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New Special Instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 30 Jul 2014 15:58:21 +0200 |
| parents | 22d6a63640c6 |
| children |
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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 Ioannis Baptistae Baliani</title> <date>1638</date> <place>Genua</place> <translator/> <lang>la</lang> <cvs_file>balia_demot_076_la_1638.xml</cvs_file> <cvs_version/> <locator>076.xml</locator> </info> <text> <front> <pb xlink:href="076/01/001.jpg"/> <section> <p type="head"> <s id="s.000001"> De MOTV<lb/> NATVRALI,<lb/> GRAVIVM SOLIDORYVM<lb/> IOANNIS BAPTISTAE BALIANI<lb/> PATRITII GENVENSIS.</s> </p> <p type="head"> <s id="s.000002"> GENVUAE,<lb/> EX Typographia Io: Mariae Farroni, Nicolai Pe&longs;agnij,<lb/> et Petri Franci&longs;ci Barberij, &longs;oc. <lb/>MDCXXXVIII.<lb/> SVPERIORUM PERMISSV. </s> </p> </section> <pb xlink:href="076/01/002.jpg"/> <pb xlink:href="076/01/003.jpg"/> <section> <p type="head"> <s id="s.000003">PRAEFATIO</s> </p> <p> <s id="s.000004">Mihi quoque, sicut et caeteris hominibus,<lb/> inest sciendi cupiditas, nec grave fuit, usque<lb/> à primis annis, et aliorum scripta percur-<lb/>rere, et naturales effectus observare, qui<lb/> facile mihi persuaserim, ex bisce fontibus,<lb/> tum scientiam, tum sapientiam in animum<lb/> derivare, si tandem ex effectibus diligentius perspectis, non<lb/> modo ad inde consequentes, sed etiam ad causas, usque ad<lb/> primam deveniat intellectus. </s> <s id="s.000005">Statui igitur apud me ipsum<lb/> non acquiescere soli relationi plurimorum, etiam doctiorum;<lb/> potuisse siquidem contingere existimavi, ut aliqua laterent,<lb/> etiam in plurimis oculatissimos, vel non plene ab eis explica-<lb/>rentur; & ratus sum non inutilem laborem futurum, si ex<lb/> accuratiori naturae rerum investigatione, & ex affectionum<lb/> inde resultantium deductione, circa quod omnis demonstra-<lb/>tiva scientia versatur, aut scitis adderem aliqua, aut doctiori-<lb/>bus acuerem desiderium addendi plura: hinc factum est, ut exci-<lb/>tata mens ex praecognitis legendo, ad ea, quae se offerebant,<lb/> secundum privatas, aut publicas occupationes pervestiganda,<lb/> converteretur studiosus. </s> <s id="s.000006">Inter alia dum anno millesimo sex-<lb/>centesimo undecimo, per paucos menses, ex patriae legis prae-<lb/>scripto, Praefectum Arcis Savonae agerem, ex militaribus<lb/> observationibus quae occurrebant, illud maxime depraehendi,<lb/> ferreos, & lapideos tormentorum bellicorum globos, & sic<lb/> corpora gravia, seu eiusdem, seu diversae speciei, in inaequali<lb/> satis Mole, & gravitate, per idem spatium, aequali tempore,<lb/> & motu, naturaliter descendere, idque ita uniformiter, ut<lb/> repetitis experimentis mihi plane constiterit, duos ex prae-<lb/>dictis globis, vel ferreos ambos, vel alterum lapideum al-<lb/>terum plumbeum, eodem plane momento temporis dimissos<lb/> sibi, per spatium quincaginta pedum, etiam si unus esset<lb/> librae unius tantum, alter quincaginta, in indivisibili tem-<lb/>poris momento, subjectum solum ferire, ut unus tantum am-<lb/>borum ictus sensu perciperetur. </s> <s id="s.000007">Repetebam animo sapien-<lb/><pb xlink:href="076/01/004.jpg"/>tum esse pronunciatum, gravia moveri naturali motu, se-<lb/>cundum gravitatum proportionem; Processi ulterius, & peri-<lb/>culum feci, num forte iuxta eorum sententiam contingeret,<lb/> si corpora dimissa, ejusdem fere essent molis, sed longe di-<lb/>versi ponderis, puta unum plumbeum, cereum alterum; &<lb/> expertus sum in cereo aliquam longiorem moram in descen-<lb/>su, attamen longe infra proportionem gravitatum, globus<lb/> quippe ille cereus, in data distantia quinquaginta pedum de-<lb/>scensus, uno circiter pede distabat a solo, quando plumbeus<lb/> tangebat subjectum planum, objecto aere intermedio ni fal-<lb/>lor, sensibiliter resistente, & impediente motum. </s> <s id="s.000008">Institi<lb/> adhuc, & globos in gravitate, & in materia inaequales appendi<lb/> funiculis aequalibus, & agitatos animadverti moveri tempore<lb/> aequali, & hoc servare adeo fideliter, ut globus plumbeus dua-<lb/>rum unciarum, alter librarum duarum, ferreus librarum<lb/> 34. </s> <s id="s.000009">& lapideus quadraginta circiter, nec non, & lapis in-<lb/>formis, quorum funiculi comprehensis ipsorum semidiame-<lb/>tris aequales essent, uno, & eodem temporis spatio moveren-<lb/>tur, & vibrationes easdem numero darent hinc inde, sive mo-<lb/>tus unius globi fieret per aequale spatium, sive per inaequa-<lb/>le, ita ut qui majori impetu jactabatur, & sic majus spatium<lb/> percurrebat, illud tanto velocius pertransiret. </s> <s id="s.000010">In quibus<lb/> peragendis illud praeter expectationem sese mihi obtulit,<lb/> quod quotiescunque globi penderent ex funiculis inaequalibus,<lb/> ita inaequali motu ferebantur, ut longitudines funiculorum,<lb/> durationibus motuum, in duplicata ratione responderent.<lb/> </s> </p> <p> <s id="s.000011">Porro cum ex praemissis satis superque liqueret, in naturali<lb/> motu gravium, proportionem gravitatum communiter credi-<lb/>tam, non servari; in eam descendi sententiam, ut arbitrater<lb/> fortasse, gravitatem se habere ut agens, materiam vero, seu<lb/> mavis materiale corpus, ut passum, & proinde gravia mo-<lb/>veri juxta proportionem gravitatis ad materiam, & ubi sine<lb/> impedimento naturaliter perpendiculari motu ferantur, mo-<lb/>veri aequaliter, quia ubi plus est gravitatis, plus pariter sit<lb/> materiae, seu materialis quantitatis; si vero accedat aliquid<lb/> resistantiae, regulari motum secundum excessum virtutis agen<lb/><pb xlink:href="076/01/005.jpg"/>tis supra resistentiam passi, seu impedientia motum; qui<lb/> excessus momentum noncupabitur, & quod communiter gra-<lb/>vitati attributum fuit, momento attribui debere, nimirum<lb/> ut sit momentum ad momentum, ut velocitas ad velocita-<lb/>tem; Et hinc fieri posse, ut cognoscamus qua mensura, seu<lb/> proportione corpora gravia naturali motu ferantur super su-<lb/>bjectis planis, si super eis quomodolibet inclinatis, ipsorum<lb/> gravium momenta ubique innotescant, quae majora, aut mi-<lb/>nora videntur censenda, secundum quod magis, aut minus<lb/> super plano quiescunt, & sic secundum majorem, aut mino-<lb/>rem inclinationem plani resistentis; quod demum tali pro-<lb/>portione facile fieri mihi existimandum videtur, juxta quam<lb/> reciproce momentis proportionantur lineae dictorum plano-<lb/>rum, si ambae ductae sint ab eodem puncto ad idem planum<lb/> orizontale; de quo Simon Stevinus l. p. de Statica prop.<lb/> 19. & acutissime Galileus in Mechanica manuscripta, ubi de<lb/> Cochlea, & ego æliquali experientia compertum habui. </s> <s id="s.000012">Cae-<lb/>terum si per experientiam Scienta hominibus efficitur, prae-<lb/>dicta de quibus saepius repetitis actibus expertus fui, ut prin-<lb/>cipia scientiae habenda fore censui; in quibus occultae con-<lb/>clusiones delitescant, demonstrationibus duntaxat aperiendae.<lb/> </s> <s id="s.000013">Rimari caepi; an deprehenderim aliorum erit judicium.<lb/> </s> <s id="s.000014">Subjecta paucula, quae presens aliquod otium expedire per-<lb/>misit, de motu naturali solidorum gravium, Amice lector<lb/> tibi exhibeo, mox de liquidorum, & deinceps alia plura tam<lb/> parata daturus, si haec placuerint. </s> <s id="s.000015">Placuit sane mihi, vel<lb/> paucula tibi dare, qui te ejus ingenii esse confidam, ut non<lb/> verba, sed res, easque non mole, sed pondere censeas, felicior<lb/> si de eorum genere existimaveris, quae non mole magna sunt,<lb/> quod si talia non fuerint, quo minora minus defatigabunt,<lb/> sui exilitate, auctoris partus proprios omnino esse probatura.