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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Baliani, Giovanni Baptista</author> <title>De Motu Naturali Gravium Solidorum Ioannis Baptistae Baliani</title> <date>1638</date> <place>Genua</place> <translator></translator> <lang>la</lang> <cvs_file>balia_demot_076_la_1638.xml</cvs_file> <cvs_version></cvs_version> <locator>076.xml</locator> </info> <text> <front> <pb xlink:href="076/01/001.jpg"></pb> <section> <p type="head"> <s id="s.000001"> De MOTV<lb></lb> NATVRALI,<lb></lb> GRAVIVM SOLIDORYVM<lb></lb> IOANNIS BAPTISTAE BALIANI<lb></lb> PATRITII GENVENSIS.</s> </p> <p type="head"> <s id="s.000002"> GENVUAE,<lb></lb> EX Typographia Io: Mariae Farroni, Nicolai Peſagnij,<lb></lb> et Petri Franciſci Barberij, ſoc. <lb></lb>MDCXXXVIII.<lb></lb> SVPERIORUM PERMISSV. </s> </p> </section> <pb xlink:href="076/01/002.jpg"></pb> <pb xlink:href="076/01/003.jpg"></pb> <section> <p type="head"> <s id="s.000003">PRAEFATIO</s> </p> <p> <s id="s.000004">Mihi quoque, sicut et caeteris hominibus,<lb></lb> inest sciendi cupiditas, nec grave fuit, usque<lb></lb> à primis annis, et aliorum scripta percur-<lb></lb>rere, et naturales effectus observare, qui<lb></lb> facile mihi persuaserim, ex bisce fontibus,<lb></lb> tum scientiam, tum sapientiam in animum<lb></lb> derivare, si tandem ex effectibus diligentius perspectis, non<lb></lb> modo ad inde consequentes, sed etiam ad causas, usque ad<lb></lb> primam deveniat intellectus. </s> <s id="s.000005">Statui igitur apud me ipsum<lb></lb> non acquiescere soli relationi plurimorum, etiam doctiorum;<lb></lb> potuisse siquidem contingere existimavi, ut aliqua laterent,<lb></lb> etiam in plurimis oculatissimos, vel non plene ab eis explica-<lb></lb>rentur; & ratus sum non inutilem laborem futurum, si ex<lb></lb> accuratiori naturae rerum investigatione, & ex affectionum<lb></lb> inde resultantium deductione, circa quod omnis demonstra-<lb></lb>tiva scientia versatur, aut scitis adderem aliqua, aut doctiori-<lb></lb>bus acuerem desiderium addendi plura: hinc factum est, ut exci-<lb></lb>tata mens ex praecognitis legendo, ad ea, quae se offerebant,<lb></lb> secundum privatas, aut publicas occupationes pervestiganda,<lb></lb> converteretur studiosus. </s> <s id="s.000006">Inter alia dum anno millesimo sex-<lb></lb>centesimo undecimo, per paucos menses, ex patriae legis prae-<lb></lb>scripto, Praefectum Arcis Savonae agerem, ex militaribus<lb></lb> observationibus quae occurrebant, illud maxime depraehendi,<lb></lb> ferreos, & lapideos tormentorum bellicorum globos, & sic<lb></lb> corpora gravia, seu eiusdem, seu diversae speciei, in inaequali<lb></lb> satis Mole, & gravitate, per idem spatium, aequali tempore,<lb></lb> & motu, naturaliter descendere, idque ita uniformiter, ut<lb></lb> repetitis experimentis mihi plane constiterit, duos ex prae-<lb></lb>dictis globis, vel ferreos ambos, vel alterum lapideum al-<lb></lb>terum plumbeum, eodem plane momento temporis dimissos<lb></lb> sibi, per spatium quincaginta pedum, etiam si unus esset<lb></lb> librae unius tantum, alter quincaginta, in indivisibili tem-<lb></lb>poris momento, subjectum solum ferire, ut unus tantum am-<lb></lb>borum ictus sensu perciperetur. </s> <s id="s.000007">Repetebam animo sapien-<lb></lb><pb xlink:href="076/01/004.jpg"></pb>tum esse pronunciatum, gravia moveri naturali motu, se-<lb></lb>cundum gravitatum proportionem; Processi ulterius, & peri-<lb></lb>culum feci, num forte iuxta eorum sententiam contingeret,<lb></lb> si corpora dimissa, ejusdem fere essent molis, sed longe di-<lb></lb>versi ponderis, puta unum plumbeum, cereum alterum; &<lb></lb> expertus sum in cereo aliquam longiorem moram in descen-<lb></lb>su, attamen longe infra proportionem gravitatum, globus<lb></lb> quippe ille cereus, in data distantia quinquaginta pedum de-<lb></lb>scensus, uno circiter pede distabat a solo, quando plumbeus<lb></lb> tangebat subjectum planum, objecto aere intermedio ni fal-<lb></lb>lor, sensibiliter resistente, & impediente motum. </s> <s id="s.000008">Institi<lb></lb> adhuc, & globos in gravitate, & in materia inaequales appendi<lb></lb> funiculis aequalibus, & agitatos animadverti moveri tempore<lb></lb> aequali, & hoc servare adeo fideliter, ut globus plumbeus dua-<lb></lb>rum unciarum, alter librarum duarum, ferreus librarum<lb></lb> 34. </s> <s id="s.000009">& lapideus quadraginta circiter, nec non, & lapis in-<lb></lb>formis, quorum funiculi comprehensis ipsorum semidiame-<lb></lb>tris aequales essent, uno, & eodem temporis spatio moveren-<lb></lb>tur, & vibrationes easdem numero darent hinc inde, sive mo-<lb></lb>tus unius globi fieret per aequale spatium, sive per inaequa-<lb></lb>le, ita ut qui majori impetu jactabatur, & sic majus spatium<lb></lb> percurrebat, illud tanto velocius pertransiret. </s> <s id="s.000010">In quibus<lb></lb> peragendis illud praeter expectationem sese mihi obtulit,<lb></lb> quod quotiescunque globi penderent ex funiculis inaequalibus,<lb></lb> ita inaequali motu ferebantur, ut longitudines funiculorum,<lb></lb> durationibus motuum, in duplicata ratione responderent.<lb></lb> </s> </p> <p> <s id="s.000011">Porro cum ex praemissis satis superque liqueret, in naturali<lb></lb> motu gravium, proportionem gravitatum communiter credi-<lb></lb>tam, non servari; in eam descendi sententiam, ut arbitrater<lb></lb> fortasse, gravitatem se habere ut agens, materiam vero, seu<lb></lb> mavis materiale corpus, ut passum, & proinde gravia mo-<lb></lb>veri juxta proportionem gravitatis ad materiam, & ubi sine<lb></lb> impedimento naturaliter perpendiculari motu ferantur, mo-<lb></lb>veri aequaliter, quia ubi plus est gravitatis, plus pariter sit<lb></lb> materiae, seu materialis quantitatis; si vero accedat aliquid<lb></lb> resistantiae, regulari motum secundum excessum virtutis agen<lb></lb><pb xlink:href="076/01/005.jpg"></pb>tis supra resistentiam passi, seu impedientia motum; qui<lb></lb> excessus momentum noncupabitur, & quod communiter gra-<lb></lb>vitati attributum fuit, momento attribui debere, nimirum<lb></lb> ut sit momentum ad momentum, ut velocitas ad velocita-<lb></lb>tem; Et hinc fieri posse, ut cognoscamus qua mensura, seu<lb></lb> proportione corpora gravia naturali motu ferantur super su-<lb></lb>bjectis planis, si super eis quomodolibet inclinatis, ipsorum<lb></lb> gravium momenta ubique innotescant, quae majora, aut mi-<lb></lb>nora videntur censenda, secundum quod magis, aut minus<lb></lb> super plano quiescunt, & sic secundum majorem, aut mino-<lb></lb>rem inclinationem plani resistentis; quod demum tali pro-<lb></lb>portione facile fieri mihi existimandum videtur, juxta quam<lb></lb> reciproce momentis proportionantur lineae dictorum plano-<lb></lb>rum, si ambae ductae sint ab eodem puncto ad idem planum<lb></lb> orizontale; de quo Simon Stevinus l. p. de Statica prop.<lb></lb> 19. & acutissime Galileus in Mechanica manuscripta, ubi de<lb></lb> Cochlea, & ego æliquali experientia compertum habui. </s> <s id="s.000012">Cae-<lb></lb>terum si per experientiam Scienta hominibus efficitur, prae-<lb></lb>dicta de quibus saepius repetitis actibus expertus fui, ut prin-<lb></lb>cipia scientiae habenda fore censui; in quibus occultae con-<lb></lb>clusiones delitescant, demonstrationibus duntaxat aperiendae.<lb></lb> </s> <s id="s.000013">Rimari caepi; an deprehenderim aliorum erit judicium.<lb></lb> </s> <s id="s.000014">Subjecta paucula, quae presens aliquod otium expedire per-<lb></lb>misit, de motu naturali solidorum gravium, Amice lector<lb></lb> tibi exhibeo, mox de liquidorum, & deinceps alia plura tam<lb></lb> parata daturus, si haec placuerint. </s> <s id="s.000015">Placuit sane mihi, vel<lb></lb> paucula tibi dare, qui te ejus ingenii esse confidam, ut non<lb></lb> verba, sed res, easque non mole, sed pondere censeas, felicior<lb></lb> si de eorum genere existimaveris, quae non mole magna sunt,<lb></lb> quod si talia non fuerint, quo minora minus defatigabunt,<lb></lb> sui exilitate, auctoris partus proprios omnino esse probatura.<lb></lb> </s> <s id="s.000016">Idioma latinum elegi ut communius. </s> <s id="s.000017">Praemisi aliqua na-<lb></lb>turalia principia, sine quibus naturales conclusiones aliunde<lb></lb> duci posse non video. </s> <s id="s.000018">Quae ex praedictis experimentis inno<lb></lb>tuerunt, suppositiones appellare, & a reliquis petitionibus se-<lb></lb>cernere libuit. </s> <s id="s.000019">Petitiones illas, quibus quid fieri petimus, con-<lb></lb><pb xlink:href="076/01/006.jpg"></pb>structioni deservientes, tanquam factu, & cognitu faciles,<lb></lb> & proinde supervacaneas, prudens praetermisi; ratus siqui-<lb></lb>dem nil inde incredulitatis, aut difficultatis derivaturum.