target id="marg252"/>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001211"><margin.target id="marg253"/>Per 5. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/165.jpg"/> <subchap1 n="7" type="proposition"> <p type="head"> <s id="s.001212">PROPOSITIO VII. PROBL. III.</s> </p> <subchap2 n="7" type="statement"> <p type="main"> <s id="s.001213">Secto foramine vasis in partes a recta orizon­<lb/>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 id="s.001214">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 id="s.001215">Quaeritur proportio F ad G.</s> </p> <p type="main"> <s id="s.001216">Si ED foramen minus non mensurat CE, repe­<lb/>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 id="s.001217"><margin.target id="marg254"/>Per 3. decimi.</s> </p> <p type="main"> <s id="s.001218">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 id="s.001219">Quod etc.</s> </p> <p type="margin"> <s id="s.001220"><margin.target id="marg255"/>Per 6. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/166.jpg"/> <subchap1 n="8" type="proposition"> <p type="head"> <s id="s.001221">PROPOSITIO VIII. PROBL. IV.</s> </p> <subchap2 n="8" type="statement"> <p type="main"> <s id="s.001222">Datis foraminibus inaequalibus super eadem <lb/>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 id="s.001223">Sint foramina AB, & CD super orizontali <lb/>BD.</s> </p> <p type="main"> <s id="s.001224">Quaerenda proportio aquarum ex eis eodem tem­<lb/>pore fluentium.</s> </p> <p type="main"> <s id="s.001225">Producatur CE FG parallela DB.</s> </p> <p type="main"> <s id="s.001226">Quoniam nota est ratio aquarum fluentium ex <lb/>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 id="s.001227">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 id="s.001228">Quod etc.</s> </p> <p type="margin"> <s id="s.001229"><margin.target id="marg256"/>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001230"><margin.target id="marg257"/>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/167.jpg"/> <subchap1 n="9" type="proposition"> <p type="head"> <s id="s.001231">PROPOSITIO IX. PROBL. V.</s> </p> <subchap2 n="9" type="statement"> <p type="main"> <s id="s.001232">Datis foraminibus, quorum unum superius, <lb/>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 id="s.001233">Dentur foramina AB, CD inter parallelas <lb/>AC, & DB.</s> </p> <p type="main"> <s id="s.001234">Venanda ratio aquarum ex eis, aequo tempore, <lb/>fluentium.</s> </p> <p type="main"> <s id="s.001235">Concipiatur BC tanquam foramen.</s> </p> <p type="main"> <s id="s.001236">Quoniam nota est ratio aquarum fluentium ex CD, <lb/>& 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 id="s.001237">Quod etc.</s> </p> <p type="margin"> <s id="s.001238"><margin.target id="marg258"/>Per 7. huius.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/168.jpg"/> <subchap1 n="10" type="proposition"> <p type="head"> <s id="s.001239">PROPOSITIO X. PROBL. VI.</s> </p> <subchap2 n="10" type="statement"> <p type="main"> <s id="s.001240">Datis foraminibus venari aquas.<figure id="id.064.01.168.1.jpg" xlink:href="064/01/168/1.jpg"/></s> </p> </subchap2> <subchap2 n="10" type="proof"> <p type="main"> <s id="s.001241">Data sint foramina AD, EH.</s> </p> <p type="main"> <s id="s.001242">Oportet reperire rationem aquarum per <lb/>illa aequo tempore fluentium.</s> </p> <p type="main"> <s id="s.001243">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 id="s.001244">Item quaeratur ratio IL ad EH,<arrow.to.target n="marg260"/>, & sit ut N ad O.</s> </p> <p type="margin"> <s id="s.001245"><margin.target id="marg259"/>Per 9. huius.</s> </p> <p type="margin"> <s id="s.001246"><margin.target id="marg260"/>Per 2. huius.</s> </p> <p type="main"> <s id="s.001247">Dico M ad O esse rationem aquarum per AD, HE.</s> </p> <p type="main"> <s id="s.001248">Quoniam ut M ad N ita est AD ad IL, & ut <lb/>N ad O, ita IL ad EH per constr. </s> <s id="s.001249">Erit ex <lb/>aequo ut M ad O, ita AD ad EH<arrow.to.target n="marg261"/>. </s> <s id="s.001250">Quod etc.</s> </p> <p type="margin"> <s id="s.001251"><margin.target id="marg261"/>Per 22. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/169.jpg"/> <subchap1 n="11" type="proposition"> <p type="head"> <s id="s.001252">PROPOSITIO XI. PROBL. VII</s> </p> <subchap2 n="11" type="statement"> <p type="main"> <s id="s.001253">Dato foramine, & linea orizontali intermi­<lb/>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 id="s.001254">Dato foramine AB, & orizontali CD.</s> </p> <p type="main"> <s id="s.001255">Describendum sit foramen super CD, a <lb/>quo effluat aqua ut per AB.</s> </p> <p type="main"> <s id="s.001256">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 id="s.001257">Dico per LG foramen fluere aquam ut per AB.</s> </p> <p type="main"> <s id="s.001258">Quoniam aqua LG ad aquam FC est ut HI ad <lb/>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 id="s.001259">Quod etc.