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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Baldi, Bernardino</author> <title>In mechanica Aristotelis problemata exercitationes, old version (214 pages)</title> <date>1621</date> <place>Mainz</place> <translator></translator> <lang>la</lang> <cvs_file>baldi_mecha_007_la_1621.xml</cvs_file> <cvs_version></cvs_version> <locator>007.xml</locator> <echodir>/permanent/archimedes/baldi_mecha_007_la_1621</echodir> </info> <text> <front> <section> <pb xlink:href="007/01/001.jpg"></pb> <p type="head"> <s id="s.000001">BERNARDINI BALDI VRBINATIS <lb></lb>GVASTALLÆ AB<lb></lb>BATIS <lb></lb><emph type="italics"></emph>IN<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.000002">MECHANICA ARISTOTE<lb></lb>LIS PROBLEMATA <lb></lb>EXERCITATIONES:</s> </p> <p type="head"> <s id="s.000003"><emph type="italics"></emph>ADIECTA SVCCINCTA NAR<lb></lb>ratione de autoris vita & ſcriptis.<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.000004"><emph type="italics"></emph>MOGVNTIAE.<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.000005">Typis & Sumptibus Viduæ Ioannis Albini.</s> </p> <p type="head"> <s id="s.000006"><lb></lb>M. D C. XXI.</s> </p> <pb xlink:href="007/01/002.jpg"></pb> </section> <section> <pb xlink:href="007/01/003.jpg"></pb> <p type="head"> <s id="s.000007"><emph type="italics"></emph>NOBILISSIMO AC GENE<lb></lb>ROSO DOMINO<emph.end type="italics"></emph.end> D. ADAMO PHILIP<lb></lb>PO BARONI A CRON<lb></lb>BERG, EQVITI, SACRÆ CÆSA<lb></lb>REÆ MAIESTATIS, ET SERENISSIMI Principis Archiducis Alberti Camerario intimo & c. <lb></lb></s> <s id="s.000008">Domino meo gratioſiſſimo.</s> </p> <p type="main"> <s id="s.000009">Opportune ſub hoc ipſum tem<lb></lb>pus, quo in Belgium ad Sere<lb></lb>niſſimos Principes iter ador<lb></lb>nat.</s> <s id="s.000010"> Nobiliſſima & Generoſa <lb></lb>Dom. V.^{ra}, prodit noſtris for<lb></lb>mis in publicum editus Com<lb></lb>mentarius Bernardini Baldi Vrbinatis Gua<lb></lb>ſtallæ Abbatis in Ariſtotelis Mechanica. </s> <s id="s.000011">Is <lb></lb>vir in omni ſcientiæ genere, at maxime in Ma<lb></lb>thematicis diſciplinis fuit verſatiſſimus, quod <lb></lb>multa ab eo præclare ſcripta teſtantur opera, <lb></lb>ex quibus paucula edita, reliqua vero ſpera<pb xlink:href="007/01/004.jpg"></pb>mus ſuo tempore in publicam lucem produ<lb></lb>cenda. </s> <s id="s.000012"> Cum vero nemini ſit obſcurum Nobi<lb></lb>liſſimæ ac Generoſæ Dom. V.^{ræ} id ſemper <lb></lb>extitiſſe familiariſſimum, vt tum domeſticum <lb></lb>otium, tum maxime peregrinationes, quibus <lb></lb>totam pæne Europam ſumma cum laude <lb></lb>circumſcripſit, tum variarum linguarum per<lb></lb>fecto vſu, tum Mathematicarum diſciplina<lb></lb>rum notitia & exercitio redderet <expan abbr="iucũdiores">iucundiores</expan>, <lb></lb>nulla me tenet dubitatio quin & Baldum Vr<lb></lb>binatem noſtris typis loquentem in hoc iti<lb></lb>nere, quod à Deo feliciſſimum Nobiliſſimæ <lb></lb>ac Generoſæ Dom. V.^{ræ} precor, in ſuum comi<lb></lb>tatum ac tutelam beneuolo animo ſit admiſ<lb></lb>ſura.</s> <s id="s.000013"> Id rogo humillime ſimulque precor, vt. <lb></lb></s> <s id="s.000014">hanc meam typographiam plurimis iam re<lb></lb>tro annis de inclytæ familiæ Cronbergicæ tu<lb></lb>tela gloriantem, ſuo fauore proſequatur, vi<lb></lb>duæque afflictæ fortunis beneuole adſpiret.</s> <s id="s.000015"> <lb></lb>Sic Deus Nobiliſſ. & Generoſam Dom. V.^{ram} <lb></lb>illuſtret omnibus bonis, eamque R.^{mo} & Ill.^{mo} <lb></lb>Principi ac Domino meo Clementiſſimo, D. <lb></lb>Ioanni Suicardo Archiepiſcopo Mogunti<lb></lb>no Principi Electori ac per <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ermaniam Ar-<pb xlink:href="007/01/005.jpg"></pb>chicancellario &c. </s> <s id="s.000016">patruo ſuo optatiſſimo <lb></lb>ſaluo florentique redhibeat ſaluum ſimili<lb></lb>ter florentem ac incolumem.</s> <s id="s.000017"> Moguntiæ è <lb></lb>typographeio Viduæ Albinianæ, honori No<lb></lb>biliſſimæ ac <emph type="italics"></emph>G<emph.end type="italics"></emph.end>eneroſæ Dom. Veſtræ perpe<lb></lb>tuum dicato. </s> <s id="s.000018">Anno 1621.26.Martij. </s> </p> <pb xlink:href="007/01/006.jpg"></pb> </section> <section> <p type="head"> <s id="s.000019">PRAEFATIO.</s> </p> <p type="main"> <s id="s.000020"><emph type="italics"></emph>Diligenter legenti mihi quæſtiones il<lb></lb>las, in quibus ea quæ ad Mecha<lb></lb>nicam facultatem pertinent, expli<lb></lb>cantur, multa in mentem venie<lb></lb>bant; & primum quidem eorum, quæ ibi dispu<lb></lb>tantur, vtilitatem, ſubtilitatem, copiam admi<lb></lb>rabar: Tum ex animo dolebam, aureum hunc li<lb></lb>bellum propè negligi, & ab iis qui pulcherrimis <lb></lb>hiſce ſtudiis dant operam, assiduè præ manibus <lb></lb>non haberi: Multas autem Auctori ipſi haben<lb></lb>das referendasque eſſe gratias, qui tam egregiam, <lb></lb>vtilem & probè inſtructam ſupellectilem Archi<lb></lb>tectis, Mechanicis, & omnibus ferè Artificibus <lb></lb>ſuppeditauerit. </s> <s id="s.000021">Ariſtotelis nomini aſcribitur <lb></lb>Commentarius, licet nonnulli, ſitne Philoſophi <lb></lb>illius præclarissimi & acutissimi labor, an non, <lb></lb>adfirmare ſubdubitauerint. </s> <s id="s.000022">Ariſtotelis tamen <lb></lb>eſſe omnes ferè meliores conſentiunt: Idque tum <lb></lb>ex phraſi, & explicatione, quæ Ariſtotelem ſa<lb></lb>piunt, tum iudicio ſubtilitatis & rationum, qui-<pb xlink:href="007/01/007.jpg"></pb>bus quæſtiones ipſæ ingenioſissimè diluuntur. </s> <s id="s.000023">Vi<lb></lb>detur autem mihi, rem accuratius exploranti, ſa<lb></lb>tis veriſimile (nullum enim habeo opinionis hu<lb></lb>ius aſſertorem,) ſectionem eſſe hanc, & partem <lb></lb>quandam eius operis nobilissimi, quod idem au<lb></lb>ctor De Problematibus edidit, & hanc, neſcio <lb></lb>quam ob cauſam; niſi fortè quod tractatio merè <lb></lb>Phyſica non ſit, à reliquo corpore diſtractam at<lb></lb>que reuulſam. </s> <s id="s.000024">Id certè quod ad rem facit, probè <lb></lb>nouimus, Diogenem Laërtium inter cætera Ari<lb></lb>ſtotelici ingenij monumenta Mechanica quoque <lb></lb>adnumeraſſe. </s> <s id="s.000025">Quibus conſideratis magnopere <lb></lb>ſubit mirari, cur ij qui poſt Ariſtotelem floruêre <lb></lb>atque vixere, Mechanici, Archimedes, Athenæus, <lb></lb>Heron, Pappus, & cæteri, nullam huius libelli fe<lb></lb>cerint commemorationem: & ſanè debuerunt; <lb></lb>neque enim à vero est dissimile, ipſos per hunc ali<lb></lb>quatenus profeciſſe. </s> <s id="s.000026">Verum enim uero cum inge<lb></lb>nui illi fuerint homines, & nullatenus obtrecta<lb></lb>tores, credendum potius est, Comment ariolum i<lb></lb>ſtud, eorum æuo, paucis cognitum, alicubi in Bi<lb></lb>bliothecis latuiſſe: etenim cætera quoque Ariſtote<lb></lb>lis ſcripta, poſt vetuſta illa tempora, ante Ale<lb></lb>xandrum Aphrodiſienſem, à multis fuiſſe igno-<pb xlink:href="007/01/008.jpg"></pb>rata non dubitamus. </s> <s id="s.000027">Habemus ſiquidem, Stra<lb></lb>bone teſte, lib. 13. Ariſtotelis, & Theophraſti bi<lb></lb>bliothecam, poſt ipſius Theophraſti deceſſum, ad <lb></lb>Neleum quendam Scepſium, Coriſci filium, qui <lb></lb>eius fuerat auditor, perueniſſe; poſt hæc libros, <lb></lb>blattis olim, & humore corruptos, Apelliconi Te<lb></lb>io venditos, & ab eo Athenas translatos, tum <lb></lb>Athenis captis in Syllæ poteſtatem deueniſſe, eoſ<lb></lb>que tandem à Sylla acceptos, Tyrannionem <lb></lb>Grammaticum, vt potuit meliùs emendatos, <lb></lb>promulgaſſe. </s> <s id="s.000028">Ex quibus colligimus, mirum non <lb></lb>eſſe, Archimedi, Heroni, & alijs qui ante Syllam <lb></lb>vixêre, fuiſſe incognitos. </s> <s id="s.000029">quicquid ſit, illud cer<lb></lb>tum est, Ariſtotelem eorum omnium quidem Me<lb></lb>chanicis commentaria edidere, eſſe longè vetu<lb></lb>ſtissimum. </s> <s id="s.000030">Pappus enim Herone iunior, Athe<lb></lb>næus Archimedi æqualis, vterque enim ſub Mar<lb></lb>cello, cui Athenæus ſuum de bellicis Machinis <lb></lb><expan abbr="libellū">libellum</expan> dedicauit. </s> <s id="s.000031">Archimedes verò circa CXL. <lb></lb></s> <s id="s.000032">Olympiadem floruit, quamobrem poſt Ariſtote<lb></lb>lem Olympiadas XL. hoc est, annos ferè CLX. <lb></lb></s> <s id="s.000033">Iſthæc autem conſiderantibus, facile eſt cognoſce<lb></lb>re facultatis huius nobilitatem, atque dignitatem; <lb></lb>quippe quod ſummus Philoſophus non modo eam <pb xlink:href="007/01/009.jpg"></pb>probauerit, ſed etiam ſuis acutissimis lucubra<lb></lb>tionibus illuſtrauerit. </s> <s id="s.000034">Hanc porro tractationem <lb></lb>ſubiecto quidem Phyſicam eſſe, demonſtratio<lb></lb>nibus verò Geometricam, ipſemet nos docuit <lb></lb>Ariſtoteles, cuius etiam naturæ ſunt Perſpecti<lb></lb>ua, Specularia, Muſica, & cæteræ eiuſdem <lb></lb>modi facultates, quas quidem ſubalternas Peri<lb></lb>patetici appellant. </s> <s id="s.000035">Vitruuius Architecturæ <lb></lb>membrum, vt ita dicam, & portionem quan<lb></lb>dam facit, ait enim Architecturæ partes eſſe tres, <lb></lb>Ædificationem, Gnomonicam, Machinatio<lb></lb>nem. </s> <s id="s.000036">Est autem Architecturâ quidem inferior, <lb></lb>paret enim Architecto Mechanicus; attamen ſi <lb></lb>cæteras artes ſpectes, Architectonica; hæc enim <lb></lb>omnes ferè ſedentariæ, ſellulariæue, quas banau<lb></lb>ſas Græci appellant, ordine ſubijciuntur, & ſa<lb></lb>nè latissimos iſthæc habet fines; præcipuè autem <lb></lb>circa eam verſatur cognitionem, eamque inter <lb></lb>cæteras ferè principem, quam dixere Centrobari<lb></lb>cam, quæ quidem ad Centri grauitatem, eiuſque <lb></lb>ſpeculationem pertinet: quà in ſpecie inter vete<lb></lb>res primum ſibi vindicauit locum Archimedes, <lb></lb>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"></emph.end><pb xlink:href="007/01/010.jpg"></pb><emph type="italics"></emph>tem Commandinus, qui librum de Centro gra<lb></lb>uitatis ſolidorum ſcripſit, & poſt eum G. Vbal<lb></lb>dus è Marchion. Montis, qui non modò ab<lb></lb>ſolutissimum Mechanicorum librum cum maxi<lb></lb>ma ingenij ſui laude conſcripſit, ſed & Paraphra<lb></lb>ſin in librum Æqueponderantium Archimedis <lb></lb>egregiè concinnauit Centrobaricam hanc, igno<lb></lb>tam fuiſſe Ariſtoteli, ſætis patet. </s> <s id="s.000037">nunquam enim <lb></lb>in Mechanicis demonſtrationibus, quod tamen <lb></lb>est potissimum, grauitatis centrum nominat, e<lb></lb>iuſue naturam atque vim ſpeculatur. </s> <s id="s.000038">Diuidi<lb></lb>tur autem Mechanice tota, teſte Herone apud <lb></lb>Pappum libro octauo, in Rationalem, hoc est, <lb></lb>Theoricam & Chirurgicam, id est, manu ope<lb></lb>ratricem, quam Praxim aptè dicere valemus. <lb></lb></s> <s id="s.000039">Rationalis, ſpeculationi & <expan abbr="demōſtrationibus">demonſtrationibus</expan>, ex <lb></lb>Geometricis, Arithmeticis & Phyſicis rationi<lb></lb>bus, dat operam; Chirurgica vero materiam <lb></lb>tractat, & ſeſe in varias artes diffundit, Æra<lb></lb>riam, Lignariam, Sculptoriam, Pictoriam, Æ<lb></lb>dificatoriam, Machinariam & Thaumaturgi<lb></lb>cam, cæterasque eiuſmodi. </s> <s id="s.000040">Machinatoriæ au<lb></lb>tem ſunt partes Manganaria, qua ingentia <emph.end type="italics"></emph.end><pb xlink:href="007/01/011.jpg"></pb><emph type="italics"></emph>transferuntur pondera, tum ipſa Poliorcetica, <lb></lb>quæ bellicas Machinas ad vrbium expugnatio<lb></lb>nes, quod vel ipſo nomine profitetur, ædificat. </s> <s id="s.000041">At<lb></lb>qui hac dere plura ſcribere ſuperſedemus, ne a<lb></lb>ctum agamus: quis quis enim minutè magis hæc <lb></lb>cognoſcere deſiderat, is Pappum adeat libro cita<lb></lb>to, & Guidum Vbaldum in Præfatione quam <lb></lb>ſuo Mechanicorum Operi præpoſuit. </s> <s id="s.000042">Vt autem <lb></lb>ad Ariſtotelis, de quo egimus, libellum reuerta<lb></lb>mur, pauci ſunt qui ei ante nos ſtilum & operam <lb></lb>commodauerint: Leonicenus Latinum fecit & <lb></lb>figuris tum breuissimis, & parui ſane ponderis, <lb></lb>marginalibus adnotatiunculis, inſtruxit. </s> <s id="s.000043">Poſt <lb></lb>hunc Alexander Picolomineus luculentissima <lb></lb>Paræphraſi illuſtrauit. </s> <s id="s.000044">Modo, vt audio, Simon <lb></lb>Sticinus Hollandenſis quædam edidit, quæ ad <lb></lb>nos minime peruenêre. </s> <s id="s.000045">Nos demum, omnium, <lb></lb>tum ſcientia, & ingenio, tum ætate, poſtremi huic <lb></lb>operi manum admouimus; Conſiderantes enim <lb></lb>Ariſtotelem aliis fecerint Mechanici, demonſtraſſe, <lb></lb>morem huiuſce facultatis ſtudioſis geſturos nos <lb></lb>fore arbitrati ſumus, ſi eaſdem illas quæſtiones <emph.end type="italics"></emph.end><pb xlink:href="007/01/012.jpg"></pb><emph type="italics"></emph>Mechanicis, hoc est, Archimedeis probationi<lb></lb>bus confirmaremus; dum per latissimos faculta<lb></lb>tis huius campos vagantes, alias quoque iſtis af<lb></lb>fines dubitationes introducentes ſolueremus. <lb></lb></s> <s id="s.000046"><expan abbr="quicquidautēſecerimus">quicquid autem fecerimus</expan> profecerimuſue, Lector <lb></lb>optime, boni conſule, & quia fax per manus tra<lb></lb>ditur, tu interim de me accipe, vt alijs tradas.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="007/01/013.jpg"></pb> </section> <section> <p type="head"> <s id="s.000047">DE VITA ET SCRI<lb></lb>PTIS BERNARDINI <lb></lb>BALDI VRBINATIS</s> </p> <p type="head"> <s id="s.000048"><emph type="italics"></emph>EX LITERIS FABRITII SCHAR<lb></lb>loncini ad Illuſtrissimum & Reuerendissimum <lb></lb>Dominum Lælium Ruinum Epiſcopum Bal<lb></lb>neoregienſem ex-Nuntium Apoſtolicum <lb></lb>ad Poloniæ Regem & c.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000049">Natus eſt Bern. Baldus Vrbini nobilibus <expan abbr="pa-rētibus">pa<lb></lb>rentibus</expan> poſtridie Non. </s> <s id="s.000050">Iunij anno MDLIII. <lb></lb></s> <s id="s.000051">Genus traxit, quod me ſæpè ab eo memini <lb></lb>audire, à familia Cantagallina, quæ inter <lb></lb>Peruſinas illuſtris: hoc autem cognomen, <lb></lb>Baldi accepto, vt in varietate temporum fit, <lb></lb>Abauus reliquit, à teneris vnguiculis <expan abbr="pietatē">pietatem</expan> erga Deum <lb></lb>præſetulit; nam vt mater eius narrabat, ſanctorum imagi<lb></lb>nes & Altariola non cum lætitia ſolum, ſed cum venera<lb></lb>tione anniculus intuebatur. </s> <s id="s.000052">Præceptoribus in adoleſcen<lb></lb>tia vſus fuit laudatiſſimis Io. And. Palatio, & Io. Antonio <lb></lb>Turoneo, qui altero doctior, & Paulo Manutio maxime <lb></lb>carus ob latinæ & græcæ linguæ peritiam propè ſingula<lb></lb>rem: ad illorum autem ſedulitatem tantum animi ardo<lb></lb>rem attulit, tantam ingenij ac iudicij vim, vt non tantum <lb></lb>æqualis ſed omnium vicerit expectationem. </s> <s id="s.000053">Puer adhuc <lb></lb>Arati apparitiones Italico carmme reddidit. </s> <s id="s.000054">Parens hac <lb></lb>filij laude & gloria motus anno 1573. eum ad maiorem in<lb></lb>genij cultum capeſſendum Patauium miſit. </s> <s id="s.000055">Hîc in Ema<lb></lb>nuelis Margunij familiaritatem ſtatim venit, cui porro <pb xlink:href="007/01/014.jpg"></pb>fuit in amoribus. </s> <s id="s.000056">Homeri Iliad. illo Doctore & interpre<lb></lb>te diligentius quam feciſſet antea, euoluit. </s> <s id="s.000057">priuato autem <lb></lb>ſtudio Anacreonti, Pindaro, Æſchyli, Euripidi, Sophocli <lb></lb>operam dedit, ſed præ cæteris Theocriti Bucolica triuit, <lb></lb>ad quod ſcriptionis genus natura magis ferri videbatur: <lb></lb>centenos græci alicuius poëtæ verſus memoriter tenebat, <lb></lb>ſæpeque habebat in ore, in oratoribus græcis verſandis <lb></lb>laborem ſe aliquem ſentire, in poëtis nullum. </s> <s id="s.000058">Scripſit Pa<lb></lb>tauij libellum de Tormentis Bellicis, & eorum inuentori<lb></lb>bus, & cum in Tranſalpinorum amicitias incidiſſet, ſibi <lb></lb>ducebat dedecori ipſos ſua lingua loquentes non intelli<lb></lb>gere. </s> <s id="s.000059">quare incredibili celeritate Gallicam & Germani<lb></lb>cam didicit. </s> <s id="s.000060">Peſtilentia ex eo Gymnaſio exactus in Pa<lb></lb>triam redijt, vbi quinquennium integrum Federico <expan abbr="Cō-mandino">Com<lb></lb>mandino</expan> affixus omnes Matheſeos partes perdidicit, cui <lb></lb>viro in delineandis figuris