<lb/> </s> <s id="s.000016">Idioma latinum elegi ut communius. </s> <s id="s.000017">Praemisi aliqua na-<lb/>turalia principia, sine quibus naturales conclusiones aliunde<lb/> duci posse non video. </s> <s id="s.000018">Quae ex praedictis experimentis inno<lb/>tuerunt, suppositiones appellare, & a reliquis petitionibus se-<lb/>cernere libuit. </s> <s id="s.000019">Petitiones illas, quibus quid fieri petimus, con-<lb/><pb xlink:href="076/01/006.jpg"/>structioni deservientes, tanquam factu, & cognitu faciles,<lb/> & proinde supervacaneas, prudens praetermisi; ratus siqui-<lb/>dem nil inde incredulitatis, aut difficultatis derivaturum.<lb/> </s> <s id="s.000020">Septimum postulatum ea ratione segregavi, quod illud aliquo<lb/> pacto a 22. prop. pendeat, & quod in illo etiamsi veritas<lb/> non deficiat, evidentiam tamen ut in caeteris non agnoscens,<lb/> certis dubia quo quo pacto permiscere noluerim; ut proinde<lb/> plura eorum, quae ex illa deducta sunt, & diversa Methodo<lb/> & attingendo potius, quam demonstrando subjunxerim.<lb/> </s> <s id="s.000021">Si quae demum minus probata, seu explicata, aut quo<lb/> quo pacto imperfecta reperies, velim te tribue-<lb/>re cuidam naturali meae propensioni, ad no-<lb/>va potius, qualiacumque ea sint, inve-<lb/>nienda, quam inventa<lb/> perficienda.<lb/> </s> <s id="s.000022">Vale. </s> </p> </section> <pb xlink:href="076/01/007.jpg"/> </front> <body> <chap> <p type="head"> <s id="s.000023">DEFINITIONES</s> </p> <subchap1> <p> <s id="s.000024">Pendulum dicimus pondus filo appensum.</s> </p> </subchap1> <subchap1> <p> <s id="s.000025">Pendula dicuntur aequalia, seu aequipendula,<lb/> sive inaequalia, quae, & longiora, aut brevio-<lb/>ra, quatenus fila, e quibus dependent, sunt<lb/> aequalia, longiora, aut breviora.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000026">Vibrationes pendulorum sunt eorum motus hinc inde<lb/> Vibrationes aequales dicimus, quae fiunt per spatia aequalia, & e contra inaequales.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000027">Vibrationes aeque celeres si fiant per spatia<lb/> aequalia tem-<lb/>pore aequali.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000028">Vibrationis diuturnitatem dicimus ipsius Durationem,<lb/> tempus nimirum, quo ipsa vibratio perficitur.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000029">Vibrationes aequediuturnae, sunt, quae fiunt tempore<lb/> aequali, etiamsi per spatia inaequalia, inde diuturnior<lb/> est, quae longiori perficitur tempore.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000030">Vibrationes integras dicimus eas, quae se extendunt<lb/> per integrum semicirculum, se hinc inde moventes<lb/> per circuli quadrantem.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000031">Vibrationis portio est pars arcus, quem ipsa vibratio<lb/> disignant.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000032">Vibrationum similes portiones sunt arcus ipsarum in-<lb/>tercepti inter binas lineas ductas a centro, a quo<lb/> concipiuntur pendula pendere.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000033">Vibrationis portionem priorem decimus eam mini-<lb/>mam portionem, a qua integra vibratio initium habet.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000034">Momentum est excessus virtutis moventis supra mo-<lb/>tus impedimenta.<lb/> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/008.jpg"/> <chap> <p type="head"> <s id="s.000035">SUPPOSITIONES<lb/></s> </p> <subchap1> <p> <s id="s.000036">PRIMA. Solidorum aequipendulorum cujus-<lb/>cumque gravitatis vibrationes aequales sunt aeque-<lb/>diuturnae.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000037">2 Equipendulorum eorumdem vibrationes sunt aeque-<lb/>diuturnae, etiamsi inaequales.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000038">3 Pendulorum inaequalium longitudines sunt in du-<lb/>plicata ratione diuturnitatum vibrationum, seu ut<lb/> quadrata vibrationum.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000039">4 Momentum gravis super plano inclinato est ad ip-<lb/>sius gravitatem, ut perpendicularis ad inclinatam,<lb/> si ab eodem puncto ducta sint ad idem planum<lb/> orizontale dicta perpendicularis, & dictum planum<lb/> inclinatum, & proinde tali casu proportio gravita­<lb/>tis ad momentum est reciproca proportioni linea-<lb/>rum super quibus grave movetur.<lb/> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/009.jpg"/> <chap> <p type="head"> <s id="s.000040">PETITIONES, SEU POSTULATA<lb/></s> </p> <subchap1> <p> <s id="s.000041">Pr. Pendulorum inaequalium portiones similes vibra-<lb/>tionum sunt inter se quoad diuturnitatem, ut vibra-<lb/>tiones integrae.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000042">Sint pendula AB, AC; dependentia a puncto A, & ele-<lb/>ventur ad libellam orizontis puncti A, in E, D, de-<lb/>scribentia arcus BD, CE, integrarum vibrationum, & in<lb/> arcubus BD, CE sumantur portiones similes EF, DG, seu<lb/> HI, KL ductis EA, FA, seu HA, IA. </s> <s id="s.000043">Peto mihi concedi,<lb/> esse pendulorum diuturnitates in arcubus EC, DB, ut in<lb/> portionibus EF, DG, nec non HI, KL, & ita deinceps.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000044">2. Ut est momentum ad momentum solidi gravis, ita<lb/> velocitas ad velocitatem.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000045">Hujusmodi passio communiter attribui solet gravitati sim-<lb/>pliciter, quod eum nimis clare experientiis supra expo-<lb/>sitis nullo pacto congruere possit, momentis attribuenda<lb/> esse visa est, ut in praefatione explicatum fuit.<lb/> </s> </p> </subchap1> <pb xlink:href="076/01/010.jpg"/> <subchap1> <p> <s id="s.000046">3. Portiones minimae peripheriae Circuli concipiende<lb/> sunt, ac si essent lineae rectae.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000047">Quaecumque arcus portio est circularis, attamen si est<lb/> minima portio, tam parum aberrat a linea recta, ut<lb/> non modo quo ad sensum, sed quoad quascunque physicas<lb/> passiones, perinde esse videatur, ac si esset linea recta, id-<lb/>circo ut petitionem admittendam censeo, quemadmodum in-<lb/>mechanicis admittitur illa, quod perpendiculares sunt paral-<lb/>lelae, etiamsi in centro concurrant universi, quatenus eisdem<lb/> sunt passionibus physicis subjectae, ac si vere essent parallelae.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000048">4. Data recta linea, possimus concipere circulum talis<lb/> magnitudinis, cujus portio peripheriae aequalis quo<lb/> ad sensum datae lineae, concipienda sit, ac si esset<lb/> linea recta.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000049">Haec petitio videtur concedenda, quia si concipiamus cir-<lb/>culum, ejusque portionem minimam, ut in praece-<lb/>denti, si fiat ut hujusmodi portio ad datam lineam, ita<lb/> circulus ad alium, portio hujus, datae lineae aequalis erit, &<lb/> similis omnino praedicta minimae portioni, & proinde pa-<lb/>riter concipienda ut linea recta.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000050">5. Solida perpendicula libero motu aeque velociter<lb/> feruntur, & in tali proportione, ac si essent pendula,<lb/> & moverentur in priori portione vibrationum.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000051">Quoniam prior portio non differt sensibiliter a recta, ut in<lb/> tertia petitione, nec etiamsi sit major ut in quarta, iisdem<lb/> physicis passionibus subjicitur, & exinde motibus aequalibus.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000052">6. Solida naturaliter mota super plano inclinato aeque<lb/> velociter moventur ac si essent pendula, & moveren-<lb/>tur in tali portione vibrationum, quae quoad sensum<lb/> <pb xlink:href="076/01/011.jpg"/>esset aequalis, & paralella lineae dicti plani super qua<lb/> dicta solida moverentur.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000053">Non differt a praecedente, nisi quod in illa motus est per-<lb/>pendicularis, in hac inclinatus, in reliquis est par ratio.