<lb></lb> </s> <s id="s.000020">Septimum postulatum ea ratione segregavi, quod illud aliquo<lb></lb> pacto a 22. prop. pendeat, & quod in illo etiamsi veritas<lb></lb> non deficiat, evidentiam tamen ut in caeteris non agnoscens,<lb></lb> certis dubia quo quo pacto permiscere noluerim; ut proinde<lb></lb> plura eorum, quae ex illa deducta sunt, & diversa Methodo<lb></lb> & attingendo potius, quam demonstrando subjunxerim.<lb></lb> </s> <s id="s.000021">Si quae demum minus probata, seu explicata, aut quo<lb></lb> quo pacto imperfecta reperies, velim te tribue-<lb></lb>re cuidam naturali meae propensioni, ad no-<lb></lb>va potius, qualiacumque ea sint, inve-<lb></lb>nienda, quam inventa<lb></lb> perficienda.<lb></lb> </s> <s id="s.000022">Vale. </s> </p> </section> <pb xlink:href="076/01/007.jpg"></pb> </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></lb> sive inaequalia, quae, & longiora, aut brevio-<lb></lb>ra, quatenus fila, e quibus dependent, sunt<lb></lb> aequalia, longiora, aut breviora.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000026">Vibrationes pendulorum sunt eorum motus hinc inde<lb></lb> Vibrationes aequales dicimus, quae fiunt per spatia aequalia, & e contra inaequales.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000027">Vibrationes aeque celeres si fiant per spatia<lb></lb> aequalia tem-<lb></lb>pore aequali.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000028">Vibrationis diuturnitatem dicimus ipsius Durationem,<lb></lb> tempus nimirum, quo ipsa vibratio perficitur.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000029">Vibrationes aequediuturnae, sunt, quae fiunt tempore<lb></lb> aequali, etiamsi per spatia inaequalia, inde diuturnior<lb></lb> est, quae longiori perficitur tempore.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000030">Vibrationes integras dicimus eas, quae se extendunt<lb></lb> per integrum semicirculum, se hinc inde moventes<lb></lb> per circuli quadrantem.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000031">Vibrationis portio est pars arcus, quem ipsa vibratio<lb></lb> disignant.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000032">Vibrationum similes portiones sunt arcus ipsarum in-<lb></lb>tercepti inter binas lineas ductas a centro, a quo<lb></lb> concipiuntur pendula pendere.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000033">Vibrationis portionem priorem decimus eam mini-<lb></lb>mam portionem, a qua integra vibratio initium habet.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000034">Momentum est excessus virtutis moventis supra mo-<lb></lb>tus impedimenta.<lb></lb> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/008.jpg"></pb> <chap> <p type="head"> <s id="s.000035">SUPPOSITIONES<lb></lb></s> </p> <subchap1> <p> <s id="s.000036">PRIMA. Solidorum aequipendulorum cujus-<lb></lb>cumque gravitatis vibrationes aequales sunt aeque-<lb></lb>diuturnae.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000037">2 Equipendulorum eorumdem vibrationes sunt aeque-<lb></lb>diuturnae, etiamsi inaequales.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000038">3 Pendulorum inaequalium longitudines sunt in du-<lb></lb>plicata ratione diuturnitatum vibrationum, seu ut<lb></lb> quadrata vibrationum.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000039">4 Momentum gravis super plano inclinato est ad ip-<lb></lb>sius gravitatem, ut perpendicularis ad inclinatam,<lb></lb> si ab eodem puncto ducta sint ad idem planum<lb></lb> orizontale dicta perpendicularis, & dictum planum<lb></lb> inclinatum, & proinde tali casu proportio gravita<lb></lb>tis ad momentum est reciproca proportioni linea-<lb></lb>rum super quibus grave movetur.<lb></lb> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/009.jpg"></pb> <chap> <p type="head"> <s id="s.000040">PETITIONES, SEU POSTULATA<lb></lb></s> </p> <subchap1> <p> <s id="s.000041">Pr. Pendulorum inaequalium portiones similes vibra-<lb></lb>tionum sunt inter se quoad diuturnitatem, ut vibra-<lb></lb>tiones integrae.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000042">Sint pendula AB, AC; dependentia a puncto A, & ele-<lb></lb>ventur ad libellam orizontis puncti A, in E, D, de-<lb></lb>scribentia arcus BD, CE, integrarum vibrationum, & in<lb></lb> arcubus BD, CE sumantur portiones similes EF, DG, seu<lb></lb> HI, KL ductis EA, FA, seu HA, IA. </s> <s id="s.000043">Peto mihi concedi,<lb></lb> esse pendulorum diuturnitates in arcubus EC, DB, ut in<lb></lb> portionibus EF, DG, nec non HI, KL, & ita deinceps.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000044">2. Ut est momentum ad momentum solidi gravis, ita<lb></lb> velocitas ad velocitatem.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000045">Hujusmodi passio communiter attribui solet gravitati sim-<lb></lb>pliciter, quod eum nimis clare experientiis supra expo-<lb></lb>sitis nullo pacto congruere possit, momentis attribuenda<lb></lb> esse visa est, ut in praefatione explicatum fuit.<lb></lb> </s> </p> </subchap1> <pb xlink:href="076/01/010.jpg"></pb> <subchap1> <p> <s id="s.000046">3. Portiones minimae peripheriae Circuli concipiende<lb></lb> sunt, ac si essent lineae rectae.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000047">Quaecumque arcus portio est circularis, attamen si est<lb></lb> minima portio, tam parum aberrat a linea recta, ut<lb></lb> non modo quo ad sensum, sed quoad quascunque physicas<lb></lb> passiones, perinde esse videatur, ac si esset linea recta, id-<lb></lb>circo ut petitionem admittendam censeo, quemadmodum in-<lb></lb>mechanicis admittitur illa, quod perpendiculares sunt paral-<lb></lb>lelae, etiamsi in centro concurrant universi, quatenus eisdem<lb></lb> sunt passionibus physicis subjectae, ac si vere essent parallelae.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000048">4. Data recta linea, possimus concipere circulum talis<lb></lb> magnitudinis, cujus portio peripheriae aequalis quo<lb></lb> ad sensum datae lineae, concipienda sit, ac si esset<lb></lb> linea recta.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000049">Haec petitio videtur concedenda, quia si concipiamus cir-<lb></lb>culum, ejusque portionem minimam, ut in praece-<lb></lb>denti, si fiat ut hujusmodi portio ad datam lineam, ita<lb></lb> circulus ad alium, portio hujus, datae lineae aequalis erit, &<lb></lb> similis omnino praedicta minimae portioni, & proinde pa-<lb></lb>riter concipienda ut linea recta.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000050">5. Solida perpendicula libero motu aeque velociter<lb></lb> feruntur, & in tali proportione, ac si essent pendula,<lb></lb> & moverentur in priori portione vibrationum.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000051">Quoniam prior portio non differt sensibiliter a recta, ut in<lb></lb> tertia petitione, nec etiamsi sit major ut in quarta, iisdem<lb></lb> physicis passionibus subjicitur, & exinde motibus aequalibus.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000052">6. Solida naturaliter mota super plano inclinato aeque<lb></lb> velociter moventur ac si essent pendula, & moveren-<lb></lb>tur in tali portione vibrationum, quae quoad sensum<lb></lb> <pb xlink:href="076/01/011.jpg"></pb>esset aequalis, & paralella lineae dicti plani super qua<lb></lb> dicta solida moverentur.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000053">Non differt a praecedente, nisi quod in illa motus est per-<lb></lb>pendicularis, in hac inclinatus, in reliquis est par ratio.<lb></lb> </s> </p> </subchap1> </chap> <chap> <p type="head"> <s id="s.000054">PRONUNCIATA<lb></lb></s> </p> <subchap1> <p> <s id="s.000055">P. Quae sunt aequidiuturna tertio, sunt aequidiu-<lb></lb>turna inter se.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000056">2. Quadrata datorum temporum, sunt etiam quadrata<lb></lb> aliorum datis aequalium.<lb></lb> </s> </p> </subchap1> <subchap1> <p> <s id="s.000057">3. Gravia eadem super planis aequalibus & pariter incli-<lb></lb>natis, pariter moventur.<lb></lb> </s> </p> </subchap1> </chap> <pb xlink:href="076/01/012.jpg"></pb> <chap> <p type="head"> <s id="s.000058">PROPOSITIO PRIMA.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000059">Solidi penduli naturaliter moti vibrationes quan-<lb></lb>tumvis semper minores, sunt aequidiuturnae.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000060">Sit solidum A pendulum debite applicatum filo BA, quod<lb></lb> ab altera parte elevatum naturaliter, postea faciat hinc<lb></lb> inde vibrationes semper minores, ita ut prior vibratio sit V. G.