ad Euclidis, Pappi, & Heronis <lb></lb>monumenta manum commodauit: ex eiuſdem obitu do<lb></lb>lorem vix conſolabilem ſuſtinuit, ſuſceptoque eius vitam <lb></lb>ſcribendi conſilio, ſubinde ad omnium Mathematicorum <lb></lb>vitas conſcribendas animum adplicuit, quod & duode<lb></lb>cim annorum ſpatio præſtitit feliciſſimè. </s> <s id="s.000061">cum vero Ma<lb></lb>thematicarum diſciplinarum amore torqueretur, amiſſo <lb></lb>Commandino Præceptore, amicum nactus fuit præſtan<lb></lb>tiſſimum & ſymmyſtam Guidum Vbaldum è Marchioni<lb></lb>bus Montis, in cuius ſe conſuetudinem daret: quantum <lb></lb>profeciſſet, oſtendunt ij commentarij quos anno 1582. in <lb></lb>Ariſt. Mechanica ſcripſit. </s> <s id="s.000062">Vt poſtea à grauioribus ſtudijs <lb></lb>ad amœniora animum abduceret, de re nautica poëma I<lb></lb>talicè confecit. </s> <s id="s.000063">quo abſoluto Paradoxa multa Mathema<lb></lb>tica explicauit. </s> <s id="s.000064">Fama de Baldi virtutibus diſſipata Ferran<lb></lb>dus Gonzaga Molfetræ Princeps & Guaſtallæ Dominus <lb></lb>cœpit de illo in ſuam familiam aſciſcendo cogitare, vt qui <lb></lb>ijſdem caperetur artibus, quibus excellere Baldus inci-<pb xlink:href="007/01/015.jpg"></pb>piebat: Itaque opera Curtij Arditij honorifice fuit in au<lb></lb>lam euocatus, dum vitam non aulicam viueret totus in <lb></lb>litteras abditus precibus Veſpaſiani Gonzagæ Sablonetæ <lb></lb>Ducis ad explanandos Vitruuij libros adactus fuit. </s> <s id="s.000065">quare <lb></lb><expan abbr="tūc">tunc</expan> natus de <expan abbr="Verborū">Verborum</expan> Vitruuianorum ſignificatione com<lb></lb>mentarius; in quo minime mirandum ſi minuta quædam <lb></lb>proſequutus fuit, quæ viro magno minus eſſe digna vi<lb></lb>deantur:illi enim Principi morem geſſit. </s> <s id="s.000066">ſcio dixiſſe ali<lb></lb>quando Adrianum Romanum è Polonia reuerſum, vbi <lb></lb>Vitruuium Palatino cuidam explicauerat, ſi commen<lb></lb>tarium Baldi in Polonia adhibere potuiſſem, aurum quod <lb></lb>mecum attuli emunxiſſem, quia ſatis feciſſem muneri la<lb></lb>bore nullo. </s> <s id="s.000067">Cum Ferrando hero ſuo obueniſſet neceſſi<lb></lb>tas Hiſpanias adeundi, illud iter ſine Baldo facere ſe poſ<lb></lb>ſe non putabat, non tam vt haberet, qui erudito eloquio <lb></lb>viæ tæ dium leuaret, quam cui poſſet arcana committere, <lb></lb>atque adeo à quo iuuaretur conſilio. </s> <s id="s.000068">Vix viæ ſe dederant <lb></lb>cum Baldus grauem in morbum delapſus itinere cogitur <lb></lb>deſiſtere: Mediolanum proinde diuertit, vbi à S. Carolo <lb></lb>Borromæo & benignè exceptus, & tamdiu detentus do<lb></lb>nec valetudinem recuperaret. </s> <s id="s.000069">Guaſtallam poſtea ſe re<lb></lb>cepit, vbi cum abſente Domino liberiori otio frueretur, <lb></lb>libros ſex de Aula eruditiſſimos methodo analytica con<lb></lb>ſcripſit. </s> <s id="s.000070">alios non commemoro, quod cum otium erit, o<lb></lb>mnium ſyllabum dabo. </s> <s id="s.000071">Anno 1586. ipſo nihil poſtulante <lb></lb>eligitur Guaſtallæ Abbas, à quo tempore Iuri Can. </s> <s id="s.000072">Con<lb></lb>cilijs, & SS.Patribus totum ſe dedit. </s> <s id="s.000073">Hebreæ & Chaldææ <lb></lb>linguarum diſcendarum triennium poſuit. </s> <s id="s.000074">Anno 1593. no<lb></lb>uæ Gnomonices libros quinque compoſuit. </s> <s id="s.000075">inſequenti <lb></lb>Chaldæam Onkeli paraphraſin in Pentateuchum vertit <lb></lb>& commentarios adiunxit; quo exant lato labore in Iob <lb></lb>ex Heb. fonte paraphraſin texuit, quam & ſcholijs illu<lb></lb>ſtrauit. </s> <s id="s.000076">Tabulam Etruſcam Eugubinam interptetatus <pb xlink:href="007/01/016.jpg"></pb>fuit:in ea autem diuinatione, vt aiebat, ſubciſiuas vnius <lb></lb>menſis horas conſumpſit. </s> <s id="s.000077">De Firmamento & aquis egre<lb></lb>gie ſcripſit. </s> <s id="s.000078">Oeconomiam Tropologicam in S.Matthæum <lb></lb>Card. Baronius, qui non alia Baldi vidit, vehementer pro<lb></lb>babat. </s> <s id="s.000079">Romæ dum viueret, fere neſciuit quid gereretur <lb></lb>in Aulis: Arabicæ enim linguæ cum Io. Baptiſta Raimon<lb></lb>do diligentiſſime ſtuduit, & arcana induſtria Slauonicæ, <lb></lb>quam perfecte callebat. </s> <s id="s.000080">Ex Arabico vertit Hortum Geo<lb></lb>graphicum Anonymi, quem ante ſexcentos annos flo<lb></lb>ruiſſe arbitrabatur. </s> <s id="s.000081">Hunc vero extruſiſſet, vt alios Baldi <lb></lb>libros, Marcus Velſerus IIvir Aug. ſi eo paulo longior <lb></lb>huius lucis vſura contigiſſet. </s> <s id="s.000082">Compoſuit & Dictionarium <lb></lb>Arabicum. </s> <s id="s.000083">atque cum beatiſſimam illam vbertatem in<lb></lb>genij aſſidue diffundi neceſſe eſſet, anno 1603. orbem vni<lb></lb>uerſum deſcribere aggreſſus fuit; atque ita quidem, vt <lb></lb>tam quæ ad Hiſtoriam, quam quæ ad Geographiam per<lb></lb>tinerent complecteretur: Neque illuſtrare ſolum voluit <lb></lb>quæ nouerunt antiqui, quemadmodum viſum Ortelio, <lb></lb>ſed vel oppidula omnia & pagos, de quibus aliqua in po<lb></lb>ſtremis ſcriptoribus mentio. </s> <s id="s.000084">& profecto totum opus ad <lb></lb>vmbilicum perduxit: non digeſſit tamen vniuerſum. </s> <s id="s.000085">qua<lb></lb>tuor aut ni fallor quinque tantum Tomi fuerunt ordine <lb></lb>Alphabetico diſpoſiti:ſupereſſent ſeptem aut octo diſpo<lb></lb>nendi, quantum ex chartarum & faſciculorum mole con<lb></lb>ijcere licet. </s> <s id="s.000086">Anno 1617. quarto Idus Octob. </s> <s id="s.000087">poſtea quam <lb></lb>dies 40. vehementi deſtillatione vexatus fuiſſet, ſpiritum <lb></lb>Deo reddidit Sacramentis Eccleſiæ omnibus rite muni<lb></lb>tus. </s> <s id="s.000088">Statura procerus fuit, facie oblonga & acribus oculis, <lb></lb>colore ſubfuſco. </s> <s id="s.000089">Membrorum ei fuit decens habitudo, & <lb></lb>compactum corpus. </s> <s id="s.000090">Diebus feſtis omnibus ſacrum facie<lb></lb>bat, ieiunabat bis in hebdomada, eleemoſyniſque paupe<lb></lb>res ſubleuabat. </s> <s id="s.000091">In ſtudijs ſic aſſiduus fuit, vt ſæpe & legeret <lb></lb>& comederet. </s> <s id="s.000092">S.Auguſtini libros de Ciuitate Dei ter in-<pb xlink:href="007/01/017.jpg"></pb>ter prandium euoluit. </s> <s id="s.000093">Statim à noctis meridie dum ei vi<lb></lb>res firmiores eſſent ad lucubrandum ſurgebat. </s> <s id="s.000094">à prandio <lb></lb>Euclidem Arabice editum, vel libellum aliquem germa<lb></lb>nicum aut gallicum in manus ſumebat. </s> <s id="s.000095">Suauitate morum <lb></lb>& modeſtia, etiam ſi ceteræ dotes abfuiſſent, quemlibet <lb></lb>ad amorem ſui allicere potuiſſet. </s> <s id="s.000096">Sermo modicus ei fuit, <lb></lb>itemque cultus. </s> <s id="s.000097">Nullos vnquam honores petijt, qui à <lb></lb>Clem. 8. ampliſſimi promiſſi fuerant; nullum emolumen<lb></lb>tum quæſiuit ſuo cenſu contentus. </s> <s id="s.000098">facile parcendum eſſe <lb></lb>dicebat, ijs maxime qui in re leui impegiſſent, quoniam ſi <lb></lb>quos cenſemus optimos, nudos conſpiceremus, nullum <lb></lb>eorum non iudicaremus multis dignum verberibus. </s> <s id="s.000099">Bi<lb></lb>bliothecam habuit non locupletem, ſed ſelectis <expan abbr="inſtructã">inſtructam</expan> <lb></lb>codicibus. </s> <s id="s.000100">Verum ire per ſingula longum eſſet. </s> <s id="s.000101">Satis mihi <lb></lb>de incomparabili Baldi doctrina, & ſumma innocentia, ô <lb></lb>rarum connubium, pauca dixiſſe, quæ forſitan ad imitan<lb></lb>dum nimis multa. </s> </p> </section> <section> <p type="head"> <s id="s.000102">SYLLABVS LIBRORVM</s> </p> <p type="head"> <s id="s.000103">omnium B.Abb.Baldi.</s> </p> <p type="main"> <s id="s.000104">Arati apparitiones è gr.in Ital. vertit. </s> </p> <p type="main"> <s id="s.000105">De Tormentis Bellicis & eorum Inuentoribus lib. Heronis automata vertit. </s> </p> <p type="main"> <s id="s.000106">Vitas omnium Mathematicorum ſcripſit, & trib. </s> <s id="s.000107">in Tom. <lb></lb>2.1.P^{s}.à Thalete ad Chriſtum.2.à Chriſto ad ſua tem<lb></lb>pora. </s> </p> <p type="main"> <s id="s.000108">Earumdem vitarum Epitomen Chronologicum confecit. </s> </p> <p type="main"> <s id="s.000109">In Ariſtot. Mechan. Commentar. </s> </p> <p type="main"> <s id="s.000110">De Re nautica Poëmation. </s> </p> <p type="main"> <s id="s.000111">Paradoxorum Mathematicorum liber. </s> </p> <p type="main"> <s id="s.000112">Deſcriptio Palatij Ducum Vrbinarum quod eſt Vrbini. </s> </p> <p type="main"> <s id="s.000113">Poema cui titulus, Lamus. </s> </p> <pb xlink:href="007/01/018.jpg"></pb> <p type="main"> <s id="s.000114">Carmina pia, quæ inſcribuntur, Anni Corona. </s> </p> <p type="main"> <s id="s.000115">De Verborum Vitruuianorum ſignificatione. </s> </p> <p type="main"> <s id="s.000116">Carmina varia & eclogæ mixtæ. </s> </p> <p type="main"> <s id="s.000117">Apologi centum, quos ſcripſit æmulatus Leonem Bapt. <lb></lb>Albertum. </s> </p> <p type="main"> <s id="s.000118">De Humanitate Dialogus qui inſcribitur Goſelinus. </s> </p> <p type="main"> <s id="s.000119">Comparatio Vitæ Monaſticæ cum ſeculari. </s> </p> <p type="main"> <s id="s.000120">De Aula libri ſex. </s> </p> <p type="main"> <s id="s.000121">De felicitate Principis Dialogus. </s> </p> <p type="main"> <s id="s.000122">De Dignitate Dial. </s> </p> <p type="main"> <s id="s.000123">Carmina Romana. </s> </p> <p type="main"> <s id="s.000124">Moſæi fabulam vertit. </s> </p> <p type="main"> <s id="s.000125">De Italici carminis natura Dial. qui inſcribitur Taſſus. </s> </p> <p type="main"> <s id="s.000126">De vniuerſali Diluuio poemation. </s> </p> <p type="main"> <s id="s.000127">Nouæ Gnomonices lib. quin que. </s> </p> <p type="main"> <s id="s.000128">Hieremiæ Threnos vertit, & ex Heb. fonte annotat. </s> <s id="s.000129">ad<lb></lb>iecit. </s> </p> <p type="main"> <s id="s.000130">Poemation inſcriptum, Deiphobe, quod ſcripſit æmula<lb></lb>tus Lycophonem in Caſſandra. </s> </p> <p type="main"> <s id="s.000131">Scala cœleſtis.1.Sermones pij & carmina. </s> </p> <p type="main"> <s id="s.000132">Onkeli paraphraſin Chaldæam in Pentateuchum ver<lb></lb>tit & vberes commentarios adiecit. </s> </p> <p type="main"> <s id="s.000133">In Iob Paraphraſis latina ex fonte Heb. additis Scholijs. </s> </p> <p type="main"> <s id="s.000134">De ſcamillis imparibus Vitruuij. </s> </p> <p type="main"> <s id="s.000135">De firmamento & aquis. </s> </p> <p type="main"> <s id="s.000136">Quincti Calabri Paralipomena vertit. </s> </p> <p type="main"> <s id="s.000137">Tabulæ Etruſcæ Eugubinæ Interpretatio. </s> </p> <p type="main"> <s id="s.000138">Oeconomía Tropologicain S.Matthæum. </s> </p> <p type="main"> <s id="s.000139">Vrbini encomium. </s> </p> <p type="main"> <s id="s.000140">Horti geographici ex Arab. verſio. </s> </p> <p type="main"> <s id="s.000141">Aduerſus Aulam Carmina. </s> </p> <p type="main"> <s id="s.000142">Luciani de miſerijs.Aulicorum verſio. </s> </p> <p type="main"> <s id="s.000143">Oratio ad Romæ conſeruatores pro antiquitatum eius <lb></lb>Vrbis cuſtodia. </s> </p> <pb xlink:href="007/01/019.jpg"></pb> <p type="main"> <s id="s.000144">Vniuerſi orbis geographica & Hiſtorica deſcriptio con<lb></lb>texta ex ſeptingentis & eo amplius ſcriptoribus. </s> </p> <p type="main"> <s id="s.000145">Federici Vrbini Ducis Vita. </s> </p> <p type="main"> <s id="s.000146">Guidi Vbaldi Vrbini Ducis Vita. </s> </p> <p type="main"> <s id="s.000147">Epigrammaton & Odarum libri tres. </s> </p> <p type="main"> <s id="s.000148">Aliorum Carminum liber. </s> </p> <p type="main"> <s id="s.000149">Sententiarum moralium liber. </s> </p> <p type="main"> <s id="s.000150">Dictionarium Arabicum. </s> </p> <p type="main"> <s id="s.000151">Pro Procopio contra Flauium Blondum. </s> </p> <p type="main"> <s id="s.000152">Horographium vniuerſale. </s> </p> <p type="main"> <s id="s.000153">Epigrammata alia. </s> </p> <p type="main"> <s id="s.000154">Heronis lib. de Balliſtis conuerſio. </s> </p> <p type="main"> <s id="s.000155">Exercitationes in Ariſtotelis Mechan. </s> </p> <p type="main"> <s id="s.000156">Templi Ezechielis noua deſcriptio. </s> </p> <p type="main"> <s id="s.000157">Antiquitatum Guaſtallenſium liber. </s> </p> <p type="main"> <s id="s.000158">Hiſtoriæ ſcribendæ leges. </s> </p> <p type="main"> <s id="s.000159">Et alia quædam. </s> </p> <pb xlink:href="007/01/020.jpg"></pb> <pb xlink:href="007/01/021.jpg"></pb> </section> </front> <body> <chap> <p type="head"> <s id="s.000160">IN MECHANICA ARISTOTE<lb></lb>LIS PROBLEMATA EXERCITATIONES.</s> </p> <subchap1> <p type="head"> <s id="s.000161"><emph type="italics"></emph>Mechanices deſcriptio, natura, finis.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000162">MECHANICE, facultas quædam eſt, quæ <lb></lb>naturali materiâ, Geometricisque; demon<lb></lb>ſtrationibus vſa, ex centrobaricâ, & <expan abbr="eorū">eorum</expan> <lb></lb>quæ ad vectem & libram rediguntur, ſpe<lb></lb>culatione; humanæ conſulens neceſſitati, <lb></lb>commoditatiqueue, ſuapte vi, Naturam i<lb></lb>pſam vel ſecundans, vel ſuperans, varia, eaque mirabilia <lb></lb>operatur. </s> <s id="s.000163">Hac diffinitione deſcriptionéue breuiter ea fe<lb></lb>rè omnia complexi ſumus, quæ fuſiſſimè ab Ariſtotele, <lb></lb>Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000164"><emph type="italics"></emph>Mechanices Obiectum.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000165">Conſiderat autem Mechanicus Graue & Leue. </s> </p> <p type="main"> <s id="s.000166">Graue duplex, Naturâ, Violentiâ. </s> </p> <p type="main"> <s id="s.000167">Graue Naturâ dicitur, quod inſita propenſione in <lb></lb>centrum mundi fertur. </s> <s id="s.000168">Graue autem Violentiâ, quod im<lb></lb>preſſo extrinſecus pondere ab impellente pellitur. </s> </p> <p type="main"> <s id="s.000169">Leue contrà, quòd Naturâ à centro fertur. </s> </p> <p type="main"> <s id="s.000170">Cæterùm quicquid graue eſt, ſecundum punctum <lb></lb>eſt, quod Grauitatis centrum dicitur, & hoc duplex, vt <lb></lb>duplex eſt grauitas, Naturæ, Violentiæ. </s> </p> <pb xlink:href="007/01/022.jpg"></pb> <p type="main"> <s id="s.000171">Grauitatis centrum in triplici magnitudine conſi<lb></lb>derari poteſt, lineari, planà, ſolidâ. </s> </p> <p type="main"> <s id="s.000172">De centro grauitatis linearum nemo ſcripſit, ſimpli<lb></lb>ciſſimi enim illud eſt contemplationis. </s> </p> <p type="main"> <s id="s.000173">De centro grauitatis linearum egregiè tractauit Ar<lb></lb>chimedes in libro Æqueponderantium, & de quadratu<lb></lb>ra Parabole, tum in eo quem de his quæ vehuntur in<lb></lb>ſcripſit. </s> </p> <p type="main"> <s id="s.000174">De centro grauitatis ſolidorum ipſemet olim ſcri<lb></lb>pſerat Archimedes, ſed ea quæ protulit, temporis iniuriâ <lb></lb>deperdita, ſuâ diligentiâ reſtituit Federicus Commandi<lb></lb>nus. </s> </p> <p type="main"> <s id="s.000175">Eſſe autem & Leuitatis centrum in rerum natura, <lb></lb>palam eſt. </s> <s id="s.000176">Punctum enim illud eſt, ſecundum quod leuia <lb></lb>rectà à centro ſurſum feruntur. </s> <s id="s.000177">Huius autem non memi<lb></lb>nêre Mechanici, propterea quod aut nihil, aut parum ad <lb></lb>eorum rem faciat. </s> </p> <p type="main"> <s id="s.000178">Porro Grauitatis centrum ita definit Heron, & qui <lb></lb>ab Herone Pappus 1.8. Collectionum Mathematicarum. </s> </p> <p type="main"> <s id="s.000179">Centrum grauitatis <expan abbr="vniuſcuiuſq;">vniuſcuiuſque</expan> corporis eſt pun<lb></lb>ctum quoddam intra poſitum, à quo ſi graue, mente ap<lb></lb>penſum concipiatur, dum fertur, quieſcit, & ſeruat eam <lb></lb>quam in principio habuit poſitionem; neque in ipſa latio<lb></lb>ne circumuertitur. </s> <s id="s.000180">Commandinus verò in lib. de centro <lb></lb>grauitatis ſolidorum hoc pacto: Centrum grauitatis v<lb></lb>niuſcuiuſque ſolidæ figuræ, eſt punctum illud intra poſi<lb></lb>tum, circa quod vndique partes æqualium momentorum <lb></lb>adſiſtunt. </s> <s id="s.000181">Si enim per tale centrum ducatur planum, fi<lb></lb>guram quomodolibet ſecans, in partes æquè ponderantes <lb></lb>eam diuidit. </s> <s id="s.000182">Nos verò quàm breuiſſimè dicimus: <expan abbr="Centrū">Centrum</expan> <lb></lb>grauitatis, <expan abbr="vniuſcuiuſq;">vniuſcuiuſque</expan> magnitudinis punctum eſſe intra <lb></lb>extraue magnitudinem poſitum, per quod