<lb/> </s> </p> </subchap1> </chap> <chap> <p type="head"> <s id="s.000054">PRONUNCIATA<lb/></s> </p> <subchap1> <p> <s id="s.000055">P. Quae sunt aequidiuturna tertio, sunt aequidiu-<lb/>turna inter se.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000056">2. Quadrata datorum temporum, sunt etiam quadrata<lb/> aliorum datis aequalium.<lb/> </s> </p> </subchap1> <subchap1> <p> <s id="s.000057">3. Gravia eadem super planis aequalibus & pariter incli-<lb/>natis, pariter moventur.<lb/> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/012.jpg"/> <chap> <p type="head"> <s id="s.000058">PROPOSITIO PRIMA.<lb/> </s> </p> <subchap1> <p> <s id="s.000059">Solidi penduli naturaliter moti vibrationes quan-<lb/>tumvis semper minores, sunt aequidiuturnae.<lb/> </s> </p> </subchap1> <p> <s id="s.000060">Sit solidum A pendulum debite applicatum filo BA, quod<lb/> ab altera parte elevatum naturaliter, postea faciat hinc<lb/> inde vibrationes semper minores, ita ut prior vibratio sit V. G.<lb/> per spatium CD maius, posterior vero per spatium EF minus.<lb/> </s> </p> <p> <s id="s.000061">Dico quod dicta vibrationes erunt aequidiuturnae, ita ut vibra-<lb/>tio per spatium CD sit eiusdem durationis, ac vibratio per<lb/> spatium EF.<lb/> </s> </p> <p> <s id="s.000062">Sit aliud solidum G aequipendulum solido A, debite applica-<lb/>tum filo HG, quod elevetur ab una parte eodem tempore<lb/> minus quam solidum A ita ut sint minores vibrationes soli-<lb/>di G, quam, solidi A, ut sit motus penduli G in initio per<lb/> spatium IK aequale spatio EF.<lb/> </s> </p> <p> <s id="s.000063">Quoniam spatia EF, & IK, sunt aequalia ex suppositione,<lb/> sunt etiam vibrationes EF, & IK, aequidiuturnae<arrow.to.target n="marg1"/>, sed I<lb/>K, & CD sunt pariter aequidiuturnae<arrow.to.target n="marg2"/>, ergo EF, & CD<lb/> sunt etiam aequidiuturnae<arrow.to.target n="marg3"/>. </s> <s id="s.000064">Quod fuit probandum.<lb/> </s> </p> <p type="margin"> <s id="s.000065"><margin.target id="marg1"/>Per pri-<lb/>mam sup-<lb/>positionem.<lb/></s> <s id="s.000066"><margin.target id="marg2"/>Per secun-<lb/>dam sup-<lb/>positionem.<lb/></s> <s id="s.000067"><margin.target id="marg3"/>Per pr.<lb/> pron.</s> </p> </chap> <pb xlink:href="076/01/013.jpg"/> <chap> <p type="head"> <s id="s.000068">PROPOSITIO II. PROBLEMA PRIMUM. </s> </p> <subchap1> <p> <s id="s.000069">Pendula constituere, quorum diuturnitates vibra-<lb/>tionum sint in data ratione.<lb/> </s> </p> </subchap1> <p> <s id="s.000070">Data sit proportio diuturnitatum vibrationum, quam<lb/> volumus esse inter solida A,B; & sit ea, quae est inter<lb/> C, & D; quae est continuo eadem,<arrow.to.target n="marg4"/>a.<lb/></s> </p> <p> <s id="s.000071"> Venanda est longitudo filorum, quibus applicata dicta solida<lb/> producant vibrationes quaesitas. </s> </p> <p type="margin"> <s id="s.000072"><margin.target id="marg4"/>Per pr.<lb/> hujus.</s> </p> <p> <s id="s.000073">Sint E F numeri mensurantes proportionem, quae est inter C<lb/> & D, quorum quadrati numeri G & H, Fila IA, KB<lb/> fiant inter se ut G, ad H, & erunt fila quaesita, quibus si<lb/> applicentur solida A, B, producentur diuturnitates vibra-<lb/>tionum quaesita.<lb/> </s> </p> <p> <s id="s.000074">Quoniam ita est IA, ad KB, ut quadratum G numeri me-<lb/>tientis C, ad quadratum H numeri metientis D, erunt C, &<lb/> D diuturnitates vibrationum pendulorum A, & B<arrow.to.target n="marg5"/>; &<lb/> proinde in ratione data. </s> <s id="s.000075">Quod faciendum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000076"><margin.target id="marg5"/>Per 3.<lb/> suppo.</s> </p> </chap> <pb xlink:href="076/01/014.jpg"/> <chap> <p type="head"> <s id="s.000077">PROPOSITIO TERTIA.<lb/> </s> </p> <subchap1> <p> <s id="s.000078">Lineae descensus gravium, dum naturali motu perpendicula-<lb/>riter feruntur, sunt in duplicata ratione diuturnitatum.<lb/> </s> </p> </subchap1> <p> <s id="s.000079">Sint LN, KM linea descensus gravium L, K, & sint P<lb/>O ipsorum diuturnitates.<lb/> </s> </p> <p> <s id="s.000080">Dico LN, KM esse in duplicata ratione ipsarum P, O.<lb/> </s> </p> <p> <s id="s.000081">Sint pendula AH, AI, dependentia a puncto A, & eleven-<lb/>tur ad libellam ipsius A usque ad E, B, quae in elevatione<lb/> producant arcus HB, IE, & sint talis longitudinis, ut du-<lb/>cta ACF, secet arcus BC, & EF, portionis minimae, aequa-<lb/>les quo ad sensum lineis LN, KM, & sit S, quadratum<lb/> diuturnitatis P, & T quadratum O, & Q, R, diuturni-<lb/>tates vibrationum BC, & EF.<lb/> </s> </p> <p> <s id="s.000082">Quoniam diuturnitates Q, R sunt aequales diuturnitatibus<lb/> P, O<arrow.to.target n="marg6"/>; S, T, sunt etiam quadrata ipsarum Q, R<arrow.to.target n="marg7"/>, & quia<lb/> vibrationes integrae pendulorum AH, AI sunt ut qua-<lb/>dratum T ad quadratum S<arrow.to.target n="marg8"/>, portiones BC, EF, sunt pa-<lb/>riter inter se ut quadratum T ad quadratum S<arrow.to.target n="marg9"/>, sed<lb/> BC, & EF sunt aequales lineis KM, LN<arrow.to.target n="marg10"/>, ergo etiam K<lb/>M, LN sunt ut quadrata S, T<arrow.to.target n="marg11"/>, & proinde in duplicata<lb/> ratione P, O, temporum seu diuturnitatum earumdem.<lb/> </s> <s id="s.000083">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000084"><margin.target id="marg6"/>Per 5.<lb/> pet.<lb/></s> <s id="s.000085"><margin.target id="marg7"/>Per 2.<lb/> pron.<lb/></s> <s id="s.000086"><margin.target id="marg8"/>Per 3.<lb/> supposit.<lb/></s> <s id="s.000087"><margin.target id="marg9"/>Per 5.<lb/> petit.<lb/></s> <s id="s.000088"><margin.target id="marg10"/>Per 3.<lb/> petit.<lb/></s> <s id="s.000089"><margin.target id="marg11"/>Per 1.<lb/> pron.<lb/></s> </p> </chap> <pb xlink:href="076/01/015.jpg"/> <chap> <p type="head"> <s id="s.000090">PROPOSITIO QUARTA. PROBL. II.<lb/> </s> </p> <subchap1> <p> <s id="s.000091">Data diuturnitate gravis descendentis a data altitudine,<lb/> constituere altitudinem, a qua idem grave cadat in<lb/> data alia diuturnitate.<lb/> </s> </p> </subchap1> <p> <s id="s.000092">Sit A diuturnitas gravis B, dum cadit in C, & data sit<lb/> diuturnitas quaecumque D.<lb/> </s> </p> <p> <s id="s.000093">Constituenda est alia altitudo, a qua grave descendat iuxta<lb/> diuturnitatem D.<lb/> </s> </p> <p> <s id="s.000094">Fiant E, & F quadrata temporum A, D, & ut F ad E, fiat<lb/> altitudo GH, ad altitudinem datam BC; Dico GH esse al-<lb/>titudinem quaesitam.<lb/> </s> </p> <p> <s id="s.000095">Quoniam BC, & GH sunt in duplicata ratione datarum diu-<lb/>turnitatum A, D, per constructionem; per ipsas gravia B,<lb/> & G cadent in diuturnitatibus A, & D datis<arrow.to.target n="marg12"/>, unde re-<lb/>perta est altitudo GH quaesita. </s> <s id="s.000096">Quod fuit faciendum.<lb/> </s> </p> <p type="margin"> <s id="s.000097"><margin.target id="marg12"/>Per 3.<lb/> hujus.</s> </p> </chap> <pb xlink:href="076/01/016.jpg"/> <chap> <p type="head"> <s id="s.000098">PROPOSITIO V. PROBL. III.<lb/> </s> </p> <subchap1> <p> <s id="s.000099">Data altitudine, a qua descendat grave in nota diutur-<lb/>nitate; perquirere quanta sit diuturnitas, qua descen-<lb/>dat ab alia altitudine data.<lb/> </s> </p> </subchap1> <p> <s id="s.000100">Sit A altitudo per quam descendat grave diuturnitate B<lb/> nota, & data sit alia altitudo C.<lb/> </s> </p> <p> <s id="s.000101">Oportet reperire quanta sit diuturnitas, qua idem grave de-<lb/>scendat per C.<lb/> </s> </p> <p> <s id="s.000102">Fiat D quadratum diuturnitatis B, & fiat ut A ad C, ita<lb/> quadratum D ad quadratum E, cuius radix F est diutur-<lb/>nitas quaesita. </s> </p> <p> <s id="s.000103">Quoniam A, & C sunt in duplicata ratione diuturnitatum<lb/> B, & F per constructionem, per ipsas gravia descendent<lb/> in diuturnitatibus B, F,<arrow.to.target n="marg13"/> unde F est diuturnitas ipsius C<lb/> quaesita. </s> <s id="s.000104">Quod faciendum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000105"><margin.target id="marg13"/>Per 3.