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<archimedes>
<info>
<author>Bernadino Baldi</author>
<title>In Mechanica Aristotelis problemata exercitationes</title>
<date>1621</date>
<place>Mainz</place>
<editor></editor>
<publisher>Johannis Albinus</publisher>
<translator></translator>
<lang>Latin</lang>
<chunk unit="page*">page</chunk>
</info>
<text>
<front>
</front>
<body>
<chap>
<pb/>
<pb id="p.0002"/>
<p type="head">
<s>BERNARDINI</s>
</p>
<p type="head">
<s>BALDI VRBINATIS <lb/>GVASTALLÆ AB­<lb/>BATIS <lb/><emph type="italics"/>IN<emph.end type="italics"/></s>
</p>
<p type="head">
<s>MECHANICA ARISTOTE <lb/>LIS PROBLEMATA <lb/>EXER CITATIONES:</s>
</p>
<p type="head">
<s><emph type="italics"/>ADIECTA SVCCINCTA NAR­<lb/>ratione de autoris vita & &longs;criptis.<emph.end type="italics"/></s>
</p>
<figure></figure>
<p type="head">
<s><emph type="italics"/><gap/>NTIAE.<emph.end type="italics"/></s>
</p>
<p type="head">
<s>Typis & Sumptibus Viduæ Ioannis Albini.</s>
</p>
<p type="head">
<s><gap/></s>
</p>
<pb/>
<figure></figure>
<p type="head">
<s><emph type="italics"/>NOBILISSIMO AC GENE­<lb/>ROSO DOMINO<emph.end type="italics"/></s>
</p>
<p type="head">
<s>D. ADAMO PHILIP­<lb/>PO BARONI A CRON­<lb/>BERG, EQVITI, SACRÆ CÆSA­<lb/>REÆ MAIESTATIS, ET SERENISSIMI</s>
</p>
<p type="head">
<s>Principis Archiducis Alberti Camerario intimo & c. <lb/>Domino meo gratio&longs;i&longs;&longs;imo.</s>
</p>
<p type="main">
<s>Opportune &longs;ub hoc ip&longs;um tem­<lb/>pus, quo in Belgium ad Scre­<lb/>ni&longs;&longs;imos Principes iter ador­<lb/>nat. Nobili&longs;&longs;ima & Genero&longs;a <lb/>Dom. V.^{ra}, prodit no&longs;tris for­<lb/>mis in publicum editus Com­<lb/>mentarius Bernardini Baldi Vrbinatis Gua­<lb/>&longs;tallæ Abbatis in Ari&longs;totelis Mechanica. Is <lb/>virin omni &longs;cientiæ genere, at maxime inMa­<lb/>thematicis di&longs;ciplinis fuit ver&longs;ati&longs;&longs;imus, quod <lb/>multa ab eo præclare &longs;crip ta te&longs;tantur opera, <lb/>ex quibus paucula edita, reliqua vero &longs;pera­
<pb/>mus &longs;uo temporein publicam luce<gap/> produ­<lb/>cenda. Cum vero nemini&longs;it ob&longs;curum Nobi­<lb/>li&longs;&longs;imæ ac Genero&longs;æ Dom. V.^{ræ} id &longs;emper <lb/>extiti&longs;&longs;e familiari&longs;&longs;imum, vttum dome&longs;ticum <lb/>otium, tum maxime peregtinationes, quibus <lb/>totam pæne Europam &longs;umma cum laude <lb/>circum&longs;crip&longs;it, tum variarum linguarum per­<lb/>fecto v&longs;u, tum Mathematicarum di&longs;ciplina­<lb/>rum notitia & exercitio redderet <expan abbr="iucũdiores">iucundiores</expan>, <lb/>nulla me tenet dubitatio quin & Baldum Vr­<lb/>binatem no&longs;tris typis loquentem in hoc iti­<lb/>nere, quod à Deo felici&longs;&longs;imum Nobili&longs;&longs;imæ <lb/>acGenero&longs;æ Dom. V.^{ræ} precor, in &longs;uum comi­<lb/>tatum ac tutelam beneuolo animo &longs;it admi&longs;­<lb/>&longs;ura. Id rogo humillime &longs;imulque precor, vt. <lb/>hanc meam typographiam plurimis iam re­<lb/>tro annis de inclytæ familiæ Cronbergicæ tu­<lb/>tela gloriantem, &longs;uo fauore pro&longs;equatur, vi­<lb/>duæque afflictæ fortunis beneuole ad&longs;piret. <lb/>Sic Deus Nobili&longs;&longs;. & Genero&longs;am Dom. V.^{ram} <lb/>illu&longs;tret omnibus bonis, eamque R.^{mo} & Ill.^{mo} <lb/>Principiac Domino meo Clementi&longs;&longs;imo, D. <lb/>Ioanni Suicardo Archiepi&longs;copo Mogunti­<lb/>no Principi Electori ac per <emph type="italics"/>G<emph.end type="italics"/>ermaniam Ar-
<pb/>chicancellario &c. patruo &longs;uo optati&longs;&longs;imo <lb/>&longs;aluo florentique redhibeat &longs;aluum &longs;imili­<lb/>ter florentem ac incolumem. Moguntiæ è <lb/>typographeio Viduæ Albinianæ, honori No­<lb/>bili&longs;&longs;imæ ac <emph type="italics"/>G<emph.end type="italics"/>enero&longs;æ Dom. Ve&longs;træ perpe­<lb/>tuum dicato. Anno 1621.26.Martij. </s>
</p>
<pb/>
<figure></figure>
<p type="head">
<s>PRAEFATIO.</s>
</p>
<p type="main">
<s><emph type="italics"/>Diligenter legenti mihi quæ&longs;tiones il­<lb/>las, in quibus ea quæ ad Mecha­<lb/>nicam facultatem pertinent, expli­<lb/>cantur, multa in mentem venie­<lb/>bant; & primum quidem eorum, quæibi di<foreign lang="greek">c</foreign>pu­<lb/>tantur, vtilitatem, &longs;ubtilitatem, copiam admi­<lb/>rabar: Tum ex animo dolebam, aureum hunc li­<lb/>bellum propènegligi, & ab iis qui pulcherrimis <lb/>hi&longs;ce &longs;tudiis dant operam, assiduè præ manibus <lb/>non haberi: Multas autem Auctoriip&longs;ihaben­<lb/>das referendasque e&longs;&longs;e gratias, qui tam egregiam, <lb/>vtilem & probèin&longs;tructam &longs;upellectilem Archi­<lb/>tectis, Mechanicis, & omnibus ferè Artificibus <lb/>&longs;uppeditauerit. Ari&longs;toetlis nomini a&longs;cribitur <lb/>Commentarius, licet nonnulli, &longs;itne Philo&longs;ophi <lb/>illius præclarissimi & acutissimilabor, an non, <lb/>adfirmare &longs;ubdubitauerint. Ari&longs;totelis tamen <lb/>e&longs;&longs;e omnes ferè meliores con&longs;entiunt: <gap/>dquetum <lb/>ex phra&longs;i, & explicatione, quæ Ari&longs;totelem &longs;a­<lb/>piunt, tum iudicio &longs;ubtilitatis & rationum, qui-
<pb/>bus quæ&longs;tiones ip&longs;æ ingenio&longs;issimè diluuntur. Vi­<lb/>detur autem mihi, rem accuratius exploranti, &longs;a­<lb/>tis veri&longs;imile (nullum enim habeo opinionis hu­<lb/>ius a&longs;&longs;ertorem,) &longs;ectionem e&longs;&longs;e hanc, & partem <lb/>quandam eius operis nobilissimi, quod idem au­<lb/>ctor De Problematibus edidit, & hanc, ne&longs;cio <lb/>quam ob cau&longs;am; ni&longs;i fortè quod tractatio merè <lb/>Phy&longs;icanon&longs;it, àreliquo corpore di&longs;tractam at­<lb/>que reuul&longs;am. Jd certè quod ad rem facit, probè <lb/>nouimus, Diogenem Laërtium inter cætera Ari­<lb/>&longs;totelici ingenij monumenta Mechanica quoque <lb/>adnumera&longs;&longs;e. Quibus con&longs;ideratis magnopere <lb/>&longs;ubit mirari, cur ij qui po&longs;t Ari&longs;totelem floruêre <lb/>atque vixere, Mechanici, Archimedes, Athenæus, <lb/>Heron, Pappus, & cæteri, nullam huius libelli fe­<lb/>cerint commemorationem: & &longs;anè debuerunt; <lb/>neque enim à vero est dissimile, ip&longs;os per hunc ali­<lb/>quatenus profeci&longs;&longs;e. Verum enimuero cum inge­<lb/>nuiilli fuerint homines, & nullatenus obtrecta­<lb/>tores, credendum potius est, Comment ariolum i­<lb/>&longs;tud, eorum æuo, paucis cognitum, alicubi in Bi­<lb/>bliothecis latui&longs;&longs;e: etenim cætera quoque Ari&longs;tote­<lb/>lis &longs;cripta, po&longs;t vetu&longs;tailla tempora, ante Ale­<lb/>xandrum Aphrodi&longs;ien&longs;em, àmultis fui&longs;&longs;e igno-
<pb/>rata non dubitamus. Habemus &longs;iquidem, Stra­<lb/>bone te&longs;te, lib. 13. Ari&longs;totelis, & Theophra&longs;ti bi­<lb/>bliothecam, po&longs;t ip&longs;ius Theophra&longs;ti dece&longs;&longs;um, ad <lb/>Neleum quendam Scep&longs;ium, Cori&longs;cifilium, qui <lb/>eius fuerat auditor, perueni&longs;&longs;e; po&longs;t hæc libros, <lb/>blattis olim, & humore corruptos, Apelliconi Te­<lb/>io venditos, & ab eo Athenas translatos, tum <lb/>Athenis captis in Syllæ pote&longs;tatem deacni&longs;&longs;e, eo&longs;­<lb/>que tandem à Sylla acceptos, Tyrannionem <lb/>Grammaticum, vt potuit meliùs emendatos, <lb/>promulga&longs;&longs;e. Exquibus colligimus, mirum non <lb/>e&longs;&longs;e, Archimedi, Heroni, & alijs qui ante Syllam <lb/>vixêre, fui&longs;&longs;e incognitos. quicquid&longs;it, illud cer­<lb/>tumest, Ari&longs;totelem eorum omnium quide Me­<lb/>chanicis commentaria edidere, e&longs;&longs;e longè vetu­<lb/>&longs;tissimum. Pappus enim Heroneiunior, Athe­<lb/>næus Archimediæqualis, vterque enim &longs;ub Mar­<lb/>cello, cui Athenæus &longs;uum de bellicis Machinis <lb/><expan abbr="libellū">libellum</expan> dedicauit. Archimedes verò circa CXL. <lb/>Olympiadem floruit, quamobrem po&longs;t Ari&longs;tote­<lb/>lem Olympiadas XL. hoc est, annos ferè CLX. <lb/>I&longs;thæc autem con&longs;iderantibus, facile e&longs;t cogno&longs;ce­<lb/>refacultatis huius nobilitatem, atque dignit atem; <lb/>quippe quod &longs;ummus Philo&longs;ophus non modo eam
<pb/>probauerit, &longs;cd etiam &longs;uis acutissimis lucubra­<lb/>tionibus illu&longs;trauerit. Hanc porro tractationem <lb/>&longs;ubiecto quidem Phy&longs;icam e&longs;&longs;e, demon&longs;tratio­<lb/>nibus verò Geometricam, ip&longs;emet nos docuit <lb/>Ari&longs;toteles, cuius etiam naturæ &longs;unt Per&longs;pecti­<lb/>ua, Specularia, Mu&longs;ica, & cæteræ eiu&longs;dem <lb/>modifacultates, quas quidem &longs;ubalternas Peri­<lb/>patetici appellant. Vitruuius Architecturæ <lb/>membrum, vt ita dicam, & portionem quan­<lb/>dam facit, ait enim Architecturæ partes e&longs;&longs;e tres, <lb/>Ædificationem, Gnomonicam, Machinatio­<lb/>nem. Estautem Architectur â quideminferior, <lb/>paret enim Architecto Mechanicus; attamen &longs;i <lb/>cæteras artes &longs;pectes, Architectonica; hæc enim <lb/>omnesferè &longs;edentariæ, &longs;ellulariæue, quas banau­<lb/>&longs;as Græci appellant, ordine &longs;ubijciuntur, & &longs;a­<lb/>nè latissimos i&longs;ihæc habetfines; præcipuè autem <lb/>circa eam ver&longs;atur cognitionem, eamque inter <lb/>cæterasferè principem, quam dixere Centrobari­<lb/>cam, quæ quidem ad Centri grauitatem, eiu&longs;que <lb/>&longs;peculationem pertinet: quà in &longs;pecie inter vete­<lb/>res primum &longs;ibi vindicauit locum Archimedes, <lb/>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"/>
<pb/><emph type="italics"/>tem Commandinus, qui librum de Centro gra­<lb/>uitatis &longs;olidorum &longs;crip&longs;it, & po&longs;t eum G. Vbal­<lb/>dus è Marchion. Montis, qui non modò ab­<lb/>&longs;olutissimum Mechanicorum librum cum maxi­<lb/>maingenij &longs;ui laude con&longs;crip&longs;it, &longs;ed & Paraphra­<lb/>&longs;in in librum Æqueponderantium Archimedis <lb/>egregiè concinnauit Centrobaricam hanc, igno­<lb/>tam fui&longs;&longs;e Ari&longs;toteli, &longs;ætis patet. nunquam enim <lb/>in Mechanicis demon&longs;trationibus, quod tamen <lb/>est potissimum, grauitatis centrum nominat, e­<lb/>iu&longs;ue naturam atque vim &longs;peculatur. Diuidi­<lb/>tur autem Mechanice tota, te&longs;te Herone apud <lb/>Pappum libro octauo, in Rationalem, hoc est, <lb/>Theoricam & Chirurgicam, id est, manu ope­<lb/>ratricem, quam Praxim aptè dicere valemus. <lb/>Rationalis, &longs;peculationi & <expan abbr="demō&longs;trationibus">demon&longs;trationibus</expan>, ex <lb/>Geometricis, Arithmeticis & Phy&longs;icis rationi­<lb/>bus, dat operam; Chirurgica vero materiam <lb/>tractat, & &longs;e&longs;ein varias artes diffundit, Æræ­<lb/>riam, Lignariam, Sculptoriam, Pictoriam, Æ­<lb/>dificatoriam, Machinariam & Thaumaturgi­<lb/>cam, cæterasque eiu&longs;modi. Machinatoriæ au­<lb/>tem &longs;unt partes Manganaria, qua ingentiæ<emph.end type="italics"/>
<pb/><emph type="italics"/>transferuntur pondera, tum ip&longs;a Poliorcetica, <lb/>quæ bellicas Machinas ad vrbium expugnatio­<lb/>nes, quod velip&longs;o nomine profitetur, ædificat. At­<lb/>qui hac dere plura &longs;cribere &longs;uper&longs;edemus, ne a­<lb/>ctum agamus: quis quis enim minutè magis hæc <lb/>cogno&longs;cere de&longs;iderat, is Pappum adeat libro cita­<lb/>to, & Guidum Vbaldumin Præfatione quam <lb/>&longs;uo Mechanicorum Operi præpo&longs;uit. Vt autem <lb/>ad Ari&longs;totelis, de quo egimus, libellum reuerta­<lb/>mur, pauci &longs;unt qui ei ante nos &longs;tilum & operam <lb/>commodauerint: Leonicenus Latinum fecit & <lb/>figuris tum breuissimis, & parui &longs;ane ponderis, <lb/>marginalibus adnotatiunculis, in&longs;truxit. Po&longs;t <lb/>hunc Alexander Picolomineus luculentissima <lb/>Paræphra&longs;i illu&longs;trauit. Modo, vt audio, Simon <lb/>Sticinus Hollanden&longs;is quædam edidit, quæ ad <lb/>nos minime peruenêre. Nos demum, omnium, <lb/>tum &longs;cientia, & ingenio, tum ætate, po&longs;tremi huic <lb/>operi manum admouimus; Con&longs;iderantes enim <lb/>Ari&longs;totelem æly<gap/>s principijs v&longs;um, ac probatissi­<lb/>mipo&longs;t eum fecerint Mechanici, demon&longs;tra&longs;&longs;e, <lb/>morem huiu&longs;ce facultatis &longs;tudio&longs;is ge&longs;turos nos <lb/>fore arbitrati&longs;umus, &longs;i ea&longs;dem illas quæ&longs;tiones<emph.end type="italics"/>
<pb/><emph type="italics"/>Mechanicis, hocest, Archimedeis probationi­<lb/>bus confirmaremus; dum per latissimos faculta­<lb/>tis huius campos vægantes, alias quoque i&longs;tis af­<lb/>fines dubitationes introducentes &longs;olueremus. <lb/><expan abbr="quicquidautē&longs;ecerimus">quicquidauten&longs;ecerimus</expan> profecerimu&longs;ue, Lector <lb/>optime, boni con&longs;ule, & quia fax per manus tra­<lb/>ditur, tu interim de me accipe, vt alijs tradas.<emph.end type="italics"/></s>
</p>
<pb/>
<p type="head">
<s>DE VITA ET SCRI­<lb/>PTIS BERNARDINI <lb/>BALDI VRBINATIS</s>
</p>
<p type="head">
<s><emph type="italics"/>EX LITERIS FABRITII SCHAR­<lb/>loncini ad Illu&longs;trissimum & Reuerendissimum <lb/>Dominum Lælium Ruinum Epi&longs;copum Bal­<lb/>neoregien&longs;em ex-Nuntium Apo&longs;tolicum <lb/>ad Poloniæ Regem & c.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Natus e&longs;t Bern. Baldus Vrbini nobilibus <expan abbr="pa-rētibus">pa­<lb/>rentibus</expan> po&longs;tridie Non. Iunij anno MDLIII. <lb/>Genus traxit, quod me &longs;æpè ab eomemini <lb/>audire, à familia Cantagallina, quæ inter <lb/>Peru&longs;inas illu&longs;tris: hocautem cognomen, <lb/>Baldiaccepto, vtin varietate temporum fit, <lb/>Abauus reliquit, à teneris vnguiculis <expan abbr="pietatē">pietatem</expan> erga Deum <lb/>præ&longs;etulit; nam vt mater eius narrabat, &longs;anctorum imagi­<lb/>nes & Altariola non cum lætitia &longs;olum, &longs;ed cum venera­<lb/>tione anniculus intuebatur. Præceptoribus in adole&longs;cen­<lb/>tia v&longs;us fuit laudati&longs;&longs;imis Io. And. Palatio, & Io. Antonio <lb/>Turoneo, qui altero doctior, & Paulo Manutio maxime <lb/>carus ob latinæ & græcæ linguæ peritiam propè &longs;ingula­<lb/>rem: adillorum autem &longs;edulitatem tantum animi ardo­<lb/>rem attulit, tantam ingenij aciudicij vim, vt non tantum <lb/>æqualis &longs;ed omnium vicerit expectationem. Puer adhuc <lb/>Aratiapparitiones Italico carmmered didit. Parens hac <lb/>filij laude & gloria motus anno 1573. eum ad maiorem in­<lb/>genij cultum cape&longs;&longs;endum Patauium mi&longs;it. Hîcin Ema­<lb/>nuelis Margunij familiaritatem &longs;tatim venit, cui porro
<pb/>fuit in amotibus. Homeri Iliad. illo Doctore & interpre­<lb/>te diligentius quam feci&longs;&longs;et antea, euoluit. priuato autem <lb/>&longs;tudio Anacreonti, Pindaro, Æ&longs;chyli, Euripidi, Sophocli <lb/>operam dedit, &longs;ed præ cæteris Thcocriti Bucolica triuit, <lb/>ad quod &longs;criptionis genus natura magis ferri videbatur: <lb/>centenos græci alicuius poëtæver&longs;us memoriter tenebat, <lb/>&longs;æpeque habebat in ore, in oratoribus græcis ver&longs;andis <lb/>laborem &longs;ealiquem &longs;entire, in poëtis nullum. Scrip&longs;it Pa­<lb/>tauij libellum de Tormentis Bellicis, & eoruminuentori­<lb/>bus, & cum in Tran&longs;alpinorum amicitias incidi&longs;&longs;et, &longs;ibi <lb/>ducebat dedecori ip&longs;os &longs;ua lingua loquentes non intelli­<lb/>gere. quare incredibili celeritate Gallicam & Germani­<lb/>cam didicit. Pe&longs;tilentia ex co Gymna&longs;io exactus in Pa­<lb/>triam redijt, vbi quin quennium integrum Federico <expan abbr="Cō-mandino">Con­<lb/>mandino</expan> affixus omnes Mathe&longs;eos partes perdidicit, cui <lb/>viro in delinean dis figuris ad Euclidis, Pappi, & Heronis <lb/>monumenta manum commodauit: ex eiu&longs;dem obitu do­<lb/>lorem vix con&longs;olabilem &longs;u&longs;tinuit, &longs;u&longs;ceptoque eius vitam <lb/>&longs;cribendi con&longs;ilio, &longs;ubinde ad omnium Mathematicorum <lb/>vitas con&longs;cribendas animum adplicuit, quod & duode­<lb/>cim annorum &longs;patio præ&longs;titit felici&longs;&longs;imè. cum vero Ma­<lb/>thematicarum di&longs;ciplinarum amore torqueretur, ami&longs;&longs;o <lb/>Commandino Præceptore, amicum nactus fuit præ&longs;tan­<lb/>ti&longs;&longs;imum & &longs;ymmy&longs;tam Guidum Vbaldum è Marchioni­<lb/>bus Montis, in cuius &longs;e con&longs;uetudinem daret: quantum <lb/>profeci&longs;&longs;et, o&longs;tendunt ij commentarij quos anno 1582. in <lb/>Ari&longs;t. Mechanica &longs;crip&longs;it. Vt po&longs;tea à grauioribus &longs;tudijs <lb/>ad amœniora animum abduceret, de re nautica poëma I­<lb/>talicè <gap/>onfecit. quo ab&longs;oluto Paradoxa multa Mathema­<lb/>tica explicauit. Fama de Baldi virtutibus di&longs;&longs;ipata Ferran­<lb/>dus Gonzaga Molfetræ Princeps & Gua&longs;tallæ Dominus <lb/>cœpit deillo in &longs;uam familiam a&longs;ci&longs;cendo cogitare, vt qui <lb/>ij&longs;dem caperetur artibus, quibus excellere Baldus inci-
<pb/>piebat: Itaque opera Curtij Arditij honorifice fuit in au­<lb/>lam euocatus, dum vitam non aulicam viueret totus in <lb/>litteras abditus precibus Ve&longs;pa&longs;iani Gonzagæ Sablonetæ <lb/>Ducis ad explanandos Vitruuij libros adactus fuit. quare <lb/><expan abbr="tūc">tunc</expan> natus de <expan abbr="Verborū">Verborum</expan> Vitruuianorum &longs;ignificatione com­<lb/>mentarius; in quo minime miran dum &longs;i minuta quæ dam <lb/>pro&longs;equutus fuit, quæ viro magno minus e&longs;&longs;e digna vi­<lb/>deantur:illi enim Principi morem ge&longs;&longs;it. &longs;cio dixi&longs;&longs;e ali­<lb/>quando Adrianum Romanum è Polonia reuer&longs;um, vbi <lb/>Vitruuium Palatino cuidam explicauerat, &longs;i commen­<lb/>tarium Baldi in Polonia adhibere potui&longs;&longs;em, aurum quod <lb/>mecum attuli emunxi&longs;&longs;em, quia &longs;atis feci&longs;&longs;em muneri la­<lb/>borenullo. Cum Ferrando hero &longs;uo obueni&longs;&longs;et nece&longs;&longs;i­<lb/>tas Hi&longs;panias adeundi, illud iter &longs;ine Baldo facere &longs;e po&longs;­<lb/>&longs;e non putabat, non tam vt haberet, qui erudito cloquio <lb/>viæ tæ dium leuaret, quam cui po&longs;&longs;et arcana committere, <lb/>atque adeo à quo iuuaretur con&longs;ilio. Vix viæ &longs;e dederant <lb/>cum Baldus grauem in morbum delap&longs;us itinere cogitur <lb/>de&longs;i&longs;tere: Mediolanum proinde diuertit, vbi à S. Carolo <lb/>Borromæo & benignè exceptus, & tamdiu detentus do­<lb/>necvaletudinem recuperaret. Gua&longs;tallam po&longs;tea &longs;e re­<lb/>cepit, vbi cum ab&longs;ente Domino liberiori otio fruerctur, <lb/>libros &longs;ex de Aula eruditi&longs;&longs;imos methodo analytica con­<lb/>&longs;crip&longs;it. alios non commemoro, quod cum otium erit, o­<lb/>mnium &longs;yllabum dabo. Anno 1586. ip&longs;o nihil po&longs;tulante <lb/>eligitur Gua&longs;tallæ Abbas, à quo tempore Iuri Can. Con­<lb/>cilijs, & SS.Patribustotum &longs;e dedit. Hebreæ & Chaldææ <lb/>linguarum di&longs;cendarum triennium po&longs;uit. Anno 1593. no­<lb/>uæ Gnomonices libros quinque compo&longs;uit. in&longs;equenti <lb/>Chaldæam Onkeli paraphra&longs;in in Pentateuchum vertit <lb/>& commentarios adiunxit; quo exantlato labore in Iob <lb/>ex Heb. fonte paraphra&longs;in texuit, quam & &longs;cholijs illu­<lb/>&longs;trauit. Tabulam Etru&longs;cam Eugubinam interptetatus
<pb/>fuit:in ca autem diuinatione, vt aiebat, &longs;ubci&longs;iuas vnius <lb/>men&longs;is horas con&longs;ump&longs;it. De Firmamento & aquis egre­<lb/>gic &longs;crip&longs;it. Oeconomiam Tropologicam in S.Matthæum <lb/>Card. Baronius, qui non alia Baldi vidit, vehementer pro­<lb/>babat. Romæ dum viueret, fere ne&longs;ciuit quid gereretur <lb/>in Aulis: Arabicæ enim linguæ cum Io. Bapti&longs;ta Raimon­<lb/>do diligenti&longs;&longs;ime &longs;tuduit, & arcana indu&longs;tria Slauonicæ, <lb/>quam perfecte callebat. Ex Arabico vertit Hortum Geo­<lb/>graphicum Anonymi, quem ante &longs;excentos annos flo­<lb/>rui&longs;&longs;e arbitrabatur. Hunc vero extru&longs;i&longs;&longs;et, vtalios Baldi <lb/>libros, Marcus Vel&longs;erus IIvir Aug. &longs;i eo paulo longior <lb/>huius lucis v&longs;ura contigi&longs;&longs;et. Compo&longs;uit & Dictionarium <lb/>Arabicum. atque cum beati&longs;&longs;imam illam vbertatem in­<lb/>genij a&longs;&longs;idue diffundi nece&longs;&longs;e e&longs;&longs;et, anno 1603. orbem vni­<lb/>uer&longs;um de&longs;eribere aggre&longs;&longs;us fuit; atque ita quidem, vt <lb/>tam quæ ad Hi&longs;toriam, quam quæ ad Geographiam per­<lb/>tinerent complecteretur: Neque illu&longs;trare &longs;olum voluit <lb/>quæ nouerunt antiqui, quemadmodum vi&longs;um Ortelio, <lb/>&longs;ed vel oppidula omnia & pagos, de quibus aliquain po­<lb/>&longs;tremis &longs;criptoribus mentio. & profecto totum opus ad <lb/>vmbilicum perduxit: non dige&longs;&longs;it tamen vniuer&longs;um. qua­<lb/>tuor aut ni fallor quinque tantum Tomi fuerunt ordine <lb/>Alphabetico di&longs;po&longs;iti:&longs;upere&longs;&longs;ent &longs;eptem aut octo di&longs;po­<lb/>nendi, quantum ex chartarum & fa&longs;ciculorum mo<gap/>e con­<lb/>ijcere licet. Anno 1617. quarto Idus Octob. po&longs;teaquam <lb/>dies 40. vehementi de&longs;tillatione vexatus fui&longs;&longs;et, &longs;piritum <lb/>Deo reddidit Sacramentis Eccle&longs;iæ omnibus rite muni­<lb/>tus. Statura procerus fuit, facie oblonga & acribus oculis, <lb/>colore &longs;ubfu&longs;co. Membrorum ei fuit decens habitudo, & <lb/>compactum corpus. Diebus fe&longs;tis omnlbus &longs;acrum facie­<lb/>bat, ieiunabat bis in hebdomada, eleemo&longs;yni&longs;que paupe­<lb/>res &longs;ubleuabat. In&longs;tudijs &longs;ica&longs;&longs;iduus fuit, vt &longs;æpe & legeret <lb/>& comederet. S.Augu&longs;tinilibros de Ciuitate Dei ter in-
<pb/>ter prandium euoluit. Statim à noctis meridie dum ei vi­<lb/>res firmiores e&longs;&longs;ent ad lucubrandum &longs;urgebat. à prandio <lb/>Euclidem Arabice editum, vel libellum aliquem germa­<lb/>nicum aut gallicum in manus &longs;umebat. Suauitate morum <lb/>& mode&longs;tia, etiam &longs;i ceteræ dotes abfui&longs;&longs;ent, quemlibet <lb/>ad amorem &longs;ui allicere potui&longs;&longs;et. Sermo modicus ei fuit, <lb/>itemque cultus. Nullos vnquam honores petijt, qui à <lb/>Clem. 8. ampli&longs;&longs;imi promi&longs;&longs;i fuerant; nullum emolumen­<lb/>tum quæ&longs;iuit &longs;uo ceniu contentus. facile parcendum e&longs;&longs;e <lb/>dicebat, ijs maxime qui in re leui impegi&longs;&longs;ent, quoniam &longs;i <lb/>quos cen&longs;emus optimos, nudos con&longs;piceremus, nullum <lb/>eorum non iudicaremus multis dignum verberibus. Bi­<lb/>bliothecam habuit non locupletem, &longs;ed &longs;electis <expan abbr="in&longs;tructã">in&longs;tructam</expan> <lb/>codicibus. Verum ire per &longs;ingula longum e&longs;&longs;et. Satis mihi <lb/>de incomparabili Baldi doctrina, & &longs;umma innocentia, ô <lb/>rarum connubium, pauca dixi&longs;&longs;e, quæ for&longs;itan ad imitan­<lb/>dum nimis multa. </s>
</p>
<p type="head">
<s>SYLLABVS LIBRORVM</s>
</p>
<p type="head">
<s>omnium B.Abb.Baldi.</s>
</p>
<p type="main">
<s>Arati apparitiones è gr.in Ital. vertit. </s>
</p>
<p type="main">
<s>De Tormentis Bellicis & eorum Inuentoribus lib. </s>
</p>
<p type="main">
<s>Heronis automata vertit. </s>
</p>
<p type="main">
<s>Vitas omnium Mathematicorum &longs;crip&longs;it, & trib. in Tom. <lb/>2.1.P^{s}.à Thalete ad Chri&longs;tum.2.àChri&longs;toad &longs;ua tem­<lb/>pora. </s>
</p>
<p type="main">
<s>Earumdem vitarum Epitomen Chronologicum confecit. </s>
</p>
<p type="main">
<s>In Ari&longs;tot. Mechan. Commentar. </s>
</p>
<p type="main">
<s>De Renautica Poëmation. </s>
</p>
<p type="main">
<s>Paradoxorum Mathematicorum liber. </s>
</p>
<p type="main">
<s>De&longs;criptio Palatij Ducum Vrbinarum quod e&longs;t Vrbini. </s>
</p>
<p type="main">
<s>Poema cui titulus, Lamus. </s>
</p>
<pb/>
<p type="main">
<s>Carmina pia, quæ in&longs;cribuntur, Anni Corona. </s>
</p>
<p type="main">
<s>De Verborum Vitruuianorum &longs;ignificatione. </s>
</p>
<p type="main">
<s>Carmina varia & eclogæ mixtæ. </s>
</p>
<p type="main">
<s>Apologi centum, quos &longs;crip&longs;it æmulatus Leonem Bapt. <lb/>Albertum. </s>
</p>
<p type="main">
<s>De Humanitate Dialogus qui in&longs;cribitur Go&longs;elinus. </s>
</p>
<p type="main">
<s>Compatatio Vitæ Mona&longs;ticæ cum &longs;eculari. </s>
</p>
<p type="main">
<s>De Aula libri &longs;ex. </s>
</p>
<p type="main">
<s>De felicitate Principis Dialogus. </s>
</p>
<p type="main">
<s>De Dignitate Dial. </s>
</p>
<p type="main">
<s>Carmina Romana. </s>
</p>
<p type="main">
<s>Mo&longs;æi fabulam vertit. </s>
</p>
<p type="main">
<s>De Italici carminis natura Dial. qui in&longs;cribitur Ta&longs;&longs;us. </s>
</p>
<p type="main">
<s>De vniuer&longs;ali Diluuio poemation. </s>
</p>
<p type="main">
<s>Nouæ Gnomonices lib. quin que. </s>
</p>
<p type="main">
<s>Hieremiæ Threnos vertit, & ex Heb. fonte annotat. ad­<lb/>iecit. </s>
</p>
<p type="main">
<s>Poemation in&longs;eriptum, Deiphobe, quod &longs;erip&longs;it æmula­<lb/>tus Lycophonem in Ca&longs;&longs;andra. </s>
</p>
<p type="main">
<s>Scala cœle&longs;tis.1.Sermones pij & carmina. </s>
</p>
<p type="main">
<s>Onkeli paraphra&longs;in Chaldæam in Pentateuchum ver­<lb/>tit & vberes commentarios adiecit. </s>
</p>
<p type="main">
<s>In Iob Paraphra&longs;is latina ex fonte Heb. additis Scholijs. </s>
</p>
<p type="main">
<s>De &longs;camillis imparibus Vitruuij. </s>
</p>
<p type="main">
<s>De firmamento & aquis. </s>
</p>
<p type="main">
<s>Quincti Calabri Paralipomena vertit. </s>
</p>
<p type="main">
<s>Tabulæ Etru&longs;cæ Eugubinæ Interpretatio. </s>
</p>
<p type="main">
<s>Oeconomía Tropologicain S.Matthæum. </s>
</p>
<p type="main">
<s>Vrbini encomium. </s>
</p>
<p type="main">
<s>Horti geographici ex Arab. ver&longs;io. </s>
</p>
<p type="main">
<s>Aduer&longs;us Aulam Carmina. </s>
</p>
<p type="main">
<s>Luciani de mi&longs;erijs.Aulicorum ver&longs;io. </s>
</p>
<p type="main">
<s>Oratio ad Romæ con&longs;eruatores pro antiquitatum eius <lb/>Vrbis cu&longs;todia. </s>
</p>
<pb/>
<p type="main">
<s>Vniuer&longs;i orbis geographica & Hi&longs;torica de&longs;criptio con­<lb/>texta ex &longs;eptingentis & eo amplius &longs;criptoribus. </s>
</p>
<p type="main">
<s>Fede ici Vrbini Ducis Vita. </s>
</p>
<p type="main">
<s>Guidi Vbaldi Vibini Ducis Vita. </s>
</p>
<p type="main">
<s>Epigrammaton & Odarum libritres. </s>
</p>
<p type="main">
<s>Aliorum Carminum liber. </s>
</p>
<p type="main">
<s>Sententiarum moralium liber. </s>
</p>
<p type="main">
<s>Dictionarium Arabicum. </s>
</p>
<p type="main">
<s>Pro Procopio contra Flauium Blondum. </s>
</p>
<p type="main">
<s>Horographium vniucr&longs;ale. </s>
</p>
<p type="main">
<s>Epigrammata alia. </s>
</p>
<p type="main">
<s>Heronis lib. de Balli&longs;tis conuer&longs;io. </s>
</p>
<p type="main">
<s>Exercitationes in Ari&longs;totelis Mechan. </s>
</p>
<p type="main">
<s>Templi Ezechielis noua de&longs;criptio. </s>
</p>
<p type="main">
<s>Antiquitatum Gua&longs;tallen&longs;ium liber. </s>
</p>
<p type="main">
<s>Hi&longs;toriæ &longs;cribendæ leges. </s>
</p>
<p type="main">
<s>Etalia quædam. </s>
</p>
<pb pagenum="1"/>
<figure></figure>
<p type="head">
<s>IN MECHANICA ARISTOTE­<lb/>LIS PROBLEMATA</s>
</p>
<p type="head">
<s>EXER CITATIONES.</s>
</p>
<p type="head">
<s><emph type="italics"/>Mechanices de&longs;criptio, natura, finis.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>MECHANICE, facultas quædam e&longs;t, quæ <lb/>naturalimateriâ, Geometricisque; demon­<lb/>&longs;trationibus v&longs;a, ex centrobaricâ, & <expan abbr="eorū">eorum</expan> <lb/>quæ ad vectem & libram rediguntur, &longs;pe­<lb/>culatione; humanæ con&longs;ulens nece&longs;&longs;itati, <lb/>commoditatiqueue, &longs;uapte vi, Naturam i­<lb/>p&longs;am vel &longs;ecundans, vel &longs;uperans, varia, caqueue mirabilia <lb/>operatur. Hac diffinitione de&longs;criptionéue brcuiter ca fe­<lb/>rè omnia complexi &longs;umus, quæ fu&longs;i&longs;&longs;imè ab Ari&longs;totele, <lb/>Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. </s>
</p>
<p type="head">
<s><emph type="italics"/>Mechanices Obiectum.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Con&longs;ideratautem Mechanicus Graue & Leue. </s>
</p>
<p type="main">
<s>Graue duplex, Naturâ, Violentiâ. </s>
</p>
<p type="main">
<s>Graue Naturâ dicitur, quod in&longs;ita propen&longs;ione in <lb/>centrum mundifertnr. Graue autem Violentiâ, quod im­<lb/>pre&longs;&longs;o extrin&longs;ecus pondere ab impellente pellitur. </s>
</p>
<p type="main">
<s>Leue contrà, quòd Naturâ à centro fertur. </s>
</p>
<p type="main">
<s>Gæterùm quicquid graue e&longs;t, &longs;ecundum punctum <lb/>e&longs;t, quod Grauitatis centrum dicitur, & hoc duplex, vt <lb/>duplex e&longs;t grauitas, Naturæ, Violentiæ. </s>
</p>
<pb pagenum="2"/>
<p type="main">
<s>Grauitatis centrum in triplici magnitudine con&longs;i­<lb/>deraripote&longs;t, lineari, planà, &longs;olidâ. </s>
</p>
<p type="main">
<s>De centro grauitatis linearum nemo &longs;crip&longs;it, &longs;impli­<lb/>ci&longs;&longs;imi enim illud e&longs;t contemplationis. </s>
</p>
<p type="main">
<s>De centro grauitatis linearum egregiè tractauit Ar­<lb/>chimedes in libro Æ queponderantium, & de quadratu­<lb/>ra Parabole, tum in co quem de his quæ vehuntur in­<lb/>&longs;crip&longs;it. </s>
</p>
<p type="main">
<s>De centro grauitatis &longs;olidorum ipíemet olim &longs;cri­<lb/>p&longs;erat Archimedes, &longs;ed ea quæ protulit, temporis iniuriâ <lb/>deperdita, fuâ diligentiâ re&longs;tituit Iedericus Commandi­<lb/>nus. </s>
</p>
<p type="main">
<s>E&longs;&longs;e autem & Leuitatis centrum in rerum natura, <lb/>palam e&longs;t. Punctum enim illud e&longs;t, &longs;ecundum quod lcuia <lb/>rectà à centro &longs;ur&longs;um feruntur. Huius autem non memi­<lb/>nêre Mechanici, propterea quod aut nihil, aut parum ad <lb/>eorum rem faciat. </s>
</p>
<p type="main">
<s>Porro Grauitatis centrum ita definit Heron, & qui <lb/>ab Herone Pappus 1.8. Collectionum Mathematicarum. </s>
</p>
<p type="main">
<s>Centrum grauitatis <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> corporis e&longs;t pun­<lb/>ctum quod dam intra po&longs;itum, à quo &longs;i graue, mente ap­<lb/>pen&longs;um concipiatur, dum fertur, quie&longs;cit, & &longs;eruat eam <lb/>quam in prin cipio habuit po&longs;itionem; neque in ip&longs;a latio­<lb/>ne circumuertitur. Commandinus verò in lib. de centro <lb/>grauitatis &longs;olidorum hoc pacto: Centrum grauitatis v­<lb/>niu&longs;cuiu&longs;que &longs;olidæ figuræ, e&longs;t punctum illud intra po&longs;i­<lb/>tum, circa quod vndique partes æqualium momentorum <lb/>ad&longs;i&longs;tunt. Sienim per tale centrum ducatur planum, fi­<lb/>guram quomodolibet &longs;ecans, in partes æ què ponderantes <lb/>eam diuidit. Nos verò quàm breui&longs;&longs;imè dicimus: <expan abbr="Centrū">Centrum</expan> <lb/>grauitatis, <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> magnitudinis punctum e&longs;&longs;e intra <lb/>extraue magnitudinem po&longs;itum, per quod &longs;i plano linea <lb/>punctoue diuidatur, in partes &longs;ecatur æqueponderantes. </s>
</p>
<pb pagenum="3"/>
<figure></figure>
<p type="main">
<s>Diximus, Magnitudinis vtlineæ, plani &longs;olidique; cen­<lb/>trum complecteremur. Eritigitur, vt in præ&longs;enti figura, li­<lb/>neæ quidem centrum A, plani B, &longs;olidi verò C. quod &longs;i ob­<lb/>ijciat qui&longs;piam, lineam & &longs;uperficiem nullam habere gra­<lb/>uitatem; is &longs;ciat, <expan abbr="neq;">neque</expan> corpora Mathematica grauitatem <lb/>habere, Mechanicum verò funes, ha&longs;tas, vectes pro lineis <lb/>&longs;umere; tabulas verò, & eiu&longs;modi plana ad &longs;uperficierum <lb/>naturam referre. </s>
</p>
<p type="main">
<s>Diximus in&longs;uper, intra extraue. Aliquando enim <lb/>grauitatis centrum extra molem corporis cuius corporis <lb/>centrum e&longs;t, cadit, vtin &longs;equenti figura. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to corpus aliquod <lb/>&longs;uperficiesue ABCDE, <lb/>ducatur linea CF, <expan abbr="diuidēs">diuidens</expan> <lb/>figuras in partes hinc inde <lb/>æqueponderantes ABC, <lb/>EDC. Ducatur & GH. <lb/>diuídens item in partes æ­<lb/>queponderantes GCH, & GAB, EDH. &longs;ecentautem <lb/>&longs;eip&longs;as in I. eritigitur centrum I extra figuræ terminos & <lb/>molemip&longs;am. Attamen licet hoc verum &longs;it, intra e&longs;&longs;e dici <lb/>pote&longs;t, quippe quod imaginario quodam, & vtita dicam, <lb/>virtuali ambitu ACDA contineatur. </s>
</p>
<p type="main">
<s>Dicebamus, duplex e&longs;&longs;e grauitatis centrum, Natu­
<pb pagenum="4"/>râ, Violentià: a&longs;firmamus modò, hæc re quidem vnum e&longs;­<lb/>&longs;e, & ratione &longs;olum, non autem reip&longs;a ac&longs;i duo e&longs;&longs;ent con­<lb/>&longs;iderari. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim grauitatis na­<lb/>turalis centrum B, corporis A, <lb/>&longs;ecundum quod dimi&longs;&longs;um, &longs;ua­<lb/>pte naturâ cadet in C, &longs;i verò <lb/>corpus violenter impellatur in <lb/>D, aiiud acquiret centrum gra­<lb/>uitatis ex violentia &longs;ecundum <lb/>quam fertur, motum, in D, <expan abbr="idē">idem</expan> <lb/>autem &longs;untre, nempe vnum B, <lb/>duo autem &longs;i violentia & natura &longs;eor&longs;um con&longs;ideren­<lb/>tur. </s>
</p>
<p type="main">
<s>Hæc centra, duo motus &longs;equuntur, rectus vterque, <lb/>Naturalis videlicet, & Violentus. Tertius ex his mixtus, & <lb/>is quidem non rectus, &longs;ed curuus. </s>
</p>
<figure></figure>
<p type="main">
<s>Proijciatur enim violen­<lb/>ter corpus graue A &longs;uperante <lb/>igitur violentia, rectà feretur <lb/>in B; ea autem elangue&longs;cente <lb/>paullatim per curuam & mi­<lb/>xtam <expan abbr="lineã">lineam</expan> &longs;ecetur in C, qua­<lb/>tenus enim ad anteriora fer­<lb/>tur, violentia e&longs;t; quatenus ve­<lb/>rò ad inferiores partes, naturæ. Vbi verò peruenit in C, <lb/>violentiâ ce&longs;&longs;ante, naturâ verò manente, rectà deor&longs;um <lb/>fertur DCD. </s>
</p>
<p type="main">
<s>Cæteiùm hæc centra, hiqueue motus, naturalis nem­<lb/>pe, & violentus diuer&longs;imode &longs;e habent adinuicem. Sie­<lb/>nim graue corpus externâ vi adhibita, centrum mundi <lb/>ver&longs;us impellatur, adiuuabunt &longs;e inuicem Natura, Vio­<lb/>lentia, Si autem contra, altera alteri re&longs;i&longs;ter, in motibus
<pb pagenum="5"/>autem ad latus, eo magis pugnabunt, quo magis ab infe­<lb/>rioribus ad &longs;uperiora fiet motus. </s>
</p>
<p type="head">
<s><emph type="italics"/>Mechanices præcipua in&longs;trumenta.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Hic ira con&longs;titutis dicimus, in&longs;trumenta, quibusad <lb/>varias operationes Mechanici vtuntur, e&longs;&longs;e inter &longs;e qui­<lb/>dem diuer&longs;a, multiplicia, & &longs;i varietatem &longs;pectes, penè in­<lb/>numerabilia; quod quamuis verum&longs;it, ea omnia Ari&longs;tote­<lb/>les ad vectem re ducit, & libram: quod etiam G. Vbaldus <lb/>in libris Mechanicoiumfecit. Cæterum qui po&longs;t Ari&longs;to­<lb/>telem floruere Mechanici, omnia ad quinque, quas ap­<lb/>pellant, Potentias, redegêre. Sunt autem ex Herone, Pap­<lb/>po, Guido Vbaldo, qui eos &longs;ecutus e&longs;t, Vectis, Trochlea, <lb/>Axis in Peritrochio, Cuneus, Cochlèa. Videtur autemi­<lb/>p&longs;e G. Vbaldus &longs;extam addere, nempe Libram, de qua & <lb/>primus ip&longs;e Mechanicorum tractatum in &longs;tituit. Verum <lb/>enimuero idem ferè &longs;unt Vectis & Libra, ni&longs;i forte quod <lb/>Libra tunc dicitur, cum brachia &longs;unt æ qualia. Vectis vero <lb/>quomodocun que ea &longs;e habeant; quinque harum <expan abbr="Poten-tiarū">Poten­<lb/>tiarum</expan> imagines ita ob oculos ponimus. Vectis A. Trochlea <lb/>B, Axisin Peritrochio C. Cuneus D. Cochlea vero E. </s>
</p>
<pb pagenum="6"/>
<figure></figure>
<p type="main">
<s>Porro, Cuneum ad libram reducere conatur Ari­<lb/>&longs;toteles, quod facit & G. Vbaldus, qui cò refert & Co­<lb/>chleam, quippe quod nihil aliud &longs;it Cochlea, quàm Cu­<lb/>neus Cylindro inuolutus. Nos autem duas tantùm Po­<lb/>tentias ad vectem reduci po&longs;&longs;e arbitramur, Trochleam <lb/>nempe, & Axem in Peritrochio. Nequaquam autem Cu­<lb/>neum & Cochleam. quod latiùs quidem o&longs;tendemus, <lb/>cùm de Cuneo eritnobis &longs;ermo peculiaris. </s>
</p>
<p type="head">
<s><emph type="italics"/>De Vecte & Libra &longs;ecundum Ari­<lb/>&longs;totelem.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ari&longs;toteles in ip&longs;o Mechanicorum ingre&longs;&longs;uita &longs;cri­<lb/>bit, Mirum videri ab exigua virtute magnum pondus mo-
<pb pagenum="7"/>ueri, addito nimirum ponderi pondere, &longs;iquidem & vectis <lb/>e&longs;t pondus. Duplex ergo illi admiratio, &longs;cilicet quòd exi­<lb/>gua potentia moucat ingens pondus, idqueue etiam addito <lb/>vectis ip&longs;ius pondere, fiat. Hoc &longs;ecundum adieci&longs;&longs;e vide­<lb/>tur, amplificationis alicuius gratiâ. Erenim quatenus <lb/>ad rem pertinet, &longs;i mouendis ponderibus vectis ip&longs;ius <lb/>pondus compares, nullius ferè e&longs;&longs;e momenti proculdu­<lb/>bio affirmaueris. Sed & illud quoque notandum, aliquan­<lb/>do vectis pondus mouenti auxilium ferre, quod fit vbi <lb/>fulcimento inter potentiam mouentem, & pondus ip&longs;um <lb/>collocato, vectis pars quæ à fulcimento ad potentiam e&longs;t, <lb/>premitur. Tunc enim, vt dicebamus, vectis pondere &longs;uo <lb/>potentiam adiuuat. Contra verò accidit, cum pondus i­<lb/>p&longs;um inter fulcimentum e&longs;t & potentiam vel potentia i­<lb/>p&longs;a inter fulcimentum & pondus. tunc cnim vectis vnâ <lb/>cum pondere attollitur. quæ licet vera &longs;int, non tamen in­<lb/>de &longs;e quitur, vectis pondus, quicquam quod curandum &longs;it, <lb/>in operatione efficere, aut impedire. </s>
</p>
<p type="main">
<s>Porrò vectem ita finire po&longs;&longs;umus, longitudinem e&longs;­<lb/>&longs;e quandam inflexibilem, quæ fulcimento dato, datâ po­<lb/>tentiâ datum pondus mouetur. </s>
</p>
<p type="main">
<s>Ip&longs;a quoque Libra, vt diximus, vectis e&longs;t: eius autem <lb/>naturæ, vt&longs;emper fulcimentum medium obtineat locum <lb/>inter pondus & pondus. Statera autem merus e&longs;t vectis, &longs;i <lb/>&longs;par&longs;um pro fulcimento; appendiculum verò currens pro <lb/>potentia mouente deputaueris. </s>
</p>
<p type="head">
<s><emph type="italics"/>De Circulo eiusque natura Ari&longs;totelis doctri­<lb/>na examinata.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ari&longs;toteles, quicquid mirum in Mechanicis opera­<lb/>tur, id totum admirabili circuli naturæ e&longs;&longs;e tribuen dum <lb/>arbitratur. Aitautem, ab&longs;urdum nullatenus e&longs;&longs;e, &longs;i ex re <lb/>mirabili mirandum quippiam oriatur. In circulo autem
<pb pagenum="8"/>quatuorinueniri qualitates admiratione dignas. <expan abbr="Primã">Primam</expan>, <lb/>quod ex contrarijs con&longs;tituatur, mouente videlicet & <lb/>moto. Secundam, quòd contraria in eius circumferentia <lb/>inueniantur, quippe quæ cum vnica linea &longs;it, concaua &longs;i­<lb/>mul e&longs;t & conuexa. Tettiam, quod contrarijs feratur mo­<lb/>tionibus, antror&longs;um nimirum, retror&longs;um, &longs;ur&longs;um, atque <lb/>deor&longs;um. Quartam, quod vnicâ exi&longs;tente &longs;emidiametro, <lb/>nullum in ca punctum &longs;umi po&longs;&longs;it, æqualis alteri, in latio­<lb/>ne, velocitatis. Sit enim circulus AB, cuius centrum C, <lb/>&longs;emidiameter AC, &longs;umatur autem in ea punctum D, i­<lb/>temqueue punctum E. Erit itaque in ip&longs;a circulatione D <lb/>tardius E, ip&longs;um verò E tardius A, & ita citius id feretur <lb/>&longs;emper, quod remotius à mouente termino accipitur. </s>
</p>
<figure></figure>
<p type="main">
<s>Hæc ex illo, quibus ne vltro a&longs;­<lb/>&longs;en&longs;um præbeamus non vnica de cau­<lb/>&longs;a cohibemur. Dicimus igitur, videri <lb/>nobis, circulum non ex contrarijs <expan abbr="cō-&longs;titui">con­<lb/>&longs;titui</expan>, puta ex manente & moto, &longs;ed ex <lb/>moto &longs;impliciter. Nulla e&longs;t enim &longs;e­<lb/>midiametri pars, quæ non moueatur. <lb/>Punctum autem, quod &longs;tat, &longs;emidia­<lb/>metri pars nulla e&longs;t. Et &longs;anè cur moto <lb/><expan abbr="&longs;emidiamētro">&longs;emidiamentro</expan> fiat circulus, non ideo accidit, quod <expan abbr="alterū">alterum</expan> <lb/>extremum &longs;tet, alterum verò moueatur:led ideo quòd &longs;e­<lb/>midiameter perpetuò candem &longs;eruct longitudinem. Elli­<lb/>p&longs;is &longs;anè centrum habet, &longs;ed ab eo ad circumferentiam <lb/>quatuor tantùm &longs;emidiametri quomodolibet &longs;umpti du­<lb/>cuntur æ quales. Si quis igitur &longs;emidiametrum daret pro­<lb/>portione cre&longs;centem & decre&longs;centem, &longs;tante altero ex­<lb/>tremoru<gap/> Ellip&longs;is de&longs;criberetur. Præterea & &longs;piralis li­<lb/>nea, quæ mixta e&longs;t, altero &longs;emidiametri extremo manen­<lb/>te, altero vero moto producitur. Legem itaque circulo
<pb pagenum="9"/>prælcribit, non quidem quòd hæc extremitas &longs;ter, illa ve­<lb/>rò moueatur, &longs;ed quod &longs;ua circulatione &longs;em per &longs;emidia­<lb/>meter eandem &longs;eruet longitudinem, quod vel ex ip&longs;a cir­<lb/>culi definitionc colligitur. </s>
</p>
<p type="main">
<s>Ad &longs;ecundum miraculum, &longs;cilicet, quòd in circulo <lb/>circum ferentia, quæ vacua linea e&longs;t, concaua &longs;imul&longs;it, & <lb/>conuexa. Diceret qui&longs;piam id, &longs;i modò mirabile e&longs;t non <lb/>circulari tantum, &longs;ed cui ibet curuæ lineæ primo compe­<lb/>tere, etenim & Elhp&longs;is & Hyperbole, & Parabolc, & &longs;pi­<lb/>ra, tum Cy&longs;&longs;ois, Conchois, & infinitæ aliæ irregulares <lb/>concauæ &longs;imul &longs;unt & conuexæ. Sed & hæcin &longs;uperficie­<lb/>bus quoque de&longs;iderantur. </s>
</p>
<p type="main">
<s>Ad tertium, quod contrarijs feratur lationibus, an­<lb/>tror&longs;um, retror&longs;um, &longs;ur&longs;um & deor&longs;um. Dicimus, facilè <lb/>&longs;olui, Nullus enim, re bene per&longs;pectâ, affirmauerit circu­<lb/>lum contrarijs lationibus moueri. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim circulus ABCD, <lb/>circa centrum E; ponamus ro­<lb/>tari, & A ver&longs;us B, exempli gra­<lb/>tiâ, antror&longs;um, mouebitur <expan abbr="autē">autem</expan> <lb/>& B ver&longs;us C, & C ver&longs;us D, tum <lb/>D ver&longs;us A. Non puto <expan abbr="quenquã">quenquam</expan> <lb/>dicturum, circulum hunc an­<lb/>tror&longs;um codem tempore, & re­<lb/>tror&longs;um ferri nec &longs;ur&longs;um aut de­<lb/>or&longs;um, &longs;i enim qui&longs;piam per eius circuli circumferentiani <lb/>ambularet, is certè centrum ip&longs;um &longs;emper ad dexteram <lb/>haberet, vel ad &longs;ini&longs;tram, &longs;i ad dexteram, antror&longs;um ibit, &longs;i <lb/>ad &longs;ini&longs;tram, tetror&longs;um. Sed nec &longs;ur&longs;um vel deor&longs;um, e&longs;t <lb/>manife&longs;tum. Nihil autem prohibet eundem motum va­<lb/>rio re&longs;pectu contrarium dici po&longs;&longs;e, id tamen profectò fie­<lb/>rinequaquam pote&longs;t, nempe A moueriver&longs;us B, hoc e&longs;t,
<pb pagenum="10"/>antro r&longs;um, & eandem codem tempore ver&longs;us B, id e&longs;t, re­<lb/>tror&longs;um; repugnat enim naturæ. </s>
</p>
<p type="main">
<s>De quarto circuli miraculo, ibi erit nobis &longs;ermo, vbi <lb/>ca perpenderimus primò, quæ Philo&longs;ophus de Circuli <lb/>productione di&longs;&longs;erens in medium profert. Sunt autem e­<lb/>iu&longs;modi: </s>
</p>
<p type="main">
<s>Circulum quidem duplici notione produci, Natu­<lb/>rali videlicet altera, & altera quæ e&longs;t præter naturam, & <lb/>ideo circularem lineam in ter mixtas computari. </s>
</p>
<p type="main">
<s>Motus mixtus ait, vel proportione &longs;eruata fit, aut <lb/>non; Si proportione &longs;eruatâ, rectam lineam; ea verò non <lb/>&longs;eruata, circularem lineam produci. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim rectangu­<lb/>lum ABCD, cuius late­<lb/>ra in datâ &longs;int proportio­<lb/>ne, AD cum AB. Mo­<lb/>ueatur A, duplici motu, <lb/>Altero quidem tendens <lb/>in B, altero vero ad mo­<lb/>tum lineæ AB, feratur <lb/>ver&longs;us D, &longs;eruata inte­<lb/>rim laterum proportione. Itaque ponatur ex motu ab A <lb/>ver&longs;us B, perueni&longs;&longs;e in E, ex motu autem quo proportio­<lb/>naliter fertur cum linea AB, facta ip&longs;a AB, in FH, perue­<lb/>ni&longs;&longs;e in G, & EG connectatur. Eritigitur Parallelogram­<lb/>mum AEGF, Parallelogrammo ABCD proportiona­<lb/>le &longs;imile, & circa eandem diametrum AGC. Semperigi­<lb/>tur punctum A &longs;i duabus lationibus feratur, laterum pro­<lb/>portione &longs;eruata, lineam producet rectam, diametrum <lb/>nempe AGC. Et hoc &longs;anè nullam habet dubitationem, <lb/>ex ijs quæ docet Euclides 1. 6. prop. 24. </s>
</p>
<p type="main">
<s>His ita demon&longs;tratis hac vti videtur Philo&longs;ophus
<pb pagenum="11"/>argumentatione: Si mixtus motus proportione &longs;emotâ, <lb/>rectam producir, &longs;i nun quam &longs;emota, efficiet circulum; &longs;i <lb/>enim modo &longs;eruaretur, modo non, partim recta partim <lb/>non recta produceretur. Ingenio&longs;a quidem argumenta­<lb/>tio, ni vitium contineret. non enim mixtus motus, qui <lb/>nun quam &longs;eruatâ proportione fit, &longs;emper ci, culum pro­<lb/>ducit, &longs;ed & Elli &longs;im pote&longs;t, & quamlibet aliam lineam, <lb/>cuius nulla pars &longs;it recta. Hanc difficultatem vidit Pico­<lb/>lomineus in &longs;ua Paraphra&longs;i, & eam &longs;oluere conatus e&longs;t, <lb/>&longs;ed quàm bene, aliorum e&longs;to <gap/>dicium. Cæterùm fal&longs;um <lb/>e&longs;t, a&longs;&longs;erere circulum ex mixto motu nun quam &longs;eruatâ <lb/>proportione produci. &longs;eruat enim a&longs;&longs;iduè mixtus motus <lb/>quo producitur (&longs;i cum mixto motu producere velimus) <lb/>aliquam proportionem, &longs;ed non eandem. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim recta AB, cui ad rectos <lb/>angulos AC. Moueatur autem A, ver­<lb/>&longs;us C per lineam AC, & eodem tempo­<lb/>re linea AC, ver&longs;us B, ita tamen, vt &longs;em­<lb/>per ip&longs;i AB, &longs;it perpendicularis. feratur <lb/>autem eâ lege, vt quam proportionem <lb/>habet motus lineæ AC ver&longs;us B, ad mo­<lb/>tum puncti A ve, &longs;us C, eandem habeat <lb/>ip&longs;e motus ab A ver&longs;us C, ad re&longs;iduum <lb/>lineæ AB, demptâ nempe ea parte quam <lb/>peragrauit linea AC mota ver&longs;us B. Sit <lb/>autem, cum AC &longs;uo motu peruenerit <lb/>in D, punctum A, &longs;imiliter &longs;uo motu per eam latum perue­<lb/>nitle in E erit eigo ex mixto motu, non quidem in D, nec <lb/>in E, &longs;ed in F, eritque punctum F in circum ferentia circu­<lb/>li, cuius e&longs;t diameter ip&longs;a linea AB, quod quidem demon­<lb/>&longs;tratur ex conuer&longs;a propo&longs;. 13. lib. 6. Elem. E&longs;t enim AE <lb/>hoce&longs;t DF media proportionalis inter EF, hoc e&longs;t, AD, <lb/>& DB. Iterum &longs;i &longs;iat motus AC in GH, ad motum H per
<pb pagenum="12"/>lineam AC, v&longs;que in C, vt &longs;e habet proportio AG ad <lb/>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb/>eiu&longs;dem circuli circum ferentia AFHB. ex quibus ha­<lb/>bemus, circulum ex mixto motu fieri po&longs;&longs;e proportioni­<lb/>bus quidem mediarum &longs;eruatis, &longs;ed nun quam ij&longs;dem. </s>
</p>
<p type="main">
<s>Vera hæc pro culdubio &longs;unt; nihilominus, veluti ad <lb/>rectam producendam mixtus motus non e&longs;t nece&longs;&longs;arius, <lb/>licet mixto motu produci po&longs;&longs;it, ita ne que ad circularem, <lb/>& ideo verum non e&longs;&longs;e quod a&longs;&longs;erebat Philo&longs;ophus, cir­<lb/>culum ex mixto motu proportione nun quam &longs;eruatâ ne­<lb/>ce&longs;&longs;ariò produci. </s>
</p>
<p type="main">
<s>Conatur po&longs;t hæc Ari&longs;toteles rationem afferre, cur <lb/>circuli partes, quò propiores centro fuerint, eo &longs;int tar­<lb/>diores. Ait autem; &longs;i duobus ab eadem potentia latis hoc <lb/>quidem plus repellatur, illud verò minus, æquum e&longs;t tar­<lb/>diùs id moueri quod plus repellitur, eo quod minus. De­<lb/>trahi autem plus lineam, cuius extremum prepius e&longs;t cen­<lb/>tro illa quæ &longs;uum habet terminum à centro remotiorem. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to, inquit, circulus <lb/>BCDE & alter in eo minor <lb/>MNOP circa idem centrum <lb/>A. Ducanturque; Diametrima­<lb/>ioris quidem CD, EB, mino­<lb/>ris verò MO, NP. Itaque vbi <lb/>AB circulata eò peruenerit <lb/>vnde e&longs;t gre&longs;&longs;a, ip&longs;a quoque <lb/>AM eo vnde moueri cœpe­<lb/>rat, perueniet. Tardiùs antem <lb/>fertur AM, quam AD, pro­<lb/>pterea quòd AM à centro <lb/>magis retrahatur quàm ip&longs;a AB. Ducatur igitur ALF & <lb/>à puncto L, ip&longs;i AB perpendicularis L q, cadens in mino-
<pb pagenum="13"/>ri circulo, & rur&longs;us ab codem L ip&longs;i AB, parailela duca­<lb/>tur LS, Ab S verò eidem perpendicularis ST, & ab F i­<lb/>tem FX. Sunt ergo q L, ST, quidem æquales, nempeillæ, <lb/>per qua<gap/>, &longs;ecundum naturam, mouentur puncta BM. Mo­<lb/>tu verò retractionis ad centrum, hoc e&longs;t, præter naturam, <lb/>plus motum e&longs;t M quàm B. Maior enim e&longs;t M q, ip&longs;a BT, <lb/>quod, ceu notum, &longs;uppo&longs;uit Ari&longs;toteles. nos autem inf. à <lb/>demon&longs;trabimus. Si igitur fiat vt motus præter naturam <lb/>ad motum præter naturam, ita motus <expan abbr="&longs;ecūdum">&longs;ecundum</expan> naturam, <lb/>ad motum &longs;ecundum naturam, punctum B; cum M fuerit <lb/>in L, non eritin S, &longs;ed in F. tunc enim, vt e&longs;t FX motus &longs;e­<lb/>cundùm naturam ad XB, præter naturam, ita e&longs;t q L &longs;e­<lb/>cundum naturam ad q M præter naturam; &longs;ed BF maior <lb/>e&longs;t ML, ergo proportione &longs;eruatâ, velociùs mouetur B <lb/>quàm M circa idem centrum A. Hæc autem &longs;umma e&longs;t <lb/>eorum quæ præfert Ari&longs;toteles. Cæterùm nos parallelo­<lb/>grammum, quod in figura eius habetur prætermi&longs;imus, <lb/>quippe quod nihil ad eam quæ affertur, demon&longs;tratio­<lb/>nem faciat. </s>
</p>
<p type="main">
<s>Modò quod pollicebamur, nempe minorem e&longs;&longs;e <lb/>BT, quàm q M, ita demon&longs;tramus. <expan abbr="quoniã">quoniam</expan> ST. ex prop. 13. <lb/>1. 6. media proportionalis e&longs;t inter BT & TE, erit qua­<lb/>dratum TS æquale <expan abbr="parallelogrãmo">parallelogrammo</expan> &longs;eu rectangulo BT, <lb/>TE, item, quoniam q L media proportionalis e&longs;t inter <lb/>M q, & q O. erit quadratum q L æquale rectangulo M q, <lb/>q O, æqualia ergo &longs;unt rectangula BTE, M q O, itaque <lb/>reciprocalatera habent proportionalia. quare, vt TE, ad <lb/>q O, ita M q ad TB, &longs;ed TE maior e&longs;t ip&longs;a q O, quippe <lb/>quòd pars &longs;it q O ip&longs;ius TE, maior ergo & M q ip&longs;a TB, <lb/>quod o&longs;tendendum fuerat. </s>
</p>
<p type="main">
<s>Cæterùm &longs;ubtilia & ingenio&longs;a i&longs;thæc e&longs;&longs;e non nega­<lb/>mus, & longè faciliori & explicatiori modo veritas hæc <lb/>demon&longs;trari pote&longs;t, reiectis nem peillis, &longs;ecundùm, & prae­
<pb pagenum="14"/>ter naturam motibus, qui <expan abbr="quidē">quidem</expan> in &longs;implici circulo nece&longs;­<lb/>&longs;ario non cadunt: caderent autem forta&longs;&longs;e, &longs;i de circulo <lb/>res e&longs;&longs;et à <expan abbr="pōderibus">ponderibus</expan> circumlatis ex &longs;tabili centro de&longs;eri­<lb/>pto, qua de re agit G. Vbaldus in Mechanicis ttactatu de <lb/>libra. tunc enim dici pote&longs;t, pondus quod aliâs rectà ad <lb/>mundi centrum tenderet, à circuli centro in circulatio­<lb/>ne retrahi, &longs;ed hæc ad circuli naturam, quatenus circulus <lb/>e&longs;t, ne quaquam &longs;pectant. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur circum ferentia <lb/>AFBH, cuius centrum C, dia­<lb/>meter ACB, &longs;emidiameter AC. <lb/>&longs;umatur in AC punctum quod­<lb/>libet, D, & centro C, &longs;patio CD, <lb/>circumferentia de&longs;cribatur <lb/>DGEI. Dico punctum A velo­<lb/>cius moueri puncto D eâdem <lb/>circulatione rotato. etenim vt <lb/>diameter ad diametrum, & &longs;emidiameter ad &longs;emidiame­<lb/>trum, ita circumferentia ad circumferentiam: igitur vt <lb/>AC ad CD, ita circumferentia AFHB ad circumferen­<lb/>tiam DGEI. At mota linea CA circa centrum C mo­<lb/>uetur &longs;imul & CD, eodem igitur tempore rotationem <lb/>complent puncta AD, maius ergo &longs;patium eodem tem­<lb/>pore metitur A, ip&longs;a D, quare velocior. Ita igitur &longs;e ha­<lb/>bet velocitas ad velocitatem, vt circumferentia ad cir­<lb/>cumferentiam, & diameter ad diametrum, quare id quod <lb/>mouetur in puncto à centro remotiori, velocius illo mo­<lb/>uetur quod ab eo di&longs;tat minus, quod fuerat <lb/>demon&longs;trandum. </s>
</p>
<pb pagenum="15"/>
<p type="head">
<s>QVÆSTIONES <lb/>MECHANICÆ.</s>
</p>
<p type="head">