<lb></lb> per spatium CD maius, posterior vero per spatium EF minus.<lb></lb> </s> </p> <p> <s id="s.000061">Dico quod dicta vibrationes erunt aequidiuturnae, ita ut vibra-<lb></lb>tio per spatium CD sit eiusdem durationis, ac vibratio per<lb></lb> spatium EF.<lb></lb> </s> </p> <p> <s id="s.000062">Sit aliud solidum G aequipendulum solido A, debite applica-<lb></lb>tum filo HG, quod elevetur ab una parte eodem tempore<lb></lb> minus quam solidum A ita ut sint minores vibrationes soli-<lb></lb>di G, quam, solidi A, ut sit motus penduli G in initio per<lb></lb> spatium IK aequale spatio EF.<lb></lb> </s> </p> <p> <s id="s.000063">Quoniam spatia EF, & IK, sunt aequalia ex suppositione,<lb></lb> sunt etiam vibrationes EF, & IK, aequidiuturnae<arrow.to.target n="marg1"></arrow.to.target>, sed I<lb></lb>K, & CD sunt pariter aequidiuturnae<arrow.to.target n="marg2"></arrow.to.target>, ergo EF, & CD<lb></lb> sunt etiam aequidiuturnae<arrow.to.target n="marg3"></arrow.to.target>. </s> <s id="s.000064">Quod fuit probandum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000065"><margin.target id="marg1"></margin.target>Per pri-<lb></lb>mam sup-<lb></lb>positionem.<lb></lb></s> <s id="s.000066"><margin.target id="marg2"></margin.target>Per secun-<lb></lb>dam sup-<lb></lb>positionem.<lb></lb></s> <s id="s.000067"><margin.target id="marg3"></margin.target>Per pr.<lb></lb> pron.</s> </p> </chap> <pb xlink:href="076/01/013.jpg"></pb> <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></lb>tionum sint in data ratione.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000070">Data sit proportio diuturnitatum vibrationum, quam<lb></lb> volumus esse inter solida A,B; & sit ea, quae est inter<lb></lb> C, & D; quae est continuo eadem,<arrow.to.target n="marg4"></arrow.to.target>a.<lb></lb></s> </p> <p> <s id="s.000071"> Venanda est longitudo filorum, quibus applicata dicta solida<lb></lb> producant vibrationes quaesitas. </s> </p> <p type="margin"> <s id="s.000072"><margin.target id="marg4"></margin.target>Per pr.<lb></lb> hujus.</s> </p> <p> <s id="s.000073">Sint E F numeri mensurantes proportionem, quae est inter C<lb></lb> & D, quorum quadrati numeri G & H, Fila IA, KB<lb></lb> fiant inter se ut G, ad H, & erunt fila quaesita, quibus si<lb></lb> applicentur solida A, B, producentur diuturnitates vibra-<lb></lb>tionum quaesita.<lb></lb> </s> </p> <p> <s id="s.000074">Quoniam ita est IA, ad KB, ut quadratum G numeri me-<lb></lb>tientis C, ad quadratum H numeri metientis D, erunt C, &<lb></lb> D diuturnitates vibrationum pendulorum A, & B<arrow.to.target n="marg5"></arrow.to.target>; &<lb></lb> proinde in ratione data. </s> <s id="s.000075">Quod faciendum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000076"><margin.target id="marg5"></margin.target>Per 3.<lb></lb> suppo.</s> </p> </chap> <pb xlink:href="076/01/014.jpg"></pb> <chap> <p type="head"> <s id="s.000077">PROPOSITIO TERTIA.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000078">Lineae descensus gravium, dum naturali motu perpendicula-<lb></lb>riter feruntur, sunt in duplicata ratione diuturnitatum.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000079">Sint LN, KM linea descensus gravium L, K, & sint P<lb></lb>O ipsorum diuturnitates.<lb></lb> </s> </p> <p> <s id="s.000080">Dico LN, KM esse in duplicata ratione ipsarum P, O.<lb></lb> </s> </p> <p> <s id="s.000081">Sint pendula AH, AI, dependentia a puncto A, & eleven-<lb></lb>tur ad libellam ipsius A usque ad E, B, quae in elevatione<lb></lb> producant arcus HB, IE, & sint talis longitudinis, ut du-<lb></lb>cta ACF, secet arcus BC, & EF, portionis minimae, aequa-<lb></lb>les quo ad sensum lineis LN, KM, & sit S, quadratum<lb></lb> diuturnitatis P, & T quadratum O, & Q, R, diuturni-<lb></lb>tates vibrationum BC, & EF.<lb></lb> </s> </p> <p> <s id="s.000082">Quoniam diuturnitates Q, R sunt aequales diuturnitatibus<lb></lb> P, O<arrow.to.target n="marg6"></arrow.to.target>; S, T, sunt etiam quadrata ipsarum Q, R<arrow.to.target n="marg7"></arrow.to.target>, & quia<lb></lb> vibrationes integrae pendulorum AH, AI sunt ut qua-<lb></lb>dratum T ad quadratum S<arrow.to.target n="marg8"></arrow.to.target>, portiones BC, EF, sunt pa-<lb></lb>riter inter se ut quadratum T ad quadratum S<arrow.to.target n="marg9"></arrow.to.target>, sed<lb></lb> BC, & EF sunt aequales lineis KM, LN<arrow.to.target n="marg10"></arrow.to.target>, ergo etiam K<lb></lb>M, LN sunt ut quadrata S, T<arrow.to.target n="marg11"></arrow.to.target>, & proinde in duplicata<lb></lb> ratione P, O, temporum seu diuturnitatum earumdem.<lb></lb> </s> <s id="s.000083">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000084"><margin.target id="marg6"></margin.target>Per 5.<lb></lb> pet.<lb></lb></s> <s id="s.000085"><margin.target id="marg7"></margin.target>Per 2.<lb></lb> pron.<lb></lb></s> <s id="s.000086"><margin.target id="marg8"></margin.target>Per 3.<lb></lb> supposit.<lb></lb></s> <s id="s.000087"><margin.target id="marg9"></margin.target>Per 5.<lb></lb> petit.<lb></lb></s> <s id="s.000088"><margin.target id="marg10"></margin.target>Per 3.<lb></lb> petit.<lb></lb></s> <s id="s.000089"><margin.target id="marg11"></margin.target>Per 1.<lb></lb> pron.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/015.jpg"></pb> <chap> <p type="head"> <s id="s.000090">PROPOSITIO QUARTA. PROBL. II.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000091">Data diuturnitate gravis descendentis a data altitudine,<lb></lb> constituere altitudinem, a qua idem grave cadat in<lb></lb> data alia diuturnitate.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000092">Sit A diuturnitas gravis B, dum cadit in C, & data sit<lb></lb> diuturnitas quaecumque D.<lb></lb> </s> </p> <p> <s id="s.000093">Constituenda est alia altitudo, a qua grave descendat iuxta<lb></lb> diuturnitatem D.<lb></lb> </s> </p> <p> <s id="s.000094">Fiant E, & F quadrata temporum A, D, & ut F ad E, fiat<lb></lb> altitudo GH, ad altitudinem datam BC; Dico GH esse al-<lb></lb>titudinem quaesitam.<lb></lb> </s> </p> <p> <s id="s.000095">Quoniam BC, & GH sunt in duplicata ratione datarum diu-<lb></lb>turnitatum A, D, per constructionem; per ipsas gravia B,<lb></lb> & G cadent in diuturnitatibus A, & D datis<arrow.to.target n="marg12"></arrow.to.target>, unde re-<lb></lb>perta est altitudo GH quaesita. </s> <s id="s.000096">Quod fuit faciendum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000097"><margin.target id="marg12"></margin.target>Per 3.<lb></lb> hujus.</s> </p> </chap> <pb xlink:href="076/01/016.jpg"></pb> <chap> <p type="head"> <s id="s.000098">PROPOSITIO V. PROBL. III.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000099">Data altitudine, a qua descendat grave in nota diutur-<lb></lb>nitate; perquirere quanta sit diuturnitas, qua descen-<lb></lb>dat ab alia altitudine data.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000100">Sit A altitudo per quam descendat grave diuturnitate B<lb></lb> nota, & data sit alia altitudo C.<lb></lb> </s> </p> <p> <s id="s.000101">Oportet reperire quanta sit diuturnitas, qua idem grave de-<lb></lb>scendat per C.<lb></lb> </s> </p> <p> <s id="s.000102">Fiat D quadratum diuturnitatis B, & fiat ut A ad C, ita<lb></lb> quadratum D ad quadratum E, cuius radix F est diutur-<lb></lb>nitas quaesita. </s> </p> <p> <s id="s.000103">Quoniam A, & C sunt in duplicata ratione diuturnitatum<lb></lb> B, & F per constructionem, per ipsas gravia descendent<lb></lb> in diuturnitatibus B, F,<arrow.to.target n="marg13"></arrow.to.target> unde F est diuturnitas ipsius C<lb></lb> quaesita. </s> <s id="s.000104">Quod faciendum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000105"><margin.target id="marg13"></margin.target>Per 3.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/017.jpg"></pb> <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></lb> ratione, ut temporibus aequalibus descendant per spa-<lb></lb>tia semper maiora, iuxta proportionem quam ha-<lb></lb>bent impares numeri ab unitate inter se.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000108">Sit grave A quod descendat per lineam ABC, & tempus<lb></lb> quo descendit ab A in B sit aequale tempori, quo de-<lb></lb>scendit a B in C, & a C in D.<lb></lb> </s> </p> <p> <s id="s.000109">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &<lb></lb> sic deinceps.