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/017.jpg"/> <chap> <p type="head"> <s id="s.000106">PROPOSITIO VI. </s> </p> <subchap1> <p> <s id="s.000107">Gravia naturali motu descendunt semper velocius ea<lb/> ratione, ut temporibus aequalibus descendant per spa-<lb/>tia semper maiora, iuxta proportionem quam ha-<lb/>bent impares numeri ab unitate inter se.<lb/> </s> </p> </subchap1> <p> <s id="s.000108">Sit grave A quod descendat per lineam ABC, & tempus<lb/> quo descendit ab A in B sit aequale tempori, quo de-<lb/>scendit a B in C, & a C in D.<lb/> </s> </p> <p> <s id="s.000109">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &<lb/> sic deinceps.<lb/> </s> </p> <p> <s id="s.000110">Sit G numerus mensurans tempus, quo A descendit in B, &<lb/> H, quo descendit a B in C, & I, quo descendit a C in D,<lb/> quae tempora sunt ex suppositione aequalia, & sit K qua-<lb/>dratum ipsius G, & L quadratum GH, & M quadratum<lb/> totius GHI. </s> </p> <p> <s id="s.000111">Quoniam quadrata K, L, N sunt ut AB, AC, AD<arrow.to.target n="marg14"/>, quae<lb/> quadrata sunt ut 1, 4, 9, sunt itidem AB, AC, AD, ut<lb/> 1. 4. 9. & dividendo AB, BC, CD, ut 1. 3. 5. & sic dein-<lb/>ceps. </s> <s id="s.000112">Quod probandum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000113"><margin.target id="marg14"/>Per 3.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/018.jpg"/> <chap> <p type="head"> <s id="s.000114">PROPOSITIO VII.<lb/> </s> </p> <subchap1> <p> <s id="s.000115">Lineae descensus gravium super plano inclinato mo-<lb/>torum, sunt in duplicata ratione diuturnitatum.<lb/> </s> </p> </subchap1> <p> <s id="s.000116">Sint AB, CD plana pariter inclinata, super quibus mo-<lb/>veantur gravia A, C, & sint EF ipsorum diuturnitates.<lb/> </s> </p> <p> <s id="s.000117">Dico AB, CD, esse in duplicata ratione ipsarum E, F.<lb/> </s> </p> <p> <s id="s.000118">Secetur AB bifariam in G, & erecta GH, perpendiculari<lb/> longissima, fiant pendula HI, HK, quae sint inter se ut A<lb/>B, CD, & eleventur in L, M, describentia arcus LI, KM,<lb/> secantes GH in N, O, & ab N hinc inde secentur arcus N<lb/>P, NQ aequales quo ad sensum rectis GA, GB, & ductis P<lb/>H, QH, secetur pariter arcus LI, in R, S, & intelligan-<lb/>tur arcus PQ, RS, tam parvae curvitatis, ob maximam<lb/> longitudinem pendulorum HI, HK, ut pro rectis habean-<lb/>tur, puta portionis minimae, & proinde aequales rectis A<lb/>B, CD: sit Z quadratum diuturnitatis E, & V, diuturnitatis<lb/> F, & sint XY diuturnitates vibrationum PQ, RS.<lb/> </s> </p> <pb xlink:href="076/01/019.jpg"/> <p> <s id="s.000119">Quoniam diuturnitates X, Y, sunt aequales diuturnitatibus<lb/> E, F,<arrow.to.target n="marg15"/> sunt etiam Z, V, quadrata ipsarum X, Y<arrow.to.target n="marg16"/>; & quia<lb/> vibrationes integrae pendulorum HI, HK sunt inter se, ut<lb/> quadratum V, ad quadratum Z<arrow.to.target n="marg17"/>, portiones RS, PQ erunt<lb/> etiam inter se ut quadratum V ad quadratum Z<arrow.to.target n="marg18"/>; sed R<lb/>S, PQ aequantur rectis CD, AB,<arrow.to.target n="marg19"/>, ergo, & CD, AB<lb/> sunt ut quadratum V, ad quadratum Z<arrow.to.target n="marg20"/>, & proinde, in<lb/> duplicata ratione ipsarum EF. </s> <s id="s.000120">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000121"><margin.target id="marg15"/>Per 6.<lb/> petit.<lb/></s> <s id="s.000122"><margin.target id="marg16"/>Per 2.<lb/> pron.<lb/></s> <s id="s.000123"><margin.target id="marg17"/>Per 3.<lb/> hujus.<lb/></s> <s id="s.000124"><margin.target id="marg18"/>Per pr.<lb/> pet.<lb/></s> <s id="s.000125"><margin.target id="marg19"/>Per 3.<lb/> petit.<lb/></s> <s id="s.000126"><margin.target id="marg20"/>Per 2.<lb/> pron.<lb/></s> </p> <p> <s id="s.000127">Corolarium.<lb/> </s> </p> <p> <s id="s.000128">Hinc patet esse longitudines planorum per quae gravia fe-<lb/>runtur ut quadrata temporum, & tempora ut radices<lb/> longitudinum planorum.<lb/> </s> </p> </chap> <pb xlink:href="076/01/020.jpg"/> <chap> <p type="head"> <s id="s.000129">PROPOSITIO VIII. PROB. IV.<lb/> </s> </p> <subchap1> <p> <s id="s.000130">Dato plano inclinato, super quo per spatium datum<lb/> grave moveatur in nota diuturnitate, determinare in<lb/> eodem plano spatium per quod dictum grave mo-<lb/>veatur in quavis alia diuturnitate data.<lb/> </s> </p> </subchap1> <p> <s id="s.000131">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb/>no inclinato BC, & data diuturnitas D.<lb/> </s> </p> <p> <s id="s.000132">Praescribendum est aliud spatium in eodem plano BC, per<lb/> quod idem grave pertranseat in diuturnitate D.<lb/> </s> </p> <p> <s id="s.000133">Fiant E, F quadrata temporum A, D, & ut F ad E fiat BG ad<lb/> BC, Dico BG esse spatium quaesitum.<lb/> </s> </p> <p> <s id="s.000134">Quoniam BC, & BG sunt in duplicata ratione datorum<lb/> temporum A, D per constructionem, per ipsa cadet grave<lb/> B diuturnitatibus A, D datis<arrow.to.target n="marg21"/>, ergo reperta est BG quae-<lb/>sita. </s> <s id="s.000135">Quod faciendum erat.<lb/> </s> </p> <p type="margin"> <s id="s.000136"><margin.target id="marg21"/>Per 6.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/021.jpg"/> <chap> <p type="head"> <s id="s.000137">PROPOSITIO IX. PROB. V.<lb/> </s> </p> <subchap1> <p> <s id="s.000138">Dato plano inclinato, super quo per spatium datum gra-<lb/>ve moveatur nota diuturnitate; & dato alio spatio<lb/> quocumque; reperire diuturnitatem, qua grave per<lb/> ipsum descendat.<lb/> </s> </p> </subchap1> <p> <s id="s.000139">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb/>no inclinato BC, & dato alio spatio BG.<lb/> </s> </p> <p> <s id="s.000140">Querendum quanta sit diuturnitas gravis in BG.<lb/> </s> </p> <p> <s id="s.000141">Fiat E quadratum diuturnitatis A, & ut BC ad BG fiat ut<lb/> quadratum E ad quadratum F, cuius radix D erit diutur-<lb/>nitas ipsius BG quaesita.<lb/> </s> </p> <p> <s id="s.000142">Quoniam BC, & BG sunt in duplicata ratione diuturnita-<lb/>tum A, D per constructionem; per ipsa cadunt gravia diu-<lb/>turnitatibus A, D<arrow.to.target n="marg22"/>, unde D est diuturnitas per spatium<lb/> BG quaesita. </s> <s id="s.000143">Quod faciendum erat.<lb/> </s> </p> <p type="margin"> <s id="s.000144"><margin.target id="marg22"/>Per 7.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/022.jpg"/> <chap> <p type="head"> <s id="s.000145">PROPOSITIO X.<lb/> </s> </p> <subchap1> <p> <s id="s.000146">Gravia descendunt super planis inclinatis per spatia<lb/> semper maiora, iuxta rationem, quam habent im-<lb/>pares numeri successive inter se.<lb/> </s> </p> </subchap1> <p> <s id="s.000147">Sit grave A, quod descendat super plano ABC inclinato,<lb/> & tempus quo descendit ab A in B sit aequale tempo-<lb/>ri, quo descendit a B in C, & a C in D.<lb/> </s> </p> <p> <s id="s.000148">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &. sic<lb/> deinceps. </s> </p> <p> <s id="s.000149">Sit E numerus mensurans tempus, quo A descendit in B, & F<lb/> quo descendit a B in C, & G quo descendit a C in D, quae<lb/> tempora sunt ex suppositione aequalia, & sit H quadratum<lb/> ipsius E, & I quadratum EF, & K quadratum totius EFG.<lb/> </s> </p> <p> <s id="s.000150">Quoniam quadrata HIK sunt ut AB, AC, AD<arrow.to.target n="marg23"/>, quae<lb/> quadrata sunt ut 1. 4. 9. sunt pariter AB, AC, AD, ut<lb/> 1. 4. 9. & dividendo AB, BC, CD, sunt ut 1. 3. 5. & sic<lb/> deinceps. </s> <s id="s.000151">Quod probandum erat.<lb/> </s> </p> <p type="margin"> <s id="s.000152"><margin.target id="marg23"/>Per 7.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/023.jpg"/> <chap> <p type="head"> <s id="s.000153">PROPOSITIO XI.