<s>QVÆSTIO I.</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur maiores libræ exactiores &longs;int mi­<lb/>noribus?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Prioríbus, ceu fundamentis quibu&longs;dam iactis, oppor­<lb/>tunè ad quæ&longs;tiones proponendas, eas queue diluendas &longs;e <lb/>confert Ari&longs;toteles. Porro in propo&longs;ita quæ&longs;tione vide­<lb/>tur prima fi onte cau&longs;&longs;am quæri de re quæ non e&longs;t: etenim <lb/>quis affirmauerit vnquam, lances quibus Apothecarij & <lb/>Macellarij vtuntur, magnas eas quidem, illis exactiores <lb/>e&longs;&longs;e quibus Gemmatij, atque Argentarij &longs;iliquis, & &longs;eru­<lb/>pulis minuti&longs;&longs;ima appendunt, quæ tamen perexiguæ &longs;unt, <lb/>& &longs;i illis comparentur minimæ? Veruntamen, ita pror&longs;us <lb/>res habet, vt a&longs;&longs;erit Ari&longs;toteles. Non enim propterea <lb/>quòd illæ magnæ &longs;int, hæ verò exiguæ, hæ &longs;unt illis exa­<lb/>ctiores; &longs;ed quoniam magnæ, rudes &longs;unt, minores verò ex­<lb/>qui&longs;ita diligentia elaboratæ, & à materiæ pertina cia libe­<lb/>riores. Cæteris ergo paribus, exactiores e&longs;&longs;e maiores, ex <lb/>Philo&longs;ophimente, ita docebimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to libra maior AB, <lb/>cuius fulcimentum C. <lb/>Minor verò libra DE, <lb/>circa idem <expan abbr="fulcimētum">fulcimentum</expan> <lb/>C, vnà cum maiori, ima­<lb/>ginatione, conuer&longs;a. Ap­<lb/>ponatur queduis pon­<lb/>dus maiori libræ in A, <lb/>de clinetque; exempli gratiâ in F, erit queue minor libra in G, <lb/>in eadem enim linea &longs;unt CGF. <expan abbr="Vnaq;">Vnaque</expan> igitur ex eodem
<pb pagenum="16"/>centro C portionem circuli de&longs;cribet GD, AF, eritqueue <lb/>ACF &longs;ector circuli, cuius diameter AB, &longs;ed DCG &longs;e­<lb/>ctor circuli, cuius diameter DE. Itaque vt diameter ad <lb/>diametrum, ita portio ad portionem: maior autem dia­<lb/>meter AB diametro DE: maior ergo portio AF, portio­<lb/>ne DG. quod autem maius e&longs;t, minus obtutum fallit, ex­<lb/>qui&longs;itius ita que tractum ex maiori AB quàm ex ip&longs;a mi­<lb/>nori DE cogno &longs;cemus, quod fuerat o&longs;tendendum. </s>
</p>
<p type="main">
<s>Cæterùm hac eadem de cau&longs;&longs;a, A&longs;tronomica in­<lb/>&longs;trumenta, puta A&longs;trolabia, Armillæ, & alia eiu&longs;modi, <lb/>quo ampliora eò ex qui&longs;itiora, & certiora probantur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim A­<lb/>&longs;trolabium magnum, <lb/>cuius diameter AB, <lb/>paruum autem CD, <lb/>circa idem centrum <lb/>E. Ducatur à centro <lb/>recta EF tangens ma­<lb/>iorem circulum in F, <lb/><expan abbr="minorē">minorem</expan> verò <expan abbr="&longs;ecãs">&longs;ecans</expan> in <lb/>G, vt igitur GD ad to­<lb/>tum circulum GCD, <lb/>ita FB. ad totum cir­<lb/>culum FAB, vt ergò <lb/>GD ad FB, ita gradus <lb/>&longs;ignati in GD, ad eos qui &longs;ignantur in BF, maiores ergo <lb/>&longs;unt qui in FB, & minutarum partium capaciores. Hinc <lb/>itaque apparet, <expan abbr="in&longs;trumēta">in&longs;trumenta</expan> quælibet quò maiora fuerint, <lb/>eò e&longs;&longs;e & exqui&longs;itiora, quod propo&longs;uerat Ari&longs;toteles, in <lb/>hac quæ&longs;tione de Libra. </s>
</p>
<p type="main">
<s>Quod autem addit de fraudibus Purpurariorum, <lb/>inquiens; quamobrem machinánturij qui purpuram ven­<lb/>dunt, vt <expan abbr="pēdendo">pendendo</expan> defraudent, dum ad medium, &longs;partum,
<pb pagenum="17"/>non ponentes; tum plumbum in alterutram libræ partem <lb/>infundentes; aut ligni quod ad radicem vergebat, in eam <lb/>quam deferri volunt partem con&longs;tituentes, aut &longs;i nodum <lb/>habucrit, ligni enim grauior ea e&longs;t pars, in qua e&longs;t radix, <lb/>nodus verò radix quæ dam e&longs;t. Hinc quæri po&longs;&longs;et: </s>
</p>
<p type="head">
<s><emph type="italics"/>Vtrum libræ quæ ponderibus vacuæ æquilibrant, <lb/>omni pror&longs;us careant fraude?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Videri cuipiam po&longs;&longs;et, libras, quæ ponderibus va­<lb/>cuæ, æquilibrant, omm pror&longs;us fraude carere, verunta­<lb/>men ita non e&longs;t, quod diligentiùs (res enim magni mo­<lb/>menti e&longs;t) di&longs;quiremus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim libra AB, ita diui&longs;a <lb/>in C, vt AC &longs;it partium IS, CB ve­<lb/>rò carundem &longs;it 10. apponatur parti <lb/>A lanx ponderans 10, parti vero B <lb/>lanx ponderans 15. ex permutata i­<lb/>gitur proportione libra &longs;u&longs;pen&longs;a in <lb/>C, aequè ponderabit; &longs;i autem appo­<lb/>natur lanci B &longs;acoma vnciarum 6, & in lance A con&longs;titua­<lb/>tur purpura, quæ ita &longs;e habeat ad vncias 6, vt 10 ad 15, ite­<lb/>rum æqueponderabit, &longs;ed vt 10 ad 15, ita 4 ad 6. Purpura­<lb/>rius ergo fraudulentus, ponens in lance A vncias purpuræ <lb/>4, facto æquilibrio petet pretium vnciarum 6, & ita em­<lb/>ptorem decipiet, quod &longs;anè innuerat, non autem demon­<lb/>&longs;trauerat Ari&longs;toteles. Hæc autem faciliora fient ex ijs, <lb/>quæ in &longs;equentibus quæ &longs;tionibus, vbi de vecte agetur, ex­<lb/>plicabuntur. </s>
</p>
<p type="main">
<s>Detegitur autem fraus, &longs;i alternatim &longs;acoma in pon­<lb/>derando, modo huic, modò illi lanci apponatur. Si enim <lb/>in lance A con&longs;tituatur &longs;acoma, in B verò purpura non fit <lb/>æquilibrium. </s>
</p>
<pb pagenum="18"/>
<p type="head">
<s>QVÆSTIO II.</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur, &longs;i &longs;ur &longs;um libræ fulcimentum &longs;it, appo&longs;ito ad alteram partem <lb/>pondere, de &longs;cendat libra, & eo amoto, iterum a&longs;cendat, & ad æqui­<lb/>librium reuertatur. Si verò deor &longs;um fulcimentum fuerit, de­<lb/>pre&longs;&longs;a ad æquilibrium non reuertatur?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Bimembrem proponit Philo&longs;ophus quæ&longs;tionem, quam <lb/>trimembrem debuit, triplici &longs;i quidem loco fulcimen­<lb/>tum aptari pote&longs;t, &longs;uperiori, medio, inferiori. Nos de o­<lb/>mnibus ver ba faciemus. </s>
</p>
<p type="head">
<s>Prima Quæ&longs;tionis pars.</s>
</p>
<p type="head">
<s><emph type="italics"/>De Libra &longs;ur &longs;um fulcimentum habente.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ari&longs;toteles primam quæ&longs;tionis partem ita &longs;oluit: An <lb/>quia &longs;ur&longs;um parte quidem exi&longs;tente, plus libræ extra per­<lb/>pend culum &longs;it? Spartum enim perpendiculum e&longs;t: quare <lb/>nece&longs;&longs;<gap/> e&longs;t deor&longs;um ferriid quod plus e&longs;t, donec a&longs;cendat <lb/>qua bifariam libram diuidit ad ip&longs;um perpendiculum, <lb/>cum onus in cum bat ad libræ partem &longs;ur&longs;us raptam. </s>
</p>
<figure></figure>
<p type="main">
<s>Sit libra recta (hoc e&longs;t, in æquilibrio con&longs;tituta) BC, <lb/>&longs;partum autem AD, <lb/>fulcimentum autem <lb/>D, de&longs;uper: &longs;parto au­<lb/>tem deor&longs;um proie­<lb/>cto ad M perpendicu­<lb/>laris erit vbi ADM. <lb/>Si igitur in ip&longs;o B po­<lb/>natur onus, erit B qui­<lb/>dem vbi E, C autem <lb/>vbi H, quamobrem <lb/>ea quæ bifariam <expan abbr="librã">libram</expan> <lb/>&longs;ecat, primo quidem erit DM, ip&longs;ius perpendiculi; in <expan abbr="cū-bente">cun­<lb/>bente</expan> <expan abbr="autē">autem</expan> onere, erit DG. quare libræ ip&longs;ius EH, quod
<pb pagenum="19"/>extra perpendiculum, e&longs;t AM, vbi e&longs;t q P maius e&longs;t dimi­<lb/>dio. Si igitur amoueatur onus ab E, nece&longs;&longs;e e&longs;t deor&longs;um <lb/>ferri H, minus e&longs;t enim E: &longs;iquidem igitur habuerit &longs;par­<lb/>tum &longs;ur&longs;um, propter hoc a&longs;cendit libra. </s>
</p>
<p type="main">
<s>Pe&longs;&longs;imè omnes &longs;chema hoc lineârunt, ita vt difficil­<lb/>limum &longs;it auctoris inde &longs;en&longs;um a&longs;&longs;equi. Nos autem cla­<lb/>rius rem ob oculos ponimus. Id ergo &longs;ibi vult Ari&longs;toteles, <lb/>propterea quòd pars iugi HDG maior e&longs;t parte ED q, <lb/>eam eleuatam nece&longs;&longs;e e&longs;t de&longs;cendere, & iterum à perpen­<lb/>diculari ADM bifariam diui&longs;am ad æquilibrium reuer­<lb/>ti, Po&longs;&longs;umus nos idem &longs;impliciori figura demon&longs;trare. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to libra AB, bi­<lb/>fariam, diui&longs;a in G, <lb/><expan abbr="fulcimentū">fulcimentum</expan> verò &longs;ur­<lb/>&longs;um vbi D, prod<gap/>ca­<lb/>tur perpendicularis <lb/>DC in E. Stante igi­<lb/>tur libra AB, in æqui­<lb/>librio æqualis e&longs;t pars <lb/>CH, ip&longs;i parti CB <lb/>apponatur pondus in <lb/>B. Declinabit igitur <lb/>libra mota circa centrum D, fiat autem in FG, &longs;ecetqueue <lb/>perpendicularem in I. Punctum vero C eodem motu cir­<lb/>ca idem centrum D erit in H. amoueatur pondus appo&longs;i­<lb/>tum: Dico libram à &longs;itu FG declinaturam & iterum re­<lb/>uer&longs;uram in &longs;itum pri&longs;tinum ACB. quoniam enim parti <lb/>GH, quæ æqualis e&longs;t parti HF, additur pars IH, quæ à <lb/>perpendiculari e&longs;t v&longs;que ad H, ip&longs;i verò HF eadem pars <lb/>detrahitur, erit IF minor GI. Superabiturita que IF à <lb/>GI, de&longs;cendetque FI, a&longs;cendet verò IF, doneciterum li­
<pb pagenum="20"/>bra ín partes æquales, vt antea, diuidatur in C, &longs;iat que æ­<lb/>quili brium. </s>
</p>
<p type="main">
<s>Hæc Philo&longs;ophi demon &longs;tratio e&longs;t vera illa quidem, <lb/>&longs;ed non ex Mechanicis principijs, hoc e&longs;t, ex centri graui­<lb/>tatis &longs;pe culatione; nos igitur clariùs rem exponemus, his <lb/>quæ &longs;equuntur con&longs;ideratis. </s>
</p>
<p type="main">
<s>Si pondus circa &longs;tabile centrum conuertatur, dimi&longs;­<lb/>&longs;um non &longs;tabit, ni&longs;i &longs;ecundum grauitatis centrum fuerit <lb/>in perpendiculari, quæ per centrum, circa quod conuer­<lb/>titur, ad mundi centrum cadit. Stabit autem in ea per­<lb/>pendiculari in duobus punctis, altero à centro mundi <lb/>remoti&longs;&longs;imo; altero verò cidem quantum licuerit pro­<lb/>ximo. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to corpus A, cuius graui­<lb/>tatis centrum B, nixum lineae in­<lb/>flexibili BC, cum qua liberè <lb/>conuertatur circa centrum C. <lb/>Ducatur autem per mundi cen­<lb/>trum perpendicularis BCD. <lb/>Sit igitur primò pondus A <expan abbr="&longs;ecū-dum">&longs;ecun­<lb/>dum</expan> gracilis B centrum, in per­<lb/>pendiculari ip&longs;a &longs;upra centrum <lb/>C, puta in B. Moueatur & <expan abbr="de&longs;cē-dat">de&longs;cen­<lb/>dat</expan> in E. Po&longs;t hæc verò in F, hoc <lb/>e&longs;t iterum in ip&longs;a perpendiculari <lb/>infra centrum C. De&longs;cribet er­<lb/>go circulum ex centro C, nem­<lb/>pe BEF &longs;ecantem perpendicu­<lb/>larem in duobus punctis oppo­<lb/>&longs;itis BF, dico, pondus libe è di-
<pb pagenum="21"/>mi&longs;&longs;um in duobus tantum punctis &longs;uapte naturâ perman­<lb/>&longs;urum, BF, in B, primò, quoniam cum corpus ip&longs;um A à <lb/>perpendiculari, quæ &longs;upei ficiei loco intelligitur ABCD <lb/>per centrum grauitatis diuidatur, in partes diuiditur æ­<lb/>queponderantes, quare in neutram partem inclinabit. <lb/>Stabit igitur erectum, lineæ ip&longs;i fultum, inflexibili BC, <lb/>quæ nititur puncto C. In E verò non &longs;tabit, quippe quod <lb/>eo &longs;itu centrum ip&longs;um grauitatis &longs;it extra perpendicula­<lb/>rem, & ideo extra fulcimentum &longs;tabile C. In F verò ite­<lb/>rum &longs;tabit, pendens à centro C, propterea quòd & ibi ab <lb/>eadem perpendiculari diuidatur per grauitatis centrum <lb/>in partes æqueponderantes. E&longs;t igitur re&longs;pectu B, ip&longs;um <lb/>punctum C, ful cimentum deor&longs;um, re&longs;pectu verò F, ful­<lb/>cimentum &longs;ur&longs;um. At quia linea DFCB, à centro mundi, <lb/>quod e&longs;t extra circulum, BEF, circulum ip&longs;um per cen­<lb/>trum C &longs;ecat, erit pars eius DF quidem breui&longs;&longs;ima, ip&longs;a <lb/>verò DB longi&longs;&longs;ima, ex propo&longs;. 8. lib. 3. Elem. Pondus igi­<lb/>tur A conuer&longs;um &longs;eu liberè motum circa centrum C, in <lb/>duobus tantum locis perpendicularis lineæ &longs;tabit remo­<lb/>ti&longs;&longs;imo altero, vt e&longs;t B, altero verò cidem quamproximo, <lb/>vt e&longs;t F. </s>
</p>
<p type="main">
<s>Hoc idem egregiè demon&longs;trauit G. Vbald. in &longs;uis <lb/>Mechanicis, Tractatu de Libra prop.1.<gap/></s>
</p>
<p type="main">
<s>Ad hæc autem dubitare quis po&longs;&longs;et, cur experientiâ <lb/>docente, pondera quæ infra fulcimentum habent, vt lan­<lb/>cea &longs;ari&longs;&longs;aue ad planum horizontis perpendiculariter e­<lb/>recta, licet eo ca&longs;u grauitatis centrum in ip&longs;a perpendicu­<lb/>lari con&longs;tituatur, non &longs;tet quidem, &longs;ed altrin&longs;ecus ca­<lb/>dat? </s>
</p>
<pb pagenum="22"/>
<figure></figure>
<p type="main">
<s>Sit enim horizontis <lb/>planum AB, cui in puncto <lb/>C perpendiculariter ere­<lb/>cta &longs;tatuatur &longs;ari&longs;&longs;a DC, <lb/>cuius grauitatis centrum <lb/>E, in ip&longs;a perpendiculari. <lb/>Stabit ergo, ex præmi&longs;&longs;is, <lb/>& certè &longs;tare debuit, &longs;ta­<lb/>retqueue, ni vitium ob&longs;taret <lb/>materiæ; non &longs;tat autem, <lb/>quia difficillimum e&longs;t gra­<lb/>uitatis centrum, &longs;uapte naturâ indiui&longs;ibile, ita ad amu&longs;&longs;im <lb/>&longs;i&longs;tere, vt in neutram partem à perpendiculari declinet. <lb/>Hæc igitur ex ijs &longs;peculationibus e&longs;t, quæ ad praxim, ma­<lb/>teriæ vitio impediente, aut vix aut nun quam rediguntur. </s>
</p>
<p type="main">
<s>Hinc autem ea quæ&longs;tio &longs;oluitur, Cur ij qui &longs;ari&longs;&longs;am <lb/>erectam digito &longs;ummo &longs;u&longs;tinere conantur, non &longs;tent qui­<lb/>dem, &longs;ed digiti motu, &longs;ari&longs;&longs;æ motum &longs;equantur. </s>
</p>
<p type="main">
<s>Id certè agit, qui nutantis &longs;ari&longs;&longs;æ, digito, motum &longs;e­<lb/>quitur; vt in ip&longs;o motu digitum a&longs;&longs;iduè centro grauitatis <lb/>&longs;ari&longs;&longs;æ &longs;upponat, vnde &longs;it vt nun quam extra fulcimentum <lb/>permanens, nun quam cadat. </s>
</p>
<p type="main">
<s>Similis huic alia quo que dubitatio &longs;oluitur: Nempe, <lb/>Cur turbines, quibus pueri ludunt, dum quidem rotan­<lb/>tur, &longs;tent erecti, rotationevero ce&longs;&longs;ante, cadant. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim Turbo AB, cu­<lb/>ius grauitatis centrum C, planum <lb/>horizontis DE, linea Horizonti <lb/>perpendicularis ABC, tran&longs;iens <lb/>per centrum grauitatis C, &longs;it au­<lb/>tem fulcimentum in B. <expan abbr="Itaq;">Itaque</expan> cum <lb/>centrum grauitatis C &longs;it in ip&longs;a <lb/>perpendiculari, &longs;tabit ex demon-
<pb pagenum="23"/>&longs;tratis, at ex vitio materiæ non &longs;tabit. Modò, vt a&longs;&longs;olet, ra­<lb/>pido motu rotetur. Dico, Turbinem, motu &longs;eu rotatione <lb/>durante &longs;tare. ea autem paullatim elangue&longs;cente ín ca­<lb/>&longs;um vergere; ce&longs;&longs;ante verò penitus cadere. fit enim ex in­<lb/>æqualitate materiæ, vel operis ruditate, vel aliâ quauis <lb/>ex cau&longs;&longs;a, grauitatis centrum non e&longs;&longs;e in C, &longs;ed exempli <lb/>gratiâ vbi F, notentur autem hinc inde Turbinis latera <lb/>notis GH. Vtique cum F extra perpendicularem fuerit, <lb/>cadet Turbo ad partem G; at id ne &longs;iat, efficitur velocita­<lb/>te motus, quo centrum F transfertur in contrariam par­<lb/>tem, vbi I. non autem cadit ver&longs;us H, quoniam eadem ve­<lb/>locitate iterum transfertur in F, quamobrem cum huius­<lb/>cemodi centri a&longs;&longs;idua circa perpendicularem fiat trans­<lb/>latio, ad nullam partem Turbo cadere pote&longs;t; elangue­<lb/>&longs;cente verò motu rotans, paullatim in cipit inclinari, do­<lb/>nec eo penitus ce&longs;&longs;ante, ad eam partem cadit, ad quam à <lb/>per pendiculari grauitatis centrum vergit. De&longs;cribit au­<lb/>tem in rotatione grauitatis centrum, quod in medio non <lb/>e&longs;t paruum circulum, per cuius centrum ip&longs;a perpendi­<lb/>cula<gap/>is pertingit. </s>
</p>
<p type="main">
<s>Modò redeuntes ad libram, cuius ful cimentum e&longs;t <lb/>&longs;ur&longs;um, alio principio, nempe Mechanico, cur depre&longs;&longs;a <lb/>ad æqualitatem reuertatur, demon&longs;trabimus. </s>
</p>
<pb pagenum="24"/>
<figure></figure>
<p type="main">
<s>Sit igitur, vt &longs;u­<lb/>periùs, libra AB, cu­<lb/>ius centrum grauita­<lb/>tis C, fulcimentum, <lb/>verò &longs;ur&longs;um, in D li­<lb/>bræ quidem in C per­<lb/>pendiculariter con­<lb/>iunctum. Perpendi­<lb/>cularis verò quæ per <lb/>fulcimentum, & gra­<lb/>uitatis <expan abbr="cētrum">centrum</expan> tran&longs;­<lb/>iens ad mundi cen­<lb/>trum tendit DLE. &longs;tante igitur librâ in &longs;ua æqualitate, e­<lb/>rit centrum grauitatis C in ip&longs;a perpendiculari infra qui­<lb/>dem fulcimentum D. Loco verò, mundi centro quàm <lb/>proximo. Pondus po&longs;t hæc apponatur in B, Declinabit au­<lb/>tem pars CB, in HF, eleuatâ interim parte AC, in GH. <lb/>Mota igitur libra tota, circa fulcimentum D mouebitur <lb/>circa idem centrum, & grauitatis centrum C, de&longs;cribens <lb/>portionem circuli CH, fi etque; C in H, & quoniam H, hoc <lb/>e&longs;t C, extra per pendicularem fit, amoto pondere, ex lan­<lb/>ce B, cuius pre&longs;&longs;ione libra declinauerat, centrum grauita­<lb/>tis per eandem circulì portionem HC, ad perpendicula­<lb/>rem de&longs;cendet, donec iterum in ea quie&longs;cat, quo ca&longs;u li­<lb/>bra AB ad æquilibrium reuertetur: quod fuerat demon­<lb/>&longs;trandum. </s>
</p>
<p type="main">
<s>His ita declaratis, o&longs;tendemus, (quod nullus ante <lb/>nos animaduertit) harum librarum, quæ fulcimentum <lb/>habent &longs;ur&longs;um, eam e&longs;&longs;e naturam, vt non à quouis ponde­<lb/>re appo&longs;ito moueantur, vel penitus declinent. </s>
</p>
<p type="main">
<s>Ij&longs;dem enim &longs;tantibus, addatur quoduis pondus lan­<lb/>ci B; Itaque &longs;i tale fuerit quod &longs;uperet re&longs;i&longs;tentiam, quam
<pb pagenum="25"/>illi facit centrum grauitatis contra naturam elatum in H <lb/>mouebitur quædam libra. Sin autem tam parui momenti <lb/>&longs;it, vt eam re&longs;i&longs;tentiam non vincat, &longs;tante circa locum in­<lb/>fimum centro C, non mouebitur aut &longs;altem parum, ip&longs;a <lb/>libra. </s>
</p>
<p type="main">
<s>Hinc colligimus &longs;ieri po&longs;&longs;e, libras illas, quæ non <gap/><lb/>quouis, quantumuis paruo pondere declinant, cas fulci-<gap/><lb/>mentum habere &longs;ur&longs;um. </s>
</p>
<p type="main">
<s>His ad dimus, cæteris paribus, re&longs;i&longs;tentiam eò e&longs;&longs;e <lb/>maiorem, quo minus grauitatis centrum di&longs;tat à fulci­<lb/>mento &longs;ur&longs;um, circa quod ip&longs;a libra aduertitur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to libra AB, cuius gra­<lb/>uitatis centrum C, & primò <lb/>quidem eius ful cimentum <lb/>&longs;ur&longs;um &longs;it vbi D, itaque &longs;i ap­<lb/>po&longs;ito pondere de clinauerit <lb/>libra ad partes B, punctum <lb/>C, dum a&longs;cendet de&longs;cribet <lb/>portionem circuli CE. fulciatur iterum &longs;ur&longs;um puncto F, <lb/>& iterum declinet ad partes B, & iterum punctum C, dum <lb/>a&longs;cendet, circuli portionem de&longs;cribet CG. E&longs;t autem <lb/>minor angulus contactus ACE, angulo ACG, magis er­<lb/>go &longs;ur&longs;um, hoc e&longs;t, ad naturam &longs;ui feretur C, per CG, ex <lb/>centro F, quàm per CE, ex centro D, quod fuerat de­<lb/>mon&longs;trandum. </s>
</p>
<p type="main">
<s>Hæc autem re&longs;i&longs;tentia ex eodem fulcimento & eo­<lb/>dem pondere eo faciliùs &longs;uperabitur, quo longius bra­<lb/>chium libræ fuerit. </s>
</p>
<p type="main">
<s>E&longs;to enim iterum libra AB, cuius fulcimentum D, <lb/>centrum grauitatis C, &longs;it & alia libra, cuius brachia bre­<lb/>uiora EF, idem habens centrum C, & eidem puncto &longs;u­<lb/>&longs;pen&longs;a D. Dico igitur, eodem pondere appo&longs;ito, faciliùs
<pb pagenum="26"/>
<arrow.to.target n="fig1"></arrow.to.target><lb/>declinaturam libram ad <lb/>partes B, quàm &longs;i idem ap­<lb/>poneretur in F. Demit­<lb/>tatur enim, à puncto B <lb/>horizonti perpendicula­<lb/>ris BG, & ab F item per­<lb/>pendicularis FH, Tum <lb/>iuncta DB, centro D, eo­<lb/>dem vero &longs;patio DB, circuli portio de&longs;cribatur BI, item <lb/>iuncta DF eodem centro D, &longs;patio DF, portio circuli de­<lb/>&longs;cribatu: FK. e&longs;t autem maior DB ip&longs;a DF ex propo&longs;. <lb/>21. lib. 1. Elem. quare maiotis circuli portio e&longs;t BI quàm <lb/>FK. Obliquior autem, hoc e&longs;t, à perpendiculariremotior <lb/>e&longs;t motus per FK quàm per BI. maior &longs;i quidem e&longs;t angu­<lb/>lus KFH angulo IBG. quod nos ita probamus. Ducatur <lb/>perpendicularis ip&longs;i DF linea LF contingens circulum <lb/>FK in F, item ip&longs;i DB, perpendicularis MB, contingens <lb/>circulum BI in B, & quia angulus contingentiæ maioris <lb/>circuli minor e&longs;t angulo contingentiæ minoris, erit KFL <lb/>maior IBM, R ectiautem &longs;unt DFL, DBM, minor ergo <lb/>DFK re&longs;idua ip&longs;o DBI re&longs;iduo. Maior autem DFC ex <lb/>iam citata propo&longs;. <expan abbr="quã">quam</expan> DBC, erit igitur re&longs;iduum CFK, <lb/>multo minus re&longs;iduo FBI, &longs;ed recti &longs;unt CFH, FBG, ex <lb/>quibus &longs;i detra hantur CFK, FBI, erit re&longs;iduum KFH, <lb/>maius re&longs;iduo IBG, plus ergo retra hitur à perpendicula­<lb/>ri po<gap/>dus de&longs;cendens per FK quàm per BI, minus igitur <lb/>præ<gap/>alebit re&longs;i&longs;tentiæ in C pondus appen&longs;um in F, quàm <lb/>&longs;i appendatur in B. quod fuerat demon&longs;trandum. </s>
</p>
<figure id="fig1"></figure>
<p type="main">
<s>Po&longs;&longs;<gap/>mus & idem quoque aliter o&longs;tendere. </s>
</p>
<p type="main">
<s>Sint enim &longs;eor&longs;um duæ libræ, maior AB, mïnor EF, <lb/>quàm commune grauitatis centrum C, fulcimentum ve­<lb/>rò &longs;ur&longs;um D. Producatur perpendicularis DC, in G & fiat <lb/>CG æqualis CB, CH verò æqualis CF. Sunt igitur duo
<pb pagenum="27"/>
<arrow.to.target n="fig2"></arrow.to.target><lb/>vectes DG, DH, quo­<lb/>rum quidem commu­<lb/>ne fulcimentum D, <lb/>pondus verò C, poten­<lb/>tiæ vbi HG. Sunt au­<lb/>tem hi vectes cius na­<lb/>turæ, in quibus <expan abbr="pōdus">pondus</expan> <lb/>e&longs;t inter fulcimentum <lb/>& potentiam, itaque <lb/>vt &longs;e habet DC, ad <lb/>DG, ita potentia in G <lb/>ad pondus in C, item vt DC ad DH ita potentia in H ad <lb/>idem pondus C, &longs;ed minor e&longs;t propo&longs;itio DC, ad DG <lb/>quàm DC ad DH. minor ergo potentia requiritur in G, <lb/>hoc e&longs;t, in B, quàm in H, hoc e&longs;t in F. Data igitur ponderis <lb/>æqualitate faciliùs &longs;uperabitur re&longs;i&longs;tentia C in B, quàm <lb/>in F: quod o&longs;tendendum fuerat. </s>
</p>
<figure id="fig2"></figure>
<p type="main">
<s>Ad huius libræ naturam illæ quoque rediguntur, <lb/>quarum iugum non rectum quidem, &longs;ed curuum, vel ex <lb/>rectis &longs;ur&longs;um in angulum ad fulcimentum detinentibus, <lb/>nec refert vtrum curuitas &longs;it circuli portio quælibet, aut <lb/>ellip&longs;is &longs;ecundum alterum diametrorum; quod ita de­<lb/>mon&longs;tramus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to libra, cuius iugum <lb/>curuum <expan abbr="angulatūue">angulatunue</expan> ABC, <lb/>cuius fulcimentum B, æqua­<lb/>lia autem brachia AB, BC, <lb/>& pondera item <expan abbr="vtrinq;">vtrinque</expan> ap­<lb/>pen&longs;a æqualia. Demittatur <lb/>ex puncto B ad mundi cen­<lb/>trum perpendicularis BD. <lb/>Stante igitur libra ABC in <lb/>æquilibrio, erit eius graui­
<pb pagenum="28"/>tatis centrum in ip&longs;a perpendiculari BD, puta in E. Ap­<lb/>ponatur pondus in C, declinabit autem libra, &longs;it autem <lb/>iuxta po&longs;itionem FBG. Centrum igitur grauitatis E per <lb/>portionem EH, erit in H. A&longs;cendit ergo centrum graui­<lb/>tatis in H, hoc e&longs;t, &longs;ur&longs;um, id e&longs;t, contra cius naturam; a­<lb/>moto igitur pondere ex C, grauitatis centrum extra per­<lb/>pendicularem con&longs;titutum rur&longs;us de&longs;cendet, & iterum <lb/>libra ABC ad æquilibrium reuertetur. Hoc idem egre­<lb/>giè o&longs;tendit G. Vbald. in tractatu de libra, propo&longs;. 4. </s>
</p>
<p type="main">
<s>Hinc ratio pendet earum imaguncularum, quas ex <lb/>contu&longs;a papyro ligneaue leui materia compingunt, per­<lb/>queue manus earum ambas, ferreum filum trajicientes, v­<lb/>trinque plumbea appendunt pondera æqualia, ea <expan abbr="quidē">quidem</expan> <lb/>lege, vt centrum grauitatis infra pedes imaguncula &longs;ta­<lb/>tuatur. Tunc enim exten &longs;o filo imponentes ceu funam­<lb/>bulos per illud, vltrò citroque; decurrere faciunt, imagun­<lb/>cula interim erecta & in neutram partem cadente, quod <lb/>vt figurâ clarius fiat; </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to imaguncu­<lb/>la AB, per cuius ma­<lb/>nus traij ciatur filum <lb/>ferreum curuum <expan abbr="cū">cum</expan> <lb/>æ qualibus ponderi­<lb/>bus hinc inde <expan abbr="appē-&longs;is">appen­<lb/>&longs;is</expan> CD. Nitatur au­<lb/>tem pedibus filo HI <lb/>in <emph type="italics"/>B<emph.end type="italics"/>, &longs;itque; totìus ma­<lb/>chinæ grauitatis <expan abbr="cē-trum">cen­<lb/>trum</expan> E, &longs;itque <expan abbr="per-pēdicularis">per­<lb/>pendicularis</expan> per gra­<lb/>uitatis <expan abbr="centrū">centrum</expan> tran&longs;i­<lb/>ens A<emph type="italics"/>B<emph.end type="italics"/> E. Itaque in­<lb/>clinata imaguncula, & conuer&longs;a circa punctum <emph type="italics"/>B<emph.end type="italics"/>, &longs;i de-
<pb pagenum="29"/>clinet ad partes I, centrum grauitatis eleuabitur in F. Si <lb/>verò ad partes H eleuabitur in G. quare cum FG loca <lb/>&longs;intremotiora à mundi centro, quàm &longs;it E, non &longs;tabit gra­<lb/>uitatis centrum in punctis FG, &longs;ed ad infimum locum re­<lb/>uertecur, hoc e&longs;t, in ip&longs;a perpendiculari in E, & imagun­<lb/>cula ad perpendiculum ip&longs;i H<emph type="italics"/>B<emph.end type="italics"/>E filo, hoc e&longs;t, ip&longs;i hori­<lb/>zonti reuertetur. </s>
</p>
<p type="main">
<s>Hinc etiam Arictum, T e&longs;tudinun<gap/> queue demolito­<lb/>riatum Machinarum vis pendet, nempe ex ratione libra­<lb/>rum, quæ fulcimentum habent &longs;ur&longs;um. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim Aries A<emph type="italics"/>B<emph.end type="italics"/><lb/>funi appen&longs;us CD, cu­<lb/>ius grauitatis centrum, <lb/>D, perpendicularis verò <lb/>quæ ad mundi centrum <lb/>ip&longs;a CDE. Stante igitur <lb/>in æquilibrio machina, <lb/>centrum grauitatis erit <lb/>in ip&longs;a perpendiculari. <lb/>Applicetur alicubi po­<lb/>tentia retropellens, eleuabitur igitur centrum grauitatis <lb/>per circuli portionem DF, cuius &longs;emidiameter e&longs;t CD, <lb/>&longs;i etqueue iuxta po&longs;itionem CF. Aries verò in GFH. Di­<lb/>mi&longs;&longs;a itaque Machina centrum F vtpote graue, non &longs;tabit, <lb/>&longs;ed &longs;uapte naturâ reuertetur in D. Quadruplici autem <lb/>de cau&longs;&longs;a motus Arietis violenti&longs;&longs;imus e&longs;t ex vi naturalis <lb/>ponderis, quo deor&longs;um fertur, tum velo citate naturalis <lb/>motus in de&longs;cendendo auctæ, tum ex vi pote<gap/> tiæ impel­<lb/>lentis, & naturalem motum adiuuantis, tum ex velocita­<lb/>te ex motu violento deor&longs;um & antror&longs;um impellente <lb/>acqui&longs;itâ. Id etiam addimus, eo validiores fore ictus, quò <lb/>grauior fuerit Machina, & maius &longs;patium, quo retrotra­
<pb pagenum="30"/>hitur, grauitate ip&longs;a & &longs;patio tum virium vnione opera<gap/><lb/>tionem mirum in modum adiuuantibus. </s>
</p>
<p type="main">
<s>Hæc nos de Libra &longs;ur&longs;um fulcimentum habente, dí­<lb/>cta voluimus, nunc de ea, cuius fulcimentum deor&longs;um, <lb/>e&longs;t, verba faciemus. </s>
</p>
<p type="head">
<s>Altera quæ&longs;tionis pars:</s>
</p>
<p type="head">
<s><emph type="italics"/>De Libra cuius fulcimentum deor&longs;um e&longs;t.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Si deor&longs;um fuerit, inquit Ari&longs;toteles, id quod &longs;ub­<lb/>&longs;tat, contrarium facit illi quæ &longs;ur&longs;um habet, nempe ad æ­<lb/>quilibrium non reuertitur. Plus enim, ait, dimidio fit li­<lb/>bræ, quæ deor&longs;um e&longs;t pars, quàm quod perpendiculum <lb/>&longs;ecet, quapropter non a&longs;cendit. eleuata enim pars leuior <lb/>e&longs;t. </s>
</p>
<p type="main">
<s>Hæc ille, qui &longs;chemate quo que rem aperit, at eo a­<lb/>pud interpretes, & Picolomineum Paraphra&longs;tem, ita <expan abbr="mē-dosè">men­<lb/>dosè</expan> lineato, vt inde ob&longs;curitas lucis loco, legentibus of­<lb/>fundatur. Nos, quod & &longs;uprà quo que fecimus, no&longs;tra fi­<lb/>gurâ, &longs;ole ip&longs;o clariorem, ex Ari&longs;to telis ip&longs;ius mente rem <lb/>totam efficiemus. </s>
</p>
<figure></figure>
<p type="main">
<s>Sit libra recta, (hoc <lb/>e&longs;t, in æquilibrio con­<lb/>&longs;tituta) vbi NG. Per­<lb/>pendiculum autem (id <lb/>e&longs;t, perpendicularis <lb/>quæ ad mundi <expan abbr="centrū">centrum</expan>) <lb/>KLM. Bifariam igitur <lb/>&longs;ecatur NG. impo&longs;ito <lb/>po&longs;thæc onere in ip&longs;o <lb/>N, erit quidem N, vbi <lb/>O. ip&longs;um autem G vbi <lb/>R. KL autem vbi LP.
<pb pagenum="31"/>quare maius e&longs;t KO, quam LR, ip&longs;a parte PKL. Amoto <lb/>igitur onere nece&longs;&longs;e e&longs;t manere. Incumbit enim onus ex­<lb/>ce&longs;&longs;us medietatis eius, vbi e&longs;t F. Sen&longs;us e&longs;t igitur, idcirco <lb/>partem iugi KLO inclinatam, ad æquilibrium non re­<lb/>uerti, propterea quòd maior &longs;it ip&longs;a KLO pars quæ tra­<lb/>hit, ip&longs;a<gap/> RKL, quæ trahitur & eleuatur. </s>
</p>
<figure></figure>
<p type="main">
<s>Pote&longs;t hoc idem longè <lb/>&longs;impliciori themate demon­<lb/>&longs;trari. E&longs;to enim libra AB, <lb/>cuius centrum C, fulcimen­<lb/>tum vero deor&longs;um D, Per­<lb/>pendicularis per centrum & <lb/>fulcimentum tran&longs;iens EF. <lb/>Apponatur pondus in B, de­<lb/>clinabitque; puta ad GH, cen­<lb/>trum verò C, ex &longs;tabili fulci­<lb/>mento D, circuli portionem de&longs;cribet CI, libra autem <lb/>&longs;ecabit EF perpendicularem in K. Æquales autem &longs;unt <lb/>IG, IH, at ex parte HI de&longs;umpta e&longs;t KI, addita queue ip&longs;i <lb/>IG, maior e&longs;t ergo tota KG, torâ KH. Non igitur KH <lb/>habet KG, &longs;ed libra, ni&longs;i impedita fuerit, cum centro C <lb/>de&longs;cendente per <gap/>in M, ad ip&longs;am perpendicularem dela­<lb/>ta, ad in feriorem partem, mutatis vicibus quie&longs;cet, facto <lb/>nempe fulcimento &longs;ur&longs;um, fietque; horizonti æque di&longs;tans <lb/>iuxta po&longs;itionem LMN. </s>
</p>
<p type="main">
<s>Demon&longs;tratio <expan abbr="quidē">quidem</expan> e&longs;t hæc, &longs;ed non ex proprijs prin­<lb/>cipijs Mechanicis, <expan abbr="nēpe">nempe</expan> ex ratione <expan abbr="cēt">cent</expan><gap/>i grauitatis petitâ. <lb/>Ii&longs;dem enim &longs;tantibus, <expan abbr="cū">cum</expan> centrum grauitatis C fiat extra <lb/>perpendicularem, de&longs;cendens ad I, nun quam reuert<gap/> tur <lb/>in C, a&longs;cen deret enim; &longs;ed &longs;i liberè circa centrum D con­<lb/>uerteretur, de&longs;cendens vt dictum e&longs;t per circulum CIM <lb/>pondus B, fieret in L, A vero in N adepta po&longs;itione <lb/>LMN. </s>
</p>
<pb pagenum="32"/>
<p type="main">
<s>Cur autem huius libræ, quæ aliàs inutilis e&longs;t, memi­<lb/>nerit Philo&longs;ophus, ea videtur cau&longs;&longs;a, quòd inde vectis vir­<lb/>tutem eliciat, vt &longs;uo loco videbimus. Id autem valde mi­<lb/>rum, hominem acuti&longs;&longs;imum nihil pror&longs;us de ea libra egi&longs;­<lb/>&longs;e, quæ fulcimentum nec &longs;ur&longs;um habet, nec deor&longs;um, &longs;ed <lb/>in ip&longs;o exqui&longs;itè medio, ita vt centrum grauitatis in ip&longs;o­<lb/>met fulcimento con&longs;i&longs;tat. Nos igitur de hac quod operæ <lb/>pretium fuerit, & ad rem, qua de agimus, vtile, in medium <lb/>proferemus. </s>
</p>
<p type="head">
<s><emph type="italics"/>De libra cuius fulcimentum est in medio.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Dicimus itaque, libram, cuius fulcimentum nec &longs;ur­<lb/>&longs;um e&longs;t, nec deor&longs;um, &longs;ed pror&longs;us in medio, nempe in ip&longs;o <lb/>grauitatis centro, vbi brachia & pondera vtrinque appo­<lb/>&longs;ita fuerint æqualia, &longs;i ab æquilibrio mouentur, quomo­<lb/>docunque po&longs;ita, &longs;tare nec ab eo, quem adepta e&longs;t, &longs;itu di­<lb/>moueri. </s>
</p>
<p type="main">
<s>Quæ&longs;tionem hanc perperam tractârunt recentio­<lb/>res quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij <lb/>nonnulli, qui Iordani Nemoracij a&longs;&longs;ertiones &longs;unt &longs;ecuti, <lb/>quorum demon&longs;trationes vel paralogi&longs;mos potiùs egre­<lb/>giè confutauit in libr. Mechanicor. Tractatu de libra pro­<lb/>po&longs;. 4. Guid. Vbald. ad cuius probati&longs;&longs;ima &longs;cripta Lecto­<lb/>rem ablegamus. fu&longs;i&longs;&longs;imè enim ibi hac de re & ab&longs;oluti&longs;&longs;i­<lb/>mè agit. Nos autem quidem paucis ea, quæ ad hanc co­<lb/>gnitionem pertinent, explicabimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim libra A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>cuius brachia æqualia, <lb/>& centrum grauitatis <lb/>in C, brachijs verò <lb/>AC, C<emph type="italics"/>B<emph.end type="italics"/> æqualibus, æ­<lb/>qualia pondera hinc <lb/>inde <expan abbr="apponãtur">apponantur</expan>. Tum
<pb pagenum="33"/>fulcimento in medio, hoc e&longs;t, vbi grauitatis centrum C <lb/>applicato per centrum ip&longs;um C ducatur perpendicularis, <lb/>quæ ad mundi centrum, DCE, &longs;itque primum libra æ­<lb/>quedi&longs;tans horizonti, con&longs;tituta. Tum ex altera parte <lb/>pre&longs;&longs;a moueatur & fiat iuxta po&longs;itionem FCG. Dico eam <lb/>dimi&longs;&longs;am permanere, etenim cum grauitatis centrum &longs;it <lb/>in ip&longs;a perpendiculari, in neutram partem verget, &longs;ed nec <lb/>vergere pote&longs;t, quippe quod non circa fulcimentum ceu <lb/>centrum motus, moueatur grauitatis centrum, &longs;ed in ip&longs;o <lb/>&longs;it ful cimento; &longs;itum ergo non mutat. Præterea cum per­<lb/>pendicularis DCE per grauitatis centrum ducatur, cor­<lb/>pus ip&longs;um ex ponderibus & libra con&longs;tans ab ea in partes <lb/>çque ponderantes &longs;ecatur, & ideo ex centri grauitatis dif­<lb/>finitione, quam protulit Pappus, corpus ip&longs;um centro <lb/>grauitatis appen&longs;um, dum fertur quie&longs;cit, & &longs;eruat eam, <lb/>quam à principio habuit po&longs;ition<gap/>. Et &longs;anè &longs;i partes quo­<lb/>modo libet librâ per grauitatis centrum diuisâ, &longs;untæ­<lb/>queponderantes nec trahent inuicem, nec trahentur, &longs;ta­<lb/>bit ergo libra, & quam adepta fuerat po&longs;itionem, eam &longs;er­<lb/>uabit. Id tamen non negamus, difficile e&longs;&longs;e libras eiu&longs;ce­<lb/>modi ex materia fabricare, quippe quod non omnia quæ <lb/>vera &longs;unt, & euidenti&longs;&longs;imis demon&longs;trationibus patent, <lb/>commodè ad praxim, ex artis & materiæ imperfectione, <lb/>reducuntur. </s>
</p>
<p type="main">
<s>Cæterùm harum librarum ea e&longs;t virtus, vt vel mini­<lb/>mo pondere altrin&longs;ecus appo&longs;ito, declinet; quod illis quæ <lb/>centrum &longs;u &longs;um habent, non euenire, demon&longs;trauimus. </s>
</p>
<p type="main">
<s>Circa hæc po&longs;&longs;et cuipiam oriri Dubium, num chor­<lb/>dulæ, quibus lances appenduntur, variationem aliquam <lb/>circa ea quæ demon&longs;trata &longs;unt, inducere valeant. </s>
</p>
<p type="main">
<s>Dicimus nullam inde fieri: E&longs;to enim libra AB, cu­<lb/>ius centrum & fulcimentum C, ab cuius extremitate A <lb/>dependeat, funiculus AD, ab alia verò <emph type="italics"/>B<emph.end type="italics"/>, funiculus <emph type="italics"/>B<emph.end type="italics"/>E,
<pb pagenum="34"/>
<arrow.to.target n="fig3"></arrow.to.target><lb/>quibus appen&longs;æ &longs;int æ­<lb/>qualis ponderis lances <lb/>DE. Moueatur libra, <lb/>fiatque in ICH, funi­<lb/>culi verò in lancibus in <lb/>IK, HL. &longs;ecet autem fu­<lb/>niculus IK libram A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>in M, LH verò produ­<lb/>catur & eandem &longs;ecer <lb/>in N. quoniam igitur <lb/>IC, æqualis e&longs;t CH, pa­<lb/>rallelæ autem KI, LN æquales <expan abbr="erūt">erunt</expan> alterni anguli MIC, <lb/>NHC, &longs;ed & anguli ad verticem ICH, BCH æquales <lb/>&longs;unt, quare triangulum IMC, æquale triangulo HNC, <lb/>& latera lateribus, quæ æqualibus angulis &longs;ubtenduntur. <lb/>Æqualis e&longs;t igitur linea MC lineæ NC. Itaque &longs;i ponde­<lb/>ra lancesue, KL mente concipiantur appen&longs;æ in punctis <lb/>MN, ex brachiorum & ponderum æqualitate æquepon­<lb/>derabunt. quod fuerat demon&longs;trandum. </s>
</p>
<figure id="fig3"></figure>
<p type="head">
<s>QVÆSTIO III.</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur exiguæ vires (quod etiam à principio dixerat) vecte magna <lb/>mouent pondera, vectes in&longs;uper onus accipientes, cum facilius <lb/>&longs;it, minorem mouere grauitatem, minor est au­<lb/>tem &longs;ine vecte?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ari&longs;toteles ita quæ&longs;tionem proponit, vt eam R heto­<lb/>rico quodam fuco admirabiliorem f<gap/>ciat. Soluit au­<lb/>tem hoc pacto, <expan abbr="inquiēs">inquiens</expan>, fieri po&longs;&longs;e eam e&longs;&longs;e cau&longs;&longs;am, quod <lb/>vectis &longs;it libra, eius nempe generis quod fulcimentum ha­<lb/>bet deor&longs;um, atque id circo in ip&longs;a pre&longs;&longs;ione in partes in­<lb/>æquales vectem diuidi. </s>
</p>
<pb pagenum="35"/>
<figure></figure>
<p type="main">
<s>Figura quam ex­<lb/>hibet, vix ferè quid &longs;i­<lb/>bi velit explicat. Nos <lb/>ad eius <expan abbr="mētem">mentem</expan> aliam <lb/>proponemus <expan abbr="eamq;">eamque</expan> <lb/>longè clariorem. </s>
</p>
<p type="main">
<s>E&longs;to vectis A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>cuius fulcimentum, <lb/>deor&longs;um in C, pon­<lb/>dus D, potentia ex vecte, pondus &longs;u&longs;tinens E. Perpendi­<lb/>cularis per fulcimentum FCG. Itaque quoniam poten­<lb/>tia in E non &longs;uperat pondus D, nec ab eo &longs;uperatur, &longs;tat <lb/>vectis cum potentia Horizonti æquidi&longs;tans, hoc e&longs;t, in æ­<lb/>quilibrio, vectis autem in puncto C diuiditur in partes æ­<lb/>queponderantes. Modo præualeat potentia ponderi, & <lb/>vectem deprimat, fiat autem in LCH, erit igitur <emph type="italics"/>B<emph.end type="italics"/>, in L, <lb/>A in H, D in K, & CF, quæ vectem in partes æque ponde­<lb/>rantes diuidebat, in CI. Iam igitur non æqueponderant <lb/>partes, &longs;i quidem pars vectis FCI, aufertur parti HCI, & <lb/>adiungitur parti ICL, quæ ideo &longs;it pondero&longs;ior, vnde & <lb/>potentia ad ponderis eleuationem adiuuatur. Eadem i­<lb/>gitur vtitur hic demon&longs;tratione, quam in explicando ef­<lb/>fectu libræ, cuius fulcimentum deor&longs;um e&longs;t, adhibuerat. <lb/>Nec alia de cau&longs;&longs;a, vt &longs;uprà notauimus, videtur eius libræ <lb/>in &longs;uperiori quæ&longs;tione, con&longs;iderationem introduxi&longs;&longs;e. Et <lb/>&longs;anè verum e&longs;t quod concludit, Veruntamen minimi e&longs;t <lb/>momenti ad tantam vim parua illa adiectio, quæ parti ve­<lb/>ctis depre&longs;&longs;æ in ip&longs;a depre&longs;&longs;ione adiungitur. Aliunde igi­<lb/>tur tantæ rei cau&longs;&longs;a e&longs;t petenda, quod & nos deinceps fa­<lb/>ciemus. Videtur autem ip&longs;e quoque Ari&longs;toteles non &longs;ibi <lb/>pror&longs;us in a&longs;&longs;ignata ratione &longs;atis feci&longs;&longs;e, & ideo &longs;ubiungit: <lb/>quoniam ab æquali pondere celerius mouetur maior ca­<lb/>rum quæ à centro &longs;unt duo verò pondera; quod mouet &
<pb pagenum="36"/>quod mouetur, quod igitur motum pondus ad mouens <lb/>longitudo patitur ad longitudinem, &longs;emper autem <expan abbr="quã-tum">quan­<lb/>tum</expan> ab hypomoch&longs;io (id e&longs;t, fulcimento) di&longs;tabit magis, <lb/>tanto facilius mouebit. Cau&longs;&longs;a autem <gap/>, quæ retro com­<lb/>memorata e&longs;t, quoniam quæ plus à centro di&longs;tat <expan abbr="maiotē">maiotem</expan> <lb/>de&longs;cribit circulum. quare ab eadem potentia plus &longs;upera­<lb/>biturid quod mouetur, quæ plus à fulcimento di&longs;&longs;at. H&uncedil;c <lb/>ille, qui a&longs;&longs;erit duo pondera in vecte con&longs;iderari, Pondus <lb/>nempe motum, & mouentem Potentiam (hanc enim <expan abbr="pō-deris">pon­<lb/>deris</expan> habere vim <expan abbr="atq;">atque</expan> rationem certum e&longs;t) Vires autem <lb/>potentiam acquirere ex brachij longitudine, & ex inde <lb/>con&longs;equenti velocitate, quo enim brachia longiora, eo <lb/>in extremitate velociora, atque idcirco ita &longs;e habere mo­<lb/>tum pondus ad potentiam mouentem, vt brachij longi­<lb/>tudo ad brachij longitudinem: brachia autem vocamus, <lb/>partes illas vectis, quæ à fulcimento ad vtranque vectis <lb/>extremitatem pertingunt, & ideo quantum à fulcimento <lb/>potentia di&longs;tabit magis, eo faciliùs pondus mouebit. </s>
</p>
<p type="main">
<s>Vera vtique & explorati&longs;&longs;ima hæc a&longs;&longs;ertio e&longs;t. Ve­<lb/>runtamen, cau&longs;&longs;am huiu&longs;ce mirabilis effectus, e&longs;&longs;e velo­<lb/>citatem, quæ brachij longitudinem con&longs;equitur, non af­<lb/>firmamus. quæ enim velocitas in re &longs;tante? Stant autem <lb/>vectis, & libra dum manent in æquilibrio, & nihilo &longs;ecius <lb/>parua potentia ingens &longs;u&longs;tinet pondus. </s>
</p>
<p type="main">
<s>Dicet ad hæc qui&longs;piam, velocitatem in longiori bra­<lb/>chio &longs;i non actu, &longs;altem potentiâ e&longs;&longs;e maiorem. At quæ&longs;o <lb/>quid in re quæ e&longs;t actu, momenti habet potentia? actu e­<lb/>nim &longs;u&longs;tinet, &longs;u&longs;tinens. Con&longs;equìtur, (id vtique fatemur) <lb/>nece&longs;&longs;ariò velocitas maior motu brachij maioris; non ta­<lb/>men cau&longs;&longs;a e&longs;t cur vis loco vbi velocitas maior &longs;it, appo&longs;i­<lb/>ta magis moueat. Sanè ex velocitate, dum mouentur, <expan abbr="pō-dus">pon­<lb/>dus</expan> acquirere corpora, tum proiecta, tum cadentia cer­<lb/>tum e&longs;t, quod etiam in quæ&longs;tione 19. cum Philo&longs;opho <expan abbr="cō-">con-</expan>
<pb pagenum="37"/>&longs;i derabimus. Sed hoc ex velocitate & motu &longs;it, quæ &longs;unt <lb/>actu. At brachia in ip&longs;o æquilibrio &longs;u&longs;tinent actu quidem, <lb/>&longs;ed non mouentur. Cæterum videtur A riftoteles id &longs;ub­<lb/>odora&longs;&longs;e, quod po&longs;tea Archimedes, Mechanicorum prin­<lb/>ceps, in propo&longs;. 6. primi Æqueponderantium explicitè <lb/>protulit & probauit: nempe in æquilibrio ita e&longs;&longs;e pondus <lb/>ad pondus, vt brachium ad brachium, ratione permutata. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim vectis <lb/>AB, quomodolibet <lb/>fulcimento diui&longs;us in <lb/>C. <expan abbr="appēdatur">appendatur</expan> autem <lb/>in A, pondus D, in B <lb/>verò pondus E, ita &longs;e <lb/>habens ad pondus D, vtip&longs;a AC ad CB. Stabit igitur ve­<lb/>ctis, & neutram in partem verget, erit enim centrum gra­<lb/>uitatis in C, diui&longs;o nempe ibi vecte in partes æque ponde­<lb/>rantes. Hoc po&longs;t Archimedem, & in&longs;ignes illos veteres <lb/>Mechanicos præclari&longs;&longs;imè demon&longs;trauit G. Vbaldus in <lb/>Mechanicis, Tractatu de Libra propo&longs;. 6. nec non de Ve­<lb/>cte propo&longs;. 4. </s>
</p>
<p type="main">
<s>Cæterùm vt aliquid interim, quod no&longs;trum &longs;it, affe­<lb/>ramus, liceat nobis egregios illos viros interrogare, quæ­<lb/>nam mirabilis eius effectionis &longs;it cau&longs;&longs;a? Dicent permu­<lb/>tatam proportionem. Teneo, at nondum acquie&longs;co: pe­<lb/>tam enim, Cur ea rationis permutatio mirabilem illum <lb/>effectum pariat. Hoc quod illi non do cent, puto nos, i­<lb/>gnorantiæ &longs;omno &longs;epultos, &longs;omnia&longs;&longs;e. </s>
</p>
<figure></figure>
<p type="main">
<s>Æqualitatem &longs;tatus <lb/>e&longs;&longs;e cau&longs;&longs;am, nemo, vt <lb/>puto, inficiabitur. res e&longs;t <lb/>enim per &longs;e clara. E&longs;to &longs;i­<lb/>quidem linea quæpiam AB, applicetur extremitati A po­
<pb pagenum="38"/>tentia quæ dam quæ lineam ad &longs;e trahat ad partes nempe <lb/>A, Tum in B quædam alia potentia ip&longs;i quæ in A potentiae, <lb/>æqualis, quæ lin eam trahat &longs;imili modo ad partes B. Datâ <lb/>igitur harum potentiarum æqualitate, linea AB, nec ad <lb/>partes A, nec ad partes B transferetur, &longs;ed pror&longs;us immo­<lb/>bilis &longs;tabit. </s>
</p>
<p type="main">
<s>His ita con&longs;titutis, Dico vecte quomodolibet diui&longs;o, <lb/>ponderibu&longs;que vtrinque appo&longs;itis, permutatâ propor­<lb/>tione &longs;ibi inuicem re&longs;pondentibus, rem e&longs;&longs;e redactam ad <lb/>æqualitatem, & inde &longs;tatum fieri, hoc e&longs;t, æquilibrium. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim vectis AB, quo modo libet diui&longs;us in C, & <lb/>ip&longs;i quidem C fulcimentum &longs;upponatur. Appendantur <lb/>quo que vtrinque pondera ex ratione brachiorum AC, <lb/>CB, &longs;ibi inuicem permutatim re&longs;pondentia, &longs;intque; DE. <lb/>Dico vectem ex æqualitate, in neutram partem <expan abbr="inclina-turū">inclina­<lb/>turum</expan>, &longs;ed perman&longs;urum in æquilibrio. quoniam enim <expan abbr="Pō-dus">Pon­<lb/>dus</expan> D idem pote&longs;t quod brachium CB, addatur in dire­<lb/>ctum ip&longs;i AC, recta AF æqualis ip&longs;i CB, item quoniam <lb/>Pondus E id pote&longs;t quod brachium AC, rectæ CB ad­<lb/>datur in directum BG, ip&longs;i AC æqualis. Igitur cum par­<lb/>tes CA, AF totius FC, æquales &longs;int partibus CB, BG, <lb/>totius CG, erit totum FC, toti CG æquale. Diui&longs;us ita-
<pb pagenum="39"/>que erit vectis FG in partes æquales FC, CG in puncto <lb/>fulcimenti C. Et quoniam æquale in æquale non agit, <lb/>&longs;tabit vectis & in neutram partem inclinabit. Rur&longs;um <lb/>quoniam ad partem FC, duæ &longs;unt brachiorum potentiæ <lb/>FA, HC, appendantur puncto F, duo pondera H, I, ip&longs;is <lb/>DE æqualia, item puncto G, alia duo pondera ij&longs;dem DE <lb/>æqualia KL, iterum æqueponderabit, quippe quod æ­<lb/>quahbus brachijs FCCG æqualia appen&longs;a &longs;int pondera <lb/>HI KL. Cur igitur &longs;eruata permutatim brachiorum & <lb/>ponderum proportione fiat æquilibrium, ex his quæ de­<lb/>mon&longs;trauimus, clarè patet. </s>
</p>
<p type="main">
<s>Sed forte dicet qui&longs;piam, &longs;i brachia, pondera &longs;unt, <lb/>vel ponderibus æquipollentia, &longs;u&longs;tinenti duplicabitur <lb/>pondus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim vectis AB, <lb/>ita diui&longs;us in C, vt pars <lb/>maior CB minori AC &longs;it <lb/>in proportione quintu­<lb/>pla. Appendatur autem <lb/>in A pondus D, <expan abbr="quintuplū">quintuplum</expan> <lb/>ponderi E appen&longs;o in B. Si <lb/>igitur brachio AC, quod <lb/>e&longs;t vnum, ad datur pondus <lb/>D, quod e&longs;t quinque, fi ent &longs;ex, item &longs;i brachio CB, quod <lb/>e&longs;t quinque, addatur pondus E, quod e&longs;t vnum, fient &longs;ex. <lb/>Fulcimentum igitur &longs;u&longs;tinebit duodecim, quod e&longs;t ab­<lb/>&longs;urdum ex ijs quæ clarè demon&longs;trauit G. Vbald. in Me­<lb/>chan. tractatu de Libra propo&longs;. 5. His re&longs;pondemus, bra­<lb/>chia quidem operari non pondere, &longs;ed potentiâ, quæ vis <lb/>quædam e&longs;t, non autem pondus. Et&longs;i & illud verum &longs;it, da­<lb/>to vecte pondero&longs;o, fulcimentum rum ponderum appen­<lb/>&longs;orum, tum vectis ip&longs;ius pondus &longs;u&longs;tinere. </s>
</p>
<p type="main">
<s>Iacta huiu&longs;cemodi, quam diximus, æqualitate, &longs;e-
<pb pagenum="40"/>quitur nece&longs;&longs;ariò, centrum grauitatis ip&longs;ius vectis cum <lb/>appen&longs;is ponderibus, ac &longs;i vnum idemqueue e&longs;&longs;et corpus <lb/>cadere in perpen diculari quæ per centrum ip&longs;um & ful­<lb/>cimentum tran&longs;iens ad mundi centrum pertingit. </s>
</p>
<p type="head">
<s>QVÆSTIO IV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quærit hic Ari&longs;toteles, cur ij qui in nauis medio &longs;unt remiges ma­<lb/>ximè nauem moueant?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ait, ideo forta&longs;&longs;e fieri, quò dremus vectis &longs;it, fulcimen­<lb/>tum verò &longs;calmus, &longs;tat enim. Pondus autem marei­<lb/>p&longs;um, quod à remo propellitur, mouens verò ip&longs;um remi­<lb/>gem, &longs;emper autem plus mouere ponderis quimouet, <lb/>quo magis di&longs;tatà fulcimento. Ita enim maiorem fieri <lb/>quæ ex centro; Scalmum verò centrum e&longs;&longs;e. Cæterùmin <lb/>medio nauis plurimum remi intus e&longs;&longs;e. Ibi enim nauem <lb/>e&longs;&longs;e lati&longs;&longs;imam. Moueri autem nauim, quoniam <expan abbr="appellē-te">appellen­<lb/>te</expan> mariremo, <expan abbr="extremū">extremum</expan> illius quod intus e&longs;t anterius pro­<lb/>mouctur, cuius motum nauis &longs;equitur, cui &longs;calmus alliga­<lb/>tur. Vbiautem plurimum maris diuidit remus, eo maximè <lb/>nece&longs;&longs;e e&longs;&longs;e propelli. Plurimum autem diuidi vbi plurima <lb/>pars remi à &longs;calmo e&longs;t. Rem facilem, eo quod verbis potu­<lb/>erit, &longs;chemate non declarauit, nos autem apponemus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim nauis AB, mare CD, <lb/>remorum alter, quiad proram EF, cu­<lb/>ius &longs;calmus G, alterverò in medio na­<lb/>uis, HI, circa &longs;calmum K. Ait igitur, <lb/>remos e&longs;&longs;e vectes, &longs;calmos verò fulci­<lb/>menta, pondus quod remo, ceu vecte, <lb/>mouetur mare ip&longs;um. Itaque quoniam <lb/>nauis lata e&longs;t in medio vbi Scalmus K <lb/>maior pars KH intra nauim e&longs;t, minor <lb/>verò KI, extra. Contra autem remiad <lb/>proram, nempe EF pars minor EG
<pb pagenum="41"/>intra nauim, pars verò maior GF extra nauim e&longs;t. Pondus <lb/>autem cò faciliùs mouctur, quo maior e&longs;t vectis pars, quæ <lb/>à fulcimento e&longs;t ad mouentem potentiam. </s>
</p>
<p type="main">
<s>Acutè &longs;anè Philo&longs;ophus. Ego autem &longs;i per mode&longs;tiam <lb/>liceret, dicerem, non quidem e&longs;&longs;e fulcimentum <expan abbr="&longs;calmū">&longs;calmum</expan>, <lb/>&longs;ed mare ip&longs;um, pondus vero nauim, ad locum &longs;calmi, <expan abbr="nē-pe">nen­<lb/>pe</expan> inter mouentem potentiam, & fulcimentum po&longs;itum, <lb/>etenim & eo pacto po&longs;&longs;umus vti vecte, quod ob&longs;eruat & <lb/>demon&longs;trat G. Vbaldus tractatu de vecte propo&longs;. 2. Erunt <lb/>igitur in de&longs;cripta figura puncta FI, quæ in mari&longs;unt, ful­<lb/>cimenta, quibus remorum extrema in ip&longs;aim pul&longs;ione ni­<lb/>tuntur, pondera verò &longs;eu pondus pluribus vectibus & po­<lb/>tentijs impul&longs;um nauis ip&longs;a, quæ &longs;calmis e&longs;t annexa. Re&longs;i­<lb/>&longs;tente igitur mari, cedente autem impul&longs;ionibus &longs;calmo, <lb/>nauis eo transfertur, quo &longs;calmi ab ip&longs;a potentia mouen­<lb/>te in anteriorem partem pelluntur. quoniam autem vt <lb/>FG ad FE ita potentia mouens in E ad pondus motum <lb/>in G. item vt IK ad IH ita potentia mouens in H ad pon­<lb/>dus motum in K, maior autem e&longs;t proportio FG ad FE <lb/>quàm proportio IK ad IH. Maiori indiget potentia vt <lb/>pellatur pondus in G quàm pondus in K. </s>
</p>
<p type="main">
<s>Hæc certè vti diximus ita &longs;e habent. Philo&longs;ophi au­<lb/>tem ratio tunc procederet, &longs;i &longs;tante naui immobili, vt fit <lb/>vbi à Remoræ occulta vi aut ab alio impedimento reti­<lb/>netur, remiges in ip&longs;o remigandi actu mare pul&longs;arent, <lb/>Tunc enim verè &longs;calmus fieret fulcimentum, mare autem <lb/>pondus, remex verò ip&longs;e mouens. </s>
</p>
<p type="main">
<s>Addimus, fal&longs;um videri quod a&longs;&longs;erit Ari&longs;toteles, <lb/>nempeillos qui in media naui &longs;unt, remiges, maximè na­<lb/>uim mouere; facilius, melius dixi&longs;&longs;et. Si enim maximè, <lb/>quod ait, denorat, maximo &longs;patio, & velocius pror&longs;us fal­<lb/>&longs;um, etenim tardius mouent & minori &longs;patio, quod nos i­<lb/>ta demon&longs;tramus. </s>
</p>
<pb pagenum="42"/>
<figure></figure>
<p type="main">