<lb></lb> </s> </p> <p> <s id="s.000110">Sit G numerus mensurans tempus, quo A descendit in B, &<lb></lb> H, quo descendit a B in C, & I, quo descendit a C in D,<lb></lb> quae tempora sunt ex suppositione aequalia, & sit K qua-<lb></lb>dratum ipsius G, & L quadratum GH, & M quadratum<lb></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"></arrow.to.target>, quae<lb></lb> quadrata sunt ut 1, 4, 9, sunt itidem AB, AC, AD, ut<lb></lb> 1. 4. 9. & dividendo AB, BC, CD, ut 1. 3. 5. & sic dein-<lb></lb>ceps. </s> <s id="s.000112">Quod probandum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000113"><margin.target id="marg14"></margin.target>Per 3.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/018.jpg"></pb> <chap> <p type="head"> <s id="s.000114">PROPOSITIO VII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000115">Lineae descensus gravium super plano inclinato mo-<lb></lb>torum, sunt in duplicata ratione diuturnitatum.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000116">Sint AB, CD plana pariter inclinata, super quibus mo-<lb></lb>veantur gravia A, C, & sint EF ipsorum diuturnitates.<lb></lb> </s> </p> <p> <s id="s.000117">Dico AB, CD, esse in duplicata ratione ipsarum E, F.<lb></lb> </s> </p> <p> <s id="s.000118">Secetur AB bifariam in G, & erecta GH, perpendiculari<lb></lb> longissima, fiant pendula HI, HK, quae sint inter se ut A<lb></lb>B, CD, & eleventur in L, M, describentia arcus LI, KM,<lb></lb> secantes GH in N, O, & ab N hinc inde secentur arcus N<lb></lb>P, NQ aequales quo ad sensum rectis GA, GB, & ductis P<lb></lb>H, QH, secetur pariter arcus LI, in R, S, & intelligan-<lb></lb>tur arcus PQ, RS, tam parvae curvitatis, ob maximam<lb></lb> longitudinem pendulorum HI, HK, ut pro rectis habean-<lb></lb>tur, puta portionis minimae, & proinde aequales rectis A<lb></lb>B, CD: sit Z quadratum diuturnitatis E, & V, diuturnitatis<lb></lb> F, & sint XY diuturnitates vibrationum PQ, RS.<lb></lb> </s> </p> <pb xlink:href="076/01/019.jpg"></pb> <p> <s id="s.000119">Quoniam diuturnitates X, Y, sunt aequales diuturnitatibus<lb></lb> E, F,<arrow.to.target n="marg15"></arrow.to.target> sunt etiam Z, V, quadrata ipsarum X, Y<arrow.to.target n="marg16"></arrow.to.target>; & quia<lb></lb> vibrationes integrae pendulorum HI, HK sunt inter se, ut<lb></lb> quadratum V, ad quadratum Z<arrow.to.target n="marg17"></arrow.to.target>, portiones RS, PQ erunt<lb></lb> etiam inter se ut quadratum V ad quadratum Z<arrow.to.target n="marg18"></arrow.to.target>; sed R<lb></lb>S, PQ aequantur rectis CD, AB,<arrow.to.target n="marg19"></arrow.to.target>, ergo, & CD, AB<lb></lb> sunt ut quadratum V, ad quadratum Z<arrow.to.target n="marg20"></arrow.to.target>, & proinde, in<lb></lb> duplicata ratione ipsarum EF. </s> <s id="s.000120">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000121"><margin.target id="marg15"></margin.target>Per 6.<lb></lb> petit.<lb></lb></s> <s id="s.000122"><margin.target id="marg16"></margin.target>Per 2.<lb></lb> pron.<lb></lb></s> <s id="s.000123"><margin.target id="marg17"></margin.target>Per 3.<lb></lb> hujus.<lb></lb></s> <s id="s.000124"><margin.target id="marg18"></margin.target>Per pr.<lb></lb> pet.<lb></lb></s> <s id="s.000125"><margin.target id="marg19"></margin.target>Per 3.<lb></lb> petit.<lb></lb></s> <s id="s.000126"><margin.target id="marg20"></margin.target>Per 2.<lb></lb> pron.<lb></lb></s> </p> <p> <s id="s.000127">Corolarium.<lb></lb> </s> </p> <p> <s id="s.000128">Hinc patet esse longitudines planorum per quae gravia fe-<lb></lb>runtur ut quadrata temporum, & tempora ut radices<lb></lb> longitudinum planorum.<lb></lb> </s> </p> </chap> <pb xlink:href="076/01/020.jpg"></pb> <chap> <p type="head"> <s id="s.000129">PROPOSITIO VIII. PROB. IV.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000130">Dato plano inclinato, super quo per spatium datum<lb></lb> grave moveatur in nota diuturnitate, determinare in<lb></lb> eodem plano spatium per quod dictum grave mo-<lb></lb>veatur in quavis alia diuturnitate data.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000131">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb></lb>no inclinato BC, & data diuturnitas D.<lb></lb> </s> </p> <p> <s id="s.000132">Praescribendum est aliud spatium in eodem plano BC, per<lb></lb> quod idem grave pertranseat in diuturnitate D.<lb></lb> </s> </p> <p> <s id="s.000133">Fiant E, F quadrata temporum A, D, & ut F ad E fiat BG ad<lb></lb> BC, Dico BG esse spatium quaesitum.<lb></lb> </s> </p> <p> <s id="s.000134">Quoniam BC, & BG sunt in duplicata ratione datorum<lb></lb> temporum A, D per constructionem, per ipsa cadet grave<lb></lb> B diuturnitatibus A, D datis<arrow.to.target n="marg21"></arrow.to.target>, ergo reperta est BG quae-<lb></lb>sita. </s> <s id="s.000135">Quod faciendum erat.<lb></lb> </s> </p> <p type="margin"> <s id="s.000136"><margin.target id="marg21"></margin.target>Per 6.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/021.jpg"></pb> <chap> <p type="head"> <s id="s.000137">PROPOSITIO IX. PROB. V.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000138">Dato plano inclinato, super quo per spatium datum gra-<lb></lb>ve moveatur nota diuturnitate; & dato alio spatio<lb></lb> quocumque; reperire diuturnitatem, qua grave per<lb></lb> ipsum descendat.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000139">Sit A diuturnitas gravis B, dum descendit in C super pla-<lb></lb>no inclinato BC, & dato alio spatio BG.<lb></lb> </s> </p> <p> <s id="s.000140">Querendum quanta sit diuturnitas gravis in BG.<lb></lb> </s> </p> <p> <s id="s.000141">Fiat E quadratum diuturnitatis A, & ut BC ad BG fiat ut<lb></lb> quadratum E ad quadratum F, cuius radix D erit diutur-<lb></lb>nitas ipsius BG quaesita.<lb></lb> </s> </p> <p> <s id="s.000142">Quoniam BC, & BG sunt in duplicata ratione diuturnita-<lb></lb>tum A, D per constructionem; per ipsa cadunt gravia diu-<lb></lb>turnitatibus A, D<arrow.to.target n="marg22"></arrow.to.target>, unde D est diuturnitas per spatium<lb></lb> BG quaesita. </s> <s id="s.000143">Quod faciendum erat.<lb></lb> </s> </p> <p type="margin"> <s id="s.000144"><margin.target id="marg22"></margin.target>Per 7.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/022.jpg"></pb> <chap> <p type="head"> <s id="s.000145">PROPOSITIO X.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000146">Gravia descendunt super planis inclinatis per spatia<lb></lb> semper maiora, iuxta rationem, quam habent im-<lb></lb>pares numeri successive inter se.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000147">Sit grave A, quod descendat super plano ABC inclinato,<lb></lb> & tempus quo descendit ab A in B sit aequale tempo-<lb></lb>ri, quo descendit a B in C, & a C in D.<lb></lb> </s> </p> <p> <s id="s.000148">Dico quod lineae AB, BC, CD sunt inter se ut 1. 3. 5. &. sic<lb></lb> deinceps. </s> </p> <p> <s id="s.000149">Sit E numerus mensurans tempus, quo A descendit in B, & F<lb></lb> quo descendit a B in C, & G quo descendit a C in D, quae<lb></lb> tempora sunt ex suppositione aequalia, & sit H quadratum<lb></lb> ipsius E, & I quadratum EF, & K quadratum totius EFG.<lb></lb> </s> </p> <p> <s id="s.000150">Quoniam quadrata HIK sunt ut AB, AC, AD<arrow.to.target n="marg23"></arrow.to.target>, quae<lb></lb> quadrata sunt ut 1. 4. 9. sunt pariter AB, AC, AD, ut<lb></lb> 1. 4. 9. & dividendo AB, BC, CD, sunt ut 1. 3. 5. & sic<lb></lb> deinceps. </s> <s id="s.000151">Quod probandum erat.<lb></lb> </s> </p> <p type="margin"> <s id="s.000152"><margin.target id="marg23"></margin.target>Per 7.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/023.jpg"></pb> <chap> <p type="head"> <s id="s.000153">PROPOSITIO XI.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000154">Si Duo gravia descendant alterum super linea perpen-<lb></lb>diculari, alterum vero super inclinata; proportio ve-<lb></lb>locitatum est reciproca proportioni linearum.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000155">Sit ABC planum normaliter erectum super lineam ori-<lb></lb>zontalem BC, cuius latus AB sit perpendiculare, &<lb></lb> AC, inclinatum.<lb></lb> </s> </p> <p> <s id="s.000156">Dico quod proportio velocitatum solidorum gravium moto-<lb></lb>rum secundum lineam AB perpendicularem, & AC in-<lb></lb>clinatum, est ut proportio longitudinis inclinatae AC ad<lb></lb> longitudinem perpendicularis AB; videlicet ita est longi-<lb></lb>tudo AB ad longitudinem AC, ut velocitas super AC ad<lb></lb> velocitatem in AB.<lb></lb> </s> </p> <p> <s id="s.000157">Quoniam est ut AC ad AB, ita momentum in AB, ad mo-<lb></lb>mentum in AC<arrow.to.target n="marg24"></arrow.to.target>; & ut momentum in AB ad momentum<lb></lb> in AC, ita velocitas in AB ad velocitatem in AC<arrow.to.target n="marg25"></arrow.to.target>; er-<lb></lb>go est etiam ut AC ad AB, ita velocitas in AB ad velo-<lb></lb>citatem in AC. </s> <s id="s.000158">Quod fuit probandum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000159"><margin.target id="marg24"></margin.target>Per 4.