<lb/> </s> </p> <subchap1> <p> <s id="s.000154">Si Duo gravia descendant alterum super linea perpen-<lb/>diculari, alterum vero super inclinata; proportio ve-<lb/>locitatum est reciproca proportioni linearum.<lb/> </s> </p> </subchap1> <p> <s id="s.000155">Sit ABC planum normaliter erectum super lineam ori-<lb/>zontalem BC, cuius latus AB sit perpendiculare, &<lb/> AC, inclinatum.<lb/> </s> </p> <p> <s id="s.000156">Dico quod proportio velocitatum solidorum gravium moto-<lb/>rum secundum lineam AB perpendicularem, & AC in-<lb/>clinatum, est ut proportio longitudinis inclinatae AC ad<lb/> longitudinem perpendicularis AB; videlicet ita est longi-<lb/>tudo AB ad longitudinem AC, ut velocitas super AC ad<lb/> velocitatem in AB.<lb/> </s> </p> <p> <s id="s.000157">Quoniam est ut AC ad AB, ita momentum in AB, ad mo-<lb/>mentum in AC<arrow.to.target n="marg24"/>; & ut momentum in AB ad momentum<lb/> in AC, ita velocitas in AB ad velocitatem in AC<arrow.to.target n="marg25"/>; er-<lb/>go est etiam ut AC ad AB, ita velocitas in AB ad velo-<lb/>citatem in AC. </s> <s id="s.000158">Quod fuit probandum.<lb/> </s> </p> <p type="margin"> <s id="s.000159"><margin.target id="marg24"/>Per 4.<lb/> supp.<lb/></s> <s id="s.000160"><margin.target id="marg25"/>Per 2.<lb/> pet.<lb/></s> </p> </chap> <pb xlink:href="076/01/024.jpg"/> <chap> <p type="head"> <s id="s.000161">PROPOSITIO XII.<lb/> </s> </p> <subchap1> <p> <s id="s.000162">Gravia descendunt super plana diverse inclinata tali<lb/> proportione, ut si velocitas ad velocitatem recipro-<lb/>ca longitudinibus planorum ductorum ab eodem<lb/> puncto, ad idem planum orizontale.<lb/> </s> </p> </subchap1> <p> <s id="s.000163">Sint F, D plana inclinata ducta ad idem planum orizon-<lb/>tale.<lb/> </s> </p> <p> <s id="s.000164">Dico esse ut planum D ad planum F, ita velocitatem gravis<lb/> ducti super F, ad velocitatem eiusdem ducti super D.<lb/> </s> </p> <p> <s id="s.000165">Ducatur perpendicularis E, & sint B, A, C velocitates gra-<lb/>vium latorum super perpendiculari, & super planis F, D.<lb/> </s> </p> <p> <s id="s.000166">Quoniam est A ad B, ut E ad F, item, & B ad C, ut D, ad<lb/> E<arrow.to.target n="marg26"/>, erit A ad C ut D ad F<arrow.to.target n="marg27"/>, scilicet velocitas gravis su-<lb/>per F ad velocitatem gravis super D, ut longitudo pla-<lb/>ni D ad longitudinem plani F. </s> <s id="s.000167">Quod fuit probandum.<lb/> </s> </p> <p type="margin"> <s id="s.000168"><margin.target id="marg26"/>Per 11.<lb/> hujus.<lb/></s> <s id="s.000169"><margin.target id="marg27"/>Per 23.<lb/> Quinti.<lb/></s> </p> </chap> <pb xlink:href="076/01/025.jpg"/> <chap> <p type="head"> <s id="s.000170">PROPOSITIO XIII. PROBL. VI.<lb/> </s> </p> <subchap1> <p> <s id="s.000171">Reperire inclinationem plani, super quo grave movea-<lb/>tur tali velocitate quae cum alia super diversa incli-<lb/>natione sit in ratione data.<lb/> </s> </p> </subchap1> <p> <s id="s.000172">Moveatur grave A super recta AB, seu perpendicula-<lb/>ri, seu inclinata, & data sit proportio C ad D.<lb/> </s> </p> <p> <s id="s.000173">Oportet reperire aliud planum inclinatum, ita ut velocitas<lb/> gravis moti super AB ad velocitatem alterius moti<lb/> super illo reperiendo, sit ut D ad C.<lb/> </s> </p> <p> <s id="s.000174">Producatur BA; & fiat ut C ad D ita BA, ad AE; &<lb/> centro A, intervallo AE describatur circulus, secans BF<lb/> in F; ni secet, problema insolubile est; si secat, ducatur<lb/> AF, quam dico esse planum quaesitum.<lb/> </s> </p> <p> <s id="s.000175">Quoniam ut C ad D, ita AB ad AE, seu AF per constructio-<lb/>nem, erit C velocitas super AF, & D super AB<arrow.to.target n="marg28"/>, unde velo-<lb/>citates super ipsis sunt in ratione data. </s> <s id="s.000176">Quod faciendum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000177"><margin.target id="marg28"/>Per 12.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/026.jpg"/> <chap> <p type="head"> <s id="s.000178">PROPOSITIO XIV. PROBL. VII.<lb/> </s> </p> <subchap1> <p> <s id="s.000179">Data linea perpendiculari, per quam grave descendat,<lb/> cui annectatur linea, seu planum declinans; in decli-<lb/>nante reperire punctum, quo grave perveniat eo<lb/> tempore, quo pertransiverit perpendicularem.<lb/> </s> </p> </subchap1> <p> <s id="s.000180">Sit triangulum ABC orthogonaliter erectum super pla<lb/>no orizontali BC, cuius latus AB intelligatur sit linea perpendicu-<lb/>laris, per quam grave descendat, & latus AC sit planum<lb/> inclinatum.<lb/> </s> </p> <p> <s id="s.000181">Oportet in plano AC reperire punctum quo grave perveniat<lb/> eodem tempore, quo in B.<lb/> </s> </p> <p> <s id="s.000182">Fiat ut AC ad AB, ita AB ad tertiam AD, & D erit pun-<lb/>ctum quaesitum.<lb/> </s> </p> <p> <s id="s.000183">Quoniam ut AC ad AD, ita quadratum AC ad quadra-<lb/>tum AB<arrow.to.target n="marg29"/>, & ut AC ad AD, ita quadratum temporis A<lb/>C ad quadratum temporis AD<arrow.to.target n="marg30"/>, ergo ut quadratum A<lb/>C ad quadratum AB, ita quadratum temporis AC ad qua-<lb/>dratum temporis AD<arrow.to.target n="marg31"/>, ergo ut AC ad AB, ita tempus<lb/> AC ad tempus AD<arrow.to.target n="marg32"/>, sed ut AC ad AB, ita tempus AC ad<lb/> tempus AB<arrow.to.target n="marg33"/>, ergo tempus AB est aequale tempori AD.<lb/> </s> <s id="s.000184">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000185"><margin.target id="marg29"/>Per 19.<lb/> Sexti.<lb/></s> <s id="s.000186"><margin.target id="marg30"/>Per cor.<lb/> 7. hujus.<lb/></s> <s id="s.000187"><margin.target id="marg31"/>Per 11.<lb/> Quinti.<lb/></s> <s id="s.000188"><margin.target id="marg32"/>Per 22.<lb/> Sexti.<lb/></s> <s id="s.000189"><margin.target id="marg33"/>Per 11.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/027.jpg"/> <chap> <p type="head"> <s id="s.000190">PROPOSITIO XV.<lb/> </s> </p> <subchap1> <p> <s id="s.000191">Linea connectens puncta, ad quae duo gravia ab eo-<lb/>dem puncto digressa, quorum alterum perpenden-<lb/>ter, alterum super plano declinante descendat, simul<lb/> perveniunt, est perpendicularis dicto plano declinanti.<lb/> </s> </p> </subchap1> <p> <s id="s.000192">Descendant simul duo gravia a puncto A primum per-<lb/>pendiculariter in B, secundum super plano inclinato<lb/> AC, tali lege, ut simul perveniant ad puncta BD,<lb/> & ducta sit linea BD.<lb/> </s> </p> <p> <s id="s.000193">Dico quod dicta linea BD est perpendicularis ad AD.<lb/> </s> </p> <p> <s id="s.000194">Fiat AF aequalis datae AB, & AE aequalis AD, & duca-<lb/>tur EF.<lb/> </s> </p> <p> <s id="s.000195">Quoniam ut AD ad AB, ita AB ad AC<arrow.to.target n="marg34"/>, & AD,<lb/> AE, item AB, AF sunt aequales per constructionem, se-<lb/>quitur quod AE ad AF est ut AB ad AC, ergo EF, BC<lb/> sunt parallelae<arrow.to.target n="marg35"/>, unde triangulum AEF, & proin-<lb/>de ABD est simile triangulo ABC<arrow.to.target n="marg36"/>, unde anguli AB<lb/>C, ADB simul recti, & BD perpendicularis ad AD.<lb/> </s> <s id="s.000196">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000197"><margin.target id="marg34"/>Per 13.<lb/> hujus.<lb/></s> <s id="s.000198"><margin.target id="marg35"/>Per 2.<lb/> Sexti.<lb/></s> <s id="s.000199"><margin.target id="marg36"/>Per 4.<lb/> Sexti.<lb/></s> </p> </chap> <pb xlink:href="076/01/028.jpg"/> <chap> <p type="head"> <s id="s.000200">PROPOSITIO XVI. PROBL. VIII.<lb/> </s> </p> <subchap1> <p> <s id="s.000201">Data linea perpendiculari, & plano declinante; reperire<lb/> in perpendiculari producta punctum, quo perveniat<lb/> grave eo tempore, quo pertransit planum inclinatum.<lb/> </s> </p> </subchap1> <p> <s id="s.000202">Data sit perpendicularis AB, cui connexum planum<lb/> inclinatum AD.<lb/> </s> </p> <p> <s id="s.000203">Oportet in AB producta reperire punctum, quo perveniat<lb/> grave eo tempore, quo pervenit in puncto D.