<s>E&longs;to enim Remus AB <lb/>qui marí fulcitur in B, Scal­<lb/>mus remi qui ad <expan abbr="prorã">proram</expan> pup­<lb/>pimue C, qui in media naui <lb/>D, maior autem remi pars <lb/>e&longs;t à &longs;calmo Dad A quami­<lb/>p&longs;ius C 2d A, Pellantur remi & &longs;tante ceu centro BA, in <lb/>E. eodem igitur tempore C eritin F, & D in G, &longs;ed maiu<gap/><lb/>e&longs;t &longs;patium CF &longs;patio DG, Ergo vnica impul&longs;ione, plus <lb/>mouit &longs;calmum, hoc e&longs;t, nauim, potentia ad puppim pro­<lb/>ramue remigans, quàm ea quæ operatur in media naui vt <lb/>&longs;entire vid<gap/>batur (&longs;i modo is e&longs;t eius &longs;en&longs;us) Ari&longs;toteles. <lb/>Nece&longs;&longs;arium igitur e&longs;t, quodait, maximè intelligendum, <lb/>faciliùs, Veritatem hanc cogno&longs;centes Triremium præ­<lb/>fecti robu&longs;tiores quidem remiges ad proram & puppim, <lb/>inualidiores vcrò circa mediam triremem collocant. </s>
</p>
<p type="head">
<s>QVÆSTIO V.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitat<gap/>r, Cur paruum exi&longs;tens gubernaculum, & in extremo <lb/>nauigio tantas habeat vires, vt ab exiguo temone, & ab hominis <lb/>vnius viribus alioqui modicè vtentis magnæ nauigiorum <lb/>moueantur moles?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>AN, inquit, quoniam gubernaculum vectis e&longs;t, onus <lb/>autem mare, Gubernator vero mouens e&longs;t? Non au­<lb/>tem &longs;ecundùm latitudinem veluti remus, mare accipit <lb/>gubernaculum; non enim in ante nauigium mouet, &longs;edi­<lb/>p&longs;um commotum mare accipiens inclinat obliquè. quo­<lb/>niam enim pondus e&longs;t mare contrario innixum modo na­<lb/>uem inclinat. fulcimentum enimin contrarium ver&longs;atur, <lb/>mare vetò interius, & illud exterius. illud autem &longs;equitur <lb/>nauis quæ illi e&longs;t alligata & remus quidem &longs;ecundum la­<lb/>titudinem onus propellens & ab eodem repul&longs;us in re-
<pb pagenum="43"/>ctum propellit, Gubernaculum verò, vt obliquum iacet <lb/>hinc inde in obliquum motionem facit. in extremo <expan abbr="autē">autem</expan>, <lb/>non in medio iacet, quoniam mouenti fa cillimum e&longs;t mo­<lb/>tum moucre: prima enim pars celerrimè fertur, & quo­<lb/>niam, quemadmodum in ijs quæ feruntur in fine deficit <lb/>latio, &longs;ic ip&longs;ius continui in finem, imbecillima e&longs;t latio. <lb/>Imbecillima autem ad expellendum e&longs;t facilis. Propter <lb/>hæc igitur in puppi gubernaculum ponitur, nec minus, <lb/>quoniam paruaibi motione facta, multo maior fit in vlti­<lb/>mo, quia æqualis angulus &longs;emper maiorem ad&longs;pectat, <expan abbr="tã-to">tan­<lb/>to</expan> queue magis, quanto maiores fuerint illæ, quæ continent. <lb/>Exijs ctiam manife&longs;tum e&longs;t, quam ob cau&longs;&longs;am magis in <lb/>contrarium procedit nauigium, quam remi ip&longs;ius palmu­<lb/>la, eadem enim magnitudo ij&longs;dem mota viribus in aëre <lb/>plus quàm in aqua progreditur. Hæc Philo&longs;ophus, qui <lb/>haudquaquam ex more &longs;uo, quod duobus ferè poterat, <lb/>&longs;excentis verbis expo&longs;uit. Licebat enimid tantum dicere, <lb/>Gubernaculum (ita vocatid totum quod gubernaculo & <lb/>temone con&longs;tat) e&longs;&longs;e ceuremum, quo nauis non antror­<lb/>&longs;um, &longs;ed obliquè & ad latus mouetur. quamobrem omnia <lb/>ferè quæ de Temone dicenda fuerant, de remo loquens <lb/>proponit. Aitautem. </s>
</p>
<figure></figure>
<p type="main">
<s>Sit remus AB, <lb/>&longs;calmus vero C, remi <lb/>in nauigio <expan abbr="principiū">principium</expan> <lb/>A, palmula autem, <lb/>quæ in mari B. Si igi­<lb/>tur A, vbi D transla­<lb/>tum e&longs;t, non erit B v­<lb/>bi E. æqualis enim, <lb/>BE ip&longs;i AD, æquale <lb/>igitur translatum erit, &longs;ed erat minus. eritigitur vbi F, mi­<lb/>nor enim BF, ip&longs;a AD, quareip&longs;o GF ip&longs;a DG. Hæc
<pb pagenum="44"/>demon&longs;tratio licet vera videatur, rei ta men, de qua e&longs;t <lb/>&longs;ermo, minimè aptatur. Si enim aptaretur in ip&longs;ius remi <lb/>motu, cum palmula e&longs;&longs;et in F, &longs;calmus ficret in G, excur­<lb/>reretergo vel &longs;calmus per remum, vel remus per <expan abbr="&longs;calmū">&longs;calmum</expan>, <lb/>facta nempe ciu&longs;modi translatione de C in G, & &longs;ic intra <lb/>nauim modo e&longs;&longs;et pars remi DC, modò verò GD, quod <lb/>tamen non &longs;ieri ipsâ experientia docemur. Illud quoque <lb/>fal&longs;um e&longs;t, nauim ip&longs;am tantum moueri in aëre, quantum <lb/>e&longs;t &longs;patium AD, hoc e&longs;t, remi extremum quod e&longs;t in naui, <lb/>&longs;iquidem &longs;calmi motu, non autem manubrij remi, nauis <lb/>agatur. Aliter igitur res &longs;e habet, & forte hoc pacto. </s>
</p>
<figure></figure>
<p type="main">
<s>Sit remus AB, cuíus <lb/>manubrium A, palmula <lb/>B, &longs;calmus C. Pellatur an­<lb/>tror&longs;us A, fiatque; in D, tunc <lb/>&longs;i æqualiter mouerentur <lb/>manubrium & palmula, i­<lb/>p&longs;a palmula ficret in G, at <lb/>minus mouetur: fiet ergo <lb/>in E. ip&longs;e verò &longs;calmus C <lb/>translatus erit in F, motaque; erit nauis à C in F, non autem <lb/>ab A in D. P o&longs;uitautem Ari&longs;toteles &longs;calmum ad medium <lb/>remi, &longs;ed non ad medium collocari &longs;olet, maior enim pars <lb/>in mare propendet puta HB, quo ca&longs;u translationis &longs;pa­<lb/>tium fit maius, nempe ab H in I. fit autem motus &longs;calmi ex <lb/>centris qui &longs;unt in &longs;patio ip&longs;o BE, quatenus autem ad te­<lb/>monem pertinet, quem remum ait, obliquè puppim ip&longs;am <lb/>propellentem, ita &longs;e res habet. </s>
</p>
<p type="main">
<s>E&longs;to nauis carina AB, prora A, puppis B, Temonis <lb/>ala BC, gubernaculum BD, cardo verò fulcimentumue <lb/>B; factaitaque impul&longs;ione obliquâ gubernaculi à D in E, <lb/>minor fiet motus in mari à C in F, eritqueue temo vbi EGF,
<pb pagenum="45"/>
<arrow.to.target n="fig4"></arrow.to.target><lb/>cardo verò vbi G, translata igitur e­<lb/>rit eo motu, puppis ip&longs;a à B in G. facta <lb/>itaque paruâ motione puppis ex B in <lb/>G, prora ip&longs;a quæ longè di&longs;tat à pup­<lb/>pi B maiori &longs;patio &longs;uperato translata <lb/>erit in H facta proræ in contrariam <lb/>partem ab ea quæ facta e&longs;t guberna­<lb/>culi motione. Porrò quod & in præ­<lb/>cedente quæ&longs;tione a dnotauimus, <expan abbr="lō-gè">lon­<lb/>gè</expan> meliùs procedet demon&longs;tratio &longs;i <lb/><expan abbr="fulcimentū">fulcimentum</expan> mare intelligatur, quàm <lb/>&longs;calmus, neque enim mare ceu pon­<lb/>dus, &longs;ed &longs;calmus ip&longs;e Temonisuecardo, ponderum in&longs;tar <lb/>transferuntur. </s>
</p>
<figure id="fig4"></figure>
<p type="main">
<s>Cæterùm in hac &longs;peculatione liceat nobis aliquan­<lb/>tulum à Philo&longs;opho di&longs;&longs;entire. Certè &longs;i breuitas Temo­<lb/>nis, è puppi eminentis, re&longs;pectu longitudinis totius nauis <lb/>con&longs;ideretur, & parua motio, quæ temone guberna culo­<lb/>ue moto fit, nullius ferè momenti erit ad eam quæ in pro. <lb/>ra fit translationem. aliter ergo &longs;e rem habere non dubi­<lb/>tamus, & quæ&longs;tionis &longs;olutionem aliunde petendam. Na­<lb/>uinon currentenullum ferè, aut qui vix curandus &longs;it ex <lb/>gubernaculi conuer&longs;ione nauis ad dextram &longs;ini&longs;tramue <lb/>motum fieri. at eâ currente maximum, experientiâ doce­<lb/>mur. Obliqui igitur motus qui validèin puppi &longs;it, cau&longs;&longs;a <lb/>e&longs;t non quidem ex conuer&longs;ione temonis percu&longs;&longs;io maris, <lb/>&longs;ed mare ip&longs;um, cuius fluctus naui currente obliquam te­<lb/>monis alam ad eam partem quæ mari obuertitur, impel­<lb/>lentes temonem cum puppiad contrariam partem vali­<lb/>di&longs;&longs;imè transferunt. </s>
</p>
<p type="main">
<s>E&longs;to nauis carina AB, prora B, puppis A, Temo AC, <lb/>gubernaculum AD; Itaque currentenaui, Temone in­<lb/>terim & guberna culo in eadem carinæ linea exi&longs;tentibus,
<pb pagenum="46"/>
<arrow.to.target n="fig5"></arrow.to.target><lb/>Temo quidem mare &longs;ecat, nulla fa­<lb/>ctâ in puppi, nauis ad &longs;ini&longs;tram dex­<lb/>tramue translatione. Si verò mouea­<lb/>tur gubernaculum à D in E, co moto <lb/>mouebitur aliquantulum & puppis <lb/>ad partes E, quod voluit Ari&longs;toteles. <lb/>Sedminimi, vt diximus, ea res ad tan­<lb/>tum effectum e&longs;t momenti. Temone <lb/>autem in obliquum <expan abbr="cō&longs;tituto">con&longs;tituto</expan> vt AF, <lb/>naui interim, ventorum aut remorum <lb/>vi pul&longs;a proram ver&longs;us currente te­<lb/>monis latus à fluctibus obliquam par­<lb/>tem alamue in ip&longs;o cur&longs;u ferientibus, <lb/>in contrariam partem transfertur, ad <lb/>eam nempe, ad quam ip&longs;um gubernaculum vergit. facta i­<lb/>gitur nauis ceu circa centrum centraue quæ in carina in­<lb/>ter puppim proramue con&longs;i derantur A, fertur in G, prora <lb/>verò in H. ex quibus manife&longs;tè apparet, duo ad nauis ex <lb/>temone in puppi conuer&longs;ione motionem e&longs;&longs;e ne ce&longs;&longs;aria; <lb/>Temonis nempe obliquationem, & nauis cur&longs;um, <expan abbr="quorū">quorum</expan> <lb/>&longs;i alterum &longs;ine altero adhibeatur, nullam fieri quæ alicu­<lb/>ius momenti &longs;it, nauis conuer&longs;ionem. Illud quoque nota­<lb/>mus, carinam in nauis conuer&longs;ione vectis in&longs;tar &longs;e habere, <lb/>cuius pars mota ad puppim, & mouens potentia e&longs;t; fulci­<lb/>mentum verò circa proram, potentia autem mouens ma­<lb/>reip&longs;um, temonem in nauis cur&longs;u oblique feriens. Vnde <lb/>colligimus naues, quo longiores &longs;unt in mouente ad Te­<lb/>monem adhibita maiori facilitate ad dextram &longs;ini&longs;tram­<lb/>ue propelli: quod &longs;anè ip&longs;emet con&longs;iderauit Ari&longs;toteles, <lb/>quì idcirco inquir, in extremo, non autem in medio temo­<lb/>nem poni eo quod mouenti facilimum &longs;it ab extremo <lb/>motum mouere. </s>
</p>
<figure id="fig5"></figure>
<p type="main">
<s>Ex hac no&longs;trâ &longs;peculatione ratio habetur eius ma-
<pb pagenum="47"/>chinationis, quâ in magnis fluminibus, ceu Pado, Abdua <lb/>& &longs;imilibus, Portitores, equos, currus, viatore&longs;que; ip&longs;os, è <lb/>ripa in ripam transferunt. Pulcherrima enim res e&longs;t, & <lb/>nobis per&longs;pecti&longs;&longs;ima, qui Gua&longs;tallâ re&longs;identiæ olim no­<lb/>&longs;træ oppido ad Padum, Mantuam pergentes &longs;æpi&longs;&longs;imè ad <lb/>Ca&longs;trum B<gap/>rgi Iu&longs;is ea qua diximus machinatione lati&longs;­<lb/>&longs;imum eiu&longs;dem Padi aluum tran&longs;ie cimus. Habet autem <lb/>&longs;e hoc pacto. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to fluminis citerior <lb/>ripa AB, vlterior CD. Pon­<lb/>tones duo tabulis &longs;trati, & v­<lb/>nà firmiter juncti EF, Temo <lb/>inter eorum puppes extans <lb/>GH, locus in ripa &longs;tabilis A, <lb/>funis, quo pontones, & ma­<lb/>china tota continetur AI. <lb/>fluuij decur&longs;us ver&longs;us BD, <lb/>&longs;tantibus itaque pontonibus <lb/>ad ripam citeriorem AB, Te­<lb/>mone in <expan abbr="neutrã">neutram</expan> partem pul­<lb/>&longs;o, cum aqua decurrens eum <lb/>re&longs;i&longs;tentem non inueniat, <lb/>&longs;cinditur quidem ab eo, &longs;ed <lb/>non propellit, eo autem con­<lb/>uer&longs;o & in GK con&longs;tituto, a­<lb/>la eius GK ab aqua defluente propul&longs;a machinam &longs;ecum <lb/>trahit ver&longs;us ripam CD, factâ motione circa centrum &longs;eu <lb/>&longs;tabilem locum A, otio&longs;is interim portitoribus, donec per <lb/>circuli portionem ML deuenerit ad vlteriorem ripam in <lb/>L. Vnde iterum temone in contrariam partem conuer&longs;o, <lb/>aquâ &longs;imiliter temonem propellente, per eandem circuli <lb/>portionem ad ripam citeriorem reuertitur, à qua paullo <lb/>antè di&longs;ce&longs;&longs;erat. Ex quibus apparet, motus cau&longs;&longs;am non
<pb pagenum="48"/>e&longs;&longs;e &longs;olam cam, quæ ab ala temonis fit, aquæ <expan abbr="percu&longs;&longs;ionē">percu&longs;&longs;ionem</expan>, <lb/>vt &longs;en&longs;erat Ari&longs;toteles, &longs;ed currentis a quæ temonis alam <lb/>ferientis impul&longs;ion<gap/>m: nihil autem referre, vtrum &longs;tante <lb/>naui a qua currat, vel câ currente a qua &longs;tet, vt in mari fit, <lb/>idem enim vtroque modo temo patitur. Vt autem machi­<lb/>næ huius & totius negotij &longs;pecies facilius animo concipia­<lb/>tur, &longs;chema hoc &longs;tudio &longs;orum oculis &longs;ubijciemus. </s>
</p>
<figure></figure>
<p type="main">
<s>Lembi nauiculæueideo appo&longs;itæ &longs;unt, vt oblongum <lb/>funem &longs;u&longs;tineant; id etenim nî fieret, aquæ immer&longs;us a­<lb/>quam &longs;cindens machinæ motum impediret, ideo etiam <lb/>apponuntur, ne funis madens celeriter maceretur & pu­<lb/>tre&longs;cat. </s>
</p>
<p type="main">
<s>Huic &longs;peculationi affinis e&longs;t ea, velorum eorum, <lb/>quæ obliquè ventum, excipientia frumentarijs molis <lb/>dant motum, item verticillorum ex papyro, quibus con­<lb/>tra ventum currentes per lu&longs;um pueri vtuntur. vnicum
<pb pagenum="49"/>enim horum emnium principium, & eadem, ratio. </s>
</p>
<p type="main">
<s>Diximus enim, Temonem currente naui, lateraliter <lb/>conuer&longs;um obuios fluctus ex cipientem puppim ip&longs;am ob­<lb/>liquè in alteram partem transferre. Porrò ea vela, de qui­<lb/>bus loquimur, ventorum flatibus obliquè oppo&longs;ita can­<lb/>dem ob cau&longs;&longs;am circulariter agitantur, quodvt figurâ eui­<lb/>dentius fiat, </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to velum AB, brachio <lb/>CE obliquè affixum, ita vt <lb/>angulus ACE maior &longs;it an­<lb/>gulo BCE, ventus obliquè <lb/>velum feriens FG. <expan abbr="Itaq;">Itaque</expan> quo­<lb/>niam ventus in velum obli. <lb/>quum incidit, elabiturvelum, <lb/>& circa centrum E vnà cum <lb/>brachio circumuertitur, in <lb/>cuius locum &longs;uccedit velum <lb/>HI, ex qua a&longs;&longs;idua velorum <lb/>&longs;ucce&longs;&longs;ione, brachiorum & a­<lb/>xis cui adhærent, rotatio fit <lb/>perpetua. Sed enim de Te­<lb/>mone agentes non e&longs;t interim cur de caudis auium pi&longs;ci­<lb/>umque taceamus, in&longs;tar enim remonum &longs;unt à Naturai­<lb/>p&longs;a opportunis animalium partibus, po&longs;tremis videlicet, <lb/>appo&longs;iti, quanquam nec&longs;olum Temonis v&longs;um præ&longs;tent, <lb/>vt videbimus. </s>
</p>
<p type="main">
<s>E&longs;to pi&longs;cis AB, cuius caput A, cauda verò CB. Hac <lb/>igitur neutram in partem reflexâ, pi&longs;cis pinnarum motu <lb/>rectâ in anteriorem partem progreditur. Si autem nece&longs;­<lb/>&longs;e ei fuerit ad dextram &longs;ini&longs;tramqueue conuerti non pote­<lb/>rit, ni&longs;i cauda ip&longs;a iuuetur. Omnis enim motus progre&longs;&longs;i­<lb/>uus quiete indiget, nec <expan abbr="ab&longs;q;">ab&longs;que</expan> &longs;tabili fulcimento progredi
<pb pagenum="50"/>
<arrow.to.target n="fig6"></arrow.to.target><lb/>pote&longs;t, quod in libris de ani­<lb/>malium ince&longs;&longs;u docetip&longs;e­<lb/>met Philo&longs;ophus. Sit igitur, <lb/>pi&longs;cem conuerti velle, & fie­<lb/>ri capite in D, deflectet illi­<lb/>co caudam in E, caque; aquam <lb/>ceu &longs;tabile quippiam <expan abbr="feriēs">feriens</expan> <lb/>eiqueue quod<gap/>mmodo fultus, <lb/>reliquum corpus CA refle­<lb/>ctet in D, &longs;i autem conuerti <lb/>velit in F, caudam defle ctet in G, & eadem ratione <gap/> cte­<lb/>tur in F. Sed & Temonis quoque v&longs;um præ&longs;tat natatili­<lb/>bus & volatilibus cauda. Sit enim rectus pi&longs;cis, hoc e&longs;t, re­<lb/>ctâ pergens IKL, caudam obliquet in KM itaque ex a­<lb/>quæ in ip&longs;o motu colli&longs;ione, eius po&longs;teriora pellentur vbi <lb/>INO. Hæc itaque nos de Temone, quatenus ad hanc <lb/>quæ&longs;tionem pertinet, con&longs;idera&longs;&longs;e &longs;it &longs;atis. </s>
</p>
<figure id="fig6"></figure>
<p type="head">
<s>QVÆSTIO VI.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, Cur quanto Antenna &longs;ublimior fuerit, ÿ&longs;dem velis, & <lb/>vento eodem celeriùs ferantur nauigia?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit Philo&longs;ophus, inquiens: An quia malus quidem <lb/>&longs;it vectis, fulcimentum verò mali &longs;edes, in qua colloca­<lb/>tur, pondus autem quod moueri debet, ip&longs;um nauigium: <lb/>mouens verò is, qui vela tendit &longs;piritus? Si igitur quanto <lb/>remotior fuerit fulcimentum facilius cadem potentia, & <lb/>citiùs idem mouet pondus, altius certè &longs;ublatâ antennâ, <lb/>velum à mali &longs;ede, quae fulcimentum e&longs;t remotius faciens, <lb/>id efficiet. Hæcille. quæ &longs;ic figurâ explicamus. </s>
</p>
<pb pagenum="51"/>
<figure></figure>
<p type="main">
<s>E&longs;to nauis AB, malus CD, <lb/>mali &longs;edes D, locus antennæ <lb/>&longs;ublimior C, depre&longs;&longs;ior E: ita­<lb/>que quoniam CD vectis e&longs;t, <lb/>quo mouens remotior fuerit à <lb/>fulcimento D, co citiùs & vio­<lb/>lentiùs pellet, velocius ergo <lb/>nauis mouebitur antenna in <lb/>C, quàm in E, con&longs;tituta. </s>
</p>
<p type="main">
<s>Plau&longs;ibilia &longs;unt hæc, at certè per veritatem ip&longs;am, <lb/>non vera. Rogo, Si fulcimentum dum vectis mouetur, <expan abbr="cē-trum">cen­<lb/>trum</expan> e&longs;t, centrum vtique motus erit D. &longs;pirante igitur va­<lb/>lidè vento inclinabitur malus, fietque; vbi FGD, quæ qui­<lb/>dem in clinatio vio lentius fiet, vento pellentein F q uàm <lb/>in G, vtpote puncto à fulcimento remotiore. Impul&longs;o ma­<lb/>lo, duo nece&longs;&longs;ariò <expan abbr="cō&longs;equentur">con&longs;equentur</expan>, vel enim ad ip&longs;am &longs;edem <lb/>D. frangetur vel puppis ip&longs;a circa D punctum conuer&longs;a, <lb/>vt mali &longs;e quatur motum eleuabitur. Prora verò &longs;ubmer­<lb/>getur facta naui in HDI. Quod &longs;i qui&longs;piam funem ad ma­<lb/>li &longs;ummitatem annexam ad ip&longs;am puppim alligauerit in <lb/>B, impe dietur &longs;anè mali in clinatio ad partes F, & ideo nul­<lb/>la vis pror&longs;us fiet in D ex vectis ratione. Attamen nihilo <lb/>&longs;ecius, quo &longs;ublimior fuerit antenna, eo faciliùs à &longs;pirante <lb/>vento puppis eleuabitur. quatenus igitur malus vectis <lb/>e&longs;t, hoc tantum quod dicimus operatur. Quod &longs;i contrà <lb/>obiectum fuerit, experientiam docere, quo &longs;ublimior an­<lb/>tenna fuerit, eo citiùs nauigium, &longs;piritu flante moueri. <lb/>Re&longs;pon&longs;io facilis, nempe, mirum non e&longs;&longs;e, &longs;i mali pars &longs;ub­<lb/>limior validius à vento feriatur. Videmus enim, & turres <lb/>quo &longs;ublimiores fuerint, eo magis à ventorum impetuo&longs;is <lb/>flatibus infe&longs;tari, quod &longs;anè ad vectis longitudinem refer­<lb/>re, e&longs;&longs;et ridiculum. Cætcrùm quod ad puppis faciliorem <lb/>eleuationem ex mali ip&longs;ius altitudine pertinet, ad vectis
<pb pagenum="52"/>contemplationem reducimus. e&longs;t enim quæ dam vectium <lb/>&longs;pecies ab alijs non con&longs;iderata, cuius brachia in angu­<lb/>lum de&longs;inunt, vtip&longs;e angulus in operatione &longs;it fulcimen­<lb/>tum. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim vectis, de quo agimus, <lb/>ABC, cuius brachia AB, BC. iuncta <lb/>ad angulum B, &longs;itqueue B in operatione <lb/>fulc mentum. Nec quicquam refert <lb/>quatenus ad v&longs;um pertinet, vtrum an­<lb/>gulus ip&longs;e rectus &longs;it, acutus vel obtu­<lb/>&longs;us. &longs;it autem modò rectus. Ponaturi­<lb/>gitur pondus aliquod in C, tum po­<lb/>tentia quædam applicetur in A, quae i­<lb/>p&longs;am vectis extremitatem A propel­<lb/>lat in D. erit igitur AB in DB & an­<lb/>gulo &longs;eruato BC in BE. Pondus igi­<lb/>tur cum parte vectis BC eleuabitur in E. In hoc autem <lb/>vectis genere attenditur proportio quam habet AB ad <lb/>BC. Si enim potentia quæ applicatur in A ita &longs;e habet ad <lb/>pondus in C vt CB, ip&longs;i BA, fiet æ quilibrium. Si maior <lb/>autem fuerit proportio potentiæ in A, ad pondus in C, ea <lb/>quam habet AB ad BC, &longs;uperatâ ponderis re&longs;i&longs;tentiâ fiet <lb/>motus. Res autem haud aliter &longs;e habet, ac &longs;i producta in <lb/>F, fieret BF æqualis BC. Tunc enim vectis ad rectitudi­<lb/>nem, &longs;eruatâ proportione, redigeretur, & ita potentia in <lb/>A, fulcimento B operaretur in F, vt operabatur in C. </s>
</p>
<p type="main">
<s>Ad huius vectis naturam referuntur fabrorum mal­<lb/>lei, quibus clauos reuellunt, forcipes item quæ tenaci <lb/>mor&longs;u clauorum capita vmbellasue apprendentes, vio­<lb/>lenterè tabulis extrahunt. In malleo itaque &longs;ubtili, vt in <lb/>figura videre e&longs;t, AB vectis e&longs;t pars quæ à fulcimento ad <lb/>potentiam; ac verò quæ à fulcimento ad pondus, ponderi
<pb pagenum="53"/>
<arrow.to.target n="fig7"></arrow.to.target><lb/>&longs;iquidem æquiparatur re&longs;i­<lb/>&longs;tentia quae fit in C. I dem ob­<lb/>&longs;eruamus in forcipe, in quo <lb/>duo quidem brachia AD, <lb/>CB, quatenus ad appren&longs;io­<lb/>nem pertinet, fulcimentum, <lb/>habentin ip&longs;o <expan abbr="cētro">centro</expan> &longs;eu ver­<lb/>rebra, & ideo quo longiores <lb/>fuerint, eo tenaciùs appre­<lb/>hendunt & retinent. quate­<lb/>nus autem ad extractionem, <lb/>facit, pro vnico forceps totus habetur vecte, cuius <expan abbr="quidē">quidem</expan> <lb/>pars à potentia ad fulcimentum AB. quæ verò à <expan abbr="fulcimē-to">fulcimen­<lb/>to</expan> ad hoc e&longs;t clauum ip&longs;um qui reuellitur AC. Violenti&longs;­<lb/>&longs;imè autem extrahunt forcipes, propterea quod maxima <lb/>&longs;it proportio longitudinis brachij BA, ad eam quæ e&longs;t ab <lb/>A ad C. </s>
</p>
<figure id="fig7"></figure>
<p type="main">
<s>His igitur hoc pacto examinatis, ad nauim & malum <lb/>reuertentes, dicimus, tunc facillimam fieri puppis eleua­<lb/>tionem, proræ verò demer&longs;ionem, cum maxima fuerit <lb/>proportio, quam habet altitudo mali, ad eam nauis <expan abbr="partē">partem</expan> <lb/>quæ à malo ad ip&longs;am puppis extremitatem, pertingit. <lb/>Quamobrem prudentes nauium fabri, vt huic difficultati <lb/>occurrant, malum non in medio quidem nauis, &longs;ed in ter­<lb/>tia ferè parte longitudinis quæ à prora e&longs;t, puppim ver&longs;us <lb/>con&longs;tituunt. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim nauis AB; cuius <lb/>malus CD: prora A: puppis B; <expan abbr="vē-to">ven­<lb/>to</expan> igitur velum impellente, <expan abbr="malū">malum</expan> <lb/>ad partem contrariam vergit, pu­<lb/>ta in FD. At <expan abbr="quoniã">quoniam</expan> ca<gap/>che&longs;ium <lb/>funi ad puppim vnitur in B, nauim, <lb/>hoc e&longs;t, ip&longs;am puppim trahatne­
<pb pagenum="54"/>ce&longs;&longs;e e&longs;t. non pote&longs;t autem; quoniam &longs;uburræ grauitas & <lb/>onera, quæ naui impo&longs;ita inter D. & <emph type="italics"/>B.<emph.end type="italics"/> grauitatis centrum <lb/>circa punctum E con&longs;tituunt, quod quidem vi ventorum <lb/>inclinante malo ab E, in G eleuaretur, quo igitur minor <lb/>fuerit proportio CD ad DE & maius pondus ip&longs;um cu­<lb/>ius grauitatis centrum in E minus præualebit potentia <lb/>pellens in C ad eleu<gap/>tionem partis nauigij, quæ à mali &longs;e­<lb/>de ad puppim intercedit, An igitur malus &longs;it vectis, pesve­<lb/>rò fulcimentum, pondus autem quodvecte mouetur, <expan abbr="ipsū">ipsum</expan> <lb/>nauigium, vt placuit Ari&longs;toteli, & qua item ratione malus <lb/>in nauim vt vectis operetur, exijs quae dicta &longs;unt, facilè pa­<lb/>tet. </s>
</p>
<p type="head">
<s>QVÆSTIO VII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quaritur, Cur quando ex puppi nauigare voluerint, non flante ex <lb/>puppi vento, veli quidem partem, quæ ad gubernatorem vergit, <lb/>con&longs;tringunt; illam verò quæ proram ver&longs;us e&longs;t, pedem <lb/>facientes, relaxant?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Mirabilis huius effe ctionis cau&longs;&longs;am explicat Ari&longs;tote­<lb/>les. inquit enim, An quia retrahere quidem multo <lb/>exi&longs;tente vento gubernaculum non pote&longs;t, pauco autem <lb/>pote&longs;t, quem con&longs;tringunt? propellit igitur quidem ip&longs;e <lb/>ventus, in puppim verò illum con&longs;tituit gubernaculum, <lb/>retrahens, & mare compellens: &longs;imul & nautæ ip&longs;i cum <lb/>vento contendunt; in contrariam enim &longs;e reclinant par­<lb/>tem. Hæcille. </s>
</p>
<p type="main">
<s>Cuius &longs;en&longs;um breuitate &longs;ubob&longs;curum, mirâ facilita­<lb/>te explicat Picolomineus. Nos autem vt rem lucidiorem <lb/>faciamus, &longs;chema, quod necip&longs;e fecit, nec Philo&longs;ophus, <lb/>proponemus. </s>
</p>
<p type="main">
<s>E&longs;to nauis A <emph type="italics"/>B<emph.end type="italics"/>, cuius prora A, puppis verò D, guber­<lb/>naculum C<emph type="italics"/>B<emph.end type="italics"/>, temonis ala <emph type="italics"/>B<emph.end type="italics"/>D, veli &longs;inus EF, velum vero <lb/>ita con&longs;titutum, vt directè ex puppi flantem ventum exci-
<pb pagenum="55"/>
<arrow.to.target n="fig8"></arrow.to.target><lb/>piat. Hoc vbi euenerit, naui­<lb/>gium, rectâ è puppi mouetur <lb/>in proram; Si autem ventus la­<lb/>teraliter &longs;pirat, puta à parte <lb/>G ver&longs;us H & nihilo &longs;ecius na­<lb/>uigium, ac &longs;i ventus ex pup­<lb/>pi e&longs;&longs;et antror&longs;um propelle­<lb/>re volunt, velum quidem obli­<lb/>quant partem cius infimam, <lb/>pedem nempe, quæ e&longs;t in F <lb/>contrahentes, Cornu verò <lb/>antennæ vbi E, proram ver&longs;us <lb/>laxantes ventumque; ip&longs;um obliquè ex cipientesid <expan abbr="efficiūt">efficiunt</expan>, <lb/>vt ventus minus violenter feriat, & minori &longs;ui parte <expan abbr="velū">velum</expan> <lb/>impleat, & quoniam ventus velum pellit in partem con­<lb/>trariam, nempe in H, ip&longs;ivt vento re&longs;i&longs;tant conuer&longs;o gu­<lb/>bernaculo ex C in L, & temone <emph type="italics"/>B<emph.end type="italics"/>D, in <emph type="italics"/>B<emph.end type="italics"/>M compellunt <lb/>proram ad partem à qua ventu<gap/> ip&longs;e &longs;pirat. Sit igitur inter <lb/>ventum & temonem pugna, illo proram in dextram, hoc <lb/>verò eandem in &longs;ini&longs;tram pellente, <expan abbr="itaq;">itaque</expan> cum neuter præ­<lb/>ualeat, nece&longs;&longs;ario nauis mediam viam, quæ inter <expan abbr="vtramq;">vtramque</expan> <lb/>e&longs;t, &longs;uo cur&longs;u tenet. Nautæ autem ideo in partem nauis <lb/>AE<emph type="italics"/>B<emph.end type="italics"/>, quæ ver&longs;us ventum e&longs;t, &longs;e conferunt, vt vento æqui­<lb/>librium faciant, ne &longs;cilicetnaui in <expan abbr="cōtrariam">contrariam</expan> partem pel­<lb/>lente &longs;piritu, eam demergat. Cæterùm quod nec Ari&longs;to­<lb/>teles nec Picolomineus animaduerterunt, velum obli­<lb/>què con&longs;titutum à vento in anteriora impellitur eandem <lb/>ob cau&longs;&longs;am, quam retulimus, vbi de temone & velis, qui­<lb/>bus farin ariæ molæ <expan abbr="cōuertuntur">conuertuntur</expan>, verba faceremus. Quod <lb/>autem addit Picolomineus rem ad vectem reduci po&longs;&longs;e, <lb/>non e&longs;t cur &longs;ub &longs;ilentio prætereamus. Ventus, in quit, pon­<lb/>deris gubernaculum mouentis vicem obtinet; centrum <lb/>verò (fulcimentum intelligit) in medio nauis e&longs;t, quod ta-
<pb pagenum="56"/>men ad proram vergit, vt faciliùs ip&longs;i vento re&longs;i&longs;tere po&longs;­<lb/>&longs;it. Tunc enim in rectum mouebitur nauis, cum &longs;ibi inui­<lb/>cem æ quatæ vires, qua&longs;i libramentum con&longs;tituerint. Hæc <lb/>ille, cuius &longs;en&longs;um figurâ propo&longs;itâ facilè aperiemus. </s>
</p>
<figure id="fig8"></figure>
<figure></figure>
<p type="main">
<s>E&longs;to carina AB, cuius prora <lb/>A, puppis, B temo BC, ventus verò <lb/>obliquè feriens H. Conuer&longs;us ita­<lb/>que temo vt in BC vndarum vi cur­<lb/>rente naui repul&longs;us &longs;it in EF ten­<lb/>dens ver&longs;us I, quo ca&longs;u prora con­<lb/>uertitur in D, nempe contra <expan abbr="ventū">ventum</expan> <lb/>qui &longs;pirat ex H. fit autem conuer­<lb/>&longs;io circa punctum G, quod fulcimenti locum obtinet. <expan abbr="Vē-tus">Ven­<lb/>tus</expan> verò ad contrariam <expan abbr="partē">partem</expan> proram impellit, repugnans <lb/>Temonis violentiæ contra ip&longs;am proram dirigentis. E&longs;t i­<lb/>gitur AB, &longs;eu DE carina, in&longs;tar vectis, cuius fulcimentum <lb/>G, vis mouens mare quo temo EF repellitur, pondus ve­<lb/>ro, ventus premens in D; quo igitur remotior erittemo à <lb/>fulcimento G, D autem vbi pondus ei vicinius, eo magis <lb/>temo venti vim&longs;uperabit. Hæc Picolominei ratio, quam <lb/>explicauimus, &longs;anè ingenio&longs;a e&longs;t, verum enimuero, quo­<lb/>niam fulcimentum &longs;ui naturâ &longs;tare debet, hic verò <expan abbr="nullã">nullam</expan> <lb/>habeat &longs;ta bilitatem, difficultatem patitur. </s>
</p>
<p type="head">
<s>QVÆSTIO VIII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur ex figuris omnibus rotundæ faciliùs <lb/>moueantur?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Trifariam, in quit Ari&longs;toteles, circulum rotari contin­<lb/>git; Aut &longs;ecundum ab&longs;idem <expan abbr="cētro">centro</expan> &longs;imul moto, quem­<lb/>admodum plau&longs;tri vertitur rota; aut circa manens cen­<lb/>trum, velutitrochleæ puteorum, &longs;tante centro: Autin pa­<lb/>uimento manente centro, &longs;icuti figuli rota conuertitur.
<pb pagenum="57"/>Cau&longs;&longs;am verò explicans, ait, celerrima eiu&longs;modi corpora <lb/>e&longs;&longs;e, eo quod paruâ &longs;ui parte planum contingunt, vti cir­<lb/>culus &longs;e cundum punctum, item quoniam non offen&longs;ant: <lb/>Non offen&longs;andi vero e&longs;&longs;e cau&longs;&longs;am, quod &longs;emotum à terra <lb/>habeant angulum. Item propterea quod corpus, cui fiunt <lb/>obuiam, &longs;ecundum pu&longs;illum tangunt. Rectilineo autem <lb/>aliter euenire, quippe quod rectitudine &longs;uâ, multum pla­<lb/>ni contingat. Ad hæc, quo nutat pondus eo mouentem <lb/>mouere. </s>
</p>
<p type="main">
<s>Hæc ferè Philo&longs;ophus, cuius rationes ad eum &longs;olum­<lb/>modo circularem motum faciunt, qui fit &longs;e cundum ab&longs;i­<lb/>dem, vt in carrorum rotis v&longs;u venit, nec aptantur rotis fi­<lb/>gulorum trochlei&longs;queue, cuiu&longs;modi &longs;unt illæ, quæ &longs;upra <lb/>puteos appenduntur. Nos igitur, ad Ari&longs;totelis mentem, <lb/>primam rotationis &longs;peciem, quæ e&longs;t &longs;ecundum ab&longs;idem, <lb/>examinabimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to rota &longs;phæ­<lb/>raue AB, cuius cen­<lb/>trum C; Horizontis <lb/>planum DE; conta­<lb/>ctus circuli in plano <lb/>B. <expan abbr="perpē">perpem</expan> dicularis ho­<lb/>rizonti à puncto <expan abbr="cō-tactus">con­<lb/>tactus</expan> B ip&longs;a <emph type="italics"/>B<emph.end type="italics"/>CA, <lb/>tran&longs;iens per <expan abbr="centrū">centrum</expan> <lb/>C, partes rotæ circa <lb/>perpendicularem AF<emph type="italics"/>B<emph.end type="italics"/>, AG<emph type="italics"/>B<emph.end type="italics"/>, angulus contactus G<emph type="italics"/>B<emph.end type="italics"/>E. <lb/>Primo itaque id con&longs;tat, circulum in puncto planum, &longs;eu <lb/>lineam contingere. At quoniam, vt Mechanici, de circulis <lb/>roti&longs;queue &longs;eu &longs;phæris agimus materialibus, rectè Philo&longs;o­<lb/>phus non in puncto planum præ cisè tangere dixit, &longs;ed &longs;e­<lb/>cundum partem &longs;ui minimam. Angulum porro, quem à <lb/>terra &longs;emotum dicit, ip&longs;e angulus e&longs;t contingentiae. cleua­
<pb pagenum="58"/>tur enim ex <emph type="italics"/>B<emph.end type="italics"/> in G. Si autem corpus quodpiam in plano <lb/>fuerit, puta HI in puncto illud tanget ci culus ei occur­<lb/>rens, exempli gratiâ in K. Hæc igitur accidunt circulari <lb/>figuræ. In lateratis autem &longs;ecus fit, quippe quænec in <expan abbr="pū-cto">pun­<lb/>cto</expan> &longs;eu &longs;ecundum paruam &longs;ui partem, planum tangunt, <lb/>nec &longs;emotum vt circulus à plano habent angulum, nec <lb/>impingentes offen diculum in puncto tangunt. Cæterùm <lb/>poti&longs;&longs;imam facilitatis motus in rotatione quæ fit &longs;ecun­<lb/>dum ab&longs;idem, e&longs;&longs;e cau&longs;&longs;am dixit, nempe quò nutat pon­<lb/>dus cò à mouente impelli ac moueri. Primò igitu circu­<lb/>laris &longs;phæricaue figura in æquilibrio &longs;tat; æquales enim <lb/>&longs;unt partes quæ circa perpendicularem: ceu &longs;unt AF<emph type="italics"/>B<emph.end type="italics"/>, <lb/>AG<emph type="italics"/>B.<emph.end type="italics"/> &longs;i enim impul&longs;us fiat ex parte F, pars oppo&longs;ita nuta­<lb/>bit, & propendet in pa<gap/>tem G, & &longs;uo nutu motuque; &longs;ecum <lb/>trahet partem AF<emph type="italics"/>B<emph.end type="italics"/>, fietqueue progre&longs;&longs;us. Si enim ducatur <lb/>FCG diameter, ip&longs;i horizonti æ que di&longs;tans, erit veluti li­<lb/>bra, cuius pondera vtrinque AF<emph type="italics"/>B<emph.end type="italics"/>, AG<emph type="italics"/>B<emph.end type="italics"/>, brachia verò <lb/>æqualia CF, CG. Potentia autem quâ trahitur pellitur­<lb/>ue ad in&longs;tar ponderis &longs;e habet, quo addito partium alteri, <lb/>facto queue rece&longs;&longs;u ab æquilibrio, &longs;equetur motus. Putauêre <lb/>quidam, vt refert Philo&longs;ophus, <expan abbr="circularē">circularem</expan> lineam, ita per­<lb/>peti motu ver&longs;atumiri, vt manentia, propte<gap/> contrarium <lb/>nixum, manent, neque enim circulus in plano contrarium <lb/>nixum habet, cum &longs;it, veluti dicebamus, in æquilibrio & <lb/>facilis in vtramuis partem moueri. Veruntamen perpe­<lb/>tuum e&longs;&longs;e non po&longs;&longs;e horum corporum motum, ea e&longs;t cau&longs;­<lb/>&longs;a, quod violentum accidat naturæ, & ideo non durabile. <lb/>Ad hæc, addit Philo&longs;ophus, Maiores circulos ad minores <lb/>nutum habere <expan abbr="quēdam">quendam</expan>; & nutum maioris ad minoris nu­<lb/>tum, &longs;e habere vt angulos ad angulos, & <expan abbr="diametrū">diametrum</expan> ad dia­<lb/>metrum. Angulos autem hî c&longs;ectores ip&longs;os vocat; oportet <lb/>enim circulos tum maiores tum minores circa idein cen­<lb/>trum e&longs;&longs;e con&longs;titutos. Hæc autem non ab&longs;imili ab eo <lb/>quod &longs;uprà po&longs;uimus &longs;chemate explicantur. </s>
</p>
<pb pagenum="59"/>
<figure></figure>
<p type="main">
<s>E&longs;to enim circulus <lb/>AB circa centrum, C, <lb/>Horizontis planum DE, <lb/>tangens circulum in B, <lb/>linea verò perpendicu­<lb/>laris per centrum BCA. <lb/>Sit autem circa idem <expan abbr="cē-trum">cen­<lb/>trum</expan> C, minor circulus <lb/>FG, ducatur queue CH &longs;e­<lb/>cu<gap/> minorem circulum in I, tangens verò maiorem in H, <lb/>con&longs;tituen&longs;queue cum AC linea angulum ACH, duos an­<lb/>gulos, ex Ari&longs;totelis mente comprehendentem, hoc e&longs;t, <lb/>duos &longs;ectores ACH, FCI. quoniam igitur &longs;ector &longs;eu an­<lb/>gulus ACH, &longs;uo &longs;patio &longs;uperat angulum &longs;eu &longs;ectorem <lb/>FGI, facilè ex nutu quem maior &longs;upra minorem habet, <lb/>maior ip&longs;e mìnorem mouet. Videtur autem tacitè Philo­<lb/>&longs;ophus hæc ad vectis naturam referre, cuius altera extre­<lb/>mitatum in centro &longs;it, altera verò in ab &longs;ide, & ita &longs;e habe­<lb/>renutum maioris &longs;upra minorem, vt vectis ad vectem, hoc <lb/>e&longs;t, &longs;emid<gap/>ameter ad &longs;emidiametrum, &longs;eu &longs;ector ad &longs;ecto­<lb/>rem, quos quidem &longs;ectores, vt vidimus, angulos appellat. <lb/>Hæc autem quæ de nutu refert, licet &longs;ubtilia &longs;int, vera e&longs;­<lb/>&longs;e non videntur. Si enim in figura producatur ad oppo&longs;i­<lb/>tam partem &longs;emidiameter HC in K &longs;ecans minorem cir­<lb/>culum in L, duos alios &longs;ectores angulosue habebimus, <expan abbr="nē-pe">nen­<lb/>pe</expan> KCB, LCG, ip&longs;is ACHFCI æ quales. <expan abbr="Itaq;">Itaque</expan> quan­<lb/>tum adiuu at motum anguli ACH maioris nutus, in de­<lb/>&longs;cendendo ad partes B, tantundem retardat anguli item <lb/>maioris KCB, contra nutus (vtita appellem) in <expan abbr="a&longs;cendē-do">a&longs;cenden­<lb/>do</expan> ad partes A. & &longs;anè quatenus ad reinaturam pertinet <lb/>& ad ip&longs;um æquilibrium, non differunt maiores circuli à <lb/>minoribus, nec &longs;unt maiores minoribus mobiliores, imo <lb/>ex ali quaratione minores videntur fore ad motum faci­
<pb pagenum="60"/>liores, tum quia data materiæ æqualitate &longs;unt leuiores, <lb/>tum etiam quod maior e&longs;t angulus contactus ad planuin <lb/>circum ferentiae minoris quàm maioris circuli, vt in &longs;ubie­<lb/>
<arrow.to.target n="fig9"></arrow.to.target><lb/>cta figura angulus ABC maior <lb/>e&longs;t angulo DBC, in materiali i­<lb/>gitur circulo rotaue maiore &longs;ui <lb/>parte tanget planum DB circu­<lb/>lus, ip&longs;o AB. quicquid tamen fit, <lb/>mobiliores &longs;unt maiores circuli <lb/>non quidem ex natura circuli, <lb/>quæ tam in maioribus quàm in <lb/>ip&longs;is minoribus e&longs;t par, &longs;ed alijs de cau&longs;&longs;is, quas &longs;uo loco <lb/>examin abimus. </s>
</p>
<figure id="fig9"></figure>
<p type="main">
<s>Cæterùm vt aliquid de motu qui &longs;e cundum ab&longs;idem <lb/>fit, ex no&longs;tro penu promamus, Dicimus, Circulos, rota&longs;ue, <lb/>quæ hoc pacto mouentur, vel per horizontis planum mo­<lb/>ueri, vel per accliue, aut decliue. Siautem perhorizontis <lb/>planum, ideo facilem e&longs;&longs;e motum, quòd nunquam, cæte­<lb/>ris paribus, centrum grauitatis ip&longs;ius corporis à centro <lb/>mundi, in ip&longs;a rotatione, fiat remotius. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim planum, <lb/>horizontis AB, cui circu­<lb/>lus in&longs;i&longs;tat AD, circa cen­<lb/>trum C, diui&longs;us per <expan abbr="centrū">centrum</expan> <lb/>ip&longs;um à perpendiculari <lb/>ACD; Ducatur autem per <lb/>centrum C recta linea ho­<lb/>rizonti æquidi&longs;tans, ECFG: dum diuidatur circulus vt­<lb/><gap/>unque in partes AH, HF, FI, ID, & CI, CH iungan­<lb/>tur. Po&longs;th æcintelligatur circulum &longs;ecundum ab&longs;idem <lb/>moueri ad partes G, erit igitur aliquando punctum H, <lb/>rangens horizontis planum, tangat autem in K, tum F in
<pb pagenum="61"/>L, I in N. D verò in O. Ducanturqueue KP, LQ, NR, OS <lb/>ip&longs;i AC parallelæ horizonti autem perpen diculares. <lb/>Centrum ergo circuli, quod idem & grauitatis e&longs;t <expan abbr="centrū">centrum</expan>, <lb/>feretur per rectam CPQRS, &longs;unt enim KP, LQ, NR, <lb/>OS ip&longs;i AC &longs;emidiam etro æquales, <expan abbr="nūquam">nunquam</expan> igitur cen­<lb/>trum ip&longs;um C in circuli rotatione ab horizontis plano e­<lb/>leuabitur, nec à mundi centro fietremotius. </s>
</p>
<p type="main">
<s>Hoc autem longè aliter cæteris figuris contingit, <lb/>quarum motus ideo in æ qualis, quòd non &longs;em per in rota­<lb/>tione centium grauitatis eandem &longs;eruet à mundi centro <lb/>di&longs;tantiam. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim Ellip&longs;is <lb/>ABCD, cuius <expan abbr="cētrum">centrum</expan> <lb/>E, diameter longior <lb/>BED, breuior AEC, <lb/>Horizontis planum, <lb/>FCG. locus contactus <lb/>C perpendicularis à <lb/>contactu per centrum i­<lb/>p&longs;a CEA diuidens El­<lb/>lip&longs;im in partes æquales, & æqueponderantes ABC, <lb/>ADC. Sumantur in quadrante CD, <expan abbr="pūcta">puncta</expan> HI, tum EH, <lb/>HI iungantur, eritautem EH longior ip&longs;a EC, tum EI, <lb/>ip&longs;a EH & ED, p&longs;a EI. Rotetur ellip&longs;is &longs;ecun dum ab&longs;i­<lb/>dem, fiet igitur punctum H in K, & à puncto K horizonti <lb/>perpendicularis erigatur KL, quæ fiat æ qualis EH. P o&longs;t <lb/>hæc punctum I eritin M, & ab M perpen dicularis, æqua­<lb/>lis EI. rui&longs;us D fiat in O, & ip&longs;i ED, æqualis perpendicu­<lb/>laris OP. Mota igitur ellip&longs;ià C in K, haud ita difficilis e­<lb/>rit motus, quippe quod haud multum EH &longs;uperet EC, at <lb/>difficilior erit translatio in M, difficillima verò in O. Val<gap/><lb/>de enim à &longs;itu E, ibi attollitur grauitatis centrum, a&longs;cen­<lb/>dens nempe vbi P. Videmus igitur ex his eandem poten­
<pb pagenum="62"/>tiam in mouendo ellip&longs;im, haud pariter &longs;e habere, vt in <lb/>mouendo circulum. ibi enim centrum grauitatis fertur <lb/>per æquidi&longs;tantem horizonti, hic verò modò attollitur, <lb/>modò deprimitur, quod &longs;anè mole&longs;tiam & difficultatem <lb/>facit. Sed idem alijs figuris contingere, & maximè latera­<lb/>tis, ita docebimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim triangulum <lb/>æquilaterum ABC, cuius <lb/>grauitatis centrum E hori­<lb/>zontis planum BD. Demit­<lb/>tatur à vertice A perpendi­<lb/>cularis horizonti AF tran&longs;­<lb/>ibit autem per centrum E, <lb/>& bifariam diuidet ba&longs;im <lb/>BC in F. Sunt autem trianguli ABF, ACF, æquales & <lb/>æqueponderantes. angulus verò AFC rectus. lungatur <lb/>EC, erit igitur maior EC, ip&longs;a EF. Rotetur iraque trian­<lb/>gulum circa punctum C, fiatque; EC horizonti perpendi­<lb/>cularis, &longs;itqueue GH, & per E horizonti parallela ducatur <lb/>EK, moto igitur triangulo, centrum grauitatis E transla­<lb/>tum erit in H, &longs;ed KC æqualis e&longs;t EF, minor autem ip&longs;a <lb/>CH, eleuatur ergo centrum grauitatis ab Ein H, nempe <lb/>&longs;upra K, totum &longs;patium KH. ex qua eleuatione fit in mo­<lb/>tu difficultas. Idem pror&longs;us eadem demon&longs;tratione o&longs;ten­<lb/>deretur fieri in quadrato & alijs lateratis figuris. Curigi­<lb/>tur in plano horizontis facillimè circularia, difficile <expan abbr="autē">autem</expan> <lb/>laterata & quæ inæquales habent &longs;emidiametros, mo­<lb/>ueantur, ex dictis clarè patet. </s>
</p>
<p type="main">
<s>Ad hanc quæ&longs;tionem illud quoque facit, cur per de­<lb/>cliue planum grauiora corpora, & rotunda maximè; ma­<lb/>gno impetu dimi&longs;&longs;a, delabantur. </s>
</p>
<p type="main">
<s>E&longs;to enim rota &longs;phæraue aut Cylindrus CD, cuius <lb/>centrum E, tangens decliue planum AB in D, quæritur
<pb pagenum="63"/>cur dimi&longs;&longs;a hæc magno impetu deferantur ad partes B, <lb/>Ducatur per grauitatis centrum E ad horizontem, BK <lb/>perpendicularis FEL &longs;ecans decliue planum in G, cir­<lb/>cum ferentiam verò in H. opponitur autem EG angulo <lb/>recto EDG, maior ergo EG ip&longs;a ED, hoc e&longs;t, EH, inter <lb/>
<arrow.to.target n="fig10"></arrow.to.target><lb/>circumferentiam igitur & pla­<lb/>num decliue, &longs;patium interce­<lb/>dit HG. Ducatur item DI ip&longs;i <lb/>FG æquidi&longs;tans. non tran&longs;ibit <lb/>igitur per centrum E. minor e­<lb/>rit igitur diametro CD, quare <lb/>circulum in partes inæquales <lb/>&longs;ecabit, & non per grauitatis <lb/>centrum, quod idem cum ma­<lb/>gnitudinis &longs;eu figuræ centro &longs;upponitur. Dimi&longs;&longs;a igitur <lb/>rota, contingit quidem planum decliue in puncto D. At <lb/>centrum grauitatis premit &longs;e cun dam per lineam perpen­<lb/>dicularem FG, non &longs;u&longs;tentatur autem in H, quippe quod <lb/>inter planum & circum <expan abbr="ferentiã">ferentiam</expan> intercedat &longs;patium HG, <lb/>nec H locum habeat cui innitatur, corpus autem ita per <lb/>lineam DI e&longs;t diui&longs;um, vt longè maior &longs;it pars IFCHD <lb/>ip&longs;a DI, & centrum in ea parte eadat quæ non fulcitur. i. <lb/>taque &longs;uopte nutu, cum extra ful cimentum &longs;it D & per­<lb/>pendicularem DI ad inferiores partes rapidè rotans de­<lb/>labitur. Ducatur autem perpen dicularis GL, parallela <lb/>MN, & quoniam BN breuior e&longs;t BL, erit MN ip&longs;a GL <lb/>breuior. E&longs;t igitur punctum M mundi centro propius <lb/>quàm D & G, quare eò non impedita rota ip&longs;a &longs;uo nutu <lb/>feretur, nec&longs;tabit donec in fimum <expan abbr="locū">locum</expan> vbi quie&longs;catnan. <lb/>ci&longs;catur. Po&longs;&longs;umus etiam Rota &longs;phæraue in plano decliui <lb/>collocata, datam potentiam inuenire, quæ extremitati <lb/>diametri ad eam partem quavergit applicata ip&longs;am rotam <lb/>&longs;phæramue impediatne delabatur. </s>
</p>
<pb pagenum="64"/>
<figure id="fig10"></figure>
<figure></figure>
<p type="main">
<s>E&longs;to planum in clinatum <lb/>AB, cui Rota &longs;phæraue in&longs;i­<lb/>&longs;tat tangatque; illud in C. Rota <lb/>verò ip&longs;a &longs;phæraue DC, cu­<lb/>ius centrum E, diameter ve­<lb/>rò DEC ip&longs;i BA ad <expan abbr="punctū">punctum</expan> <lb/>contactus C, perpendicula­<lb/>ris. Ducatur per C ip&longs;i hori­<lb/>zonti perpendiculatis FCG <lb/>circulum <expan abbr="&longs;ecãs">&longs;ecans</expan> in G tum per <lb/>E ip&longs;i CG perpendicularis, ip&longs;i verò BF horizonti æ qui­<lb/>di&longs;tans HEI ceu vectis, cuius fulcimentum I re&longs;pondens <lb/>ip&longs;i C, pondus verò in E, vbi grauitatis e&longs;t centrum. Ap­<lb/>plicata igitur potentia in H erit pondus inter fulcimen­<lb/>tum & potentiam, quare vt IE ad IH ita potentla &longs;u&longs;ti­<lb/>nens in H ad pondus in E, quod demon&longs;trandum fuerat. </s>
</p>
<p type="main">
<s>Quippiam &longs;imile o&longs;ten dit Pappus 1. 8. prop. 9. alijs <lb/>tamen &longs;uppo&longs;itis & con&longs;ideratis. Dico præterea, ij&longs;dem <lb/>&longs;tantibus angulum ECI æqualem e&longs;&longs;e angulo inclinatio­<lb/>nis CBF. Producatur HI concurrens cum ip&longs;a AB in K, <lb/>concurret autem propterea, quod CIK rectus &longs;it, ICA <lb/>minorrecto, & quoniam HK parallela e&longs;t horizonti BF <lb/>alterni anguli IKC, CBF, æquales erunt. Similes autem <lb/>&longs;unt ECI, ECK, trianguli, e&longs;tqueue ECI angulus æqualis <lb/>angulo EKC, hoc e&longs;t, ip&longs;i CBF. vnde &longs;equitur, quo mi­<lb/>nor fuerit inclinationis angulus, eo facilius rotam &longs;phæ­<lb/>ramue in piano inclinato &longs;u&longs;tineri. quo enim minor fuerit <lb/>angulus ECI, eo minus latus EI & minor proportio EI <lb/>ad IH, & ideo minor potentia &longs;u&longs;tinens requiratur in H. <lb/>Cæterùm accliue & decliue planum nihil differunt ni&longs;i <lb/>re&longs;pectu. </s>
</p>
<p type="main">
<s>His ita con&longs;ideratis, admonetnos locus, vt pulcher­<lb/>rimam dubitationem diluamus. Quæritur, Cur maiores
<pb pagenum="65"/>rotae impingentes, facilius offendicula &longs;uperent quàm mi­<lb/>nores. Neque enim &longs;atisfacere videtur quod ait Ari&longs;tote­<lb/>les, ex contactu in puncto eo anguli à plano eleuationeid <lb/>fieri, alijs ergo principijs dubitatio &longs;oluitur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to rota quidem maior <lb/>AB, circa centrum C minor <lb/>vero DB circa centrum, E, <lb/><expan abbr="tãgentes">tangentes</expan> horizontis planum <lb/>in B. Diameter maioris AB, <lb/>minoris DB, offen diculum, <lb/>horizonti perpendiculare <lb/>FG. Ducatur per F horizonti <lb/>parallela FK &longs;ecans minoris <lb/>rotæ peripheriam in H, dia­<lb/>metrum verò AB in K, & à <lb/>puncto H ad <expan abbr="planū">planum</expan> horizon­<lb/>tis perpendicularis demittatur HI: erit autem HI æqua­<lb/>lis ip&longs;i offendiculo FG, & iungantur BH, BF. <expan abbr="Itaq;">Itaque</expan> quo­<lb/>niam BH ab extremo B cadit in triangulum KFB, erit <lb/>KHB angulus maior angulo KFB. Parallelæ autem &longs;unt <lb/><emph type="italics"/>K<emph.end type="italics"/>F, BG, pares ergo anguli <emph type="italics"/>K<emph.end type="italics"/>HB, HBG, pares item <emph type="italics"/>K<emph.end type="italics"/>FB, <lb/>FBG, Maior ergo HBI, ip&longs;o FBC. At minoris rotæ gra­<lb/>uitatis centrum mouetur &longs;ecundum lineam BH, maius <lb/>verò &longs;ecun dum literam BF, difficilius ergo mouebitur, & <lb/>&longs;uperabit offen diculum minorrota, quàm maior: quod <lb/>fuerat demon&longs;trandum. </s>
</p>
<p type="main">
<s>Po&longs;&longs;umus idem o&longs;tendere magis mechanicè, hoc <lb/>e&longs;t, tem ad vectem reducendo. E&longs;to horizontis planum <lb/>AB, rota maior CD planum tangens in D. rotæ verò ma­<lb/>ioris centrum E. Rota verò minor FD, tangens itidem <lb/>planum in D. rotæ autem centrum G, offendiculi verò re­<lb/>ctitudo DH. Ducatur per Hip&longs;i AB horizonti æquidi­<lb/>&longs;tans HI &longs;ecans minorem circulum in K, maiorem verò
<pb pagenum="66"/>
<arrow.to.target n="fig11"></arrow.to.target><lb/>in I. Ducantur etiam dia­<lb/>metri maioris quidem <lb/>LEM, minoris NGO, <lb/>Tum à puncto K perpen­<lb/>dicularis ducatur ad <lb/>GO, ip&longs;a KP, item à pun­<lb/>cto I ad EM perpendi­<lb/>cularis <expan abbr="Iq.">Ique</expan> Dico EQ ad <lb/>QL, minorem habere <lb/>proportionem quam GP, <lb/>ad PN. Connectatur <lb/>GK, & ei per E parallela <lb/>ducatur ER, &longs;ecans maiorem circulum in R, & ab Rip&longs;i <lb/>EM perpen dicularis ducatur RS. quoniam igitur ER <lb/>parallela e&longs;t ip&longs;i GK, erit GER angulus HGK angulo <lb/>æqualis. Recti autem &longs;unt HGP, GES reliqui ergo KGP, <lb/>RES ad inuicem &longs;unt æquales. Sed & ESR, GPK recti <lb/>&longs;unt, quare ERSGKP anguli æquales &longs;unt, & trianguli <lb/>GPKESR, per pr. diff. 1.6. &longs;imiles. Vtergo GK hoc e&longs;t <lb/>GN ad GP, ita ER hoc e&longs;t EL ad ES. Componendo igi­<lb/>tur vt NP ad PG, ita LS ad SE. quamobrem &longs;i fulcimen­<lb/>tum e&longs;&longs;etin S, pondus in E, <expan abbr="potētia">potentia</expan> in L, idem &longs;ieret ac fiat <lb/>fulcimento in P, pondere in G, potentia verò in N con&longs;ti­<lb/>tuta. & id quidem &longs;i eiu&longs;dem ponderis vtraque rota &longs;up­<lb/>ponatur. Rur&longs;us quoniam vt DK ad totum circulum DF, <lb/>ita DR ad totum DC. Minor e&longs;t autem proportio DI ad <lb/>totum circulum DC, ergo minor e&longs;t DI ip&longs;a DR. Maior <lb/>ergo MI ip&longs;a MR, maior ergo QI ip&longs;a SR, propius ergo <lb/>centro E e&longs;t Q ip&longs;o puncto S, minor e&longs;t igitur proportio <lb/>EG ad LQ quàm ES ad SL. Minor ergo potentia requi­<lb/>ritur in L ad &longs;u&longs;tinendum pondus E ex fulcimento Q hoc <lb/>e&longs;t I, quàm requiratur in N ad &longs;u&longs;tinendum pondus G ex <lb/>fulcimento P, hoc e&longs;t K. Minor ergo potentia requiritur
<pb pagenum="67"/>ad transferendam maiorem retam CD vltra offendicu­<lb/>lum IV, hoc e&longs;t, DH, quàm requiratur ad trans ferendam <lb/>minorem vltra offendiculum KT, hoc e&longs;t HD, quod fue­<lb/>rat o&longs;ten dendum. </s>
</p>
<figure id="fig11"></figure>
<p type="main">
<s>Ad hæc, quæri pote&longs;t, quo pacto plau&longs;trorum rotæ <lb/>in ip&longs;a plau&longs;tri conuer&longs;ione &longs;e habeant, nempe quæ &longs;it li­<lb/>neailla curua, quam in conuer&longs;ione de&longs;cribunt. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to rotarum in <lb/>plano orbita, <expan abbr="dū">dum</expan> plau­<lb/>&longs;trum rectâ procedit <lb/>AB, CD, Sunt autemi­<lb/>p&longs;æ lineæ, quod o&longs;ten­<lb/>demus po&longs;tea, æquedi­<lb/>&longs;tantes. Sit itaque pun­<lb/>ctum. B illud in quod <lb/>rota quæ per AB &longs;er­<lb/>tur, eò delata planum <lb/>tangit. D verò alterius rotæ at que plani contactus. Igitur <lb/>dum plau&longs;tri fit conuer&longs;io, punctum D conuer&longs;ionis fit <lb/>centrum. Stat enim interim rota & circa lineam conuer­<lb/>titur, quæ å puncto contactus D per rotæ centrum ducta <lb/>horizontis plano e&longs;t perpendicularis. ea autem &longs;tante, ro­<lb/>ta quæ in B circa centrum D <expan abbr="&longs;emicirculū">&longs;emicirculum</expan> pertran&longs;it DEF, <lb/>vbi autem rota B, peruenerit in F, plau&longs;tro iam in oppo&longs;i­<lb/>tam partem conuer&longs;o, rota quæ e&longs;t in D per lineam DC, <lb/>quæ verò in F per rectam FG mouetur, plau&longs;triqueue fit re­<lb/>gre&longs;&longs;us. Et quoniam vel D in ip&longs;a conuer&longs;ione &longs;tat omnino <lb/>nec quicquam progreditur, vt in prima figura, vel non &longs;tat <lb/>vt in &longs;ecunda, quo ca&longs;u portionem parui circuli de&longs;cribit, <lb/>ip&longs;i maiori circulo & exteriori concentricam. Vnde col­<lb/>ligimus, Plau&longs;trorum conuer&longs;iones flexione<gap/>queue &longs;emper <lb/>circa centrum, & con centricorum circulorum portiones <lb/>fieri, <emph type="italics"/>H<emph.end type="italics"/>inc etiam di&longs;cimus, cur veteres, vt ex antiquis co­
<pb pagenum="68"/>gno&longs;cimus ve&longs;tigijs, circos in quibus cur&longs;us quadrigarum <lb/>fiebant ea forma quæ apparet, efformauerint. Hoc etiam <lb/>theorema probamus. </s>
</p>
<p type="main">
<s>Cylindros, quorum ba&longs;es axi &longs;unt perpendiculares, <lb/>dum in æquato plano conuoluuntur, rectâ incedere & <lb/>per parallelas, quarum di&longs;tantia axis &longs;eu latoris longitudi­<lb/>ne præfinitur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim Cylin­<lb/>drus ABCD, cuius a­<lb/>xis GH, <expan abbr="horizōtis">horizontis</expan> pla­<lb/>no in&longs;i&longs;tens &longs;ecundum <lb/>latus AB, cui latus op­<lb/>po&longs;itum & aequale CD. <lb/>Moueatur Cylindrus <lb/>rotans, donec latus <lb/>CD, in plano &longs;it vbi EF. De&longs;cribat autem circuli CB <expan abbr="lineã">lineam</expan> <lb/>BF. Circulo verò AD lineam AE. Dico eas rectas e&longs;&longs;e, & <lb/>parallelas. Si enim &longs;uperficies ba&longs;ium DA, CB, extendan­<lb/>tur ita vt horizontis planum &longs;ecent, illud &longs;eca bunt iuxta <lb/>lineas AE BF, recta ergo e&longs;t vtraque. Sed & parallelas e&longs;&longs;e <lb/>ad inuicem ita o&longs;tendimus. quoniam &longs;emicirculus AD, <lb/>æqualis e&longs;t &longs;emicirculo BC, erit linea AE, æqualis lineæ <lb/>BF, &longs;ed & AB, æqualis e&longs;t ip&longs;i DC, quare & ip&longs;i EF. Oppo­<lb/>&longs;ita igitur quadrilateri figura ABFE latera æqualia &longs;unt, <lb/>quare EF æquedi&longs;tat ip&longs;i AB, tum AE ip&longs;i BF, quod fue­<lb/>rat demon&longs;trandum. </s>