<lb></lb> supp.<lb></lb></s> <s id="s.000160"><margin.target id="marg25"></margin.target>Per 2.<lb></lb> pet.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/024.jpg"></pb> <chap> <p type="head"> <s id="s.000161">PROPOSITIO XII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000162">Gravia descendunt super plana diverse inclinata tali<lb></lb> proportione, ut si velocitas ad velocitatem recipro-<lb></lb>ca longitudinibus planorum ductorum ab eodem<lb></lb> puncto, ad idem planum orizontale.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000163">Sint F, D plana inclinata ducta ad idem planum orizon-<lb></lb>tale.<lb></lb> </s> </p> <p> <s id="s.000164">Dico esse ut planum D ad planum F, ita velocitatem gravis<lb></lb> ducti super F, ad velocitatem eiusdem ducti super D.<lb></lb> </s> </p> <p> <s id="s.000165">Ducatur perpendicularis E, & sint B, A, C velocitates gra-<lb></lb>vium latorum super perpendiculari, & super planis F, D.<lb></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></lb> E<arrow.to.target n="marg26"></arrow.to.target>, erit A ad C ut D ad F<arrow.to.target n="marg27"></arrow.to.target>, scilicet velocitas gravis su-<lb></lb>per F ad velocitatem gravis super D, ut longitudo pla-<lb></lb>ni D ad longitudinem plani F. </s> <s id="s.000167">Quod fuit probandum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000168"><margin.target id="marg26"></margin.target>Per 11.<lb></lb> hujus.<lb></lb></s> <s id="s.000169"><margin.target id="marg27"></margin.target>Per 23.<lb></lb> Quinti.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/025.jpg"></pb> <chap> <p type="head"> <s id="s.000170">PROPOSITIO XIII. PROBL. VI.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000171">Reperire inclinationem plani, super quo grave movea-<lb></lb>tur tali velocitate quae cum alia super diversa incli-<lb></lb>natione sit in ratione data.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000172">Moveatur grave A super recta AB, seu perpendicula-<lb></lb>ri, seu inclinata, & data sit proportio C ad D.<lb></lb> </s> </p> <p> <s id="s.000173">Oportet reperire aliud planum inclinatum, ita ut velocitas<lb></lb> gravis moti super AB ad velocitatem alterius moti<lb></lb> super illo reperiendo, sit ut D ad C.<lb></lb> </s> </p> <p> <s id="s.000174">Producatur BA; & fiat ut C ad D ita BA, ad AE; &<lb></lb> centro A, intervallo AE describatur circulus, secans BF<lb></lb> in F; ni secet, problema insolubile est; si secat, ducatur<lb></lb> AF, quam dico esse planum quaesitum.<lb></lb> </s> </p> <p> <s id="s.000175">Quoniam ut C ad D, ita AB ad AE, seu AF per constructio-<lb></lb>nem, erit C velocitas super AF, & D super AB<arrow.to.target n="marg28"></arrow.to.target>, unde velo-<lb></lb>citates super ipsis sunt in ratione data. </s> <s id="s.000176">Quod faciendum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000177"><margin.target id="marg28"></margin.target>Per 12.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/026.jpg"></pb> <chap> <p type="head"> <s id="s.000178">PROPOSITIO XIV. PROBL. VII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000179">Data linea perpendiculari, per quam grave descendat,<lb></lb> cui annectatur linea, seu planum declinans; in decli-<lb></lb>nante reperire punctum, quo grave perveniat eo<lb></lb> tempore, quo pertransiverit perpendicularem.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000180">Sit triangulum ABC orthogonaliter erectum super pla<lb></lb>no orizontali BC, cuius latus AB intelligatur sit linea perpendicu-<lb></lb>laris, per quam grave descendat, & latus AC sit planum<lb></lb> inclinatum.<lb></lb> </s> </p> <p> <s id="s.000181">Oportet in plano AC reperire punctum quo grave perveniat<lb></lb> eodem tempore, quo in B.<lb></lb> </s> </p> <p> <s id="s.000182">Fiat ut AC ad AB, ita AB ad tertiam AD, & D erit pun-<lb></lb>ctum quaesitum.<lb></lb> </s> </p> <p> <s id="s.000183">Quoniam ut AC ad AD, ita quadratum AC ad quadra-<lb></lb>tum AB<arrow.to.target n="marg29"></arrow.to.target>, & ut AC ad AD, ita quadratum temporis A<lb></lb>C ad quadratum temporis AD<arrow.to.target n="marg30"></arrow.to.target>, ergo ut quadratum A<lb></lb>C ad quadratum AB, ita quadratum temporis AC ad qua-<lb></lb>dratum temporis AD<arrow.to.target n="marg31"></arrow.to.target>, ergo ut AC ad AB, ita tempus<lb></lb> AC ad tempus AD<arrow.to.target n="marg32"></arrow.to.target>, sed ut AC ad AB, ita tempus AC ad<lb></lb> tempus AB<arrow.to.target n="marg33"></arrow.to.target>, ergo tempus AB est aequale tempori AD.<lb></lb> </s> <s id="s.000184">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000185"><margin.target id="marg29"></margin.target>Per 19.<lb></lb> Sexti.<lb></lb></s> <s id="s.000186"><margin.target id="marg30"></margin.target>Per cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000187"><margin.target id="marg31"></margin.target>Per 11.<lb></lb> Quinti.<lb></lb></s> <s id="s.000188"><margin.target id="marg32"></margin.target>Per 22.<lb></lb> Sexti.<lb></lb></s> <s id="s.000189"><margin.target id="marg33"></margin.target>Per 11.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/027.jpg"></pb> <chap> <p type="head"> <s id="s.000190">PROPOSITIO XV.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000191">Linea connectens puncta, ad quae duo gravia ab eo-<lb></lb>dem puncto digressa, quorum alterum perpenden-<lb></lb>ter, alterum super plano declinante descendat, simul<lb></lb> perveniunt, est perpendicularis dicto plano declinanti.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000192">Descendant simul duo gravia a puncto A primum per-<lb></lb>pendiculariter in B, secundum super plano inclinato<lb></lb> AC, tali lege, ut simul perveniant ad puncta BD,<lb></lb> & ducta sit linea BD.<lb></lb> </s> </p> <p> <s id="s.000193">Dico quod dicta linea BD est perpendicularis ad AD.<lb></lb> </s> </p> <p> <s id="s.000194">Fiat AF aequalis datae AB, & AE aequalis AD, & duca-<lb></lb>tur EF.<lb></lb> </s> </p> <p> <s id="s.000195">Quoniam ut AD ad AB, ita AB ad AC<arrow.to.target n="marg34"></arrow.to.target>, & AD,<lb></lb> AE, item AB, AF sunt aequales per constructionem, se-<lb></lb>quitur quod AE ad AF est ut AB ad AC, ergo EF, BC<lb></lb> sunt parallelae<arrow.to.target n="marg35"></arrow.to.target>, unde triangulum AEF, & proin-<lb></lb>de ABD est simile triangulo ABC<arrow.to.target n="marg36"></arrow.to.target>, unde anguli AB<lb></lb>C, ADB simul recti, & BD perpendicularis ad AD.<lb></lb> </s> <s id="s.000196">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000197"><margin.target id="marg34"></margin.target>Per 13.<lb></lb> hujus.<lb></lb></s> <s id="s.000198"><margin.target id="marg35"></margin.target>Per 2.<lb></lb> Sexti.<lb></lb></s> <s id="s.000199"><margin.target id="marg36"></margin.target>Per 4.<lb></lb> Sexti.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/028.jpg"></pb> <chap> <p type="head"> <s id="s.000200">PROPOSITIO XVI. PROBL. VIII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000201">Data linea perpendiculari, & plano declinante; reperire<lb></lb> in perpendiculari producta punctum, quo perveniat<lb></lb> grave eo tempore, quo pertransit planum inclinatum.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000202">Data sit perpendicularis AB, cui connexum planum<lb></lb> inclinatum AD.<lb></lb> </s> </p> <p> <s id="s.000203">Oportet in AB producta reperire punctum, quo perveniat<lb></lb> grave eo tempore, quo pervenit in puncto D.<lb></lb> </s> </p> <p> <s id="s.000204">In puncto D perpendicularis erigatur ad AD, & protraha-<lb></lb>tur usquequo coeat cum AB producta in E, & E est pu-<lb></lb>nctum quaesitum. </s> </p> <p> <s id="s.000205">Quoniam triangula & ADE, AEC sint aequiangula,<lb></lb> cum anguli ADE, AEC sint aequales, nempe recti, &<lb></lb> BAD communis<arrow.to.target n="marg37"></arrow.to.target>, sunt etiam similia<arrow.to.target n="marg38"></arrow.to.target>, ergo ut AC<lb></lb> ad AE, ita AE ad AD<arrow.to.target n="marg39"></arrow.to.target>, sed ut AC ad AD, ita qua-<lb></lb>dratum AC ad quadratum AE<arrow.to.target n="marg40"></arrow.to.target>, & ut AC ad AD,<lb></lb> ita quadratum temporis AC ad quadratum temporis A<lb></lb>D<arrow.to.target n="marg41"></arrow.to.target>, ergo ut quadratum AC ad quadratum AE ita qua-<lb></lb>dratum temporis AC ad quadratum temporis AD<arrow.to.target n="marg42"></arrow.to.target>, er-<lb></lb>go ut AC ad AE, ita tempus AC ad tempus AD<arrow.to.target n="marg43"></arrow.to.target>, sed<lb></lb> ut AC ad AE, ita tempus AC ad tempus AE<arrow.to.target n="marg44"></arrow.to.target>, ergo<lb></lb> tempora AE, & AD sunt aequalia. </s> <s id="s.000206">Quod &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000207"><margin.target id="marg37"></margin.target>Per 32.