<lb/> </s> </p> <p> <s id="s.000204">In puncto D perpendicularis erigatur ad AD, & protraha-<lb/>tur usquequo coeat cum AB producta in E, & E est pu-<lb/>nctum quaesitum. </s> </p> <p> <s id="s.000205">Quoniam triangula & ADE, AEC sint aequiangula,<lb/> cum anguli ADE, AEC sint aequales, nempe recti, &<lb/> BAD communis<arrow.to.target n="marg37"/>, sunt etiam similia<arrow.to.target n="marg38"/>, ergo ut AC<lb/> ad AE, ita AE ad AD<arrow.to.target n="marg39"/>, sed ut AC ad AD, ita qua-<lb/>dratum AC ad quadratum AE<arrow.to.target n="marg40"/>, & ut AC ad AD,<lb/> ita quadratum temporis AC ad quadratum temporis A<lb/>D<arrow.to.target n="marg41"/>, ergo ut quadratum AC ad quadratum AE ita qua-<lb/>dratum temporis AC ad quadratum temporis AD<arrow.to.target n="marg42"/>, er-<lb/>go ut AC ad AE, ita tempus AC ad tempus AD<arrow.to.target n="marg43"/>, sed<lb/> ut AC ad AE, ita tempus AC ad tempus AE<arrow.to.target n="marg44"/>, ergo<lb/> tempora AE, & AD sunt aequalia. </s> <s id="s.000206">Quod &c.<lb/> </s> </p> <p type="margin"> <s id="s.000207"><margin.target id="marg37"/>Per 32.<lb/> prim.<lb/></s> <s id="s.000208"><margin.target id="marg38"/>Per 4.<lb/> sexti.<lb/></s> <s id="s.000209"><margin.target id="marg39"/>Per 4.<lb/> sexti.<lb/></s> <s id="s.000210"><margin.target id="marg40"/>Per 19.<lb/> Sexti.<lb/></s> <s id="s.000211"><margin.target id="marg41"/>Per Cor.<lb/> 7. hujus.<lb/></s> <s id="s.000212"><margin.target id="marg42"/>Per 11.<lb/> Quinti.<lb/></s> <s id="s.000213"><margin.target id="marg43"/>Per 22.<lb/> sexti.<lb/></s> <s id="s.000214"><margin.target id="marg44"/>Per 11.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/029.jpg"/> <chap> <p type="head"> <s id="s.000215">PROPOSITIO XVII. PROBL. IX.<lb/> </s> </p> <subchap1> <p> <s id="s.000216">Dato plano declinante, super quo grave descendat, &<lb/> dato alio plano minus declinante, in hoc reperire<lb/> punctum, quo perveniat mobile eo tempore, quo<lb/> pertransit dictum planum magis declinans.<lb/> </s> </p> </subchap1> <p> <s id="s.000217">Sint plana AB, AC quorum AC minus inclinatum.<lb/> </s> </p> <p> <s id="s.000218">Oportet in AC reperire punctum, quo grave perveniat,<lb/> quando pervenit in B.<lb/> </s> </p> <p> <s id="s.000219">Fiat ut AC ad AB ita AB ad AD, & dico D esse punctum<lb/> quaesitum. </s> </p> <p> <s id="s.000220">Quoniam ut AC ad AD ita est quadratum AC ad quadra-<lb/>tum AB<arrow.to.target n="marg45"/>, & ut AC ad AD ita quadratum temporis<lb/> AC ad quadratum temporis AD<arrow.to.target n="marg46"/>, ergo ut quadratum A<lb/>C ad quadratum AB, ita quadratum temporis AC ad<lb/> quadratum temporis AD<arrow.to.target n="marg47"/>, Unde AC ad AB ut tempus<lb/> AC ad tempus AD<arrow.to.target n="marg48"/>, sed ut AC ad AB, ita tempus A<lb/>C ad tempus AB<arrow.to.target n="marg49"/>, ergo tempora AB, AD, sunt aequa-<lb/>lia. </s> <s id="s.000221">Quod, &c. </s> </p> <p type="margin"> <s id="s.000222"><margin.target id="marg45"/>Per 19.<lb/> sexti.<lb/></s> <s id="s.000223"><margin.target id="marg46"/>Per cor.<lb/> 7. hujus.<lb/></s> <s id="s.000224"><margin.target id="marg47"/>Per 11.<lb/> Quinti.<lb/></s> <s id="s.000225"><margin.target id="marg48"/>Per 22.<lb/> sexti.<lb/></s> <s id="s.000226"><margin.target id="marg49"/>Per 11.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/030.jpg"/> <chap> <p type="head"> <s id="s.000227">PROPOSITIO XVIII. PROBL. X.<lb/> </s> </p> <subchap1> <p> <s id="s.000228">Datis planis declinantibus ortis ab eodem puncto, re-<lb/>perire in magis declinante punctum quo grave per-<lb/>veniat eo tempore, quo pertransit planum minus<lb/> declinans. </s> </p> </subchap1> <p> <s id="s.000229">Datum sit planum minus declinans AC, & magis A<lb/>D, terminantia super plano orizontali BD.<lb/> </s> </p> <p> <s id="s.000230">Oportet in AD producta reperire punctum, quo perveniat<lb/> grave eo tempore, quo pertransivit planum minus decli-<lb/>nans AC.<lb/> </s> </p> <p> <s id="s.000231">Fiat ut AD ad AC ita AC ad dictam AD productam in<lb/> E, quod est punctum quaesitum.<lb/> </s> </p> <p> <s id="s.000232">Quoniam ut AE ad AD ita est quadratum AC ad quadra-<lb/>tum AD<arrow.to.target n="marg50"/>, sed AE ad AD est ut quadratum temporis<lb/> AE, ad quadratum temporis AD<arrow.to.target n="marg51"/>, ergo ut quadratum<lb/> AC ad quadratum AD, ita quadratum temporis AE ad<lb/> quadratum temporis AD<arrow.to.target n="marg52"/>, unde AC ad AD ut tempus<lb/> AE ad tempus AD<arrow.to.target n="marg53"/>, sed AC ad AD est ut tempus AC<lb/> ad tempus AD<arrow.to.target n="marg54"/>, ergo tempora AE, AC sunt aequalia.<lb/> </s> <s id="s.000233">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000234"><margin.target id="marg50"/>Per 19.<lb/> sexti.<lb/></s> <s id="s.000235"><margin.target id="marg51"/>Per cor.<lb/> 7. hujus.<lb/></s> <s id="s.000236"><margin.target id="marg52"/>Per 11.<lb/> Quinti.<lb/></s> <s id="s.000237"><margin.target id="marg53"/>Per 22.<lb/> sexti.<lb/></s> <s id="s.000238"><margin.target id="marg54"/>Per 11.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/031.jpg"/> <chap> <p type="head"> <s id="s.000239">PROPOSITIO XIX. PROBL. XI.<lb/> </s> </p> <subchap1> <p> <s id="s.000240">Dato motus naturali gravis quomodocumque ad pun-<lb/>ctum datum, reperire seu in perpendiculari, seu in<lb/> plano quomodolibet inclinato punctum, a quo di-<lb/>gressum, perveniat ad idem punctum quo prius, tem-<lb/>pore aequali. </s> </p> </subchap1> <p> <s id="s.000241">Sit AB linea quomodocumque aut perpendicularis, seu<lb/> planum inclinatum; super qua grave descendat in B, &<lb/> data sit quaecumque linea BC, aut perpendicularis, aut<lb/> quomodolibet inclinata, quae cum AB, coeat in B.<lb/> </s> </p> <p> <s id="s.000242">Oportet in BC reperire punctum, a quo grave digressum per-<lb/>veniat in B tempore quo pervenit ab A in idem B.<lb/> </s> </p> <p> <s id="s.000243">Ducatur AC orizontalis, & fiat BD tertia proportiona-<lb/>lis ad CB AB<arrow.to.target n="marg55"/>, & D est punctum quaesitum. </s> <s id="s.000244">Quod ut probetur.<lb/> </s> </p> <p type="margin"> <s id="s.000245"><margin.target id="marg55"/>Per 11.<lb/> Sexti.<lb/></s> </p> <p> <s id="s.000246">Fiat iterum rectae AC paralella, & aequalis BE, & ducta<lb/> EA, secetur recta BF parallela ipsi AD.<lb/> </s> </p> <p> <s id="s.000247">Quoniam AF, BD sunt pariter inclinatae, & aequales<arrow.to.target n="marg56"/>, gra-<lb/>via per ipsas aequali tempore moventur<arrow.to.target n="marg57"/>, ergo aequali tem-<lb/>pore ut per AB<arrow.to.target n="marg58"/>, quod, &c. </s> </p> <p type="margin"> <s id="s.000248"><margin.target id="marg56"/>Per 33.<lb/> Primi.<lb/></s> <s id="s.000249"><margin.target id="marg57"/>Per 3.<lb/> pronun.<lb/></s> <s id="s.000250"><margin.target id="marg58"/>Per pr.<lb/> pron.<lb/></s> </p> </chap> <pb xlink:href="076/01/032.jpg"/> <chap> <p type="head"> <s id="s.000251">PROPOSITIO XX. PROBL. XII.<lb/> </s> </p> <subchap1> <p> <s id="s.000252">Datis duobus planis diverse inclinatis longitudinis no-<lb/>tae; & nota diuturnitate gravis moti super uno, re-<lb/>perire diuturnitatem si moveatur super alio.<lb/> </s> </p> </subchap1> <p> <s id="s.000253">Sint plana AB, CD inclinata, & sit data diuturnitas E<lb/> plani AB.<lb/> </s> </p> <p> <s id="s.000254">Oportet reperire diuturnitatem plani CD.<lb/> </s> </p> <p> <s id="s.000255">Fiat AF, paralella, & aequalis datae CD, in qua reperiatur<lb/> punctum G quo perveniat grave, tempore quo in B<arrow.to.target n="marg59"/>, unde<lb/> E est etiam diuturnitas spatii AG, quo dato, & spatio<lb/> AF perquiratur eias diuturnitas, quae sit H<arrow.to.target n="marg60"/>, & dico<lb/> H esse diuturnitatem quae grave descendit in CD.<lb/> </s> </p> <p type="margin"> <s id="s.000256"><margin.target id="marg59"/>Per 17.<lb/> hujus.<lb/></s> <s id="s.000257"><margin.target id="marg60"/>Per 9.<lb/> hujus.