</p>
<p type="main">
<s>Probabimus etiam &longs;i cylindri ba&longs;es axi perpendicu­<lb/>lares non fuerint, & ideo ellip&longs;es in ip&longs;a rotatione perpla­<lb/>num, parallelas quidem de&longs;cribere, &longs;ed non rectas. </s>
</p>
<p type="main">
<s>E&longs;to enim Cylindrus ABCD, cuius ba&longs;es ellip&longs;es <expan abbr="inuicē">inuicem</expan> <lb/><expan abbr="æquedi&longs;tãtes">æquedi&longs;tantes</expan>, quarum axes longiores AB, CD, Commu­<lb/>nis autem &longs;ectio cylindri & plani ad axem & horizontem <lb/>planum perpendicularis EHF. Diuidatur autem &longs;emicir-
<pb pagenum="69"/>culus EHF in partes æquales quatuor FI, IH, HG, GE. <lb/>
<arrow.to.target n="fig12"></arrow.to.target><lb/>Tum per diui&longs;ionum puncta lateri parallelae, rectæ ducan­<lb/>tur KGL, M<emph type="italics"/>H<emph.end type="italics"/>N, OIP, quæ quidem <expan abbr="cū">cum</expan> ba&longs;es AMB, DNC <lb/>parallelæ &longs;int, eruntinuicem æ quales, cumqueue circum­<lb/>ferentia E<emph type="italics"/>H<emph.end type="italics"/>F æquales, eosqueue rectos angulos <expan abbr="cō&longs;tituent">con&longs;tituent</expan>. <lb/>Ducatur po&longs;t hæc &longs;eor&longs;um recta QR, & eidem perpendi­<lb/>cularis ST eam &longs;ecans in V. applicetur autem rectæ ST <lb/>æqualis Cylindri lateri BC, ip&longs;a <foreign lang="greek">hz</foreign>. ita tamen vt punctum <lb/>E congruat puncto V, &longs;itqueue V<foreign lang="greek">h</foreign> æqualis EB, V<foreign lang="greek">z</foreign>verò æ­<lb/>qualis EC. Tum fiant VX, XY, YZ, Z<foreign lang="greek">a</foreign> æ quales ip&longs;is EG, <lb/>G<emph type="italics"/>H<emph.end type="italics"/>, <emph type="italics"/>H<emph.end type="italics"/>I, IF, & per puncta X, Y, Z, <foreign lang="greek">a</foreign> & paralleli ip&longs;i ST du­<lb/>cantur <foreign lang="greek">o a p, n *z c, l g m, k x q</foreign>, tum & his ex altera parte re­<lb/>&longs;pondentes parallelæ per puncta <foreign lang="greek">b, g, d, e</foreign>. Sit autem <foreign lang="greek">o a</foreign> æ­<lb/>qualis AF, <foreign lang="greek">a</foreign> <11> æqualis FD, item <foreign lang="greek">e</foreign> <10>, æqualis EC, <foreign lang="greek">e s</foreign> æqualis <lb/>EB, &longs;ed & <foreign lang="greek">n *z</foreign> aequalis OI, <foreign lang="greek">*z c</foreign> ip&longs;i P, <foreign lang="greek">l</foreign>yi<gap/> &longs;i MH, y <foreign lang="greek">m</foreign> verò ip&longs;i <lb/>HN, <expan abbr="tū">tum</expan> <foreign lang="greek">k x</foreign> ip&longs;i KG. & <foreign lang="greek">x q</foreign>, ip&longs;i GL & ip&longs;is æquales & aequa­<lb/>liter po&longs;itæ ad partes R, aliæ paralle læ <expan abbr="aptētur">aptentur</expan> per <foreign lang="greek">b, g, d, c</foreign>,
<pb pagenum="70"/>quibusita di&longs;po&longs;itis per puncta <foreign lang="greek">o, n, l, k, h</foreign>, item per <foreign lang="greek">p, c, m, q, z</foreign>. <lb/>ducantur lineæ <foreign lang="greek">oh, pz</foreign>, curuæ quidern & codem pacto a­<lb/>liæ curuæ illis re&longs;pondentes <foreign lang="greek">h <10>, zs</foreign>, Erunt igitur <foreign lang="greek">o, h, <10>, <lb/>p, z, s</foreign>, parallelæ quidem eo quod lincae quæ inter ip&longs;as du­<lb/>cuntur, parallelæ &longs;int & æquales, non tamen rectæ illæ, <lb/>&longs;ed curuæ. Moto igitur Cylindro circulus EHF rectam <lb/>de&longs;cribet<foreign lang="greek">ae</foreign>, ellip&longs;is verò AMB, curuam <foreign lang="greek">ohr</foreign>, ellip&longs;is au­<lb/>rem DNC, ip&longs;am curuam <foreign lang="greek">pzs</foreign>. In hoc <expan abbr="autē">autem</expan> Cylindri mo­<lb/>tuillud mirabile, velociores nempe, in ip&longs;a rotatione e&longs;&longs;e <lb/>ellip&longs;es ip&longs;o circulo EHF. Ducatur enim recta<foreign lang="greek">o<10></foreign> quæ oc­<lb/>currat ip&longs;i VS in S, & <foreign lang="greek">oh</foreign> iungatur, fietqueue triangulum <lb/><foreign lang="greek">oh</foreign>S. c&longs;t autem, angulus <foreign lang="greek">o</foreign> S <foreign lang="greek">h</foreign> rectus, maior erg. <foreign lang="greek">oh</foreign> i­<lb/>p&longs;a <foreign lang="greek">o</foreign> S, &longs;ed recta <foreign lang="greek">o</foreign> S æqualis e&longs;t ip&longs;i<foreign lang="greek">an</foreign>, hoce&longs;t, &longs;emicircu­<lb/>lo FHE. multo maior e&longs;t autem curua, <foreign lang="greek">o, n, l, k, h</foreign>, ip&longs;a recta <lb/><foreign lang="greek">oh</foreign>, &longs;ed eodem tempore quo &longs;emicirculus EHF conficit <lb/>in rotatione <expan abbr="&longs;patiū">&longs;patium</expan> <foreign lang="greek">a</foreign> V, eodem dimidia ellip&longs;is BMA me­<lb/>titur curuam <foreign lang="greek">onlkh</foreign>. velocior igitur e&longs;t ellip&longs;is ip&longs;o cir­<lb/>culo. </s>
</p>
<figure id="fig12"></figure>
<p type="main">
<s>Hæc quo que &longs;peculatio ad motum qui &longs;ecundum <lb/>ab&longs;idem fit, manife&longs;tè pertinet. Coni, quorum ba&longs;es cir­<lb/>culi &longs;unt, &longs;i in plano &longs;ecundum latus rotentur, ba&longs;i circu­<lb/>lum de&longs;cribunt, cuius centrum immobile coni ip&longs;ius e&longs;t <lb/>vertex, &longs;emidiameter verò ip&longs;um latus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to conus ABC cu­<lb/>ius vertex C ba&longs;is AB, axis <lb/>DC, ba&longs;is verò centrum, <lb/>D, latus quo planum tan­<lb/>git BC, &longs;ecatur itaque Co­<lb/>nus per latus BC & axem <lb/>DE à plano horizonti per­<lb/>pendiculari, cuius & coni <lb/>communis &longs;ectio e&longs;t ABC <lb/>triangulum, & quoniam coni grauitatis centrum e&longs;t in
<pb pagenum="71"/>axe ip&longs;o, conus in partes æque <expan abbr="pōderantes">ponderantes</expan> &longs;ecatur AEBC, <lb/>AFBC, &longs;tat ergo conus &longs;ibimet æquili bris. Si autem à po­<lb/>tentia quadam moueatur, puta ab A ver&longs;us F, trahitur &longs;e­<lb/>micirculus BEA, à &longs;emicirculo AFB, & ita fit rotatio. Ita­<lb/>que &longs;i imaginemur, in finitos v&longs;que ad verticem parallelos <lb/>ba&longs;i cir culos, eorum &longs;emicirculi in ip&longs;o motu & trahent & <lb/>trahentur; at cum ad verticem circuli de&longs;inant, nec ibi &longs;e­<lb/>micirculi &longs;int qui trahant & trahantur, motus rotationis <lb/>pror&longs;us ce&longs;lat & vertex ip&longs;e immobilis fit rotationis cen­<lb/>trum. Quoniam igitur lateris BC, punctum C &longs;tat, B verò <lb/>circa ip&longs;um mouetur, in ip&longs;o motu circulus de&longs;cribitur <lb/>BHIK, cuius &longs;emidiameter BC, & eodem pacto alij cir­<lb/>culi in cono, qui ba&longs;i HEBF &longs;unt æquedi&longs;tantes, circulos <lb/>in plano circa idem centrum de&longs;cribent, vt facile videre <lb/>e&longs;t in obiecto &longs;chemate. Huic &longs;imilem demon &longs;trationem <lb/>affert Heron in libello Automatum, quem nos Tyrones <lb/>adhuc vernacule è Græco translatum, Ven<gap/>e tijs prælo <lb/>&longs;ubiecimus. </s>
</p>
<p type="main">
<s>Porrò &longs;i conus rotundus pro ba&longs;i ellip&longs;im habeat, <lb/>&longs;ectionem videlicet per planum axi non perpendiculare, <lb/>in ip&longs;a rotatione, &longs;tante vertice, ellip&longs;is ba&longs;is, ellip&longs;im de­<lb/>&longs;cribit in plano, cuius maior diameter à puncto quod co­<lb/>nivertex e&longs;t, ita diuiditur, vt diametri pars maior æqualis <lb/>&longs;it lateri maximo; minor verò æqualis lateri minimo. Sed <lb/>hæc ad aliam pertinent &longs;peculationem. </s>
</p>
<p type="main">
<s>His ita que de moturotundorum, qui circa ab&longs;idem <lb/>fit, con&longs;ideratis, reliquum e&longs;&longs;et de motu trochlearum, qui <lb/>circa centrum &longs;it, opportunè agere, &longs;ed cùm in &longs;equenti <lb/>quæ&longs;tione de hoc &longs;ermonem faciat Philo&longs;ophus, ad ea <lb/>quæ ibi di&longs;puta buntur, lectorem ablegamus. </s>
</p>
<p type="main">
<s>Modò de tertia motus &longs;pecie nobis erit &longs;ermo; in <lb/>qua quidem &longs;pecienonnulla perpendemus, quæ omi&longs;it A­<lb/>ri&longs;toteles. Agitur autem hîc de rotundorum corporum
<pb pagenum="72"/>motu, qui fit çirca axem horizonti perpendicularem, axis <lb/>altera extremitate in codem horizontis plano manente, <lb/>vti videre e&longs;t in ip&longs;is figulorum rotis. </s>
</p>
<p type="main">
<s>Hanc motus &longs;peciem in extrema quæ&longs;tionis parte <lb/>cum duabus alijs &longs;peciebus comparans ait, cam quæ in <lb/>obliquo fit motionem (ita enim hanc, de qua agimus, ap­<lb/>pellat) ip&longs;am impellere mouentem, hoc e&longs;t, nullum ex&longs;e <lb/>ad motum propen&longs;ionem habere, nutumue, & omnia illi <lb/>e&longs;&longs;e à motore, &longs;ecundum verò eam motionem, quæ &longs;upra <lb/>diametrum e&longs;t, &longs;e ip&longs;um mouere circulum. Dixerat enim, <lb/>ea referens quæ &longs;uperiùs circa principium de circulo ver­<lb/>ba faciens, examinauerat, circulum ex duabus fieri latio­<lb/>nibus, altera præter, altera verò &longs;ecundum naturam, & <lb/>ideo hanc &longs;emper nutum habere, & ceu continuo motam <lb/>ab eo moueri quimouet. Videtur autem clarè profiteri, <lb/>ideo difficiliorem e&longs;&longs;e huius terriæ &longs;peciei motum, eo <lb/>quòd nutu <gap/>areat proprio & t<gap/>m ab alieno, vt ita di­<lb/>cam, motore, moueatur. </s>
</p>
<p type="main">
<s>Veruntamen motum hunc facilitate alijs illis duo­<lb/>bus nequaquam cedere, facilè ex &longs;equentibus o&longs;tende­<lb/>mus. </s>
</p>
<p type="main">
<s>Primo, quia pondus totum rotati corporis, ex graui­<lb/>tatis centro quodin ip&longs;o axe e&longs;t à plano cuinititur, &longs;u&longs;ti­<lb/>netur: minima quidem &longs;ui parte axe ip&longs;o tangente <expan abbr="planū">planum</expan> <lb/>vndefit, nullam ferè dum rotatur corpus, circa centrum <lb/>vbi nititur, frictionem partium fieri. Præterea grauitatis <lb/>centrum &longs;emper &longs;tat, nec minimum quidem in ip&longs;a rota­<lb/>tione attollitur, quod &longs;anè cum naturæ &longs;it repugnans, dif­<lb/>ficultatem facit. Ad hæc circa axem ita libratur rota, vt <lb/>quantumuis exigua potentia alteri parti applicetur, alte­<lb/>raillico &longs;uperata moueatur. Licet enim propliè ea <expan abbr="tantū">tantum</expan> <lb/>corpora æquilibrare dicantur, quæ ob ponderis hinc in de
<pb pagenum="73"/>æqualitatem horizonti fiunt æquidi&longs;tantes, nihilominus <lb/>& hic aliquam e&longs;&longs;e æquilibrij &longs;imilitudinem patebit. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim rota ABCD, <lb/>cuius axis horizonti perpendi­<lb/>cularis FEG tran&longs;iens per cen­<lb/>trum E, tangens autem planum <lb/>in puncto G. Ducatur diame­<lb/>ter BED, Itaque &longs;i per diame­<lb/>trum BED, & axem FEG cor­<lb/>pus diuidatur, eo quòd <expan abbr="centrū">centrum</expan> <lb/>grauitatis in axe inueniatur, <lb/>corpus ip&longs;um in duas partes <expan abbr="tū">tum</expan> <lb/>mole tum <expan abbr="pōdere">pondere</expan> æquales &longs;ecabitur, nempe BAD, BCD. <lb/>Nullaigitur adhibita vi extranea &longs;tabit corpus in <expan abbr="quodã">quodam</expan>, <lb/>vt diximus, æquilibrio. At alteri partium potentiâ quauis <lb/>licet exigua appo&longs;itâ, puta in C, præualebit pars BCD, & <lb/>partem BAD vel impellet vel rapiet, alterâ interim eius <lb/>motui ob&longs;equente. Potentia igitur quæ in C, nullam rem <lb/>quæ impediat inueniens, velo ci&longs;&longs;imè rotam mouet, quod <lb/>eo faciliùs velocius queue fit, quo magis rota e&longs;t in motu, e­<lb/>ius verò diameter maior & potentia mouens à centro re­<lb/>motior, & &longs;anè motus <expan abbr="facilitatē">facilitatem</expan> inde cogno&longs;cimus, quòd <lb/>ip&longs;o impul&longs;ore ab impul&longs;u ce&longs;&longs;ante, diuti&longs;<gap/>è rota im­<lb/>pre&longs;&longs;um motum &longs;eruet, nec ni&longs;i po&longs;t longam rotationem <lb/>omnino quie&longs;cat. </s>
</p>
<p type="main">
<s>Cæterùm quia &longs;icco, vtaiunt, pede Ari&longs;toteles quæ <lb/>ad hunc motum <expan abbr="pertinēt">pertinent</expan> pertran&longs;ijt, nos quædam quæ ad <lb/>hancrem faciunt, diligentiùs expendemus. </s>
</p>
<p type="main">
<s>Quærimus igitur primò; Cur ea quæ hoc pacto <expan abbr="ro-tãtur">ro­<lb/>tantur</expan>, in ip&longs;a rotatione locum non mutent, ni&longs;i extrin&longs;eca <lb/>aliqua id fiat ex cau&longs;&longs;a. </s>
</p>
<p type="main">
<s>E&longs;to enim rota aut aliud quippiam rotundum ccu <lb/>Turbines &longs;unt, quibus pueri ludunt, quod circa axem ho­
<pb pagenum="74"/>
<arrow.to.target n="fig13"></arrow.to.target><lb/>rizonti perpendicularem mo­<lb/>ueatur, ABCD, cuius centrum <lb/>E, Diameter AEC. Modò circa <lb/>centrum E in finiti imagin entur <lb/>circuli, alij alijs minores v&longs;que <lb/>ad <expan abbr="centrū">centrum</expan> ip&longs;um, vti &longs;unt FGH; <lb/>ibi enim circuli e&longs;&longs;e de&longs;inunt, <lb/>vbi nullum amplius e&longs;t &longs;patium. <lb/>Applicetur itaque potentia in <lb/>B, quæ rotam v. geat ver&longs;us A. <lb/>codem igitur tempore & in&longs;imul A ver&longs;us D, D ver&longs;us C, <lb/>& Cver&longs;us B mouebitur. quantum enim &longs;emicirculorum <lb/>à parte CBA tran&longs;it vltra diametrum AEC, tantundem <lb/>&longs;emicir culorum, qui &longs;unt ad partem ADC, tran&longs;ibit ad <lb/>partes CBA. At vbi de&longs;ierit motus, ibi de&longs;init rotatio; vbi <lb/>autem de&longs;init &longs;patium, de&longs;init motus, &longs;ed vbi de&longs;inunt cir­<lb/>culi, de&longs;init &longs;patium, quare in centro cum non &longs;int circuli, <lb/>nec &longs;patium ibi de&longs;init motus. nulla enim ade&longs;t ratio, cur <lb/>ip&longs;um corpus alio à loco in quo e&longs;t, ex rotatione transfe­<lb/>ratur. Statergo rotans, quod fuerat demon&longs;trandum. E&longs;t <lb/>autem hæc demon&longs;tratio ei &longs;imilis, quam &longs;uprà retuli­<lb/>mus de coni in plano circa verticem rotatione, quam ab <lb/>Herone in Automatis excogitatam diximus. </s>
</p>
<figure id="fig13"></figure>
<p type="main">
<s>Addimus in hoc rotationis genere corpus in ip&longs;o­<lb/>motu fieri leuius, idqueue eo magis, quo rotatio velocior. <lb/>Cau&longs;&longs;a e&longs;t, quod lateralis motus eum motum aliqualiter <lb/>impedit, qui ex naturali grauitate fit ad centrum, idcirco <lb/>experientiâ docemur, leui&longs;&longs;imos e&longs;&longs;e turbines, quibus pu­<lb/>eri ludunt, &longs;i manus teneantur palmâ, dum citi&longs;&longs;ima rota­<lb/>tione mouentur. </s>
</p>
<p type="main">
<s>Ad hæc alia proponitur, & &longs;oluitur quæ&longs;tio, Cur ro­<lb/>tunda corpora huic motionis generi &longs;int aptiora. </s>
</p>
<p type="main">
<s>Explorati&longs;&longs;imum e&longs;t, corporum, quæ ita mouentur,
<pb pagenum="75"/>partes eo e&longs;&longs;e velo ciores, quo magis à centro, circa quod <lb/>mouentur, fuerint remotiores. maius enim eodem tem­<lb/>pore &longs;patium pertran&longs;eunt. quo igitur figura ijs partibus, <lb/>quæ longius à centro ab&longs;unt, abundauerit magis, eo faci­<lb/>lius, & velocius in circulum rotata mouebitur. Modò o­<lb/>&longs;tendemus, circularem cæteras omnes ea qua diximus <lb/>partium à centro remoti&longs;&longs;imarum copiâ abundare. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to triangulum puta æqui­<lb/>iaterum, ABC circa centrum D. <lb/>Ducantur Catheti per centrum ab <lb/>oppo&longs;itis angulis ad oppo&longs;ita late­<lb/>ra ADG, BDF, CDE, erunt autem <lb/>lateribus perpendiculares. <expan abbr="quoniã">quoniam</expan> <lb/>igitur latera AD, DB, DC, rectis <lb/>angulis &longs;ubtenduntur, maiora <expan abbr="erūt">erunt</expan> <lb/>lateribus DE, DF, DG. tres igitur <lb/>lineæ in hoc triangulo &longs;unt longi&longs;&longs;imæ DA; DB, DC. tres <lb/>verò breui&longs;&longs;imæ DE, DG, DF, quamobrem rotato &longs;uper <lb/>centrum D triangulo, tres tantum partes eius ABC velo­<lb/>ci&longs;&longs;imæ erunt, tres verò tardi&longs;&longs;imæ E, G, F. Minus igitur a­<lb/>pta e&longs;t motui huic triangularis figura, quam quadrata, in <lb/>qua partes à centroremoti&longs;&longs;imè, & ideo veloci&longs;&longs;imè &longs;unt <lb/>quatuor. <expan abbr="Itaq;">Itaque</expan> quo <gap/>agis laterata figura angulis abunda­<lb/>bit, eo magis erit ad hunc, & cæteros omnes circulares <lb/>motus aptior. At circulus infinitas, vt ita dicam, partes à <lb/>centro remoti&longs;&longs;imas habet, itaque nulla figura e&longs;t circu­<lb/>lari, in ip&longs;a rotatione, commodior atque velocior. Alia <lb/>quoque de cau&longs;&longs;a id fit, quod dum circularis figura mo­<lb/>uetur, nullis eminentibus angulis aërem verberet <expan abbr="circū-&longs;tãtem">circun­<lb/>&longs;tantem</expan>, ex qua verberatione motus impeditus &longs;it tardior. <lb/>Quæri etiam pote&longs;t, Num axe in clinato, rotæ motus ali­<lb/>qualiter impediatur? Nos negatiuam partem amplecti­<lb/>mur. </s>
</p>
<pb pagenum="76"/>
<figure></figure>
<p type="main">
<s>E&longs;to enim tota ABCD, cuius cen­<lb/>trum E axis inclin itus, circa quem <lb/>conuertitur EGF. Duobus aute pun­<lb/>ctis fulcitur GF. Sit autem tum gra­<lb/>uius tum figuræ centrum E, Perpen­<lb/>dicularis vero per inferius fulcimen­<lb/>tum tran&longs;iens HFI. Conuer&longs;a igitur <lb/>rota, grauitatis centrum &longs;tabit nec à <lb/>&longs;uo &longs;itu &longs;ur&longs;um deor&longs;umue mouebi­<lb/>tur. E&longs;t autem axis FEG, ceu vectis in <lb/>quo pondus in E, potentiæ &longs;u&longs;tinentes GF; non enim hic <lb/>vt in axe perpendiculari pondus totum ab inferiori fulci­<lb/>mento &longs;u&longs;tinetur. quo igitur minor erit proportio FE ad <lb/>FG, eo minori in digebit potentiâ is qui pondus &longs;u&longs;tinet in <lb/>G. Et hæc &longs;anè ita &longs;e habent, grauitatis çentro in axe ip&longs;o <lb/>con&longs;tituto, &longs;i enim extra fuerit motus impeditur & moto­<lb/>re ce&longs;&longs;ante citò quie&longs;cit. E&longs;to enim grauitatis centrum in <lb/>K. Dum igitur circa axem fit motus, centrum circulatum <lb/>aliquando erit in L; Secetautem rotæ diameter AC per­<lb/>pendicularem Hl in M. Porrò à punctis LK ad ip&longs;am <expan abbr="per-pēdicularem">per­<lb/>pendicularem</expan> ducantur ad rectos angulos lineæ LN, KO. <lb/>Maior e&longs;t autem MK ip&longs;a ML, maior ergo MO, ip&longs;a MN. <lb/>magisigitur à mundi centro di&longs;tat punctum N puncto O. <lb/>Centrum ergo grauitatis K &longs;i liberè dimittatur, requie&longs;cet <lb/>in K & contranaturam transferetur in L. Ce&longs;&longs;ante igitur <lb/>violentiâ & præualentenaturâ citò rota &longs;uâ &longs;ponte quie­<lb/>&longs;cet, quod fuerat o&longs;tendendum. </s>
</p>
<p type="head">
<s>QVÆSTIO IX.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur ea quæ per maiores cir culos tolluntur, & trahuntur <lb/>faciliùs, & celeriùs moueri contingat, veluti maioribus tro­<lb/>chleis, & &longs;cytalis &longs;imiliter?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Re&longs;pondet ad hæc Philo&longs;ophus, forteid cuenire, quo-
<pb pagenum="77"/>niam quanto maior fucrit illa quæ à centro e&longs;t, in æquali <lb/>tempore maius mouetur &longs;patium. quamobrem æquali <lb/>exi&longs;tente onere idem faciet. Ita enim dixerat de <expan abbr="librarū">librarum</expan> <lb/>natura, & differentijs agens, maiores minoribus exactio­<lb/>res e&longs;&longs;e. Circulos verò jibras, in quibus centrum &longs;partum, <lb/>&longs;emidiametri hinc in de æqualia brachia. </s>
</p>
<p type="main">
<s>Quod vltimo loco affirmauit, trochleas e&longs;&longs;e in&longs;tar <lb/>librarum, verum e&longs;t. Quo d autem dixit, faciliùs & cele­<lb/>rius mouere maiores libras ijs quæ minores &longs;unt, &longs;i &longs;inipli­<lb/>citer intelligatur, fal&longs;um, quippe quod facilitas motus, in <lb/>tractorijs machinis velo citati &longs;it contraria, quod demon­<lb/>&longs;trauit Guid. Vbald. in tractatu de Trochlea in 2. Corol­<lb/>lario propo&longs;itione vltima. </s>
</p>
<p type="main">
<s>Ad id autem quod dixit, quo <expan abbr="maiorēs">maiorens</expan> fuerint tro­<lb/>chleæ, eo faciliùs mouere, non e&longs;t, vt dicebamus, &longs;impli­<lb/>citer verum, quod facilè o&longs;tendemus. </s>
</p>
<figure></figure>
<p type="main">
<s>Efto enim trochlea AB circa centrum C, appen&longs;a in <lb/>puncto D, perpendicularis quæ ad mundi centrum DCE, <lb/>pondera æqualia vtrinque appen&longs;a FG. E&longs;to item alia <lb/>Trochlea, eaque; maior HI, circa centrum K appen&longs;a in L, <lb/>perpen dicularis, quæ ad mundi centrum LKM, æqualia
<pb pagenum="78"/>pondera vtrinque appen&longs;a N, O. Dico maiorem Hl ip&longs;a <lb/>minori DE facilius pondera non mouere, eo quòd &longs;it ma­<lb/>ior, illa verò difficiliùs, propterea quòd &longs;it minor. Etcnim, <lb/>quoniam vtraque trochlea per centrum graultatis à per­<lb/>pendiculari diui ditur, erunt partes DAE, DBE, æque <expan abbr="pō-derantes">pon­<lb/>derantes</expan>. Eadem ratione ip&longs;æ quoque LHM, L<gap/>M æquè <lb/>ponderabunt. Itaque &longs;i quantumuis pu&longs;illa pondera ad­<lb/>das, <expan abbr="vtriq;">vtrique</expan> earum ad alteram partem tolletur <expan abbr="æquilibriū">æquilibrium</expan>, <lb/>nec minus requiritur pondus vt recedat ab æquilibrio <lb/>Trochlea minor, quàm maior. Vnico autem verbo con­<lb/>cludi pote&longs;t di&longs;putatio, <expan abbr="tã">tam</expan> in minori quàm in maiori, bra­<lb/>chia &longs;iqui dem bifariam diuiduntur, ergo in <expan abbr="vtri&longs;q;">vtri&longs;que</expan> eadem <lb/>brachiorum proportio, & eadem ponderum ratio. </s>
</p>
<p type="main">
<s>Explorati&longs;&longs;ima &longs;unt hæc. Veruntamen cùm res ip&longs;a <lb/>doceat, verum e&longs;&longs;e quod &longs;cribit Ari&longs;toteles, huius effe­<lb/>ctus cau&longs;&longs;a aliunde à nobis, nempe à mechanicis princi­<lb/>pijs, e&longs;t mutuanda. Dico igitur, Axium, circa quos tro­<lb/>chleæ rotæue conuertuntur ad rotas ip&longs;as, varias habere <lb/>proportiones. O&longs;tendemus autem <expan abbr="rotã">rotam</expan> illam, trochleam­<lb/>ue faciliùs moueri, & mouere pondera, quo rotæ diame­<lb/>ter ad axis diametrum maiorem habuerit proportionem, <lb/>& ideo fieri po&longs;&longs;e rotam maiorem ad &longs;uum axem <expan abbr="minorē">minorem</expan> <lb/>habere proportionem quam rotam minorem ad &longs;uum. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim <lb/>trochlea AB cir­<lb/>ca centrum C, <lb/>cuius diameter <lb/>DCE &longs;it in ip&longs;a <lb/>quæ ad mundi <lb/>centrum <expan abbr="perpē-diculari">perpen­<lb/>diculari</expan>: &longs;it au­<lb/>tem appen&longs;a in D. Alia &longs;imiliter ei æqualis &longs;it trochlea F <lb/>G circa centrum H, cuius diameter IHK, conueniens
<pb pagenum="79"/>cum perpen diculari quæ ad mundi centrum. appendatur <lb/>autem in I. Habeant autem & axes, circa quos conuertan­<lb/>tur. Hi&longs;i æquales fuerint, proportione non mutatâ idem <lb/>operabuntur. Modò ponanturinæquales, &longs;itqueue axis ro­<lb/>tç AB, cra&longs;&longs;ior axe rotæ FG, &longs;itqueue cra&longs;&longs;ioris quidem &longs;emi­<lb/>diameter CL, &longs;ubtilioris autem HM. Dico per tro chleam <lb/>FG facilius attolli pondera æqualia quàm per AB, licet <lb/>altera tro chlearum alteri &longs;it æqualis. Quoniam enim me­<lb/>chanica corpora &longs;ine materia & pondere non &longs;unt, onera <lb/><expan abbr="appē&longs;a">appen&longs;a</expan> & trochlearum ip&longs;arum grauitas ex &longs;uperiori par­<lb/>te prement axes, vbi puncta L, M, quæres, &longs;ecutâ in uicem <lb/>corporum &longs;olidorum fricatione, motum ip&longs;um trochlea­<lb/>rum difficiliorem & a&longs;periorem facit. Succedit igitur im­<lb/>pedimentum loco ponderis. Duos igitur habemus vectes <lb/>DC, IH, quorum fulcimenta contra ip&longs;a C, H. Pondera <lb/>verò inter fulcimenta & potentiasin L, M. Intelligantur <lb/>autem potentiæ applicatæ punctis DI. Igitur ex natura e­<lb/>iu&longs;modi vectis, in quo pondus inter fulcimentum e&longs;t & <lb/>potentiam erit vt CL, ad CD, ita potentia in D ad <expan abbr="pōdus">pondus</expan>, <lb/>hoc e&longs;t, re&longs;i&longs;tentiam fricationis, quæ fit in L. Sed maior <lb/>e&longs;t proportio CL ad CD quàm HM ad HI. Maior igitur <lb/>ad &longs;uperandum idem &longs;eu æquale impedimentum poten­<lb/>tia requiritur in D, qu<gap/>m in I. Itaque cum vis tota in rota­<lb/>rum & axium, diametrorum proportione con&longs;i&longs;tat, fieri <lb/>pote&longs;t, quod dicebamus, minorem trochleam dari, quæ <lb/>maiorem habeat proportionem ad &longs;uum axem, quàm, <lb/>maior ad &longs;uum, quo ca&longs;u minor rota facilius imp edimen­<lb/>tum, quod diximus, ip&longs;a maiori rota &longs;eu trochlea &longs;upera­<lb/>bit. Veruntamen quoniam ex materia fiunt tum axes tum <lb/>rotæ, nec rei natura patitur axes &longs;ubtiles, & imbecilles <lb/>magna <expan abbr="pōdera">pondera</expan> &longs;u&longs;tinere po&longs;&longs;e, idcirco cra&longs;&longs;iores fiunt, quç <lb/>cra&longs;&longs;itudo cum proportione magis à magnarum rotarum <lb/>diametris &longs;uperetur; fit hinc maiores rotas datâ axium pa-
<pb pagenum="80"/>ritate facilius impe dimentum &longs;uperare quàm minores, & <lb/>hoc videtur &longs;en&longs;i&longs;&longs;e Philo&longs;ophus in ip&longs;a quæ&longs;tionis huius <lb/>propo&longs;itione, Hinc aurigæ vulgo axungiâ (quæ inde no­<lb/>men trahit) axium a&longs;peritates mitigant, vt minor in rotan. <lb/>do, ex fricatione fiat re&longs;i&longs;tentia. Concludimus igitur, fa­<lb/>cillimè trochleam illam pondus trahere, quæ cum maxi­<lb/>ma &longs;it, axem habet minimum, cumqueue axungiâ aliaue vn­<lb/>ctuo &longs;a materia perfu&longs;um. De manubrijs, quæ rotarum a­<lb/>xibus aptantur, nemo ferè verba fecit; nos igitur de his a­<lb/>liquid; &longs;iquidem res ad &longs;peculationem, qua de agimus, <expan abbr="nē-pe">nen­<lb/>pe</expan> Mechanicam pertinet. </s>
</p>
<p type="main">
<s>Manubria vectes &longs;unt, & ad vectium naturam redu­<lb/>cuntur, corum &longs;cilicet, in quibus fulcimentum e&longs;tinter <lb/>pondus & potentiam. In his autem attenditur proportio, <lb/>quam habet manubrij longitudo ad ip&longs;um axis &longs;emidia­<lb/>metrum, eo enim faciliùs mouent, quo eorum longitudo <lb/>ad axium &longs;emidia metros proportionem, habuerit ma­<lb/>iorem. Duabus autem partibus con&longs;tant, alterâ, quæ ab <lb/>axe ad angulum; quæ verè vectis e&longs;t; alterâ, cui manusi­<lb/>p&longs;a admouetur, ex qua res tota manubrium dicitur. Fiunt <lb/>autem manubria hæc vt plurimum amouibilia, &longs;unt <expan abbr="tamē">tamen</expan> <lb/>ceu rotarum ip&longs;arum partes, & rotis ip&longs;is commodè affi­<lb/>gerentur, ni&longs;i in rotatione à tran&longs;uer&longs;arijs, quibus rotæ &longs;u­<lb/>&longs;tinentur, impedimentum fieret. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim rota AB, cu­<lb/>ius axis E, terebretur autem <lb/>in F, ibiqueue paxillus affigatur <lb/>FK. Sit & alia rota CD, cu­<lb/>ius axis G, manubrium axi <lb/>appo&longs;itum GHI. Sint autem <lb/>rotæ æquales & axes æqua­<lb/>les. Sint etiam æqualia ip&longs;a <lb/>&longs;patia EF, GH, hoc e&longs;t, ma-
<pb pagenum="81"/>nubrij GHI longitudo. Dico, câdem facilitate moueri AB <lb/>rotam à potentia in FK, quâ mouetur CB, à potentia po­<lb/>&longs;ita in HI, datis ip&longs;i nempe potentijs æqualibus. Produca­<lb/>tur enim IH, v&longs;que adrotæ CD latus in L, & LG ducatur, <lb/>& FE in rota AB iungatur. Erunt igitur FE LG inter &longs;e æ­<lb/>quales. Sunt autem eorum circulorum &longs;emidiametri, qui <lb/>à punctis FL, in ip&longs;a rotatione de&longs;cribuntur. Ita igitur &longs;e <lb/>habebit potentia applicata in L ad diametrum &longs;emidia­<lb/>metrumue axis rotæ CD, vt &longs;e habet potentia applicata <lb/>in F, ad diametrum &longs;emidiame trumue axis E rotæ AB, &longs;ed <lb/>&longs;patia &longs;unt æqualia & potentiæ æquales, quare nihil re­<lb/>fert, vtrum manubrium lateria ffigatur, vel axi à latere ro­<lb/>tæ &longs;eparatum applicetur. </s>
</p>
<figure></figure>
<p type="main">
<s>Duplex autem e&longs;t ma­<lb/>nubriorum forma; altera e­<lb/>nim rectis partibus con&longs;tat, <lb/>altera verò curua e&longs;t tota, <lb/>&longs;ed rectis vtimur vt mani­<lb/>bus apprendamus, curuis <lb/>verò vt locum illis appona­<lb/>mus, & pedis pre&longs;&longs;ione ceu <lb/>in molis lapideis, quibus <lb/>gladij acuuntur, fieri a&longs;&longs;olet, conuertantnr. Cur autem <lb/>manubria hæc curua fiant, ea videtur ratio, ne videlicet <lb/>manubrij capite &longs;upra centrum in linea quæ per centrum <lb/>tran&longs;it, <expan abbr="cō&longs;tituto">con&longs;tituto</expan>, factâ interim pre&longs;&longs;ione motus à centro, <lb/>ad quod directè fieret pre&longs;&longs;io, impediretur. Curuitas <expan abbr="autē">autem</expan> <lb/>facilitatem quan dam habet, ex qua factâ modicâ flexione <lb/>axis caput, dum premitur ab ip&longs;a perpendiculari linea le­<lb/>niter abducitur. quæ cum ce&longs;&longs;ent in manubrijs quæ manu <lb/>aguntur, ideo alia forma, nempe ex rectis partibus pa&longs;&longs;im <lb/>fiunt. E&longs;to igiturillud quod ex rectis partibus AB, curuum <lb/>verò CD, linea verò, &longs;ecundum quam pede fit pre&longs;&longs;io
<pb pagenum="82"/>CDE. Hæc itaque de manubrijs &longs;eu vectibus nos con&longs;i­<lb/>dera&longs;&longs;e &longs;it &longs;atis. </s>
</p>
<p type="main">
<s>Quæriinterim po&longs;&longs;et, Cur duabus datis rotis æqua­<lb/>lis magnitudinis in æqualis ponderis, circa æquales axes <lb/>con&longs;titutis leuior faciliùs moueatur & citiùs quie&longs;cat; <lb/>grauior verò di&longs;ficilius moucatur & tardiùs ce&longs;&longs;et à mo­<lb/>tu, ea videtur ratio, quod grauior re&longs;i&longs;tens magis cum &longs;u­<lb/>peratur impre&longs;&longs;am vim &longs;u&longs;cipit, & diutiùs retinet, quod <lb/>ce&longs;&longs;at in leuiore. </s>
</p>
<p type="head">
<s>QVÆSTIO X.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitat Ari&longs;toteles, Cur faciliùs, quando &longs;ine pondere est, mouca­<lb/>tur libra, quàm cum pondus habet. Simili modo rota, & eiu&longs;modi­<lb/>quidpiam, quod grauius quidem est, item quod maius & <lb/>grauius minori, & leutori?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Breuiter autem &longs;oluit, ait enim, An quia non &longs;olum in <lb/>contrarium quod graue e&longs;t, &longs;ed in obliquam etiam dif­<lb/>ficulter mouetur? In contrarium enim ei ad quod vergit <lb/>onus mouere difficile e&longs;t, quo autem vergit, e&longs;t facile. In <lb/>obliquum autem haudqua quam vergit. Nos quod ip&longs;e <lb/>non fecit figurâ ip&longs;a appo&longs;itâ rem clariorem faciemus. </s>
</p>
<figure></figure>
<p type="main">
<s>Efto libra AB, cuius ful­<lb/>cimentum C, pondera vtrin­<lb/>que appen&longs;a AB, quorum v­<lb/>trumque ponderet 10. Item <lb/>libra DE, cuius fulcimentum <lb/>F pondere vero appen&longs;a D, E, <lb/>ip&longs;is A, B, dimidio leuiora, <expan abbr="nē-pe">nen­<lb/>pe</expan> S. Addatur ponderi B pon­<lb/>dus G, & ponderi E pondus <lb/>H, quorum &longs;imiliter <expan abbr="vtrumq;">vtrumque</expan> <lb/>ponderet S, nutabunt igitur <lb/>libræ ponderibus appo&longs;itis, &
<pb pagenum="83"/>BG &longs;ecetur in K, EH verò in N, grauius e&longs;t autem GB, e&longs;t <lb/>enim IS, ip&longs;o EH, quod e&longs;t 10. Difficiliùs autem de&longs;cen­<lb/>det BG, quàm EH. hoc autem ex doctrina Ari&longs;totelis, <lb/>quia non &longs;olum in contrarium quod graue e&longs;t, &longs;ed in obli­<lb/>quum etiam difficulter mouetur, in contrarium enim ei <lb/>ad quod vergit onus mouere difficile e&longs;t, quò autem ver­<lb/>git facilè in obliquum autem puta per lineas BK, EN non <lb/>vergit onus. Difficiliùs ergo in obliquum mouebitur pon­<lb/>dus BG ip&longs;o pondere EH. vtrumque autem in de&longs;cen&longs;u <lb/>retrahitur nempe à perpendicularibus BI, EM & retra­<lb/>ctionis quidem anguli &longs;unt æquales & æquales ip&longs;æ retra­<lb/>ctioncs. Sedgrauius e&longs;t pondus GB. quod autem grauius <lb/>e&longs;t, violentius <expan abbr="de&longs;cēdit">de&longs;cendit</expan> eo quod e&longs;t leuius. maiori igitur ni­<lb/>&longs;u atque impetu cum cætera paria &longs;int, de&longs;cendet pondus <lb/>BG, ip&longs;o EH, quod è diametro Ari&longs;totelis a&longs;&longs;ertioni e&longs;t <lb/>contrarium. ex alijs igitur principijs veritas ip&longs;a e&longs;t eruen­<lb/>da. Dicimus autem id ex proportionum fieri inæqualita­<lb/>te; quia enim is ad 10. proportionem habet &longs;e&longs;quialteram, <lb/>10. verò ad 5. duplam, maiorem proportionem habet EH <lb/>ad oppo&longs;itum pondus D, quàm BG ad pondus A, facilius <lb/>ergo trahet libra DE lcuior pondus D, quàm ip&longs;a AB, gra­<lb/>uior pondus A, quod vtique fuerat o&longs;tendendum. Alia <lb/>quoque cau&longs;&longs;a & hæc accidentalis ad hunc effectum pa­<lb/>riendum concurrit, axium nempe ad fulcimenta, in qui­<lb/>bus rotantur, fricatio. quo enim maius e&longs;t pondus cæteris <lb/>paribus, quod nos in præ cedente quæ&longs;tione demon&longs;tra­<lb/>uimus, eò maiìor fit ip&longs;a collifio. </s>
</p>
<p type="main">
<s>Porrò huius <expan abbr="quoq;">quoque</expan> &longs;peculationis e&longs;t, Cur æqualia & <lb/>&longs;imilia corpora in æqualibus &longs;imilibu&longs;queue bafibus con&longs;ti­<lb/>tuta eodem &longs;imiliqueue plano fulta, ponderibus tamen in­<lb/>æqualia, non eâdem facilitate euertantur, &longs;ed horum gra­<lb/>uiora difficilius. </s>
</p>
<pb pagenum="84"/>
<figure></figure>
<p type="main">
<s>Sit enim Pri&longs;ma &longs;eu <lb/>Cylindrus ABCD, cuius <lb/>grauitatis centrum E in <lb/>plano Cl, ba&longs;i fultus CD. <lb/>Sit & alter Cylindrus <lb/>FGHI, cuius grauitatis <lb/>centrum K fultus ba&longs;i HI <lb/>æqualis quidem & &longs;imilis <lb/>ip&longs;i AD. Sit autem grauior FGHI, ip&longs;o ABCD. Dico, pari <lb/>potentiâ vtrumque impellente, facilius euer&longs;um iri Cy­<lb/>lindrum AD, ip&longs;o Fl. Ducantur EC, KH, & æquales po­<lb/>tentiæ applicentur punctis BG, pellentes Cylindros ad <lb/>partes AF. Euer&longs;io autem non fiet donec facta corporis <lb/>conuer&longs;ione circa puncta CH, grauitatis centra E, K <expan abbr="trãs-ferunturin">trans­<lb/>ferunturin</expan> L, M, in ip&longs;is &longs;cilicet <expan abbr="perpēdicularibus">perpendicularibus</expan> ACFH. <lb/>Demit tantur EN, KO, perpendiculares ip&longs;is CD, HF. Et <lb/>quoniam CNE, HOK anguli recti &longs;unt, erunt EC KH i­<lb/>p&longs;is EN, KO, maiores, quare & LC, MH ip&longs;is EN KO, ma­<lb/>iores atto lluntur ergo in ip&longs;a cuer&longs;ione, grauitatum cen­<lb/>tra E in L, K in M. At quod grauius e&longs;t, difficilius contra <lb/>&longs;ui naturam mouetur, ideo difficilius euertetur corpus <lb/>FI, ip&longs;o AD, quod fuerat demon&longs;trandum. </s>
</p>
<p type="head">
<s>QVÆSTIO XI.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitat Philo&longs;ophus, Cur &longs;uper &longs;cytalas facilius portentur onera <lb/>quàm &longs;uper currus, cum tamen ij magnas habeant rotas, <lb/>illæ verò pu&longs;illas?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Optimè re&longs;pondet dubitationi. An, in quiens, quoniam <lb/>in &longs;cytalis nulla e&longs;t offen&longs;atio; in curribus verò axis <lb/>e&longs;t, ad quem offen&longs;ant. De&longs;uper enim illum premunt, & <lb/>à lateribus. quod autem e&longs;t in &longs;cytalis ad i&longs;thæc duo mo­<lb/>uetur & inferiori &longs;ub&longs;trato &longs;patio, & onere &longs;uperimpo&longs;i-
<pb pagenum="85"/>to, in vtri&longs;que enim ijs <gap/>euoluitur locis circulus, & motus <lb/>impellitur. Tam appo&longs;itè paucis verbis veritatem expli­<lb/>cauit, vt ferè quicquid in&longs;uper ad datur, &longs;uperuacaneum <lb/>videri po&longs;&longs;it. quicquid tamen &longs;it, ad maiorem claritatem <lb/>aliquantulum in hac ip&longs;a quæ&longs;tione immorabimur. </s>
</p>
<p type="main">
<s>Rotatas &longs;cytalas proponit hîc Ari&longs;toteles. Coniun­<lb/>ctasautem e&longs;&longs;e rotas ip&longs;is &longs;cytalis e&longs;t intelligen dum, nem­<lb/>pe, vt &longs;imul rotæ cum &longs;cytalis conuertantur. Secus enim <lb/>axium & Rotarum fieret offen&longs;atio, cuius offen&longs;ationis <lb/>vim & effectum cum nouerit Ari&longs;toteles, vel hoc ip&longs;o lo­<lb/>co te&longs;te, mirum e&longs;t, nihil de ea egi&longs;&longs;e quæ&longs;tione 9. vbi nos <lb/>hac de re fu&longs;i&longs;&longs;imè tractauimus. </s>
</p>
<p type="main">
<s>Cæterùm quod de rotatis &longs;cytalis &longs;cribit Philo&longs;o­<lb/>phus, notandum, à Pappo quidem lib. 8. & à no&longs;tris Me­<lb/>chanicis pa&longs;&longs;im ab&longs;que rotis Cylindrica &longs;implici videli­<lb/>cet, & tereti formâ ad v&longs;um adhiberi. E&longs;to igitur Ari­<lb/>
<arrow.to.target n="fig14"></arrow.to.target><lb/>&longs;totelis quidem &longs;cytala <lb/>AB, Pappi verò &longs;eu vul­<lb/>garis, & communis CD. <lb/>His non modò lapicidæ <lb/>pa&longs;&longs;im, &longs;ed & nautæ na­<lb/>uiumqueue fabri &longs;ubdu­<lb/>cendis & mari inducen­<lb/>dis nauibus vtuntur, quod varare dicunt vernaculè, Hi­<lb/>&longs;panico, vt arbitror, vocabulo. ca enim natio teres lignum <lb/>baculumue appellat Varam. </s>
</p>
<figure id="fig14"></figure>
<p type="main">
<s>Quæriautem po&longs;&longs;et, vtra harum formatum &longs;it vti­<lb/>lior atque commodior? Nos rotatas laudamus magis in <lb/>plano duroqueue &longs;olo, minus enim tangunt & minus offen­<lb/>&longs;ant; in molliori autem & minus duro proponimus non <lb/>rotatas, &longs;iquidem rotæ &longs;ui naturâ pondere pre&longs;&longs;æ &longs;olum, <lb/>facillimè &longs;cindunt & ab&longs;orbentur. </s>
</p>
<p type="main">
<s>Quatenus autem ad v&longs;um pertinet. E&longs;to horizontis
<pb pagenum="86"/>
<arrow.to.target n="fig15"></arrow.to.target><lb/>planum AB, &longs;cytalae duae<lb/>CD, EF, Pondus verò <lb/>eis impo&longs;itum G, tan­<lb/>gens ip&longs;as in <expan abbr="pūctis">punctis</expan> CE, <lb/>&longs;cytalæ autem planum <lb/>in punctis D, F, Pellatur <lb/>à potentia quapiam <expan abbr="pō-dus">pon­<lb/>dus</expan> Gad anteriora, <expan abbr="nē-pe">nen­<lb/>pe</expan> ad partes E. rotabuntur igitur &longs;cytalæ & pars quædam <lb/>&longs;cytalæ D, in qua &longs;it contactus a&longs;cendet in I, C verò de­<lb/>&longs;cendetin H, nulla re motum impediente, quippe quòd <lb/>nulla ponderis &longs;cytalarum, & plani ad inuicem fiat offen­<lb/>&longs;atio. Præterea cum &longs;cytalarum centra ab horizontis pla­<lb/>no æqualiter di&longs;tent, pondus quidem horizonti æquidi­<lb/>&longs;tanter mouetur, & ideo cius centrum grauitatis nequa­<lb/>quam, in motu qui &longs;it, eleuatur. </s>
</p>
<figure id="fig15"></figure>
<p type="main">
<s>Cæterùm materiæ imperfectione remota nihil re­<lb/>fert ad facilitatem, vtrum maioris minorisue diametri <lb/>&longs;int&longs;cytalæ, vt ea po&longs;ita eo quod maiores circuli faciliùs <lb/>offendicula &longs;uperent, quod demon&longs;tratum e&longs;t in quæ&longs;tio­<lb/>ne 8. eo vtiliores &longs;unt &longs;cytalæ, quo cra&longs;&longs;iores. Quatenus <lb/>autem ad plau&longs;tri naturam &longs;pectat, cuius ad &longs;cytalas Phi­<lb/>lo&longs;ophus fecit comparationem, vt o&longs;ten da mus difficilius <lb/>ex eo moueri pondera. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to plau&longs;tri rota <lb/>KL, cuius centrum M, a­<lb/>xis verò NO circa quem <lb/>rotaip&longs;a conuertiturKL. <lb/>Funis quo rota ex axis <lb/>centro M trahitur MP, <lb/>pondus vero QR. Quo­<lb/>niam igitur pondus axem <lb/>premit in N, axis autem rotæ modiolum in O, & codem,
<pb pagenum="87"/>tempore potentia quæ trahitin P, axem admouet modio­<lb/>lo in parte V. duplex itaque fit ex fricatione &longs;eu offen&longs;a­<lb/>tione impedimentum, in fra nempe, vbi O, & ad latus vbi <lb/>V. quæ quidem offen&longs;iones currus motum reddunt diffi­<lb/>ciliorem, quæ quidem difficultas eo maior erit, quo ma­<lb/>ior fuerit pondus axem premens, & minor proportio &longs;e­<lb/>midiametri rotæ KM, ad axis &longs;emidiametrum MO. Cur <lb/>igitur &longs;cytalis facilius pondera transferantur quam plau­<lb/>&longs;tris, apertè ex dictis ad Ari&longs;to telis mentem demon&longs;tra­<lb/>uimus. </s>
</p>
<p type="main">
<s>Cætetùm quod ip&longs;e reticuit, n<gap/>s dicemus, nempe <lb/>validi&longs;&longs;imè enormia pondera per &longs;cytalas moueri, &longs;i &longs;cy­<lb/>talis ip&longs;is vectes adiungantur. Et &longs;anè motus erit tar di&longs;&longs;i­<lb/>mus, veruntamen tar ditas ip&longs;a facilitate, quæ in de fit, v­<lb/>berrimè compen&longs;atur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur horizontis planum AB, &longs;cytalæ CD, fo­<lb/>ramina in &longs;cytalis EFGH, vectes foraminibus in&longs;erti IE, <lb/>KF, LG, MH. Pondus vero &longs;cytalis impo&longs;itum N. Appli­<lb/>catis igìtur quatuor potentijs extremitatibus vectium I, <lb/><emph type="italics"/>K<emph.end type="italics"/>, L, M, ij&longs;que in anteriora propul&longs;is, fiet &longs;cytalarum rota-
<pb pagenum="88"/>tio, & ponderis N translatio ad anteriores partes B. E&longs;to <lb/>item &longs;eor&longs;um &longs;cytala PR, cuius centrum Q, vectis eidem <lb/>per centrum in&longs;ertus O, P, Q, R. facto igitur vectis motu <lb/>O P Q R fiet ex O; centro <expan abbr="autē">autem</expan> Q circuli quadrans O T. <lb/>exi&longs;tente igitur O in T erit P in S. facta quartæ partisip&longs;ius <lb/>&longs;cytalæ rotatione. Et quoniam ex eodem centro &longs;unt qua­<lb/>drantes PSOT. erit vt OQ ad QP. ita quadrans OT, ad <lb/>quadrantem PS. Maxima autem e&longs;t proportio OQ, ad <lb/>QP. Maxima igitur proportio OT ad PS. Ex magno igitur <lb/>motu O ad T, paruus &longs;it &longs;cytalæ motus à P in S. tardius i­<lb/>gitur progreditur &longs;cytala, quæ longioribus vectibus rota­<lb/>tur, vis tamen maxima, quippe quod vt &longs;o habet QP, hoc <lb/>e&longs;t, QR ad QO, ita potentia in O ad pondus quod premit <lb/>in P vel in V. Facillimè ita que pondera vectibus & &longs;cyta­<lb/>lis per horizontis planum transferri, exi&longs;tis patet. </s>
</p>
<p type="head">
<s>QVAESTIO XII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur Mi&longs;&longs;ilia longius funda mittantur quam manu, <lb/>præ&longs;ertim cum proijcienti fundæ pondus addatur lapidis &longs;eu mi&longs;&longs;i­<lb/>lis ponderi: & minus mi&longs;&longs;ili, manu proiecto, com­<lb/>prehendatur?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit Philo&longs;ophus, inquiens, fortè ita fieri, quòd fun­<lb/>ditor mi&longs;&longs;ile proijciatiam ex funda commotum, &longs;iqui­<lb/>dem fundam circulo &longs;ubinderotans, iaculatur, ex manu <lb/>autem à quiete e&longs;t initium. Oinnia autem cum in motu <lb/>&longs;unt, quàm cum quie&longs;cunt, facilius mouentur. Addit præ­<lb/>terea, An & ob eam cau&longs;&longs;am e&longs;t, &longs;ed nec minus etiam, quia <lb/>in fundç v&longs;a manus quidem fit centrum, funda verò quod <lb/>à centro exit? quantò igitur productius fuerit quod à cen­<lb/>tro e&longs;t, tanto citiùs mouetur; iactus autem, qui manu fit, <lb/>fundæ re&longs;pectu breuior e&longs;t. </s>
</p>
<p type="main">
<s>Hæc Philo&longs;ophus. Et &longs;anè perquàm appo&longs;itè, <expan abbr="itaq;">itaque</expan>
<pb pagenum="89"/>illi pror&longs;us a&longs;&longs;entirer, ni&longs;i pro comperto habercm, in la ctu <lb/>qui fundâ fit, non e&longs;&longs;e manum ip&longs;am motus centium, &longs;ed <lb/>potius partem illam brachij, quæ humero iungitur, & id­<lb/>co motum eo fieri velociotem, quo longior e&longs;t linea quæ <lb/>ab humero ad &longs;ummitatem fundæ e&longs;t, ea quæ ab humero <lb/>ad manum ip&longs;am. Illud quo que mirabilc e&longs;t, quod non <lb/>ob&longs;etuat Ari&longs;toteles, nempe à funditoribus in ip&longs;o eiacu­<lb/>landi actu, tardam fieri circa caput fundæ rotationem. <lb/>Quamobrem con&longs;iderandum e&longs;t, quo pacto fiat à tardi­<lb/>tate velocitas. Re&longs;pondemus, velocitatem acquirinon ex <lb/>&longs;implici, quæ circa funditoris caput &longs;it, rotatione, &longs;ed ex <lb/>co impetu qui fit in ip&longs;a lapidis emi&longs;&longs;ione, qui quidem im­<lb/>petus &longs;i ante vel po&longs;t illud tempus fiat, quod à funditore <lb/>captatur, ca&longs;&longs;a pror&longs;us & inualida fit ip&longs;a iaculatio. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to funda AB, manus <lb/>B, brachium BC. Vt igitur&longs;e <lb/>habet CH, ad CB, ita veloci­<lb/>tas AD ad velocitatem, BE; <lb/>Vidimus nos pueros, arundi­<lb/>ni ad caput &longs;ci&longs;&longs;æ, paruos la­<lb/>pides in&longs;erentes, arundinem­<lb/>queue manu rotantes longi&longs;&longs;i­<lb/>mè lapides ip&longs;os proijcere; A­<lb/>rundo FG, lapis F, manus G, <lb/>brachium GH. </s>
</p>
<p type="head">
<s>QVÆSTIO XIII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur circa idemiugum, maiores collopes (vectes &longs;unt, <lb/>quos alij &longs;cytalas appellant, vt Pappus & Heron) faciliùs quàm mi­<lb/>nores mouentur: & item &longs;uculæ, quæ graciliores &longs;unt eadem <lb/>vi quam cra&longs;&longs;iores?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ideo hoc fieri po&longs;&longs;e docet Philo&longs;ophus, quòd <expan abbr="tamiugū">tamiugum</expan> <lb/>quam &longs;ucula <expan abbr="cētrum">centrum</expan> &longs;it, prominenres autem collopum
<pb pagenum="90"/>longitudines eæ lineæ quæ &longs;unt à centro. Celeriùs autem <lb/>moueri & plus ab eadem vi quæ maiorum &longs;unt <expan abbr="circulorū">circulorum</expan> <lb/>quàm quæ minorum. quippe quod ab ea dem vi plus <expan abbr="trã&longs;-feratur">tran&longs;­<lb/>feratur</expan> illud extremum quod longius à centro di&longs;tat. In <lb/>gracilioribus verò &longs;uculis datâ collopum paritate plus e&longs;­<lb/>&longs;e id quod à ligno di&longs;tat. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to iugum &longs;ucu­<lb/>laue maior, AB circa <lb/>centrum C, minor verò <lb/>circa idem <expan abbr="centrū">centrum</expan> DE. <lb/>Collops <expan abbr="autē">autem</expan> AF, pon­<lb/>dus quod per iugum at­<lb/>tollitur G. A it igitur A­<lb/>ri&longs;toteles, &longs;uculas, iu­<lb/>gaue AB, DE ceu cen­<lb/>tra e&longs;&longs;e, à quibus extat colops AB, ex maiori quidem, totâ <lb/>&longs;ui parte BF, ex minori autem EF. quo igitur, ait, longior <lb/>fuerit collops extans, eo maior, & deo velocior ad <expan abbr="partē">partem</expan> <lb/>F per maiorem circulum FH, fiet collopis motus & pon­<lb/>deris eleuatio, at maior e&longs;t collops EF ip&longs;o BF, facil. us er­<lb/>go mouebitur pondus per &longs;uculam DE, ex collope EF, ab <lb/>cadem vi, quam per &longs;uculam AB, & collopem BF. </s>
</p>
<p type="main">
<s>Hæc &longs;en&longs;i&longs;&longs;e videtur Ari&longs;to teles, qui cra&longs;&longs;a, vt aiunt, <lb/>Minerua rem pulchram & &longs;ubtilem e&longs;t pro&longs;equutus. Di­<lb/>cimus igitur primò, in&longs;trumentum illud quod Latini &longs;u­<lb/>culam, id e&longs;t, &longs;ero&longs;ulam, à &longs;tridore arbitror qui in conuer­<lb/>&longs;ione fit, appellauere, Græci verò <foreign lang="greek">o)/non</foreign>, id e&longs;t, A &longs;inum, quip­<lb/>pe quod ceu A &longs;inus pondera &longs;u&longs;tineat portetque. Hanc <lb/>eandem Machinam veteres Mechanici vocauere Axem <lb/>in Peritrochio, cuius nos imaginem, è Pàppo in 8. Col­<lb/>lect. Mathematicarum de&longs;umptam in ip&longs;o huius no&longs;trio­<lb/>peris initio, inter quinque Potentias propo&longs;uimus. Huius <lb/>vim inter antiquos diligenti&longs;&longs;ime examinauêre Heron, &
<pb pagenum="91"/>ip&longs;emet Pappus, inter iuniores verò Guilibaldus co Tra­<lb/>ctatu quem hac de Potentia Mechanicis &longs;uis in&longs;eruit. <lb/>Summa e&longs;t, hanc Machinam ad vectem reduci. Nec ve­<lb/>rum e&longs;t quod &longs;cribit Ari&longs;to teles, iugum &longs;uculamue cen­<lb/>tra e&longs;&longs;e, hæc enim centrum habent, quod in figura &longs;upe­<lb/>rius po&longs;ita notatur &longs;igno C. igitur vt &longs;e habet FC, ad CA, <lb/>ita pondus G ad potentiam in F; e&longs;t autem maior propor­<lb/>tio FC ad CD, quàm FC, ad CA. faciliùs ergo mouebit <lb/>potentia quæ in F, pondus in D, quàm eadem potentia F, <lb/>pondus in A, hoc e&longs;t, G. Huius naturæ &longs;unt quo que Erga­<lb/>tæ, quas machinas no&longs;tr<gap/>, Græco luxato vocabulo Arga­<lb/>nos appellant. Suculæ enim reuera &longs;unt, po&longs;itione <expan abbr="tantū">tantum</expan> <lb/>ab eis differentes, non enim plano horizontis ergatæ æ­<lb/>quidi&longs;tant, ceu &longs;uculæ & Axis in Peritrochio, &longs;ed eidem <lb/>fiunt perpendiculares. Cæterùm facilitatem à velocitate <lb/>non oriri &longs;uperius demon&longs;trauimus. </s>
</p>
<p type="head">
<s>QVAESTIO XIV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Proponitur dubitatio, Cur eiu&longs;dem magnitudinis lignum facilius <lb/>genu frangatur &longs;i qui&longs;piam æque diductis manibus extrema com­<lb/>prehendens fregerit, quàm &longs;i iuxta genu. Et &longs;i terræ applicans pede <lb/>&longs;uperpo&longs;ito manu hinc inde diducta confregerit <lb/>quàm propè.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluitur à Philo&longs;opho paucis verbis, An quia ibi genu <lb/>centrum e&longs;t, hic verò ip&longs;e pe? quanto autem remotius <lb/>à centro fuerit, facilius mouetur quodcunque: Moueri <lb/>autem quod frangitur nece&longs;&longs;e e&longs;t. </s>
</p>
<p type="main">
<s>E&longs;to lignum quod frangi debet AB, genu vel pedis <lb/>locus C, manuum latè diductarum &longs;itus DE, minus didu­<lb/>ctarum FG; ltaque quoniam DE magis à centro C di&longs;tant <lb/>quàm FG, velo cius mouebuntur puncta DE ip&longs;is FG, er­<lb/>go inde facilius fiet fractio quam ex FG. Hæcille ex &longs;uis
<pb pagenum="92"/>
<arrow.to.target n="fig16"></arrow.to.target><lb/>principijs. Nos dili­<lb/>gentius, &longs;i fieri poterit, <lb/>effectus huius cau&longs;&longs;am <lb/>per&longs;crutemur. E&longs;to igi­<lb/>tur in &longs;ecunda figura <lb/>lignum oblongum AB, <lb/>cuius medium C, linea <lb/>ducatur CD perpen­<lb/>dicularis ip&longs;i AB. Ad­<lb/>moucatur genu <expan abbr="pūcto">puncto</expan> <lb/>C, manus verò diuari­<lb/>centur in AB, facta i­<lb/>gitur vtrinque impre&longs;­<lb/>&longs;ione, lignum non <expan abbr="frã-getur">fran­<lb/>getur</expan>, ni&longs;i partium in CD coniunctarum &longs;eparatio fiat, <lb/>&longs;itqueue altera in E, altera verò in F, fractum ergo erit <expan abbr="lignū">lignum</expan>, <lb/>& centro C immobili permanente, partes facto angulo <lb/>GCH erunt in GC, HC: Modò lignum &longs;uæ integritati re­<lb/>&longs;tituetur, & denuò admoto genu puncto C, manus didu­<lb/>cantur in I, K, quæ lo ca viciniora &longs;intip&longs;i C, quam AB, Di­<lb/>co hinc difficilius fractionem fieri quam ex AB. Con&longs;ide­<lb/>ramus enim in integro ligno AB, duos vectes ACD, BCD, <lb/>quorum anguli concurrunt in commune fulcimentum C, <lb/>Sunt autem vectes angulati, & eius naturæ, quam exami­<lb/>nauimus in quæ&longs;tiones. E&longs;t igitur re&longs;i&longs;tentia, qua ligni <lb/>partes vniuntur in D, loco ponderis: &longs;uperanda hæc e&longs;t, vt <lb/>ligni fiat fractio. Dico id facilius ce&longs;&longs;urum, &longs;i fiat ex pun­<lb/>ctis A, B, remotioribus quam ex IK, ip&longs;i puncto C propio­<lb/>ribus: etenim vt AC, ad CD, itare&longs;i&longs;tentia quæ fit in Dad <lb/>potentiam in A, item vt &longs;e habet IC ad CD, ita re&longs;i&longs;tentia <lb/>in Dad potentiam in I, &longs;ed minor e&longs;t proportio IC ad CD, <lb/>quam AC ad CD. ergo facilius potentia quæ e&longs;t in A, re­<lb/>&longs;i&longs;tentiam &longs;uperabit, quæ e&longs;t in D, quam ea quæ e&longs;t in I,
<pb pagenum="93"/>quod fuerat demon&longs;trandum. Idem autem <expan abbr="intelligendū">intelligendum</expan> <lb/>e&longs;t de parte CB; eadem enim e&longs;t ratio. Curigitur longio­<lb/>ra & graciliora ligna facilè frangantur, ex i&longs;tis clare patet: <lb/>nempe quia maxima e&longs;t proportio longitudinis ad cra&longs;&longs;i­<lb/>tudinem, cuius quidem cra&longs;&longs;itudinis &longs;patium loco partis <lb/>illius in vecte &longs;uccedit, quæ pertingit à fulcimento ad <expan abbr="pō-dus">pon­<lb/>dus</expan>, hoc e&longs;t, ad ip&longs;am re&longs;i&longs;tentiam. Sed nos hac eadem de <lb/>re nonnulla in declaranda quæ&longs;tione 16. perpendemus. </s>
</p>
<figure id="fig16"></figure>
<p type="head">