<lb></lb> prim.<lb></lb></s> <s id="s.000208"><margin.target id="marg38"></margin.target>Per 4.<lb></lb> sexti.<lb></lb></s> <s id="s.000209"><margin.target id="marg39"></margin.target>Per 4.<lb></lb> sexti.<lb></lb></s> <s id="s.000210"><margin.target id="marg40"></margin.target>Per 19.<lb></lb> Sexti.<lb></lb></s> <s id="s.000211"><margin.target id="marg41"></margin.target>Per Cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000212"><margin.target id="marg42"></margin.target>Per 11.<lb></lb> Quinti.<lb></lb></s> <s id="s.000213"><margin.target id="marg43"></margin.target>Per 22.<lb></lb> sexti.<lb></lb></s> <s id="s.000214"><margin.target id="marg44"></margin.target>Per 11.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/029.jpg"></pb> <chap> <p type="head"> <s id="s.000215">PROPOSITIO XVII. PROBL. IX.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000216">Dato plano declinante, super quo grave descendat, &<lb></lb> dato alio plano minus declinante, in hoc reperire<lb></lb> punctum, quo perveniat mobile eo tempore, quo<lb></lb> pertransit dictum planum magis declinans.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000217">Sint plana AB, AC quorum AC minus inclinatum.<lb></lb> </s> </p> <p> <s id="s.000218">Oportet in AC reperire punctum, quo grave perveniat,<lb></lb> quando pervenit in B.<lb></lb> </s> </p> <p> <s id="s.000219">Fiat ut AC ad AB ita AB ad AD, & dico D esse punctum<lb></lb> quaesitum. </s> </p> <p> <s id="s.000220">Quoniam ut AC ad AD ita est quadratum AC ad quadra-<lb></lb>tum AB<arrow.to.target n="marg45"></arrow.to.target>, & ut AC ad AD ita quadratum temporis<lb></lb> AC ad quadratum temporis AD<arrow.to.target n="marg46"></arrow.to.target>, ergo ut quadratum A<lb></lb>C ad quadratum AB, ita quadratum temporis AC ad<lb></lb> quadratum temporis AD<arrow.to.target n="marg47"></arrow.to.target>, Unde AC ad AB ut tempus<lb></lb> AC ad tempus AD<arrow.to.target n="marg48"></arrow.to.target>, sed ut AC ad AB, ita tempus A<lb></lb>C ad tempus AB<arrow.to.target n="marg49"></arrow.to.target>, ergo tempora AB, AD, sunt aequa-<lb></lb>lia. </s> <s id="s.000221">Quod, &c. </s> </p> <p type="margin"> <s id="s.000222"><margin.target id="marg45"></margin.target>Per 19.<lb></lb> sexti.<lb></lb></s> <s id="s.000223"><margin.target id="marg46"></margin.target>Per cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000224"><margin.target id="marg47"></margin.target>Per 11.<lb></lb> Quinti.<lb></lb></s> <s id="s.000225"><margin.target id="marg48"></margin.target>Per 22.<lb></lb> sexti.<lb></lb></s> <s id="s.000226"><margin.target id="marg49"></margin.target>Per 11.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/030.jpg"></pb> <chap> <p type="head"> <s id="s.000227">PROPOSITIO XVIII. PROBL. X.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000228">Datis planis declinantibus ortis ab eodem puncto, re-<lb></lb>perire in magis declinante punctum quo grave per-<lb></lb>veniat eo tempore, quo pertransit planum minus<lb></lb> declinans. </s> </p> </subchap1> <p> <s id="s.000229">Datum sit planum minus declinans AC, & magis A<lb></lb>D, terminantia super plano orizontali BD.<lb></lb> </s> </p> <p> <s id="s.000230">Oportet in AD producta reperire punctum, quo perveniat<lb></lb> grave eo tempore, quo pertransivit planum minus decli-<lb></lb>nans AC.<lb></lb> </s> </p> <p> <s id="s.000231">Fiat ut AD ad AC ita AC ad dictam AD productam in<lb></lb> E, quod est punctum quaesitum.<lb></lb> </s> </p> <p> <s id="s.000232">Quoniam ut AE ad AD ita est quadratum AC ad quadra-<lb></lb>tum AD<arrow.to.target n="marg50"></arrow.to.target>, sed AE ad AD est ut quadratum temporis<lb></lb> AE, ad quadratum temporis AD<arrow.to.target n="marg51"></arrow.to.target>, ergo ut quadratum<lb></lb> AC ad quadratum AD, ita quadratum temporis AE ad<lb></lb> quadratum temporis AD<arrow.to.target n="marg52"></arrow.to.target>, unde AC ad AD ut tempus<lb></lb> AE ad tempus AD<arrow.to.target n="marg53"></arrow.to.target>, sed AC ad AD est ut tempus AC<lb></lb> ad tempus AD<arrow.to.target n="marg54"></arrow.to.target>, ergo tempora AE, AC sunt aequalia.<lb></lb> </s> <s id="s.000233">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000234"><margin.target id="marg50"></margin.target>Per 19.<lb></lb> sexti.<lb></lb></s> <s id="s.000235"><margin.target id="marg51"></margin.target>Per cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000236"><margin.target id="marg52"></margin.target>Per 11.<lb></lb> Quinti.<lb></lb></s> <s id="s.000237"><margin.target id="marg53"></margin.target>Per 22.<lb></lb> sexti.<lb></lb></s> <s id="s.000238"><margin.target id="marg54"></margin.target>Per 11.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/031.jpg"></pb> <chap> <p type="head"> <s id="s.000239">PROPOSITIO XIX. PROBL. XI.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000240">Dato motus naturali gravis quomodocumque ad pun-<lb></lb>ctum datum, reperire seu in perpendiculari, seu in<lb></lb> plano quomodolibet inclinato punctum, a quo di-<lb></lb>gressum, perveniat ad idem punctum quo prius, tem-<lb></lb>pore aequali. </s> </p> </subchap1> <p> <s id="s.000241">Sit AB linea quomodocumque aut perpendicularis, seu<lb></lb> planum inclinatum; super qua grave descendat in B, &<lb></lb> data sit quaecumque linea BC, aut perpendicularis, aut<lb></lb> quomodolibet inclinata, quae cum AB, coeat in B.<lb></lb> </s> </p> <p> <s id="s.000242">Oportet in BC reperire punctum, a quo grave digressum per-<lb></lb>veniat in B tempore quo pervenit ab A in idem B.<lb></lb> </s> </p> <p> <s id="s.000243">Ducatur AC orizontalis, & fiat BD tertia proportiona-<lb></lb>lis ad CB AB<arrow.to.target n="marg55"></arrow.to.target>, & D est punctum quaesitum. </s> <s id="s.000244">Quod ut probetur.<lb></lb> </s> </p> <p type="margin"> <s id="s.000245"><margin.target id="marg55"></margin.target>Per 11.<lb></lb> Sexti.<lb></lb></s> </p> <p> <s id="s.000246">Fiat iterum rectae AC paralella, & aequalis BE, & ducta<lb></lb> EA, secetur recta BF parallela ipsi AD.<lb></lb> </s> </p> <p> <s id="s.000247">Quoniam AF, BD sunt pariter inclinatae, & aequales<arrow.to.target n="marg56"></arrow.to.target>, gra-<lb></lb>via per ipsas aequali tempore moventur<arrow.to.target n="marg57"></arrow.to.target>, ergo aequali tem-<lb></lb>pore ut per AB<arrow.to.target n="marg58"></arrow.to.target>, quod, &c. </s> </p> <p type="margin"> <s id="s.000248"><margin.target id="marg56"></margin.target>Per 33.<lb></lb> Primi.<lb></lb></s> <s id="s.000249"><margin.target id="marg57"></margin.target>Per 3.<lb></lb> pronun.<lb></lb></s> <s id="s.000250"><margin.target id="marg58"></margin.target>Per pr.<lb></lb> pron.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/032.jpg"></pb> <chap> <p type="head"> <s id="s.000251">PROPOSITIO XX. PROBL. XII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000252">Datis duobus planis diverse inclinatis longitudinis no-<lb></lb>tae; & nota diuturnitate gravis moti super uno, re-<lb></lb>perire diuturnitatem si moveatur super alio.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000253">Sint plana AB, CD inclinata, & sit data diuturnitas E<lb></lb> plani AB.<lb></lb> </s> </p> <p> <s id="s.000254">Oportet reperire diuturnitatem plani CD.<lb></lb> </s> </p> <p> <s id="s.000255">Fiat AF, paralella, & aequalis datae CD, in qua reperiatur<lb></lb> punctum G quo perveniat grave, tempore quo in B<arrow.to.target n="marg59"></arrow.to.target>, unde<lb></lb> E est etiam diuturnitas spatii AG, quo dato, & spatio<lb></lb> AF perquiratur eias diuturnitas, quae sit H<arrow.to.target n="marg60"></arrow.to.target>, & dico<lb></lb> H esse diuturnitatem quae grave descendit in CD.<lb></lb> </s> </p> <p type="margin"> <s id="s.000256"><margin.target id="marg59"></margin.target>Per 17.<lb></lb> hujus.<lb></lb></s> <s id="s.000257"><margin.target id="marg60"></margin.target>Per 9.<lb></lb> hujus.<lb></lb></s> </p> <p> <s id="s.000258">Quoniam E, H sunt diuturnitates gravium descendentium<lb></lb> in AG, seu AB, & AF, per constructionem, & AF<lb></lb> est aequalis, & paralella datae CD per constructionem,<lb></lb> sunt etiam E, H diuturnitates ipsarum AB, & CD<arrow.to.target n="marg61"></arrow.to.target>, unde<lb></lb> reperta est diuturnitas ipsius CD. </s> <s id="s.000259">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000260"><margin.target id="marg61"></margin.target>Per 3.