ſi plano linea <lb></lb>punctoue diuidatur, in partes ſecatur æqueponderantes. </s> </p> <pb xlink:href="007/01/023.jpg"></pb> <figure id="id.007.01.023.1.jpg" xlink:href="007/01/023/1.jpg"></figure> <p type="main"> <s id="s.000183">Diximus, Magnitudinis vt lineæ, plani ſolidique; cen<lb></lb>trum complecteremur. </s> <s id="s.000184">Erit igitur, vt in præſenti figura, li<lb></lb>neæ quidem centrum A, plani B, ſolidi verò C. quod ſi ob<lb></lb>ijciat quiſpiam, lineam & ſuperficiem nullam habere gra<lb></lb>uitatem; is ſciat, <expan abbr="neq;">neque</expan> corpora Mathematica grauitatem <lb></lb>habere, Mechanicum verò funes, haſtas, vectes pro lineis <lb></lb>ſumere; tabulas verò, & eiuſmodi plana ad ſuperficierum <lb></lb>naturam referre. </s> </p> <p type="main"> <s id="s.000185">Diximus inſuper, intra extraue. </s> <s id="s.000186">Aliquando enim <lb></lb>grauitatis centrum extra molem corporis cuius corporis <lb></lb>centrum eſt, cadit, vt in ſequenti figura. </s> </p> <figure id="id.007.01.023.2.jpg" xlink:href="007/01/023/2.jpg"></figure> <p type="main"> <s id="s.000187">Eſto corpus aliquod <lb></lb>ſuperficiesue ABCDE, <lb></lb>ducatur linea CF, <expan abbr="diuidēs">diuidens</expan> <lb></lb>figuras in partes hinc inde <lb></lb>æqueponderantes ABC, <lb></lb>EDC. </s> <s id="s.000188">Ducatur & GH. <lb></lb>diuídens item in partes æ<lb></lb>queponderantes GCH, & GAB, EDH. ſecent autem <lb></lb>ſeipſas in I. erit igitur centrum I extra figuræ terminos & <lb></lb>molem ipſam. </s> <s id="s.000189">Attamen licet hoc verum ſit, intra eſſe dici <lb></lb>poteſt, quippe quod imaginario quodam, & vt ita dicam, <lb></lb>virtuali ambitu ACDA contineatur. </s> </p> <p type="main"> <s id="s.000190">Dicebamus, duplex eſſe grauitatis centrum, Natu<pb xlink:href="007/01/024.jpg"></pb>râ, Violentià: affirmamus modò, hæc re quidem vnum eſ<lb></lb>ſe, & ratione ſolum, non autem re ipſa ac ſi duo eſſent con<lb></lb>ſiderari. </s> </p> <figure id="id.007.01.024.1.jpg" xlink:href="007/01/024/1.jpg"></figure> <p type="main"> <s id="s.000191">Eſto enim grauitatis na<lb></lb>turalis centrum B, corporis A, <lb></lb>ſecundum quod dimiſſum, ſua<lb></lb>pte naturâ cadet in C, ſi verò <lb></lb>corpus violenter impellatur in <lb></lb>D, aliud acquiret centrum gra<lb></lb>uitatis ex violentia ſecundum <lb></lb>quam fertur, motum, in D, <expan abbr="idē">idem</expan> <lb></lb>autem ſunt re, nempe vnum B, <lb></lb>duo autem ſi violentia & natura ſeorſum conſideren<lb></lb>tur. </s> </p> <p type="main"> <s id="s.000192">Hæc centra, duo motus ſequuntur, rectus vterque, <lb></lb>Naturalis videlicet, & Violentus. </s> <s id="s.000193">Tertius ex his mixtus, & <lb></lb>is quidem non rectus, ſed curuus. </s> </p> <figure id="id.007.01.024.2.jpg" xlink:href="007/01/024/2.jpg"></figure> <p type="main"> <s id="s.000194">Proijciatur enim violen<lb></lb>ter corpus graue A ſuperante <lb></lb>igitur violentia, rectà feretur <lb></lb>in B; ea autem elangueſcente <lb></lb>paullatim per curuam & mi<lb></lb>xtam <expan abbr="lineã">lineam</expan> ſecetur in C, qua<lb></lb>tenus enim ad anteriora fer<lb></lb>tur, violentia eſt; quatenus ve<lb></lb>rò ad inferiores partes, naturæ. </s> <s id="s.000195">Vbi verò peruenit in C, <lb></lb>violentiâ ceſſante, naturâ verò manente, rectà deorſum <lb></lb>fertur DCD. </s> </p> <p type="main"> <s id="s.000196">Cæterùm hæc centra, hiqueue motus, naturalis nem<lb></lb>pe, & violentus diuerſimode ſe habent adinuicem. </s> <s id="s.000197">Si e<lb></lb>nim graue corpus externâ vi adhibita, centrum mundi <lb></lb>verſus impellatur, adiuuabunt ſe inuicem Natura, Vio<lb></lb>lentia, Si autem contra, altera alteri reſiſtet, in motibus <pb xlink:href="007/01/025.jpg"></pb>autem ad latus, eo magis pugnabunt, quo magis ab infe<lb></lb>rioribus ad ſuperiora fiet motus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000198"><emph type="italics"></emph>Mechanices præcipua inſtrumenta.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000199">Hic ira conſtitutis dicimus, inſtrumenta, quibus ad <lb></lb>varias operationes Mechanici vtuntur, eſſe inter ſe qui<lb></lb>dem diuerſa, multiplicia, & ſi varietatem ſpectes, penè in<lb></lb>numerabilia; quod quamuis verum ſit, ea omnia Ariſtote<lb></lb>les ad vectem re ducit, & libram: quod etiam G. Vbaldus <lb></lb>in libris Mechanicorum fecit. </s> <s id="s.000200">Cæterum qui poſt Ariſto<lb></lb>telem floruere Mechanici, omnia ad quinque, quas ap<lb></lb>pellant, Potentias, redegêre. </s> <s id="s.000201">Sunt autem ex Herone, Pap<lb></lb>po, Guido Vbaldo, qui eos ſecutus eſt, Vectis, Trochlea, <lb></lb>Axis in Peritrochio, Cuneus, Cochlèa. </s> <s id="s.000202">Videtur autem i<lb></lb>pſe G. Vbaldus ſextam addere, nempe Libram, de qua & <lb></lb>primus ipſe Mechanicorum tractatum inſtituit. </s> <s id="s.000203">Verum <lb></lb>enimuero idem ferè ſunt Vectis & Libra, niſi forte quod <lb></lb>Libra tunc dicitur, cum brachia ſunt æqualia. </s> <s id="s.000204">Vectis vero <lb></lb>quomodocunque ea ſe habeant; quinque harum <expan abbr="Poten-tiarū">Poten<lb></lb>tiarum</expan> imagines ita ob oculos ponimus. </s> <s id="s.000205">Vectis A. </s> <s id="s.000206">Trochlea <lb></lb>B, Axis in Peritrochio C. </s> <s id="s.000207">Cuneus D. </s> <s id="s.000208">Cochlea vero E. </s> </p> <pb xlink:href="007/01/026.jpg"></pb> <figure id="id.007.01.026.1.jpg" xlink:href="007/01/026/1.jpg"></figure> <p type="main"> <s id="s.000209">Porro, Cuneum ad libram reducere conatur Ari<lb></lb>ſtoteles, quod facit & G. Vbaldus, qui eò refert & Co<lb></lb>chleam, quippe quod nihil aliud ſit Cochlea, quàm Cu<lb></lb>neus Cylindro inuolutus. </s> <s id="s.000210">Nos autem duas tantùm Po<lb></lb>tentias ad vectem reduci poſſe arbitramur, Trochleam <lb></lb>nempe, & Axem in Peritrochio. </s> <s id="s.000211">Nequaquam autem Cu<lb></lb>neum & Cochleam. </s> <s id="s.000212">quod latiùs quidem oſtendemus, <lb></lb>cùm de Cuneo erit nobis ſermo peculiaris. </s> </p> <p type="head"> <s id="s.000213"><emph type="italics"></emph>De Vecte & Libra ſecundum Ari<lb></lb>ſtotelem.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000214">Ariſtoteles in ipſo Mechanicorum ingreſſu ita ſcri<lb></lb>bit, Mirum videri ab exigua virtute magnum pondus mo-<pb xlink:href="007/01/027.jpg"></pb>ueri, addito nimirum ponderi pondere, ſiquidem & vectis <lb></lb>eſt pondus. </s> <s id="s.000215">Duplex ergo illi admiratio, ſcilicet quòd exi<lb></lb>gua potentia moueat ingens pondus, idqueue etiam addito <lb></lb>vectis ipſius pondere, fiat. </s> <s id="s.000216">Hoc ſecundum adieciſſe vide<lb></lb>tur, amplificationis alicuius gratiâ. </s> <s id="s.000217">Etenim quatenus <lb></lb>ad rem pertinet, ſi mouendis ponderibus vectis ipſius <lb></lb>pondus compares, nullius ferè eſſe momenti procul du<lb></lb>bio affirmaueris. </s> <s id="s.000218">Sed & illud quoque notandum, aliquan<lb></lb>do vectis pondus mouenti auxilium ferre, quod fit vbi <lb></lb>fulcimento inter potentiam mouentem, & pondus ipſum <lb></lb>collocato, vectis pars quæ à fulcimento ad potentiam eſt, <lb></lb>premitur. </s> <s id="s.000219">Tunc enim, vt dicebamus, vectis pondere ſuo <lb></lb>potentiam adiuuat. </s> <s id="s.000220">Contra verò accidit, cum pondus i<lb></lb>pſum inter fulcimentum eſt & potentiam vel potentia i<lb></lb>pſa inter fulcimentum & pondus. </s> <s id="s.000221">tunc enim vectis vnâ <lb></lb>cum pondere attollitur. </s> <s id="s.000222">quæ licet vera ſint, non tamen in<lb></lb>de ſequitur, vectis pondus, quicquam quod curandum ſit, <lb></lb>in operatione efficere, aut impedire. </s> </p> <p type="main"> <s id="s.000223">Porrò vectem ita finire poſſumus, longitudinem eſ<lb></lb>ſe quandam inflexibilem, quæ fulcimento dato, datâ po<lb></lb>tentiâ datum pondus mouetur. </s> </p> <p type="main"> <s id="s.000224">Ipſa quoque Libra, vt diximus, vectis eſt: eius autem <lb></lb>naturæ, vt ſemper fulcimentum medium obtineat locum <lb></lb>inter pondus & pondus. </s> <s id="s.000225">Statera autem merus eſt vectis, ſi <lb></lb>ſparſum pro fulcimento; appendiculum verò currens pro <lb></lb>potentia mouente deputaueris. </s> </p> <p type="head"> <s id="s.000226"><emph type="italics"></emph>De Circulo eiusque natura Ariſtotelis doctri<lb></lb>na examinata.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000227">Ariſtoteles, quicquid mirum in Mechanicis opera<lb></lb>tur, id totum admirabili circuli naturæ eſſe tribuendum <lb></lb>arbitratur. </s> <s id="s.000228">Ait autem, abſurdum nullatenus eſſe, ſi ex re <lb></lb>mirabili mirandum quippiam oriatur. </s> <s id="s.000229">In circulo autem <pb xlink:href="007/01/028.jpg"></pb>quatuor inueniri qualitates admiratione dignas. </s> <s id="s.000230"><expan abbr="Primã">Primam</expan>, <lb></lb>quod ex contrarijs conſtituatur, mouente videlicet & <lb></lb>moto. </s> <s id="s.000231">Secundam, quòd contraria in eius circumferentia <lb></lb>inueniantur, quippe quæ cum vnica linea ſit, concaua ſi<lb></lb>mul eſt & conuexa. </s> <s id="s.000232">Tertiam, quod contrarijs feratur mo<lb></lb>tionibus, antrorſum nimirum, retrorſum, ſurſum, atque <lb></lb>deorſum. </s> <s id="s.000233">Quartam, quod vnicâ exiſtente ſemidiametro, <lb></lb>nullum in ea punctum ſumi poſſit, æqualis alteri, in latio<lb></lb>ne, velocitatis. </s> <s id="s.000234">Sit enim circulus AB, cuius centrum C, <lb></lb>ſemidiameter AC, ſumatur autem in ea punctum D, i<lb></lb>temq́ue punctum E. </s> <s id="s.000235">Erit itaque in ipſa circulatione D <lb></lb>tardius E, ipſum verò E tardius A, & ita citius id feretur <lb></lb>ſemper, quod remotius à mouente termino accipitur. </s> </p> <figure id="id.007.01.028.1.jpg" xlink:href="007/01/028/1.jpg"></figure> <p type="main"> <s id="s.000236">Hæc ex illo, quibus ne vltro aſ<lb></lb>ſenſum præbeamus non vnica de cau<lb></lb>ſa cohibemur. </s> <s id="s.000237">Dicimus igitur, videri <lb></lb>nobis, circulum non ex contrarijs <expan abbr="cō-ſtitui">con<lb></lb>ſtitui</expan>, puta ex manente & moto, ſed ex <lb></lb>moto ſimpliciter. </s> <s id="s.000238">Nulla eſt enim ſe<lb></lb>midiametri pars, quæ non moueatur. <lb></lb></s> <s id="s.000239">Punctum autem, quod ſtat, ſemidia<lb></lb>metri pars nulla eſt. </s> <s id="s.000240">Et ſanè cur moto <lb></lb><expan abbr="ſemidiamētro">ſemidiametro</expan> fiat circulus, non ideo accidit, quod <expan abbr="alterū">alterum</expan> <lb></lb>extremum ſtet, alterum verò moueatur:sed ideo quòd ſe<lb></lb>midiameter perpetuò eandem ſeruet longitudinem. </s> <s id="s.000241">Elli<lb></lb>pſis ſanè centrum habet, ſed ab eo ad circumferentiam <lb></lb>quatuor tantùm ſemidiametri quomodolibet ſumpti du<lb></lb>cuntur æquales. </s> <s id="s.000242">Si quis igitur ſemidiametrum daret pro<lb></lb>portione creſcentem & decreſcentem, ſtante altero ex<lb></lb>tremorum Ellipſis deſcriberetur. </s> <s id="s.000243">Præterea & ſpiralis li<lb></lb>nea, quæ mixta eſt, altero ſemidiametri extremo manen<lb></lb>te, altero vero moto producitur. </s> <s id="s.000244">Legem itaque circulo <pb xlink:href="007/01/029.jpg"></pb>præſcribit, non quidem quòd hæc extremitas ſter, illa ve<lb></lb>rò moueatur, ſed quod ſua circulatione ſemper ſemidia<lb></lb>meter eandem ſeruet longitudinem, quod vel ex ipſa cir<lb></lb>culi definitione colligitur. </s> </p> <p type="main"> <s id="s.000245">Ad ſecundum miraculum, ſcilicet, quòd in circulo <lb></lb>circum ferentia, quæ vacua linea eſt, concaua ſimul ſit, & <lb></lb>conuexa. </s> <s id="s.000246">Diceret quiſpiam id, ſi modò mirabile eſt non <lb></lb>circulari tantum, ſed cuilibet curuæ lineæ primo compe<lb></lb>tere, etenim & Ellipſis & Hyperbole, & Parabole, & ſpi<lb></lb>ra, tum Cyſſois, Conchois, & infinitæ aliæ irregulares <lb></lb>concauæ ſimul ſunt & conuexæ. </s> <s id="s.000247">Sed & hæc in ſuperficie<lb></lb>bus quoque deſiderantur. </s> </p> <p type="main"> <s id="s.000248">Ad tertium, quod contrarijs feratur lationibus, an<lb></lb>trorſum, retrorſum, ſurſum & deorſum. </s> <s id="s.000249">Dicimus, facilè <lb></lb>ſolui, Nullus enim, re bene perſpectâ, affirmauerit circu<lb></lb>lum contrarijs lationibus moueri. </s> </p> <figure id="id.007.01.029.1.jpg" xlink:href="007/01/029/1.jpg"></figure> <p type="main"> <s id="s.000250">Eſto enim circulus ABCD, <lb></lb>circa centrum E; ponamus ro<lb></lb>tari, & A verſus B, exempli gra<lb></lb>tiâ, antrorſum, mouebitur <expan abbr="autē">autem</expan> <lb></lb>& B verſus C, & C verſus D, tum <lb></lb>D verſus A. </s> <s id="s.000251">Non puto <expan abbr="quenquã">quenquam</expan> <lb></lb>dicturum, circulum hunc an<lb></lb>trorſum eodem tempore, & re<lb></lb>trorſum ferri nec ſurſum aut de<lb></lb>orſum, ſi enim quiſpiam per eius circuli circumferentiam <lb></lb>ambularet, is certè centrum ipſum ſemper ad dexteram <lb></lb>haberet, vel ad ſiniſtram, ſi ad dexteram, antrorſum ibit, ſi <lb></lb>ad ſiniſtram, retrorſum. </s> <s id="s.000252">Sed nec ſurſum vel deorſum, eſt <lb></lb>manifeſtum. </s> <s id="s.000253">Nihil autem prohibet eundem motum va<lb></lb>rio reſpectu contrarium dici poſſe, id tamen profectò fie<lb></lb>ri nequaquam poteſt, nempe A moueri verſus B, hoc eſt, <pb xlink:href="007/01/030.jpg"></pb>antrorſum, & eandem eodem tempore verſus B, id eſt, re<lb></lb>trorſum; repugnat enim naturæ. </s> </p> <p type="main"> <s id="s.000254">De quarto circuli miraculo, ibi erit nobis ſermo, vbi <lb></lb>ea perpenderimus primò, quæ Philoſophus de Circuli <lb></lb>productione diſſerens in medium profert. </s> <s id="s.000255">Sunt autem e<lb></lb>iuſmodi: </s> </p> <p type="main"> <s id="s.000256">Circulum quidem duplici notione produci, Natu<lb></lb>rali videlicet altera, & altera quæ eſt præter naturam, & <lb></lb>ideo circularem lineam in ter mixtas computari. </s> </p> <p type="main"> <s id="s.000257">Motus mixtus ait, vel proportione ſeruata fit, aut <lb></lb>non; Si proportione ſeruatâ, rectam lineam; ea verò non <lb></lb>ſeruata, circularem lineam produci. </s> </p> <figure id="id.007.01.030.1.jpg" xlink:href="007/01/030/1.jpg"></figure> <p type="main"> <s id="s.000258">Eſto enim rectangu<lb></lb>lum ABCD, cuius late<lb></lb>ra in datâ ſint proportio<lb></lb>ne, AD cum AB. </s> <s id="s.000259">Mo<lb></lb>ueatur A, duplici motu, <lb></lb>Altero quidem tendens <lb></lb>in B, altero vero ad mo<lb></lb>tum lineæ AB, feratur <lb></lb>verſus D, ſeruata inte<lb></lb>rim laterum proportione. </s> <s id="s.000260">Itaque ponatur ex motu ab A <lb></lb>verſus B, perueniſſe in E, ex motu autem quo proportio<lb></lb>naliter fertur cum linea AB, facta ipſa AB, in FH, perue<lb></lb>niſſe in G, & EG connectatur. </s> <s id="s.000261">Erit igitur Parallelogram<lb></lb>mum AEGF, Parallelogrammo ABCD proportiona<lb></lb>le ſimile, & circa eandem diametrum AGC. </s> <s id="s.000262">Semper igi<lb></lb>tur punctum A ſi duabus lationibus feratur, laterum pro<lb></lb>portione ſeruata, lineam producet rectam, diametrum <lb></lb>nempe AGC. </s> <s id="s.000263">Et hoc ſanè nullam habet dubitationem, <lb></lb>ex ijs quæ docet Euclides 1. 