<s>QVAESTIO XV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Proponitur inueftigandum, Cur litterales crocæ (glareas dicunt <lb/>Latini, velcalculos, quos vmbilicos appellat Cicero lib. 2. de Orat.) <lb/>rotundâ&longs;int figurâ, cum aliquando ex magnis &longs;int la­<lb/>pidibus te&longs;tisue?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>A It Philo&longs;ophus, ideo forta&longs;&longs;e fieri, quòd ca quæ à me­<lb/>dio magis recedunt, in motionibus, celerius feran­<lb/>tur; me dium e&longs;&longs;e centrum, interuallum vero quæ à cen­<lb/>tro, &longs;emper autem maiorem ab æ quali motione maiorem <lb/>de&longs;cribere circulum; quod autem maius in æquali tem­<lb/>pore &longs;patium tran&longs;it, celerius ferri; quæ autem celerius ex <lb/>æquali feruntur &longs;patio vehementius impetere, quæ <expan abbr="autē">autem</expan> <lb/>impetunt, impeti magis, & ideo quæ magis à centro di­<lb/>&longs;tant, nece&longs;&longs;e e&longs;&longs;e confringi, quod cum glareæ &longs;eu croc æ <lb/>patiantur, nece&longs;&longs;ariò rotundas fieri. Hactenus ille, & &longs;anè <lb/>p<gap/>obabiliter. Verum enimuerò aliter &longs;eres habere vide­<lb/>tur: &longs;iquidem enim à rotatione ex maiori à centro di&longs;tan­<lb/>tia id fieret, maiores quidem glareæ crocæue e&longs;&longs;ent ro­<lb/>tundiores, at nos non maximas modò, &longs;ed & minimas, <lb/>ea&longs;queuemagis angulis carere, & ad rotunditatem accede­<lb/>revidemus. Præterea non moueri eas circa centrum pa­<lb/>lam e&longs;t, imò vt varia &longs;unt figura, ita varijs quo que motio­<lb/>nibus, ex agitatione moueri. Id &longs;anè explorati&longs;&longs;imum e&longs;t,
<pb pagenum="94"/>angulos omnes, & emin entias quaslibet in corporibus e&longs;­<lb/>&longs;e infirmiores, offen&longs;ionibus enim expo&longs;itæ &longs;unt, necre&longs;i­<lb/>&longs;ten di habent facultatem. Itaque in attritione quæ fit in <lb/>eorum agitatione perpetua, eminentiæ contunduntur, & <lb/>&longs;uperficies ip&longs;a paullatim leuigatur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to angulatus lapis ABCD. <lb/>Dum igitur perpeti motione <expan abbr="atq;">atque</expan> <lb/>a&longs;&longs;iduâ ver&longs;atione agitatur, fer­<lb/>turqueue, eminentiæ anguliqueue, vt­<lb/>pote debiles & imbecilli, conte­<lb/>runtur, & inde figura fit quædam <lb/>irregularis, ad primam quidem la­<lb/>pidis <expan abbr="formã">formam</expan> accedens, leuistamen <lb/>& quouis angulo carens, qualis e&longs;t E remotis ABCD, an­<lb/>gularibus eminentijs. </s>
</p>
<p type="main">
<s>Hanc eandem ob cau&longs;&longs;am, &longs;culptores antequam mar­<lb/>moribusvltimum læuorem inducant, dentato malleo pri­<lb/>mum quidem vtuntur, tum demum eminentiores parti­<lb/>culas radula facilè amouentes &longs;uperficiem ip&longs;am læuem <lb/>& adæquatam reddunt. </s>
</p>
<p type="main">
<s>Hinc etiam no&longs;trates Architecti, in arcium propu­<lb/>gnaculis efformandis a cutos angulos <expan abbr="deuitãt">deuitant</expan>, vtpote de­<lb/>biliores, & magis offen&longs;ionibus obnoxios. quod nec Vi­<lb/>truuium latuit, qui ideo lib. 1. cap. 5. ita &longs;cribit: <emph type="italics"/>Turresitaque<lb/>rotundæ aut polygoniæ &longs;unt faciendæ, quadrat as enim machinæ <lb/>celerius di&longs;&longs;ipant; & angulos, Arietes tundendo frangunt, inro­<lb/>tundationibus autem, vti cuneos adcentrum adigendo lædere non <lb/>po&longs;&longs;unt.<emph.end type="italics"/> Hæcille. Cur autem no&longs;tri rotundas figuras alias <lb/>vtiles reijciant, ab ijs petendum qui in ea facultate ver­<lb/>&longs;antur. Porrò quod ad hanc eandem &longs;peculationem facit, <lb/>videmus, antiquas &longs;tatuas, vt &longs;æpius auribus, na&longs;o, digitis, <lb/>manibu&longs;ue atque pedibus carere, quippe quodimbecillæ <lb/>&longs;int partes, & facilè quouis occur&longs;u mutilentur. Quæo-
<pb pagenum="95"/>m<gap/>a cùm vera &longs;int, nemo, vt arbitror, dixerit, ab&longs;olutè, <lb/>quod voluit Ari&longs;toteles, id ex rotatione velociori & par­<lb/>tium à centro remotione, fieri. </s>
</p>
<p type="head">
<s>QVAESTIO XVI.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, quare, quò longiora &longs;unt ligna, <gap/><expan abbr="ãto">anto</expan> imbecilliora fiant, <lb/>& &longs;itolluntur, inflectuntur magis: tamet&longs;i quod breue est ceu bi­<lb/>cubit umfuerit, tenue, quod verò cubitorum cen­<lb/>tum cra&longs;&longs;um?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ex &longs;uis principijs &longs;oluit Ari&longs;totelcs. Inquit enim: An <lb/>quia & vectis & ont s & hypomochlium, id e&longs;t, fulci­<lb/>mentum in leuando, fit ip&longs;a ligni proceritas? Prior <expan abbr="namq;">namque</expan> <lb/>illius pars ceu hypomochlium fit, quod verò in extremo <lb/>e&longs;t, pondus: quamobrem quanto exten&longs;ius fuerit id quod <lb/>à fulcimento e&longs;t, in flectinece&longs;&longs;e e&longs;t magis; quo enim plus <lb/>à fulcimento di&longs;tat, eo magis incuruarinece&longs;&longs;e e&longs;t. Ne­<lb/>ce&longs;&longs;ariò igitur extrema vectis eleuantur. Si igitur flexilis <lb/>fuerit vectis, ip&longs;um inflectimagis cum extollitur nece&longs;&longs;e <lb/>e&longs;t, quod longis accidit lignis, in breuibus autem quod vl­<lb/>timum e&longs;t, quie&longs;centi hypomochlio depropè fit. Hæc <lb/>&longs;ubiectâ figurâ ob oculos ponimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to longum ac fle­<lb/>xile lignum AB, manu ele­<lb/>uetur in A, fle ctetur <expan abbr="itaq;">itaque</expan> <lb/>in B, & declinabit in C. et­<lb/>enim manus quæ &longs;u&longs;tin et <lb/>in A, fulcimenti loco &longs;uccedit: longitudo vero AB ponde­<lb/>ris vices refert, at que vectis, quare quo longius abfuerit à <lb/>fulcimento, id e&longs;t, manu extremum B, eo magis flectetur; <lb/>&longs;i autem lignum breuius fuerit, nempe terminatum in D, <lb/>nequaquam fle ctetur, eò quòd eius extremum D minus à <lb/>fulcimento quod e&longs;t in A &longs;it remotum. Hæcigitur e&longs;t <expan abbr="mēs">mens</expan>
<pb pagenum="96"/>Ari&longs;totelis, cuius quidem &longs;ententiam non damnamus; <lb/>quippiam tamen addimus. Dicimus autem materiam, <lb/>quatenus ad hanc contemplationem &longs;pectat, in duplici <lb/>e&longs;&longs;e differentia. aut enim rarefactionis & con&longs;tipationis <lb/>e&longs;t incapax, vt in chalybe videmus, nitro, metallo, mar­<lb/>more, aut capax quidem, & hæc duplex: Vel enim natura <lb/>nata e&longs;t ad rectitudinem quandam, vt ar borum flagella <lb/>virgæque, autnon item, ceu &longs;tannum, plumbum, & cæte­<lb/>ra eiu&longs;modi. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to primò vitreum <lb/>corpus gracile, procerum, <lb/>teres AB, manu capiaturin <lb/>A, <expan abbr="itaq.">itaque</expan> pondere ip&longs;ius cor­<lb/>poris præualente ad partes <lb/>B, quia in C puncto, quod <lb/>circa medium e&longs;t, ex parte <lb/>&longs;uperiori non fit rarefactio, <lb/>nec in in feriori con&longs;tipatio, <lb/>nec interim datur penetra­<lb/>tio corporum, fit fractio à <lb/>&longs;uperiori parte, & pars CB à <lb/>reliqua parte AC, auul&longs;a & <lb/>&longs;eparata cadit in D, &longs;uccedit autem ip&longs;a &longs;eparatio rarefa­<lb/>ctioni. Porrò quod materias ha&longs;ce non flexibiles diximus, <lb/>&longs;ed frangibiles, non ideo negamus vel &longs;en&longs;u docente, ali­<lb/>quam inijs fieri flexionem. Si autem lignea fuerit mate­<lb/>ria, caque; flexibilis, vt EF, &longs;i manu eleuetur in E, præualen­<lb/>te pondere in F flectetur vbi G. ibi enim à parte &longs;uperiori <lb/>fitrarefactio, ab in feriori verò con&longs;tipatio, & pars GF de­<lb/>clinabitin H, quæ declinatio eò v&longs;que procedet, quo ra­<lb/>refactio & con&longs;tipatio competens naturæ illius materiæ, <lb/>quæ flectitur ad &longs;ummam inten&longs;ionem deuenerint; tunc <lb/>&longs;ivis maioringruerit, frangetur omnino: &longs;i &longs;ecus factaibi
<pb pagenum="97"/>re&longs;i&longs;tentia, vbi rarefactio fit & con&longs;tipatio pe&longs;t inclina­<lb/>tionem &longs;ur&longs;um ferctur pars in clinata & nutans, tum in <lb/>contrariam partem tendens reflectetur, vt videre e&longs;t in <lb/>virga IN. Declinans enim in KL, repellente ea quæ infra <lb/>K fit materiæ conden&longs;atione, impetu ex de&longs;cen&longs;u a cqui­<lb/>&longs;ito facta reflexione a&longs;cendit in KM, donec paullatim cir­<lb/>ca pri&longs;tinam rectitudinem reuertatur, & hic quidem mo­<lb/>tus vibratio dicitur, agitatioue. Si autem virga plumbea <lb/>fuerit, naturâ non factâ ad rectitudinem, puta OP, pro­<lb/>prio vincente pondere, ad partes declinabit QS, fietque; in <lb/>QR rarefacta, nempe &longs;uperiori parte ea con&longs;tipata infe­<lb/>riori in Q, nec reflectetur, quippe quòd eius natura con­<lb/>den&longs;ationem & rarefactionem commodè patiatur, nec <lb/>facta &longs;it ad rectitudinem. </s>
</p>
<p type="main">
<s>Porrò tripliciter fieri pote&longs;t horum oblongorum <lb/>corporum eleuatio, nempe vel extremorum alteio, aut &longs;i <lb/>ambobus, &longs;i vtrinque &longs;u&longs;pen datur, vel alicubi inter extre­<lb/>ma. De priori modo iam egimus. Modò &longs;u&longs;pendatur in <lb/>medio vt AB, in C. eo igitur ca&longs;u cum fulcimentum &longs;it in <lb/>C, <expan abbr="vtrinq;">vtrinque</expan> fit flexio in D, & E, & id quidem &longs;i materia fle­<lb/>xionem patitur: &longs;in minus, fractio fit in C. Si autem ab ex­<lb/>
<arrow.to.target n="fig17"></arrow.to.target><lb/>tremis fiat &longs;u&longs;pen&longs;io, vt in <lb/>AB, tunc ceu duo vectes <lb/>fient, quorum fulcimenta in <lb/>extremis AB. Pondera au­<lb/>tem communia in medio vbi <lb/>Cremoti&longs;&longs;ima enim ea pars e&longs;t ab extremis AB. Cedente <lb/>
<arrow.to.target n="fig18"></arrow.to.target><lb/>igitur materia &longs;uomet pon­<lb/>deri, &longs;iquidein in flexibilis fu­<lb/>erit, frangetur, & fiet <expan abbr="partiū">partium</expan> <lb/>&longs;eparatio in C, duoque in de <lb/>corpora AD, BE. Si autem fle­<lb/>xionis capax, vt AB in po&longs;tre­
<pb pagenum="98"/>ma figura, facta ex contrario, nempe in in feriori parte cir­<lb/>ca C rarefactione, in &longs;uperiori verò conden&longs;atione, pon­<lb/>dere præualente curuabitur, fietque; lignum quidue aliud <lb/>huiu&longs;modi, vt ADB, nec amplius pondere &longs;uapte naturâ <lb/>inferiùs vergente ad rectitudinem reuertetur. </s>
</p>
<figure id="fig17"></figure>
<figure id="fig18"></figure>
<p type="main">
<s>Cæterùm cur oblonga & graciliora corpora facilius <lb/>illis, quæ contrario &longs;e habent modo, frangantur, ex me­<lb/>chanicis principijs in quæ&longs;tione 14. apertè demon&longs;traui­<lb/>mus. Modò vt ex hac contemplatione, quæ aliàs inutilis <lb/>videtur, aliquam vtilitatem capiamus, & ex his quæ con­<lb/>templabimur, Architecti prudentiotes fiant, i&longs;thæcip&longs;a, <lb/>de quibus agimus, ad rem ædificatoriam commodè apta­<lb/>bimus. Transferamus igitur cogitationem ad eam <expan abbr="trabiū">trabium</expan> <lb/>comp gem, quæ ad tecta &longs;u&longs;tinenda ex tran&longs;uer&longs;ario ar­<lb/>rectarioque; &longs;it, & duobus cauterijs, quam no&longs;trià Latinis <lb/>detorto vocabulo Bi&longs;cauterium dicunt. Per&longs;crutabimur <lb/>enim, vnde illi tanta ad &longs;u&longs;tin endum vis, & quæ compa­<lb/>gem hanc con&longs;equantur pa&longs;&longs;iones. quamuis enim fabri <lb/>meræ praxi, quod vtile e&longs;t efficiant, nos meliorum inge­<lb/>niorum gratiâ, rei ip&longs;ius cau&longs;&longs;as diligenter examinatas in <lb/>medium proferemus; nec de hac re tantùm agemus, &longs;ed <lb/>de Cameris quoque, fornicibus eorumqueue vitijs & virtu­<lb/>tibus quatenus ad Mechanicum pertinet, &longs;ermonem ha­<lb/>bebimus. Quærimus primo, cur perpendiculariter erectae<lb/>trabes &longs;uperimpo&longs;ita pondera validi&longs;&longs;ime &longs;u&longs;tineant? Et <lb/>&longs;ane hoc omnes norunt, &longs;ed non per cau&longs;&longs;as. </s>
</p>
<p type="main">
<s>E&longs;to horizontis planum, illudqueue &longs;olidi&longs;&longs;imum, & <lb/>impenetrabile AB, trabs eidem adperpendiculum erecta <lb/>CD fulta ba&longs;i vbi C grauitatis centrum F. pondus &longs;uper­<lb/>impo&longs;itum FG, cuius grauitatis centrum H: Sint autem <lb/>H & E in eadem perpendiculari, quæ ad mun di centrum <lb/>HEC. Itaque eo quod tum penderis tum trabis centra <lb/>grauitent in perpendiculari, illa verò fulciatur in C, to-
<pb pagenum="99"/>
<arrow.to.target n="fig19"></arrow.to.target><lb/>tius ponderis moles recumbet <lb/>in C: non de&longs;cendet autem in I, <lb/>propterea quod &longs;upponatur i­<lb/>p&longs;um planum AB, impenetrabi­<lb/>le. Igitur vt pondus H de&longs;cen­<lb/>dat in C, alterum duorum e&longs;t <lb/>nece&longs;&longs;arium, nempe vel trabem <lb/>&longs;ubiectam comminui, aut eius <lb/>partes &longs;e&longs;e penetrare, & plura <lb/>corpora e&longs;&longs;e in eodem loco, pu­<lb/>ta KC, quorum hoc &longs;ecun dum <lb/>naturæ penitus repugnat, illud <lb/>vero primum, penè impo&longs;&longs;ibile. Diuidatur enim trabs in <lb/>partes æquales tres, lineis KL, ip&longs;a igitur KC infima &longs;u&longs;ti­<lb/>net mediam KL, hæc verò &longs;upremam LD, hæc autem <expan abbr="pō-dus">pon­<lb/>dus</expan>, ip&longs;um &longs;uperpo&longs;itum in H. Seigitur &longs;u&longs;tinent partes. <lb/>Sed illud totum partibus con&longs;tat. ergo pondus totum à <lb/>trabe tota, hoc e&longs;t, à &longs;e toto &longs;u&longs;tinetur. </s>
</p>
<figure id="fig19"></figure>
<p type="main">
<s>Præterea in præcedenti quæ&longs;tione mon&longs;trauimus <lb/>tunc facilem e&longs;&longs;e gracilis & oblongi ligni fractionem, <expan abbr="cū">cum</expan> <lb/>maxima e&longs;t longitudinis ad cra&longs;&longs;itudinem proportio. Hîc <lb/>verò contrà accidit, etenim MD pars vectis quæ à fulci­<lb/>mento e&longs;t ad potentiam minimam habet proportionem <lb/>ad rectam DC, quæ à fulcimento ad locum fractionis ex­<lb/>tenditur, vbi C, quod vt euidentius pateat, </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to &longs;eor&longs;umtrabs AB, <lb/>cuius medium C. Sit autem <lb/>pondus D impo&longs;itum pun­<lb/>cto C. facilè igitur frange­<lb/>tur lignum AB, propterea <lb/>quòd maxima &longs;it proportio <lb/>AC ad CE; re&longs;i&longs;tentia verò <lb/>fiat in E, addatur vniatu que;
<pb pagenum="100"/>ligno AB lignum FH. Cra&longs;&longs;ius igitur e&longs;t totum AL, ip&longs;o <lb/>A<gap/>, & ideo minor proportio AC ad CG quàm AC, ad <lb/>CE. Addavur adhuc & IM. Longè itaque difficilius fran­<lb/>getur in K propterea quòd longè minor &longs;it proportio AC <lb/>ad CK quàm ciuidem ad CE & CG. His igitur con&longs;ide­<lb/>ratis, & demon&longs;tratis concludimus, impo&longs;libile e&longs;&longs;e ere­<lb/>ctam trabem ponderi cedere, & frangi. </s>
</p>
<p type="main">
<s>Dicet autem qui&longs;piam, haec&longs;i vera &longs;unt, quo gracilius <lb/>fuerit fulcrum, eo validiùs &longs;u&longs;tinebit, & frangetur minus, <lb/>quod oppido fal&longs;um e&longs;t. Re&longs;pondemus, id non ex propor­<lb/>tionum naturâ, &longs;ed ex materiæ ip&longs;ius infirmitate fieri. Ita <lb/>quoque invecte non materiam, quatenus ad vim pertinet, <lb/>&longs;ed proportiones partium con&longs;ideramus. Vtiumqueigi­<lb/>turrequiritur ad fulcti validitatem proportio longitudi­<lb/>nis ad cra&longs;&longs;itudinem debita, & materiæ ip&longs;ius robur & <lb/>fortitudo. Præterea, quoniam pondus, cuifulcrum re&longs;i­<lb/>&longs;tit, vel ex natura premit, vel ex violentia, illud quidem <lb/>per lincam perpen dicularem, quæ ad mundi <expan abbr="cētrum">centrum</expan>, hoc <lb/>autem lateraliter & diuer&longs;imodè, varia fit fulcrorum di&longs;­<lb/>po&longs;itio. Cuius rei &longs;umma hæce&longs;t, vt &longs;emper contra impe­<lb/>tum &longs;upponantur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim horizontis planum <lb/>AB, <expan abbr="eidē">eidem</expan> perpendiculares CADB, <lb/>ítaque &longs;i naturaliter pondus pre­<lb/>matex C, fulcrum &longs;upponetur AE. <lb/>Siautem ex F ip&longs;um GE, &longs;i verò ex <lb/>H, &longs;upponaturiuxta BE. Si verò &longs;e­<lb/>cundum I ponderi opponatur KE. <lb/>Hæc nos de arrectarijs fulcrisue; <lb/>nunc de tran&longs;uer&longs;arijs, & inclinatis agemus, & primum <lb/>de tran&longs;uer&longs;arijs, quatenus ad tectorum trabeationes &longs;pe­<lb/>ctat. </s>
</p>
<p type="main">
<s>E&longs;to tran&longs;uer&longs;aria trabs AB, muris <expan abbr="vtrinq;">vtrinque</expan> fulta CD,
<pb pagenum="101"/>
<arrow.to.target n="fig20"></arrow.to.target><lb/>cuius grauitatis centrum <lb/>E, in <expan abbr="perpēdiculari">perpendiculari</expan> FEG, <lb/>quæ quidem ad mundi <lb/>centrum vergit. <expan abbr="Itaq;">Itaque</expan> eo­<lb/>dem tendente grauitatis <lb/>contro, &longs;i pondus quod <lb/>premit in E, non præua­<lb/>leat vnioni <expan abbr="partiū">partium</expan> ip&longs;ius <lb/>materiæ quæ e&longs;t in E, re&longs;i&longs;tet trabs &longs;uomet ponderi, nec <lb/>frangetur. Si autem vel in firmitate materiæ, aut vitio, vel <lb/>maxima exiftente proportione AF ad FE, fractio fiet in E, <lb/>& &longs;ecutâ partium &longs;epaiatione duæ fient vtrin que trabes <lb/>AH, Bl, quorum grauitatis centra KL. Erunt igitur duo <lb/>vectes AE, BE, quorum fulcimenta MN, quamobrem &longs;i <lb/>proportio EM ad MH ita præualeat, vt pondus quod e &longs;t <lb/>in E, &longs;uperet pondus muri O &longs;uperimpo&longs;iti, & item muri <lb/>P, corruent quidem trabes, & murorum fiet hinc inde di&longs;­<lb/>&longs;ipatio. Si autem non præualuerit ea, quam diximus, pro­<lb/>portio, &longs;u&longs;pen&longs;æ remanebunt vtrin que trabes vt AHBI. </s>
</p>
<figure id="fig20"></figure>
<p type="main">
<s>Huic difficultati egregiè occurrunt Architecti, ali­<lb/>quando autem hoc modo: </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to tran&longs;uer&longs;aria <lb/>trabs &longs;uâ gracilitate, alia­<lb/>ue de cau&longs;&longs;a imbecilla <lb/>AB, muri quibus <expan abbr="vtrinq;">vtrinque</expan> <lb/>&longs;u&longs;tinetur CD, Trabis i­<lb/>p&longs;ius grauitatis centrum <lb/>G. Itaque adpactis trabi <lb/>lignis EF, caprcolos ad­<lb/>dunt muro vtrinque ful­<lb/>tos CE, DF, corum capita adpactis lignis admouentes EF, <lb/>&longs;ed & tunc validi&longs;&longs;ima fit colligatio, &longs;i inter E & F capreo­<lb/>lorum capita inte grum lignum trabi &longs;upponatur EF. Ra­
<pb pagenum="102"/>tio autem validitatis patet; premente enim grauitatis <expan abbr="cē-tro">cen­<lb/>tro</expan> in G, fulcra hincinde &longs;uccurrunt CE, DF, quæ cum &longs;e­<lb/>ip&longs;is fieri non valeant breuiora, ne corpori detur penetra­<lb/>tio, re&longs;i&longs;tunt & robu&longs;ti&longs;&longs;imè ip&longs;i ponderi &longs;uperimpo&longs;ito <lb/>contranituntur. Videntur autem in hoc opere duo con­<lb/>&longs;iderari vectes, GH, GB, quorum fulcimenta EF, potentia <lb/>premens vtrinque G. Pondera autem parietum partes ca­<lb/>pitibus trabis impo&longs;itæ in A & B. Quoniam igitur parua <lb/>e&longs;t proportio GE ad EH, parua potentia premens in G, <lb/>maximè autem pondus in A, fieri non pote&longs;t trabem fran­<lb/>gi aut muros vtrin que di&longs;&longs;ipare in AB. Po&longs;&longs;unt etiam to­<lb/>tius trabis tres partes con&longs;iderari AE, EF, FB, quarum ful­<lb/>cimenta quatuor A, E, F, B, Diui&longs;o igitur pondere & mul­<lb/>tiplicatis fulcimentis impo&longs;&longs;ibile e&longs;t trabem conuelli & <lb/>vitium facere. </s>
</p>
<p type="main">
<s>Sed & tectorum contignationes imbecillaque; tran&longs;­<lb/>uer&longs;aria Mechanici corroborare &longs;olent, additis nempe <lb/>arrectaria trabe atque cauterijs. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim tran&longs;­<lb/>uer&longs;aria trabs AB <lb/>parietibus vtrinque <lb/>fulta I, K, <expan abbr="arrectariū">arrectarium</expan> <lb/>CD. Cauterij vtrin­<lb/>que AD, BD, ita <lb/>tran&longs;uer&longs;ariæ trabi <lb/>in AB, & arrectario <lb/>in D in&longs;erti, vt ne­<lb/>quaquam inde ela­<lb/>bi valeant. Tum ferrea fa&longs;cia EF mediam tran&longs;uer&longs;ariam <lb/>trabem AB, à parte inferiori ip&longs;i arrectario connectens, <lb/>Debet autem arrectarij pes vbi C, aliquantulum à tran&longs;­<lb/>uer&longs;aria trabe di&longs;tare, ne deor&longs;um ex pondere vergente <lb/>paululum arrectario ip&longs;am tran&longs;uer&longs;ariam premat. His i-
<pb pagenum="103"/>gitur ita con&longs;titutis pondus quidem tran&longs;uer&longs;ariæ trabis, <lb/>quod &longs;uapte naturâ premit in medio vbi C, ferrea fa&longs;cia, <lb/>arrectariæ trabi affixa di&longs;tinetur, Arrectariam cauterij &longs;u­<lb/>&longs;tinent, hos verò tran&longs;uer&longs;ariæ capita AB, quibus indun­<lb/>tur. Tota igitur eiu&longs;cemodi operis vis in eo con&longs;i&longs;tit, vt <lb/>probè cauterij tran&longs;uerlariæ & arrectariæ trabi in&longs;eran­<lb/>tur. fixis enim cauteriorum pedibus in AB, non <expan abbr="de&longs;cendēt">de&longs;cendent</expan> <lb/>à partibus &longs;eu capitibus D, ijs verò &longs;tantibus &longs;tabit & arre­<lb/>ctarium, quo inde &longs;u&longs;pen&longs;o tran&longs;uer&longs;aria trabs ei ex ferrea <lb/>fa&longs;cia alligata nequaquam pendebit. Stabit ergo compa­<lb/>ges tota & &longs;uapte vi robu&longs;ti&longs;&longs;imè connexa totius tecti <lb/>pondus &longs;u&longs;tinebit. </s>
</p>
<p type="main">
<s>Quoniam autem v&longs;u venire &longs;olet, cauterios nimia <lb/>longitudine debiles, aliquando tum proprio tum extra­<lb/>neo cedentes ponderi deor&longs;um vergentes pandare, Ar­<lb/>chitecti capreolis hinc inde &longs;uppo&longs;itis, ceu fulcris, huic <lb/>medentur infirmitati. </s>
</p>
<figure></figure>
<p type="main">
<s>Sint enim cauterij <lb/>debiles hinc inde AB, <lb/>AC, media trabs arre­<lb/>ctaria, quam <expan abbr="Monachū">Monachum</expan> <lb/>dicimus AD. Cauterio­<lb/>rum mediæ partes E, F, <lb/>in punctis igitur EF, vtpote maximè ab extremis di&longs;tanti­<lb/>bus debiles cauterij val de laborant. Itaque &longs;uppo&longs;itis v­<lb/>trin que arrectariolis EH, Fl, eorum capitibus E, F, duos <lb/>cauteriolos &longs;ibi ip&longs;is ad pedem arrectarij in D, re&longs;i&longs;tentes <lb/>apponunt. quibus ita con&longs;titutis nec E, nec F ad partes H, <lb/>I, de&longs;cendere valent. Capiatur enim inter EH, quoduis <lb/>punctum G, & BG, DG, connectantur, erunt autem BG, <lb/>DG ip&longs;is BE ED breuiores ex 21. primi elem. Tuncigitur <lb/>punctum E fiet in G cum BE, ED fient in BG, DG, quod <lb/>non cedentibus B, D, & &longs;ibi ip&longs;is breuioribus factis parti-
<pb pagenum="104"/>bus BE, ED, pror&longs;us e&longs;t impo&longs;&longs;ibile. &longs;tabuntigitur in co­<lb/>rum rectitudine cauterij AB, AC, nec pandabunt, quod <lb/>fieri querebatur. </s>
</p>
<p type="main">
<s>Hîc autem damnandi veniunt ij, quitran&longs;uer&longs;ariæ <lb/>quidem trabis capitibus cauteriorum pedes non <expan abbr="in&longs;erūt">in&longs;erunt</expan>, <lb/>&longs;ed ea vice tran&longs;uer&longs;ariolo quodam medios cauterios v­<lb/>trin que connectunt ad in&longs;tar elementi A, quam compa­<lb/>gem, capram, appellant. Sint enim cauterij hinc inde AB, <lb/>AC, quorum medias partes connectit tran&longs;uer&longs;ariolum, <lb/>DE. Dico igitur colligationem i&longs;tam magnopere impro­<lb/>bandam. Sunt enim AB, AC vectes, quorum commune <lb/>fulcimentum A, potentiæ hinc inde diuaricantes B, C, <lb/>pondera inter fulcimentum & potentias DE. quoniami­<lb/>gitur vt DH ad AB, ita potentia in B, ad pondus in D, par­<lb/>ua quidem potentia, pondus in D di&longs;trahet & &longs;uperabit: <lb/>facillimaque; in de fiet tran&longs;uer&longs;ariolì à capreolis ip&longs;is vtrin­<lb/>que reuul&longs;io: Et quoniam centrum quidem e&longs;t A, fact, in <lb/>D, E, parua diuaricatione, maxima fit in BC, vtpote parti­<lb/>bus ab ip&longs;o centro A quam remotis. Calcitrant igitur li­<lb/>beri prope cauteriorum pedes, & muros ip&longs;os &longs;ummos, <lb/>non &longs;ine magno operis totius vitio, &longs;ua calcitratione pro­<lb/>pellunt. </s>
</p>
<p type="main">
<s>Hæc nos de trabeationibus, modò ad fornicum ca­<lb/>merarumque; naturam &longs;tilum transferemus; id enim &longs;uadet <lb/>vtilitas, imo & nece&longs;&longs;itas ip&longs;a. Pauci enim ante nos hæc <lb/>tractarunt, & &longs;anè his probè non cognitis aut neglectis, <lb/>Architecti fabriqueue ingentes per&longs;æpe incurrunt, & inex­<lb/>plicabiles difficultates. Dicimus igitur primò, coctiles la­<lb/>teres, & non cuneatos lapides ad rectam lineam di&longs;po&longs;i. <lb/>tos, non &longs;tare. </s>
</p>
<p type="main">
<s>Sint enim muri vtrinque AC, BD. Ducatur hori­<lb/>zonti æquidi&longs;tans CD, iuxta quam lateres lapide&longs;ue non <lb/>cuneati, &longs;eriatim collocentur EF. Dicimus amoto arma-
<pb pagenum="105"/>
<arrow.to.target n="fig21"></arrow.to.target><lb/>mento, hoc e&longs;t, pro­<lb/>hibente ip&longs;o lateres <lb/>ruere. Producantur <lb/>enim AC in G, BD <lb/>verò in H, cum ip&longs;is <lb/>CG, DH, æquales <lb/>fiant CI, DK, & recta <lb/>IK iungatur, crit igi­<lb/>tur GD &longs;patium ip&longs;i <lb/>CK &longs;patio &longs;imile qui­<lb/>dem & æquale, quod <lb/>cùm ita &longs;it, nihil prohibet quin tota laterum GD moles in <lb/>&longs;patium CK transferatur, & corruat. </s>
</p>
<figure id="fig21"></figure>
<p type="main">
<s>Si autem cunei ip&longs;i latere&longs;ue, cuneatim di&longs;po&longs;iti, ita <lb/>&longs;int vt ad vnum centrum tendant, licet ad rectam lineam <lb/>collocentur, non delabentur, &longs;ed &longs;tabunt; quod ita o&longs;ten­<lb/>demus. </s>
</p>
<figure></figure>
<p type="main">
<s>Sint cunci latere&longs;ue <lb/>cuneatim di&longs;po&longs;iti ABCD, <lb/>tendentes ad centrum, &longs;eu <lb/>commune punctum E, Du­<lb/>cantur CAE, DBE, &longs;intqueue <lb/>muri vtrinque ponderire&longs;i­<lb/>&longs;tentes CL, DM, Demitta­<lb/>tur perpendicularis, quæ ad <lb/>mundi centrum FGE &longs;ecans AB, in G. Tum fiat GK aequa­<lb/>lis GF & per K ip &longs;i AGB parallela ducatur, HKI claudens <lb/>&longs;patium AHIB. Quoniam igitur vt EC, ad EA, ita CD ad <lb/>AB per 4. propo&longs;. lib. 6. maior erit CD ip&longs;a AB, & eâdem <lb/>de cau&longs;&longs;a maior AB, ip&longs;a HI, & idcirco maius ABDC &longs;pa­<lb/>tium, &longs;patio AHIB. Non igitur pote&longs;t linea CD, fieri in <lb/>AB, neque AB, in HI, neque &longs;patium totum CABD, tran&longs;­<lb/>ferri in &longs;patium AHIB non data (quod naturæ ip&longs;i repu­
<pb pagenum="106"/>gnat) corporum penetratione. Stabunt ergo cunei, quod <lb/>fuerat demon&longs;trandum. </s>
</p>
<p type="main">
<s>Verumenimuero, debilis hæc &longs;tructura e&longs;t, & eo de­<lb/>bilior, quo vani latitudo fucrit maior, cuneorum verò al­<lb/>titudo minor. Idem enim patitur quod epi&longs;tylia in &longs;pecie <lb/>Aræos&longs;tyla, quæ, vt&longs;cribit Vitruuius lib. 3. c. 2. propter in­<lb/>teruallorum magnitudinem franguntur. Id quoque ha­<lb/>bet vitij, quod cunei ita di&longs;po&longs;iti &longs;uo pondere in cumbas <lb/>vtrinque violenti&longs;&longs;imè pellant. Vtilis tamen e&longs;&longs;c pote&longs;t <lb/>ad portarum & fene&longs;trarum, quæ in medijs muris &longs;unt, & <lb/>mediocri vano aperiuntur, &longs;uperliminaria. </s>
</p>
<p type="main">
<s>Si verò ad minorem circuli portionem curuetur Ca­<lb/>mera, vtilior quidem erit &longs;tructura ea ip&longs;a, de qua locuti <lb/>&longs;umus; non tamen omninò &longs;ine vitio. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to fornix ex minori <lb/>circuli portione AB, cuius in­<lb/>cumbæ AF, BH muris fultæ <lb/>AC, BD. Con&longs;tet autem vel <lb/>ex lapidibus cuneatis, vel ex <lb/>coctilibus lateribus ad E <expan abbr="cē-trum">cen­<lb/>trum</expan> tendentibus. Sitque; for­<lb/>nicis linea exterior FGH, in­<lb/>terior AIB. Ducantur EA, <lb/>ED, & producantur in M, N. <lb/>Quoniam igitur vt EM ad EA, ita MGN ad AIB, maior e­<lb/>rit MGN lineaip&longs;a AIB, quamobrem fieri non pote&longs;t vt <lb/>aptetur lineæ AIB, & in eius locum de&longs;cendat. Stabit igi­<lb/>tur, in cumbis vtrinque non cedentibus. Validè autem <lb/>&longs;peciem hanc, loca quibus incumbit, propellere, ita o­<lb/>&longs;tendemus. </s>
</p>
<p type="main">
<s>Producatur in eadem figura CA in K, & DB in L. <lb/>Partes igitur quæ muris ad perpendiculum fulciuntur, <lb/>&longs;unt AKF, BLH, minimæ illæ quidem, maxima verò pars
<pb pagenum="107"/>e&longs;t extra fulcimenta, nempetota AKLB quæ idcircó &longs;uo­<lb/>pte pondere deor&longs;um vergens & in incumbas <expan abbr="vtrinq;">vtrinque</expan> pel­<lb/>lens aperitur, & facillimè vitium facit. Eiu&longs;dem ferè na­<lb/>turæ ea &longs;pecies e&longs;t, quæ vel ex media, vel ex minori ellip&longs;is <lb/>&longs;ecundum maiorem diametrum fit &longs;egmento. Vtilior ta­<lb/>men hæc e&longs;t, præcipuè circa incumbas, propterea quod <lb/>partes habeat erectiores, & circulari illa de qua egimus, <lb/>magis fultas. circa medium autem pote&longs;t videri debilior, <lb/>quippe quod ellip&longs;isibi circulo curuetur minus. </s>
</p>
<p type="main">
<s>Ea verò forma, qua mirum in modum delectati &longs;unt <lb/>Barbari, qui declinante imperio Italiam inua&longs;erunt, & <lb/>bonam emendati&longs;&longs;imamqueue antiquorum ædificandi ra­<lb/>tionem deturparunt, ex duobus con&longs;tat circuli portioni­<lb/>bus, quamobrem Albertus lib. 3. ho&longs;ce arcus, compo&longs;itos, <lb/>appellat. Circinantur autem hoc pacto, diui&longs;a nempe <lb/>&longs;ubten&longs;a, in partes tres, ea&longs;que æquales, ponitur circini <lb/>pes in altero diui&longs;ionum puncto & pars circuli de&longs;cribi­<lb/>tur, mox in altero puncto circini pede collocato alia cir­<lb/>culi portio lineatur, quibus arcus ip&longs;e integratur. Appel­<lb/>lant autem tertium acutum, eo quod ex &longs;ubten&longs;a in tres <lb/>partes diui&longs;a, arcus non fiatrotundus, &longs;ed in acutum an­<lb/>gulum ex duabus circuli portionibus de&longs;inens. </s>
</p>
<figure></figure>
<p type="main">
<s>Sint igitur muri <lb/>AC, BD, in quibus v­<lb/>trinque incumbæ KA, <lb/>BI. Ducatur itaque &longs;ub­<lb/>ten&longs;a horizonti æquidi­<lb/>&longs;tans AP, quæ in tres æ­<lb/>quales partes diuidatur <lb/>punctis E, F, tum centris <lb/>EF, circulorum portio­<lb/>nes de&longs;cribantur hinc <lb/>AG, HK, inde verò BG,
<pb pagenum="108"/>IH, ex quibus arcus totus integratur. Vtilis hæc quidem <lb/>&longs;pecies e&longs;t, licet inuenu&longs;ta, propterea quod haud violen­<lb/>ter incumbas vtrinque repellat, & in &longs;ummo magnis &longs;u&longs;ti­<lb/>nendis oneribus &longs;it apta. Producantur CH in N, DB verò <lb/>in O, &longs;itqueue centrum grauitatis AG in L, partis vero BG <lb/>in M. Quoniam igitur centra hæc ob elatam portionum <lb/>con&longs;titutionem quam proxima lineis AN, BO, fulcimen­<lb/>torum fiunt, maximè <expan abbr="&longs;u&longs;tinētur">&longs;u&longs;tinentur</expan>, & deor&longs;um potius quam <lb/>lateraliter in cumbas ip&longs;as premunt. Si quid tamen <expan abbr="habēt">habent</expan> <lb/>vitij, illud e&longs;t quod grauitatis centra momentum haben­<lb/>tia ad interiorem partem ver&longs;us PQ vim faciant, & nifi <lb/>partes magno &longs;uperimpo&longs;ito pondere comprimantur, <lb/>partes quæ &longs;unt circa HG, &longs;ur&longs;um pellentes aliquali &longs;ibi <lb/>rectitudine comparata corruunt, facta nempe circa L, M, <lb/>coniunctarum partium &longs;eparatione. </s>
</p>
<p type="main">
<s>His hoc pacto explicatis de &longs;emicirculari fornice a­<lb/>gemus, quæ cæteris omnibus vtilior e&longs;t, & longè pulcher­<lb/>rima, quamobrem Antiquis Architectis omnibus inpri­<lb/>mis admodum familiaris: </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to vanum <lb/>ABCD, muris v­<lb/>trinque clau&longs;um. <lb/>Ducatur per <expan abbr="sū-mitates">sun­<lb/>mitates</expan> <expan abbr="murorū">murorum</expan> <lb/>horizonti æqui­<lb/>di&longs;tans recta AD, <lb/>hac bifariam &longs;e­<lb/>cta in E, eodem <lb/>centro E, &longs;patio <lb/>verò EA &longs;emicir­<lb/>culus de&longs;cribatur <lb/>AFD, concaua <lb/>nempeip&longs;ius for-
<pb pagenum="109"/>nicis pars; tum eodem centro, &longs;patio verò EG, circinetur <lb/>GHI eiu&longs;dem fornicis pars conuexa. Po&longs;t hæc productis <lb/>lineis BH, CD, in OP, &longs;ecetur fornix tota in tres æquales <lb/>partes AGKM, MNLK, NDIL, & KME, LNE iungantur, <lb/>&longs;int autem partium ip&longs;arum grauitatis centra QRS. E&longs;t <lb/>autem R in ip&longs;a perpendiculari HE. Quoniam igitut <lb/>partium AGKM, DILN, quæ <expan abbr="vtrinq;">vtrinque</expan> &longs;unt grauitatis cen­<lb/>tra QS, in ip&longs;is &longs;unt fulcimentorum lineis OH PD, &longs;uâ <lb/>&longs;ponte fulcimentis eas &longs;u&longs;tinentibus partes ip&longs;æ &longs;tabunt. <lb/>Pars autem media KMNL deor&longs;um vergente perip&longs;am <lb/>HE lineam grauitatis centro, &longs;i parumper vel incumbæ <lb/>vel partes vtrinque AG<emph type="italics"/>K<emph.end type="italics"/>M, DILN cedant, vtpote quæ à <lb/>fulcimentis e&longs;t remoti&longs;&longs;ima, magno impetu &longs;uopte pon­<lb/>dere deor&longs;um feretur. quæ igitur in his &longs;emicircularibus <lb/>fornicibus partes &longs;tabiliores &longs;int, quæ verò ca&longs;ibus obno­<lb/>xiæ, ex his quæ diximus, clarè patet. </s>
</p>
<p type="main">
<s>Cæterùm cur incumbis manentibus fornix &longs;tet, ea <lb/>cau&longs;&longs;a e&longs;t, quod partes exteriores G<emph type="italics"/>K<emph.end type="italics"/>, <emph type="italics"/>K<emph.end type="italics"/>L, LI, maiores &longs;int <lb/>in ferioribus & oppo&longs;itis AM, MN, NG; quod &longs;uprà de­<lb/>mon&longs;trauimus. </s>
</p>
<p type="main">
<s>Si quid autem vitij in hac &longs;pecie e&longs;t, illud quidem <lb/>e&longs;t, quod &longs;ummapars <emph type="italics"/>K<emph.end type="italics"/>MNL deor&longs;um vergens magnâ vi <lb/>partes, quæ vtrinque &longs;unt, repellat, ex qua re &longs;olidarum <lb/>partium fit &longs;olutio, & inde ruina. </s>
</p>
<p type="main">
<s>Huic difficultati vt occurrerent peritiores Archite­<lb/>cti, plura excogitârunt remedia. Primum enim parietes <lb/>hinc inde ita &longs;olidos, cra&longs;&longs;os & firmos faciunt, vt &longs;uapte vi <lb/>re&longs;i&longs;tentes dimoueri loco nequeant, vel para&longs;tatas <expan abbr="addūt">addunt</expan> <lb/>vtin figura TX, VY. Præterea & ferrea claui ex incumba <lb/>in incumbam ducta & vtrinque firmata contrarias partes <lb/>validi&longs;&longs;imè connectunt, quæ calcitrantes (ita enim lo­<lb/>quuntur no&longs;trates <emph type="italics"/>A<emph.end type="italics"/>rchitecti,) fornicis pedes cohibent, & <lb/>&longs;olidum ne &longs;oluatur impediunt. qua in &longs;pecie dubitan <expan abbr="dū">dum</expan>
<pb pagenum="110"/>e&longs;&longs;et, an optimo loco &longs;it a &longs;it clauis, quæ per centrum? Et <lb/>&longs;anè videtur, quippe quod circa incumbas impetus fiat <lb/>maior. Ego autem vtiliusibi poni arbitror, vbi <expan abbr="punctaq.">punctaque</expan> <lb/>5. hoc e&longs;t, in medio tertiarum illarum partium, quæ vtrin­<lb/>que incumbis in&longs;i&longs;tunt, propterea quod primus impul&longs;us <lb/>ex media parte quæ impendet, ibi fiat. Rarò tamen boni <lb/>Architecti eo loco aptare &longs;olent, eo quòd eiu&longs;modi cla­<lb/>ues vel pulcherrimis ædifi cijs minuant gratiam. Vnde fit <lb/>vt nunquam &longs;atis laudetur Lucianus ille Benuerardus <lb/>Lauranen&longs;is Dalmata, qui nullibi apparentes eas po&longs;uit <lb/>in admirabili illa Vrbini Aula, quam Federico Feltrio, fe­<lb/>lici&longs;&longs;imo æquè & inuicti&longs;&longs;imo Duci, ædificauit. </s>
</p>
<p type="main">
<s>Tertio denique modo huic infirmitati me dentur, <lb/>vt videre e&longs;t in &longs;equenti figura, in qua vanum ADBC, mu­<lb/>ri vtrinque AF, BH, fornix verò FGH. Itaque dum muros <lb/>
<arrow.to.target n="fig22"></arrow.to.target><lb/>ex&longs;truunt, arre­<lb/>ctarias trabes, ro­<lb/>bore aliaue mate­<lb/>ria firmi&longs;&longs;ima, illis <lb/>in&longs;erunt, quales <lb/>&longs;unt IF<emph type="italics"/>K<emph.end type="italics"/> LHM, <lb/>ea proceritate vt <lb/>futuri fornicis &longs;u­<lb/>perent &longs;ummita­<lb/>tem. Con&longs;umma­<lb/>to enim fornice, <lb/>nondum tamen, <lb/>exarmato, tran&longs;­<lb/>uer&longs;ariam <expan abbr="trabē">trabem</expan> à <lb/>&longs;ummo fornicis <lb/>dor&longs;o parumper <lb/>eminentem in punctis I, L, arrectarijs trabibus validi&longs;&longs;i­<lb/>mis clauibus connectunt, tum punctis NP, Oq, capreolos
<pb pagenum="111"/>tran&longs;uer&longs;ario, & arrectarijs ferreis, clauis affigunt. Qui­<lb/>bus ita concinnatis, facta fornicis validâ pre&longs;&longs;ione in G, <lb/>in cumbi&longs;queue F, H, ad exteriora repul&longs;is, AB &longs;patium non <lb/>fit maius. Repul&longs;is enim in cumbis & muros propelli ne­<lb/>ce&longs;&longs;e e&longs;t, & cum muris ip&longs;as in&longs;ertas trabes, I<emph type="italics"/>K<emph.end type="italics"/>, LM. At va­<lb/>ricari non po&longs;&longs;unt, nî &longs;ecum trahant puncta PQ, quod fie­<lb/>ri non pote&longs;t, propterea quod in punctis N, O, validè di&longs;­<lb/>tineantur. Itaque &longs;patio AB non dilatato nulla fit ip&longs;ius <lb/>fornicis di&longs;&longs;olutio, quod vtique à principio ceu propo&longs;i­<lb/>tus finis quærebatur. Sed dicet qui&longs;piam, Nonne pende­<lb/>bit tran&longs;uer&longs;aria trabs in ip&longs;a di&longs;tractione arrectariorum, <lb/>pre&longs;&longs;a in punctis N, O? aut parum dicimus, aut nihil. Cum <lb/>enim PQ proxima &longs;int punctis FH, quæ cum arrectarijs à <lb/>muro di&longs;tinentur, magna in ijs fit vtrobique re&longs;i&longs;tentia. </s>
</p>
<figure id="fig22"></figure>
<p type="main">
<s>Rebus igitur ita &longs;e habentibus cum ob&longs;erua&longs;&longs;ent Ar­<lb/>chitecti, ob enormitatem ponderis fornices in tertia illa <lb/>
<arrow.to.target n="fig23"></arrow.to.target><lb/>parte quæ &longs;umma e&longs;t <lb/>laborare, <expan abbr="quãtum">quantum</expan> ter­<lb/>tijs vtrinque partibus <lb/>&longs;oliditatis addunt, tan­<lb/>tundem ex illa parte <lb/>&longs;uprema demere <expan abbr="&longs;olēt">&longs;olent</expan>, <lb/>vt videre e&longs;t in &longs;ubie­<lb/>cta figura, in qua par­<lb/>tes A, B, &longs;olidæ & cra&longs;­<lb/>&longs;iores, quibus hærent <lb/>partes, quæ CE, DG <lb/>cra&longs;&longs;æ quidem & illæ, <lb/>tum vero &longs;umma EFG, <lb/>alijs &longs;ubtilior. Minus <lb/>igitur grauante ponde­<lb/>re in F, minor fit ad in cumbas pre&longs;&longs;io, aut &longs;i qua fit, à <expan abbr="partiū">partium</expan> <lb/>ACE, BDG &longs;oliditate haud inualidè &longs;u&longs;tinetur. </s>
</p>
<pb pagenum="112"/>
<figure id="fig23"></figure>
<p type="main">
<s>Cæterùm admonet nos locus, vt aliquid de forni­<lb/>cum di&longs;&longs;olutionibus in medium afferamus: cau&longs;&longs;is enim <lb/>morborum cognitis, facilius periti medici adhibere &longs;o­<lb/>lent remedia. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim &longs;emicircula­<lb/>ris fornix ABC, cuius cen­<lb/>trum E, perpendicularis ve­<lb/>rò quæ per centrum DBE, &longs;e­<lb/>micirculi ABC, diameter <lb/>AEC, incumbæ <expan abbr="vtrinq;">vtrinque</expan> A, C. <lb/>Itaque &longs;i nulla fiat incumba­<lb/>rum repul&longs;io, &longs;tabit fornix; &longs;i verò fiat, ruinam facict. </s>
</p>
<p type="main">
<s>Pellanturitaque ad exteriores partes, vt in &longs;ecunda <lb/>
<arrow.to.target n="fig24"></arrow.to.target><lb/>figura, H in F, & C in G, <lb/>ex qua pul&longs;ione cum ma­<lb/>ius fiat &longs;patium quod in­<lb/>tegro fornice impleba­<lb/>tur, iam di&longs;tractis <expan abbr="vtrinq;">vtrinque</expan> <lb/>fornicis partibus <expan abbr="nō">non</expan> im­<lb/>pletur, Diuiditur igitur <lb/>locus maior factus in tres partes, quarum hincinde duas <lb/>replent fornicis partes, tertiam verò quæ media e&longs;t, re­<lb/>plet in&longs;ertus, ne vacuum detur, aër, vt in figura videre e&longs;t, <lb/>in qua &longs;olutæ vtrinque fornicis partes HIKF, PMNG, aër <lb/>autem medius &longs;patium replens IKMN. Diuidantur &longs;in­<lb/>guli qua drantes FK, GN, in partes tres, quarum duæ &longs;int <lb/>hincinde FQ, GR, & à centris, quæ &longs;eparatis quadranti­<lb/>bus facta &longs;untin ST, rectæ ducantur SQV. TRX. Quo­<lb/>niam igitur tertiæ partes vtrinque VIKQ MNRX pro­<lb/>pria grauitate depre&longs;&longs;æ, nullum quo &longs;u&longs;tineantur fulci­<lb/>mentum habent, corruent quidem. Ducantur autem re­<lb/>ctæ QI, RM, con&longs;tituentes cum ip&longs;is QV, RX pares an­<lb/>gulos VQI MRX. Itaque centris QR partes QIRM ad
<pb pagenum="113"/>inferiores partes deuoluentur, fientqueue QI, RM, vbi QZ, <lb/>RZ. Siautem QI, RM perpendicularibus quæ à punctis <lb/>QR ad perpendicularem DE ducuntur, fuerint maiores <lb/>conuenient alicubi in ip&longs;a perpendiculari, & altera alte­<lb/>ram &longs;u&longs;tinebit; &longs;i autem æquales tangent &longs;e & nihilomi­<lb/>nus fiet ruina, &longs;i minores nec &longs;e inuicem tangent, & nullà <lb/>re prohibente deor&longs;um corruent. tangant tautem &longs;e in <expan abbr="pū-cto">pun­<lb/>cto</expan> Z. quo pacto igitur fornices incumbis cedentibus in <lb/>medio aperti, <expan abbr="di&longs;&longs;oluãtur">di&longs;&longs;oluantur</expan> & ruinam faciant, ex i&longs;tis patet. </s>
</p>
<figure id="fig24"></figure>
<p type="main">
<s>Ex demon&longs;tratis qua&longs;i ex con&longs;ectario habemus for­<lb/>nices quo fuerint cra&longs;&longs;iores dato pari in cum barum &longs;ece&longs;­<lb/>&longs;u, ruinæ minus e&longs;&longs;e obnoxios quàm tenuiores, hoc e&longs;t, <lb/>maiori aperitione indigere ad ruinam cra&longs;&longs;iores quam te­<lb/>nuiores, quod licet ex iam dictis re&longs;ultet, nos tamen cla­<lb/>rius ex &longs;ubiecto &longs;chemate demon&longs;trabimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim cra&longs;&longs;ioris <lb/>fornicis pars <expan abbr="quidē">quidem</expan> ABCD, <lb/>tenuioris EFCD circa <expan abbr="idē">idem</expan> <lb/>centrum R. Ducatur au­<lb/>tem RM, &longs;ecans CD in G. <lb/>EF in H AB, in M. Centro <lb/>igitur G fiet euer&longs;io portio­<lb/>num fornicum, MD, HD, <lb/>Ducantur GA, GE & producta AD in N ip&longs;i AN perpen­<lb/>dicularis ducatur GN. quoniam igitur GE cadit in trian­<lb/>gulo AGN erit ex 21. propo&longs;. lib. 1. elem. GA, maior GE. <lb/>Corruente igitur maioris fornicis portione MD, recta <lb/>GA centro G punctum A de&longs;cribet portionem AI, mino­<lb/>ris interim ex GE, de&longs;cribente EL, at cadenti angulo A <lb/>occurrit in perpendiculari IK in puncto I angulus oppo­<lb/>&longs;itæ portionis, O, ip&longs;i autem E cadenti per EL non occur­<lb/>ret punctum P, cadens per Pq eo quod neutrum eorum <lb/>pertingat ad perpendicularem IK. Tenuioris ergo forni­
<pb pagenum="114"/>cis partes è &longs;uis locis auul&longs;æ ex eadem aperitione ruinam <lb/>facient, quod non contingit partibus cra&longs;&longs;ioris. quod &longs;a­<lb/>nè fuerat de clarandum. </s>
</p>
<p type="main">
<s>Quæritur adhuc, quare grauiores fornices in &longs;um­<lb/>mis ædificijs non &longs;ine vitio fiant? </s>
</p>
<p type="main">
<s>E&longs;to ædificium ABGH, cuius <expan abbr="vtrinq;">vtrinque</expan> muri ABCD, <lb/>EFGH, maiorum &longs;ummitates AD, EH, mediæ murorum <lb/>partes KL, fornicum &longs;ummus quidem DIE, medius verò <lb/>
<arrow.to.target n="fig25"></arrow.to.target><lb/>KML. Dico, magis cedere pul­<lb/>&longs;os muros &longs;ummos circa DE, <lb/>quam in medio circa KL. Sunt <lb/>enim muri BA, GH ceu vectes <lb/>quidam, <expan abbr="quorū">quorum</expan> extremis par­<lb/>tibus à fulcimentis BG remo­<lb/>ti&longs;&longs;imis potentia admouetur, <lb/>hoc e&longs;t, ip&longs;ius fornicis DIE ad <lb/>DE in cumbans repul&longs;io; lon­<lb/>gior e&longs;t autem pars à <expan abbr="fulcimē-to">fulcimen­<lb/>to</expan> ad potentiam AB, ip&longs;a BK. <lb/>Data igitur paritate potentia­<lb/>rum plus operabitur ea quæ in <lb/>D, illa quæ K. facilius crgo re­<lb/>pellentur muri in DE quàm in <lb/>KL. Alia quo que ratio intercedit, &longs;iquidem pondus muri <lb/>&longs;uperioris ADK, premens inferiorem murum KBC, cum <lb/>&longs;ua grauitate firmiorem, & pul&longs;ionibus minus obnoxium <lb/>reddit. Difficilius enim propellitur id quod graue e&longs;t <expan abbr="quã">quam</expan> <lb/>quod leue, vt nos quæ&longs;tione 10. demon&longs;trauimus. </s>
</p>
<figure id="fig25"></figure>
<p type="head">
<s>QVÆSTIO XVII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quærit Ari&longs;toteles, Cur paruo exi&longs;tente cuneo magna &longs;cindantur <lb/>pondera & corporum moles, validaque, fiat impre&longs;&longs;io?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>In parua re magnum negotium. Etenim quæ&longs;tio hæc
<pb pagenum="115"/>clari&longs;&longs;imorum virorum ingenia magnopere fatigauit. Ex <lb/>quibus Ari&longs;toteles inter veteres, Guid. Vbald. inter re­<lb/>centiores ad vectis naturam (ne quid in Mechanicis ad <lb/>vectem non reduci putaretur) cuneum ip&longs;um trahere co­<lb/>
<arrow.to.target n="fig26"></arrow.to.target><lb/>nati &longs;unt. Nos autem pro <lb/>veritate certantes, &longs;i in <lb/>horum &longs;ententiam vltrò <lb/>non tran&longs;ierimus, multa <lb/>venia digni à non iniquo <lb/>iudice exi&longs;timabimur. A­<lb/>ri&longs;totelis mentem clarè <lb/>& fusè explicat G. V­<lb/>bald. in Mechan. vbi de <lb/>Cuneo peculiariter a­<lb/>git. </s>
</p>
<figure id="fig26"></figure>
<p type="main">
<s>E&longs;to igitur &longs;cindendum quippiam ABCD, Cuneus <lb/>EFG, cuius pars HFI &longs;ci&longs;&longs;uræ in&longs;erta HI, facta igitur vali­<lb/>da percu&longs;&longs;ione in EG, fiet vt cum EG fuerit in NO, H &longs;it v­<lb/>bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit <lb/>&longs;ci&longs;&longs;io NSO, toti nempe cuneo EFG, æqualis. Vultigitur <lb/>Ari&longs;toteles, duos in cunco vectes con&longs;iderari EF, GF, quo­<lb/>rum alterius, nempe EF, fulcimentum &longs;it in H, pondus ve­<lb/>ro in F; alterius autem, hoc e&longs;t, GF fulcimentum quidem <lb/>&longs;it in I, pondus verò itidem &longs;it in F. His nequaquam con­<lb/>&longs;entiens G. Vbald. aliam viam ingreditur. Ait enim EHF <lb/>vectes quidem e&longs;&longs;e, quorum commune fulcimentum F, <lb/>potentias verò mouentes in EG. Pondera vtrinque inter <lb/>fulcimenta & potentias, vbi HI, idemque; e&longs;&longs;e ac &longs;i EF, GF, <lb/><gap/>eor&longs;um à cuneo con&longs;iderati in puncto F, adinuicem fulti <lb/>atque di&longs;tracti pondera pellerent H in NP, I verò in O, <lb/><expan abbr="q.">que</expan> Verumenimuerò quoniam cunei angulus non muta­<lb/>tur, nec vertex ip&longs;e centri vllum pror&longs;us præbet v&longs;um, nec <lb/>eius latera vtrinque di&longs;tracta ad contrarias partes didu­
<pb pagenum="116"/>cuntur, vectes in cuneo hoc pacto con&longs;iderare videtur à <lb/>veritate alienum. Ari&longs;totelis autem &longs;olutionem fal&longs;am e&longs;­<lb/>&longs;e, clarè patet. quo pacto enim F pellet ex fulcimento Hi­<lb/>p&longs;am ligni partem OS, & idem F ex fulcimento I pellet <lb/>oppo&longs;itam partem NS, &longs;i inuicem contendentes extremæ <lb/>vectium partes in F, altera alteri ne quicquam operentur, <lb/>e&longs;t impedimento? Et &longs;anè opinionis fal&longs;itas inde patet, <lb/>quòd videamus materiæ partes &longs;ci&longs;&longs;as, in ip&longs;o &longs;ei&longs;&longs;ionis a­<lb/>ctu facta di&longs;tractione à cunei vertice nequaquam tangi. <lb/>At eiu&longs;modi operationes per contactum fieri nulli e&longs;t i­<lb/>gnotum. Solutio igitur i&longs;ta mco iudicio, tanto Philo&longs;o­<lb/>pho pror&longs;us videtur indigna. </s>
</p>
<p type="main">
<s>Porrò G. Vbald. ijs quæ de diuaricatis vectibus in <lb/>medium adduxerat non acquie&longs;cens alias quærit cau&longs;&longs;as, <lb/>cur cuneus minoris anguli validiùs &longs;cindat. Idque; ex quo­<lb/>dam lemmate demon&longs;trare conatur, figura autem eius ita <lb/>ferè &longs;e habet. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to cuneus ABC, <lb/>item alius DEF. <expan abbr="Demō-&longs;trauit">Demon­<lb/>&longs;trauit</expan> igitur ex a&longs;&longs;um­<lb/>pto, quo acutior fuerit <lb/>angulus BIM, eo facilius <lb/>pondera moueri, & ideo <lb/>facilius ceu vecte AB <lb/>moueri pondus I quàm <lb/>vecte DE pondus <expan abbr="q.">que</expan> In­<lb/>geniosè quidem. At ma­<lb/>gnam hæc apud me ha<gap/><lb/>bent difficultatem. Si e­<lb/>nim ita &longs;e habet AB, ad BI, vt DE, ad EQ (ip&longs;æ enim DE, <lb/>EQ &longs;upponuntur æquales) ergo eadem æquali&longs;ue poten­<lb/>tia æqualiter mouebit pondera I & <expan abbr="q.">que</expan> quod ip&longs;i eiu&longs;dem <lb/>demon&longs;trationi pror&longs;us concludit contrarium. Nec meo
<pb pagenum="117"/>quidem iudicio id &longs;equi videtur, propterea quod ex Pap­<lb/>po ea quæ in planis inclinatis mouentur, redigantur ad li­<lb/>bram. Ratio enim valde e&longs;t diuer&longs;a, &longs;iquidem pondera <lb/>quæ in planis inclinatis mouentur, certa habent fulci­<lb/>menta & determinatas tum brachiorum tum ponderum <lb/>proportiones, quæ omnia in cuneo, nec quidem mente <lb/>concipi po&longs;&longs;e, clarè patet. </s>
</p>
<p type="main">
<s>His igitur difficultatibus con&longs;ideratis, Nos cunei <lb/>vim, ad alia e&longs;&longs;e principia referendam pro comperto ha­<lb/>bemus. Ordimur igitur hoc pacto. Cuneo quidem res di­<lb/>uidi certum e&longs;t. Cæ terùm quæ natura diuidere apta &longs;unt, <lb/>tria &longs;unt, punctum, linea, &longs;uperficies, Puncto enim linea, <lb/>lineâ &longs;uperficies, &longs;uperficie autem corpus ip&longs;um diuidi­<lb/>tur. quæ omnia à Mathematico ab&longs;que materia con&longs;ide­<lb/>rantur. De diui&longs;ione autem quæ fit ex puncto, nihil agit <lb/>Mechanicus, qui corporibus quidem vtitur, ad cuius na­<lb/>turam non trahitur punctum, cuius partes &longs;unt nullæ. At <lb/>non lineis & &longs;uperficiebus modò corpora diuiduntur, &longs;ed <lb/>etiam corporibus, quod verum e&longs;t, at ea corpora ad linea­<lb/>rum & &longs;uperficierum naturam quodammodo aptari faci­<lb/>lè docebimus. Dicimus igitur, duplicem e&longs;&longs;e Cuneorum <lb/>&longs;peciem, linearem vnam, &longs;uperficialem alteram. linearem <lb/>appello, quæ ad lineæ naturam magnopere accedit. Tales <lb/>&longs;unt orbiculares illæ cu&longs;pides, quibus ad perforandum v­<lb/>timur, & ideo vernaculè Pantirolos vocamus. Acus item <lb/>&longs;utorij, & cætera quæ nen &longs;ecus ac linea in punctum de&longs;i­<lb/>nunt, & imagina<gap/>iam quandam lineam ceu axem in eo <lb/>puncto de&longs;inen<gap/>em continent. Ad lineam quo que refe­<lb/>runtur lateratæ cu&longs;pides oblongæ, & &longs;ubtiles ceu&longs;ubulæ, <lb/>claui, en&longs;es, pugiones, & his &longs;imilia, quæ cum adacta vali­<lb/>dam faciant partium &longs;eparationem ad cunei naturam <expan abbr="nō">non</expan> <lb/>referre magnæ videretur dementiæ. Ettunc quantoma­<lb/>gis corpora hæc ad linearem naturam accedunt, eo ma­
<pb pagenum="118"/>gis penetrant. Sed & hocidem in rebus non ab arte, &longs;ed <lb/>ab ip&longs;anatura productis facile e&longs;t cogno&longs;cere. Quis enim <lb/>non experitur, quàm validè culex, infirmi&longs;&longs;imum animal, <lb/>& ea paruitate qua e&longs;t, hominum & cæterorum <expan abbr="animaliū">animalium</expan>, <lb/>cutes aculeata probo&longs;cide penetret? Id vtique non alia de <lb/>cau&longs;&longs;a fit, quod ad imaginariæ lineæ &longs;ubtilitatem quam, <lb/>proximè accedat. Ve&longs;pæ quoque, Apes, Scorpiones a­<lb/>culeis i&longs;tis ceu linearibus cuneis vtuntur. Nec refert, vt <lb/>diximus, vt um laterati &longs;int, ceu &longs;ubulæ, & claui, vel ro­<lb/>tundi & vtrum plura paucioraue latera habeant, dummo­<lb/>do in punctum & aculeatam aciem de&longs;inant. Altera por­<lb/>ro cuneorum &longs;pecies &longs;uperficiei naturam &longs;apit, acie &longs;iqui­<lb/>dem in lineam de&longs;init, quæ &longs;uperficiei e&longs;t terminus, <expan abbr="quã">quam</expan>. <lb/>obrem huc ea omnia referuntur, quæ acie ipsâ &longs;cindunt, <lb/>ceu &longs;unt cunei propriè dicti, de quibus hoc loco e&longs;t &longs;er­<lb/>mo, cultra, en&longs;es, a&longs;ciæ, &longs;ecures, &longs;calpra lata, & cætera e­<lb/>in&longs;modi, quibus corpora acie &longs;cinduntur. Quidam his ad­<lb/>dunt &longs;erras, quibus haud pror&longs;us a&longs;&longs;entimur. Etenim alia <lb/>ratione diuidunt, &longs;icut & limæ &longs;olent, deterendo enim, <expan abbr="nō">non</expan> <lb/>&longs;cindendo ferri, ligni, & marmorum duritiem diuidunt & <lb/>domant. His igitur <expan abbr="cō&longs;ideratis">con&longs;ideratis</expan>, &longs;i daretur ex materia qua­<lb/>piam in frangibili cuneus, qui maximè ad &longs;uperfi ciei natu­<lb/>ram accederet, vel paruo labore tenaci&longs;&longs;ima ligna validi&longs;­<lb/>&longs;imè &longs;cinderet, & ideo optimè res gladijs illis diuiditur, <lb/>qui magis ad &longs;uperficiei naturam accedunt. Ex quibus o­<lb/>mnibus, nî fallimur, clarè patet, curacutiores angulo cu­<lb/>nei obtu&longs;ioribus facilius &longs;cindant, quæ quidem ratio lon­<lb/>gè ab ea di&longs;tat, ex qua cæteri ferè omnes Cuneum ad ve­<lb/>ctis naturam referre hactenus contenderunt. </s>
</p>
<p type="main">
<s>Cæterùm vtramque eorum quos diximus, <expan abbr="cuneorū">cuneorum</expan> <lb/>&longs;peciem &longs;olerti&longs;&longs;ima cognouit Natura, & ideo quoniam <lb/>res vel contu&longs;ione vel perforatione, vel &longs;ecatione con&longs;i­<lb/>ciuntur, triplicem dentium qualitatem dentatis animali-
<pb pagenum="119"/>
<arrow.to.target n="fig27"></arrow.to.target><lb/>bus dedit, Molares, <lb/>qui & Maxillares ap­<lb/>pellantur, quibus <lb/>cibus contunditur, <lb/>Canini, quibus fit <lb/>perforatio, Anterio­<lb/>res, quibus cibus <lb/>&longs;cinditur, quos ideo <lb/><foreign lang="greek">temnikou\s</foreign>, id e&longs;t, &longs;ecan­<lb/>tes appellant Graeci. </s>