<lb></lb> pron.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/033.jpg"></pb> <chap> <p type="head"> <s id="s.000261">PROPOSITIO XXI. PROBL. XIII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000262">Datis duabus diuturnitatibus, quarum prior sit gravis<lb></lb> moti super plano dato longitudinis notae, & dato<lb></lb> alio plano diversimode declinante; reperiendum est<lb></lb> in eo punctum, quo grave perveniat in secunda<lb></lb> diuturnitate data.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000263">Dato plano declinante AB, super quo grave A movea<lb></lb>tur diuturnitate C, & dato alio plano D declinationis<lb></lb> quae sit dissimilis declinationi datae AB; data itidem diu-<lb></lb>turnitate E.<lb></lb> </s> </p> <p> <s id="s.000264">Oportet reperire in D punctum quo grave perveniat in<lb></lb> diuturnitate E.<lb></lb> </s> </p> <p> <s id="s.000265">Ducatur AF parallela ipsi D, in eaque reperiatur pun-<lb></lb>ctum F, quo grave perveniat tempore quo in B<arrow.to.target n="marg62"></arrow.to.target>, & prae-<lb></lb>scribatur in eadem spatium AG per quod moveatur in<lb></lb> diuturnitate E<arrow.to.target n="marg63"></arrow.to.target>, & fiat DH aequalis ipsi AG, & dico H<lb></lb> esse punctum quaesitum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000266"><margin.target id="marg62"></margin.target>Per 17.<lb></lb> hujus.<lb></lb></s> <s id="s.000267"><margin.target id="marg63"></margin.target>Per 8.<lb></lb> hujus.<lb></lb></s> </p> <p> <s id="s.000268">Quoniam diuturnitates in AB, AF sunt aequales per con-<lb></lb>structionem, & C, E sunt diuturnitates super planis AF,<lb></lb> AG per constructionem, sunt etiam diuturnitates super<lb></lb> AB, AG, & proinde super DH ipsi AG aequali, &<lb></lb> paralellae, quod, &c.<lb></lb> </s> </p> </chap> <pb xlink:href="076/01/034.jpg"></pb> <chap> <p type="head"> <s id="s.000269">PROPOSITIO XXII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000270">Si duo gravia descendunt alterum quidem perpendicu-<lb></lb>lariter, alterum vero super plano declinante, perve-<lb></lb>niunt ad idem planum Orizontale tali ratione, ut sit<lb></lb> eadem proportio inter diuturnitates eorum, quae in-<lb></lb>ter perpendicularem, & declinantem.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000271">Sit linea AB perpendiculariter erecta super plano Ori-<lb></lb>zontali BC, & AC planum declinans.<lb></lb> </s> </p> <p> <s id="s.000272">Dico quod diuturnitates gravium descendentium per AB, &<lb></lb> per AC, sunt ut AB ad AC.<lb></lb> </s> </p> <p> <s id="s.000273">Ducatur BD normalis ad AC.<lb></lb> </s> </p> <p> <s id="s.000274">Quoniam est ut AD ad AC ita quadratum temporis AD<lb></lb> ad quadratum temporis AC<arrow.to.target n="marg64"></arrow.to.target>, & tempora AD, & AB<lb></lb> sunt aequalia<arrow.to.target n="marg65"></arrow.to.target>, & proinde eorum quadrata<arrow.to.target n="marg66"></arrow.to.target>, ergo ut A<lb></lb>D, ad AC ita quadratum temporis AB ad quadratum tem-<lb></lb>poris AC, sed ut AD ad AC ita quadratum AB ad qua-<lb></lb>dratum AC<arrow.to.target n="marg67"></arrow.to.target>, ergo ut quadratum temporis AB ad qua-<lb></lb>dratum temporis AC, ita quadratum AB ad quadratum<lb></lb> AC<arrow.to.target n="marg68"></arrow.to.target>, sed quia latera sunt inter se ut eorum quadrata<arrow.to.target n="marg69"></arrow.to.target>, est<lb></lb> ut AB ad AC ita tempus AB ad tempus AC. </s> <s id="s.000275">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000276"><margin.target id="marg64"></margin.target>Per cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000277"><margin.target id="marg65"></margin.target>Per 15.<lb></lb> hujus.<lb></lb></s> <s id="s.000278"><margin.target id="marg66"></margin.target>Per 2.<lb></lb> pron.<lb></lb></s> <s id="s.000279"><margin.target id="marg67"></margin.target>Per 19.<lb></lb> Sexti.<lb></lb></s> <s id="s.000280"><margin.target id="marg68"></margin.target>Per 22.<lb></lb> Quinti.<lb></lb></s> <s id="s.000281"><margin.target id="marg69"></margin.target>Per 24.<lb></lb> Sexti.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/035.jpg"></pb> <chap> <p type="head"> <s id="s.000282">PROPOSITIO XXIII.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000283">Duo gravia descendentia super planis diversa ratione<lb></lb> declinantibus, perveniunt ad idem planum orizon-<lb></lb>tale ea ratione, ut sit eadem proportio inter diutur-<lb></lb>nitates, quae inter dicta plana si ab eodem puncto ad<lb></lb> idem planum orizontale producta sint.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000284">Datis planis AB, AC declinantibus, ductis ab eodem<lb></lb> puncto A ad planum orizontale BC.<lb></lb> </s> </p> <p> <s id="s.000285">Dico quod diuturnitates gravium descendentium per AB, AC<lb></lb> sint ut AB ad AC.<lb></lb> </s> </p> <p> <s id="s.000286">Fiat ut AC ad AB ita AB ad AD, ita ut grave perveniat<lb></lb> in D eodem tempore quo pervenit in B<arrow.to.target n="marg70"></arrow.to.target>.<lb></lb> </s> </p> <p type="margin"> <s id="s.000287"><margin.target id="marg70"></margin.target>Per 13.<lb></lb> hujus.<lb></lb></s> </p> <p> <s id="s.000288">Quoniam est ut AD ad AC, ita quadratum temporis AD<lb></lb> ad quadratum temporis AC<arrow.to.target n="marg71"></arrow.to.target>, & tempora AD, AB<lb></lb> sunt aequalia<arrow.to.target n="marg72"></arrow.to.target>, & proinde eorum quadrata; ergo ut AD<lb></lb> ad AC ita quadratum temporis AB, ad quadratum tem-<lb></lb>poris AC<arrow.to.target n="marg73"></arrow.to.target>, sed ut AD ad AC, ita quadratum AB ad qua-<lb></lb>dratum AC<arrow.to.target n="marg74"></arrow.to.target>, ergo ut quadratum temporis AB ad quadra-<lb></lb>tum temporis AC, ita quadratum AB ad quadratum AC,<lb></lb> ergo ut tempus AB ad tempus AC, ita AB ad AC<arrow.to.target n="marg75"></arrow.to.target>.<lb></lb> </s> <s id="s.000289">Quod fuit probandum.<lb></lb> </s> </p> <p type="margin"> <s id="s.000290"><margin.target id="marg71"></margin.target>Per Cor.<lb></lb> 7. hujus.<lb></lb></s> <s id="s.000291"><margin.target id="marg72"></margin.target>Per 17.<lb></lb> hujus.<lb></lb></s> <s id="s.000292"><margin.target id="marg73"></margin.target>Per 2.<lb></lb> pronun.<lb></lb></s> <s id="s.000293"><margin.target id="marg74"></margin.target>Per 19.<lb></lb> sexti.<lb></lb></s> <s id="s.000294"><margin.target id="marg75"></margin.target>Per 22.<lb></lb> sexti.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/036.jpg"></pb> <chap> <p type="head"> <s id="s.000295">PROPOSITIO XXIV<lb></lb></s> </p> <subchap1> <p> <s id="s.000296">Datis planis, & perpendiculari ad eadem linea orizon-<lb></lb>tali egressis, quae coeant infra in eodem puncto, gra-<lb></lb>via super ipsis mota procedunt ea ratione, ut sit ea-<lb></lb>dem proportion inter diuturnitates, quae inter longi-<lb></lb>tudines planorum, & dictam perpendicularem.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000297">Data sit linea orizontalis AB, in qua initium sumant<lb></lb> plana declinantia AC, DC, nec non perpendicula-<lb></lb>ris BC coeuntia in puncto C.<lb></lb> </s> </p> <p> <s id="s.000298">Dico quod diuturnitates gravium super ipsis motorum, sunt<lb></lb> ut AC, DC, BC.<lb></lb> </s> </p> <p> <s id="s.000299">Ducatur CE paralella ipsi AB, & a puncto A ducantur<lb></lb> paralellae ipsis CB, CD, & sint AE, AF.<lb></lb> </s> </p> <p> <s id="s.000300">Quoniam diuturnitates super planis AF, AC, sunt ut A<lb></lb>F, AC<arrow.to.target n="marg76"></arrow.to.target>, & super planis eisdem, & perpendiculari A<lb></lb>E, sunt ut AF, seu AC ad AE<arrow.to.target n="marg77"></arrow.to.target>, & AE, AF sunt paralellae<lb></lb> ipsis CD, CB, & eisdem aequales<arrow.to.target n="marg78"></arrow.to.target>, sequitur quod etiam<lb></lb> super AC, DC, BC diuturnitates sunt iuxta propor-<lb></lb>tiones longitudinum<arrow.to.target n="marg79"></arrow.to.target>, Quod probandum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000301"><margin.target id="marg76"></margin.target>Per 23.<lb></lb> hujus.<lb></lb></s> <s id="s.000302"><margin.target id="marg77"></margin.target>Per 15.<lb></lb> hujus.<lb></lb></s> <s id="s.000303"><margin.target id="marg78"></margin.target>Per 33.<lb></lb> prim.<lb></lb></s> <s id="s.000304"><margin.target id="marg79"></margin.target>Per 3.<lb></lb> pron.