6. prop. 24. </s> </p> <p type="main"> <s id="s.000264">His ita demonſtratis hac vti videtur Philoſophus <pb xlink:href="007/01/031.jpg"></pb>argumentatione: Si mixtus motus proportione ſemotâ, <lb></lb>rectam producit, ſi nunquam ſemota, efficiet circulum; ſi <lb></lb>enim modo ſeruaretur, modo non, partim recta partim <lb></lb>non recta produceretur. </s> <s id="s.000265">Ingenioſa quidem argumenta<lb></lb>tio, ni vitium contineret. </s> <s id="s.000266">non enim mixtus motus, qui <lb></lb>nun quam ſeruatâ proportione fit, ſemper ci, culum pro<lb></lb>ducit, ſed & Ellipſim poteſt, & quamlibet aliam lineam, <lb></lb>cuius nulla pars ſit recta. </s> <s id="s.000267">Hanc difficultatem vidit Pico<lb></lb>lomineus in ſua Paraphraſi, & eam ſoluere conatus eſt, <lb></lb>ſed quàm bene, aliorum eſto iudicium. </s> <s id="s.000268">Cæterùm falſum <lb></lb>eſt, aſſerere circulum ex mixto motu nunquam ſeruatâ <lb></lb>proportione produci. </s> <s id="s.000269">ſeruat enim aſſiduè mixtus motus <lb></lb>quo producitur (ſi cum mixto motu producere velimus) <lb></lb>aliquam proportionem, ſed non eandem. </s> </p> <figure id="id.007.01.031.1.jpg" xlink:href="007/01/031/1.jpg"></figure> <p type="main"> <s id="s.000270">Eſto enim recta AB, cui ad rectos <lb></lb>angulos AC. </s> <s id="s.000271">Moueatur autem A, ver<lb></lb>ſus C per lineam AC, & eodem tempo<lb></lb>re linea AC, verſus B, ita tamen, vt ſem<lb></lb>per ipſi AB, ſit perpendicularis. </s> <s id="s.000272">feratur <lb></lb>autem eâ lege, vt quam proportionem <lb></lb>habet motus lineæ AC verſus B, ad mo<lb></lb>tum puncti A ve, ſus C, eandem habeat <lb></lb>ipſe motus ab A verſus C, ad reſiduum <lb></lb>lineæ AB, demptâ nempe ea parte quam <lb></lb>peragrauit linea AC mota verſus B. </s> <s id="s.000273">Sit <lb></lb>autem, cum AC ſuo motu peruenerit <lb></lb>in D, punctum A, ſimiliter ſuo motu per eam latum perue<lb></lb>nitle in E erit ergo ex mixto motu, non quidem in D, nec <lb></lb>in E, ſed in F, eritque punctum F in circum ferentia circu<lb></lb>li, cuius eſt diameter ipſa linea AB, quod quidem demon<lb></lb>ſtratur ex conuerſa propoſ. </s> <s id="s.000274">13. lib. 6. Elem. </s> <s id="s.000275">Eſt enim AE <lb></lb>hoc eſt DF media proportionalis inter EF, hoc eſt, AD, <lb></lb>& DB. </s> <s id="s.000276">Iterum ſi fiat motus AC in GH, ad motum H per <pb xlink:href="007/01/032.jpg"></pb>lineam AC, vſque in C, vt ſe habet proportio AG ad <lb></lb>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb></lb>eiuſdem circuli circum ferentia AFHB. ex quibus ha<lb></lb>bemus, circulum ex mixto motu fieri poſſe proportioni<lb></lb>bus quidem mediarum ſeruatis, ſed nunquam ijſdem. </s> </p> <p type="main"> <s id="s.000277">Vera hæc procul dubio ſunt; nihilominus, veluti ad <lb></lb>rectam producendam mixtus motus non eſt neceſſarius, <lb></lb>licet mixto motu produci poſſit, ita ne que ad circularem, <lb></lb>& ideo verum non eſſe quod aſſerebat Philoſophus, cir<lb></lb>culum ex mixto motu proportione nunquam ſeruatâ ne<lb></lb>ceſſariò produci. </s> </p> <p type="main"> <s id="s.000278">Conatur poſt hæc Ariſtoteles rationem afferre, cur <lb></lb>circuli partes, quò propiores centro fuerint, eo ſint tar<lb></lb>diores. </s> <s id="s.000279">Ait autem; ſi duobus ab eadem potentia latis hoc <lb></lb>quidem plus repellatur, illud verò minus, æquum eſt tar<lb></lb>diùs id moueri quod plus repellitur, eo quod minus. </s> <s id="s.000280">De<lb></lb>trahi autem plus lineam, cuius extremum propius eſt cen<lb></lb>tro illa quæ ſuum habet terminum à centro remotiorem. </s> </p> <figure id="id.007.01.032.1.jpg" xlink:href="007/01/032/1.jpg"></figure> <p type="main"> <s id="s.000281">Eſto, inquit, circulus <lb></lb>BCDE & alter in eo minor <lb></lb>MNOP circa idem centrum <lb></lb>A. Ducanturque; Diametri ma<lb></lb>ioris quidem CD, EB, mino<lb></lb>ris verò MO, NP. </s> <s id="s.000282">Itaque vbi <lb></lb>AB circulata eò peruenerit <lb></lb>vnde eſt greſſa, ipſa quoque <lb></lb>AM eo vnde moueri cœpe<lb></lb>rat, perueniet. </s> <s id="s.000283">Tardiùs autem <lb></lb>fertur AM, quam AD, pro<lb></lb>pterea quòd AM à centro <lb></lb>magis retrahatur quàm ipſa AB. </s> <s id="s.000284">Ducatur igitur ALF & <lb></lb>à puncto L, ipſi AB perpendicularis L q, cadens in mino-<pb xlink:href="007/01/033.jpg"></pb>ri circulo, & rurſus ab eodem L ipſi AB, parallela duca<lb></lb>tur LS, Ab S verò eidem perpendicularis ST, & ab F i<lb></lb>tem FX. </s> <s id="s.000285">Sunt ergo q L, ST, quidem æquales, nempe illæ, <lb></lb>per quæ, ſecundum naturam, mouentur puncta BM. </s> <s id="s.000286">Mo<lb></lb>tu verò retractionis ad centrum, hoc eſt, præter naturam, <lb></lb>plus motum eſt M quàm B. </s> <s id="s.000287">Maior enim eſt M q, ipſa BT, <lb></lb>quod, ceu notum, ſuppoſuit Ariſtoteles. </s> <s id="s.000288">nos autem inf. </s> <s id="s.000289">à <lb></lb>demonſtrabimus. </s> <s id="s.000290">Si igitur fiat vt motus præter naturam <lb></lb>ad motum præter naturam, ita motus <expan abbr="ſecūdum">ſecundum</expan> naturam, <lb></lb>ad motum ſecundum naturam, punctum B; cum M fuerit <lb></lb>in L, non erit in S, ſed in F. tunc enim, vt eſt FX motus ſe<lb></lb>cundùm naturam ad XB, præter naturam, ita eſt q L ſe<lb></lb>cundum naturam ad q M præter naturam; ſed BF maior <lb></lb>eſt ML, ergo proportione ſeruatâ, velociùs mouetur B <lb></lb>quàm M circa idem centrum A. </s> <s id="s.000291">Hæc autem ſumma eſt <lb></lb>eorum quæ præfert Ariſtoteles. </s> <s id="s.000292">Cæterùm nos parallelo<lb></lb>grammum, quod in figura eius habetur prætermiſimus, <lb></lb>quippe quod nihil ad eam quæ affertur, demonſtratio<lb></lb>nem faciat. </s> </p> <p type="main"> <s id="s.000293">Modò quod pollicebamur, nempe minorem eſſe <lb></lb>BT, quàm q M, ita demonſtramus. </s> <s id="s.000294"><expan abbr="quoniã">quoniam</expan> ST. ex prop. 13. <lb></lb>1. 6. media proportionalis eſt inter BT & TE, erit qua<lb></lb>dratum TS æquale <expan abbr="parallelogrãmo">parallelogrammo</expan> ſeu rectangulo BT, <lb></lb>TE, item, quoniam q L media proportionalis eſt inter <lb></lb>M q, & q O. erit quadratum q L æquale rectangulo M q, <lb></lb>q O, æqualia ergo ſunt rectangula BTE, M q O, itaque <lb></lb>reciproca latera habent proportionalia. </s> <s id="s.000295">quare, vt TE, ad <lb></lb>q O, ita M q ad TB, ſed TE maior eſt ipſa q O, quippe <lb></lb>quòd pars ſit q O ipſius TE, maior ergo & M q ipſa TB, <lb></lb>quod oſtendendum fuerat. </s> </p> <p type="main"> <s id="s.000296">Cæterùm ſubtilia & ingenioſa iſthæc eſſe non nega<lb></lb>mus, & longè faciliori & explicatiori modo veritas hæc <lb></lb>demonſtrari poteſt, reiectis nempe illis, ſecundùm, & prae<pb xlink:href="007/01/034.jpg"></pb>ter naturam motibus, qui <expan abbr="quidē">quidem</expan> in ſimplici circulo neceſ<lb></lb>ſario non cadunt: caderent autem fortaſſe, ſi de circulo <lb></lb>res eſſet à <expan abbr="pōderibus">ponderibus</expan> circumlatis ex ſtabili centro deſcri<lb></lb>pto, qua de re agit G. Vbaldus in Mechanicis tractatu de <lb></lb>libra. </s> <s id="s.000297">tunc enim dici poteſt, pondus quod aliâs rectà ad <lb></lb>mundi centrum tenderet, à circuli centro in circulatio<lb></lb>ne retrahi, ſed hæc ad circuli naturam, quatenus circulus <lb></lb>eſt, nequaquam ſpectant. </s> </p> <figure id="id.007.01.034.1.jpg" xlink:href="007/01/034/1.jpg"></figure> <p type="main"> <s id="s.000298">Eſto igitur circumferentia <lb></lb>AFBH, cuius centrum C, dia<lb></lb>meter ACB, ſemidiameter AC. <lb></lb>ſumatur in AC punctum quod<lb></lb>libet, D, & centro C, ſpatio CD, <lb></lb>circumferentia deſcribatur <lb></lb>DGEI. </s> <s id="s.000299">Dico punctum A velo<lb></lb>cius moueri puncto D eâdem <lb></lb>circulatione rotato. </s> <s id="s.000300">etenim vt <lb></lb>diameter ad diametrum, & ſemidiameter ad ſemidiame<lb></lb>trum, ita circumferentia ad circumferentiam: igitur vt <lb></lb>AC ad CD, ita circumferentia AFHB ad circumferen<lb></lb>tiam DGEI. </s> <s id="s.000301">At mota linea CA circa centrum C mo<lb></lb>uetur ſimul & CD, eodem igitur tempore rotationem <lb></lb>complent puncta AD, maius ergo ſpatium eodem tem<lb></lb>pore metitur A, ipſa D, quare velocior. </s> <s id="s.000302">Ita igitur ſe ha<lb></lb>bet velocitas ad velocitatem, vt circumferentia ad cir<lb></lb>cumferentiam, & diameter ad diametrum, quare id quod <lb></lb>mouetur in puncto à centro remotiori, velocius illo mo<lb></lb>uetur quod ab eo diſtat minus, quod fuerat <lb></lb>demonſtrandum. </s> </p> <pb xlink:href="007/01/035.jpg"></pb> </subchap1> </chap> <chap> <p type="head"> <s id="s.000303">QVÆSTIONES <lb></lb>MECHANICÆ.</s> </p> <subchap1> <p type="head"> <s id="s.000304">QVÆSTIO I.</s> </p> <p type="head"> <s id="s.000305"><emph type="italics"></emph>Cur maiores libræ exactiores ſint mi<lb></lb>noribus?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000306">Prioríbus, ceu fundamentis quibuſdam iactis, oppor<lb></lb>tunè ad quæſtiones proponendas, eas queue diluendas ſe <lb></lb>confert Ariſtoteles. </s> <s id="s.000307">Porro in propoſita quæſtione vide<lb></lb>tur prima fronte cauſſam quæri de re quæ non eſt: etenim <lb></lb>quis affirmauerit vnquam, lances quibus Apothecarij & <lb></lb>Macellarij vtuntur, magnas eas quidem, illis exactiores <lb></lb>eſſe quibus Gemmatij, atque Argentarij ſiliquis, & ſcru<lb></lb>pulis minutiſſima appendunt, quæ tamen perexiguæ ſunt, <lb></lb>& ſi illis comparentur minimæ? </s> <s id="s.000308">Veruntamen, ita prorſus <lb></lb>res habet, vt aſſerit Ariſtoteles. </s> <s id="s.000309">Non enim propterea <lb></lb>quòd illæ magnæ ſint, hæ verò exiguæ, hæ ſunt illis exa<lb></lb>ctiores; ſed quoniam magnæ, rudes ſunt, minores verò ex<lb></lb>quiſita diligentia elaboratæ, & à materiæ pertinacia libe<lb></lb>riores. </s> <s id="s.000310">Cæteris ergo paribus, exactiores eſſe maiores, ex <lb></lb>Philoſophi mente, ita docebimus. </s> </p> <figure id="id.007.01.035.1.jpg" xlink:href="007/01/035/1.jpg"></figure> <p type="main"> <s id="s.000311">Eſto libra maior AB, <lb></lb>cuius fulcimentum C. <lb></lb></s> <s id="s.000312">Minor verò libra DE, <lb></lb>circa idem <expan abbr="fulcimētum">fulcimentum</expan> <lb></lb>C, vnà cum maiori, ima<lb></lb>ginatione, conuerſa. </s> <s id="s.000313">Ap<lb></lb>ponatur quoduis pon<lb></lb>dus maiori libræ in A, <lb></lb>declinetque; exempli gratiâ in F, erit queue minor libra in G, <lb></lb>in eadem enim linea ſunt CGF. <expan abbr="Vnaq;">Vnaque</expan> igitur ex eodem <pb xlink:href="007/01/036.jpg"></pb>centro C portionem circuli deſcribet GD, AF, eritqueue <lb></lb>ACF ſector circuli, cuius diameter AB, ſed DCG ſe<lb></lb>ctor circuli, cuius diameter DE. </s> <s id="s.000314">Itaque vt diameter ad <lb></lb>diametrum, ita portio ad portionem: maior autem dia<lb></lb>meter AB diametro DE: maior ergo portio AF, portio<lb></lb>ne DG. quod autem maius eſt, minus obtutum fallit, ex<lb></lb>quiſitius itaque tractum ex maiori AB quàm ex ipſa mi<lb></lb>nori DE cognoſcemus, quod fuerat oſtendendum. </s> </p> <p type="main"> <s id="s.000315">Cæterùm hac eadem de cauſſa, Aſtronomica in<lb></lb>ſtrumenta, puta Aſtrolabia, Armillæ, & alia eiuſmodi, <lb></lb>quo ampliora eò exquiſitiora, & certiora probantur. </s> </p> <figure id="id.007.01.036.1.jpg" xlink:href="007/01/036/1.jpg"></figure> <p type="main"> <s id="s.000316">Eſto enim A<lb></lb>ſtrolabium magnum, <lb></lb>cuius diameter AB, <lb></lb>paruum autem CD, <lb></lb>circa idem centrum <lb></lb>E. </s> <s id="s.000317">Ducatur à centro <lb></lb>recta EF tangens ma<lb></lb>iorem circulum in F, <lb></lb><expan abbr="minorē">minorem</expan> verò <expan abbr="ſecãs">ſecans</expan> in <lb></lb>G, vt igitur GD ad to<lb></lb>tum circulum GCD, <lb></lb>ita FB. ad totum cir<lb></lb>culum FAB, vt ergò <lb></lb>GD ad FB, ita gradus <lb></lb>ſignati in GD, ad eos qui ſignantur in BF, maiores ergo <lb></lb>ſunt qui in FB, & minutarum partium capaciores. </s> <s id="s.000318">Hinc <lb></lb>itaque apparet, <expan abbr="inſtrumēta">inſtrumenta</expan> quælibet quò maiora fuerint, <lb></lb>eò eſſe & exquiſitiora, quod propoſuerat Ariſtoteles, in <lb></lb>hac quæſtione de Libra. </s> </p> <p type="main"> <s id="s.000319">Quod autem addit de fraudibus Purpurariorum, <lb></lb>inquiens; quamobrem machinántur ij qui purpuram ven<lb></lb>dunt, vt <expan abbr="pēdendo">pendendo</expan> defraudent, dum ad medium, ſpartum, <pb xlink:href="007/01/037.jpg"></pb>non ponentes; tum plumbum in alterutram libræ partem <lb></lb>infundentes; aut ligni quod ad radicem vergebat, in eam <lb></lb>quam deferri volunt partem conſtituentes, aut ſi nodum <lb></lb>habuerit, ligni enim grauior ea eſt pars, in qua eſt radix, <lb></lb>nodus verò radix quæ dam eſt. </s> <s id="s.000320">Hinc quæri poſſet: </s> </p> <p type="head"> <s id="s.000321"><emph type="italics"></emph>Vtrum libræ quæ ponderibus vacuæ æquilibrant, <lb></lb>omni prorſus careant fraude?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000322">Videri cuipiam poſſet, libras, quæ ponderibus va<lb></lb>cuæ, æquilibrant, omni prorſus fraude carere, verunta<lb></lb>men ita non eſt, quod diligentiùs (res enim magni mo<lb></lb>menti eſt) diſquiremus. </s> </p> <figure id="id.007.01.037.1.jpg" xlink:href="007/01/037/1.jpg"></figure> <p type="main"> <s id="s.000323">Eſto enim libra AB, ita diuiſa <lb></lb>in C, vt AC ſit partium IS, CB ve<lb></lb>rò earundem ſit 10. apponatur parti <lb></lb>A lanx ponderans 10, parti vero B <lb></lb>lanx ponderans 15. ex permutata i<lb></lb>gitur proportione libra ſuſpenſa in <lb></lb>C, aequè ponderabit; ſi autem appo<lb></lb>natur lanci B ſacoma vnciarum 6, & in lance A conſtitua<lb></lb>tur purpura, quæ ita ſe habeat ad vncias 6, vt 10 ad 15, ite<lb></lb>rum æqueponderabit, ſed vt 10 ad 15, ita 4 ad 6. Purpura<lb></lb>rius ergo fraudulentus, ponens in lance A vncias purpuræ <lb></lb>4, facto æquilibrio petet pretium vnciarum 6, & ita em<lb></lb>ptorem decipiet, quod ſanè innuerat, non autem demon<lb></lb>ſtrauerat Ariſtoteles. </s> <s id="s.000324">Hæc autem faciliora fient ex ijs, <lb></lb>quæ in ſequentibus quæſtionibus, vbi de vecte agetur, ex<lb></lb>plicabuntur. </s> </p> <p type="main"> <s id="s.000325">Detegitur autem fraus, ſi alternatim ſacoma in pon<lb></lb>derando, modo huic, modò illi lanci apponatur. </s> <s id="s.000326">Si enim <lb></lb>in lance A conſtituatur ſacoma, in B verò purpura non fit <lb></lb>æquilibrium. </s> </p> <pb xlink:href="007/01/038.jpg"></pb> </subchap1> <subchap1> <p type="head"> <s id="s.000327">QVÆSTIO II.</s> </p> <p type="head"> <s id="s.000328"><emph type="italics"></emph>Cur, ſi ſurſum libræ fulcimentum ſit, appoſito ad alteram partem <lb></lb>pondere, deſcendat libra, & eo amoto, iterum aſcendat, & ad æqui<lb></lb>librium reuertatur. </s> <s id="s.000329">Si verò deorſum fulcimentum fuerit, de<lb></lb>preſſa ad æquilibrium non reuertatur?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000330">Bimembrem proponit Philoſophus quæſtionem, quam <lb></lb>trimembrem debuit, triplici ſi quidem loco fulcimen<lb></lb>tum aptari poteſt, ſuperiori, medio, inferiori. </s> <s id="s.000331">Nos de o<lb></lb>mnibus verba faciemus. </s> </p> <p type="head"> <s id="s.000332">Prima Quæſtionis pars.</s> </p> <p type="head"> <s id="s.000333"><emph type="italics"></emph>De Libra ſurſum fulcimentum habente.