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<figure id="fig27"></figure>
<p type="main">
<s>Molares KK, <lb/>CaniniL, L, Temni­<lb/>ci &longs;eu &longs;ecantes M. Cuneus orbicularis lineari&longs;queue AB, in <lb/>quo axis linea e&longs;t, ad cuius naturam accedit AB cuneus <lb/>&longs;uperficialis CD, accedens ad &longs;uperficiei naturam, quam <lb/>vitro imaginamur EFGD, in aciem cunei de&longs;inentem, <lb/>GD, Lateratus lineari&longs;que cuneus, clauus HI. </s>
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<p type="main">
<s>Cunei autem omnes dupliciter &longs;unt efficaces, vel e­<lb/>nim malleo, vt in ijs fit, quibus lìgna &longs;cin duntur & &longs;calpris <lb/>fieri &longs;olet, adiguntur, vel impul&longs;u & pre&longs;&longs;ione, vt in gla­<lb/>dijs fit, pugionibus, cælatorum &longs;calpris, &longs;ubulis, & cæteris <lb/>eiu&longs;modi. Quidam etiam &longs;unt, qui licet mallei ictu non <lb/>adigantur, malleum coniunctum habent, ceu &longs;unt &longs;ecu­<lb/>res, ligones, A&longs;ciæ, & his &longs;imilia, quæ ex percu&longs;&longs;ione &longs;e­<lb/>metip&longs;a &longs;cindendis rebus in&longs;erunt & validè penetrant. <lb/>De vi autem & efficacia ictus &longs;eu percu&longs;&longs;ionis hic &longs;uper­<lb/>&longs;ed emus aliquid, ea de re, in &longs;equenti quæ&longs;tione verba fa­<lb/>cturi. </s>
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<p type="main">
<s>Multa hîc addere potui&longs;&longs;emus ad Cochleam &longs;pe­<lb/>ctantia, quippe quòd Cochlea cuneus &longs;it Cylindro inuo­<lb/>lutus, qui quidem ad mallei, &longs;ed vectis virtute &longs;ibi adiun­<lb/>ctâ, validi&longs;&longs;imè operatur, & &longs;excentis in&longs;eruit v&longs;ibus. Ve­<lb/>runtamen cùm de hac &longs;pecie egregiè di&longs;&longs;erat G. Vbaldus,
<pb pagenum="120"/>con&longs;ultò hanc di&longs;putationem omittimus; idque hac quo­<lb/>que de cau&longs;&longs;a, quod nihil de cochlea, ac &longs;i eam non noui&longs;­<lb/>&longs;et, locutus &longs;it Ari&longs;toteles. </s>
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<p type="main">
<s>Po&longs;&longs;umus autem in actu &longs;ci&longs;&longs;ionis, quæ cuneo fit, a­<lb/>liâ tamen ratione vectem con&longs;iderare, nempe non in cu­<lb/>neo quidem, &longs;ed in ip&longs;a re quæ &longs;cinditur. </s>
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<figure></figure>
<p type="main">
<s>E&longs;to enim quip­<lb/>piam &longs;ci&longs;&longs;ile ABCD, <lb/>cui alteri extremita­<lb/>tum, puta BD, cuneus <lb/>adigatur EFG, <expan abbr="fiatq;">fiatque</expan> <lb/>&longs;ci&longs;&longs;io per longitudi­<lb/>nem &longs;ecundum <expan abbr="lineã">lineam</expan> <lb/>EH. facta igitur ex <lb/>cunei ingre&longs;&longs;u <expan abbr="partiū">partium</expan> &longs;eparatione B, expelletur in I, D ve­<lb/>rò in K. fient igitur materiæ &longs;ci&longs;&longs;æ partes AIBH, CKDH, <lb/>ceu duo vectes, quorum hinc inde in corpore ip&longs;o fulci­<lb/>menta L, M potentiæ vtrinque dilatantes BD, pondus ve­<lb/>rò materiæ re&longs;i&longs;tentia, in &longs;eparationis loco vbi N. Duca­<lb/>tur NL, quanto itaque BN maiorem habebit proportio­<lb/>nem ad LN, eo faciliùs re&longs;i&longs;tentia quæ in N, &longs;uperabitur. <lb/>Mutatur <expan abbr="autē">autem</expan> a&longs;&longs;iduè in ip&longs;a &longs;ci&longs;&longs;ione fulcimentum, & <expan abbr="cū">cum</expan> <lb/>fulcimento ip&longs;a proportio. Pertingente enim &longs;ci&longs;&longs;ione in <lb/>O, <expan abbr="fulcimētum">fulcimentum</expan> fit in P. quo ca&longs;u &longs;ci&longs;&longs;ura e&longs;t facilior, quip­<lb/>pe quod maiorem habeat proportionem BO ad OP, <expan abbr="quã">quam</expan> <lb/>BN ad NL. Hoc autem experiuntur materiarij, qui primis <lb/>ictibus, &longs;ecuriculâ nondum probè adactâ, & nondum fa­<lb/>ctâ notabili &longs;ci&longs;&longs;ione difficultatem &longs;entiunt, mox <expan abbr="factaiã">factaiam</expan> <lb/>&longs;eparatione faoillima paullatim fit materiæ totius &longs;epara­<lb/>tio. Hocidem & nos ab&longs;que cunei v&longs;u experimur, cum ba­<lb/>culum aut quippiam tale manibus diductis &longs;cin dimus. à <lb/>principio enim difficultatem &longs;entimus, deinde ex ea <expan abbr="quã">quam</expan> <lb/>diximus proportionc &longs;ci&longs;&longs;io ip&longs;a fit apprime facilis. Vti-
<pb pagenum="121"/>mur etiam vecte cuneato ad &longs;cindendum & aperiendum: <lb/>adacto enim &longs;ci&longs;&longs;uræ cuneo, idqueue manu malleoue, tum <lb/>ab altera extremitate pre&longs;&longs;o, valida fit ex vectis vi <expan abbr="cōtinui">continui</expan> <lb/>
<arrow.to.target n="fig28"></arrow.to.target><lb/>corporis &longs;eparatio. Ma­<lb/>teria &longs;ci&longs;&longs;ilis AB <expan abbr="&longs;calprū">&longs;calprum</expan> <lb/>ceu vectis cuneatus CD, <lb/>cuius fulcimentum, E, <lb/>pondus verò vbi C, po­<lb/>tentia vbi D, quo ca&longs;u <lb/>quo maior e&longs;t proportio <lb/>DE ad EC, eo e&longs;t ip&longs;a &longs;ci&longs;&longs;io leuior & facilior. </s>
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<figure id="fig28"></figure>
<p type="head">
<s>QV AESTIO XVIII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quærit hic Ari&longs;toteles, Cur per Trochleas ab exiguapotentia in­<lb/>gentia moueantur pondera?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>De Trochlea Pappus, & veteres: inter recentiores e­<lb/>gregiè admodum, vt omnia examinauit in Mechani­<lb/>cis G. Vbaldus. Nos tamen interim po&longs;t clari&longs;&longs;imos illos <lb/>viros aliquid quod nouitatem & &longs;ubtilitatem &longs;apiat, de <lb/>no&longs;tro penu promemus. Et &longs;anè inuentis quidem addere <lb/>res e&longs;t fa cilis, at quod inuentis addas inuenire haud adeo <lb/>facile. Sed nos primum Philo&longs;ophi ip&longs;ius dicta ad <expan abbr="trutinã">trutinam</expan> <lb/>reuocemus. Ita autem quæ&longs;tionem proponit; Cur &longs;i qui&longs;­<lb/>piam Trochleas componens duas, in &longs;ignis duobus, ad &longs;e <lb/>inuicem iunctis contrario ad Trochleas modo circulo fu­<lb/>nem circumduxerit, cuius alterum quidem caput tigno­<lb/>rum appendatur alteri, alterum verò Trochleis &longs;it <expan abbr="innixū">innixum</expan> <lb/>& à funis initio trahere cœperit, magna trahit pondera, li­<lb/>cetimbecillium fuerit virium? </s>
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<p type="main">
<s>Ob&longs;euri&longs;&longs;ima expo&longs;itio, & nî res e&longs;&longs;et vulgò per &longs;e <lb/>nota, dequeue ea Vitruuius & Mechanici non egi&longs;&longs;ent, diffi­<lb/>cile vtique e&longs;&longs;et ex eius verbis &longs;en&longs;um a&longs;&longs;equi. </s>
</p>
<pb pagenum="122"/>
<p type="main">
<s>Tigna &longs;anè voca&longs;&longs;e videtur ea ligna, quæ à Vitruuio <lb/>Rechami dicuntur, in quibus nempe ip&longs;i in&longs;eruntur orbi­<lb/>culi. Et&longs;i de tignis eiu&longs;mo di aliud quippiam &longs;entire videa­<lb/>tur Picolomineus. Græea lectio pro tignis habet <foreign lang="greek">cu/la</foreign>, id <lb/>e&longs;t, ligna; item vbi Leoniceni ver&longs;io legit, ad &longs;e inuicem <lb/>iunctis, textus habet <foreign lang="greek">snm<gap/>ai/nousin e(autoi_s e)ranti/ws</foreign>, hoc e&longs;t, in­<lb/>uicem ex oppo&longs;ito concurrunt. Certè locum totum ita <lb/>redderem: Cur &longs;i quis duas Trochleas fecerit, in duobus <lb/>lignis &longs;ibi ex oppo&longs;ito concurrentibus, ei&longs;queue Trochleis <lb/>circumpo&longs;uerit funem, cuius alterum caput alteri ligno­<lb/>rum &longs;it annexum, alterum verò Trochleis cohæreat, vel <lb/>apponatur. Si quis alterum funis principium trahat, ma­<lb/>gna trahat pondera, et&longs;i trahens potentia &longs;it exigua? Nos <lb/>verbis figuram, & figurâ verba ip&longs;a elucidabimus. </s>
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<figure></figure>
<p type="main">
<s>Sint duo ligna ex oppo&longs;ito concurrentia, <lb/>in quibus Trochleæ, hoc e&longs;t, orbiculi AB, fu­<lb/>nis ductarius DABC, cuius alterum caput re­<lb/>ligatum e&longs;t ligno trochleæ A, vbi e&longs;t C. Tro­<lb/>chlea A loco &longs;tabili commendata, vbi E. Pon­<lb/>dus alteri ligno Trochleæ appen&longs;um F. Tra­<lb/>cto itaque fune DABC, eleuatur & trahitur <lb/>pondus F. Ex quibus clarè patet, <expan abbr="Philo&longs;ophū">Philo&longs;ophum</expan> <lb/>propo&longs;ui&longs;&longs;e Trochleam duobus tantum orbi­<lb/>culis munitam, quod vtique &longs;atis erat ad ex­<lb/>plicationem. Inquit autem, faciliùs vecte <expan abbr="quã">quam</expan> <lb/>manu pondus moueri. Trochleam vero (id <lb/>e&longs;t, orbiculum; ita enim e&longs;t intelligendum) e&longs;­<lb/>&longs;e vectem, aut vectis virtute operari. Ita autem <lb/>videtur argumentari. Si vnicâ Trochleâ plus trahitur <lb/>quàm manu, multo faci ius & velocius id fiet duobus, <lb/>quibus plus, vt ip&longs;e ait, quàm in duplici velocitate pon­<lb/>dus leuabitur. Summa dictorum e&longs;t, ex multiplicatione <lb/>orbiculorum pondus ip&longs;um imminui, & minori difficul-
<pb pagenum="123"/>tate leuari, quod &longs;anè verum e&longs;t. Nos tamen nonnulla <expan abbr="cō-&longs;iderabimus">con­<lb/>&longs;iderabimus</expan>. quod ait, vecte facilius moueri pondera <lb/>quam manu, &longs;emper non e&longs;t verum. Si enim vectis pars <lb/>quæ à fulcimento ad manum breuior fuerit illâ, quæ à <lb/>fulcimento ad pondus difficilius vecte pondus mouebi­<lb/>tur quam manu. Idem quoque accidet, &longs;i eo modo vecte <lb/>vtamur, quem ob&longs;eruat Guidus Vbald. Tract. de Vecte <lb/>prop. 3. Po&longs;ita nempe inter fulcimentum & pondus &longs;u&longs;ti­<lb/>nente potentiâ. Præterea quod a&longs;&longs;eruit Ari&longs;toteles, Tro­<lb/>chleas ad vectem reduci, verum quidem e&longs;t, &longs;ed aptius di­<lb/>xi&longs;&longs;et ad libram, etenim vectis vtcunque à ful cimento di­<lb/>uiditur. Libra verò quod & orbiculis ex centro accidit, <lb/>&longs;emper bifariam. Ad hæc videtur ille ad orbiculorum <lb/>multiplicitatem Trochlearum vim referre. Si enim, ait, <lb/>vnicâ Trochleâ pondus facile trahitur, id multo validius <lb/>pluribus fiet. Veruntamen non ab&longs;olutè ex orbiculorum <lb/>multiplicationeid fieriita o&longs;tendemus. </s>
</p>
<figure></figure>
<p type="main">
<s>Sint duæ op­<lb/>po&longs;itæ lineae rectae, <lb/>vtpote trabes AB, <lb/>CD, <expan abbr="inuicē">inuicem</expan> æqui­<lb/>di&longs;tantes & ip&longs;æ <lb/>&longs;tabiles: &longs;uperiori <lb/>tres appendantur <lb/>orbiculi ex <expan abbr="pūctis">punctis</expan> <lb/>E, F, G, <expan abbr="nēpe">nempe</expan> ML, <lb/>PQ, TV. inferiori <lb/><expan abbr="autē">autem</expan> duobus pun­<lb/>ctis IH, nempe <lb/>NO, RS. Erunti­<lb/>gitur invniuer&longs;um <lb/>quinque, indatur pereos funis ductarius KLMNOP <lb/>QRSTVX, ex cuius extremitate pendeat pondus X,
<pb pagenum="124"/>Trahatur funis in K. Dico ex multiplicatione <expan abbr="orbiculorū">orbiculorum</expan>, <lb/>trahentipondus nequaquam minui. Sint autem orbicu­<lb/>lorum diametri, LM, NO, PQ, RS, TV, applicetur poten­<lb/>tîa in S. Erit igitur ad hoc vt &longs;u&longs;tineat æqualis ponderi X, <lb/>orbiculi enim TV &longs;emidiametri &longs;unt æquales. Transfe­<lb/>ratur <expan abbr="potētia">potentia</expan> in q, & ita deinceps donec perueniatur in K, <lb/>vbi funis ip&longs;ius e&longs;t principium, Idem e&longs;t igitur &longs;eruata &longs;em­<lb/>per &longs;emidiametrorum æqualitate ac &longs;i potentia quæ e&longs;t in <lb/>K, applicata intelligatur in T vel in V. vbicunque enim <lb/>collocetur, ponderi erit æqualis. Nihil igitur rebus ita <lb/>di&longs;po&longs;itis, orbiculorum multiplicatio ad facilitatem ope­<lb/>ratur. Alia itaque ratio quærenda e&longs;t, quam non &longs;atis ex­<lb/>plica&longs;&longs;e videtur Ari&longs;toteles. Probabimus autem, nullam <lb/>ex &longs;uperioribus orbiculis fieri ponderum imminutionem, <lb/>&longs;ed totam vim in inferioribus con&longs;i&longs;tere. At nos interim <lb/>quippiam quod ad rem faciat, proponamus. </s>
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<figure></figure>
<p type="main">
<s>E&longs;to punctum A, cuirectæ ap­<lb/>pendantur lineæ BAC, diui&longs;æ qui­<lb/>dem in A, &longs;it autem lineæ BA caput <lb/>B, ip&longs;ius verò CA caput C. Modò <lb/>intelligantur vnitæ in A, &longs;itqueue vni­<lb/>ca linea à puncto A ceu funiculus <lb/>dependens BAC; Appendatur capi­<lb/>ti B pondus B. Capiti vero C, <expan abbr="pōdus">pondus</expan> <lb/>C, inter &longs;e æqualia. Potentia igitur <lb/>in A, duo &longs;u&longs;tinebit pondera BC. <lb/>Pondera verò ex æqualitate æque­<lb/>ponderabunt. Quod &longs;i B potentia <lb/>dicatur &longs;u&longs;tinens pondus C, aut C <lb/>potentia &longs;u&longs;tinens pondus D, vel <lb/>duæ potentiæ inter &longs;e æquales, nihil <lb/>refert. Vtcunque enim id &longs;it, fiet æquilibrium. Habemus <lb/>igitur ex i&longs;tis ad &longs;u&longs;tinendum pondus ex &longs;uperiori parte
<pb pagenum="125"/>appen&longs;um potentiam requiri ip&longs;i ponderi æqualem. Ani­<lb/>mo po&longs;thæc concipiatur alia recta linea DEF, cuius inte­<lb/>gra longitudo &longs;i exten deretur, e&longs;&longs;et DE, EF. Appendatur <lb/>in E pondus E æ quale alteri ponderum B vel, C, &longs;int autem <lb/>duæ potentiæ pondus E &longs;u&longs;tinentes D, F. Vtraque igitur <lb/>dimidium &longs;u&longs;tinebit ponderis E, &longs;ed potentia quæ &longs;u&longs;ti­<lb/>nebat pondus B, in C erat ip&longs;i B æqualis, vbi appen&longs;io pon­<lb/>deris erat in &longs;uperiori parte in A, hîc autem, vbi appen&longs;io <lb/>e&longs;t in parte in feriori, vtraque potentia dimidium &longs;u&longs;tinet <lb/>appen&longs;i ponderis. Videmus igitur illam appen&longs;ionem <lb/>quidem pondus nullatenus imminuere, hanc verò pon­<lb/>dus ip&longs;um, bifariam diui&longs;um, &longs;u&longs;tinentibus potentijs im­<lb/>partiri. Hæ cin lineis, Mathematicâ v&longs;i ab&longs;tractione, con­<lb/>&longs;iderauimus, nunc verò eadem mechanicè perpenda­<lb/>mus. </s>
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<figure></figure>
<p type="main">
<s>Sit igitur <lb/>punctum A, vt <lb/>in &longs;e quenti figu­<lb/>ra clauus paxil­<lb/>lu&longs;ue, cui appen­<lb/>&longs;us funiculus <lb/>BAC, & funicu­<lb/>li capitibus pon­<lb/>dera BC, &longs;it quo­<lb/>que anulus D, <lb/>per quem traìe­<lb/>ctus funiculus <lb/>EDF. Anulo au­<lb/>tem <expan abbr="cōiunctum">coniunctum</expan> <lb/>pondus G. His igiturita con&longs;titutis, eadem demon&longs;tra­<lb/>buntur quæ &longs;uperius, nempe oportere vt fiat æquilibrium <lb/>B, C, e&longs;&longs;e æqualia, tum potentias, quæ &longs;unt in EF pondus <lb/>G inter eas diui&longs;um &longs;u&longs;tinere. Porrò volentes Mechanici
<pb pagenum="126"/>funiculos circa paxillum, & anulum ad attollenda & de­<lb/>primenda pondera mouere incommodè illis vtique &longs;uc­<lb/>cedebat, clauo & anulo motum difficilem facientibus. <lb/>Quamobrem vt difficultati occurrerent, ad locum claui <lb/>clauo ip&longs;i orbiculum circumpo&longs;uerunt, & anuli itidem <lb/>loco orbiculum aptauerunt. Hæc autem agentes reii­<lb/>p&longs;ius naturam non mutauerunt, &longs;ed &longs;ibi, vt diximus, ex or­<lb/>biculis maximam commoditatem <expan abbr="atq;">atque</expan> facilitatem com­<lb/>parârunt. </s>
</p>
<p type="main">
<s>Ex his principîjs tota Trochlearum ratio pendet, <lb/>quæ tamen alia quoque con&longs;ideratione in idem tenden­<lb/>te examinari pote&longs;t, quod quidem fecere veteres, & ip&longs;e, <lb/>qui veteres optim è imitatus e&longs;t, Guid. Vbaldus. </s>
</p>
<p type="main">
<s>Vidimus vtique nos, à potentia quæ e&longs;t in B, pondus <lb/>par &longs;u&longs;tineri in C, Potentiam autem quæ e&longs;t in E <expan abbr="dimidiū">dimidium</expan> <lb/>&longs;u&longs;tinere ponderis quod e&longs;t in G. Nos igiturij&longs;dem in&longs;i­<lb/>&longs;tentes adiecta libra, vecteue, bifariam diui&longs;o rem ip&longs;am <lb/>ex &longs;ubiecto diagrammate lucidiorem faciemus. </s>
</p>
<p type="main">
<s>E&longs;to linea quædam &longs;tabilis ceu trabs horizontiæ­<lb/>quedi&longs;tans AB, cui in A funiculus annectatur AC, cuius <lb/>extremum C vecti cuidam alligetur CD, in medio diui&longs;o <lb/>vbi E, tum alteri vectis eiu&longs;dem extremitati D, funiculus <lb/>nectatur DG, & à puncto E pondus appendatur F. puta li­<lb/>brarum mille, Tum puncto G in medio vectis HI, funis re­<lb/>ligetur DG, & ex altero vectis extremo alligato fune HK <lb/>commendetur lo co &longs;tabili in K, & ab alio capite vectis vbi <lb/>Iad medium vectis MN, vbi L, funis annectatur lL, tum <lb/>ex vectis capite M, funis commendetur MO, loco &longs;tabili <lb/>in O, & alteri capiti N, funis, NP, qui alligetur medio ve­<lb/>cti QR in P, & ex Q, funis QS. Commendetur loco &longs;tabili <lb/>in S, & alteri vectis extremo R funis alligetur RT, cui <lb/>quidem potentia &longs;u&longs;tinens applicetur in T. Dico igitur,
<pb pagenum="127"/>
<arrow.to.target n="fig29"></arrow.to.target><lb/>rebus ita di&longs;po&longs;itis, <lb/>potentiam in T ita <lb/>&longs;e habere ad pondus <lb/>F, vt vnum ad &longs;ex de­<lb/>cim, hoc e&longs;t, in pro­<lb/>portione e&longs;&longs;e &longs;ub­<lb/>&longs;exdecupla. Sunt <lb/>autem, hic vectes <lb/>quatuor in feriorum <lb/>cubiculorum, loco, <lb/>CD, HI, MN, QR, <lb/>qucrum, centra E, <lb/>G, L, P. quoniam e­<lb/>nim A hoc e&longs;t, C, v­<lb/>nà cum potentia G, <lb/>hoc e&longs;t, D, &longs;u&longs;tinet <lb/>pondus F alterum, <lb/>ponderis dimidium <lb/>&longs;u&longs;tinebit C, <expan abbr="alterū">alterum</expan> <lb/>vero D. erunt igitur <lb/>vtrinque librae quin­<lb/>gentæ. Tum potentia in K, hoc e&longs;t, in H, vna cum poten­<lb/>tia in L, hoc e&longs;t, in I &longs;u&longs;tinebunt quingenta. Quare <expan abbr="vtraq;">vtraque</expan> <lb/>ducenta quin quaginta, &longs;ed hoc totum bifariam diuiditur <lb/>inter potentias, O, id e&longs;t, M, & P, id e&longs;t H. erunt igitur v­<lb/>trinque centum viginti quinque. Ea autem &longs;umma <expan abbr="iterū">iterum</expan> <lb/>bifariam diuìditur, hoc e&longs;t, inter potentias S, id e&longs;t, Q & <lb/>T, id e&longs;t, R, quare vtraque &longs;u&longs;tinet &longs;exaginta duo cum di­<lb/>midio. Sed numerus i&longs;te ad Millenarium ita &longs;e habet vt v­<lb/>num ad &longs;exdecim. Hinc colligimus, pondus totum inter <lb/>loca &longs;tabilia diuidi, nempe A, K, O, S, & ip&longs;am potentiam <lb/>quæ &longs;u&longs;tinet in T, & locis ip&longs;is &longs;tabilibus quindecim par­<lb/>tes integri ponderis, potentia verò T &longs;extam decimam
<pb pagenum="128"/>tantùm commendari. Itaque &longs;i ex puncto V appendere­<lb/>tur AB, in X potentia, quæ in X &longs;u&longs;tineret mille, minus <lb/>&longs;exaginta duo cum dimidio, quod quidem à potentia in <lb/>T &longs;u&longs;tinetur; quod &longs;i alius adderetur orbiculus, & fierent <lb/>quinque, potentia in T &longs;u&longs;tiner et trige&longs;imam &longs;ecundam <lb/>partem integri ponderis, hoce&longs;t, dimidium librarum &longs;e­<lb/>vaginta duarum cum dimidio, nempe triginta & vnam <lb/>cum quarta parte, &longs;i item textus adderetur, potentia in T <lb/>&longs;exage&longs;imam partem &longs;u&longs;tineret integri ponderis, hoc e&longs;t, <lb/>libras quindecim & <gap/> libræ vnius. Vnde patet clarè pon­<lb/>deris diminutionem fieri ex orbiculis inferioribus, non <lb/>autem ex&longs;uperioribus, &longs;uperiores autem addi non nece&longs;­<lb/>&longs;itatis quidem, &longs;ed commoditatis gratiâ: neque enim ab&longs;­<lb/>que &longs;uperioribus vnico ductario fune fieri po&longs;&longs;et attractio <lb/>& ponderis ip&longs;ius eleuatio. Hactenus igitur nobis i&longs;thæc <lb/>de Trochleæ natura & vi po&longs;t alios, con&longs;idera&longs;&longs;e &longs;it &longs;atis. </s>
</p>
<figure id="fig29"></figure>
<p type="head">
<s>QVÆSTIO XIX.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitat Philo&longs;ophus, Cur &longs;i quis &longs;uper lignum magnam imponat <lb/>&longs;ecurim, de&longs;uperque magnum adijciat pondus, ligni quippiam quod <lb/>cur andum &longs;it, non diuidit; &longs;i verò &longs;ecurim extollens percutiat, illud <lb/>&longs;cindit, cum alioquin multo minus habeat ponder is id quod <lb/>percutit, quam illud quod&longs;uperiacet <lb/>& premit?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Poterat Ari&longs;toteles, nî fallimur, rem breuius & vniuer­<lb/>&longs;alius proponere. Scilicet cur motus ponderi addat <lb/>pondus & efficacius ex motu quam ex immoto pondere <lb/>mota res operetur. Soluitautem. An, inquiens, ideo fit, <lb/>quia omnia cum motu fiunt, & graue ip&longs;um grauitatis ma­<lb/>gis a&longs;&longs;umit motum, dum mouetur quam dum quie&longs;cit? <lb/>Incumbens igitur connatam graui motionem non moue­<lb/>tur, motum verò & &longs;ecundum hanc mouetur & &longs;ecun-
<pb pagenum="129"/>dum eam quæ e&longs;t <expan abbr="percutiētis">percutientis</expan>? Hæc præclarè quidem, cæ­<lb/>tera autem, quæ de cuneoiterat, nempe ad vectem eiuslo­<lb/>perationem referri &longs;uperius confutauimus. Porrò effe­<lb/>ctus huius, de quo agitur, di&longs;putatio illuc &longs;pectat, videli­<lb/>cet ad cadentium atque proiectorum naturam. Ad maio­<lb/>rem autem rei euidentiam hæc addimus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to libra AB, cu­<lb/>ius centrum C, libra­<lb/>ta æqualibus ponde­<lb/>ribus DE, apponatur <lb/>ponderi E pondus F, <lb/>item ponderi D pon­<lb/>dus G ip&longs;i ponderi F <lb/>æquale, æquilibrabit <lb/>itidem, Modò non apponatur &longs;impliciter pondus G &longs;ex <lb/>ex H in lancem A dimittatur, tunc &longs;anè non æquilibrabit, <lb/>&longs;ed libram deprimet. Duo enim in pondere dimi&longs;&longs;o con­<lb/>&longs;iderantur pondera; naturale &longs;cilicet, & quod motu ip&longs;i <lb/>moto, ponderi e&longs;t acqui&longs;itum. Itaque quo motus fuerit <lb/>maior, puta &longs;i cadat ex I, grauitas ex maiori motu fiet ma­<lb/>ior. quod vtique efficacius fieret &longs;i pondus G non dimit­<lb/>tetur modo remoto prohibente, &longs;ed proijceretur. Tunc <lb/>enim tria concurrerent, grauitas naturalis, grauitas ac­<lb/>qui&longs;ita ex naturali motu, & ea quæ naturali adij citur ex <lb/>violentia. Pondus igitur &longs;ecuri impo&longs;itum & &longs;ecuris ip&longs;ius <lb/>naturalis grauitas naturali tantum grauitate operantur, <lb/>& ideo minus efficaciter. Hucautem ea ferè pertinent <lb/>quæ nos à principio de duobus centris retulimus, natura­<lb/>lis nempe grauitatis, & acqui&longs;itæ. </s>
</p>
<p type="main">
<s>Cæterùm cur mallei & &longs;ecuris ictus &longs;it violenti&longs;&longs;i­<lb/>mus, ideo fit quod non ex vnico neque duplici, &longs;ed ex tri­<lb/>plici grauitate operetur. E&longs;to enim &longs;ecuris A, cuius manu­<lb/>brium AB, brachium vero &longs;ecuri vtentis BC, erit igitur C
<pb pagenum="130"/>
<arrow.to.target n="fig30"></arrow.to.target><lb/>locus vbi humero <lb/>brachium iungi­<lb/>tur, motus ip&longs;ius <lb/>centrum, attollit <lb/>autem &longs;ecurim is <lb/>qui percutit, & re­<lb/>tro ad &longs;capulas re­<lb/>ducens totis viri­<lb/>bus ex centro C <lb/>&longs;ecurim vibrat, <lb/>portionem circuli <lb/>de&longs;cribens ADE <lb/>ictumqueue faciens <lb/>in E. Vires igitur acquirit &longs;ecuris, tum ex naturali grauita­<lb/>te, cadens ex D, in E, tum ex proprio pondere, tum etiam <lb/>ex violentia eidem à percutiente impre&longs;&longs;a. Fiunt autem <lb/>motus tam naturalis quàm violentus eo validiores, quo <lb/>maius e&longs;t &longs;patium, quo res mota mouetur, idqueue praecipuè <lb/>cum violentia ip&longs;am &longs;ecundat naturam. Itaque maior fit <lb/>ictus in E quàm in F, & in F maior quàm in D. Item violen­<lb/>tius feriret percutiens, &longs;imanubrium e&longs;&longs;et longius, puta <lb/>BG. Tunc enim maior e&longs;&longs;et circulus GH, & motus tum <lb/>prolixior, tum velocior. quo igitur longiora habet bra­<lb/>chia is qui &longs;ecuri malleoue vtitur, data virium paritate, ex <lb/>eadem ratione validius percellit. E&longs;t autem &longs;ecuris, vel <lb/>malleus cuneatus, vel cuneus malleatus manubrio in&longs;er­<lb/>tus. An autem operetur efficacius cuneus malleo percu&longs;­<lb/>&longs;us, aut cum manubrio motus, vt fit in &longs;ecuri, data aciei & <lb/>ponderis æqualitate, difficile e&longs;t determinare. Certè va­<lb/>lidius, & certius fieri &longs;ci&longs;&longs;ionem ex cuneo & malleo, ea ra­<lb/>tio e&longs;t, quod cuneus adactus, nec inde remotus eam inte <lb/>rim &longs;eruat, quam antea fecerat partium &longs;eparationem,
<pb pagenum="131"/>quod quidem &longs;ecuri non accidit, quæ adacta ad nouam <lb/>percu&longs;&longs;ionem faciendam extrahitur. </s>
</p>
<figure id="fig30"></figure>
<p type="main">
<s>Hoc etiam con&longs;ideramus, &longs;ecuris in circulo motum, <lb/>ex A in D, e&longs;&longs;e videndum, id e&longs;t, non &longs;ecundum naturam, <lb/>&longs;ur&longs;um enim fertur quod e&longs;t graue, ex D verq in F <expan abbr="mixtū">mixtum</expan>: <lb/>magis autem ad naturalem accedere qui fit ex F in E. Tar­<lb/>dior ergo ex A in D, velocior ex D, in F, veloci&longs;&longs;imus ex F <lb/>in E; quæ dam quæ ad hanc rem faciunt, egregiè con&longs;ide­<lb/>rat Guid, Vbald. in calce Tractatus, De Cuneo; ip&longs;um <lb/>con&longs;ule. </s>
</p>
<p type="main">
<s>Ad hæc &longs;uccurrit nobis pulcherrima quæ&longs;tio. Du­<lb/>bitari enim pote&longs;t, vtrumictus ex en&longs;e e&longs;ficacior &longs;it à par­<lb/>te quæ e&longs;t circa aciem, aut circa medium en&longs;em, vel pro­<lb/>pe manubrium capulumue; etenim hinc inde &longs;unt ra­<lb/>tiones. </s>
</p>
<p type="main">
<s>E&longs;to quidem en&longs;is AB, cuius capulus A, &longs;piculum ve <lb/>rò B, centrum grauitatis C, pars capulo proxima D. Libra­<lb/>to itaque gladio tres fiunt circulorum portiones BE, CF, <lb/>DG, quæritur quo loco ictus &longs;it validior, nempe in E, in F, <lb/>velin G. Videtur validiorem futurum in E, quippe quod <lb/>ex maiori &longs;emidiametro AB, maioris &longs;it circuli portio BE, <lb/>& ideo velociormotus ex B in E. Contra efficaciorem <lb/>futurum apparet in F, propterea quod ibi ex centro C to­<lb/>tius fiat grauitatis impre&longs;&longs;io, fieri autem validi&longs;&longs;imum in <lb/>G, licetibi motus &longs;it tardior inde videtur, quod &longs;icon&longs;ide­<lb/>retur en&longs;is vt vectis, cuius fulcim entum e&longs;t A, potentia <lb/>premens in B, ponderis vero loco re&longs;i&longs;tentia rei quæ per­<lb/>cutitur in D. Maior e&longs;t autem proportio BA, ad AD, quam <lb/>BA ad AC, & ideo violentior fiet pre&longs;&longs;io ex ictu in D, <expan abbr="quã">quam</expan> <lb/>in C. Hi&longs;ce hoc pacto con&longs;ideratis, putarem ictum effica­<lb/>ciorem fieri in F ex medio C, quam ex extremis & oppo­<lb/>&longs;itis partibus EG. Licet enim in B velocitas &longs;it maior, dee&longs;t <lb/>ibi pondus. Si enim en&longs;is iterum vt vectis con&longs;ideretur, e­
<pb pagenum="132"/>runt AB. duo fulcimenta &longs;u&longs;tinentía pondus in C, vbi gra­<lb/>uitatis e&longs;t centrum. Si igitur paria fuerint &longs;patia BC, CA, <lb/>
<arrow.to.target n="fig31"></arrow.to.target><lb/>in B erit dìmidium <lb/>ponderis C, quantum <lb/>ergo velocitate præ­<lb/>ualetictus in B, <expan abbr="tantū">tantum</expan> <lb/>ponderis amittit. D <lb/>verò plus quidem de <lb/>pondere participat, <lb/>&longs;ed velocitatis habet <lb/>minimum, in C verò <lb/>velocitas e&longs;t medio­<lb/>cris, tota tamen ip&longs;ius <lb/>ex grauitatis centro <lb/>ponderis fit impre&longs;­<lb/>&longs;io. </s>
</p>
<figure id="fig31"></figure>
<p type="main">
<s>Quidam, quod huc pertinet, vt ex acieip&longs;a quæ lon­<lb/>gius à capulo abe&longs;t, violenti&longs;&longs;imum facerent ictum, Ar­<lb/>gentum viuum, quod &longs;ui naturâ graui&longs;&longs;imum quidem e&longs;t <lb/>& mobili&longs;&longs;imum in canali à manubrio ad verticem exca­<lb/>uato infundunt, quo in gladij de&longs;cen&longs;u ad verticem velo­<lb/>ci&longs;&longs;imè delato illuc transfert grauitatem totam, quare <lb/>tum velocitate tum grauitate concurrentibus ictus fit <lb/>violenti&longs;&longs;imus & longè validi&longs;&longs;imus. </s>
</p>
<p type="head">
<s>QVAESTIO XX.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, Cur &longs;tatera qua carnes ponderantur, paruo appendicu­<lb/>lo, magnatrutinet onera, cum alioqui tota, dimidiata exi&longs;tat <lb/>libra, altera vero parte &longs;ola&longs;it <lb/>&longs;tatera?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit Philo&longs;ophus, inquiens, &longs;tateram &longs;imul, & vectem <lb/>e&longs;&longs;e & libram, ip&longs;ius verò libræ centra &longs;eu fulcimenta
<pb pagenum="133"/>e&longs;&longs;e ibi vbi fit &longs;u&longs;pen&longs;io. Pondera verò hinc in de in lance <lb/>& appendiculo, loco &longs;cilicet æquipondij, appendiculo <lb/>&longs;uccedente. Reducit autem demon&longs;trationem ad ea quæ <lb/>&longs;tatuit ip&longs;e Mechanica principia; nem pe ad circulum & <lb/>circuli virtutem. Ait igitur, appendiculum licet parui <expan abbr="pō-deris">pon­<lb/>deris</expan> &longs;it, ideo maiori ponderi virtute æquari, quod lon­<lb/>gius à centro, hoc e&longs;t, ab ip&longs;o fulcimento &longs;i&longs;tatur. quic­<lb/>quid tamen &longs;it, &longs;tateram e&longs;&longs;e vectem, res e&longs;t explorati&longs;­<lb/>&longs;ima. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur &longs;tatera AB, <lb/>cuius appendiculum cur­<lb/>rens F, fulcimentum cen­<lb/>trumue C, lanx quæ cate­<lb/>na &longs;u&longs;penditur E &longs;patium <lb/>à loco fulcimenti ad ap­<lb/>pendiculum CF. quod ve­<lb/>rò à fulcimento ad cate­<lb/>nam, ex qua lanx appen­<lb/>ditur AC. Intelligatur autem & aliud fulcimentum D, &longs;it­<lb/>queue maius &longs;pacium AD, quam AC. Porrò ita &longs;e habeat <lb/>pondus in E ad appendiculi F pondus, vt CF &longs;patium, ad <lb/>&longs;patium AC, quo ca&longs;u &longs;eruata, permutatim, ponderum & <lb/>brachiorum proportione, fiet ae quilibrium. Si autem pon­<lb/>deribus ita con&longs;titutis iterum &longs;u&longs;pendatur in D, non fiet <lb/>æquilibrium, propterea quod minor &longs;it proportio DF ad <lb/>DA, ea quæ e&longs;t FC ad CA. Minor ergo e&longs;t proportio FD <lb/>ad DA, quam ponderis E ad pondus F, & idcirco facta <lb/>&longs;u&longs;pen&longs;ione præualebit pondus E ponderi F. Ita que vt it e­<lb/>rum fiat æquilibrium, nece&longs;&longs;e e&longs;t <expan abbr="iterū">iterum</expan> proportiones bra­<lb/>chiorum &longs;eu &longs;patiorum proportionibus ponderum æqua­<lb/>re. Transferatur igitur (lancis interim immoto pondeie) <lb/>ip&longs;um appendiculum in B, fiatque vt FC ad CA, ita BD ad <lb/>DA. Stabitautem iterum &longs;tatera ad eam redacta quam
<pb pagenum="134"/>diximus brachiorum & ponderum permutatam propor­<lb/>tionem. </s>
</p>
<p type="main">
<s>Nos &longs;tateris vtimur ex duplici fulcimento, altero <lb/>propiori, altero à lance &longs;eu loco, vbi lanx appenditur, re­<lb/>motiori, illa grauiora appendimus pondera, & non per <lb/>vncias & libras, &longs;ed per libras tantum & &longs;elibra ponde­<lb/>ramus; & hoc &longs;tateræ latus eo quod minus minutè &longs;it di­<lb/>ui&longs;um; vulgo no&longs;trates Gro&longs;&longs;um, hoc e&longs;t, rude & cra&longs;&longs;um <lb/>appellant. Aliud verò, cum fulcimentum e&longs;t loco appen­<lb/>&longs;ionis lancis vicinius, & per libras, &longs;elibras & vncias diui­<lb/>ditur, quo quidem minora appendimus pondera, cò quod <lb/><expan abbr="exqui&longs;itiorē">exqui&longs;itiorem</expan> contineat diui&longs;ionem, &longs;ubtile dicunt. Rectè <lb/>igitur dicebat Philo&longs;ophus, in &longs;tatera plures e&longs;&longs;e libras, <lb/>quanquam & ea quoque de cau&longs;&longs;a dici po&longs;&longs;it, quod, quot <lb/>&longs;unt appen diculi, è locoin locum translationes, totidem <lb/>ex proportionum variatione fiant libræ. Et hoc quidem <lb/>&longs;en&longs;i&longs;&longs;e videtur Ari&longs;toteles. </s>
</p>
<figure></figure>
<p type="main">
<s>Po&longs;&longs;emus & alio <lb/>modo &longs;tatera vti, nempe <lb/>&longs;tabili appendiculo, mo­<lb/>bili autem fulcimento. <lb/>E&longs;to enim &longs;tatera AB, <lb/>cuius lanx C appen&longs;a in <lb/>A, appendiculum verò <lb/>&longs;tabile D, appen&longs;um in <lb/>B, Apponatur ip&longs;i l&acedil;nci <lb/>C, pondus E. Vnicum ergo fiet corpus CEABD con&longs;tans <lb/>ex lance, libra & ponderibus. Habet ergo hoc totum gra­<lb/>uitatis &longs;uæ centrum, quod quidem vbi &longs;it e&longs;t ignotum. Ex <lb/>illo autem inuento &longs;i corpus totum appendatur, partes æ­<lb/>queponderabunt. Appendatur autem, puta in G, &longs;it <expan abbr="autē">autem</expan> <lb/>grauitatis centrum in H. Quoniam igitur He&longs;t extra ful­<lb/>cimentum G, declinabit &longs;tateræ pars GA, centro G per
<pb pagenum="135"/>circuli portionem Hl, à centro grauitatis in ip&longs;a de&longs;cen­<lb/>&longs;ione de&longs;criptam. Siautem grauitatis centrum fuerit vbi <lb/>K, eo quodibi quoque &longs;it extra fulcimentum G, de&longs;cen­<lb/>detpars GB, de&longs;cribente interim grauitatis centro K, cir­<lb/>culi portionem KL. ltaque &longs;i &longs;tateram totam eum ponde­<lb/>ribus trahamus <expan abbr="pellamu&longs;q;">pellamu&longs;que</expan> vltro citroque;, immoto appen­<lb/>diculo eritaliquando fulcimentum in ea linea perpendi­<lb/>culari velloco ip&longs;o, vbi e&longs;t grauitatis centrum, quo ca&longs;u <lb/>&longs;tatera&longs;tabit, & tuncita erit diui&longs;a, vt fiat brachiorum & <lb/>ponderum eadem ratio, ordine permutato. Hicautem <lb/>modusideo non e&longs;t in v&longs;u, quod mole&longs;tum &longs;it libram &longs;eu <lb/>&longs;tateram cum ponderibus vltro citroqueue transferre, quæ <lb/>difficultas commodè appendiculi mobilitate vitatur. </s>
</p>
<p type="head">
<s>QVAESTIO XXI.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur facilius dentes extrahunt Chirurgi, denti forcipis <lb/>onere adiecto, quam &longs;i&longs;ola manu vtantur?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Re&longs;pondet Philo&longs;ophus, An quia ex manu, magis quam <lb/>ex dentiforcipe lubrius elabitur dens? An ferro id po­<lb/>tius accidit quam digitis, quoniam vndique dentem non <lb/>comprehendunt, quod mollis facit digitorum caro; ad­<lb/>hæret enim & complectitur magis. Hæc &longs;ecunda ratio <lb/>videtur primam de&longs;truere, & contrarium pror&longs;us &longs;enten­<lb/>tiæ, quæ in problemate proponitur, a&longs;&longs;erere. Si Græca ad <lb/>verbum reddas ita habent: An magisip&longs;a manu labile e&longs;t <lb/>ferrum, & ip&longs;um vndique (dentemnempe) non comple­<lb/>ctitur, caro autem digitorum cum mollis &longs;it, adhæret ma­<lb/>gis, & vndique congruit. Certè vt &longs;ententia non &longs;it con­<lb/>traria propo&longs;itioni, Græca ver&longs;io ita videtur concinnan­<lb/>da: Vel magis è m n<gap/>bitur, mollis enim e&longs;t digitorum <lb/>caro, ferrum autem circumplectitur, & haeret magis. quic­<lb/>quid &longs;it, Græcam lectionem contrarium ei quod quæri-
<pb pagenum="136"/>tur, affirmare certum e&longs;t. Picolomineus, Ideo, inquit, di­<lb/>gitorum caro mollis minus aptè extrahit, quod dentem <lb/>totum comprehendere non pote&longs;t, quod ferrum ob &longs;uam <lb/>durítiem & con&longs;tantiam commodi&longs;&longs;imè facit. Sen&longs;um ex <lb/>mente reddidit, quod ex verbis non poterat. Subiungit <lb/>denique Ari&longs;toteles, An quia dentiforcipes &longs;int duo con­<lb/>trarij vectes vnicum habentes fulcimentum, ip&longs;am &longs;cili­<lb/>cet in &longs;trumenti partium connexionem. Hoc igitur ad ex­<lb/>tractionem vtuntur^{**}, vt facilius moueant. Figuram hoc <lb/>pactto proponit Philo&longs;ophus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to dentiforcipis alterum <lb/>quidem extremum vbi A, alte­<lb/>rum autem quod extrahit B, ve­<lb/>ctis vbi ADF, alter vectis, vbi <lb/>BCE, fulcimentum verò CGD <lb/>connexio vbi G. Densautem pondus: vtroque igitur ve­<lb/>cte B, & F &longs;imul comprehendentes mouent, Hæcille. At­<lb/>tamen rem ip&longs;am &longs;ubtilius con&longs;iderantibus aliter videtur <lb/>habere, acip&longs;e a&longs;&longs;erat. Et&longs;anè dentisforcipis brachia ve­<lb/>ctes e&longs;&longs;e, quorum commune fulcimentum e&longs;t in ip&longs;o cen­<lb/>tro vbi vertebra, nemo negauerit. Dentem autem e&longs;&longs;e <lb/>pondus, ego quidem ab&longs;olute non dixerim. Pondus <expan abbr="autē">autem</expan> <lb/>hîcproprie e&longs;t ip&longs;a dentis durities, cuius re&longs;i&longs;tentia eo fa­<lb/>cilius &longs;uperatur, quo maior e&longs;t proportio brachiorum à <lb/>manu ad vertebram, ad partem illam quæ à vertebra e&longs;t <lb/>ad dentem. At dentis ex con&longs;trictione fractio nihil facit <lb/>pror&longs;us ad extractionem: id tamen operatur brachio­<lb/>rum longitudine dentiforceps, quod valide ex vectium <lb/>oppo&longs;itorum vi dentes con&longs;tringit & extra ctioni commo­<lb/>dum reddit & facilem. Neque enim totus Dentiforceps <lb/>hic ceu vectis vnicus operatur, quod fit in forcipibus quas <lb/>Tenaleas vocamus, quibus è tabulis claui reuelluntur, <lb/>qua de re nos quae&longs;tione 6. verba fecimus. Quo pacto <expan abbr="autē">autem</expan>
<pb pagenum="137"/>dentis ex Dentiforcipe extractio ad vectem reducatur, <lb/>&longs;ubtilius e&longs;t perpendendum, neque enim res e&longs;t in propa­<lb/>tulo. </s>
</p>
<p type="main">
<s>Dicimus igitur, tum dentem ip&longs;um, tum dentifor­<lb/>cipem vectes e&longs;&longs;e, varia tamen ratione & &longs;atis &longs;ane diuer­<lb/>&longs;a. Dens enim fit vectis eius nempe naturæ quæ fulcimen­<lb/>tum habet in angulo, quo ca&longs;u ip&longs;ius Dentiforcipis <expan abbr="partiū">partium</expan>, <lb/>quibus Dens apprehenditur, ea quæ longior e&longs;t poten­<lb/>tiæ mouentis loco &longs;uccedit, breuior vero fulcimentum <lb/>facit, Dentis vero re&longs;i&longs;tentia ponderis vices refert. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim dens qui­<lb/>dem A, cuius diameter <lb/>BC, longitudo v&longs;que ad <lb/>extremas radices CD, <lb/>pars dentiforcipis breui­<lb/>or CG, longior BG. Fit <lb/>ergo vectis BCD, habens <lb/>&longs;ulcimentum in C. Den­<lb/>teigitur apprehen&longs;o in BC, & manu dentiforcipe ceu ve­<lb/>cte ad inferiora compre&longs;&longs;o C, fit fulcimentum centrum­<lb/>ue. Stante enim puncto C, trahente autem potentia quæ <lb/>e&longs;t in B, fit motus ip&longs;ius B, per circuli portionem BE, radi­<lb/>cis vero D, fit motus per DF, & inde ip&longs;ius dentis extra­<lb/>ctio facilis. Quibus con&longs;ideratis vt rem ad proportiones <lb/>quatenus fieri pote&longs;t reducamus, dicimus, quo maior fu­<lb/>erit proportio BC, ad CD, hoc e&longs;t, partis vectis, quæ à ful­<lb/>cimento ad potentiam ad eam quæ à fulcimento e&longs;t ad <lb/>pondus, eo facilius fieri dentis auul&longs;ionem, quod vtique <lb/>demon&longs;trandum fuerat. </s>
</p>
<p type="main">
<s>Porro quod in calce quæ&longs;tionis addit Philo&longs;ophus, <lb/>Dentes commotos facilius manu extrahi quam in&longs;tru­<lb/>mento, nulla ratione probat. Ego autem arbitror, huc <lb/>pertinere ea verba, quæ &longs;uperius habentur, videlicet fer­
<pb pagenum="138"/>rum quidem non vndique dentem <expan abbr="comprehēdere">comprehendere</expan>, quod <lb/>mollisfacit digitorum caro, quæ id circo adhæret & <gap/>om­<lb/>plectitur magis. An autem ita &longs;it, alij videant, nobis enim <lb/>digito rem o&longs;tendi&longs;&longs;e fuerit &longs;atis. </s>
</p>
<p type="head">
<s>QVÆSTIO XXII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Hîc quærit Ari&longs;toteles, Cur nuces ab&longs;que ictu facile confringuntur <lb/>in&longs;trumentis quæ ad eum faciunt v&longs;um, & hoc licet multum aufe­<lb/>ratur virium, ce&longs;&longs;ante motu & violentia, quod accidit dum mal­<lb/>leo confringuntur. Addit præterea, citius fieri confractionem <lb/>graui, & duro in&longs;trumento ferreo vide­<lb/>licet quàm ligneo.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit, inquiens, id fieri quod in&longs;trumentum duobus <lb/>vectibus con&longs;tet, coëuntibus in connexione &longs;eu verte­<lb/>bra, & idcirco eo violentius fieri confractionem, quo mi­<lb/>nus e&longs;t &longs;patium à nuce, quæ frangitur, ad vertebram. ma­<lb/>ius verò quod à vertebra ad extremitates, quæ confrin­<lb/>gentis manu comprimuntur. Ait igitur, & id quam oppo­<lb/>fite, vim ex vectibus ictus loco &longs;uccedere & idem operari. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur in &longs;trumentum, <lb/>de quo agimus CDBF, ex duo­<lb/>bus vectibus con&longs;tans, quorum <lb/>alter CAF, alter vero DAB ver­<lb/>tebra &longs;eu connexio A locus v­<lb/>bi nux frangitur K, manubria <lb/>vero BF. quo igitur prolixiores <lb/>erunt AB, AF, breuiores vero ACAD, violentius fiet <expan abbr="cō-fractio">con­<lb/>fractio</expan>. Erit autem nucis re&longs;i&longs;tentia loco ponderis A, ful­<lb/>cimentum BF loco potentiæ. Itaque nî maior &longs;it propor­<lb/>tio potentiæ ad re&longs;i&longs;tentiam, quam brachij à potentia ad <lb/>ful cimentum ad eam partem quæ à fulcimento e&longs;t ad nu­<lb/>cem, non fiet confractio. eo autem magis &longs;up erabit, quo
<pb pagenum="139"/>maior fuerit pars vectis quæ à potentia ad fulcimentum. </s>
</p>
<p type="main">
<s>Quod autem addit Ari&longs;toteles, eo maiorem fieri <lb/>vectium eleuationem, hoc e&longs;t, in&longs;trumenti aperitionem, <lb/>quo magis nux quæ frangitur, fuerit propior fulcimento, <lb/>hoc e&longs;t, ip&longs;i vertebræ, facile o&longs;tenditur ex conuer&longs;a 21. <lb/>propo&longs;. lib. 1. Elem. &longs;i enim ab extremitatibus vnius line æ <lb/>ad ea&longs;dem partes con&longs;tituantur duæ line æ maiores con­<lb/>currentes in angulo, & ab ij&longs;dem extremitatibus duæ a­<lb/>liæ minores, quæ intra triangulum à maioribus con&longs;titu­<lb/>tum cadant, maiorem angulum continebunt. At talis e&longs;t <lb/>angulus qui fit in in &longs;trumento, cum partes vectis à verte­<lb/>bra adnucem fuerint breuiores. magìs ergo dilatantur <lb/>vectes, & magis dilatati magis comprimuntur, magis au­<lb/>tem compre&longs;li validius frangunt, quod dixerat Ari&longs;to­<lb/>teles. </s>
</p>
<p type="main">
<s>Cæterum & illud quod &longs;cribit, ex grauiori & durio­<lb/>ri materia in&longs;trumentum citius fractionem facere, quam <lb/>ex leuiori & minus dura, ex parte quidem materiæ verum <lb/>e&longs;t, nec pertinet ad proportionem, quæ &longs;ane in <expan abbr="huiu&longs;modī">huiu&longs;modim</expan> <lb/>in&longs;trumentis formæ ferè habent rationem. Nos hi&longs;ce in­<lb/>&longs;trumentisnon vtimur. Sunt autem &longs;imilia in&longs;trumentis <lb/>illis, quibus figuli cretaceas pilas ad chirobali&longs;tarum v&longs;um <lb/>facere & efformare con&longs;ueuerunt. </s>
</p>
<p type="head">
<s>QVÆSTIO XXIII.</s>
</p>
<p type="main">
<s>Pvlcherrimam proponit hoc loco Philo&longs;ophus con­<lb/>templationem, eamque ad mixtos motus <expan abbr="pertinētem">pertinentem</expan>. <lb/>Mixtorum autem motuum &longs;peculationem antiquis Me­<lb/>chanicis fui&longs;&longs;e tum vtilem tum etiam familiarem, norunt <lb/>ij qui norunt quæ de lineis &longs;piralibus Helici&longs;ue, cy&longs;&longs;oidi­<lb/>bus, conchoidibus & alijs eiu&longs;cemo di &longs;cripta & contem­<lb/>plata reperiuntur, quibus tum ad duarum mediarum pro­
<pb pagenum="140"/>portionalium inuentionem, tum ad circuli quadratio­<lb/>nem vti&longs;olent. Quod autem hîc quærit Ari&longs;toteles, ita &longs;e <lb/>habet. </s>
</p>
<p type="head">
<s><emph type="italics"/>Cur &longs;i duo extrema in Rhombo punct a duabus ferantur lationibus, <lb/>haudquaquam æqualem vtrumque eorum pertran&longs;it rectam, &longs;ed <lb/>multo plus alteram? Item cur quod &longs;uper latus fertur, minus per­<lb/>tran&longs;eat quam ip&longs;um latus. Illudenim diametrum pertran&longs;ire <lb/>certum est, hoc vero maius lat us, licet hoc vnica, illud au­<lb/>tem duabus feratur lationibus?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Difficile hoc intellectu prima fronte, & &longs;ane admi­<lb/>rabile, itaque in tentam con templationem requirit. Nos <lb/>primo cum Ari&longs;totele, rem totam explicabimus, tum ali­<lb/>quid forta&longs;&longs;e non pœnitendum no&longs;tro de promptuario <lb/>proferemus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to itaque Rhombus ABCD, <lb/>cuius latera AB, BD, DC, CA, diame­<lb/>trorum maior AD, minor BC, &longs;ecan­<lb/>tes &longs;e inuicem in puncto &longs;eu figuræ <lb/>centro K. Sunt <expan abbr="autē">autem</expan> ex ip&longs;ius Rhom­<lb/>binatura latera æqualia & parallela, <lb/>Angulorum vero qui maiori diame­<lb/>tro opponuntur, recto maiores, qui <lb/>vero minori minores. His igitur con­<lb/>&longs;ideratis, intelligatur punctum A mo­<lb/>ueri peculiari & &longs;im plici motu, per li­<lb/>neam AB, ab A ver&longs;us B, & eodem <expan abbr="tē-pore">ten­<lb/>pore</expan> moueri totam lineam AB, ver&longs;us lineam DC, hac ta­<lb/>men lege, vt &longs;emper eidem DC feratur parallela, & eius <lb/>alterum extremorum feratur per AC, alterum vero per <lb/>BD, Intelligatur etiam punctum B moueri eodem tem­<lb/>pore proprio motu, eoque &longs;implici, per eandem rectam <lb/>BA, ver&longs;us A, & cum eadem, vt dictum e&longs;t, mota; ferri ver-
<pb pagenum="141"/>&longs;us CD. Erunt autem &longs;emper AB puncta in eadem linea <lb/>quæ mouetur, &longs;ibi inuicem ex contrarijs partibus occur­<lb/>rentia. Itaque cum ex duobus motibus &longs;emper propor­<lb/>tionalibus, hoc e&longs;t, laterum proportione &longs;eruata, recta <lb/>producatur, vt demon&longs;tratum e&longs;t à principio, vbi produ­<lb/>ctio circuli ex Philo&longs;ophi mente e&longs;t declarata, <expan abbr="vtraq;">vtraque</expan> pun­<lb/>cta quæ ean dem laterum proportionem &longs;eruantia <expan abbr="mouē-tur">mouen­<lb/>tur</expan>, rectas lineas <expan abbr="producēt">producent</expan> A quidem AD, B autem ip&longs;am <lb/>BC. Feraturigitur A, tum mixto tum &longs;implici motu per <lb/>diametrum AD. B vero quoque tum mixto, tum proprio <lb/>per diametrum BC, &longs;upponitur autem motus omnes &longs;im­<lb/>plices, tum punctorum, tum etiam line ae, à qua puncta ip&longs;a <lb/>feruntur, æquali velocitate fieri. Illud igitur mirabile e&longs;t, <lb/>cuius etiam ratio quæritur, quo pacto eodem tempore ea­<lb/>dem que velocitate latum A quidem totam percurrat AD <lb/>maiorem, B vero totam BC, eamque longe minorem? <lb/>Porro nece&longs;&longs;e fuit rem in Rhombo &longs;peculari, non autem <lb/>in quadrato & altera parte longiori rectangulo, in quibus <lb/>diametri (quod Rhombo non accidit) &longs;unt æquales. Ima­<lb/>ginemur igitur A, proprio motu percurri&longs;&longs;e &longs;patium AE, <lb/>nempe ip&longs;ius AB line æ dimidium. Erit igitur in E, item li­<lb/>neam totam AB eodem tempore pertran&longs;i&longs;&longs;e dimidia op­<lb/>po&longs;itarum linearum, ACBD, & e&longs;&longs;e translatam, vbi FKG. <lb/>Quoniam igitur æquali celeritate lineæ AB extremitas <lb/>A, translata e&longs;t in F & A, punctum per eam motum in E, e­<lb/>rit &longs;patium AE, æquale &longs;patio AF. Ductis igitur lineis <lb/>FKG, EKH lateribus AB, AC æquidi&longs;tantibus, erit figura <lb/>AEKF. Rhombus &longs;imilis quidem Rhombo ABCD, recta <lb/>igitur FK æqualis erit oppo&longs;itæ AE. quare A punctum <lb/>translatum erit ex mixto motu in K. Eodem pacto <expan abbr="quoniã">quoniam</expan> <lb/>punctum B. eadem velocitate mouetur ver&longs;us A, & linea <lb/>AB ver&longs;us CD, cum B fuerit in E extremum line æ motæ <lb/>BA, <expan abbr="nēpe">nempe</expan> B eritin G. æquales ergo &longs;unt BE, BG & Rhom­
<pb pagenum="142"/>bus EBGK, circa diametrum BKC ip&longs;i Rhombo ABCD <lb/>&longs;imilis, & ideo GK æqualis oppo&longs;itæ BE & BG æqualis <lb/>EK. Cum ergo B confecerit &longs;patium BE, erit ex mixto <lb/>motu in K, &longs;uperato nempe &longs;patio BK, idque eodem tem­<lb/>pore quo A percurrerat totum &longs;patium AK. Ex æquali i­<lb/>gitur &longs;implicium motuum velocitate, in æqualia &longs;patia <lb/>AB puncta pertran&longs;ierunt, quæ res miraculo, cuius dilu­<lb/>tio quæritur, præbet occa&longs;ionem. </s>
</p>
<p type="main">
<s>Porro quod de dimidijs diametris demon&longs;tratum <lb/>e&longs;t, po&longs;&longs;umus & de totis eadem ratione concludere, quip­<lb/>pe quod eadem &longs;it proportio partium ad partes, quæ to­<lb/>tius ad totum. Hæcigitur prima e&longs;t pars propo&longs;itæ quæ­<lb/>&longs;tionis. Secunda vero dubitatio ita habet; Nempe mirum <lb/>videri punctum B, cum peruenerit in C, extremum lineæ <lb/>BA, videlicet ip&longs;um B, translatum e&longs;&longs;e in D, licet æquali­<lb/>ter moueantur linea BA, per lineam BD, & punctum B per <lb/>lineam BA. &longs;itque BC ip&longs;a BD maior. Primam dubitatio­<lb/>nem hoc pacto &longs;oluit Philo&longs;ophus; A fertur tum proprio, <lb/>tum alieno motu, hoc e&longs;t, line æ AB ver&longs;us oppo&longs;itam par­<lb/>tem CD, Itaque cum vterque motus deor&longs;um vergat, mo­<lb/>tus fit velocior. Contra vero B proprio quidem motu fer­<lb/>tur ver&longs;us A, hoc e&longs;t, &longs;ur&longs;um, alieno vero, hoc e&longs;t, line æ BA <lb/>ver&longs;us D, hoc e&longs;t, deor&longs;um, qui motus cum inuicem aduer­<lb/>&longs;entur, motus ip&longs;e fit tardior, non igitur e&longs;t mirum, A eo­<lb/>dem tempore maius &longs;patium pertran&longs;ire quam B. </s>
</p>
<p type="main">
<s>Hæc &longs;olutio non modo vera videtur, &longs;ed mirabilis <lb/>& ip&longs;omet Philo&longs;opho digni&longs;&longs;ima, cui quidem <expan abbr="temerariū">temerarium</expan> <lb/>iudicaremus contradicere, nîin genere ver&longs;aremur, in <lb/>quo non probabilia quæruntur, &longs;ed demon&longs;trata, &longs;ed ve­<lb/>ra. Futilem igitur e&longs;&longs;e rationem hanc ip&longs;ius Ari&longs;totelis <lb/>pace, hoc pacto o&longs;tendemus. </s>
</p>
<p type="main">
<s>E&longs;to quadratum ABCD, cuius diametri ACBD &longs;e­<lb/>cantes &longs;e&longs;e in E, moueatur eodem pacto BA, ver&longs;us CD,
<pb pagenum="143"/>
<arrow.to.target n="fig32"></arrow.to.target><lb/>item A, ver&longs;us B, & B ver&longs;us A, ita­<lb/>que punctum A tum proprio tum <lb/>alieno, hoc e&longs;t lineæ illud <expan abbr="deferē-tis">deferen­<lb/>tis</expan> motu deor&longs;um trudet, hoc e&longs;t, <lb/>ver&longs;us CD. Motus ergo velocior <lb/>erit motu puncti B, quod lationi­<lb/>bus fertur ferè contrarijs, hoc e&longs;t, <lb/>ex B ver&longs;us A &longs;ur&longs;um, cum linea <lb/>autem BA ver&longs;us C deor&longs;um. Ve­<lb/>locius tamen non mouetur, quip­<lb/>pe quod æquali tempore æquale <lb/>&longs;patium vtrum que punctum conficiat. Stante igitur cau&longs;­<lb/>&longs;a &longs;equi debui&longs;&longs;et effectus; non &longs;equitur autem, Ari&longs;tote­<lb/>lis igitur cau&longs;&longs;a non e&longs;t cau&longs;&longs;a. Rhombo quoque inuer&longs;o <lb/>idem clarius o&longs;tendemus hoc pacto: Sit Rhombus ABCD, <lb/>
<arrow.to.target n="fig33"></arrow.to.target><lb/>cuius diametri AC, BD &longs;ecan­<lb/>tes &longs;e&longs;e in E. Mota igitur linea <lb/>AB ver&longs;us CD, nempe deor&longs;um <lb/>& A quoque deor&longs;um ver&longs;us B, <lb/>contra vero B quidem &longs;ur­<lb/>&longs;um ver&longs;us A, deor&longs;um vero <lb/>ver&longs;us C, erit B tardior A, &longs;ed <lb/>contrarium fit, quippe quod <lb/>longior &longs;it BD, per quam mouetur B ip&longs;a AC, per quam <lb/>mouetur A. </s>
</p>
<figure id="fig32"></figure>
<figure id="fig33"></figure>
<p type="main">
<s>His igitur non &longs;atisfacientibus veriorem &longs;i perim­<lb/>becillitatem no&longs;tram licuerit, huius effectus cau&longs;&longs;am in­<lb/>ue&longs;tigabimus. Rationibus igitur & veritate contra aucto­<lb/>ritatem & probabilitatem e&longs;t nobis pugnandum: quod & <lb/>intrepide faciemus. </s>
</p>
<p type="main">
<s>Dicimus igitur, in quouis parallelogrammo &longs;itillud <lb/>qua dratum aut altera parte longius, vel idem Rhombus <lb/>Rhomboi&longs;ue &longs;emper mixtos motus proportione &longs;eruata
<pb pagenum="144"/>fieri per diametros. Cæterum díametrorum ad latera <lb/>proportiones e&longs;&longs;e varias (quadratis exceptis, in quibus ea­<lb/>dem e&longs;t &longs;emper) explorati&longs;&longs;imum. Illud quoque certum <lb/>e&longs;t, in rectangulis nunquam dari po&longs;&longs;e diametros lateri­<lb/>bus vtcunque captis æquales, &longs;emper enim diametri re­<lb/>ctis angulis &longs;ubtruduntur. In Rhombis vero & Rhombo­<lb/>idibus diametrorum ad latera proportiones variant. Dari <lb/>enim po&longs;&longs;unt diametri lateribus longiores item æquales, <lb/>& lateribus quoque ip&longs;is breuiores. </s>
</p>
<p type="main">