<lb/></s> </p> <p> <s id="s.000258">Quoniam E, H sunt diuturnitates gravium descendentium<lb/> in AG, seu AB, & AF, per constructionem, & AF<lb/> est aequalis, & paralella datae CD per constructionem,<lb/> sunt etiam E, H diuturnitates ipsarum AB, & CD<arrow.to.target n="marg61"/>, unde<lb/> reperta est diuturnitas ipsius CD. </s> <s id="s.000259">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000260"><margin.target id="marg61"/>Per 3.<lb/> pron.<lb/></s> </p> </chap> <pb xlink:href="076/01/033.jpg"/> <chap> <p type="head"> <s id="s.000261">PROPOSITIO XXI. PROBL. XIII.<lb/> </s> </p> <subchap1> <p> <s id="s.000262">Datis duabus diuturnitatibus, quarum prior sit gravis<lb/> moti super plano dato longitudinis notae, & dato<lb/> alio plano diversimode declinante; reperiendum est<lb/> in eo punctum, quo grave perveniat in secunda<lb/> diuturnitate data.<lb/> </s> </p> </subchap1> <p> <s id="s.000263">Dato plano declinante AB, super quo grave A movea<lb/>tur diuturnitate C, & dato alio plano D declinationis<lb/> quae sit dissimilis declinationi datae AB; data itidem diu-<lb/>turnitate E.<lb/> </s> </p> <p> <s id="s.000264">Oportet reperire in D punctum quo grave perveniat in<lb/> diuturnitate E.<lb/> </s> </p> <p> <s id="s.000265">Ducatur AF parallela ipsi D, in eaque reperiatur pun-<lb/>ctum F, quo grave perveniat tempore quo in B<arrow.to.target n="marg62"/>, & prae-<lb/>scribatur in eadem spatium AG per quod moveatur in<lb/> diuturnitate E<arrow.to.target n="marg63"/>, & fiat DH aequalis ipsi AG, & dico H<lb/> esse punctum quaesitum.<lb/> </s> </p> <p type="margin"> <s id="s.000266"><margin.target id="marg62"/>Per 17.<lb/> hujus.<lb/></s> <s id="s.000267"><margin.target id="marg63"/>Per 8.<lb/> hujus.<lb/></s> </p> <p> <s id="s.000268">Quoniam diuturnitates in AB, AF sunt aequales per con-<lb/>structionem, & C, E sunt diuturnitates super planis AF,<lb/> AG per constructionem, sunt etiam diuturnitates super<lb/> AB, AG, & proinde super DH ipsi AG aequali, &<lb/> paralellae, quod, &c.<lb/> </s> </p> </chap> <pb xlink:href="076/01/034.jpg"/> <chap> <p type="head"> <s id="s.000269">PROPOSITIO XXII.<lb/> </s> </p> <subchap1> <p> <s id="s.000270">Si duo gravia descendunt alterum quidem perpendicu-<lb/>lariter, alterum vero super plano declinante, perve-<lb/>niunt ad idem planum Orizontale tali ratione, ut sit<lb/> eadem proportio inter diuturnitates eorum, quae in-<lb/>ter perpendicularem, & declinantem.<lb/> </s> </p> </subchap1> <p> <s id="s.000271">Sit linea AB perpendiculariter erecta super plano Ori-<lb/>zontali BC, & AC planum declinans.<lb/> </s> </p> <p> <s id="s.000272">Dico quod diuturnitates gravium descendentium per AB, &<lb/> per AC, sunt ut AB ad AC.<lb/> </s> </p> <p> <s id="s.000273">Ducatur BD normalis ad AC.<lb/> </s> </p> <p> <s id="s.000274">Quoniam est ut AD ad AC ita quadratum temporis AD<lb/> ad quadratum temporis AC<arrow.to.target n="marg64"/>, & tempora AD, & AB<lb/> sunt aequalia<arrow.to.target n="marg65"/>, & proinde eorum quadrata<arrow.to.target n="marg66"/>, ergo ut A<lb/>D, ad AC ita quadratum temporis AB ad quadratum tem-<lb/>poris AC, sed ut AD ad AC ita quadratum AB ad qua-<lb/>dratum AC<arrow.to.target n="marg67"/>, ergo ut quadratum temporis AB ad qua-<lb/>dratum temporis AC, ita quadratum AB ad quadratum<lb/> AC<arrow.to.target n="marg68"/>, sed quia latera sunt inter se ut eorum quadrata<arrow.to.target n="marg69"/>, est<lb/> ut AB ad AC ita tempus AB ad tempus AC. </s> <s id="s.000275">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000276"><margin.target id="marg64"/>Per cor.<lb/> 7. hujus.<lb/></s> <s id="s.000277"><margin.target id="marg65"/>Per 15.<lb/> hujus.<lb/></s> <s id="s.000278"><margin.target id="marg66"/>Per 2.<lb/> pron.<lb/></s> <s id="s.000279"><margin.target id="marg67"/>Per 19.<lb/> Sexti.<lb/></s> <s id="s.000280"><margin.target id="marg68"/>Per 22.<lb/> Quinti.<lb/></s> <s id="s.000281"><margin.target id="marg69"/>Per 24.<lb/> Sexti.<lb/></s> </p> </chap> <pb xlink:href="076/01/035.jpg"/> <chap> <p type="head"> <s id="s.000282">PROPOSITIO XXIII.<lb/> </s> </p> <subchap1> <p> <s id="s.000283">Duo gravia descendentia super planis diversa ratione<lb/> declinantibus, perveniunt ad idem planum orizon-<lb/>tale ea ratione, ut sit eadem proportio inter diutur-<lb/>nitates, quae inter dicta plana si ab eodem puncto ad<lb/> idem planum orizontale producta sint.<lb/> </s> </p> </subchap1> <p> <s id="s.000284">Datis planis AB, AC declinantibus, ductis ab eodem<lb/> puncto A ad planum orizontale BC.<lb/> </s> </p> <p> <s id="s.000285">Dico quod diuturnitates gravium descendentium per AB, AC<lb/> sint ut AB ad AC.<lb/> </s> </p> <p> <s id="s.000286">Fiat ut AC ad AB ita AB ad AD, ita ut grave perveniat<lb/> in D eodem tempore quo pervenit in B<arrow.to.target n="marg70"/>.<lb/> </s> </p> <p type="margin"> <s id="s.000287"><margin.target id="marg70"/>Per 13.<lb/> hujus.<lb/></s> </p> <p> <s id="s.000288">Quoniam est ut AD ad AC, ita quadratum temporis AD<lb/> ad quadratum temporis AC<arrow.to.target n="marg71"/>, & tempora AD, AB<lb/> sunt aequalia<arrow.to.target n="marg72"/>, & proinde eorum quadrata; ergo ut AD<lb/> ad AC ita quadratum temporis AB, ad quadratum tem-<lb/>poris AC<arrow.to.target n="marg73"/>, sed ut AD ad AC, ita quadratum AB ad qua-<lb/>dratum AC<arrow.to.target n="marg74"/>, ergo ut quadratum temporis AB ad quadra-<lb/>tum temporis AC, ita quadratum AB ad quadratum AC,<lb/> ergo ut tempus AB ad tempus AC, ita AB ad AC<arrow.to.target n="marg75"/>.<lb/> </s> <s id="s.000289">Quod fuit probandum.<lb/> </s> </p> <p type="margin"> <s id="s.000290"><margin.target id="marg71"/>Per Cor.<lb/> 7. hujus.<lb/></s> <s id="s.000291"><margin.target id="marg72"/>Per 17.<lb/> hujus.<lb/></s> <s id="s.000292"><margin.target id="marg73"/>Per 2.<lb/> pronun.<lb/></s> <s id="s.000293"><margin.target id="marg74"/>Per 19.<lb/> sexti.<lb/></s> <s id="s.000294"><margin.target id="marg75"/>Per 22.<lb/> sexti.<lb/></s> </p> </chap> <pb xlink:href="076/01/036.jpg"/> <chap> <p type="head"> <s id="s.000295">PROPOSITIO XXIV<lb/></s> </p> <subchap1> <p> <s id="s.000296">Datis planis, & perpendiculari ad eadem linea orizon-<lb/>tali egressis, quae coeant infra in eodem puncto, gra-<lb/>via super ipsis mota procedunt ea ratione, ut sit ea-<lb/>dem proportion inter diuturnitates, quae inter longi-<lb/>tudines planorum, & dictam perpendicularem.<lb/> </s> </p> </subchap1> <p> <s id="s.000297">Data sit linea orizontalis AB, in qua initium sumant<lb/> plana declinantia AC, DC, nec non perpendicula-<lb/>ris BC coeuntia in puncto C.<lb/> </s> </p> <p> <s id="s.000298">Dico quod diuturnitates gravium super ipsis motorum, sunt<lb/> ut AC, DC, BC.<lb/> </s> </p> <p> <s id="s.000299">Ducatur CE paralella ipsi AB, & a puncto A ducantur<lb/> paralellae ipsis CB, CD, & sint AE, AF.<lb/> </s> </p> <p> <s id="s.000300">Quoniam diuturnitates super planis AF, AC, sunt ut A<lb/>F, AC<arrow.to.target n="marg76"/>, & super planis eisdem, & perpendiculari A<lb/>E, sunt ut AF, seu AC ad AE<arrow.to.target n="marg77"/>, & AE, AF sunt paralellae<lb/> ipsis CD, CB, & eisdem aequales<arrow.to.target n="marg78"/>, sequitur quod etiam<lb/> super AC, DC, BC diuturnitates sunt iuxta propor-<lb/>tiones longitudinum<arrow.to.target n="marg79"/>, Quod probandum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000301"><margin.target id="marg76"/>Per 23.<lb/> hujus.<lb/></s> <s id="s.000302"><margin.target id="marg77"/>Per 15.<lb/> hujus.<lb/></s> <s id="s.000303"><margin.target id="marg78"/>Per 33.<lb/> prim.<lb/></s> <s id="s.000304"><margin.target id="marg79"/>Per 3.<lb/> pron.<lb/></s> </p> </chap> <pb xlink:href="076/01/037.jpg"/> <chap> <p type="head"> <s id="s.000305">PROPOSITIO XXV. </s> </p> <subchap1> <p> <s id="s.000306">In circulo Orthogonaliter erecto, si a summitate ad<lb/> puncta peripheriae ducantur plana, quo tempore gra-<lb/>ve perpendiculariter inde pervenit ad planum ori-<lb/>zontale; si descendat per dicta plana, eodem perve-<lb/>niet respective ad quodlibet dictorum punctorum<lb/> peripheriae.