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000334">Ariſtoteles primam quæſtionis partem ita ſoluit: An <lb></lb>quia ſurſum parte quidem exiſtente, plus libræ extra per<lb></lb>pendiculum ſit? </s> <s id="s.000335">Spartum enim perpendiculum eſt: quare <lb></lb>neceſſe eſt deorſum ferri id quod plus eſt, donec aſcendat <lb></lb>qua bifariam libram diuidit ad ipſum perpendiculum, <lb></lb>cum onus incumbat ad libræ partem ſurſus raptam. </s> </p> <figure id="id.007.01.038.1.jpg" xlink:href="007/01/038/1.jpg"></figure> <p type="main"> <s id="s.000336">Sit libra recta (hoc eſt, in æquilibrio conſtituta) BC, <lb></lb>ſpartum autem AD, <lb></lb>fulcimentum autem <lb></lb>D, deſuper: ſparto au<lb></lb>tem deorſum proie<lb></lb>cto ad M perpendicu<lb></lb>laris erit vbi ADM. <lb></lb></s> <s id="s.000337">Si igitur in ipſo B po<lb></lb>natur onus, erit B qui<lb></lb>dem vbi E, C autem <lb></lb>vbi H, quamobrem <lb></lb>ea quæ bifariam <expan abbr="librã">libram</expan> <lb></lb>ſecat, primo quidem erit DM, ipſius perpendiculi; in<expan abbr="cū-bente">cum<lb></lb>bente</expan> <expan abbr="autē">autem</expan> onere, erit DG. quare libræ ipſius EH, quod <pb xlink:href="007/01/039.jpg"></pb>extra perpendiculum, eſt AM, vbi eſt q P maius eſt dimi<lb></lb>dio. </s> <s id="s.000338">Si igitur amoueatur onus ab E, neceſſe eſt deorſum <lb></lb>ferri H, minus eſt enim E: ſiquidem igitur habuerit ſpar<lb></lb>tum ſurſum, propter hoc aſcendit libra. </s> </p> <p type="main"> <s id="s.000339">Peſſimè omnes ſchema hoc lineârunt, ita vt difficil<lb></lb>limum ſit auctoris inde ſenſum aſſequi. </s> <s id="s.000340">Nos autem cla<lb></lb>rius rem ob oculos ponimus. </s> <s id="s.000341">Id ergo ſibi vult Ariſtoteles, <lb></lb>propterea quòd pars iugi HDG maior eſt parte ED q, <lb></lb>eam eleuatam neceſſe eſt deſcendere, & iterum à perpen<lb></lb>diculari ADM bifariam diuiſam ad æquilibrium reuer<lb></lb>ti, Poſſumus nos idem ſimpliciori figura demonſtrare. </s> </p> <figure id="id.007.01.039.1.jpg" xlink:href="007/01/039/1.jpg"></figure> <p type="main"> <s id="s.000342">Eſto libra AB, bi<lb></lb>fariam, diuiſa in G, <lb></lb><expan abbr="fulcimentū">fulcimentum</expan> verò ſur<lb></lb>ſum vbi D, produca<lb></lb>tur perpendicularis <lb></lb>DC in E. </s> <s id="s.000343">Stante igi<lb></lb>tur libra AB, in æqui<lb></lb>librio æqualis eſt pars <lb></lb>CH, ipſi parti CB <lb></lb>apponatur pondus in <lb></lb>B. </s> <s id="s.000344">Declinabit igitur <lb></lb>libra mota circa centrum D, fiat autem in FG, ſecetqueue <lb></lb>perpendicularem in I. </s> <s id="s.000345">Punctum vero C eodem motu cir<lb></lb>ca idem centrum D erit in H. amoueatur pondus appoſi<lb></lb>tum: Dico libram à ſitu FG declinaturam & iterum re<lb></lb>uerſuram in ſitum priſtinum ACB. quoniam enim parti <lb></lb>GH, quæ æqualis eſt parti HF, additur pars IH, quæ à <lb></lb>perpendiculari eſt vſque ad H, ipſi verò HF eadem pars <lb></lb>detrahitur, erit IF minor GI. </s> <s id="s.000346">Superabitur ita que IF à <lb></lb>GI, deſcendetque FI, aſcendet verò IF, donec iterum li<pb xlink:href="007/01/040.jpg"></pb>bra ín partes æquales, vt antea, diuidatur in C, ſiat que æ<lb></lb>quilibrium. </s> </p> <p type="main"> <s id="s.000347">Hæc Philoſophi demonſtratio eſt vera illa quidem, <lb></lb>ſed non ex Mechanicis principijs, hoc eſt, ex centri graui<lb></lb>tatis ſpeculatione; nos igitur clariùs rem exponemus, his <lb></lb>quæ ſequuntur conſideratis. </s> </p> <p type="main"> <s id="s.000348">Si pondus circa ſtabile centrum conuertatur, dimiſ<lb></lb>ſum non ſtabit, niſi ſecundum grauitatis centrum fuerit <lb></lb>in perpendiculari, quæ per centrum, circa quod conuer<lb></lb>titur, ad mundi centrum cadit. </s> <s id="s.000349">Stabit autem in ea per<lb></lb>pendiculari in duobus punctis, altero à centro mundi <lb></lb>remotiſſimo; altero verò eidem quantum licuerit pro<lb></lb>ximo. </s> </p> <figure id="id.007.01.040.1.jpg" xlink:href="007/01/040/1.jpg"></figure> <p type="main"> <s id="s.000350">Eſto corpus A, cuius graui<lb></lb>tatis centrum B, nixum lineae in<lb></lb>flexibili BC, cum qua liberè <lb></lb>conuertatur circa centrum C. <lb></lb></s> <s id="s.000351">Ducatur autem per mundi cen<lb></lb>trum perpendicularis BCD. <lb></lb></s> <s id="s.000352">Sit igitur primò pondus A <expan abbr="ſecū-dum">ſecun<lb></lb>dum</expan> gracilis B centrum, in per<lb></lb>pendiculari ipſa ſupra centrum <lb></lb>C, puta in B. </s> <s id="s.000353">Moueatur & <expan abbr="deſcē-dat">deſcen<lb></lb>dat</expan> in E. </s> <s id="s.000354">Poſt hæc verò in F, hoc <lb></lb>eſt iterum in ipſa perpendiculari <lb></lb>infra centrum C. </s> <s id="s.000355">Deſcribet er<lb></lb>go circulum ex centro C, nem<lb></lb>pe BEF ſecantem perpendicu<lb></lb>larem in duobus punctis oppo<lb></lb>ſitis BF, dico, pondus libe è di-<pb xlink:href="007/01/041.jpg"></pb>miſſum in duobus tantum punctis ſuapte naturâ perman<lb></lb>ſurum, BF, in B, primò, quoniam cum corpus ipſum A à <lb></lb>perpendiculari, quæ ſuperficiei loco intelligitur ABCD <lb></lb>per centrum grauitatis diuidatur, in partes diuiditur æ<lb></lb>queponderantes, quare in neutram partem inclinabit. <lb></lb></s> <s id="s.000356">Stabit igitur erectum, lineæ ipſi fultum, inflexibili BC, <lb></lb>quæ nititur puncto C. </s> <s id="s.000357">In E verò non ſtabit, quippe quod <lb></lb>eo ſitu centrum ipſum grauitatis ſit extra perpendicula<lb></lb>rem, & ideo extra fulcimentum ſtabile C. </s> <s id="s.000358">In F verò ite<lb></lb>rum ſtabit, pendens à centro C, propterea quòd & ibi ab <lb></lb>eadem perpendiculari diuidatur per grauitatis centrum <lb></lb>in partes æqueponderantes. </s> <s id="s.000359">Eſt igitur reſpectu B, ipſum <lb></lb>punctum C, fulcimentum deorſum, reſpectu verò F, ful<lb></lb>cimentum ſurſum. </s> <s id="s.000360">At quia linea DFCB, à centro mundi, <lb></lb>quod eſt extra circulum, BEF, circulum ipſum per cen<lb></lb>trum C ſecat, erit pars eius DF quidem breuiſſima, ipſa <lb></lb>verò DB longiſſima, ex propoſ. 8. lib. 3. Elem. </s> <s id="s.000361">Pondus igi<lb></lb>tur A conuerſum ſeu liberè motum circa centrum C, in <lb></lb>duobus tantum locis perpendicularis lineæ ſtabit remo<lb></lb>tiſſimo altero, vt eſt B, altero verò eidem quam proximo, <lb></lb>vt eſt F. </s> </p> <p type="main"> <s id="s.000362">Hoc idem egregiè demonſtrauit G. Vbald. in ſuis <lb></lb>Mechanicis, Tractatu de Libra prop.1.</s> </p> <p type="main"> <s id="s.000363">Ad hæc autem dubitare quis poſſet, cur experientiâ <lb></lb>docente, pondera quæ infra fulcimentum habent, vt lan<lb></lb>cea ſariſſaue ad planum horizontis perpendiculariter e<lb></lb>recta, licet eo caſu grauitatis centrum in ipſa perpendicu<lb></lb>lari conſtituatur, non ſtet quidem, ſed altrinſecus ca<lb></lb>dat? </s> </p> <pb xlink:href="007/01/042.jpg"></pb> <figure id="id.007.01.042.1.jpg" xlink:href="007/01/042/1.jpg"></figure> <p type="main"> <s id="s.000364">Sit enim horizontis <lb></lb>planum AB, cui in puncto <lb></lb>C perpendiculariter ere<lb></lb>cta ſtatuatur ſariſſa DC, <lb></lb>cuius grauitatis centrum <lb></lb>E, in ipſa perpendiculari. <lb></lb></s> <s id="s.000365">Stabit ergo, ex præmiſſis, <lb></lb>& certè ſtare debuit, ſta<lb></lb>retqueue, ni vitium obſtaret <lb></lb>materiæ; non ſtat autem, <lb></lb>quia difficillimum eſt gra<lb></lb>uitatis centrum, ſuapte naturâ indiuiſibile, ita ad amuſſim <lb></lb>ſiſtere, vt in neutram partem à perpendiculari declinet. <lb></lb></s> <s id="s.000366">Hæc igitur ex ijs ſpeculationibus eſt, quæ ad praxim, ma<lb></lb>teriæ vitio impediente, aut vix aut nunquam rediguntur. </s> </p> <p type="main"> <s id="s.000367">Hinc autem ea quæſtio ſoluitur, Cur ij qui ſariſſam <lb></lb>erectam digito ſummo ſuſtinere conantur, non ſtent qui<lb></lb>dem, ſed digiti motu, ſariſſæ motum ſequantur. </s> </p> <p type="main"> <s id="s.000368">Id certè agit, qui nutantis ſariſſæ, digito, motum ſe<lb></lb>quitur; vt in ipſo motu digitum aſſiduè centro grauitatis <lb></lb>ſariſſæ ſupponat, vnde ſit vt nunquam extra fulcimentum <lb></lb>permanens, nunquam cadat. </s> </p> <p type="main"> <s id="s.000369">Similis huic alia quoque dubitatio ſoluitur: Nempe, <lb></lb>Cur turbines, quibus pueri ludunt, dum quidem rotan<lb></lb>tur, ſtent erecti, rotatione vero ceſſante, cadant. </s> </p> <figure id="id.007.01.042.2.jpg" xlink:href="007/01/042/2.jpg"></figure> <p type="main"> <s id="s.000370">Eſto enim Turbo AB, cu<lb></lb>ius grauitatis centrum C, planum <lb></lb>horizontis DE, linea Horizonti <lb></lb>perpendicularis ABC, tranſiens <lb></lb>per centrum grauitatis C, ſit au<lb></lb>tem fulcimentum in B. <expan abbr="Itaq;">Itaque</expan> cum <lb></lb>centrum grauitatis C ſit in ipſa <lb></lb>perpendiculari, ſtabit ex demon-<pb xlink:href="007/01/043.jpg"></pb>ſtratis, at ex vitio materiæ non ſtabit. </s> <s id="s.000371">Modò, vt aſſolet, ra<lb></lb>pido motu rotetur. </s> <s id="s.000372">Dico, Turbinem, motu ſeu rotatione <lb></lb>durante ſtare. </s> <s id="s.000373">ea autem paullatim elangueſcente ín ca<lb></lb>ſum vergere; ceſſante verò penitus cadere. </s> <s id="s.000374">fit enim ex in<lb></lb>æqualitate materiæ, vel operis ruditate, vel aliâ quauis <lb></lb>ex cauſſa, grauitatis centrum non eſſe in C, ſed exempli <lb></lb>gratiâ vbi F, notentur autem hinc inde Turbinis latera <lb></lb>notis GH. </s> <s id="s.000375">Vtique cum F extra perpendicularem fuerit, <lb></lb>cadet Turbo ad partem G; at id ne ſiat, efficitur velocita<lb></lb>te motus, quo centrum F transfertur in contrariam par<lb></lb>tem, vbi I. non autem cadit verſus H, quoniam eadem ve<lb></lb>locitate iterum transfertur in F, quamobrem cum huius<lb></lb>cemodi centri aſſidua circa perpendicularem fiat trans<lb></lb>latio, ad nullam partem Turbo cadere poteſt; elangue<lb></lb>ſcente verò motu rotans, paullatim incipit inclinari, do<lb></lb>nec eo penitus ceſſante, ad eam partem cadit, ad quam à <lb></lb>perpendiculari grauitatis centrum vergit. </s> <s id="s.000376">Deſcribit au<lb></lb>tem in rotatione grauitatis centrum, quod in medio non <lb></lb>eſt paruum circulum, per cuius centrum ipſa perpendi<lb></lb>cularis pertingit. </s> </p> <p type="main"> <s id="s.000377">Modò redeuntes ad libram, cuius fulcimentum eſt <lb></lb>ſurſum, alio principio, nempe Mechanico, cur depreſſa <lb></lb>ad æqualitatem reuertatur, demonſtrabimus. </s> </p> <pb xlink:href="007/01/044.jpg"></pb> <figure id="id.007.01.044.1.jpg" xlink:href="007/01/044/1.jpg"></figure> <p type="main"> <s id="s.000378">Sit igitur, vt ſu<lb></lb>periùs, libra AB, cu<lb></lb>ius centrum grauita<lb></lb>tis C, fulcimentum, <lb></lb>verò ſurſum, in D li<lb></lb>bræ quidem in C per<lb></lb>pendiculariter con<lb></lb>iunctum. </s> <s id="s.000379">Perpendi<lb></lb>cularis verò quæ per <lb></lb>fulcimentum, & gra<lb></lb>uitatis <expan abbr="cētrum">centrum</expan> tranſ<lb></lb>iens ad mundi cen<lb></lb>trum tendit DLE. ſtante igitur librâ in ſua æqualitate, e<lb></lb>rit centrum grauitatis C in ipſa perpendiculari infra qui<lb></lb>dem fulcimentum D. </s> <s id="s.000380">Loco verò, mundi centro quàm <lb></lb>proximo. </s> <s id="s.000381">Pondus poſt hæc apponatur in B, Declinabit au<lb></lb>tem pars CB, in HF, eleuatâ interim parte AC, in GH. <lb></lb></s> <s id="s.000382">Mota igitur libra tota, circa fulcimentum D mouebitur <lb></lb>circa idem centrum, & grauitatis centrum C, deſcribens <lb></lb>portionem circuli CH, fi etque; C in H, & quoniam H, hoc <lb></lb>eſt C, extra perpendicularem fit, amoto pondere, ex lan<lb></lb>ce B, cuius preſſione libra declinauerat, centrum grauita<lb></lb>tis per eandem circulì portionem HC, ad perpendicula<lb></lb>rem deſcendet, donec iterum in ea quieſcat, quo caſu li<lb></lb>bra AB ad æquilibrium reuertetur: quod fuerat demon<lb></lb>ſtrandum. </s> </p> <p type="main"> <s id="s.000383">His ita declaratis, oſtendemus, (quod nullus ante <lb></lb>nos animaduertit) harum librarum, quæ fulcimentum <lb></lb>habent ſurſum, eam eſſe naturam, vt non à quouis ponde<lb></lb>re appoſito moueantur, vel penitus declinent. </s> </p> <p type="main"> <s id="s.000384">Ijſdem enim ſtantibus, addatur quoduis pondus lan<lb></lb>ci B; Itaque ſi tale fuerit quod ſuperet reſiſtentiam, quam <pb xlink:href="007/01/045.jpg"></pb>illi facit centrum grauitatis contra naturam elatum in H <lb></lb>mouebitur quædam libra. </s> <s id="s.000385">Sin autem tam parui momenti <lb></lb>ſit, vt eam reſiſtentiam non vincat, ſtante circa locum in<lb></lb>fimum centro C, non mouebitur aut ſaltem parum, ipſa <lb></lb>libra. </s> </p> <p type="main"> <s id="s.000386">Hinc colligimus fieri poſſe, libras illas, quæ non "<lb></lb>quouis, quantumuis paruo pondere declinant, eas fulci- "<lb></lb>mentum habere ſurſum. </s> </p> <p type="main"> <s id="s.000387">His addimus, cæteris paribus, reſiſtentiam eò eſſe <lb></lb>maiorem, quo minus grauitatis centrum diſtat à fulci<lb></lb>mento ſurſum, circa quod ipſa libra aduertitur. </s> </p> <figure id="id.007.01.045.1.jpg" xlink:href="007/01/045/1.jpg"></figure> <p type="main"> <s id="s.000388">Eſto libra AB, cuius gra<lb></lb>uitatis centrum C, & primò <lb></lb>quidem eius fulcimentum <lb></lb>ſurſum ſit vbi D, itaque ſi ap<lb></lb>poſito pondere declinauerit <lb></lb>libra ad partes B, punctum <lb></lb>C, dum aſcendet deſcribet <lb></lb>portionem circuli CE. fulciatur iterum ſurſum puncto F, <lb></lb>& iterum declinet ad partes B, & iterum punctum C, dum <lb></lb>aſcendet, circuli portionem deſcribet CG. </s> <s id="s.000389">Eſt autem <lb></lb>minor angulus contactus ACE, angulo ACG, magis er<lb></lb>go ſurſum, hoc eſt, ad naturam ſui feretur C, per CG, ex <lb></lb>centro F, quàm per CE, ex centro D, quod fuerat de<lb></lb>monſtrandum. </s> </p> <p type="main"> <s id="s.000390">Hæc autem reſiſtentia ex eodem fulcimento & eo<lb></lb>dem pondere eo faciliùs ſuperabitur, quo longius bra<lb></lb>chium libræ fuerit. </s> </p> <p type="main"> <s id="s.000391">Eſto enim iterum libra AB, cuius fulcimentum D, <lb></lb>centrum grauitatis C, ſit & alia libra, cuius brachia bre<lb></lb>uiora EF, idem habens centrum C, & eidem puncto ſu<lb></lb>ſpenſa D. </s> <s id="s.000392">Dico igitur, eodem pondere appoſito, faciliùs <pb xlink:href="007/01/046.jpg"></pb><figure id="id.007.01.046.1.jpg" xlink:href="007/01/046/1.jpg"></figure><lb></lb>declinaturam libram ad <lb></lb>partes B, quàm ſi idem ap<lb></lb>poneretur in F. </s> <s id="s.000393">Demit<lb></lb>tatur enim, à puncto B <lb></lb>horizonti perpendicula<lb></lb>ris BG, & ab F item per<lb></lb>pendicularis FH, Tum <lb></lb>iuncta DB, centro D, eo<lb></lb>dem vero ſpatio DB, circuli portio deſcribatur BI, item <lb></lb>iuncta DF eodem centro D, ſpatio DF, portio circuli de<lb></lb>ſcribatu: FK. eſt autem maior DB ipſa DF ex propoſ. <lb></lb></s> <s id="s.000394">21. lib. 1. Elem. quare maioris circuli portio eſt BI quàm <lb></lb>FK. </s> <s id="s.000395">Obliquior autem, hoc eſt, à perpendiculari remotior <lb></lb>eſt motus per FK quàm per BI. maior ſi quidem eſt angu<lb></lb>lus KFH angulo IBG. quod nos ita probamus. </s> <s id="s.000396">Ducatur <lb></lb>perpendicularis ipſi DF linea LF contingens circulum <lb></lb>FK in F, item ipſi DB, perpendicularis MB, contingens <lb></lb>circulum BI in B, & quia angulus contingentiæ maioris <lb></lb>circuli minor eſt angulo contingentiæ minoris, erit KFL <lb></lb>maior IBM, Recti autem ſunt DFL, DBM, minor ergo <lb></lb>DFK reſidua ipſo DBI reſiduo. </s> <s id="s.000397">Maior autem DFC ex <lb></lb>iam citata propoſ. </s> <s id="s.000398"><expan abbr="quã">quam</expan> DBC, erit igitur reſiduum CFK, <lb></lb>multo minus reſiduo FBI, ſed recti ſunt CFH, FBG, ex <lb></lb>quibus ſi detrahantur CFK, FBI, erit reſiduum KFH, <lb></lb>maius reſiduo IBG, plus ergo retrahitur à perpendicula<lb></lb>ri pondus deſcendens per FK quàm per BI, minus igitur <lb></lb>præualebit reſiſtentiæ in C pondus appenſum in F, quàm <lb></lb>ſi appendatur in B. quod fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.000399">Poſſumus & idem quoque aliter oſtendere. </s> </p> <p type="main"> <s id="s.000400">Sint enim ſeorſum duæ libræ, maior AB, mïnor EF, <lb></lb>quàm commune grauitatis centrum C, fulcimentum ve<lb></lb>rò ſurſum D. </s> <s id="s.000401">Producatur perpendicularis DC, in G & fiat <lb></lb>CG æqualis CB, CH verò æqualis CF. </s> <s id="s.000402">Sunt igitur duo <pb xlink:href="007/01/047.jpg"></pb><figure id="id.007.01.047.1.jpg" xlink:href="007/01/047/1.jpg"></figure><lb></lb>vectes DG, DH, quo<lb></lb>rum quidem commu<lb></lb>ne fulcimentum D, <lb></lb>pondus verò C, poten<lb></lb>tiæ vbi HG. </s> <s id="s.000403">Sunt au<lb></lb>tem hi vectes eius na<lb></lb>turæ, in quibus <expan abbr="pōdus">pondus</expan> <lb></lb>eſt inter fulcimentum <lb></lb>& potentiam, itaque <lb></lb>vt ſe habet DC, ad <lb></lb>DG, ita potentia in G <lb></lb>ad pondus in C, item vt DC ad DH ita potentia in H ad <lb></lb>idem pondus C, ſed minor eſt propoſitio DC, ad DG <lb></lb>quàm DC ad DH. minor ergo potentia requiritur in G, <lb></lb>hoc eſt, in B, quàm in H, hoc eſt in F. </s> <s id="s.000404">Data igitur ponderis <lb></lb>æqualitate faciliùs ſuperabitur reſiſtentia C in B, quàm <lb></lb>in F: quod oſtendendum fuerat. </s> </p> <p type="main"> <s id="s.000405">Ad huius libræ naturam illæ quoque rediguntur, <lb></lb>quarum iugum non rectum quidem, ſed curuum, vel ex <lb></lb>rectis ſurſum in angulum ad fulcimentum detinentibus, <lb></lb>nec refert vtrum curuitas ſit circuli portio quælibet, aut <lb></lb>ellipſis ſecundum alterum diametrorum; quod ita de<lb></lb>monſtramus. </s> </p> <figure id="id.007.01.047.2.jpg" xlink:href="007/01/047/2.jpg"></figure> <p type="main"> <s id="s.000406">Eſto libra, cuius iugum <lb></lb>curuum <expan abbr="angulatūue">angulatumue</expan> ABC, <lb></lb>cuius fulcimentum B, æqua<lb></lb>lia autem brachia AB, BC, <lb></lb>& pondera item <expan abbr="vtrinq;">vtrinque</expan> ap<lb></lb>penſa æqualia. </s> <s id="s.000407">Demittatur <lb></lb>ex puncto B ad mundi cen<lb></lb>trum perpendicularis BD. <lb></lb></s> <s id="s.000408">Stante igitur libra ABC in <lb></lb>æquilibrio, erit eius graui<pb xlink:href="007/01/048.jpg"></pb>tatis centrum in ipſa perpendiculari BD, puta in E. </s> <s id="s.000409">Ap<lb></lb>ponatur pondus in C, declinabit autem libra, ſit autem <lb></lb>iuxta poſitionem FBG. </s> <s id="s.000410">Centrum igitur grauitatis E per <lb></lb>portionem EH, erit in H. </s> <s id="s.000411">Aſcendit ergo centrum graui<lb></lb>tatis in H, hoc eſt, ſurſum, id eſt, contra eius naturam; a<lb></lb>moto igitur pondere ex C, grauitatis centrum extra per<lb></lb>pendicularem conſtitutum rurſus deſcendet, & iterum <lb></lb>libra ABC ad æquilibrium reuertetur. </s> <s id="s.000412">Hoc idem egre<lb></lb>giè oſtendit G. Vbald. in tractatu de libra, propoſ. 