<s>Itaque diametrorum & laterum varia adinuicem <lb/>ratione &longs;e habentibus, attentis proportionibus, <expan abbr="mixtorū">mixtorum</expan> <lb/>& &longs;implicium motuum diuer&longs;a fiet, & varia comparatio. <lb/>in quadratis motus mixtus, qui per diametros &longs;emper ve­<lb/>locior erit &longs;implici qui per latera, Idem quoque in altera <lb/>parte longiori, in quo mixti quidem motus per diametros <lb/>erunt velociores, &longs;implices vero qui per latera, tardiores <lb/><expan abbr="quidē">quidem</expan>, &longs;ed ex illis tardior qui per latus breuius. In Rhom­<lb/>bis autem mixtus motus qui fit per diametros inæqualis. <lb/>Velocior enim qui per longiorem diametrum, tardior <lb/>quiperbreuiorem. Itaque &longs;implices motus punctorum <lb/>per latera ad eum qui fit per diametrosinon eodem pacto <lb/>&longs;e habent. Porro cum Rhomboides variæ &longs;int <expan abbr="diametrorū">diametrorum</expan> <lb/>adlatera habitudines, varia quoque dari pote&longs;t propor­<lb/>tio. aliquando enim diametri dari po&longs;&longs;unt lateribus maio­<lb/>res quando que, alter eorum minor. Si autem Rhombus in <lb/>duos &longs;oluatur triangulos, alter diametrorum datur æqua­<lb/>lis æqualibus lateribus æquicrurium triangulorum; <expan abbr="itaq;">itaque</expan> <lb/>in i&longs;tis mixti motus per diametros aequeveloces erunt &longs;im­<lb/>plicibus, qui per latera longiora, velociores autem illis <lb/>qui per latera breuiora. His igitur hoc pacto non perfun­<lb/>ctoriè con&longs;ideratis, facile ex proprijs cau&longs;&longs;is, nî fallimu<gap/>, <lb/>hocce Ari&longs;totelicum & mirabile Problema &longs;oluitur. </s>
</p>
<pb pagenum="145"/>
<figure></figure>
<p type="main">
<s>E&longs;to enim Rhombus ABDC, <lb/>cuius diameter longior AD maior &longs;it <lb/>tum lateribus, tum etiam altera dia­<lb/>metro BC. &longs;ecent autem &longs;e inuicem <lb/>diametri in E. Ducatur queue ip&longs;is AB, <lb/>CD, parallela FG &longs;ecans longiorem <lb/>diametrum AD, in H, breuiorem ve­<lb/>ro BC in I. & per I ip&longs;is BD AC paral­<lb/>lela ducatur KIL, Cum ergo B mixto <lb/>motu per diametrum BC erit in I & <lb/>A per diametrum AD, mixto &longs;imili­<lb/>ter motu erit in H, & quia motus mi­<lb/>xti fiunt per diametros, vt dictum e&longs;t, <lb/>vt &longs;e habet AD ad BC, ita AE ad EB, per 15. propol. 5. elem. <lb/>item vt AE ad EB, ita per 4. propo&longs;. 6. AH ad BI. e&longs;t enim <lb/>IH ip&longs;i AB parallela. Longior e&longs;t autem AH ip&longs;a BI, quip­<lb/>pe quod AE longior &longs;it ip&longs;a EB. motus igitur mixtus pun­<lb/>cti A per diametrum AD v&longs;que ad H velocior e&longs;t motu B, <lb/>per diametrum BC v&longs;quead I. Mota igitur linea AB mo­<lb/>uebuntur communia eius & diametrorum BC, AD pun­<lb/>cta, quibus &longs;ecantur &longs;emper diametrorum proportione <lb/>&longs;eruata. Quibus ita &longs;e habentibus, nil mirum e&longs;t punctum <lb/>A motum per AD velociorem e&longs;&longs;e mixto motu puncti B, <lb/>quod per minorem diametrum fertur BC. quod fuerat <lb/>demon&longs;trandum. quatenus vero ad &longs;ecundam problema­<lb/>tis partem pertinet, dicimus Propo&longs;itionem non e&longs;&longs;e vni­<lb/>uer&longs;alem. Si enim Rhombus detur, ex duobus æquilateris <lb/>triangulis con&longs;tans, breuior diameter lateribus erit aequa­<lb/>lis, quare non mouebitur citius motu &longs;implici punctum <lb/>per latus ac faciat mixto per minorem diametrum, quod <lb/>vt mirum propo&longs;uerat A ri&longs;toteles. Si autem latus ip&longs;um <lb/>breuiori diametro &longs;it <expan abbr="lōgius">longius</expan>, nec mirum quoque erit &longs;im­<lb/>plici motu moucri velocius quam mixto, quippe quod, vt
<pb pagenum="146"/>dictum e&longs;t, motus i&longs;ti à proportionibus linearum, per quas <lb/>mouentur, legem velocitatis atque tarditatis accipiant. <lb/>Hæc igitur nos circa hoc mirabile Ari&longs;totelicum proble­<lb/>ma con&longs;iderare &longs;it &longs;atis. </s>
</p>
<p type="head">
<s>QVÆSTIO XXIV.</s>
</p>
<p type="main">
<s>Mirabilem aliam quæ&longs;tionem proponit Ari&longs;toteles, <lb/>quæ itidem ad mixtos motus pertinet. </s>
</p>
<p type="main">
<s><emph type="italics"/>Dubitatio est, quam ob cau&longs;&longs;am maior circulus æqualem minori <lb/>circulo circumuoluitur lineam, quando circa idem centrum fue­<lb/>rint po&longs;iti. Seor&longs;um autem reuoluti quemadmodum alterius ma­<lb/>gnitudo ad alterius magnitudinem &longs;e habet, ita & illorum adin­<lb/>uicem &longs;iunt line æ? Præterea vno etiam & eodem vtri&longs;que exi&longs;ten­<lb/>te centro. Aliquando quidem tanta &longs;it linea, quam conuoluuntur, <lb/>quantum minor per &longs;e conuoluitur circulus, quandoque vero quan­<lb/>tum maior.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Hæcille, qui vt prober maiorem circulum in &longs;ua ro­<lb/>tatione maiorem lineam pertran&longs;ire, minorem vero mi­<lb/>norem; ait &longs;en&longs;u cogno&longs;ci angulum maioris circuli, id e&longs;t, <lb/>eius qui maiorem habet circumferentiam, e&longs;&longs;e maiorem, <lb/>eius vero qui minorem, minorem. Ita autem &longs;e habere cir­<lb/>cumferentias vt &longs;e habent anguli, & eandem <expan abbr="proportionē">proportionem</expan> <lb/>habere per quas tum maior, tum minor circulus circum­<lb/>uoluuntur. Ad quorum clariorem intelligentiam ea re­<lb/>uocare oportet in memoriam, quæ dixit de maiorum cir­<lb/>culorum ad minores circulos nutu. Hic enim, quod ibi <lb/>quoque fecerat, &longs;ectorem ip&longs;um angulum appellauit, an­<lb/>gulum vero maiorem maioris circuli &longs;ectorem, & mino­<lb/>rem angulum minoris ip&longs;ius circuli &longs;ectorem dixit. Clau­<lb/>dit igitur dicens: quoniam circumferentiæ &longs;e habent vt <lb/>anguli, hoc e&longs;t, vt &longs;ectores, maior erit circumferentia ma­<lb/>ioris circuli, & ex con&longs;equenti maior linea, per quam cir-
<pb pagenum="147"/>cumuoluitur, ea per quam minor. Demon&longs;trationem ve­<lb/>ro ex &longs;en&longs;u petijt. Satautem erat &longs;i dixi&longs;&longs;et, ita &longs;e habere <lb/>circum ferentias vt &longs;e habent diametri &longs;eu &longs;emidiametri, <lb/>& ideo lineas in rotatione de&longs;criptas inuicem &longs;e habere vt <lb/>diametros. Ob&longs;curiu&longs;culè, hæc &longs;ua figura o&longs;tendit Ari&longs;to­<lb/>teles. Nos igitur claritatem amantibus, no&longs;tram aliquan­<lb/>to, nî fallimur, clariorem, proponemus. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to circulus <lb/>maior ABCD, mi­<lb/>nor FGHI, circa i­<lb/>dem, & commune <lb/><expan abbr="cētrum">centrum</expan> E. Circum­<lb/>uoluatur maior ad <lb/>partes D. Sint <expan abbr="autē">autem</expan> <lb/>diametri, maioris <lb/><expan abbr="quidē">quidem</expan> AEC, BED, <lb/>minoris verò FEH, <lb/>GEI, fitque CD, <lb/>quadrans maioris, <lb/>HI vero minoris circuli. Moto igitur maiori circulo <expan abbr="&longs;ecū-dum">&longs;ecun­<lb/>dum</expan> ab&longs;idem, cum D fuerit in K erit CK ip&longs;i CD æqualis, <lb/>fietque; DE ex puncto K perpendicularis ip&longs;i CK, eritque vbi <lb/>KO, & quia punctum I e&longs;t in linea DE, erit I facta <expan abbr="quadrã-tis">quadran­<lb/>tis</expan> rotatione in linea KO vbi L, centrum vero E in ip&longs;a <lb/>KO, vbi O. Reuoluto igitur qua drante maioris, & confe­<lb/>cto &longs;patio CK minoris circuli quadrans HI conficiet &longs;pa­<lb/>tium HL, quod ip&longs;i CK &longs;patio e&longs;t æquale. quod autem in <lb/>quadrantibus fit, in totis etiam fit circulis. Motus igitur <lb/>minor circulus circa centrum E, vnica rotatione æquauit <lb/>&longs;patium rotationis maioris circuli. Mirabile itaque e&longs;t mi­<lb/>norem circulum eodem tempore & circa idem centrum <lb/>circumuolutum, lineam pertran&longs;i&longs;&longs;e æqualem circum fe­<lb/>rentiæ maioris circuli. Nec &longs;ecius admirationem facitro­
<pb pagenum="148"/>tato minori circulo, maiorem vna <expan abbr="circumuolutū">circumuolutum</expan> lineam <lb/>metiri circumferentiæ minoris circuli æqualem. Rotetur <lb/>enim minoris circuli quadrans HI per rectam HL. erit i­<lb/>gitur punctum I vbi M, æquali exi&longs;tente recta HM, ip&longs;i <lb/>curuæ HI. Tuncautem facto motu centrum E erit vbi P, <lb/>exi&longs;tente EP, ip&longs;i HM æquali, demittatur autem ex P per <lb/>M, ip&longs;is HL CK perpendicularis PMN. Et quoniam in <lb/>eadem linea &longs;unt DIE, vbi E fuerit in Pleritin M, & D in <lb/>N. quamobrem rotata quarta minori<gap/> circuli parte, ma­<lb/>ioris interim circuli quadrans confecit &longs;patium CN æ­<lb/>quale ip&longs;i HM, hoc minus circuli quadranti HI, quod vti­<lb/>que e&longs;t admirabile. </s>
</p>
<p type="main">
<s>Porro cau&longs;&longs;am effectus huius mirifici diligenter quæ­<lb/>rit Philo&longs;ophus, & inuenram accurate explicat. Occur­<lb/>rit autem primo ab&longs;urdæ cuidam opinioni. Diceret enim <lb/>qui&longs;piam, ideo tardius moueri maiorem circulum, ad mo­<lb/>tum minoris, quod interim <expan abbr="dū">dum</expan> minor moucretur, aliquas <lb/>inter rotan dum moras interponeret, minor vero ad mo­<lb/>tum maioris &longs;patia aliqua tran&longs;iliret, & ita &longs;patiorum fieri <lb/>ad æquationem. Porro demon&longs;trationem aggre&longs;&longs;urus haec <lb/>a&longs;&longs;umit principia. Eandemaequalemue potentiam, <expan abbr="aliquã">aliquam</expan> <lb/>magnitudinem tardius quidem mouere, aliquam vero <lb/>celerius. quod autem natum e&longs;t aptum moueri, tardius <lb/>moueri, &longs;i &longs;imul cum non apto nato moueri, moueatur, <lb/>quam &longs;i &longs;eparatim moueretur, celerius autem &longs;i non &longs;imul <lb/>
<arrow.to.target n="fig34"></arrow.to.target><lb/>cum eo moueatur. E&longs;to enim corpus A leue <lb/>quidem & aptum natum moueri&longs;ur&longs;um, cui <lb/>connectatur B, aptum natum moueri deor­<lb/>&longs;um, Si quis igitur mouere conetur corpus A <lb/>&longs;ur&longs;um difficilius mouebit, & tardius <expan abbr="iunctū">iunctum</expan> <lb/>nempeip&longs;i B, quam &longs;i ab ip&longs;o e&longs;&longs;et <expan abbr="&longs;eiūctum">&longs;eiunctum</expan>. <lb/>Praeterea quod non&longs;uo, &longs;ed alieno motu mo­<lb/>uetur, impo&longs;&longs;ibile e&longs;&longs;e plus eo moueri qui
<pb pagenum="149"/>mouet, &longs;iquidem non &longs;uo, &longs;ed alieno motu mouctur. Mo­<lb/>to igitur &longs;uo motu maiori circulo, minor non &longs;uo mouc­<lb/>tur, &longs;ed motu maioris circuli^{1}, & ideo non plus mouetur <lb/>quam ille moueatur, mouetur autem maiori &longs;patio quam <lb/>ex &longs;e moueretur, propterea quod maior &longs;it maioris circu­<lb/>li, à quo &longs;imul defertur, circumferentia. Item &longs;i minor &longs;uo <lb/>motu circumuoluatur, maiorem feret &longs;ecum, & ideo non <lb/>plus in &longs;ua rotatione mouebitur maior, quam ip&longs;e minor <lb/>circulus moueatur. Summa rei haec e&longs;t, alterum ferriab al­<lb/>tero & latum ad ferentis &longs;patium moueri. Licet enim al­<lb/>tero moto, alter interim moueatur, nihilrefert. E&longs;t enim <lb/>ac &longs;i is qui fertur, nullam habeat motionem, aut &longs;i eam ha­<lb/>beat, ip&longs;a nequaquam vtatur. quod non fit &longs;i vterque &longs;e­<lb/>paratim circa proprium centrum moueatur, tunc enim <lb/>magnusmagnum, paruus vero paruum &longs;patium conficit. <lb/>Hinc decipi ait Ari&longs;toteles illum, qui putat vtrum que cir­<lb/>culum per &longs;e &longs;u peridem centrum in rotatione moueri, li­<lb/>cet enim videatur, re vera non e&longs;t. Id enim vtique certum <lb/>e&longs;t, cum à maiori circulo minor fertur, circa maioris cen­<lb/>trum motum fieri. Si vero maior à minori feratur circa mi­<lb/>noris circuli centrum motum fieri. Hæc ferè Philo&longs;ophi <lb/>e&longs;t mens, cuius &longs;olutionem e&longs;&longs;e certi&longs;&longs;imam, & ex veris <lb/>cau&longs;&longs;is non dubitamus. </s>
</p>
<figure id="fig34"></figure>
<p type="main">
<s>Hinc ad aliam eamqueue certam a&longs;&longs;ertionem tran&longs;i­<lb/>mus. Dicimus enim, nullam materialem <expan abbr="rotã">rotam</expan> circa axem <lb/>eidem affixum, dum rotatur, po&longs;&longs;e eundem locum &longs;eruare, <lb/>ni&longs;i cauum fiat, quod axem ip&longs;um recipiat, in tran&longs;uer&longs;a­<lb/>rijs quibus rota &longs;u&longs;tinetur & progre&longs;&longs;iuum axis motum <lb/>impediat. </s>
</p>
<p type="main">
<s>E&longs;to enim rota ABCD, cuius centrum E, diametri <lb/>AEC, BED, e&longs;to alia minor rota GH, item minor KL, tum <lb/>minor NO, & adhuc minor QR, circaidem centrum E. <lb/>Rotetur itaque &longs;ecundum ab&longs;idem integri quadrantis
<pb pagenum="150"/>
<arrow.to.target n="fig35"></arrow.to.target><lb/>&longs;patium CD, eritque <lb/>D, in F, item &longs;i ex rota <lb/>GH, ex quadrante <lb/>HT, erit T in I. Ex a­<lb/>lijs item minoribus in <lb/>M, P, S. erit <expan abbr="itaq;">itaque</expan> <expan abbr="lon-gi&longs;&longs;imū">lon­<lb/>gi&longs;&longs;imum</expan> &longs;patium CF, <lb/><expan abbr="breui&longs;&longs;imū">breui&longs;&longs;imum</expan> vero RS, <lb/>Mota igitur rota cir­<lb/>ca <expan abbr="circulū">circulum</expan> &longs;eu axem, <lb/>QR, maior rota &longs;pa. <lb/>tio mouebitur RS, <lb/>quod &longs;i intra QR, circa centrum E alij infiniti imaginen­<lb/>tur circuli, quo propio es centro fuerint, eo maioris rotæ <lb/>progre&longs;&longs;us erit minor, donec ad centrum deueniatur, vbi <lb/>cum non &longs;it circulus, nullus fiet progre&longs;&longs;iuus motus, &longs;ed <lb/>circa ip&longs;um centrum nulla facta loci mutatione rotabi­<lb/>tur. At cum nulla materialis rota circa lineam punctumue <lb/>imaginarium conuerti po&longs;&longs;it, ideo axi ferreo alteriu&longs;ue <lb/>materiæ circa quem & cum quo circumuoluatur rota, ca­<lb/>uum &longs;emirotundum incidere oportet, in quo in&longs;ertus axis <lb/>dum conuertitur à loco in quo conuertitur, non recedat. </s>
</p>
<figure id="fig35"></figure>
<p type="head">
<s>QVÆSTIO XXV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur lectulorum &longs;pondas &longs;ecundum duplam faciant pro­<lb/>portionem, hanc quidem &longs;ex pedum, vel paulo ampliorem, illam <lb/>verotrium. Item cur vectes funesue non &longs;ecundum <lb/>diametrum extendantur?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Primam quæ&longs;tionis partem ita diluit Philo&longs;ophus, for­<lb/>ta&longs;&longs;e tantæ fieri &longs;olitos magnitudinis lectulos vt corpo­<lb/>ribus &longs;int proportionem habentes, & ideo fieri &longs;ecundum <lb/>&longs;pondas dupli longitu dine nempe cubitorum quatuor, <lb/>latitudine vero duorum. </s>
</p>
<pb pagenum="151"/>
<p type="main">
<s>No&longs;trates alia vtuntur proportione, &longs;e&longs;quialtera, vi­<lb/>delicet, quam Græci Hemioliam dicunt, communiter e­<lb/>nim pedes quatuor latos faciunt plus minu&longs;ue, longos ve­<lb/>ro circiter &longs;ex. quodideo fit vt in eis duo corpora commo­<lb/>dius cubare po&longs;&longs;int. Lectuliautem, de quibus loquitur <lb/>Philo&longs;ophus, ad vnum tantummodo &longs;u&longs;tinendum facti <lb/>videntur, quicquid tamen &longs;it, nullam ferè habet res ex <lb/>hac parte dubitationem. </s>
</p>
<p type="main">
<s>Secunda quæ&longs;tionis &longs;ectio ea erat, Curnon <expan abbr="&longs;ecundū">&longs;ecundum</expan> <lb/>diametros funes extendantur? Re&longs;tium funiumue in le­<lb/>ctulis muniendis v&longs;us non e&longs;t apud nos. etenim feretra <lb/>tantum, &longs;eu &longs;andapilas, quibus defunctorum corpora ef­<lb/>feruntur, funibus ad ea &longs;u&longs;tinen da inteximus. </s>
</p>
<p type="main">
<s>Cæterum lectos tabulis &longs;eu a&longs;&longs;eribus &longs;ternimus, qui­<lb/>bus &longs;accos paleis plenos imponimus, &longs;accisvero culcitras, <lb/>& tormenta, ne tabularum durities cubantes offendat. <lb/>Atqui in re facili multum labora&longs;&longs;e videtur Ari&longs;toteles, <lb/>tum etiam ob&longs;cure & inuolute nimis quæ&longs;tionem tracta&longs;­<lb/>&longs;e. Difficilem enim apud eum habethæc explicationem, <lb/>tum ea quam diximus de cau&longs;&longs;a, tum etiam quod Græca <lb/>lectio & Latina ver&longs;io corrupta, vt apparet, præ manibus <lb/>habeantur. Sane vt veritatem hocloco vindicaret in lu­<lb/>cem, egregie laborauit Picolomineus nec parum profe­<lb/>cit. Cæterum curre&longs;tes non &longs;ecundum diametrum extru­<lb/>dantur, triplicem affert Philo&longs;ophus rationem. Prima e&longs;t <lb/>vt &longs;pondarum ligna, minus di&longs;trahantur. Secunda, vt <expan abbr="pō-dus">pon­<lb/>dus</expan> inde commodius &longs;u&longs;tineatur. Tertia, vt in ip&longs;a textura <lb/>minus re&longs;tium funiumue ab&longs;umatur. </s>
</p>
<p type="main">
<s>Ad primam, cur exten&longs;is diametraliter funibus <expan abbr="&longs;pō-dæ">&longs;pon­<lb/>dæ</expan> ip&longs;æ di&longs;trahantur di&longs;cindanturue, necillenecalij do­<lb/>cent. Ego autem demon&longs;trarem hoc pacto. </s>
</p>
<p type="main">
<s>E&longs;to &longs;ponda ABCD, cuius longitudo AB, cra&longs;&longs;itudo <lb/>AC, in ea foramen vtrin que pertinens EF, re&longs;tis per fora-
<pb pagenum="152"/>
<arrow.to.target n="fig36"></arrow.to.target><lb/>men inditus GFE, &longs;itque Epars &longs;eu ca­<lb/>put exterius, quodnodo in E di&longs;tine­<lb/>tur. Sit autem &longs;pondæ lignum iuxta <lb/>longitudinem vt natura a&longs;&longs;olet &longs;ci&longs;&longs;ile. <lb/>Vis quædam, funeita extento applice­<lb/>tur in G, quae funem ip&longs;um ad &longs;e violen­<lb/>ter trahat. non di&longs;cindetur idcirco <lb/>&longs;ponda eo quod non diametraliter fu­<lb/>nis extendatur. Modo facta capitis G <lb/>translatione in H, trahatur valide fu­<lb/>nis, fiet autem pre&longs;&longs;io valida in F. ibi e­<lb/>nìm impedimentum facit angulus, ne funisip&longs;a dum tra­<lb/>hitur, rectitudinem a&longs;&longs;equatur. Itaque vi præualente, li­<lb/>gno vero &longs;ci&longs;&longs;ili, minus re&longs;i&longs;tente, funis, a&longs;&longs;ecuta rectitudi­<lb/>ne, fiet in HIE &longs;ci&longs;&longs;a &longs;ponda ad <expan abbr="quãtitatem">quantitatem</expan> trianguli FIE, <lb/>quod fuerat demon&longs;trandum. </s>
</p>
<figure id="fig36"></figure>
<p type="main">
<s>Cur autem funes ab angulo in angulum exten&longs;æ mi­<lb/>nus commode pondus &longs;u&longs;tineant, &longs;atis patet. quo enim fu­<lb/>nis <expan abbr="lōgior">longior</expan>, eodebilior, & pre&longs;&longs;io quæ in medio fit, ea vide­<lb/>licet parte quæ ab extremis e&longs;t remoti&longs;&longs;ima, magis funem <lb/>fatigat. Longiores autem funes &longs;unt quæ diametraliter <lb/>extenduntur. </s>
</p>
<figure></figure>
<p type="main">
<s>Quatenus ad <expan abbr="tertiã">tertiam</expan> <lb/>rationem pertinet, hoc <lb/>pacto funes intexit <lb/>Philo&longs;oph^{9}. E&longs;to lectu­<lb/>lus cum &longs;uis <expan abbr="&longs;pōdis">&longs;pondis</expan> AB <lb/>CD, cuius &longs;ponda AD, <lb/>&longs;it pedum &longs;ex, AB vero <lb/><expan abbr="triū">trium</expan>, Diuidatur AD bi­<lb/>faríam in E & BC in F. item AE in tres AG, GH, HE & in <lb/>totidem ED, nempe EL, LM, MD. Similiter medietas al­<lb/>terius <expan abbr="&longs;pōdæ">&longs;pondæ</expan> BF in tres partes di&longs;tinguatur BN, NO, OF,
<pb pagenum="153"/>& FC &longs;imiliterin tres FI, IK, KC, tum alterofunis capite <lb/>in ducto per foramen A, ibique probe firmato, indatur per <lb/>F, inde per I, po&longs;tca per GHK CE, & in E probe alligetur: <lb/>Erunt igitur funis quatuor partes æquales AF, IG, HK, <lb/>EC, quibus adijciuntur particulæ cadentes extra, quæ <lb/>&longs;unt FI, GH, KC. Po&longs;t hæc alterius funis principium per <lb/>foramen traij citur, quod e&longs;t in angulo B. Deinde per E, in­<lb/>de per L, N, O, M, D, F & in F probe vincitur, & nodo fa­<lb/>cto ob&longs;irmatur. Erunt igitur aliæ quatuor alterius funis <lb/>partes, tum inter&longs;e, tum etiam &longs;upradictis æquales, nem­<lb/>pe BE, NL, OM, FD, quibus il<gap/>æ pa iter adijciuntur par­<lb/>ticulæ, quæ caduat extra, videlicet EL, NO, MD. <expan abbr="quoniã">quoniam</expan> <lb/>igitur quadratis ex BA, AE æquale e&longs;t quadratum BE, erit <lb/>BE quadratum 18. cuius latus radixue 4 1/3 quam proxime. <lb/>Sunt autem huius longitud n s funes æquales octo. Ea­<lb/>rum igitur &longs;imul &longs;umptarum longitudo erit pedum 34 2/3 vel <lb/>circiter, quibus &longs;i ad dantur pedes &longs;ex funium qui cadunt <lb/>extra, erit re&longs;tis totius longitudo expan&longs;a pedum 40 2/3 plus <lb/>minu&longs;ue. Picolomineus vero ait 34 2/3, omi&longs;it enim particu­<lb/>las illas&longs;ex, quæ, vt diximus, cadunt extra. Idem rationem <lb/>funium diametraliter exten&longs;arum in idem, ait e&longs;&longs;e longi­<lb/>tudinis pedum 40 1/2. Hic autem eas <expan abbr="quoq;">quoque</expan> particulas præ­<lb/>termittit, quæ extra cadunt. Itaque his additis clare pa­<lb/>tet, plus re&longs;tium in&longs;umi diametraliterip&longs;is, quam latera­<lb/>liter exten&longs;is. Cæterum ratio, qua Philo&longs;ophus hæc pro­<lb/>bare conatur, adeo e&longs;t mutila, inuoluta, ob&longs;cura, vt Delio <lb/>pror&longs;us, vt aiunt, indigeat natatore. Huius loci in ex plica­<lb/>bilem difficultatem, vidit Picolomineus, qui idcirco at­<lb/>te&longs;tatus e&longs;t, interpretes in hac exponenda fui&longs;&longs;e halluci­<lb/>natos. Certe Græca lectio ver&longs;ione ip&longs;a Latina non e&longs;t <lb/>clarior. Nos interim ne inutilem ferè &longs;peculationem ni­<lb/>mia diligentia, eaque forta&longs;&longs;e fru&longs;tranea pro&longs;equamur, a­<lb/>lijs difficultatem hanc di&longs;&longs;oluendam aut ceu Gordij no­
<pb pagenum="154"/>dum gladio &longs;cindendo relinquemus. Sed interim &longs;ubit <lb/>mirari, cur veteres vtiliori modo prætermi&longs;&longs;o, <expan abbr="inutilioiē">inutilioiem</expan> <lb/>fuerint amplexati. Poterant enim reticulatim hoc per li­<lb/>neas lateribus æ quidi&longs;tantes intexere. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim lectulus <lb/>eiu&longs;dem dimen&longs;ionis <lb/>ABCD, in cuius latere <lb/>AD &longs;int foramina quin­<lb/>que E, F, G, H, I, totidem <lb/>in latere oppo&longs;ito QP, <lb/>ONM. Duo vero in la­<lb/>tere breuiori AB, nempe <lb/>RS, & toti dem in oppo&longs;ito KL incipiatur exten&longs;io à fora­<lb/>mine E, per QP, F, GON, HIM & in M funis obfirmetur, <lb/>tum alterius funis caput in datur &longs;i lib et per K, & inde per <lb/>S, R, L & in L con&longs;tringatur. Sunt autem omnes EQ, FP, <lb/>GO, NN, IM, pedum quindecim, quibus &longs;i addantur KS, <lb/>RL, &longs;inguli pedum &longs;ex erunt pedum xxvii. quibus adiectis <lb/>particulis extra cadentibus QP, FG, ON, HI, & RS, erit <lb/>integra &longs;umma pedum xxxii. Vide igitur quantum hinc <lb/>minus in&longs;umatur re&longs;tium quam eo modo, quem proba­<lb/>uit, & ceu vtiliorem propo&longs;uit Ari&longs;toteles. Præterea vali­<lb/>di&longs;&longs;imum e&longs;t hoc texturæ opus nec ex eo fit vera &longs;ponda­<lb/>rum di&longs;tractio &longs;ci&longs;&longs;ioue, quibus haud parum obnoxia e&longs;t <lb/>ea ratio, quam præfertip&longs;e Philo&longs;ophus. Concludimusi­<lb/>gitur, autnos eius verba & &longs;en&longs;um non intellexi&longs;&longs;e, aut <lb/>veteresip&longs;os, quorum v&longs;um ip&longs;e explicat, rei, quam nos <lb/>proponimus, naturam & commoditatem (quod ta­<lb/>men vix credibile e&longs;t) igno­<lb/>rare. </s>
</p>
<pb pagenum="155"/>
<p type="head">
<s>QVÆSTIO XXVI.</s>
</p>
<p type="head">
<s><emph type="italics"/>Proponitur à Philo &longs;opho examinandum, Cur difficilius&longs;it, langa <lb/>ligna ab extremo &longs;uper humeros ferre, quam &longs;ecundum me­<lb/>dium, æquali exi&longs;tente pondere?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Dvo hîc con&longs;iderat, vibrationem, & pondus. Ait enim <lb/>primo fieri po&longs;&longs;c, procera ligna vibratione impedien­<lb/>te, difficilius ferri. Quærerer autem qui&longs;piam, (ip&longs;e enim <lb/>id reticet) curvibratio hæc ferenti &longs;it nocua. Nos itaque <lb/>id expliçare conabimur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur lignum <lb/>oblongum, flexile, & vt <lb/>ita dicam, vibrabile <lb/>AB, imponatur hume­<lb/>ro<gap/> eique hæreat in C, <lb/>manu vero &longs;u&longs;tineatur facta compre&longs;&longs;ione in B. Nuteti­<lb/>gitur & vibretur, in ip&longs;a vibratione, ad partem A. Sit au­<lb/>tem centrum grauitatis eius D, Lignum igitur in ip&longs;a vi­<lb/>bratione de&longs;cendet &longs;ua pre&longs;&longs;us grauitate in E, tum facta <lb/>ligni con&longs;tipatione in ea parte quæ e&longs;t inferius inter C & <lb/>D, & inde re&longs;i&longs;tentia, codem fere impetu quo de&longs;cende­<lb/>rat, repul&longs;um per D, nec enim in &longs;ua rectitudine &longs;tabit, a­<lb/>&longs;cendet in F, facta iterum materiæ con&longs;ti patione inter C <lb/>& F. Mouebitur igitur lignum &longs;ua grauitate, motu fre­<lb/>quenti&longs;&longs;imo, &longs;ur&longs;um deor&longs;um, & is interim qui lignum hu­<lb/>mero fert, procedit antror&longs;um, impedit igitur motus i&longs;te, <lb/>qui fit &longs;ur&longs;um deor&longs;um lationem, quæ fit ad anteriora; La­<lb/>torem ip&longs;um quodammodo retrahens. Siautem medio <lb/>ligno &longs;upponatur humerus, eo quod vibratio &longs;it minor. <lb/>breuiores enim partes &longs;unt, quæ à medío ad extrema mi­<lb/>nus à vibratione remorabitur ferens. </s>
</p>
<p type="main">
<s>Quoniam autem non &longs;ola vibratio in hoc lationis <lb/>modo, nempe ex ligni extremitate difficultatem facit, ait
<pb pagenum="156"/>Philo&longs;ophus, forte id fieri, quoniam licet nihil inflecta­<lb/>tur, neque multam habeat longitu dinem, difficilius <expan abbr="tamē">tamen</expan> <lb/>&longs;it ad ferendum ab extremo, eo quod facilius eleuetur ex <lb/>medio quam ab extremis, & ideo &longs;ic ferre &longs;it facilius. <lb/>Cur autem ex medio facilius eleuetur, cau&longs;&longs;am e&longs;&longs;e ait, <lb/>quod eleuato medio ligno extrema &longs;e&longs;e inuicem &longs;u&longs;pen­<lb/>dant, & altera pars alteram bene &longs;ubleuet. Medium enim <lb/>fieri velut centrum, vbi is &longs;upponit humerum qui cleuat <lb/>aut fert. Extremorum autem interim altero depre&longs;&longs;o al­<lb/>terum &longs;u&longs;tolli. Nos interim Mechanicis principijs, quod <lb/>ip&longs;e non fecit, rem clariorem efficiemus. </s>
</p>
<p type="main">
<s>E&longs;to enim oblongum lignum AB, cui humerus &longs;up­<lb/>ponatur in B, manus vero premendo &longs;u&longs;tinens in B. &longs;it au­<lb/>tem ligni pars maxima AC, minima CB, inaioris autem ad <lb/>minorem proportio exempli gratia &longs;it &longs;excupla. Ad hoc i­<lb/>gitur vt fiat æquilibrium inter potentiam &longs;u&longs;tinentem in <lb/>B, & pondus comprimens in A, ita &longs;e habere oportet po­<lb/>tentiam in B, ad pondus in A, vt &longs;e habet pars ligni AC ad <lb/>
<arrow.to.target n="fig37"></arrow.to.target><lb/>partem CD. E&longs;to igitur pon­<lb/>dus in A, puta librarum &longs;ex. <lb/>Erit igitur potentia quæ in B <lb/>ad hoc vt &longs;u&longs;tineat librarum <lb/>triginta &longs;ex, quas &longs;i addas <expan abbr="pō-deri">pon­<lb/>deri</expan> in A, fiet humerus in C <lb/>&longs;u&longs;tinens pondus librarum quadraginta duo. Siautem <lb/>humerus medio ligno, hoc e&longs;t, in D &longs;upponatur, ad hoc vt <lb/>fiat æquilibrium, nece&longs;&longs;e erit potentiam in B e&longs;&longs;e æqua­<lb/>lem ponderi in A, quod e&longs;t &longs;ex, quare humerus &longs;u&longs;tinebit <lb/>duodecim. Vnde patet, longe difficilius portari lignum <lb/>ex C extremo, quam ex D medio; quod Mechanice fue­<lb/>rat demon&longs;tran dum. </s>
</p>
<figure id="fig37"></figure>
<p type="main">
<s>Po&longs;&longs;umus & aliteridem o&longs;tendere. Intelligatur e­<lb/>nim ij&longs;dem &longs;uppo&longs;itis, vectem quidem e&longs;&longs;e AB, cuius ful-
<pb pagenum="157"/>cimentum quidem B, pondus A, potentia &longs;u&longs;tinens in C, <lb/>nempe inter fulcimentum & pondus. Res igitur ad eum <lb/>vectis v&longs;um reducitur, de quo G. Vbaldus tractatu de Ve­<lb/>cte, propo&longs;. 3. Quare vtille o&longs;ten dit, ita &longs;e habere oporter <lb/>potentiam &longs;u&longs;tinentem ad pondus, vttotus vectis ad par­<lb/>tem eius quæ à potentia ad fulcimentum. Ita igitur &longs;e ha­<lb/>bebit pre&longs;&longs;io, quæ fit in C ad pondus in A, vt totus vectis <lb/>AB ad partem eius CB, quæ à potentia ad fulcimentum. <lb/>Erit igitur potentia &longs;eptupla ponderi, & ideo &longs;u&longs;tinebit <lb/>pondus librarum quadraginta duarum. quod fuerat o­<lb/>&longs;tendendum. </s>
</p>
<p type="main">
<s>Hinc alia quæ&longs;tio huic affinis &longs;oluitur, Cur ha&longs;ta &longs;a­<lb/>ri&longs;&longs;aue &longs;olo iacens manu ad alteram extremitatum ap­<lb/>pren&longs;a di&longs;ficillime extollatur? </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to igitur &longs;ari&longs;&longs;a ha­<lb/>&longs;taue iacens AB, cuius ex­<lb/>tremitati A manus ad &longs;u­<lb/>&longs;tollendum applicetur, &longs;it <lb/>autem pars quæ digitis capitur AC, quæritur cur pars re­<lb/>liqua CB difficillime &longs;u&longs;tollatur? Facile dubitatio ex præ­<lb/>demon&longs;tratis &longs;oluitur. E&longs;t enim C fulcimentum, &longs;upponi­<lb/>tur enim loco, pugno ad &longs;u&longs;tollendum clau&longs;o, digitus in­<lb/>dex, potentia autem premens in A, vt &longs;uperet grauitatem <lb/>CB, e&longs;t manus ip&longs;ius carpus, hoc e&longs;t illa manus ip&longs;ius pars, <lb/>qua pondus facta &longs;uppre&longs;&longs;ione &longs;u&longs;tollitur. E&longs;tigitur AB <lb/>vectis, cuius fulcimentum C, pondus B, potentia A, <expan abbr="Itaq;">Itaque</expan> <lb/>quoniam maxima e&longs;t proportio BA ad AC, maximam e&longs;­<lb/>&longs;e oportet potentiam pondus &longs;u&longs;tollentem in C. </s>
</p>
<p type="main">
<s>Huc etiam illud pertinet, Cur ha&longs;ta &longs;olo iacente, &longs;i <lb/>alterum extremorum manu &longs;u&longs;tollatur, alterum vero ve­<lb/>lo ci&longs;&longs;ime &longs;ur&longs;um vibretur, & eodem tempore manus ha­<lb/>&longs;tæ &longs;ic vibratæ &longs;upponatur, haud magna difficultate ha&longs;tæ <lb/>ad perpendiculum fit erectio. </s>
</p>
<pb pagenum="158"/>
<figure></figure>
<p type="main">
<s>Sit enim ha&longs;ta AB, quæ <lb/>manu ex B capta eleuetur in <lb/>C, & fiat in AC, tum facta ex <lb/>C partis A veloci vibratione, <lb/>ip&longs;a extremitas A transferatur <lb/>in D, &longs;itque vbi CD, tum velo­<lb/><gap/>imanus depre&longs;&longs;ione extremi­<lb/>tas C transferatur in E, <expan abbr="fiatq;">fiatque</expan> <lb/>EF horizonti perpendicuiaris; <lb/>quod vbi factum fuerit, erunt <lb/>in eadem linea quæ ad centrum mundi, manus ip&longs;a quæ <lb/>&longs;u&longs;tinet, & grauitatis ip&longs;ius centrum G, quare manus ip&longs;a <lb/>facta vibratione tantum portat, quantum præci&longs;e ip&longs;ius <lb/>e&longs;t ha&longs;tæ pondus. </s>
</p>
<p type="head">
<s>QVAESTIO XXVII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, Cur &longs;i valde procerum fuerit idem pondus, difficilius <lb/>&longs;uper humeros ge&longs;tatur, etiam&longs;i medium qui&longs;piam illud fe­<lb/>rat quam &longs;i breuius &longs;it?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Qvæ&longs;tio hæc &longs;uperiori e&longs;t affinis. Ait autem Philo&longs;o­<lb/>phus, cau&longs;&longs;am non e&longs;&longs;eid, quod in præcedenti quæ­<lb/>&longs;tione dixerat, &longs;ed vibrationem: quo enim longiora &longs;unt <lb/>ligna, eo magis eorum extrema vibrantur, debiliora enim <lb/>&longs;unt & à medio remotiora, quare &longs;uopte pondere facilius <lb/>nutant. Siautem breuiora &longs;int ea cau&longs;&longs;a ce&longs;&longs;ante minor <lb/>fit aut nulla vibratio, quare breuiora feruntur facilius. <lb/>Dupliciter autem vibratione ip&longs;a, portans offenditur, <lb/>tum ex cau&longs;&longs;a quam in &longs;uperiori quæ&longs;tione con&longs;ideraui­<lb/>mus, nempe quod motus &longs;ur&longs;um deor&longs;um a&longs;&longs;iduus, pro­<lb/>gredientis motum impediat, tum etiam quod duplici <lb/>pre&longs;&longs;ione grauetur ferentis humerus, quod Philo&longs;ophus <lb/>non animaduertit. </s>
</p>
<p type="main">
<s>Sit enim oblongum lignum AB, quod humero me-
<pb pagenum="159"/>
<arrow.to.target n="fig38"></arrow.to.target><lb/>dio loco &longs;u&longs;tineatur in C. <lb/>nutabunt ergo extrema AB, <lb/>à centro C, valde remota, <lb/>cadent autem &longs;imul A m D, <lb/>& B in E trahere &longs;ecum conantes medium C, quare is qui <lb/>in C &longs;u&longs;tinet, non modo ligni &longs;u&longs;tinet pondus ex grauita­<lb/>tis centro quod e&longs;t in C, &longs;ed impetum quoque in ip&longs;a ex­<lb/>tremorum depre&longs;&longs;ione acqui&longs;itum ex ipia violentia. Illud <lb/>autem &longs;ubtiliter con&longs;ideramus, portantem ex vibratione <lb/>per inter ualla deprimi & &longs;ubleuari. fiat enim vibratum li­<lb/>gnum ex contrario motu, vbi FCG. alleuiabit igitur eo <lb/>ca&longs;u portantem, &longs;iquidem impetus ex motu ip&longs;o acqui&longs;i­<lb/>tus, medium C trahat ad &longs;uperiora. <expan abbr="Itaq;">Itaque</expan> cum e&longs;t in DCE <lb/>portans plus &longs;u&longs;tinet in ACD, æquale, in FCG minus, <lb/>quod vtique demon&longs;trandum fuerat. E&longs;t autem quæ&longs;tio <lb/>hæc illi familiaris, quam 16. loco explicauimus. </s>
</p>
<figure id="fig38"></figure>
<p type="head">
<s>QVAESTIO XXVIII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur, Cur iuxta puteos celonia faciunt eo quo vi&longs;untur mo­<lb/>do? Ligno enim plumbi adiungunt pondus, cum alioquin vas <lb/>ip&longs;um & plenum & vacuum pon­<lb/>dus habeat.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Re&longs;pondet optime Philo&longs;ophus, hauriendi opus duo­<lb/>bus temporibus diuidi, nempe dumvasip&longs;um vacuum <lb/>demittitur, dum que extrahitur plenum: Contingere au­<lb/>tem, vacuum facile demitti, plenum autem difficulter ex­<lb/>trahi. Expedire nihilominus tardius, hoc e&longs;t difficilius di­<lb/>mitti vt fa cilius extrahatur, plumbo nempe coadiuuante, <lb/>& &longs;ane Philo&longs;ophi &longs;olutio e&longs;t lucidi&longs;&longs;ima. Nos autem luci <lb/>ip&longs;i lucem aliquam adhuc afferre conabimur. </s>
</p>
<p type="main">
<s>E&longs;to Celomum (Latine Tolenonem appellant) ABC, <lb/>cuius arrectarium BD, tran&longs;uer&longs;um lignum AC, quod
<pb pagenum="160"/>
<arrow.to.target n="fig39"></arrow.to.target><lb/>conuertitur, circa <expan abbr="pūctum">punctum</expan> &longs;eu <lb/>fulcimentum B, pondus, plum­<lb/>bumue, vbi A, &longs;itula E, funi ap­<lb/>pen&longs;a CE. Dico rebus ita con­<lb/>&longs;titutis difficilem quidem e&longs;&longs;e <lb/>vacuæ &longs;itulæ demi&longs;&longs;ion em, fa­<lb/>cile vero eiu&longs;dem extractio­<lb/>nem. Vectis diui&longs;i, &longs;itulæ, ac <lb/>ponderis, ad hoc vt fiat æ quili­<lb/>brium, ca debet e&longs;&longs;e propor­<lb/>tio, vt quema dmodum &longs;e habet AB ad BC, ita &longs;e habeat <lb/>plenæ &longs;itulæ pondus E ad ip&longs;um pondus A, &longs;uperabit ergo <lb/>pondus in A &longs;itulam vacuam in E nec fiet æquilibrium, i­<lb/>taque vt vacua &longs;itula demittatur, tanta vis adhibenda c&longs;t <lb/>quantum e&longs;t ip&longs;ius aquæ, qua &longs;itulaimpl<gap/>tur pondus, quæ <lb/>vis dum apponitur difficilem, vt dicebamus, efficit &longs;itulæ <lb/>vacuæ demi&longs;&longs;ionem. Plena vero &longs;itula &longs;it æquilibrium, vn­<lb/>de quantumuis pu&longs;illa vi adhibita, &longs;itula extrahitur, qua&longs;i <lb/>ex &longs;emetip&longs;a ponderis appen&longs;i virtute a&longs;cendens. Quan­<lb/>tum igitur pondus dum vacua demittitur impedit, tan­<lb/>tundem plena dum extrahitur, adiuuat. Quae cum ita &longs;int, <lb/>&longs;i paria &longs;unt difficultas in demittendo, & facilitas in ex­<lb/>trahendo, quæ ratio hoc in negotio vtilitatis? Sane &longs;itula <lb/>vacua, manu per funem facile demittitur, plena vero dif­<lb/>ficile extrahitur, v&longs;u autem Celonij res <expan abbr="permutãtur">permutantur</expan>. Cor­<lb/>poris enim proprij pondere, dum premit, adiuuatur de­<lb/>mittens, qui per funem &longs;implicem extrahendo, ab eodem <lb/>proprij corporis pondere impediebatur. quod quidem ex <lb/>corporis pondere, auxilium, ingentem parit in extrahen­<lb/>do commoditatem. </s>
</p>
<figure id="fig39"></figure>
<p type="main">
<s>Quippiam &longs;imile accidit, aquas è puteis extrahen­<lb/>tibus v&longs;u trochleæ. Sit enim trochlea puteo imminens <lb/>ABCD, cuius centrum E &longs;u&longs;pen&longs;a quidem in A, funis, cui
<pb pagenum="161"/>&longs;itula &longs;u&longs;penditur FCABG, &longs;itula vero G. E&longs;t igitur dia­<lb/>meter CED, in&longs;tar libræ, quare vt fiat æquilibrium nece&longs;­<lb/>&longs;e e&longs;t capiti funis F, potentiam applicare, quæ &longs;it æqualis <lb/>
<arrow.to.target n="fig40"></arrow.to.target><lb/>pondere &longs;itulæ aqua plenæ, itaque extra­<lb/>hens proprijs viribus <expan abbr="corporīs">corporins</expan> pondus ad­<lb/>ijciens facile &longs;itulam aqua plenam extra­<lb/>hit, ex qua re magna extrahentibus fit <lb/>commoditas. Patet autem diuer&longs;o modo <lb/>extrahentes iuuare Celonium. & Tro­<lb/>chleam, ibi enim corporis mole adiuuatur <lb/>demittens vacuam, hic vero qui extrahit <lb/>plenam aqua &longs;itulam. </s>
</p>
<figure id="fig40"></figure>
<p type="main">
<s>Cæterum Celonij partem BC, qui à <lb/>fulcimento ad funem longe maiorem e&longs;­<lb/>&longs;e oportet, ip&longs;a AB, vt &longs;itula in profundum po&longs;&longs;it demitti, <lb/>quamobremita &longs;e debethabere pondus in A, ad pondus <lb/>&longs;itulæ plenæ, vt&longs;e habet brachium &longs;eu pars BC, ad par­<lb/>tem BA. Tunc enim ex permutata proportione efficitur <lb/>æquilibrium. </s>
</p>
<p type="main">
<s>Illud addimus, nouum non ae&longs;&longs;e Architectis Mecha­<lb/>nici&longs;que, tum hominum tum animalium vt commodius <lb/>machinas moueant, adhibere pondera corporum. Nec e­<lb/>nim alia ratione mouentur Rotæ illæ, quas ob hanc cau&longs;­<lb/>&longs;am ambulatorias vocant; quarum v&longs;us ad Mangana, ad <lb/>extrahendas è puteis aquas, & ad farinarias quoque mo­<lb/>las agitan das a dhibetur. </s>
</p>
<p type="main">
<s>Porro Tollenonem bellicam Machinam à Celonio <lb/>tum forma tum pote&longs;tate nihil differre, videre e&longs;t a pud <lb/>veteres Mechanicos, Heronem Byzantium, & alios. apud <lb/>neotericos vero hac de re agunt Daniel Barbarus in Vi­<lb/>truuium, & Iu&longs;tus Lip&longs;ius in librum quem de bellicis <lb/>machinis edidit, eleganti&longs;&longs;i­<lb/>mum. </s>
</p>
<pb pagenum="162"/>
<p type="head">
<s>QVAESTIO XXIX.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, Cur quando &longs;uper ligno, aut huiu&longs;modi quopiam, duo <lb/>portauerint homines, idem pondus non æqualiter premun­<lb/>tur, &longs;edille magis cui vicinius fuerit <lb/>pondus?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit Ari&longs;toteles, inquiens, lignum e&longs;&longs;e vectem, pon­<lb/>dus vero fulcimentum; res quæ mouetur is qui ponde­<lb/>ri e&longs;t proximior: mouens vero qui remotior. Itaque quo <lb/>magis remotus e&longs;t à pondere, hoc e&longs;t, à fulcimento is qui <lb/>mouet, eo violentius is premitur qui altera vectis parte <lb/>eaque breuiori, mouetur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to lignum AB, pondus <lb/>C appen&longs;um in E, vicinius ex­<lb/>tremo B quamip&longs;i A, &longs;it <expan abbr="autē">autem</expan> <lb/><expan abbr="portãtium">portantium</expan> alter quidem AF, <lb/>alter vero BG, Imaginemur <lb/>itaquelocum E à pondere ita <lb/>figi & deprimi, vt&longs;ur&longs;um qui­<lb/>dem ferri nequaquam po&longs;&longs;it, <lb/>circa vero punctum E, ceu <lb/>circa centrum fulcimentum­<lb/>ne ip&longs;um vectem conuerti. Lignum ergo AB vectis: mo­<lb/>uens potentia A, pars vectis à potentia ad fulcimentum <lb/>AE pars eiu&longs;dem quæ à fulcimento ad rem motam EB, & <lb/>quoniam quanto longior e&longs;t pars vectis EA ip&longs;a EB, eo fa­<lb/>cilius potentia quæ e&longs;t in A, operatur in id quod e&longs;t in B, &longs;i <lb/>res ad proportiones redigatur, erit potentia in A, ad id <lb/>quod mouetur &longs;eu premitur in B, vt pars vectis EB ad par­<lb/>tem EA, &longs;ed maior e&longs;t AEip&longs;a EB, ergo maiorem partem <lb/>&longs;u&longs;tinet ponderis, & plus premitur is qui in E, & qui mo­<lb/>uet in A. Hæc fere Philo&longs;ophi e&longs;t &longs;ententia: Picolomi­<lb/>neus vero Paraphra&longs;tes appo&longs;ite duos vectes in vnico li-
<pb pagenum="163"/>gno con&longs;iderat, alterum AB, alterum BA, in primo A e&longs;t <lb/>mouens B, motum in &longs;ecundo B, mouens A vero motum <lb/>in quibus vectibus &longs;emper idem & commune fulcimen­<lb/>tum E. Et quoniam in propo&longs;ito diagrammate breuior e&longs;t <lb/>pars vectis EB, quæ que à mouente ad fulcimentum, parte <lb/>illa quæ ab eodem fulcìmento ad rem motam, minus o­<lb/>peratur B in A, quam A in B, & ideo qui in B mouetur plus <lb/>premitur, contra vero quia maior e&longs;t pars EA ip&longs;a parte <lb/>EB, magis operatur qui in A in ip&longs;um B, quam econtra. Et <lb/>&longs;ane con&longs;ideratio hæc &longs;ubtilis e&longs;t & ingenio&longs;a, & quæ &longs;i <lb/>recte intelligatur, quatenus ad proportiones & effectum <lb/>ip&longs;um demon&longs;trandum pertinet, à veritate ip&longs;a non ab­<lb/>horret, Quicquid tamen &longs;it, Mechanice magis hoc pacto <lb/>quæ&longs;tio diluetur. Dicimus enim, pondus quidem vere e&longs;­<lb/>&longs;e pondus, non autem fulcimentum, vt &longs;ibi fingebat Ari­<lb/>&longs;toteles: lignum vero vectem, duo autem qui pondus &longs;u­<lb/>&longs;tinent pro duplici fulcimento haberi, vtri&longs;que enim ve­<lb/>ctis cum appen&longs;o pondere innititur. Pote&longs;t etiam alter <lb/>eorum pro porentia mouente, alter vero pro fulcimen­<lb/>to, & &longs;ic vici&longs;&longs;im. E&longs;t autem, quomodocunqueres accipia­<lb/>tur, pondus inter fulcimentum. & potentiam. Quare ex <lb/>ijs quæ demon&longs;trauit G. Vbald. de hoc vectis genere lo­<lb/>quens, vt &longs;e habet AE pars ad AB vectem totum, ita po­<lb/>tentia quæ &longs;u&longs;tinet in B, ad pondus appen&longs;um in E, & vt <lb/>BE ad BA ita potentia quæ &longs;u&longs;tinet in A ad pondus quod <lb/>in E. At minor e&longs;t proportio BE, ad BA, quam AE ad AB, <lb/>quare magis &longs;uperatur pondus in E à potentia quæ in A, <lb/>quam à potentia quæ in B, & ideo plus ponderis &longs;u&longs;tinet <lb/>ferens in B, quam ferens in A, quod fuerat demon&longs;tran­<lb/>dum. </s>
</p>
<p type="main">
<s>Hinc colligimus, pondere in medio vecte appen&longs;o <lb/>ferentes æqualiter &longs;u&longs;tinere, propterea quod totius vectis <lb/>ad partes ip&longs;as proportio &longs;it eadem, vel æqualis. </s>
</p>
<pb pagenum="164"/>
<p type="main">
<s>Pulchre autem dubitari pote&longs;t, an idem pror&longs;us con­<lb/>tingat, &longs;i alterum eorum qui &longs;u&longs;tinent, &longs;it &longs;tatura quidem <lb/>procerior, alterv<gap/>ro humilior. </s>
</p>
<figure></figure>
<p type="main">
<s>Sit enim vectis AB, in cuius <lb/>medio pondus H libere appen­<lb/>&longs;um ex C, alter portantium pro­<lb/>cerior AD, humilior vero BE. &longs;it <lb/>autem horizontis planum DE, <lb/>demittatur à puncto Cad <expan abbr="horizō-tem">horizon­<lb/>tem</expan> perpendicularis, ip&longs;is vero <lb/>AD, BE, æquidi&longs;tans CF. Tran&longs;i­<lb/>bit autem per ip&longs;ius ponderis, <lb/>grauitatis centrum H. Dicoigi­<lb/>tur, nil referre quatenus ad pondus &longs;u&longs;tinendum perti­<lb/>net, vtrum portantes &longs;int &longs;tatura pares velne. Ducatur e­<lb/>nim horizonti æquidi&longs;tans GB, &longs;ecans perpendicularem <lb/>CF in I. Quoniam igitur AG æquidi&longs;tans e&longs;t ip&longs;i CI erit <lb/>vt AC ad CB per 4. &longs;exti elem, ita GI ad IB. Sunt ergo GI, <lb/>IB inter &longs;e æquales. Intelligatur itaque pondus H, <expan abbr="&longs;olutū">&longs;olutum</expan> <lb/>à puncto C appen&longs;um e&longs;&longs;e libere ex puncto I, hoce&longs;t, ex <lb/>medio vectis GB, æqualiter ergo diui&longs;um erit pondus in­<lb/>ter portantes, licet alter procerior, alter vero &longs;tatura pu­<lb/>milior, quod fuerat demon&longs;trandum. </s>
</p>
<p type="main">
<s>Si autem pondus ita vecti alligatum &longs;it vt libere non <lb/>pendeat, vecte ex vna parte eleuato, ex altera vero de­<lb/>pre&longs;&longs;o, grauitatis centrum ad eam partem verget quæ <lb/>magis ab horizonte attollitur, & ad eam ip&longs;am partem <lb/>vectis à pondere ad &longs;u&longs;tinentem fit breuior. </s>
</p>
<p type="main">
<s>E&longs;to enim vectis AB, cuius medium C, pondus vecti <lb/>in C alligatum CFG, cuius grauitatis centrum H eorum <lb/>qui portant procerior AB, humilior BE, horizontis <expan abbr="planū">planum</expan> <lb/>DE. Demittatur per centrum H horizonti perpendicu­<lb/>laris IHK, &longs;ecans vectem quidem in I, horizontis vero pla-
<pb pagenum="165"/>
<arrow.to.target n="fig41"></arrow.to.target><lb/>num in K. Po&longs;t hæc intelligatur pon­<lb/>dus &longs;olutum quidem à puncto C, ap­<lb/>pen&longs;um vero ex puncto I. Stabitigitur <lb/>ex definitione centri grauitatis nec &longs;i­<lb/>tu &longs;uo mouebitur. Dico autem par­<lb/>tem AI ip&longs;a IB e&longs;&longs;e breuiorem, hoc e&longs;t, <lb/>punctum I cadere inter C & A. Si e­<lb/>nim non cadat, vel cadet in C, aut in­<lb/>ter C & B, cadat autem &longs;i fieri pote&longs;t <lb/>in C. Eritigitur CHK horizonti perpendicularis, &longs;ed ei­<lb/>dem perpendicularis AD. Eruntigitur BCK BAD anguli <lb/>inter &longs;e æquales, &longs;ed ip&longs;i BAD angulo æqualis e&longs;t CIH, <lb/>quare & BCH ip&longs;i CIH æ qualis erit. Producto igitur la­<lb/>tere IC trianguli ICH erit exterior angulus æqualis inte­<lb/>riori ex oppo&longs;ito, quod e&longs;t ab&longs;urdum. non ergo I cadet in <lb/>C. Eadem autem ratione mon&longs;trabitur non cadereinter <lb/>CB, cadet ergo inter CA, & ideo minor AI ip&longs;a IB. Itaque <lb/>vt &longs;e habet BI ad BA, ita potentia in A ad pondus in I, &longs;ed <lb/>maiorem proportionem habet BI ad BA, quam IA ad AB. <lb/>Ergo minor potentia requiretur in B quam in A, & &longs;ane <lb/>pars IB re&longs;pondet potentiæ &longs;u&longs;tinenti in A, at IA potentiæ <lb/>&longs;u&longs;tinenti in B, minor e&longs;t autem AI ip&longs;a IB, ergo maior po­<lb/>tentia requiritur in B, quam in A, quod fuerat demon­<lb/>&longs;trandum. </s>
</p>
<figure id="fig41"></figure>
<p type="main">
<s>Hocitem concludetur, &longs;i portantes &longs;tatura quidem <lb/>pares fuerint, &longs;ed per planum ambulent horizonti accliue <lb/>aut decliue. Si enim pon dus libere pendeat, vectis <expan abbr="partiū">partium</expan> <lb/>proportio non mutabitur; &longs;r autem libere non pendeat, <lb/>is magis laborabit qui in a&longs;cen&longs;u præibit, minus vero qui <lb/>in de&longs;cen&longs;u. </s>
</p>
<p type="main">
<s>Hinc quoque Carrucarum ratio pendet, quæ dupli­<lb/>ci manubrio vnica rota vulgo &longs;unt in v&longs;u, pro vecte enim <lb/>habentur, cuius fulcimentum ad contactum plani & ro­
<pb pagenum="166"/>tæ; potentiæ vero ad extremitatem duplicis manubrij. <lb/>Reducitur enim ad idem genus vectis, in quo pondus in­<lb/>ter fulcimentum e&longs;t & potentiam. quo igitur minor fue­<lb/>rit proportio partis vectis quæ à centro grauitatis ad i­<lb/>p&longs;um fulcimentum, ad totum vectem eo facilius pondus <lb/>eieuabitur. </s>
</p>
<p type="main">
<s>Cur autem difficilime hæ per accliue horizonti pla­<lb/>num pellantur, duplici fit de cau&longs;&longs;a, tum quia grauitatis <lb/>centrum ad ip&longs;um portantem &longs;eu pellentem vergit, & id­<lb/>co pars quæ a fulcimento ad centrum grauitatis ponderis <lb/>fit maior, tum etiam quoniam ip&longs;um graue contra &longs;ui na­<lb/>turam &longs;ur&longs;us pellitur ferturque. </s>
</p>
<p type="main">
<s>Quærere ad hæc qui&longs;piam po&longs;&longs;et, Cur Baiuli ma­<lb/>gna ferentes pondera, curui in cedant? Dixeritautem ali­<lb/>quis, ponderis grauitate eos deprimentis id fieri. Nos au­<lb/>tem duplici item de cau&longs;&longs;aid fieri putamus, tum ea quam <lb/>con&longs;iderauimus, tum etiam alia, nempevt grauitatis cen­<lb/>trum ip&longs;ius ponderis quod &longs;u&longs;tinent, in perpendiculari <lb/>collocent, ne &longs;i extra ponatur is qui fert à centro extra <lb/>fulcimentum po&longs;ito, ad eam partem ad quam vergit tra­<lb/>hatur, & pondere ip&longs;o opprimatur. </s>
</p>
<p type="main">
<s>Eadem de cau&longs;&longs;a fit quoque vt ij qui magna ponde­<lb/>ra &longs;ini&longs;tro ferunt humero, in dextram partem inclinentur, <lb/>qui vero dextro, contrario modo &longs;e habeant, æquatur e­<lb/>nim pondus eo pacto, & grauitatis centrum in ip&longs;a per­<lb/>pendiculari collocatur. </s>
</p>
<p type="head">
<s>QVÆSTIO XXX.</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur a&longs;&longs;urgentes omnes fœmori tibiam ad acutum angulum con&longs;ti­<lb/>tuamus & pectori thoraciue &longs;imiliter fœmur, quod nî fiat <lb/>haudquaquam &longs;urgere poterunt?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Ait Philo&longs;ophus, forte id fieri, quod æqualitas &longs;it o­<lb/>mnino quietis cau&longs;&longs;a, rectum vero angulum quietis
<pb pagenum="167"/>angulum e&longs;&longs;e, & &longs;tationem facere, nec alia de cau&longs;&longs;a &longs;tan­<lb/>tem ip&longs;i terræ e&longs;&longs;e perpendicularem, & ideo caput & pe­<lb/>des in eadem linea habere, &longs;edentem vero non item. <expan abbr="Tūc">Tunc</expan> <lb/>autem à &longs;e&longs;&longs;ione &longs;urrectionem fieri, cum caput & pedes in <lb/>vna linea collocantur, quod &longs;ane fit cum pectus & crura <lb/>acutum cum ip&longs;o fœmore angulum faciunt. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim &longs;tans AB hori­<lb/>zonti IBK perpendicularis, cù­<lb/>ius caput A, pedes vero B, &longs;edcat <lb/>modo &longs;itque eius cum capite <lb/>Thorax CD, fœmur DE, crura <lb/>EF, &longs;intque CDE, DEF anguli <lb/>recti, quibus ita con&longs;titutis non <lb/>&longs;unt in eadem linea caput C & <lb/>pedes F. Surgere ita que non po­<lb/>terit &longs;edens, propterea quod <lb/>partes omnes corporìs non &longs;int <lb/>horizonti perpendiculares. Ad <lb/>hocautem vt &longs;urrectio fiat, nece&longs;&longs;e e&longs;t vt &longs;edens retrahat <lb/>quidem pedes in H, & pectore in clinato acutum cum fœ­<lb/>more angulum con&longs;tituat GDE, quo ca&longs;u fient in eadem <lb/>recta linea, eaque horizonti perpendiculari caput in G, <lb/>& pedes in H, ex cuius &longs;itus natura commoda fiet ab ip&longs;o <lb/>&longs;edente &longs;urrectio. Hæc fere, licet alijs ab eo verbis expli­<lb/>cata, ip&longs;ius e&longs;t Philo&longs;ophi &longs;ententia; quæ licet vera &longs;it, non <lb/>tamen ex proprijs, hoc e&longs;t, Mechanicis principijs e&longs;t peti­<lb/>ta. quod quidem nos facere conabimur. </s>
</p>
<p type="main">
<s>Dicimus autem primo, &longs;edentem non ideo quie&longs;ce­<lb/>re, vt&longs;entit Ari&longs;toteles, quod rectus angulus quietis &longs;it <lb/>cau&longs;&longs;a, &longs;ed propterea quod eius thoracis tum etiam fœ­<lb/>morum pondus ab ip&longs;a &longs;ede &longs;u&longs;tineantur; crura vero & <lb/>pedes ideo non laborent, quod partim &longs;u&longs;pen&longs;a &longs;int, par­<lb/>tim &longs;olo ip&longs;i innitantur. Quare cum corpus totum nec &longs;e
<pb pagenum="168"/>&longs;u&longs;tineat, nec à pedibus &longs;u&longs;tineatur, fit quies & la&longs;&longs;itudi­<lb/>nis alleuatio. Natura autem ideo commodam hominibus <lb/>&longs;e&longs;&longs;ionem facere volui&longs;&longs;e inde apparet, quod clunes, qui­<lb/>bus tota &longs;uperior pars, & grauior nititur, carno&longs;am fece­<lb/>rit, & ceruicalis cuiu&longs;dam in&longs;tar mollem & facilem. Sed <lb/>nos ad quæ&longs;tionem. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim &longs;tans AB, cuius caput A, <lb/>Thorax AC, fœmora CD, crura DB, pe­<lb/>des vero B, centrum vero grauitatis in i­<lb/>p&longs;o Thorace E. Modo &longs;edeat, &longs;itque ca­<lb/>put in F, Thorax FG, fœmora GH, crura <lb/>HI, pedes I, grauitatis vero centrum vbi <lb/>K. Producatur recta FG in L, &longs;itque FL <lb/>horizonti perpendicularis. Centrum er­<lb/>go grauitatis K fulcitur puncto G, hoc e&longs;t, <lb/>puncto L, in quo po&longs;teriores pedes ip&longs;ius <lb/>&longs;edis &longs;olo hærent. efficit autem &longs;edens <lb/>duos rectos angulos FGH, GHI. Rebus <lb/>igitur ita di&longs;po&longs;itis &longs;eruatis rectis angulis, non fiet &longs;urre­<lb/>ctio, & id quidem non ideo quod, vt ait Philo&longs;ophus, æ­<lb/>qualitas & rectitudo angulorum quietis &longs;it cau&longs;&longs;a, &longs;ed <lb/>propterea quod centro grauitatis extra pedum <expan abbr="fulcimē-tum">fulcimen­<lb/>tum</expan> con&longs;tituto, non habet centrum &longs;tabilem locum cui in <lb/>actu &longs;urrectionis hæreat, & fulciatur, vnde fit vt &longs;i &longs;edenti <lb/>&longs;ubtrahatur &longs;edes remoto prohibente, &longs;edens pror&longs;us cor­<lb/>ruat. Modo retrahat qui &longs;edet crura, & pedes ponat in M, <lb/>à puncto autem M, horizonti perpendicularis erigatur <lb/>MN. erit ergo fulcimentum in M, &longs;ed adhuc &longs;urgere non <lb/>poterit, centro grauitatis adhuc extra lineam MN, quæ <lb/>per fulcimentum e&longs;t, con&longs;tituto. Reclinetur autem pe­<lb/>ctus ad anteriora, & cum fœmore acutum angulum faciat <lb/>&longs;itque vbi GO, erit igitur grauitatis centrum vbi P, hoc <lb/>e&longs;t, in ip&longs;a perpendiculari NM, fretigitur inde commoda