<lb/> </s> </p> </subchap1> <p> <s id="s.000307">Sit circulus cuius centrum B, & diameter AC erectus<lb/> super plano orizontali GC, & in eo ducta sint plana de-<lb/>clinantia a puncto A ad puncta peripheriae DEF, & de-<lb/>scendant gravia super dicta plana, & perpendiculariter.<lb/> </s> </p> <p> <s id="s.000308">Dico quod eodem tempore pervenient ad, D, E, F, C.<lb/> </s> </p> <p> <s id="s.000309">Ducantur DC, EC, FC.<lb/> </s> </p> <p> <s id="s.000310">Quoniam puncta praedicta sunt ea, in quae cadunt perpendi-<lb/>cularia ducta a puncto C in AD, AE, AF<arrow.to.target n="marg80"/>, eo perveniunt<lb/> gravia eodem tempore quo in C<arrow.to.target n="marg81"/>. </s> <s id="s.000311">Quod probandum fuit.<lb/> </s> </p> <p type="margin"> <s id="s.000312"><margin.target id="marg80"/>Per 30.<lb/> Tertii.<lb/></s> <s id="s.000313"><margin.target id="marg81"/>Per 16.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/038.jpg"/> <chap> <p type="head"> <s id="s.000314">PROPOSITIO XXVI.<lb/> </s> </p> <subchap1> <p> <s id="s.000315">Si in circulo erecto, a puncto inferiori ducantur plana<lb/> ad puncta peripheriae, & a dictis punctis descendant<lb/> gravia super dicta plana eodem tempore quo a puncto su-<lb/>premo descendit aliud grave perpendiculariter; perve-<lb/>nient omnia eodem instanti ad dictum punctum inferius.<lb/> </s> </p> </subchap1> <p> <s id="s.000316">Sit circulus cuius diameter ABC erectus super plano<lb/> orizontali, quod tangat in C, & a C ducantur plana C<lb/>D, CE, & a punctis, E, D gravia descendant super dicta<lb/> plana, nec non, & a puncto supremo A perpendiculariter.<lb/> </s> </p> <p> <s id="s.000317">Dico quod eodem tempore perveniunt in C.<lb/> </s> </p> <p> <s id="s.000318">A puncto A ducantur AF, AG paralellae ipsis CE, CD,<lb/> & ducantur AF, FC.<lb/> </s> </p> <p> <s id="s.000319">Quoniam in triangulis AEC, AFC anguli alterni FAC,<lb/> ACE sint aequales,<arrow.to.target n="marg82"/>, & anguli AFC, AEC sunt etiam<lb/> aequales puta recti<arrow.to.target n="marg83"/>, & basis AC communis, Triangula<lb/> sunt aequalia<arrow.to.target n="marg84"/>, & proinde AF est aequalis CE, quod idem<lb/> probabitur de reliquis, ergo cum AF, CE, & reliquae<lb/> sint paralellae, & aequales, gravia per CE, CD perve-<lb/>nient in C eodem tempore, quo digressa ab A perveniunt<lb/> ad puncta FG, sed haec eodem tempore quo perpendicularis<lb/> pervenit in C<arrow.to.target n="marg85"/>, ergo etiam ea quae per CE, CD. </s> <s id="s.000320">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000321"><margin.target id="marg82"/>Per 29.<lb/> primi.<lb/></s> <s id="s.000322"><margin.target id="marg83"/>Per 30.<lb/> Tertii.<lb/></s> <s id="s.000323"><margin.target id="marg84"/>Per 26.<lb/> primi.<lb/></s> <s id="s.000324"><margin.target id="marg85"/>Per 25.<lb/> hujus.<lb/></s> </p> </chap> <pb xlink:href="076/01/039.jpg"/> <chap> <p type="head"> <s id="s.000325">POSTULATUM VII<lb/></s> </p> <p> <s id="s.000326">Ductis planis inclinatis, & linea perpendiculari in-<lb/>ter binas paralellas orizontales, Gravia super illis<lb/> mota ubi perveniunt ad paralellam inferiorem ha-<lb/>bent aequales velocitatis gradus; & proinde si ab inde<lb/> infra sortiantur parem inclinationem, aequeveloci-<lb/>ter moventur.<lb/> </s> </p> <p> <s id="s.000327">Videtur probabile. </s> <s id="s.000328">Primo quia si diuturnitates sunt longitu-<lb/>dinibus proportionales, ut propositione 22. huius probatum<lb/> fuit, credibile est motus in fine esse aequales.<lb/> </s> </p> <p> <s id="s.000329">Secundo. Argumento ducto ab experientia pendulorum, quae<lb/> quantumvis longiora, aut breviora, & proinde circa fi-<lb/>nem magis, aut minus inclinata, pariter ascendunt, si pa-<lb/>riter descendant.<lb/> </s> </p> <p> <s id="s.000330">Tertio. Quia videmus aquam per siphones rectos, sive obli-<lb/>quos, seu inclinatos ductam, pariter ascendere, si pariter<lb/> descendat. </s> <s id="s.000331">Ceterum fateor minorem evidentiam hoc po-<lb/>stulatum caeteris praemissis prae se ferre, quae fuit causa quod<lb/> illud, ut in praefatione, segregaverim, & sequentia, alia<lb/> methodo, tangendo fere tantummodo exposuerim, & a<lb/> pluribus aliis propositionibus, quae hinc deduci facile pos-<lb/>sent, data opera abstinuerim. </s> </p> </chap> <pb xlink:href="076/01/040.jpg"/> <chap> <p type="head"> <s id="s.000332">PROPOSITIO XXVII. PROBL. XIV.<lb/> </s> </p> <subchap1> <p> <s id="s.000333">Dato gravi moto perpendiculariter per spatium datum<lb/> diuturnitate data, quod perficiat motum super plano<lb/> inclinato per spatium itidem datum; perquirere in<lb/> ipso diuturnitatem.<lb/> </s> </p> </subchap1> <p> <s id="s.000334">Moveatur grave A perpendiculariter per spatium AB<lb/> diuturnitate C, & perseveret in motu super spatio B<lb/>D in plano inclinato BD.<lb/> </s> </p> <p> <s id="s.000335">Venanda est diuturnitas eius in ipso BD.<lb/> </s> </p> <p> <s id="s.000336">Producatur DB donec concurrat cum AE orizontaliter du-<lb/>cta ab A in E, & fiat ut AB ad EB, ita diuturnitas C ad<lb/> diuturnitatem G, quae idcirco erit diuturnitas ipsius EB<arrow.to.target n="marg86"/>,<lb/> & sit H quadratum diuturnitatis G, & fiat ut EB ad ED,<lb/> ita quadratum H ad aliud quod sit I a cuius latere K, quod<lb/> est diuturnitas ipsius ED, ablata KL aequali G, erit LM<lb/> reliquum diuturnitas BD quaesita.<lb/> </s> </p> <p type="margin"> <s id="s.000337"><margin.target id="marg86"/>Per 22.<lb/> hujus.<lb/></s> </p> <pb xlink:href="076/01/041.jpg"/> <p> <s id="s.000338">Quoniam notum est triangulum AEB, cum notus sit angu-<lb/>lus AEB aequalis alterno EDF inclinationis notae, & E<lb/>AB rectus ex constructione, & notum latus AB ex hypo-<lb/>tesi, notum erit etiam latus EB, & quia diuturnitas in pla-<lb/>no BD est eadem ac si motus antecedens esset per EB<arrow.to.target n="marg87"/>, EB,<lb/> & ED sunt in duplicata ratione diuturnitatum G, K ex con-<lb/>structione; unde a K deducta KL aequali G ex constructio-<lb/>ne, remanet LM diuturnitas BD. </s> <s id="s.000339">Quod, &c.<lb/> </s> </p> <p type="margin"> <s id="s.000340"><margin.target id="marg87"/>Per 7.<lb/> post.<lb/></s> </p> <p> <s id="s.000341">Inde sequitur quod summa diuturnitatum C, & LM, est diutur-<lb/>nitas totius ABD.<lb/> </s> </p> <p> <s id="s.000342">Eadem operatione pariter reperietur diuturnitas BD si BD<lb/> sit perpendicularis, & AB inclinata.<lb/> </s> </p> <p> <s id="s.000343">Item si ambo sint plana inclinata.<lb/> </s> </p> <p> <s id="s.000344">Ducta AD facile reperietur diuturnitas in ipsa si fiat ut ED<lb/> ad AD, ita K ad aliud per 21. hujus.<lb/> </s> </p> <pb xlink:href="076/01/042.jpg"/> <p> <s id="s.000345">Ducto alio plano puta DN, reperietur eius diu-<lb/>turnitas. </s> </p> <p> <s id="s.000346">Si fiat ut ED ad OD ita diuturnitas ipsius ED puta L ad diu-<lb/>turnitatem OD, quae sit P, deinde ut OD ad ON ita<lb/> quadratum diuturnitatis P ad aliud quadratum, cuius Ra-<lb/>dix erit diuturnitas ipsius DN.<lb/> </s> </p> <p> <s id="s.000347">Ex his patet quod si addantur plura plana eadem ratione re-<lb/>perientur eius diuturnitates.<lb/> </s> </p> <pb xlink:href="076/01/043.jpg"/> <p> <s id="s.000348">Ex his etiam patet quod si in circulo dentur plura, plana verbi<lb/> gratia AB, BC, CD, DE, & data sit diuturnitas super dia-<lb/>metro, dabitur etiam diuturnitas cuiusvis dicto-<lb/>rum AB, BC, CD, DE, & etiam omnium simul.<lb/> </s> </p> <p> <s id="s.000349">Ex his facile etiam cognoscere poteris esse breviorem, diu-<lb/>turnitatem per A, B, C, C, D, E quam per AE.<lb/> </s> </p> <p> <s id="s.000350"> FINIS. <lb/></s> </p> </chap> </body> <back/> </text> </archimedes>