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/037.jpg"></pb> <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></lb> puncta peripheriae ducantur plana, quo tempore gra-<lb></lb>ve perpendiculariter inde pervenit ad planum ori-<lb></lb>zontale; si descendat per dicta plana, eodem perve-<lb></lb>niet respective ad quodlibet dictorum punctorum<lb></lb> peripheriae.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000307">Sit circulus cuius centrum B, & diameter AC erectus<lb></lb> super plano orizontali GC, & in eo ducta sint plana de-<lb></lb>clinantia a puncto A ad puncta peripheriae DEF, & de-<lb></lb>scendant gravia super dicta plana, & perpendiculariter.<lb></lb> </s> </p> <p> <s id="s.000308">Dico quod eodem tempore pervenient ad, D, E, F, C.<lb></lb> </s> </p> <p> <s id="s.000309">Ducantur DC, EC, FC.<lb></lb> </s> </p> <p> <s id="s.000310">Quoniam puncta praedicta sunt ea, in quae cadunt perpendi-<lb></lb>cularia ducta a puncto C in AD, AE, AF<arrow.to.target n="marg80"></arrow.to.target>, eo perveniunt<lb></lb> gravia eodem tempore quo in C<arrow.to.target n="marg81"></arrow.to.target>. </s> <s id="s.000311">Quod probandum fuit.<lb></lb> </s> </p> <p type="margin"> <s id="s.000312"><margin.target id="marg80"></margin.target>Per 30.<lb></lb> Tertii.<lb></lb></s> <s id="s.000313"><margin.target id="marg81"></margin.target>Per 16.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/038.jpg"></pb> <chap> <p type="head"> <s id="s.000314">PROPOSITIO XXVI.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000315">Si in circulo erecto, a puncto inferiori ducantur plana<lb></lb> ad puncta peripheriae, & a dictis punctis descendant<lb></lb> gravia super dicta plana eodem tempore quo a puncto su-<lb></lb>premo descendit aliud grave perpendiculariter; perve-<lb></lb>nient omnia eodem instanti ad dictum punctum inferius.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000316">Sit circulus cuius diameter ABC erectus super plano<lb></lb> orizontali, quod tangat in C, & a C ducantur plana C<lb></lb>D, CE, & a punctis, E, D gravia descendant super dicta<lb></lb> plana, nec non, & a puncto supremo A perpendiculariter.<lb></lb> </s> </p> <p> <s id="s.000317">Dico quod eodem tempore perveniunt in C.<lb></lb> </s> </p> <p> <s id="s.000318">A puncto A ducantur AF, AG paralellae ipsis CE, CD,<lb></lb> & ducantur AF, FC.<lb></lb> </s> </p> <p> <s id="s.000319">Quoniam in triangulis AEC, AFC anguli alterni FAC,<lb></lb> ACE sint aequales,<arrow.to.target n="marg82"></arrow.to.target>, & anguli AFC, AEC sunt etiam<lb></lb> aequales puta recti<arrow.to.target n="marg83"></arrow.to.target>, & basis AC communis, Triangula<lb></lb> sunt aequalia<arrow.to.target n="marg84"></arrow.to.target>, & proinde AF est aequalis CE, quod idem<lb></lb> probabitur de reliquis, ergo cum AF, CE, & reliquae<lb></lb> sint paralellae, & aequales, gravia per CE, CD perve-<lb></lb>nient in C eodem tempore, quo digressa ab A perveniunt<lb></lb> ad puncta FG, sed haec eodem tempore quo perpendicularis<lb></lb> pervenit in C<arrow.to.target n="marg85"></arrow.to.target>, ergo etiam ea quae per CE, CD. </s> <s id="s.000320">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000321"><margin.target id="marg82"></margin.target>Per 29.<lb></lb> primi.<lb></lb></s> <s id="s.000322"><margin.target id="marg83"></margin.target>Per 30.<lb></lb> Tertii.<lb></lb></s> <s id="s.000323"><margin.target id="marg84"></margin.target>Per 26.<lb></lb> primi.<lb></lb></s> <s id="s.000324"><margin.target id="marg85"></margin.target>Per 25.<lb></lb> hujus.<lb></lb></s> </p> </chap> <pb xlink:href="076/01/039.jpg"></pb> <chap> <p type="head"> <s id="s.000325">POSTULATUM VII<lb></lb></s> </p> <p> <s id="s.000326">Ductis planis inclinatis, & linea perpendiculari in-<lb></lb>ter binas paralellas orizontales, Gravia super illis<lb></lb> mota ubi perveniunt ad paralellam inferiorem ha-<lb></lb>bent aequales velocitatis gradus; & proinde si ab inde<lb></lb> infra sortiantur parem inclinationem, aequeveloci-<lb></lb>ter moventur.<lb></lb> </s> </p> <p> <s id="s.000327">Videtur probabile. </s> <s id="s.000328">Primo quia si diuturnitates sunt longitu-<lb></lb>dinibus proportionales, ut propositione 22. huius probatum<lb></lb> fuit, credibile est motus in fine esse aequales.<lb></lb> </s> </p> <p> <s id="s.000329">Secundo. Argumento ducto ab experientia pendulorum, quae<lb></lb> quantumvis longiora, aut breviora, & proinde circa fi-<lb></lb>nem magis, aut minus inclinata, pariter ascendunt, si pa-<lb></lb>riter descendant.<lb></lb> </s> </p> <p> <s id="s.000330">Tertio. Quia videmus aquam per siphones rectos, sive obli-<lb></lb>quos, seu inclinatos ductam, pariter ascendere, si pariter<lb></lb> descendat. </s> <s id="s.000331">Ceterum fateor minorem evidentiam hoc po-<lb></lb>stulatum caeteris praemissis prae se ferre, quae fuit causa quod<lb></lb> illud, ut in praefatione, segregaverim, & sequentia, alia<lb></lb> methodo, tangendo fere tantummodo exposuerim, & a<lb></lb> pluribus aliis propositionibus, quae hinc deduci facile pos-<lb></lb>sent, data opera abstinuerim. </s> </p> </chap> <pb xlink:href="076/01/040.jpg"></pb> <chap> <p type="head"> <s id="s.000332">PROPOSITIO XXVII. PROBL. XIV.<lb></lb> </s> </p> <subchap1> <p> <s id="s.000333">Dato gravi moto perpendiculariter per spatium datum<lb></lb> diuturnitate data, quod perficiat motum super plano<lb></lb> inclinato per spatium itidem datum; perquirere in<lb></lb> ipso diuturnitatem.<lb></lb> </s> </p> </subchap1> <p> <s id="s.000334">Moveatur grave A perpendiculariter per spatium AB<lb></lb> diuturnitate C, & perseveret in motu super spatio B<lb></lb>D in plano inclinato BD.<lb></lb> </s> </p> <p> <s id="s.000335">Venanda est diuturnitas eius in ipso BD.<lb></lb> </s> </p> <p> <s id="s.000336">Producatur DB donec concurrat cum AE orizontaliter du-<lb></lb>cta ab A in E, & fiat ut AB ad EB, ita diuturnitas C ad<lb></lb> diuturnitatem G, quae idcirco erit diuturnitas ipsius EB<arrow.to.target n="marg86"></arrow.to.target>,<lb></lb> & sit H quadratum diuturnitatis G, & fiat ut EB ad ED,<lb></lb> ita quadratum H ad aliud quod sit I a cuius latere K, quod<lb></lb> est diuturnitas ipsius ED, ablata KL aequali G, erit LM<lb></lb> reliquum diuturnitas BD quaesita.<lb></lb> </s> </p> <p type="margin"> <s id="s.000337"><margin.target id="marg86"></margin.target>Per 22.<lb></lb> hujus.<lb></lb></s> </p> <pb xlink:href="076/01/041.jpg"></pb> <p> <s id="s.000338">Quoniam notum est triangulum AEB, cum notus sit angu-<lb></lb>lus AEB aequalis alterno EDF inclinationis notae, & E<lb></lb>AB rectus ex constructione, & notum latus AB ex hypo-<lb></lb>tesi, notum erit etiam latus EB, & quia diuturnitas in pla-<lb></lb>no BD est eadem ac si motus antecedens esset per EB<arrow.to.target n="marg87"></arrow.to.target>, EB,<lb></lb> & ED sunt in duplicata ratione diuturnitatum G, K ex con-<lb></lb>structione; unde a K deducta KL aequali G ex constructio-<lb></lb>ne, remanet LM diuturnitas BD. </s> <s id="s.000339">Quod, &c.<lb></lb> </s> </p> <p type="margin"> <s id="s.000340"><margin.target id="marg87"></margin.target>Per 7.<lb></lb> post.<lb></lb></s> </p> <p> <s id="s.000341">Inde sequitur quod summa diuturnitatum C, & LM, est diutur-<lb></lb>nitas totius ABD.<lb></lb> </s> </p> <p> <s id="s.000342">Eadem operatione pariter reperietur diuturnitas BD si BD<lb></lb> sit perpendicularis, & AB inclinata.<lb></lb> </s> </p> <p> <s id="s.000343">Item si ambo sint plana inclinata.<lb></lb> </s> </p> <p> <s id="s.000344">Ducta AD facile reperietur diuturnitas in ipsa si fiat ut ED<lb></lb> ad AD, ita K ad aliud per 21. hujus.<lb></lb> </s> </p> <pb xlink:href="076/01/042.jpg"></pb> <p> <s id="s.000345">Ducto alio plano puta DN, reperietur eius diu-<lb></lb>turnitas. </s> </p> <p> <s id="s.000346">Si fiat ut ED ad OD ita diuturnitas ipsius ED puta L ad diu-<lb></lb>turnitatem OD, quae sit P, deinde ut OD ad ON ita<lb></lb> quadratum diuturnitatis P ad aliud quadratum, cuius Ra-<lb></lb>dix erit diuturnitas ipsius DN.<lb></lb> </s> </p> <p> <s id="s.000347">Ex his patet quod si addantur plura plana eadem ratione re-<lb></lb>perientur eius diuturnitates.<lb></lb> </s> </p> <pb xlink:href="076/01/043.jpg"></pb> <p> <s id="s.000348">Ex his etiam patet quod si in circulo dentur plura, plana verbi<lb></lb> gratia AB, BC, CD, DE, & data sit diuturnitas super dia-<lb></lb>metro, dabitur etiam diuturnitas cuiusvis dicto-<lb></lb>rum AB, BC, CD, DE, & etiam omnium simul.<lb></lb> </s> </p> <p> <s id="s.000349">Ex his facile etiam cognoscere poteris esse breviorem, diu-<lb></lb>turnitatem per A, B, C, C, D, E quam per AE.<lb></lb> </s> </p> <p> <s id="s.000350"> FINIS. <lb></lb></s> </p> </chap> </body> <back></back> </text> </archimedes>