</s> </p> <p type="margin"> <s id="s.001260"><margin.target id="marg262"/>Per 2. huius.</s> </p> <p type="margin"> <s id="s.001261"><margin.target id="marg263"/>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001262"><margin.target id="marg264"/>Per 11. quinti.</s> </p> <p type="margin"> <s id="s.001263"><margin.target id="marg265"/>Per nonam quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/170.jpg"/> <subchap1 n="12" type="proposition"> <p type="head"> <s id="s.001264">PROPOSITIO XII. PROBL. VIII.</s> </p> <subchap2 n="12" type="statement"> <p type="main"> <s id="s.001265">Dato foramine, & latere alterius, reperire fo­<lb/>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 id="s.001266">Datum sit foramen AB, & latere DC. </s> <s id="s.001267">Oportet describere foramen, a quo effluat <lb/>aqua ut ab AB, cuius latus sit CD.</s> </p> <p type="main"> <s id="s.001268">Ductis CE, & DF, orizontalibus; protrahatur B<lb/>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 id="s.001269"><margin.target id="marg266"/>Per 9. huius.</s> </p> <p type="main"> <s id="s.001270">Dico a foramine DM fluere aquam, ut ab AB.</s> </p> <p type="main"> <s id="s.001271">Quoniam aqua fluens per AB ad fluentem per FE <lb/>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 id="s.001272">Est foramen quaesitum. </s> <s id="s.001273">Quod etc.</s> </p> <p type="margin"> <s id="s.001274"><margin.target id="marg267"/>Per secundum huius.</s> </p> <p type="margin"> <s id="s.001275"><margin.target id="marg268"/>Per 9. quinti.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/171.jpg"/> <subchap1 n="13" type="proposition"> <p type="head"> <s id="s.001276">PROPOSITIO XIII. PROBL. IX.</s> </p> <subchap2 n="13" type="statement"> <p type="main"> <s id="s.001277">Dato foramine, reperire aliud aequale, a quo <lb/>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 id="s.001278">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 id="s.001279">Reperiendum ubi fiat aequale foramen, a quo fluat <lb/>in aequali tempore aqua E.</s> </p> <p type="main"> <s id="s.001280">Fiat ad D, E, AC quarta preportionalis AF<arrow.to.target n="marg269"/>, <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 id="s.001281">Dico G esse locum forami­<lb/>nis quaesitum.</s> </p> <p type="margin"> <s id="s.001282"><margin.target id="marg269"/>Per 12. sexti.</s> </p> <p type="margin"> <s id="s.001283"><margin.target id="marg270"/>Per 11. sexti.</s> </p> <p type="main"> <s id="s.001284">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 id="s.001285">Quod etc.</s> </p> <p type="margin"> <s id="s.001286"><margin.target id="marg271"/>Per 5. huius.</s> </p> <p type="margin"> <s id="s.001287"><margin.target id="marg272"/>Per eamdem.</s> </p> <p type="margin"> <s id="s.001288"><margin.target id="marg273"/>Per 9. quinti.</s> </p> </subchap2> <pb xlink:href="064/01/172.jpg"/> <subchap2 type="corollary"> <p type="head"> <s id="s.001289">Corollarium I.</s> </p> <p type="main"> <s id="s.001290">Parum refert sint foramina quadrata nec ne.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001291">Corollarium II.</s> </p> <p type="main"> <s id="s.001292">Idem sequitur si ambo foramina sint rotunda.</s> </p> </subchap2> </subchap1> <pb xlink:href="064/01/173.jpg"/> <subchap1 n="14" type="proposition"> <p type="head"> <s id="s.001293">PROPOSITIO XIV.</s> </p> <subchap2 n="14" type="statement"> <p type="main"> <s id="s.001294">Dato foramine, aptandum sit aliud datum <lb/>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 id="s.001295">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 id="s.001296">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 id="s.001297">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 id="s.001298">Quod si fieri <lb/>nequit problema est insolubile. </s> <s id="s.001299">Dico I esse <lb/>foramen quaesitum.</s> </p> <p type="margin"> <s id="s.001300"><margin.target id="marg274"/>Per 8. huius.</s> </p> <p type="margin"> <s id="s.001301"><margin.target id="marg275"/>Per 13. huius.</s> </p> <pb xlink:href="064/01/174.jpg"/> <p type="main"> <s id="s.001302">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 id="s.001303">Quod etc.</s> </p> <p type="margin"> <s id="s.001304"><margin.target id="marg276"/>Per 22. quinti.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001305">Corollarium I.</s> </p> <p type="main"> <s id="s.001306">Parum refert sint ne foramina quadrata, <lb/>nec ne.</s> </p> </subchap2> <subchap2 type="corollary"> <p type="head"> <s id="s.001307">Corollarium II.</s> </p> <p type="main"> <s id="s.001308">Idem sequeretur si essent ambo rotunda<arrow.to.target n="marg277"/>.</s> </p> <p type="margin"> <s id="s.001309"><margin.target id="marg277"/>Per 3. pet.</s> </p> <p type="main"> <s id="s.001310">FINIS</s> </p> </subchap2> </subchap1> </chap> <pb xlink:href="064/01/175.jpg"/> <pb xlink:href="064/01/176.jpg"/> <pb xlink:href="064/01/177.jpg"/> </body> <back/> </text> </archimedes>