4. </s> </p> <p type="main"> <s id="s.000413">Hinc ratio pendet earum imaguncularum, quas ex <lb></lb>contuſa papyro ligneaue leui materia compingunt, per<lb></lb>queue manus earum ambas, ferreum filum trajicientes, v<lb></lb>trinque plumbea appendunt pondera æqualia, ea <expan abbr="quidē">quidem</expan> <lb></lb>lege, vt centrum grauitatis infra pedes imaguncula ſta<lb></lb>tuatur. </s> <s id="s.000414">Tunc enim extenſo filo imponentes ceu funam<lb></lb>bulos per illud, vltrò citroque; decurrere faciunt, imagun<lb></lb>cula interim erecta & in neutram partem cadente, quod <lb></lb>vt figurâ clarius fiat; </s> </p> <figure id="id.007.01.048.1.jpg" xlink:href="007/01/048/1.jpg"></figure> <p type="main"> <s id="s.000415">Eſto imaguncu<lb></lb>la AB, per cuius ma<lb></lb>nus traijciatur filum <lb></lb>ferreum curuum <expan abbr="cū">cum</expan> <lb></lb>æ qualibus ponderi<lb></lb>bus hinc inde <expan abbr="appē-ſis">appen<lb></lb>ſis</expan> CD. </s> <s id="s.000416">Nitatur au<lb></lb>tem pedibus filo HI <lb></lb>in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, ſitque; totìus ma<lb></lb>chinæ grauitatis <expan abbr="cē-trum">cen<lb></lb>trum</expan> E, ſitque <expan abbr="per-pēdicularis">per<lb></lb>pendicularis</expan> per gra<lb></lb>uitatis <expan abbr="centrū">centrum</expan> tranſi<lb></lb>ens A<emph type="italics"></emph>B<emph.end type="italics"></emph.end> E. </s> <s id="s.000417">Itaque in<lb></lb>clinata imaguncula, & conuerſa circa punctum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, ſi de-<pb xlink:href="007/01/049.jpg"></pb>clinet ad partes I, centrum grauitatis eleuabitur in F. </s> <s id="s.000418">Si <lb></lb>verò ad partes H eleuabitur in G. quare cum FG loca <lb></lb>ſint remotiora à mundi centro, quàm ſit E, non ſtabit gra<lb></lb>uitatis centrum in punctis FG, ſed ad infimum locum re<lb></lb>uertetur, hoc eſt, in ipſa perpendiculari in E, & imagun<lb></lb>cula ad perpendiculum ipſi H<emph type="italics"></emph>B<emph.end type="italics"></emph.end>E filo, hoc eſt, ipſi hori<lb></lb>zonti reuertetur. </s> </p> <p type="main"> <s id="s.000419">Hinc etiam Arictum, Teſtudinumqueue demolito<lb></lb>riarum Machinarum vis pendet, nempe ex ratione libra<lb></lb>rum, quæ fulcimentum habent ſurſum. </s> </p> <figure id="id.007.01.049.1.jpg" xlink:href="007/01/049/1.jpg"></figure> <p type="main"> <s id="s.000420">Eſto enim Aries A<emph type="italics"></emph>B<emph.end type="italics"></emph.end> <lb></lb>funi appenſus CD, cu<lb></lb>ius grauitatis centrum, <lb></lb>D, perpendicularis verò <lb></lb>quæ ad mundi centrum <lb></lb>ipſa CDE. </s> <s id="s.000421">Stante igitur <lb></lb>in æquilibrio machina, <lb></lb>centrum grauitatis erit <lb></lb>in ipſa perpendiculari. <lb></lb></s> <s id="s.000422">Applicetur alicubi po<lb></lb>tentia retropellens, eleuabitur igitur centrum grauitatis <lb></lb>per circuli portionem DF, cuius ſemidiameter eſt CD, <lb></lb>ſi etqueue iuxta poſitionem CF. </s> <s id="s.000423">Aries verò in GFH. </s> <s id="s.000424">Di<lb></lb>miſſa itaque Machina centrum F vtpote graue, non ſtabit, <lb></lb>ſed ſuapte naturâ reuertetur in D. </s> <s id="s.000425">Quadruplici autem <lb></lb>de cauſſa motus Arietis violentiſſimus eſt ex vi naturalis <lb></lb>ponderis, quo deorſum fertur, tum velocitate naturalis <lb></lb>motus in deſcendendo auctæ, tum ex vi potentiæ impel<lb></lb>lentis, & naturalem motum adiuuantis, tum ex velocita<lb></lb>te ex motu violento deorſum & antrorſum impellente <lb></lb>acquiſitâ. </s> <s id="s.000426">Id etiam addimus, eo validiores fore ictus, quò <lb></lb>grauior fuerit Machina, & maius ſpatium, quo retrotra<pb xlink:href="007/01/050.jpg"></pb>hitur, grauitate ipſa & ſpatio tum virium vnione opera<lb></lb>tionem mirum in modum adiuuantibus. </s> </p> <p type="main"> <s id="s.000427">Hæc nos de Libra ſurſum fulcimentum habente, dí<lb></lb>cta voluimus, nunc de ea, cuius fulcimentum deorſum, <lb></lb>eſt, verba faciemus. </s> </p> <p type="head"> <s id="s.000428">Altera quæſtionis pars:</s> </p> <p type="head"> <s id="s.000429"><emph type="italics"></emph>De Libra cuius fulcimentum deorſum eſt.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000430">Si deorſum fuerit, inquit Ariſtoteles, id quod ſub<lb></lb>ſtat, contrarium facit illi quæ ſurſum habet, nempe ad æ<lb></lb>quilibrium non reuertitur. </s> <s id="s.000431">Plus enim, ait, dimidio fit li<lb></lb>bræ, quæ deorſum eſt pars, quàm quod perpendiculum <lb></lb>ſecet, quapropter non aſcendit. </s> <s id="s.000432">eleuata enim pars leuior <lb></lb>eſt. </s> </p> <p type="main"> <s id="s.000433">Hæc ille, qui ſchemate quoque rem aperit, at eo a<lb></lb>pud interpretes, & Picolomineum Paraphraſtem, ita <expan abbr="mē-dosè">men<lb></lb>dosè</expan> lineato, vt inde obſcuritas lucis loco, legentibus of<lb></lb>fundatur. </s> <s id="s.000434">Nos, quod & ſuprà quoque fecimus, noſtra fi<lb></lb>gurâ, ſole ipſo clariorem, ex Ariſto telis ipſius mente rem <lb></lb>totam efficiemus. </s> </p> <figure id="id.007.01.050.1.jpg" xlink:href="007/01/050/1.jpg"></figure> <p type="main"> <s id="s.000435">Sit libra recta, (hoc <lb></lb>eſt, in æquilibrio con<lb></lb>ſtituta) vbi NG. </s> <s id="s.000436">Per<lb></lb>pendiculum autem (id <lb></lb>eſt, perpendicularis <lb></lb>quæ ad mundi <expan abbr="centrū">centrum</expan>) <lb></lb>KLM. </s> <s id="s.000437">Bifariam igitur <lb></lb>ſecatur NG. impoſito <lb></lb>poſthæc onere in ipſo <lb></lb>N, erit quidem N, vbi <lb></lb>O. ipſum autem G vbi <lb></lb>R. KL autem vbi LP. <pb xlink:href="007/01/051.jpg"></pb>quare maius eſt KO, quam LR, ipſa parte PKL. </s> <s id="s.000438">Amoto <lb></lb>igitur onere neceſſe eſt manere. </s> <s id="s.000439">Incumbit enim onus ex<lb></lb>ceſſus medietatis eius, vbi eſt F. </s> <s id="s.000440">Senſus eſt igitur, idcirco <lb></lb>partem iugi KLO inclinatam, ad æquilibrium non re<lb></lb>uerti, propterea quòd maior ſit ipſa KLO pars quæ tra<lb></lb>hit, ipſa RKL, quæ trahitur & eleuatur. </s> </p> <figure id="id.007.01.051.1.jpg" xlink:href="007/01/051/1.jpg"></figure> <p type="main"> <s id="s.000441">Poteſt hoc idem longè <lb></lb>ſimpliciori themate demon<lb></lb>ſtrari. </s> <s id="s.000442">Eſto enim libra AB, <lb></lb>cuius centrum C, fulcimen<lb></lb>tum vero deorſum D, Per<lb></lb>pendicularis per centrum & <lb></lb>fulcimentum tranſiens EF. <lb></lb></s> <s id="s.000443">Apponatur pondus in B, de<lb></lb>clinabitque; puta ad GH, cen<lb></lb>trum verò C, ex ſtabili fulci<lb></lb>mento D, circuli portionem deſcribet CI, libra autem <lb></lb>ſecabit EF perpendicularem in K. Æquales autem ſunt <lb></lb>IG, IH, at ex parte HI deſumpta eſt KI, addita queue ipſi <lb></lb>IG, maior eſt ergo tota KG, torâ KH. </s> <s id="s.000444">Non igitur KH <lb></lb>habet KG, ſed libra, niſi impedita fuerit, cum centro C <lb></lb>deſcendente per I in M, ad ipſam perpendicularem dela<lb></lb>ta, ad in feriorem partem, mutatis vicibus quieſcet, facto <lb></lb>nempe fulcimento ſurſum, fietque; horizonti æque diſtans <lb></lb>iuxta poſitionem LMN. </s> </p> <p type="main"> <s id="s.000445">Demonſtratio <expan abbr="quidē">quidem</expan> eſt hæc, ſed non ex proprijs prin<lb></lb>cipijs Mechanicis, <expan abbr="nēpe">nempe</expan> ex ratione <expan abbr="cēt">cent</expan>ri grauitatis petitâ. <lb></lb></s> <s id="s.000446">Iiſdem enim ſtantibus, <expan abbr="cū">cum</expan> centrum grauitatis C fiat extra <lb></lb>perpendicularem, deſcendens ad I, nunquam reuertetur <lb></lb>in C, aſcenderet enim; ſed ſi liberè circa centrum D con<lb></lb>uerteretur, deſcendens vt dictum eſt per circulum CIM <lb></lb>pondus B, fieret in L, A vero in N adepta poſitione <lb></lb>LMN. </s> </p> <pb xlink:href="007/01/052.jpg"></pb> <p type="main"> <s id="s.000447">Cur autem huius libræ, quæ aliàs inutilis eſt, memi<lb></lb>nerit Philoſophus, ea videtur cauſſa, quòd inde vectis vir<lb></lb>tutem eliciat, vt ſuo loco videbimus. </s> <s id="s.000448">Id autem valde mi<lb></lb>rum, hominem acutiſſimum nihil prorſus de ea libra egiſ<lb></lb>ſe, quæ fulcimentum nec ſurſum habet, nec deorſum, ſed <lb></lb>in ipſo exquiſitè medio, ita vt centrum grauitatis in ipſo<lb></lb>met fulcimento conſiſtat. </s> <s id="s.000449">Nos igitur de hac quod operæ <lb></lb>pretium fuerit, & ad rem, qua de agimus, vtile, in medium <lb></lb>proferemus. </s> </p> <p type="head"> <s id="s.000450"><emph type="italics"></emph>De libra cuius fulcimentum est in medio.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000451">Dicimus itaque, libram, cuius fulcimentum nec ſur<lb></lb>ſum eſt, nec deorſum, ſed prorſus in medio, nempe in ipſo <lb></lb>grauitatis centro, vbi brachia & pondera vtrinque appo<lb></lb>ſita fuerint æqualia, ſi ab æquilibrio mouentur, quomo<lb></lb>docunque poſita, ſtare nec ab eo, quem adepta eſt, ſitu di<lb></lb>moueri. </s> </p> <p type="main"> <s id="s.000452">Quæſtionem hanc perperam tractârunt recentio<lb></lb>res quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij <lb></lb>nonnulli, qui Iordani Nemoracij aſſertiones ſunt ſecuti, <lb></lb>quorum demonſtrationes vel paralogiſmos potiùs egre<lb></lb>giè confutauit in libr. </s> <s id="s.000453">Mechanicor. Tractatu de libra pro<lb></lb>poſ. </s> <s id="s.000454">4. Guid. Vbald. ad cuius probatiſſima ſcripta Lecto<lb></lb>rem ablegamus. </s> <s id="s.000455">fuſiſſimè enim ibi hac de re & abſolutiſſi<lb></lb>mè agit. </s> <s id="s.000456">Nos autem quidem paucis ea, quæ ad hanc co<lb></lb>gnitionem pertinent, explicabimus. </s> </p> <figure id="id.007.01.052.1.jpg" xlink:href="007/01/052/1.jpg"></figure> <p type="main"> <s id="s.000457">Eſto enim libra A<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <lb></lb>cuius brachia æqualia, <lb></lb>& centrum grauitatis <lb></lb>in C, brachijs verò <lb></lb>AC, C<emph type="italics"></emph>B<emph.end type="italics"></emph.end> æqualibus, æ<lb></lb>qualia pondera hinc <lb></lb>inde <expan abbr="apponãtur">apponantur</expan>. </s> <s id="s.000458">Tum <pb xlink:href="007/01/053.jpg"></pb>fulcimento in medio, hoc eſt, vbi grauitatis centrum C <lb></lb>applicato per centrum ipſum C ducatur perpendicularis, <lb></lb>quæ ad mundi centrum, DCE, ſitque primum libra æ<lb></lb>quediſtans horizonti, conſtituta. </s> <s id="s.000459">Tum ex altera parte <lb></lb>preſſa moueatur & fiat iuxta poſitionem FCG. </s> <s id="s.000460">Dico eam <lb></lb>dimiſſam permanere, etenim cum grauitatis centrum ſit <lb></lb>in ipſa perpendiculari, in neutram partem verget, ſed nec <lb></lb>vergere poteſt, quippe quod non circa fulcimentum ceu <lb></lb>centrum motus, moueatur grauitatis centrum, ſed in ipſo <lb></lb>ſit fulcimento; ſitum ergo non mutat. </s> <s id="s.000461">Præterea cum per<lb></lb>pendicularis DCE per grauitatis centrum ducatur, cor<lb></lb>pus ipſum ex ponderibus & libra conſtans ab ea in partes <lb></lb>çque ponderantes ſecatur, & ideo ex centri grauitatis dif<lb></lb>finitione, quam protulit Pappus, corpus ipſum centro <lb></lb>grauitatis appenſum, dum fertur quieſcit, & ſeruat eam, <lb></lb>quam à principio habuit poſitione. </s> <s id="s.000462">Et ſanè ſi partes quo<lb></lb>modo libet librâ per grauitatis centrum diuisâ, ſunt æ<lb></lb>queponderantes nec trahent inuicem, nec trahentur, ſta<lb></lb>bit ergo libra, & quam adepta fuerat poſitionem, eam ſer<lb></lb>uabit. </s> <s id="s.000463">Id tamen non negamus, difficile eſſe libras eiuſce<lb></lb>modi ex materia fabricare, quippe quod non omnia quæ <lb></lb>vera ſunt, & euidentiſſimis demonſtrationibus patent, <lb></lb>commodè ad praxim, ex artis & materiæ imperfectione, <lb></lb>reducuntur. </s> </p> <p type="main"> <s id="s.000464">Cæterùm harum librarum ea eſt virtus, vt vel mini<lb></lb>mo pondere altrinſecus appoſito, declinet; quod illis quæ <lb></lb>centrum ſurſum habent, non euenire, demonſtrauimus. </s> </p> <p type="main"> <s id="s.000465">Circa hæc poſſet cuipiam oriri Dubium, num chor<lb></lb>dulæ, quibus lances appenduntur, variationem aliquam <lb></lb>circa ea quæ demonſtrata ſunt, inducere valeant. </s> </p> <p type="main"> <s id="s.000466">Dicimus nullam inde fieri: Eſto enim libra AB, cu<lb></lb>ius centrum & fulcimentum C, ab cuius extremitate A <lb></lb>dependeat, funiculus AD, ab alia verò <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, funiculus <emph type="italics"></emph>B<emph.end type="italics"></emph.end>E, <pb xlink:href="007/01/054.jpg"></pb><figure id="id.007.01.054.1.jpg" xlink:href="007/01/054/1.jpg"></figure><lb></lb>quibus appenſæ ſint æ<lb></lb>qualis ponderis lances <lb></lb>DE. </s> <s id="s.000467">Moueatur libra, <lb></lb>fiatque in ICH, funi<lb></lb>culi verò in lancibus in <lb></lb>IK, HL. ſecet autem fu<lb></lb>niculus IK libram A<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <lb></lb>in M, LH verò produ<lb></lb>catur & eandem ſecer <lb></lb>in N. quoniam igitur <lb></lb>IC, æqualis eſt CH, pa<lb></lb>rallelæ autem KI, LN æquales <expan abbr="erūt">erunt</expan> alterni anguli MIC, <lb></lb>NHC, ſed & anguli ad verticem ICH, BCH æquales <lb></lb>ſunt, quare triangulum IMC, æquale triangulo HNC, <lb></lb>& latera lateribus, quæ æqualibus angulis ſubtenduntur. <lb></lb></s> <s id="s.000468">Æqualis eſt igitur linea MC lineæ NC. </s> <s id="s.000469">Itaque ſi ponde<lb></lb>ra lancesue, KL mente concipiantur appenſæ in punctis <lb></lb>MN, ex brachiorum & ponderum æqualitate æquepon<lb></lb>derabunt. </s> <s id="s.000470">quod fuerat demonſtrandum. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000471">QVÆSTIO III.</s> </p> <p type="head"> <s id="s.000472"><emph type="italics"></emph>Cur exiguæ vires (quod etiam à principio dixerat) vecte magna <lb></lb>mouent pondera, vectes inſuper onus accipientes, cum facilius <lb></lb>ſit, minorem mouere grauitatem, minor est au<lb></lb>tem ſine vecte?