<pb pagenum="169"/>&longs;urrectio, propterea quod in eadem linea facta &longs;int, graui­<lb/>tatis centrum P, & fulcimentum ip&longs;um M. Acutum vero <lb/>angulum in &longs;urrectione nece&longs;&longs;arium e&longs;&longs;e clare patet, non <lb/>autem eff<gap/>us ip&longs;ius e&longs;&longs;e cau&longs;&longs;am, vt videtur &longs;en&longs;i&longs;&longs;e Ari­<lb/>&longs;toteles; ni<gap/>i dicamus, cau&longs;&longs;am e&longs;&longs;e cau&longs;&longs;æ, &longs;iquidem acuti <lb/>qui fiunt anguli centrum & pedes in eadem linea collo­<lb/>cant, quicquid tamen &longs;it, nos ideo &longs;urrectionem fieri dici­<lb/>mus, quod immutatis angulis centrum grauitatis &longs;upra <lb/>fulcimentum, fulcimento vero &longs;ub ip&longs;o grauitatis centro <lb/>collocetur, & hæc e&longs;t cau&longs;&longs;a proxima. Hæc nos ad Ari&longs;to­<lb/>telem. Modo qua&longs;dam alias quæ&longs;tiones, necinutiles &longs;ed <lb/>& eas non iniucundas quoque proponemus. </s>
</p>
<p type="main">
<s>Primum igitur quærimus, Curhominum & cætero­<lb/>rum animalium, quæ aliquando erecto corpore incedunt, <lb/>pedes non quidem breues &longs;int & rotundi, &longs;ed longiores <lb/>potius, & in inferiorem partem porrecti? Item cur magis <lb/>ad digitos quam ad cal caneum porrigantur? </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to <gap/>homo animalue quodpiam &longs;tans <lb/>AB, cuius pes CD, pedis pars quæ ad digitos <lb/>BC. quae vero ad cal caneum BD fœmoris ver­<lb/>tebra E, centrum vero grauitatis ip&longs;ius cor­<lb/>poris F. Primum igitur &longs;tatuendum e&longs;t, ho­<lb/>minem & cætera fere animalia à Natura fa­<lb/>cta e&longs;&longs;e vt ad anteriora moueantur, & ideo o­<lb/>mnes fere quod in &longs;enioribus manife&longs;te ap­<lb/>paret, ad anteriora ex ip&longs;a corporis di&longs;po&longs;i­<lb/>tione vergant. Itaque dum qui &longs;tat horizon­<lb/>ti pror&longs;us e&longs;t perpendicularis, grauitatis centrum F in ip&longs;a <lb/>perpendiculari con&longs;tituitur quæ ad mundi centrum AB, <lb/>& ideo corporis moles pondu&longs;que fulcitur puncto B. Mo­<lb/>do fiat ex vertebra E thoracis AE, inclinatio in anteriora, <lb/>in GE & grauitatis centrum D diluetur in H, & per H per­<lb/>pendicularis demittatur HI, non erit ** extra pedis ful­
<pb pagenum="170"/>cimentum BC. Stabit ergo qui ita in clinatur, nec corruet: <lb/>&longs;i autem a dhuc propendeat magis, fiatque in KE, centro <lb/>grauitatis con&longs;tituto in M, ducatur per M perpendicula­<lb/>ris ML, quare quoniam linea ML extra pedis fulcimen­<lb/>tum cadit, corruet qui co pacto inclinatur nec &longs;u&longs;tinebi­<lb/>tur. Cur igitur natura animalibus quae erecto corpore am­<lb/>balant, pedes in anteriora porrectos fecerit, hinc clare <lb/>patet. </s>
</p>
<p type="main">
<s>Hinc etiam ceu con&longs;ectarium habemus, cur homi­<lb/>nes &longs;i impellantur, magis ad ca&longs;um in po&longs;teriora quam in <lb/>anteriora &longs;int proni. Necnon etiam cur &longs;imiæ, vr&longs;i, & &longs;i <lb/>quæ cætera eiu&longs;modi animalia diutius erecto corpore <lb/>ambulare nequeant, nempe ideo quod eorum corporum <lb/>moles valde in anteriora propendeat, necita commodo, <lb/>vt humanis cuenit corporibus, pedum ip&longs;orum ba&longs;ibus <lb/>fulciantur. </s>
</p>
<p type="main">
<s>Quærereitem haud importune po&longs;&longs;umus, Curgral­<lb/>latores non &longs;tent erecti, ni&longs;i a&longs;&longs;idue moueantur? Solutio <lb/>facilis. grallæ etenim duobus tantum punctis &longs;olum tan­<lb/>gunt, nec porrecti beneficio, quod ambulantibus accidit, <lb/>vti po&longs;&longs;unt. quamobrem grauitatis centrum fit extra ful­<lb/>cimentum, & ideo coguntur grallatores a&longs;&longs;iduo motu <lb/>grauitatis centro fulcimentum &longs;upponere, quod dum fit, <lb/>à ca&longs;u prohibentur. </s>
</p>
<p type="main">
<s>Pote&longs;t autem id quod fulcitur, tripliciter fulciri, <expan abbr="nē-peaut">nen­<lb/>peaut</expan> puncto, aut linea, aut &longs;uperficie. </s>
</p>
<p type="main">
<s>Quod puncto fulcitur, nulla reimpediente ad quam­<lb/>uis partem cadere pote&longs;t, centrum &longs;iquidem, motus, pun­<lb/>ctum e&longs;t. </s>
</p>
<p type="main">
<s>Quod linea fulcitur ad duas tantum partes, ea&longs;que <lb/>oppo&longs;itas, habet ca&longs;um. &longs;itillud &longs;uperficies, corpu&longs;ue in <lb/>latus con&longs;titutum. </s>
</p>
<pb pagenum="171"/>
<figure></figure>
<p type="main">
<s>E&longs;to horizontis pla­<lb/>num ABCD, cui ad re­<lb/>ctos angulos in&longs;i&longs;tat &longs;u­<lb/>perficies EFGH, &longs;ecun­<lb/>dum latus FG. Sit autem <lb/>ip&longs;ius &longs;uperficiei grauita­<lb/>tis centrum I. à quo ad <lb/>horizontis planum per­<lb/>pendicularis demittatur IK. Cadet autem in lineam FG. <lb/>per propo&longs;. 38. vndecimi elem. & anguli IKG IKF recti e­<lb/>runt. Itaque &longs;uperficie EFGH circa lineam FKG ceu cir­<lb/>ca axem mota punctum I peripheriam de&longs;cribet LIM, & <lb/>&longs;iquidem cadat ad partes CD, grauitatis centrum erit vbi <lb/>M. Si vero ad partes AB, fiet vbi L. Sunt autem LKM <expan abbr="pū-cta">pun­<lb/>cta</expan> in recta LKM, quæ quidem communis &longs;ectio e&longs;t plani <lb/>horizontis, & plani per IKLM, tran&longs;euntis. </s>
</p>
<figure></figure>
<p type="main">
<s>Idem quoque de cor­<lb/>pore dicimus in latus col­<lb/>locato. E&longs;to enim cubus <lb/>LO, cuius grauitatis cen­<lb/>trum R, latus vero quo ful­<lb/>citur, NO, Si enim ita col­<lb/>locetur, vt interna &longs;uperfi­<lb/>cies LNOQ ad rectos an­<lb/>gulos horizonti &longs;it con&longs;ti­<lb/>tuta, demi&longs;&longs;a perpendicu­<lb/>laris à puncto R, ca det in S, in ip&longs;a linea NSO. Cadente i­<lb/>gitur corpore fiet motus circa lineam NO, centro graui­<lb/>tatis interim peripheriam TRV. de&longs;cribente. </s>
</p>
<p type="main">
<s>Hincanimaduertere licet, Cur prouidi&longs;&longs;ima Natu­<lb/>ranulli animantium vnicum dederit pedem, &longs;ed aut qua­<lb/>ternos, aut &longs;altem binos, & binos quidem ip&longs;os virtute <lb/>quaternos, &longs;iquidem in quolibet animantium bipedum
<pb pagenum="172"/>pede duo &longs;altem puncta con&longs;iderantur, quibus ip&longs;um ani. <lb/>mal fulcitur. </s>
</p>
<figure></figure>
<p type="main">
<s>Sint enim humani pedis ve­<lb/>&longs;tigia A, B, C, D, in vtroque igitur <lb/>duo puncta con&longs;iderantur, A, B, <lb/>C, D, illa quidem ad digitos, hæc <lb/>autem ad calcaneum. l<gap/>em quo­<lb/>que in auium pedibus ob&longs;erua­<lb/>tur, ex quibus concludimus, bi­<lb/>pedum omnium fulcimentum e&longs;­<lb/>&longs;e quadruplex. Porro quadrupe­<lb/>dia eo quod tota co<gap/>poris mole <lb/>ad in feriora vergant, quatuor ful­<lb/>cimenta, eaque di&longs;tincta, & commode ab inuicem remo­<lb/>ta eademmet Natura præparauit. </s>
</p>
<p type="main">
<s>Eadem quoque in artificialibus con&longs;ideramus. Sit <lb/>enim vas quo dpiam ABC, cuius pes vnicus, i&longs;que rotun­<lb/>dus BC, grauitatis vero centrum D. Quoniam igitur in <lb/>pedis ip&longs;ius peripheria, infinita puncta intelligantur, dici <lb/>quo dammodo pote&longs;t vas ip&longs;um infinitis fere punctis, licet <lb/>
<arrow.to.target n="fig42"></arrow.to.target><lb/>pesvnicus &longs;it, &longs;u&longs;tineri. Non­<lb/>nulla autem corpora artifi­<lb/>cialia. quatuor pedibus &longs;u­<lb/>&longs;tinentur, vt men&longs;æ <expan abbr="quædã">quædam</expan>, <lb/>nonnulla etiam tribus, vt <lb/>tripodes, qui nomen ab ip&longs;o <lb/>pedum numero &longs;ortiuntur. <lb/>Sit enim triangulum EFG, <lb/>cuius centrum grauitatis H, <lb/>nitatur autem tribus pun­<lb/>ctis I, K, L, &longs;tabit igitur. Si <lb/>autem duobus tantum; non &longs;tabit. ducta enim IK &longs;i pun­<lb/>ctis tantum IK innitatur, con&longs;tituto grauitatis centro
<pb pagenum="173"/>extra fulcimentum IK, verget cedens ver&longs;<gap/>s partes, L, Si <lb/>autem innitatur punctis IL, cadet ad partes K. Sivero ip&longs;is <lb/>KL, cadet ad partes I.Ex quibus apparet, inanimata cor­<lb/>pora aut vnico pede plurium virtutem habente, aut &longs;al­<lb/>tem tribus actu, vt &longs;u&longs;tineantur, indigere. </s>
</p>
<figure id="fig42"></figure>
<p type="main">
<s>Hinc etiam patet, cur &longs;enes, imbecilles, curui, & pe­<lb/>dibus capti, baculi baculorumue fulcimento egeant, ete­<lb/>nim cum hi debiles &longs;int, & in anteriorem partem magno­<lb/>pere propen deant, ne grauitatis centrum extra fulcimen­<lb/>tum fiat, baculo vel baculis indigent, quibus centrum i­<lb/>p&longs;um ful ciatur. </s>
</p>
<p type="main">
<s>Cæterum cur duplici genu ingeniculati difficile in <lb/>eo &longs;itu perman eant, ea cau&longs;&longs;a e&longs;t, quod grauitatis centrum <lb/>in thorace con&longs;titutum, duobus genibus fulciatur, eo&longs;­<lb/>que premat. quæ quidem genua eo quod natura apta na­<lb/>ta non &longs;int, veluti pedes, ad &longs;u&longs;tinendam corporis molem <lb/>laborant, idque eo magis, quod cum o&longs;&longs;ea &longs;int, cutem in­<lb/>ter o&longs;&longs;ium & plani duritiem con&longs;titutam, accidit arctari, <lb/>& ideo dolorem & mole&longs;tiam ingeniculatis facere. </s>
</p>
<p type="main">
<s>Siautem vnico tantum genu qui&longs;piam nitatur, dif­<lb/>ficultatem &longs;entiet longe minorem. Triplici enim fulci­<lb/>
<arrow.to.target n="fig43"></arrow.to.target><lb/>mento eo ca&longs;u ingeniculatus <lb/>fulcitur. Sit enim ingenicula­<lb/>tus ABCDE, cuius grauitatis <lb/>centrum F. dextrum vero ge­<lb/>nu, cuinititur D, &longs;ini&longs;trum ve­<lb/>ro, quod eleuatur B. Tribus ergo fulcimentis ingenicula­<lb/>tus vt diximus, &longs;u&longs;tinetur, CDE. Diuiditur itaque pondus <lb/>in tres partes, & ideo &longs;ingulæ minus fatigantur. Magis ta­<lb/>men laborat punctum D, vtpote illud, cui ad perpendicu­<lb/>lum F grauitatis centrum innititur. </s>
</p>
<figure id="fig43"></figure>
<p type="main">
<s>Vtique illud quoque mirabile e&longs;t, Aues dormientes <lb/>vnico tantum pede fulciri, & quod magis mirum e&longs;t, dor­
<pb pagenum="174"/>mientes po&longs;&longs;e, quod vel ip&longs;is vigilantibus e&longs;t difficile. Cur <lb/>id Natura docente faciant, eam puto e&longs;&longs;e cau&longs;&longs;am, quod <lb/>dum dormiunt, caput &longs;ini&longs;træ alæ, vt naturali calore iu­<lb/>uentur, &longs;upponunt, quapropter ad eam partem declinan­<lb/>tes, vt interim æquilibrium faciant, pedem &longs;ubleuant, & <lb/>eo ca&longs;u ceu inutilem retrahunt atque &longs;u&longs;pendunt: addita <lb/>item alia cau&longs;&longs;a, nempe vt pedem ip&longs;um dormientes nati­<lb/>uo calore confoueant. </s>
</p>
<p type="main">
<s>Quæritur et<gap/>am, Curij qui inclinantur, vt <expan abbr="rē">rem</expan> quam­<lb/>piam à &longs;olo &longs;u&longs;tollant, alterum crurium ad anteriora, <expan abbr="nē-pever&longs;us">nen­<lb/>pever&longs;us</expan> manum ip&longs;am, quam porrigunt, extendant? </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim qui&longs;piam ABCD, <lb/>cuius crura BC, BD, grauitatis <lb/>centrum E, vclitautem quippiam <lb/>à &longs;olo tollere quod &longs;it in F. &longs;it per­<lb/>pendicularis, quæ pergrauitatis <lb/>centrum GEH. Dumigitur ad <lb/>anteriora ínclinatur, centrum a­<lb/>mouet à perpendiculari, quam­<lb/>obrem docente Natura, crus BC <lb/>ad centrum ip&longs;um fulciendum. <lb/>ad anteriora, hoc e&longs;t, ver&longs;us rem <lb/>&longs;u&longs;tollendam porrigitur. </s>
</p>
<p type="main">
<s>Huius quoque &longs;peculationis e&longs;t inue&longs;tigare, Cur <lb/>quadrupedia dum gradiuntur, pedes diametraliter mo­<lb/>ueant. Cuius rei verba fecitip&longs;e quoque Philo&longs;ophus lib. <lb/>de animalium in ce&longs;&longs;u cap. 12. Nos autem ad maiorem de­<lb/>clarationem, quodip&longs;e Phy&longs;icis principijs fecit, mecha­<lb/>nicis demon&longs;trabimus. </s>
</p>
<p type="main">
<s>Sint duæ in plano parallelæ AB, CD, in quibus qua­<lb/>drupedis pedes E, F, B, D, quorum EF, anteriores, BD vero <lb/>po&longs;teriores. iungantur BDEF, eritque EBDF parallelo­<lb/>grammum altera parte longius, cuius diametri ducantur
<pb pagenum="175"/>
<arrow.to.target n="fig44"></arrow.to.target><lb/>ED, BF, &longs;ecantes &longs;e&longs;e in G, vbi & grauitatis <lb/>centrum. Moto igitur po&longs;teriori &longs;ini&longs;tro pe­<lb/>de B in K, &longs;i anteriorem E, eodem tempore <lb/>moueret in I, &longs;tantibus interim DF, ceu ful­<lb/>cimentis, centrum Gextra fulcimenta &longs;ieret <lb/>ad partes BE. Caderet igitur ad partes BE. Si <lb/>autem eodem tempore moueret dextros eo­<lb/>dem pacto centrum extra fulcimenta po&longs;i­<lb/>tum caderet ad partes ip&longs;as DF. Si autein <lb/>moto pede B in K, & eodem tempore F in L, <lb/>& D in H, E, in I, centrum erit in diametris HI, KL, hoc <lb/>e&longs;t, vbi M, fultum quidem ab ip&longs;is pedibus K, L, H, I. Hoc <lb/>igitur pacto transfertur vici&longs;&longs;im cum grauitatis centro &longs;i­<lb/>mul translatis fulcimentis &longs;e&longs;e diametraliter re&longs;ponden­<lb/>tibus; quod vtique demon&longs;trandum fuerat. </s>
</p>
<figure id="fig44"></figure>
<p type="main">
<s>Sane & bipedia quoque alternatim gradiendo gra­<lb/>uitatis centrum transferunt. Dum enim dextrum crus e­<lb/>leuatur, centrum &longs;ini&longs;tro fulcitur, & econtra. </s>
</p>
<p type="main">
<s>Naturalia i&longs;thæc &longs;unt; in artifi cialibus autem quæri <lb/>po&longs;&longs;et, Cur Architecti, Arcium muros non ad perpendi­<lb/>culum erectos, &longs;ed intror&longs;um inclinatos con&longs;tituant? </s>
</p>
<figure></figure>
<p type="main">
<s>Vtique hoc faciunt, vt minus <lb/>&longs;int ad ruinam proni. E&longs;to enim <lb/>murus ad interiorem partem ver­<lb/>gens ABCD, Cuius grauitatis cen­<lb/>trum E ba&longs;is BC erigatur à puncto <lb/>B horizonti perpendicularis BF, & <lb/>ad eundem à centro grauitatis E <lb/>demittatur EM, tum BE iungatur. <lb/>Po&longs;t hæc à puncto BG angulum. <lb/>cum linea horizontis BK faciens recto maiorem. Ita que <lb/>murus hoc pacto con&longs;titutus ad interiorem partem &longs;uo <lb/>pondere vergit, cadere autem non pote&longs;t, vel quod viuæ
<pb pagenum="176"/>rupi, cui forte hæret, fulciatur, vel anti&longs;tatis, quos no­<lb/>&longs;trates &longs;perones & contra fortes appellant, innitatur. Sed <lb/>nec in anteriora corruet, quandoquidem ruinam factu­<lb/>ras, nece&longs;&longs;e e&longs;t vt grauitatis centrum &longs;ecum trahat in per­<lb/>pendiculari BF, & demum in eam quæ vltra perpendicu­<lb/>larem e&longs;t BG, facta nempe circa B, ceu circa centrum, <expan abbr="cō-uer&longs;ione">con­<lb/>uer&longs;ione</expan>. Moueatur autem & ex &longs;emidiametro BE cen­<lb/>tro B portio circuli de&longs;cribatur EH, quæ &longs;ecet BG in H, <lb/>& BF in I; Et quia EM &longs;emidiametro BK perpendicularis <lb/>per B, centrum non tran&longs;it, erit EM ip&longs;a BK, hoc e&longs;t, BI <lb/>bre<gap/>ior. Ab&longs;cindatur ex BI, ip&longs;i EM æqualis LB. Eritigi­<lb/>tur punctum L infra punctum I, hoc e&longs;t, ip&longs;o I, mundi cen­<lb/>tro propius. Nece&longs;&longs;e igitur erit ad hoc vt murus corruat, <lb/>centrum grauitatis E facta circa B, conuer&longs;ione aliquan­<lb/>do fieri in I, vt demum transferri po&longs;&longs;it in H, &longs;ed I remo­<lb/>tius e&longs;t à mundi centro ip&longs;is E, L, a&longs;cendet igitur graue <lb/>contra &longs;ui naturam ex E in I, at hoc e&longs;t impo&longs;&longs;ibile; quod <lb/>fuerat demon&longs;tran dum. </s>
</p>
<p type="main">
<s>Ex his ij&longs;dem principijs alia &longs;oluitur quæ&longs;tio, Cur <lb/>&longs;cilicet Campanaria turris quæ Pi&longs;is vi&longs;itu<gap/>, nec non alia <lb/>Bononiæ in foro prope A &longs;ellorum turrim, quam à nobili <lb/>olim Cari&longs;endorum familia ex&longs;tructam, Cari&longs;endam vo­<lb/>cant, cuius meminit & Dantes Poeta &longs;ummus in &longs;ua Co­<lb/>mœdia. Propendet autem hæc in latus, & ita propendet <lb/>vt perpendicularis, quæ à &longs;ummo inclinatæ partis in &longs;o­<lb/>lum demittitur, longe cadat ab ip&longs;a, cui nititur, ba&longs;i, quod <lb/>&longs;ane mirabile videtur, muros nempe, in ruinam pronos, <lb/>ruinam non facere. </s>
</p>
<p type="main">
<s>E&longs;to enim turris ABCD, ba&longs;i fulta BC, horizontis <lb/>planum BCF latera AB, DC, centrum vero grauitatis to­<lb/>tius molis E. Propendeat autem ad partes DC ex angulo <lb/>DCF. Ita autem con&longs;tituta intelligatur vt perpendicula­<lb/>ris ab A, in planum horizontis demi&longs;&longs;a per grauitatis cen-
<pb pagenum="177"/>
<arrow.to.target n="fig45"></arrow.to.target><lb/>trum E extra ba&longs;im BC, non cadat, <lb/>cadat autem in C. Quoniam igitur <lb/>ABCD moles per E grauitatis cen­<lb/>trum diuiditur, in partes &longs;ecatur æ­<lb/>queponderantes, &longs;ed & centrum. <lb/>grauitatis extra fulcimentum non <lb/>cadit, quare nec pars ACD, trahet <lb/>partem ABC, nec centrum extra <lb/>fulcimentum po&longs;itum locum petet <lb/>centro mundi viciniorem. Cur igitur Cari&longs;enda &longs;tet, & e­<lb/>gregia illa turris campanaria quæ Pi&longs;is prope &longs;ummum <lb/>Templum marmoribus præclare ex&longs;tructa videtur, licet <lb/>ruinam minentur, &longs;tent æternum, nec cadant, ex his quæ <lb/>con&longs;iderauimus, liquido patet. </s>
</p>
<figure id="fig45"></figure>
<p type="head">
<s>QVAESTIO XXXI</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur facilius moueatur commotum quam manens, veluti currus <lb/>commotos citius agitant, quam moueri incipientes?<emph.end type="italics"/></s>
</p>
<p type="head">
<s><emph type="italics"/>Hoc quæritur.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Problema hoc e&longs;t mere Phy&longs;icum; verumtamen quo­<lb/>niam ad localem motum pertinet, de quo ip&longs;e quoque <lb/>Mechanicus agit, Hi&longs;ce quæ&longs;tionibus contemplatio hæc <lb/>inter&longs;eritur. Soluit autem Ari&longs;toteles inquiens, id forta&longs;­<lb/>&longs;e ea de cau&longs;&longs;a fieri, quod difficillimum &longs;it pondus moue­<lb/>re, quod in contrarium mouetur. Demit enim quippiam <lb/>de motoris potentia re&longs;i&longs;tens, licet mouens ip&longs;o moto &longs;it <lb/>longe potentius atque velocius. nece&longs;&longs;e enim e&longs;&longs;e id tar­<lb/>dius moueri quod repellitur. Hæc verba licet de ea po­<lb/>tentia dicta videantur, quæ rem motam in contrariam. <lb/>partem repellit, nihilominus illi quoque aptantur quæ <lb/>rem immobilem à principio mouere conatur. e&longs;t enim re­<lb/>&longs;i&longs;tentia rei quæ à &longs;tatu ad motum transfertur ceu <expan abbr="quidã">quidam</expan>
<pb pagenum="178"/>contrarius motus. Contra autem accidit illi quirem mo­<lb/>tam mouet in ip&longs;o motu: eo enim ca&longs;u mouens ab ip&longs;o rei <lb/>motu magnopere iuuatur, cooperatur enim motus moto­<lb/>ri, in ip&longs;am rem motam operanti. Auget autem res mota <lb/>quodammodo mouentis potentiam. quod enim à mouen­<lb/>te pateretur, ex &longs;e ip&longs;a agit res quæ mouetur. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to horizontis pla­<lb/>num AB, cui moles quæ­<lb/>dam in&longs;i&longs;tat, CD. Modo <lb/>potentia quædam appli­<lb/>ceturvbi E, quæ molem in <lb/>anteriora propellat, id <lb/>e&longs;t, ver&longs;us B. Primumigitur, quoniam à quiete ad motum <lb/>fit tran&longs;itus, te&longs;i&longs;tit &longs;ua quiere corpus graue, potentiæ im­<lb/>pellenti, &longs;uperata demum re&longs;i&longs;tentia moles quæ moueri <lb/>cœpit, fertur in F & mouetur, quare potentia quæ à prin­<lb/>cipio re&longs;i&longs;tentiam rei non motæ &longs;uperauerat, pellendo <lb/>rem motam pergens facilius pellit: Duo enim &longs;unt quo­<lb/>dammodo motores, mouens videlicetip&longs;e, & motus quo <lb/>res mota mouetur. facilius ergo pelletur ex F in G, quam <lb/>ex D in F, & ex G in B, quam ex F in G, & eo motus fiet in <lb/>progre&longs;&longs;u facilior atque in ip&longs;a velocitate velocior, quo <lb/>magis in ip&longs;a motione mouetur. </s>
</p>
<p type="main">
<s>Hinc &longs;oluitur ea quæ&longs;tio apud P hy&longs;icos difficillima, <lb/>Cur nempe in motu naturali velocitas v&longs;que augeatur; <lb/>etenim ibi Naturamouens e&longs;t, atque eadem in&longs;eparabilis <lb/>à remota, vrgetigitur a&longs;&longs;idue, à principio quidem tar dius, <lb/>po&longs;t hæc autem ea quam diximus, de cau&longs;&longs;a v&longs;que & v&longs;que <lb/>velocius. Motus ergo fit in motu, qui motus cum &longs;emper à <lb/>motore, & motu ip&longs;o augeatur, cre&longs;cit ex progre&longs;&longs;u in im­<lb/>men&longs;um. Certe cau&longs;&longs;am velocitatis auctæ eam e&longs;&longs;e, quod <lb/>potentia mouens rem motam in motu ip&longs;o moueat, nemo <lb/>vtarbitror, inficias ibit, acquirit enim corpus motum <expan abbr="pō-">pon-</expan>
<pb pagenum="179"/>dero&longs;itatem quan dam accidentalem, quæ cum ex motu <lb/>perinde augeatur, ip&longs;um motum faciliorem, eoque velo­<lb/>ciorem facit. Di&longs;putat hæc & Simplicius lib. 7. Phy&longs;ic. c. <lb/>11. Ari&longs;totelis de Natura libros exponens. </s>
</p>
<p type="head">
<s>QVAESTIO XXXII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quæritur hic, Cur caquæproijciuntur, ce&longs;&longs;ent <lb/>à latione?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Hocitidem problema e&longs;t mere Phy&longs;icum. Ad quod ea <lb/>pertinent quæ à Philo&longs;opho tractantur libro Natu­<lb/>ralium 8. & lib. 1. de Cœlo. Tres autem affert &longs;ubdubitan­<lb/>do rationes, An quia impellens de&longs;init potentia, vel pro­<lb/>pter retractionem, vel propter rei proiectæ in <expan abbr="clinationē">clinationem</expan>, <lb/>quando ea valentior fuerit quam proijcientis vires? </s>
</p>
<p type="main">
<s>Quicquid dicat Philo&longs;ophus, id vtique explorati&longs;­<lb/>&longs;imum e&longs;t. Proicctaideo à motu ce&longs;&longs;are, propterea quod <lb/>impre&longs;&longs;io, cuius impetu & virtute feruntur, non &longs;it proie­<lb/>ctus quidem naturalis, &longs;ed mere accidentalis & violenta, <lb/>at nullum accidentale & violentum quodque, non natu­<lb/>rale e&longs;t, perpetuum e&longs;t. Ce&longs;&longs;at ergo accidentalis illa im­<lb/>pre&longs;&longs;io, eaque paullatim ce&longs;&longs;ante proiecti motus elan­<lb/>gue&longs;cit, donec quietem pror&longs;us adipi&longs;catur. Illud quoque <lb/>notamus, quod à multis vidimus non ob&longs;eruatum, nempe <lb/>violentum mo<gap/>m violentia præualente non differre à <lb/>naturali, & ideo tardiorem e&longs;&longs;e à principio po&longs;t hæc, in i­<lb/>p&longs;o motu fieri velociotem, remittente demum paullatim <lb/>impre&longs;&longs;a violentia, tardiorem, donecimpetus, & cum im­<lb/>petu motus euane&longs;cat, & resip&longs;a mota quietem adipi&longs;ca­<lb/>tur. Vnde etiam experientia docemur, ictum ex proiectis <lb/>violentius fieri, &longs;i fiat paullo remotior à principio, & tunc <lb/>demum effe innocenti&longs;&longs;imum, cum ibi fit, vbi proie ctum <lb/>ex motu plene acqui&longs;ito, &longs;ummam adeptum e&longs;t velocita­
<pb pagenum="180"/>tem. Hin evidemus, vel pueros ip&longs;os, docente Natura <expan abbr="cū">cum</expan> <lb/>nuces, vel aliud quippiam, parieti alli&longs;um frangere <expan abbr="conã-tur">conan­<lb/>tur</expan>, à pariete moderato aliquo &longs;patio recedere. Si autem <lb/>eos interroges, curid faciant, re&longs;pondebunt, vtinde ictus <lb/>valentius fiat atque efficacius. Eleganter ex Simplicij & <lb/>Alexandri Aphrodi&longs;ien&longs;is doctrina, quæ lucidi&longs;&longs;ima e&longs;t, <lb/>quæ&longs;tionem hanc in &longs;ua Paraphra&longs;i explicat Picolomi­<lb/>neus. </s>
</p>
<p type="head">
<s>QVAESTIO XXXIII.</s>
</p>
<p type="head">
<s><emph type="italics"/>Dubitatur, Cur proiecta moueantur, licet impellens à proiectis &longs;e­<lb/>paretur; vel vt verbis Philo&longs;ophi vtar, Cur quippiam non pecu­<lb/>liarem &longs;ibi fertur lationem impul&longs;ore alioquin <lb/>non con&longs;equente?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Soluit, inquiens, an videlicet, quoniam primum, id e&longs;t, <lb/>impellens ip&longs;e, id efficit vt alterum, nempe proiectum <lb/>ip&longs;um impellat, illud vero (hoc e&longs;t proiectum) alterum <lb/>impellat, hoc e&longs;t, aërem ip&longs;um mediumue, quod à proie­<lb/>cto repelletur. Ce&longs;&longs;are autem motum, cum res eo deue­<lb/>nit, vt motus eidem à proijciente impre&longs;&longs;us, non po&longs;&longs;it <lb/>amplius rem proiectam mouere, & itidem rem ip&longs;am, aë­<lb/>rem videlicet non po&longs;&longs;it amplius repellere. Vel etiam <lb/>quando ip&longs;ius lati grauitas nutu &longs;uo declinat magis quam <lb/>impellentis in ante &longs;it potentia. Vtique res per &longs;e &longs;atis cla­<lb/>ra. etenim motus impre&longs;&longs;us a ccidentalis e&longs;t, quod vero la­<lb/>tioni violentæ re&longs;i&longs;tit principium, naturale, & ab ip&longs;o mo­<lb/>to in&longs;eparabile, vincente igitur quod natura e&longs;t, paulla­<lb/>tim remittitur quod ex accidenti e&longs;t, & indeproiecti fit <lb/>quies. E&longs;t autem & hoc quoque Problema pure phy&longs;icum, <lb/>& &longs;uperiori, de quo immediate egimus, perquam familia­<lb/>re, quamobrem ex ij&longs;dem pror&longs;us &longs;oluitur <lb/>principijs. </s>
</p>
<pb pagenum="181"/>
<p type="head">
<s>QVÆSTIO XXXIV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Cur neque parua multum, neque magna nimis longe proijci queunt, <lb/>&longs;ed proportionem quandam habere oportet proiecta ip&longs;a ad <lb/>eius vires qui proijcit?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Pvlchre dubitationem diluit, inquiens, An quia nece&longs;­<lb/>&longs;e e&longs;t quod proijcitur, & impellitur contraniti ei vnde <lb/>impellitur. Quod autem magnitudine &longs;ua nihil cedit, aut <lb/>imbecillitate nihil contranititur, non efficit <expan abbr="proiectionē">proiectionem</expan> <lb/>neque impul&longs;ionem. quod enim multo impellentis exce­<lb/>dit vires, haud quaquam cedit. Quod vero e&longs;t multo im­<lb/>becillius, nihil contranititur, & impre&longs;&longs;ionem non &longs;u&longs;ci­<lb/>pit. Aliam quoque adiungit rationem, videlicet, Tantum <lb/>ferriid quod fertur quantum aëris mouerit ad <expan abbr="profundū">profundum</expan> <lb/>(hoce&longs;t, ad eam partem aëris remotiorem, ad quam fer­<lb/>tur) etenim proiectum à principio dum fertur aërem pel­<lb/>lit, non pellit autem &longs;i nihil mouetur. Accidit igitur vt <lb/>concludit Philo&longs;ophus, proiectai&longs;thæc contrarijs ex cau­<lb/>&longs;is minus moueri. quod enim valde paruum e&longs;t nihil mo­<lb/>uet imbecillitate &longs;ua impediente. quod vero valde ma­<lb/>gnum e&longs;t, ex contraria cau&longs;&longs;a nihil mouet, nempe quod <lb/>ob magnitudinem &longs;uam nihil moueatur. Vnde fit pro­<lb/>portionem inter proiectum & proijcientem e&longs;&longs;e inprimis <lb/>ad motum, necei&longs;&longs;ariam. Hæc eadem præclare in &longs;ua Pa­<lb/>raphra&longs;i explicat Picolomineus. </s>
</p>
<p type="main">
<s>Huicnos, de proiectis quæ&longs;tioni, hæc addimus. </s>
</p>
<p type="main">
<s>Cur proiecta corpora non &longs;ibimet ip&longs;is &longs;ecun dum, <lb/>partes æ quegrauia, &longs;i fuerint irregularis figuræ in ip&longs;o mo­<lb/>tu, &longs;ecundum grauiorem partem antror&longs;us inuiolento, & <lb/>deor&longs;um in naturali ferantur, & dum in latione conuer­<lb/>tuntur, &longs;onitum edant. </s>
</p>
<p type="main">
<s>E&longs;to pila ABCD, cuius centrum E concinnata ex <lb/>di&longs;pari materia leui, nempe BCD, & graui ABD. non ergo
<pb pagenum="182"/>
<arrow.to.target n="fig46"></arrow.to.target><lb/>erit <expan abbr="centrū">centrum</expan> grauitatis & cen­<lb/>trum molis, &longs;it autem grauita­<lb/>tis centrum F. De&longs;cendat cor­<lb/>pus prohibente remoto per <lb/>rectam AG. Et quoniam gra­<lb/>uiora deor&longs;um tendunt ma­<lb/>gis, &longs;i à principio motus gra­<lb/>uior pars fuerit &longs;upra in ip&longs;o <lb/>de&longs;cen&longs;u conuertet ir pila, & <lb/>&longs;itum non &longs;eruabit donec &longs;u­<lb/>perior pars ea quæ grauior, <lb/>deor&longs;um fiat, vt videre e&longs;t in <lb/>pila HIK, cuius centrum e&longs;t G. pars grauior HIK. Si au­<lb/>tem eadem pila, laterali motu violenter feratur ver&longs;us <lb/>N, ad eam quoque partem conuertetur pars grauior. fa­<lb/>cto enim molis &longs;eu magnitudinis centro vbi L, grauior <lb/>pars fiet in MNO; quæ cunque igitur &longs;unt corporaita <expan abbr="cō-&longs;tituta">con­<lb/>&longs;tituta</expan>, vt in illis non &longs;it idem molis & grauitatis centrum <lb/>in ip&longs;a latione conuertentur, & corum pars grauior an­<lb/>tror&longs;us fiet. Sonitus porro in ip&longs;o motu editi ea e&longs;t cau&longs;&longs;a, <lb/>quod irregulare corpus à principio incipit conuerti, & in <lb/>ip&longs;a conuer&longs;ione dum fertur aërem verberat, & ab eodem <lb/>vici&longs;&longs;im reuerberatur, ex qua reuerberatione fit corporis <lb/>rotatio dum fertur, & ip&longs;e &longs;onitus, quem Græci <foreign lang="greek">roicon</foreign><lb/>Rhœzum appellant. </s>
</p>
<figure id="fig46"></figure>
<p type="main">
<s>Ad hanc quoque &longs;peculationem pertinet, Cur lapi­<lb/>des ad &longs;uperfi ciem aquæ proiecti non &longs;tatim demergan­<lb/>tur, &longs;ed aliquot vicibus a quæ &longs;uperficiem radentes, abea, <lb/>dem re&longs;iliant. </s>
</p>
<p type="main">
<s>E&longs;to aquæ &longs;uper&longs;icies AB, lapis proiectus C, tangens <lb/>aquæ &longs;uperficiem in D, & inde re&longs;iliens in E, mox iterum <lb/>eandem tangens in F, & re&longs;iliens in G, donec <expan abbr="violēto">violento</expan> mo­<lb/>tu ce&longs;&longs;ante demergatur. Vtique lapis C, proiectus in D,
<pb pagenum="183"/>
<arrow.to.target n="fig47"></arrow.to.target><lb/>ni&longs;i medio den&longs;iori, aqua vi­<lb/>delicet, repelleretur, pene­<lb/>traret per D, in H. At eo re&longs;i­<lb/>&longs;tente, & adhuc vigente im­<lb/>petu, fertur in E ad angulos <lb/>fere pares. Dico autem fere, <lb/>&longs;iquidem maior e&longs;t ADC ip&longs;o EDF, propterea quod vis <lb/>non &longs;it eadem, &longs;ed minor ea quæ ex D pellit in E. Durante <lb/>igitur impetu quo pellitur antror&longs;um, fiuntip&longs;æ re&longs;ilitio­<lb/>nes, & eo ce&longs;&longs;ante, re&longs;ilitiones ce&longs;&longs;ant, & lapis &longs;uapte gra­<lb/>uitate demergitur. </s>
</p>
<figure id="fig47"></figure>
<p type="main">
<s>Huc quoque &longs;pectat, Cur pila lu&longs;oria in horizontis <lb/>planum proiecta ad pares re&longs;iliat, angulos nempe rectos? </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to horizontis planum <lb/>AB, in quod à puncto C per <lb/>lineam perpendicularem CE <lb/>cadat proijciaturue pila DE, <lb/>cuius grauitatis centrum F. <lb/>Tangit autem planum in <expan abbr="pū-cto">pun­<lb/>cto</expan> E. Perpendicularis ergo <lb/>EC, circulum DE per <expan abbr="centrū">centrum</expan> <lb/>&longs;ecat, hoc e&longs;t, in partes æ qua­<lb/>les & æqueponderantes, &longs;ed <lb/>dum pila cadit proijciturue, <lb/>agitin planum horizontis, vbi E, & in eodem puncto re. <lb/>petitur, quare cum cadens & agens diuidatur in partes æ­<lb/>quales & æqueponderantes & item repatiens & re&longs;iliens <lb/>diuidatur item in partes æquales & æquepondetantes, ita <lb/>re&longs;ilit repatiendo, vti egerat in cadendo, hoc e&longs;t; ad angu­<lb/>los pares; quod fuerat demon&longs;trandum. Modo &longs;it <expan abbr="planū">planum</expan> <lb/>aliquod ita ad horizontem inclinatum, vt GH, & in illud <lb/>cadat proijciaturue eadem pila. Dico eam ab eodem in­<lb/>clinato plano ad pares angulos re&longs;ilire non tamen rectos.
<pb pagenum="184"/>Vtique pila cadens, planum non tanget in E. e&longs;&longs;et enim <lb/>GH, vbi AB, Tangat autem in I, & à centro F ad contin­<lb/>gentiæ punctum I, recta ducatur FI. Erit igitur FI (prop. <lb/>18. lib. 3. elem.) ip&longs;i GH plano perpendicularis. Ducatur <lb/>item peri, ip&longs;i EC, parallela IK, &longs;ecans pilæ circumferen­<lb/>tiam in K. Agit ergo & repatitur pila in puncto Inon æ. <lb/>qualiter inæquales. etenim &longs;unt partes KDLEI, & IK, eo <lb/>quod IK &longs;ecet circulum non per centrum. repellitur ergo <lb/>in repatiendo non æ qualiter, &longs;ed iuxta inæqualitatem ea­<lb/>rundem partium. Ducatur autem recta in circulo LI æ­<lb/>qualis ip&longs;i IK. Eritigitur LEI, æ qualis IK, & tota KDLI æ­<lb/>qualis toti IKDL. Vtigitur actio e&longs;t per de&longs;cen&longs;um iuxta <lb/>rectam KI, ita e&longs;t repa&longs;&longs;io per a&longs;cen&longs;um ex IL. Dico autem <lb/>angulos KIH, LIG e&longs;&longs;e æquales & &longs;ingulos recto minores. <lb/>Connectantur FL, FK. Quoniam igitur IK portio æqualis <lb/>e&longs;t portioni IEL, & recta LI æqualis rectæ IK, & LF æqua­<lb/>lis ip&longs;i FK, & FI communis, triangulum LFI, æquale e&longs;t <lb/>triangulo IFK. Quare & angulus FIL aequalis angulo FIK, <lb/>&longs;ed GIF, HIF recti &longs;unt, ergo re&longs;idui LIG, KIH æquales <lb/>&longs;untinter &longs;e comparati, & recto minores; quod fuerat o­<lb/>&longs;tendendum. </s>
</p>
<p type="main">
<s>Hinc colligimus, quo magis planum ab æquidi&longs;tan­<lb/>tia horizontis rece&longs;&longs;erit, eo pilam in eo proiectam in par­<lb/>tes in æqualiores diuidi & ad minores ip&longs;i plano angulos <lb/>re&longs;ilire. Nihil autem refert, vtrum planum, in quod pila <lb/>cadit, ad horizontem &longs;it inclinatum, vel eodem horizonti <lb/>æquedi&longs;tante pila non ad perpendiculas, &longs;ed iuxta <expan abbr="aliquē">aliquem</expan> <lb/>angulum in illud proijciatur. Hæc &longs;ane ita ex demon&longs;tra­<lb/>tione fieri o &longs;tenduntur. Veruntamen quoniam proiecta <lb/>pila materialis e&longs;t, & ideonecæqualis, nec æqueponde­<lb/>rans & &longs;ua grauitate re&longs;i&longs;tens, non ad pares ex amu&longs;&longs;i re&longs;i­<lb/>lit angulos, &longs;ed minores aliquantulum in re&longs;ilitione, re. <lb/>mittente nimirum vi in ip&longs;a reactione. Et &longs;ane fierinon
<pb pagenum="185"/>pote&longs;t, pilam à plano re&longs;ilientem eo peruenire vnde à <lb/>principio di&longs;ce&longs;&longs;erat; Id enim &longs;i daretur, æterna quoque <lb/>pilæ ip&longs;ius daretur re&longs;ilitio, & paullatim vi & impetu re­<lb/>mittente per parua interualla motus e&longs;&longs;et, donecres quæ <lb/>mouebatur, omnino quie&longs;cat. </s>
</p>
<p type="head">
<s>QVÆSTIO XXXV.</s>
</p>
<p type="head">
<s><emph type="italics"/>Quærit hoc vltimo Problemate Ari&longs;toteles, Cur eaquæin vorti­<lb/>co&longs;is feruntur aquis, ad medium tandem agan­<lb/>tur omnia?<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Tribus rationibus &longs;oluit; quarum prima e&longs;t: Quicquid <lb/>fertur, magnitudinem habet, cuius extrema in duo­<lb/>bus &longs;unt circulis, hoc in minori, illud in maiori. Et quo­<lb/>niam maior velocior e&longs;t, magnitudo media, non æquali­<lb/>ter fertur, &longs;ed à maiori quidem pellitur, à minori vero re­<lb/>trahitur, vnde transuer&longs;us fit magnitudinis motus, & ip&longs;a <lb/>magnitudo ad interiorem propellitur circulum, itaque <lb/>eodem pacto, è maiori in minorem propul&longs;a in centrum. <lb/>tantum fertur, & ibi quie&longs;cit. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to vortex AB, cuius cen­<lb/>trum C, magnitudo quæ fer­<lb/>tur AD, maior circulus AFB, <lb/>minor DHEG. Velocitas igi­<lb/>tur in A maior e&longs;t velocitate <lb/>quæ in D, magnitudinis ergo <lb/>extremum A, velocius rapitur <lb/>in A quam eiu&longs;dem extremum <lb/>inferius D, in D. Velocitas igi­<lb/>tur maioris circuli pellit Aver­<lb/>&longs;us F. tarditas vero minoris cir­<lb/>culi D retrahitad partes G. conuertitur itaque magnitu­<lb/>do inter pellentem & retrahentem circulum, donec ex­
<pb pagenum="186"/>tremitas A in circulo minori fuerit vbi H, D vero vbi I, & <lb/>ita deinceps eadem ratione vbi KL, donec paullatim fe­<lb/>raturin centrum C, facto nempe à maiori in minorem cir­<lb/>culum tran&longs;itu. </s>
</p>
<p type="main">
<s>Secunda ratio ita habet, quia quod fertur, &longs;imili &longs;e <lb/>habet modo ad omnes circulos propter centrum, hoc e&longs;t, <lb/>in quouis circulo, qui circa idem centrum fertur. Omnes <lb/>autem circuli mouentur, centrum vero &longs;tat, nece&longs;&longs;e e&longs;t à <lb/>motu tandem id quod mouetur ad quietis locum, hoc e&longs;t, <lb/>in centrum ip&longs;um peruenire. </s>
</p>
<p type="main">
<s>Tertia, quoniam circulorum, qui in vorticibus fiunt, <lb/>velocitas, & ideo impetus non e&longs;t æqualis, &longs;ed &longs;emper ex­<lb/>terior e&longs;t interiore velocior & violentior, Æqualis autem <lb/>&longs;emperin mota magnitudine, grauitas, diuer&longs;imode &longs;e <lb/>habet ad circulos, à quibus mouetur, & ideo modo vin­<lb/>citur, modovincit: vincitur autem à velocioribus circulis, <lb/>vincit autem tardiores. Ita que quoniam &longs;ua grauitatere­<lb/>&longs;i&longs;tens, maioris circuli motum pror&longs;us non &longs;equitur, ad <lb/>tardiorem reijcitur, hoc e&longs;t, interiorem, & &longs;ic deinceps, <lb/>donec tandem centrum ip&longs;umnanci&longs;catur, in quo nec &longs;u­<lb/>perans, nec &longs;uperata quie&longs;cit. </s>
</p>
<p type="main">
<s>Hæ &longs;unt rationes, licet ob&longs;curi&longs;&longs;ime propo&longs;itæ, qui­<lb/>bus, vt diximus, vtitur Ari&longs;toteles. acutæ &longs;ane illæ <expan abbr="quidē">quidem</expan>, <lb/>attamen haudqua quam vltro admittendæ. </s>
</p>
<p type="main">
<s>Primo enim fal&longs;um videtur, quod a&longs;&longs;erit, vortices <lb/>circulos e&longs;&longs;e, & circaidem centrum fieri atque rotari. Spi­<lb/>ræ enim potius &longs;unt, quæ ab exteriori parte <expan abbr="remotioreq;">remotioreque</expan> <lb/>incipientes &longs;piraliter circumuolutæ, ad intimam tandem <lb/>partem, quæ media e&longs;t & centri vices gerit, deueniunt. <lb/>qua veritate cognita, omnis pror&longs;us difficultas tollitur, <lb/>Cum enim ea quæ feruntur, ab aqua ferantur, aqua vero <lb/>feratur &longs;piraliter, ea quoque &longs;piraliter ferri, e&longs;t nece&longs;&longs;a-
<pb pagenum="187"/>rium. Hæc autem clariora erunt &longs;i quo pacto vortices <lb/>fiant, qui&longs;piam con&longs;iderauerit. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to fluminis cuiu&longs;piam curua <lb/>eademque profunda ripa ABCD. <lb/>Aquæ vero moles rapida EFDC, <lb/>quæ quidem co quod magno impe­<lb/>tu deferatur in C, ripæ ip&longs;ius <expan abbr="naturã">naturam</expan> <lb/>&longs;equens turbinatim circum uoluitur, <lb/>egre&longs;&longs;a autem extra locum &longs;euripam <lb/>B rotationis principium &longs;ecundans, <lb/>in &longs;eip&longs;am &longs;piraliter contorquetur, <lb/>& vorticem efficit GHFIK, cuius <lb/>quidem centrum e&longs;t vbi K. </s>
</p>
<p type="main">
<s>Alia quoque de cau&longs;&longs;a, ex quie&longs;cente nimirum, & <lb/>mota aqua fiunt &longs;piræ vorticesue. E&longs;to enim fluminis ripa <lb/>
<arrow.to.target n="fig48"></arrow.to.target><lb/>ABC, &longs;inum efficiens, qui a quam ex <lb/>ripæ ip&longs;ius obiectu contineat quie­<lb/>&longs;centem, Cur&longs;us vero fluminis liber & <lb/>rectus, &longs;it inter lineas AC, DE. Itaque <lb/>dum aqua AC rapide fertur ad partes <lb/>A, quie&longs;centem ABC iuxta lineam. <lb/>CA lateraliter propellit, & cius qui­<lb/>dem partem quam tangit, &longs;ecum ra­<lb/>pit, puta ex F in G. Delata igitur aqua <lb/>& currente ex F ver&longs;us G quie&longs;cens <lb/>lateraliter eidem &longs;e&longs;e aliqualiter op­<lb/>ponit, & currentem repellit ex Gin H. Cœpto <expan abbr="itaq;">itaque</expan> &longs;pirali <lb/>motu aqua circumuoluitur &longs;ecun dum lineam GHK, do­<lb/>necperueniatad centrum I, vbi circumuolutæ aquæ par­<lb/>tes &longs;e&longs;e inuicem tangunt. Porro vortices i&longs;ti &longs;piræue, quod <lb/>nos per Padum, Abduam, & magna flumina nauigantes <lb/>ob&longs;eruauimus, non eodem permanent loco, &longs;ed rapientis <lb/>aquæ motum &longs;ecundantes, paullatim in currentem <expan abbr="aquã">aquam</expan>
<pb pagenum="188"/>delati euane&longs;cunt, fiunt etiam eiu&longs;cemodi vortices nau­<lb/>tis quidem valde formidabiles etiam in mari, de quibus <lb/>Poëta libro Æneidos primo. </s>
</p>
<figure id="fig48"></figure>
<p type="main">
<s>— <emph type="italics"/>a&longs;t illam ter fluctus ibidem <lb/>Torquet agens circum, & rapidus vorat æquore vortex.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Sed & idem quoque de vorticibus, qui in fluminibus <lb/>fiunt libro 7. </s>
</p>
<p type="main">
<s>— <emph type="italics"/>hunc inter fluuio Tiberinus amœno <lb/>Vorticibus rapidis, & multa flauus arena <lb/>In mare prorumpit.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Fiunt autem in mari partim occultis de cau&longs;&longs;is, partim <lb/>etiam ex violentia aquarum &longs;ibi inuicem obuiantium a­<lb/>gitatione. Sed nos hi&longs;ce explicatis commode ad ea quæ <lb/>dixerat Ari&longs;toteles, reuertemur. </s>
</p>
<p type="main">
<s>Dicimus igitur, priman eius rationem haud magni <lb/>videri ponderis, &longs;iquidem non per circulos actu di&longs;tinctos <lb/>aqua circumfertur, &longs;ed ip&longs;amet &longs;ua mole tota &longs;imul. </s>
</p>
<figure></figure>
<p type="main">
<s>E&longs;to enim vortex AB, cu­<lb/>ius centrum C, &longs;emidiameter <lb/>CA, fiatautem rotatio totius a­<lb/>quæ CA ad partes D, in linea <lb/>autem AC, &longs;it corpus aliquod a­<lb/>quæ rotatione <expan abbr="circumlatū">circumlatum</expan> AE, <lb/>inter circulos maiorem ADB, <lb/>minorem EFG. velocius autem <lb/>mouetur ADB, ip&longs;o EFG, citius <lb/>ergo fertur pars &longs;uperior ip&longs;ius <lb/>corporis vbi A, quam inferior <lb/>vbi E. Atidnec A repellit, nec E retrahit, &longs;iquidem eodem <lb/>tempore quo A permeauit <expan abbr="circulū">circulum</expan> ADB, codem & E per­<lb/>currit circulum EFG. <expan abbr="Itaq;">Itaque</expan> A reuer&longs;o in A & E, punctum <lb/>reuer&longs;um erit in E, nulla facta corporis E quoad &longs;itum, <lb/>muratione quod voluit Ari&longs;toteles. </s>
</p>
<pb pagenum="189"/>
<p type="main">
<s>Ad &longs;ecundam vero dicimus, non ideo quod omnes <lb/>circuli æqualiter circa centrum &longs;erantur, ni&longs;i alia <expan abbr="quæpiã">quæpiam</expan> <lb/>extranea vis interce&longs;&longs;erit, quæ ea ab exterioribus circulis <lb/>pellens agat in medium. </s>
</p>
<figure></figure>
<p type="main">
<s>Tertia quoque ratio la­<lb/>borare videtur. </s>
</p>
<p type="main">
<s>E&longs;to enim vortex AB, <lb/>cuius centrum C, &longs;it autem <lb/>corpus aliquod E, cuius na­<lb/>tura apta &longs;it totationi aliqua­<lb/>tenus re&longs;i&longs;tere. Quoniam i­<lb/>gitur eius re&longs;i&longs;tentia <expan abbr="aliquã-tulum">aliquan­<lb/>tulum</expan> ab aqua rapiente &longs;u­<lb/>peratur in ip&longs;a rotatione, par­<lb/>tim aquae impetum &longs;equetur, <lb/>partim &longs;uapte natura retardabitur. Quamobrem aqua <lb/>quæ e&longs;t in A, translata in H, corpus ip&longs;um non erit in H, <lb/>&longs;ed in G. Tardius igitur corpus quam aqua ip&longs;a, rotatio­<lb/>nem complebit, non tamen propterea, ni&longs;i alia quæ piam <lb/>ad&longs;it cau&longs;&longs;a, feretur in medium. </s>
</p>
<p type="main">
<s>Cæterum horum vorticum effectum & cau&longs;&longs;am ob­<lb/>&longs;eruare licet, &longs;i va&longs;e quopiam aqua pleno aquam ip&longs;am <lb/>baculo manuue circulariter agitauerimus, fiet enim vor­<lb/>tex, & &longs;i quippiam quod leue &longs;it, in aquam motam proie­<lb/>cerimus, ea quam diximus de cau&longs;&longs;a in motum ip&longs;um, hoc <lb/>e&longs;t, vorticis &longs;piræue, centrum feretur. </s>
</p>
<p type="main">
<s>Hæc nos, vt vera proponimus, & forta&longs;&longs;e decipimur. <lb/>Certe Philo&longs;opho tantæ auctoritatis contradicere, ma­<lb/>gnæ videtur audaciæ, aut potius in&longs;aniæ. Quicquid ta­<lb/>men &longs;it, pro pulcherrima veritate labora&longs;&longs;e, à parte <lb/>aliqua laudis non fuerit pror&longs;us, vt <lb/>arbitror, alienum. </s>
</p>
<pb pagenum="190"/>
<p type="head">
<s>APPENDIX.</s>
</p>
<p type="main">
<s>Modum inueniendarum duarum mediarum propor­<lb/>tionalium non tantum vtilem e&longs;&longs;e, &longs;ed pror&longs;us nece&longs;­<lb/>&longs;arium, illi norunt, qui in Mechanicis di&longs;ciplinis vel <expan abbr="parū">parum</expan> <lb/>fuerint ver&longs;ati. Nulla enim alia ratio e&longs;t, qua corporeae ma­<lb/>gnitudines &longs;eruata figura & &longs;imilitudine augeri propor­<lb/>tionaliter imminuiue po&longs;&longs;int. Quamobrem factum e&longs;t vt <lb/>in his inueniendis tum vetu&longs;ti&longs;&longs;imo tum etiam in feriori æ­<lb/>uo, clari&longs;&longs;imi Viri magnopere laborauerint. Plato etenim, <lb/>Eudoxus (cuius modum repudiauit Eutocius) Heron A­<lb/>lexandrinus, Philon Byzantius, Apollonius, clari&longs;&longs;imi <lb/>Geometræ, Diocles, Pappus, Sporus, Menæchmus, Ar­<lb/>chytas Tarentinus, Platoni æqualls: Erato&longs;thenes, & Ni­<lb/>comedes ad has inueniendas varias rationes <expan abbr="excogitarūt">excogitarunt</expan>, <lb/>quorum omnium modos, & in&longs;trumenta, <expan abbr="demon&longs;tratio-ne&longs;q;">demon&longs;tratio­<lb/>ne&longs;que</expan> diligenti&longs;&longs;ime collegit, & in illos <expan abbr="Cōmentarios">Commentarios</expan> con­<lb/>iecit idemmet Eutocius, quos eleganti&longs;&longs;imos in Archime­<lb/>dis libros de Sphæra & Cylin dro &longs;crip&longs;it. Nos autem ijs o­<lb/>mnibus accurate per&longs;pectis, & diligenti&longs;&longs;ime ponderatis, <lb/>inuenimus eos fere omnes tentando negotium ab&longs;olue­<lb/>re, quod &longs;ane laborio&longs;um valde e&longs;t & operantibus permo­<lb/>le&longs;tum. Itaque cum modum praximue inueni&longs;&longs;emus, ex <lb/>qua is qui operatur tuti&longs;&longs;ime & facillime ad quæ &longs;itas ip&longs;as <lb/>medias manu ducitur, hunc pulcherrimæ huius facultatis <lb/>&longs;tudio &longs;is inuidere nefarium iudicaurmus. Quod &longs;i <expan abbr="qui&longs;piã">qui&longs;piam</expan> <lb/>dixerit, Balli&longs;tarum, Catapultarum, Scorpionum, & cæ­<lb/>terarum eiu&longs;cemodi Machinarum v&longs;um, olim apud nos <lb/>de&longs;ij&longs;&longs;e, & ideo Problema hoc videri &longs;uperuacaneum, Re­<lb/>&longs;pondemus, nulla alia ratione æneorum tormentorum pi­<lb/>las augeri imminuiue &longs;eruata ponderis ratione po&longs;&longs;e, in­<lb/>numeraque e&longs;&longs;e, quæ vt rite perficiantur, hæc penitus in­<lb/>digent &longs;peculatione. Nos rem Mechanicis vtilem, Me.
<pb pagenum="191"/>chanicis no&longs;tris Exercitationibus annectere, haud im­<lb/>portunum iudicauimus. Sed tempus e&longs;t, vt his breuiter <lb/>præfatis, ad rem ip&longs;am <expan abbr="explicandã">explicandam</expan> commode accedamus. </s>
</p>
<p type="head">
<s><emph type="italics"/>Datis duabus proportionalibus prima, & quarta duas inter eas <lb/>medias in continua proportione inuenire.<emph.end type="italics"/></s>
</p>
<p type="main">
<s>Esto prima datarum AB, quarta BC, inter quas <expan abbr="&longs;ecundã">&longs;ecundam</expan> <lb/>& tertiam oportetinuenire. Ducatur recta DE, cui à <lb/>puncto F, vtcunque &longs;umpto, perpendicularis demittatur <lb/>FG, Tum ab F ver&longs;us D duplicetur quarta BC, &longs;itque FH, <lb/>deinde ab H ip&longs;i FG parallela demittatur HI, & ab HF <lb/>ab&longs;cindatur HK, ip&longs;ius BC quartæ medietati æqualis. <lb/>Po&longs;thæc puncto K &longs;patio autem medietati, primæ data­<lb/>rum æquali, in linea HI notetur punctum L, & ip&longs;i HL <lb/>fiat æqualis FM, & KM iungatur. His ita con&longs;titutis pare­<lb/>tur &longs;eor&longs;um &longs;cheda regulaue quæpiam NO, in cuius late­<lb/>re accipiatur OP, æqualis medietati primæ datarum &longs;eu <lb/>ip&longs;i KL. Tum regulæ latus aptetur puncto L, extremum <lb/>vero O, feratur a&longs;&longs;idue per rectam EK, ver&longs;us K, nunquam <lb/>
<arrow.to.target n="fig49"></arrow.to.target>
<pb pagenum="192"/>interim regulæ latere ON amoto à puncto L, idque do­<lb/>nec punctum P, obuians incidat in lineam KM, puta vbi <lb/>Qextremum vero O inueniatur in R, notato igitur in li­<lb/>nea EK puncto R habebitur, quod quærebatur. Erunti­<lb/>gitur AB prima, RK &longs;ecunda, QL tertia, BC quarta. </s>
</p>
<figure id="fig49"></figure>
<p type="main">
<s>Hæc praxis ij&longs;dem prin cipijs demon&longs;tratur, quibus <lb/>&longs;uam ex Conchoide o&longs;tendit Nicomedes. Conficit ille <lb/>in&longs;trumentum, ex quo de&longs;cribit <expan abbr="Conchoidē">Conchoidem</expan>, ex qua po&longs;t­<lb/>ea duas medias venatur. Nos autem nec in&longs;trumentum <lb/>con&longs;truimus nec Conchoidem de&longs;cribimus, & duabus fe­<lb/>re lineis rem ab&longs;oluimus, vt nemo fere non dixerit, hoci­<lb/>&longs;tud quod docemus, à Nicomedea praxi e&longs;&longs;e pror&longs;us a­<lb/>lienum. </s>
</p>
<p type="main">
<s>Sed nos, vt eius, quam o&longs;tendimus, operationis de­<lb/>mon&longs;tratio habeatur; ip&longs;ius Nicomedis ex Pappi libro 3. <lb/>propo&longs;. 5. de&longs;umptam in medio afferemus, quippe quod <lb/>i&longs;thæc ea quam in &longs;uis in Archimedem commentarijs re­<lb/>fert Eutocius, &longs;it lucidior. </s>
</p>
<p type="main">
<s>Datis duabus rectis lineis CD, DA; duæ mediæ in <lb/>continua proportione hoc modo a&longs;&longs;umuntur. </s>
</p>
<p type="main">
<s>Compleatur ABCD parallelogrammum, & <expan abbr="vtraq;">vtraque</expan> <lb/>ip&longs;arum AB, BC, bifariam &longs;ecetur in punctis L, E, iuncta­<lb/>que LD producatur; & occurrat productæ CB, in G, ip&longs;i <lb/>vero BC ad rectos angulos ducatur EF, & CF iungatur, <lb/>quæ &longs;it æqualis AL. Iungatur præterea FG & ip&longs;i paralle­<lb/>la &longs;it CH, eritque angulus KCH, æqualis angulo CGF. <lb/>Tum à dato puncto F ducatur FHK, quae faciat KH æqua­<lb/>lem ip&longs;i AL vel CF. Hoc enim per lineam Conchoidem <lb/>fieri po&longs;&longs;e o&longs;tendit Nicomedes, & iuncta KD producatur, <lb/>occurratque ip&longs;i BA, productæ in puncto M. Dico vt DC <lb/>ad CK ita CK ad MA & MA ad AD. Quoniam enim BC <lb/>bifariam &longs;ecta e&longs;t in E, & ip&longs;i adijcitur CK. Rectangulum <lb/>BKC per 6. &longs;ecundi: vna cum quadrato ex CE, æquale e&longs;t
<pb pagenum="193"/>
<arrow.to.target n="fig50"></arrow.to.target><lb/>quadrato ex EK. commune apponatur ex EF quadratum, <lb/>ergo rectangulum BKC vna cum quadrato CF æquale <lb/>e&longs;t quadratis ex KE, EF, hoce&longs;t, quadrato ex FK. Et quo­<lb/>niam vt MA ad AB, ita e&longs;t MD ad DK, vt autem MD ad <lb/>DK per 2. &longs;exti, ita BC ad C<emph type="italics"/>K<emph.end type="italics"/> erit vt MA ad AB, ita BC <lb/>ad C<emph type="italics"/>K<emph.end type="italics"/>. Atque e&longs;t ip&longs;ius AB dimidi<gap/> AL, & ip&longs;ius BC, du­<lb/>pla CG, e&longs;t igitur vt MA ad AL, ita GC ad C<emph type="italics"/>K<emph.end type="italics"/>. Sed vt GC <lb/>ad C<emph type="italics"/>K<emph.end type="italics"/>, ita FH ad H<emph type="italics"/>K<emph.end type="italics"/> propter lineas parallelas GF, CH. <lb/>quare & componendo vt ML, ad LA, ita F<emph type="italics"/>K<emph.end type="italics"/> ad <emph type="italics"/>K<emph.end type="italics"/>H, &longs;ed <lb/>AL ponitur æqualis H<emph type="italics"/>K<emph.end type="italics"/>, quoniam & ip&longs;i CF, ergo & ML <lb/>per 9. lib. 5. æqualis erit F<emph type="italics"/>K<emph.end type="italics"/>, & quadratum ex ML, æquale <lb/>quadrato ex F<emph type="italics"/>K<emph.end type="italics"/>. e&longs;t autem quadrato ex ML, æquale re­<lb/>ctangulum BMA vna cum quadrato ex AL & quadrato <lb/>ex Fk æquale o&longs;ten&longs;um e&longs;t rectangulum BkC vna cum.
<pb pagenum="194"/>quadrato ex CF, quorum quidem quadratum ex AL æ­<lb/>quale e&longs;t quadrato ex CF, ponitur enim AL, ip&longs;i CF æ­<lb/>qualis, ergo reliquum BMA rectangulum æquale e&longs;t reli­<lb/>quo BkC. Vtigitur MB ad Bk, ita Ck ad MA. Sed vt MD <lb/>ad Bk, ita DC ad Ck. quare vt DC ad Ck, ita e&longs;t Ck ad <lb/>MA. vt autem MD ad Bk, ita MA, ad AD. Ergo vt DC, <lb/>prima, ad Ck &longs;ecundam, ita Ck &longs;ecunda ad MA tertiam, <lb/>& MA tertia ad AD quartam, quod fuerat demon&longs;tran­<lb/>dum. Hæc Pappus. Quod autem in no&longs;tra Praxi diximus, <lb/>QL e&longs;&longs;etertiam, earatio e&longs;t, quod LR vt in prima figura <lb/>e&longs;t, &longs;it æqualis ip&longs;i LM &longs;ecundæ figuræ, in demon&longs;tratio­<lb/>ne Pappi, ex quibus demptis QR & LA, quæ &longs;unt æqua­<lb/>les, reliqua QL primæ figuræ æqualis e&longs;t AM &longs;ecundæ fi­<lb/>guræ, hoc e&longs;t, ip&longs;i tertiæ proportionali: E&longs;t igitur, vt in pri­<lb/>ma figura dicebamus, AB prima, kR &longs;ecunda, QL tertia, <lb/>BC quarta. </s>
</p>
<figure id="fig50"></figure>
<p type="main">
<s>Vides igitur tu quilegis, nos ex Nicomedis demon­<lb/>&longs;tratione (quatenus ad praxin pertinet) &longs;uperflua re&longs;eca&longs;­<lb/>&longs;e, & ab&longs;que Conchoidis in&longs;trumento lineaue rem ip&longs;am <lb/>confeci&longs;&longs;e, idque non rentantes, vtalij, &longs;ed progre­<lb/>dientes, & qua&longs;i manuductos quæ&longs;i­<lb/>tum inue&longs;tiga&longs;&longs;e. </s>
</p>
<p type="head">
<s>FINIS.<lb/> </s>
</p>
</chap>
</body>
<back></back>
</text>
</archimedes>