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000473">Ariſtoteles ita quæſtionem proponit, vt eam Rheto<lb></lb>rico quodam fuco admirabiliorem faciat. </s> <s id="s.000474">Soluit au<lb></lb>tem hoc pacto, <expan abbr="inquiēs">inquiens</expan>, fieri poſſe eam eſſe cauſſam, quod <lb></lb>vectis ſit libra, eius nempe generis quod fulcimentum ha<lb></lb>bet deorſum, atque id circo in ipſa preſſione in partes in<lb></lb>æquales vectem diuidi. </s> </p> <pb xlink:href="007/01/055.jpg"></pb> <figure id="id.007.01.055.1.jpg" xlink:href="007/01/055/1.jpg"></figure> <p type="main"> <s id="s.000475">Figura quam ex<lb></lb>hibet, vix ferè quid ſi<lb></lb>bi velit explicat. </s> <s id="s.000476">Nos <lb></lb>ad eius <expan abbr="mētem">mentem</expan> aliam <lb></lb>proponemus <expan abbr="eamq;">eamque</expan> <lb></lb>longè clariorem. </s> </p> <p type="main"> <s id="s.000477">Eſto vectis A<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <lb></lb>cuius fulcimentum, <lb></lb>deorſum in C, pon<lb></lb>dus D, potentia ex vecte, pondus ſuſtinens E. </s> <s id="s.000478">Perpendi<lb></lb>cularis per fulcimentum FCG. </s> <s id="s.000479">Itaque quoniam poten<lb></lb>tia in E non ſuperat pondus D, nec ab eo ſuperatur, ſtat <lb></lb>vectis cum potentia Horizonti æquidiſtans, hoc eſt, in æ<lb></lb>quilibrio, vectis autem in puncto C diuiditur in partes æ<lb></lb>queponderantes. </s> <s id="s.000480">Modo præualeat potentia ponderi, & <lb></lb>vectem deprimat, fiat autem in LCH, erit igitur <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, in L, <lb></lb>A in H, D in K, & CF, quæ vectem in partes æque ponde<lb></lb>rantes diuidebat, in CI. </s> <s id="s.000481">Iam igitur non æqueponderant <lb></lb>partes, ſi quidem pars vectis FCI, aufertur parti HCI, & <lb></lb>adiungitur parti ICL, quæ ideo ſit ponderoſior, vnde & <lb></lb>potentia ad ponderis eleuationem adiuuatur. </s> <s id="s.000482">Eadem i<lb></lb>gitur vtitur hic demonſtratione, quam in explicando ef<lb></lb>fectu libræ, cuius fulcimentum deorſum eſt, adhibuerat. <lb></lb></s> <s id="s.000483">Nec alia de cauſſa, vt ſuprà notauimus, videtur eius libræ <lb></lb>in ſuperiori quæſtione, conſiderationem introduxiſſe. </s> <s id="s.000484">Et <lb></lb>ſanè verum eſt quod concludit, Veruntamen minimi eſt <lb></lb>momenti ad tantam vim parua illa adiectio, quæ parti ve<lb></lb>ctis depreſſæ in ipſa depreſſione adiungitur. </s> <s id="s.000485">Aliunde igi<lb></lb>tur tantæ rei cauſſa eſt petenda, quod & nos deinceps fa<lb></lb>ciemus. </s> <s id="s.000486">Videtur autem ipſe quoque Ariſtoteles non ſibi <lb></lb>prorſus in aſſignata ratione ſatis feciſſe, & ideo ſubiungit: <lb></lb>quoniam ab æquali pondere celerius mouetur maior ca<lb></lb>rum quæ à centro ſunt duo verò pondera; quod mouet & <pb xlink:href="007/01/056.jpg"></pb>quod mouetur, quod igitur motum pondus ad mouens <lb></lb>longitudo patitur ad longitudinem, ſemper autem <expan abbr="quã-tum">quan<lb></lb>tum</expan> ab hypomochlio (id eſt, fulcimento) diſtabit magis, <lb></lb>tanto facilius mouebit. </s> <s id="s.000487">Cauſſa autem est, quæ retro com<lb></lb>memorata eſt, quoniam quæ plus à centro diſtat <expan abbr="maiorē">maiorem</expan> <lb></lb>deſcribit circulum. </s> <s id="s.000488">quare ab eadem potentia plus ſupera<lb></lb>bitur id quod mouetur, quæ plus à fulcimento diſtat. </s> <s id="s.000489">Hųc <lb></lb>ille, qui aſſerit duo pondera in vecte conſiderari, Pondus <lb></lb>nempe motum, & mouentem Potentiam (hanc enim <expan abbr="pō-deris">pon<lb></lb>deris</expan> habere vim <expan abbr="atq;">atque</expan> rationem certum eſt) Vires autem <lb></lb>potentiam acquirere ex brachij longitudine, & ex inde <lb></lb>conſequenti velocitate, quo enim brachia longiora, eo <lb></lb>in extremitate velociora, atque idcirco ita ſe habere mo<lb></lb>tum pondus ad potentiam mouentem, vt brachij longi<lb></lb>tudo ad brachij longitudinem: brachia autem vocamus, <lb></lb>partes illas vectis, quæ à fulcimento ad vtranque vectis <lb></lb>extremitatem pertingunt, & ideo quantum à fulcimento <lb></lb>potentia diſtabit magis, eo faciliùs pondus mouebit. </s> </p> <p type="main"> <s id="s.000490">Vera vtique & exploratiſſima hæc aſſertio eſt. </s> <s id="s.000491">Ve<lb></lb>runtamen, cauſſam huiuſce mirabilis effectus, eſſe velo<lb></lb>citatem, quæ brachij longitudinem conſequitur, non af<lb></lb>firmamus. </s> <s id="s.000492">quæ enim velocitas in re ſtante? </s> <s id="s.000493">Stant autem <lb></lb>vectis, & libra dum manent in æquilibrio, & nihilo ſecius <lb></lb>parua potentia ingens ſuſtinet pondus. </s> </p> <p type="main"> <s id="s.000494">Dicet ad hæc quiſpiam, velocitatem in longiori bra<lb></lb>chio ſi non actu, ſaltem potentiâ eſſe maiorem. </s> <s id="s.000495">At quæſo <lb></lb>quid in re quæ eſt actu, momenti habet potentia? </s> <s id="s.000496">actu e<lb></lb>nim ſuſtinet, ſuſtinens. </s> <s id="s.000497">Conſequìtur, (id vtique fatemur) <lb></lb>neceſſariò velocitas maior motu brachij maioris; non ta<lb></lb>men cauſſa eſt cur vis loco vbi velocitas maior ſit, appoſi<lb></lb>ta magis moueat. </s> <s id="s.000498">Sanè ex velocitate, dum mouentur, <expan abbr="pō-dus">pon<lb></lb>dus</expan> acquirere corpora, tum proiecta, tum cadentia cer<lb></lb>tum eſt, quod etiam in quæſtione 19. cum Philoſopho <expan abbr="cō-">con-</expan><pb xlink:href="007/01/057.jpg"></pb>ſiderabimus. </s> <s id="s.000499">Sed hoc ex velocitate & motu ſit, quæ ſunt <lb></lb>actu. </s> <s id="s.000500">At brachia in ipſo æquilibrio ſuſtinent actu quidem, <lb></lb>ſed non mouentur. </s> <s id="s.000501">Cæterum videtur Ariſtoteles id ſub<lb></lb>odoraſſe, quod poſtea Archimedes, Mechanicorum prin<lb></lb>ceps, in propoſ. </s> <s id="s.000502">6. primi Æqueponderantium explicitè <lb></lb>protulit & probauit: nempe in æquilibrio ita eſſe pondus <lb></lb>ad pondus, vt brachium ad brachium, ratione permutata. </s> </p> <figure id="id.007.01.057.1.jpg" xlink:href="007/01/057/1.jpg"></figure> <p type="main"> <s id="s.000503">Eſto enim vectis <lb></lb>AB, quomodolibet <lb></lb>fulcimento diuiſus in <lb></lb>C. <expan abbr="appēdatur">appendatur</expan> autem <lb></lb>in A, pondus D, in B <lb></lb>verò pondus E, ita ſe <lb></lb>habens ad pondus D, vt ipſa AC ad CB. </s> <s id="s.000504">Stabit igitur ve<lb></lb>ctis, & neutram in partem verget, erit enim centrum gra<lb></lb>uitatis in C, diuiſo nempe ibi vecte in partes æque ponde<lb></lb>rantes. </s> <s id="s.000505">Hoc poſt Archimedem, & inſignes illos veteres <lb></lb>Mechanicos præclariſſimè demonſtrauit G. Vbaldus in <lb></lb>Mechanicis, Tractatu de Libra propoſ. </s> <s id="s.000506">6. nec non de Ve<lb></lb>cte propoſ. </s> <s id="s.000507">4. </s> </p> <p type="main"> <s id="s.000508">Cæterùm vt aliquid interim, quod noſtrum ſit, affe<lb></lb>ramus, liceat nobis egregios illos viros interrogare, quæ<lb></lb>nam mirabilis eius effectionis ſit cauſſa? </s> <s id="s.000509">Dicent permu<lb></lb>tatam proportionem. </s> <s id="s.000510">Teneo, at nondum acquieſco: pe<lb></lb>tam enim, Cur ea rationis permutatio mirabilem illum <lb></lb>effectum pariat. </s> <s id="s.000511">Hoc quod illi non docent, puto nos, i<lb></lb>gnorantiæ ſomno ſepultos, ſomniaſſe. </s> </p> <figure id="id.007.01.057.2.jpg" xlink:href="007/01/057/2.jpg"></figure> <p type="main"> <s id="s.000512">Æqualitatem ſtatus <lb></lb>eſſe cauſſam, nemo, vt <lb></lb>puto, inficiabitur. </s> <s id="s.000513">res eſt <lb></lb>enim per ſe clara. </s> <s id="s.000514">Eſto ſi<lb></lb>quidem linea quæpiam AB, applicetur extremitati A po<pb xlink:href="007/01/058.jpg"></pb>tentia quædam quæ lineam ad ſe trahat ad partes nempe <lb></lb>A, Tum in B quædam alia potentia ipſi quæ in A potentiae, <lb></lb>æqualis, quæ lineam trahat ſimili modo ad partes B. </s> <s id="s.000515">Datâ <lb></lb>igitur harum potentiarum æqualitate, linea AB, nec ad <lb></lb>partes A, nec ad partes B transferetur, ſed prorſus immo<lb></lb>bilis ſtabit. </s> </p> <p type="main"> <s id="s.000516">His ita conſtitutis, Dico vecte quomodolibet diuiſo, <lb></lb>ponderibuſque vtrinque appoſitis, permutatâ propor<lb></lb>tione ſibi inuicem reſpondentibus, rem eſſe redactam ad <lb></lb>æqualitatem, & inde ſtatum fieri, hoc eſt, æquilibrium. </s> </p> <figure id="id.007.01.058.1.jpg" xlink:href="007/01/058/1.jpg"></figure> <p type="main"> <s id="s.000517">Eſto enim vectis AB, quo modo libet diuiſus in C, & <lb></lb>ipſi quidem C fulcimentum ſupponatur. </s> <s id="s.000518">Appendantur <lb></lb>quoque vtrinque pondera ex ratione brachiorum AC, <lb></lb>CB, ſibi inuicem permutatim reſpondentia, ſintque; DE. <lb></lb></s> <s id="s.000519">Dico vectem ex æqualitate, in neutram partem <expan abbr="inclina-turū">inclina<lb></lb>turum</expan>, ſed permanſurum in æquilibrio. </s> <s id="s.000520">quoniam enim <expan abbr="Pō-dus">Pon<lb></lb>dus</expan> D idem poteſt quod brachium CB, addatur in dire<lb></lb>ctum ipſi AC, recta AF æqualis ipſi CB, item quoniam <lb></lb>Pondus E id poteſt quod brachium AC, rectæ CB ad<lb></lb>datur in directum BG, ipſi AC æqualis. </s> <s id="s.000521">Igitur cum par<lb></lb>tes CA, AF totius FC, æquales ſint partibus CB, BG, <lb></lb>totius CG, erit totum FC, toti CG æquale. </s> <s id="s.000522">Diuiſus ita-<pb xlink:href="007/01/059.jpg"></pb>que erit vectis FG in partes æquales FC, CG in puncto <lb></lb>fulcimenti C. </s> <s id="s.000523">Et quoniam æquale in æquale non agit, <lb></lb>ſtabit vectis & in neutram partem inclinabit. </s> <s id="s.000524">Rurſum <lb></lb>quoniam ad partem FC, duæ ſunt brachiorum potentiæ <lb></lb>FA, HC, appendantur puncto F, duo pondera H, I, ipſis <lb></lb>DE æqualia, item puncto G, alia duo pondera ijſdem DE <lb></lb>æqualia KL, iterum æqueponderabit, quippe quod æ<lb></lb>qualibus brachijs FCCG æqualia appenſa ſint pondera <lb></lb>HI KL. </s> <s id="s.000525">Cur igitur ſeruata permutatim brachiorum & <lb></lb>ponderum proportione fiat æquilibrium, ex his quæ de<lb></lb>monſtrauimus, clarè patet. </s> </p> <p type="main"> <s id="s.000526">Sed forte dicet quiſpiam, ſi brachia, pondera ſunt, <lb></lb>vel ponderibus æquipollentia, ſuſtinenti duplicabitur <lb></lb>pondus. </s> </p> <figure id="id.007.01.059.1.jpg" xlink:href="007/01/059/1.jpg"></figure> <p type="main"> <s id="s.000527">Eſto enim vectis AB, <lb></lb>ita diuiſus in C, vt pars <lb></lb>maior CB minori AC ſit <lb></lb>in proportione quintu<lb></lb>pla. </s> <s id="s.000528">Appendatur autem <lb></lb>in A pondus D, <expan abbr="quintuplū">quintuplum</expan> <lb></lb>ponderi E appenſo in B. </s> <s id="s.000529">Si <lb></lb>igitur brachio AC, quod <lb></lb>eſt vnum, ad datur pondus <lb></lb>D, quod eſt quinque, fient ſex, item ſi brachio CB, quod <lb></lb>eſt quinque, addatur pondus E, quod eſt vnum, fient ſex. <lb></lb></s> <s id="s.000530">Fulcimentum igitur ſuſtinebit duodecim, quod eſt ab<lb></lb>ſurdum ex ijs quæ clarè demonſtrauit G. Vbald. in Me<lb></lb>chan. tractatu de Libra propoſ. </s> <s id="s.000531">5. His reſpondemus, bra<lb></lb>chia quidem operari non pondere, ſed potentiâ, quæ vis <lb></lb>quædam eſt, non autem pondus. </s> <s id="s.000532">Etſi & illud verum ſit, da<lb></lb>to vecte ponderoſo, fulcimentum rum ponderum appen<lb></lb>ſorum, tum vectis ipſius pondus ſuſtinere. </s> </p> <p type="main"> <s id="s.000533">Iacta huiuſcemodi, quam diximus, æqualitate, ſe-<pb xlink:href="007/01/060.jpg"></pb>quitur neceſſariò, centrum grauitatis ipſius vectis cum <lb></lb>appenſis ponderibus, ac ſi vnum idemqueue eſſet corpus <lb></lb>cadere in perpendiculari quæ per centrum ipſum & ful<lb></lb>cimentum tranſiens ad mundi centrum pertingit. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000534">QVÆSTIO IV.</s> </p> <p type="head"> <s id="s.000535"><emph type="italics"></emph>Quærit hic Ariſtoteles, cur ij qui in nauis medio ſunt remiges ma<lb></lb>ximè nauem moueant?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000536">Ait, ideo fortaſſe fieri, quòd remus vectis ſit, fulcimen<lb></lb>tum verò ſcalmus, ſtat enim. </s> <s id="s.000537">Pondus autem mare i<lb></lb>pſum, quod à remo propellitur, mouens verò ipſum remi<lb></lb>gem, ſemper autem plus mouere ponderis qui mouet, <lb></lb>quo magis diſtatà fulcimento. </s> <s id="s.000538">Ita enim maiorem fieri <lb></lb>quæ ex centro; Scalmum verò centrum eſſe. </s> <s id="s.000539">Cæterùm in <lb></lb>medio nauis plurimum remi intus eſſe. </s> <s id="s.000540">Ibi enim nauem <lb></lb>eſſe latiſſimam. </s> <s id="s.000541">Moueri autem nauim, quoniam <expan abbr="appellē-te">appellen<lb></lb>te</expan> mari remo, <expan abbr="extremū">extremum</expan> illius quod intus eſt anterius pro<lb></lb>mouetur, cuius motum nauis ſequitur, cui ſcalmus alliga<lb></lb>tur. </s> <s id="s.000542">Vbi autem plurimum maris diuidit remus, eo maximè <lb></lb>neceſſe eſſe propelli. </s> <s id="s.000543">Plurimum autem diuidi vbi plurima <lb></lb>pars remi à ſcalmo eſt. </s> <s id="s.000544">Rem facilem, eo quod verbis potu<lb></lb>erit, ſchemate non declarauit, nos autem apponemus. </s> </p> <figure id="id.007.01.060.1.jpg" xlink:href="007/01/060/1.jpg"></figure> <p type="main"> <s id="s.000545">Eſto enim nauis AB, mare CD, <lb></lb>remorum alter, qui ad proram EF, cu<lb></lb>ius ſcalmus G, alter verò in medio na<lb></lb>uis, HI, circa ſcalmum K. </s> <s id="s.000546">Ait igitur, <lb></lb>remos eſſe vectes, ſcalmos verò fulci<lb></lb>menta, pondus quod remo, ceu vecte, <lb></lb>mouetur mare ipſum. </s> <s id="s.000547">Itaque quoniam <lb></lb>nauis lata eſt in medio vbi Scalmus K <lb></lb>maior pars KH intra nauim eſt, minor <lb></lb>verò KI, extra. </s> <s id="s.000548">Contra autem remi ad <lb></lb>proram, nempe EF pars minor EG <pb xlink:href="007/01/061.jpg"></pb>intra nauim, pars verò maior GF extra nauim eſt. </s> <s id="s.000549">Pondus <lb></lb>autem eò faciliùs mouetur, quo maior eſt vectis pars, quæ <lb></lb>à fulcimento eſt ad mouentem potentiam. </s> </p> <p type="main"> <s id="s.000550">Acutè ſanè Philoſophus. </s> <s id="s.000551">Ego autem ſi per modeſtiam <lb></lb>liceret, dicerem, non quidem eſſe fulcimentum <expan abbr="ſcalmū">ſcalmum</expan>, <lb></lb>ſed mare ipſum, pondus vero nauim, ad locum ſcalmi, <expan abbr="nē-pe">nem<lb></lb>pe</expan> inter mouentem potentiam, & fulcimentum poſitum, <lb></lb>etenim & eo pacto poſſumus vti vecte, quod obſeruat & <lb></lb>demonſtrat G. Vbaldus tractatu de vecte propoſ. </s> <s id="s.000552">2. Erunt <lb></lb>igitur in deſcripta figura puncta FI, quæ in mari ſunt, ful<lb></lb>cimenta, quibus remorum extrema in ipſa impulſione ni<lb></lb>tuntur, pondera verò ſeu pondus pluribus vectibus & po<lb></lb>tentijs impulſum nauis ipſa, quæ ſcalmis eſt annexa. </s> <s id="s.000553">Reſi<lb></lb>ſtente igitur mari, cedente autem impulſionibus ſcalmo, <lb></lb>nauis eo transfertur, quo ſcalmi ab ipſa potentia mouen<lb></lb>te in anteriorem partem pelluntur. </s> <s id="s.000554">quoniam autem vt <lb></lb>FG ad FE ita potentia mouens in E ad pondus motum <lb></lb>in G. item vt IK ad IH ita potentia mouens in H ad pon<lb></lb>dus motum in K, maior autem eſt proportio FG ad FE <lb></lb>quàm proportio IK ad IH. </s> <s id="s.000555">Maiori indiget potentia vt <lb></lb>pellatur pondus in G quàm pondus in K. </s> </p> <p type="main"> <s id="s.000556">Hæc certè vti diximus ita ſe habent. </s> <s id="s.000557">Philoſophi au<lb></lb>tem ratio tunc procederet, ſi ſtante naui immobili, vt fit <lb></lb>vbi à Remoræ occulta vi aut ab alio impedimento reti<lb></lb>netur, remiges in ipſo remigandi actu mare pulſarent, <lb></lb>Tunc enim verè ſcalmus fieret fulcimentum, mare autem <lb></lb>pondus, remex verò ipſe mouens. </s> </p> <p type="main"> <s id="s.000558">Addimus, falſum videri quod aſſerit Ariſtoteles, <lb></lb>nempe illos qui in media naui ſunt, remiges, maximè na<lb></lb>uim mouere; facilius, melius dixiſſet. </s> <s id="s.000559">Si enim maximè, <lb></lb>quod ait, denotat, maximo ſpatio, & velocius prorſus fal<lb></lb>ſum, etenim tardius mouent & minori ſpatio, quod nos i<lb></lb>ta demonſtramus. </s> </p> <pb xlink:href="007/01/062.jpg"></pb> <figure id="id.007.01.062.1.jpg" xlink:href="007/01/062/1.jpg"></figure> <p type="main"> <s id="s.000560">Eſto enim Remus AB <lb></lb>qui marí fulcitur in B, Scal<lb></lb>mus remi qui ad <expan abbr="prorã">proram</expan> pup<lb></lb>pimue C, qui in media naui <lb></lb>D, maior autem remi pars <lb></lb>eſt à ſcalmo Dad A quam i<lb></lb>pſius C 2d A, Pellantur remi & ſtante ceu centro BA, in <lb></lb>E. eodem igitur tempore C eritin F, & D in G, ſed maius <lb></lb>eſt ſpatium CF ſpatio DG, Ergo vnica impulſione, plus <lb></lb>mouit ſcalmum, hoc eſt, nauim, potentia ad puppim pro<lb></lb>ramue remigans, quàm ea quæ operatur in media naui vt <lb></lb>ſentire videbatur (ſi modo is eſt eius ſenſus) Ariſtoteles. <lb></lb></s> <s id="s.000561">Neceſſarium igitur eſt, quod ait, maximè intelligendum, <lb></lb>faciliùs, Veritatem hanc cognoſcentes Triremium præ<lb></lb>fecti robuſtiores quidem remiges ad proram & puppim, <lb></lb>inualidiores verò circa mediam triremem collocant. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000562">QVÆSTIO V.</s> </p> <p type="head"> <s id="s.000563"><emph type="italics"></emph>Dubitatur, Cur paruum exiſtens gubernaculum, & in extremo <lb></lb>nauigio tantas habeat vires, vt ab exiguo temone, & ab hominis <lb></lb>vnius viribus alioqui modicè vtentis magnæ nauigiorum <lb></lb>moueantur moles?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000564">AN, inquit, quoniam gubernaculum vectis eſt, onus <lb></lb>autem mare, Gubernator vero mouens eſt? </s> <s id="s.000565">Non au<lb></lb>tem ſecundùm latitudinem veluti remus, mare accipit <lb></lb>gubernaculum; non enim in ante nauigium mouet, ſed i<lb></lb>pſum commotum mare accipiens inclinat obliquè. </s> <s id="s.000566">quo<lb></lb>niam enim pondus eſt mare contrario innixum modo na<lb></lb>uem inclinat. </s> <s id="s.000567">fulcimentum enim in contrarium verſatur, <lb></lb>mare vetò interius, & illud exterius. </s> <s id="s.000568">illud autem ſequitur <lb></lb>nauis quæ illi eſt alligata & remus quidem ſecundum la<lb></lb>titudinem onus propellens & ab eodem repulſus in re-<pb xlink:href="007/01/063.jpg"></pb>ctum propellit, Gubernaculum verò, vt obliquum iacet <lb></lb>hinc inde in obliquum motionem facit. </s> <s id="s.000569">in extremo <expan abbr="autē">autem</expan>, <lb></lb>non in medio iacet, quoniam mouenti facillimum eſt mo<lb></lb>tum mouere: prima enim pars celerrimè fertur, & quo<lb></lb>niam, quemadmodum in ijs quæ feruntur in fine deficit <lb></lb>latio, ſic ipſius continui in finem, imbecillima eſt latio. <lb></lb></s> <s id="s.000570">Imbecillima autem ad expellendum eſt facilis. </s> <s id="s.000571">Propter <lb></lb>hæc igitur in puppi gubernaculum ponitur, nec minus, <lb></lb>quoniam parua ibi motione facta, multo maior fit in vlti<lb></lb>mo, quia æqualis angulus ſemper maiorem adſpectat, <expan abbr="tã-to">tan<lb></lb>to</expan> queue magis, quanto maiores fuerint illæ, quæ continent. <lb></lb></s> <s id="s.000572">Ex ijs etiam manifeſtum eſt, quam ob cauſſam magis in <lb></lb>contrarium procedit nauigium, quam remi ipſius palmu<lb></lb>la, eadem enim magnitudo ijſdem mota viribus in aëre <lb></lb>plus quàm in aqua progreditur. </s> <s id="s.000573">Hæc Philoſophus, qui <lb></lb>haudquaquam ex more ſuo, quod duobus ferè poterat, <lb></lb>ſexcentis verbis expoſuit. </s> <s id="s.000574">Licebat enim id tantum dicere, <lb></lb>Gubernaculum (ita vocat id totum quod gubernaculo & <lb></lb>temone conſtat) eſſe ceu remum, quo nauis non antror<lb></lb>ſum, ſed obliquè & ad latus mouetur. </s> <s id="s.000575">quamobrem omnia <lb></lb>ferè quæ de Temone dicenda fuerant, de remo loquens <lb></lb>proponit. </s> <s id="s.000576">Ait autem. </s> </p> <figure id="id.007.01.063.1.jpg" xlink:href="007/01/063/1.jpg"></figure> <p type="main"> <s id="s.000577">Sit remus AB, <lb></lb>ſcalmus vero C, remi <lb></lb>in nauigio <expan abbr="principiū">principium</expan> <lb></lb>A, palmula autem, <lb></lb>quæ in mari B. </s> <s id="s.000578">Si igi<lb></lb>tur A, vbi D transla<lb></lb>tum eſt, non erit B v<lb></lb>bi E. æqualis enim, <lb></lb>BE ipſi AD, æquale <lb></lb>igitur translatum erit, ſed erat minus. </s> <s id="s.000579">erit igitur vbi F, mi<lb></lb>nor enim BF, ipſa AD, quare ipſo GF ipſa DG. </s> <s id="s.000580">Hæc <pb xlink:href="007/01/064.jpg"></pb>demonſtratio licet vera videatur, rei ta men, de qua eſt <lb></lb>ſermo, minimè aptatur. </s> <s id="s.000581">Si enim aptaretur in ipſius remi <lb></lb>motu, cum palmula eſſet in F, ſcalmus fieret in G, excur<lb></lb>reret ergo vel ſcalmus per remum, vel remus per <expan abbr="ſcalmū">ſcalmum</expan>, <lb></lb>facta nempe eiuſmodi translatione de C in G, & ſic intra <lb></lb>nauim modo eſſet pars remi DC, modò verò GD, quod <lb></lb>tamen non fieri ipsâ experientia docemur. </s> <s id="s.000582">Illud quoque <lb></lb>falſum eſt, nauim ipſam tantum moueri in aëre, quantum <lb></lb>eſt ſpatium AD, hoc eſt, remi extremum quod eſt in naui, <lb></lb>ſiquidem ſcalmi motu, non autem manubrij remi, nauis <lb></lb>agatur. </s> <s id="s.000583">Aliter igitur res ſe habet, & forte hoc pacto. </s> </p> <figure id="id.007.01.064.1.jpg" xlink:href="007/01/064/1.jpg"></figure> <p type="main"> <s id="s.000584">Sit remus AB, cuíus <lb></lb>manubrium A, palmula <lb></lb>B, ſcalmus C. </s> <s id="s.000585">Pellatur an<lb></lb>trorſus A, fiatque; in D, tunc <lb></lb>ſi æqualiter mouerentur <lb></lb>manubrium & palmula, i<lb></lb>pſa palmula fieret in G, at <lb></lb>minus mouetur: fiet ergo <lb></lb>in E. ipſe verò ſcalmus C <lb></lb>translatus erit in F, motaque; erit nauis à C in F, non autem <lb></lb>ab A in D. </s> <s id="s.000586">Poſuit autem Ariſtoteles ſcalmum ad medium <lb></lb>remi, ſed non ad medium collocari ſolet, maior enim pars <lb></lb>in mare propendet puta HB, quo caſu translationis ſpa<lb></lb>tium fit maius, nempe ab H in I. fit autem motus ſcalmi ex <lb></lb>centris qui ſunt in ſpatio ipſo BE, quatenus autem ad te<lb></lb>monem pertinet, quem remum ait, obliquè puppim ipſam <lb></lb>propellentem, ita ſe res habet. </s> </p> <p type="main"> <s id="s.000587">Eſto nauis carina AB, prora A, puppis B, Temonis <lb></lb>ala BC, gubernaculum BD, cardo verò fulcimentumue <lb></lb>B; facta itaque impulſione obliquâ gubernaculi à D in E, <lb></lb>minor fiet motus in mari à C in F, eritqueue temo vbi EGF, <pb xlink:href="007/01/065.jpg"></pb><figure id="id.007.01.065.1.jpg" xlink:href="007/01/065/1.jpg"></figure><lb></lb>cardo verò vbi G, translata igitur e<lb></lb>rit eo motu, puppis ipſa à B in G. facta <lb></lb>itaque paruâ motione puppis ex B in <lb></lb>G, prora ipſa quæ longè diſtat à pup<lb></lb>pi B maiori ſpatio ſuperato translata <lb></lb>erit in H facta proræ in contrariam <lb></lb>partem ab ea quæ facta eſt guberna<lb></lb>culi motione. </s> <s id="s.000588">Porrò quod & in præ<lb></lb>cedente quæſtione adnotauimus, <expan abbr="lō-gè">lon<lb></lb>gè</expan> meliùs procedet demonſtratio ſi <lb></lb><expan abbr="fulcimentū">fulcimentum</expan> mare intelligatur, quàm <lb></lb>ſcalmus, neque enim mare ceu pon<lb></lb>dus, ſed ſcalmus ipſe Temonisuecardo, ponderum inſtar <lb></lb>transferuntur. </s> </p> <p type="main"> <s id="s.000589">Cæterùm in hac ſpeculatione liceat nobis aliquan<lb></lb>tulum à Philoſopho diſſentire. </s> <s id="s.000590">Certè ſi breuitas Temo<lb></lb>nis, è puppi eminentis, reſpectu longitudinis totius nauis <lb></lb>conſideretur, & parua motio, quæ temone guberna culo<lb></lb>ue moto fit, nullius ferè momenti erit ad eam quæ in pro. <lb></lb></s> <s id="s.000591">ra fit translationem. </s> <s id="s.000592">aliter ergo ſe rem habere non dubi<lb></lb>tamus, & quæſtionis ſolutionem aliunde petendam. </s> <s id="s.000593">Na<lb></lb>ui non currente nullum ferè, aut qui vix curandus ſit ex <lb></lb>gubernaculi conuerſione nauis ad dextram ſiniſtramue <lb></lb>motum fieri. </s> <s id="s.000594">at eâ currente maximum, experientiâ doce<lb></lb>mur. </s> <s id="s.000595">Obliqui igitur motus qui validè in puppi ſit, cauſſa <lb></lb>eſt non quidem ex conuerſione temonis percuſſio maris, <lb></lb>ſed mare ipſum, cuius fluctus naui currente obliquam te<lb></lb>monis alam ad eam partem quæ mari obuertitur, impel<lb></lb>lentes temonem cum puppi ad contrariam partem vali<lb></lb>diſſimè transferunt. </s> </p> <p type="main"> <s id="s.000596">Eſto nauis carina AB, prora B, puppis A, Temo AC, <lb></lb>gubernaculum AD; Itaque currente naui, Temone in<lb></lb>terim & guberna culo in eadem carinæ linea exiſtentibus, <pb xlink:href="007/01/066.jpg"></pb><figure id="id.007.01.066.1.jpg" xlink:href="007/01/066/1.jpg"></figure><lb></lb>Temo quidem mare ſecat, nulla fa<lb></lb>ctâ in puppi, nauis ad ſiniſtram dex<lb></lb>tramue translatione. </s> <s id="s.000597">Si verò mouea<lb></lb>tur gubernaculum à D in E, eo moto <lb></lb>mouebitur aliquantulum & puppis <lb></lb>ad partes E, quod voluit Ariſtoteles. <lb></lb></s> <s id="s.000598">Sed minimi, vt diximus, ea res ad tan<lb></lb>tum effectum eſt momenti. </s> <s id="s.000599">Temone <lb></lb>autem in obliquum <expan abbr="cōſtituto">conſtituto</expan> vt AF, <lb></lb>naui interim, ventorum aut remorum <lb></lb>vi pulſa proram verſus currente te<lb></lb>monis latus à fluctibus obliquam par<lb></lb>tem alamue in ipſo curſu ferientibus, <lb></lb>in contrariam partem transfertur, ad <lb></lb>eam nempe, ad quam ipſum gubernaculum vergit. </s> <s id="s.000600">facta i<lb></lb>gitur nauis ceu circa centrum centraue quæ in carina in<lb></lb>ter puppim proramue conſiderantur A, fertur in G, prora <lb></lb>verò in H. ex quibus manifeſtè apparet, duo ad nauis ex <lb></lb>temone in puppi conuerſione motionem eſſe ne ceſſaria; <lb></lb>Temonis nempe obliquationem, & nauis curſum, <expan abbr="quorū">quorum</expan> <lb></lb>ſi alterum ſine altero adhibeatur, nullam fieri quæ alicu<lb></lb>ius momenti ſit, nauis conuerſionem. </s> <s id="s.000601">Illud quoque nota<lb></lb>mus, carinam in nauis conuerſione vectis inſtar ſe habere, <lb></lb>cuius pars mota ad puppim, & mouens potentia eſt; fulci<lb></lb>mentum verò circa proram, potentia autem mouens ma<lb></lb>re ipſum, temonem in nauis curſu oblique feriens. </s> <s id="s.000602">Vnde <lb></lb>colligimus naues, quo longiores ſunt in mouente ad Te<lb></lb>monem adhibita maiori facilitate ad dextram ſiniſtram<lb></lb>ue propelli: quod ſanè ipſemet conſiderauit Ariſtoteles, <lb></lb>quì idcirco inquit, in extremo, non autem in medio temo<lb></lb>nem poni eo quod mouenti facilimum ſit ab extremo <lb></lb>motum mouere. </s> </p> <p type="main"> <s id="s.000603">Ex hac noſtrâ ſpeculatione ratio habetur eius ma-<pb xlink:href="007/01/067.jpg"></pb>chinationis, quâ in magnis fluminibus, ceu Pado, Abdua <lb></lb>& ſimilibus, Portitores, equos, currus, viatoreſque; ipſos, è <lb></lb>ripa in ripam transferunt. </s> <s id="s.000604">Pulcherrima enim res eſt, & <lb></lb>nobis perſpectiſſima, qui Guaſtallâ reſidentiæ olim no<lb></lb>ſtræ oppido ad Padum, Mantuam pergentes ſæpiſſimè ad <lb></lb>Caſtrum Borgi Iuſis ea qua diximus machinatione latiſ<lb></lb>ſimum eiuſdem Padi aluum tranſiecimus. </s> <s id="s.000605">Habet autem <lb></lb>ſe hoc pacto. </s> </p> <figure id="id.007.01.067.1.jpg" xlink:href="007/01/067/1.jpg"></figure> <p type="main"> <s id="s.000606">Eſto fluminis citerior <lb></lb>ripa AB, vlterior CD. </s> <s id="s.000607">Pon<lb></lb>tones duo tabulis ſtrati, & v<lb></lb>nà firmiter juncti EF, Temo <lb></lb>inter eorum puppes extans <lb></lb>GH, locus in ripa ſtabilis A, <lb></lb>funis, quo pontones, & ma<lb></lb>china tota continetur AI. <lb></lb>fluuij decurſus verſus BD, <lb></lb>ſtantibus itaque pontonibus <lb></lb>ad ripam citeriorem AB, Te<lb></lb>mone in <expan abbr="neutrã">neutram</expan> partem pul<lb></lb>ſo, cum aqua decurrens eum <lb></lb>reſiſtentem non inueniat, <lb></lb>ſcinditur quidem ab eo, ſed <lb></lb>non propellit, eo autem con<lb></lb>uerſo & in GK conſtituto, a<lb></lb>la eius GK ab aqua defluente propulſa machinam ſecum <lb></lb>trahit verſus ripam CD, factâ motione circa centrum ſeu <lb></lb>ſtabilem locum A, otioſis interim portitoribus, donec per <lb></lb>circuli portionem ML deuenerit ad vlteriorem ripam in <lb></lb>L. </s> <s id="s.000608">Vnde iterum temone in contrariam partem conuerſo, <lb></lb>aquâ ſimiliter temonem propellente, per eandem circuli <lb></lb>portionem ad ripam citeriorem reuertitur, à qua paullo <lb></lb>antè diſceſſerat. </s> <s id="s.000609">Ex quibus apparet, motus cauſſam non <pb xlink:href="007/01/068.jpg"></pb>eſſe ſolam cam, quæ ab ala temonis fit, aquæ <expan abbr="percuſſionē">percuſſionem</expan>, <lb></lb>vt ſenſerat Ariſtoteles, ſed currentis a quæ temonis alam <lb></lb>ferientis impulſionem: nihil autem referre, vtrum ſtante <lb></lb>naui a qua currat, vel câ currente a qua ſtet, vt in mari fit, <lb></lb>idem enim vtroque modo temo patitur. </s> <s id="s.000610">Vt autem machi<lb></lb>næ huius & totius negotij ſpecies facilius animo concipia<lb></lb>tur, ſchema hoc ſtudioſorum oculis ſubijciemus. </s> </p> <figure id="id.007.01.068.1.jpg" xlink:href="007/01/068/1.jpg"></figure> <p type="main"> <s id="s.000611">Lembi nauiculæue ideo appoſitæ ſunt, vt oblongum <lb></lb>funem ſuſtineant; id etenim nî fieret, aquæ immerſus a<lb></lb>quam ſcindens machinæ motum impediret, ideo etiam <lb></lb>apponuntur, ne funis madens celeriter maceretur & pu<lb></lb>treſcat. </s> </p> <p type="main"> <s id="s.000612">Huic ſpeculationi affinis eſt ea, velorum eorum, <lb></lb>quæ obliquè ventum, excipientia frumentarijs molis <lb></lb>dant motum, item verticillorum ex papyro, quibus con<lb></lb>tra ventum currentes per luſum pueri vtuntur. </s> <s id="s.000613">vnicum <pb xlink:href="007/01/069.jpg"></pb>enim horum omnium principium, & eadem, ratio. </s> </p> <p type="main"> <s id="s.000614">Diximus enim, Temonem currente naui, lateraliter <lb></lb>conuerſum obuios fluctus excipientem puppim ipſam ob<lb></lb>liquè in alteram partem transferre. </s> <s id="s.000615">Porrò ea vela, de qui<lb></lb>bus loquimur, ventorum flatibus obliquè oppoſita ean<lb></lb>dem ob cauſſam circulariter agitantur, quod vt figurâ eui<lb></lb>dentius fiat, </s> </p> <figure id="id.007.01.069.1.jpg" xlink:href="007/01/069/1.jpg"></figure> <p type="main"> <s id="s.000616">Eſto velum AB, brachio <lb></lb>CE obliquè affixum, ita vt <lb></lb>angulus ACE maior ſit an<lb></lb>gulo BCE, ventus obliquè <lb></lb>velum feriens FG. <expan abbr="Itaq;">Itaque</expan> quo<lb></lb>niam ventus in velum obli<lb></lb>quum incidit, elabitur velum, <lb></lb>& circa centrum E vnà cum <lb></lb>brachio circumuertitur, in <lb></lb>cuius locum ſuccedit velum <lb></lb>HI, ex qua aſſidua velorum <lb></lb>ſucceſſione, brachiorum & a<lb></lb>xis cui adhærent, rotatio fit <lb></lb>perpetua. </s> <s id="s.000617">Sed enim de Te<lb></lb>mone agentes non eſt interim cur de caudis auium piſci<lb></lb>umque taceamus, inſtar enim remonum ſunt à Natura i<lb></lb>pſa opportunis animalium partibus, poſtremis videlicet, <lb></lb>appoſiti, quanquam nec ſolum Temonis vſum præſtent, <lb></lb>vt videbimus. </s> </p> <p type="main"> <s id="s.000618">Eſto piſcis AB, cuius caput A, cauda verò CB. </s> <s id="s.000619">Hac <lb></lb>igitur neutram in partem reflexâ, piſcis pinnarum motu <lb></lb>rectâ in anteriorem partem progreditur. </s> <s id="s.000620">Si autem neceſ<lb></lb>ſe ei fuerit ad dextram ſiniſtramqueue conuerti non pote<lb></lb>rit, niſi cauda ipſa iuuetur. </s> <s id="s.000621">Omnis enim motus progreſſi<lb></lb>uus quiete indiget, nec <expan abbr="abſq;">abſque</expan> ſtabili fulcimento progredi <pb xlink:href="007/01/070.jpg"></pb><figure id="id.007.01.070.1.jpg" xlink:href="007/01/070/1.jpg"></figure><lb></lb>poteſt, quod in libris de ani<lb></lb>malium inceſſu docet ipſe<lb></lb>met Philoſophus. </s> <s id="s.000622">Sit igitur, <lb></lb>piſcem conuerti velle, & fie<lb></lb>ri capite in D, deflectet illi<lb></lb>co caudam in E, caque; aquam <lb></lb>ceu ſtabile quippiam <expan abbr="feriēs">feriens</expan> <lb></lb>eiqueue quoddammodo fultus, <lb></lb>reliquum corpus CA refle<lb></lb>ctet in D, ſi autem conuerti <lb></lb>velit in F, caudam deflectet in G, & eadem ratione flecte<lb></lb>tur in F. </s> <s id="s.000623">Sed & Temonis quoque vſum præſtat natatili<lb></lb>bus & volatilibus cauda. </s> <s id="s.000624">Sit enim rectus piſcis, hoc eſt, re<lb></lb>ctâ pergens IKL, caudam obliquet in KM itaque ex a<lb></lb>quæ in ipſo motu colliſione, eius poſteriora pellentur vbi <lb></lb>INO. </s> <s id="s.000625">Hæc itaque nos de Temone, quatenus ad hanc <lb></lb>quæſtionem pertinet, conſideraſſe ſit ſatis. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000626">QVÆSTIO VI.</s> </p> <p type="head"> <s id="s.000627"><emph type="italics"></emph>Dubitatur, Cur quanto Antenna ſublimior fuerit, ÿſdem velis, & <lb></lb>vento eodem celeriùs ferantur nauigia?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000628">Soluit Philoſophus, inquiens: An quia malus quidem <lb></lb>ſit vectis, fulcimentum verò mali ſedes, in qua colloca<lb></lb>tur, pondus autem quod moueri debet, ipſum nauigium: <lb></lb>mouens verò is, qui vela tendit ſpiritus? </s> <s id="s.000629">Si igitur quanto <lb></lb>remotior fuerit fulcimentum facilius eadem potentia, & <lb></lb>citiùs idem mouet pondus, altius certè ſublatâ antennâ, <lb></lb>velum à mali ſede, quae fulcimentum eſt remotius faciens, <lb></lb>id efficiet. </s> <s id="s.000630">Hæc ille. </s> <s id="s.000631">quæ ſic figurâ explicamus. </s> </p> <pb xlink:href="007/01/071.jpg"></pb> <figure id="id.007.01.071.1.jpg" xlink:href="007/01/071/1.jpg"></figure> <p type="main"> <s id="s.000632">Eſto nauis AB, malus CD, <lb></lb>mali ſedes D, locus antennæ <lb></lb>ſublimior C, depreſſior E: ita<lb></lb>que quoniam CD vectis eſt, <lb></lb>quo mouens remotior fuerit à <lb></lb>fulcimento D, eo citiùs & vio<lb></lb>lentiùs pellet, velocius ergo <lb></lb>nauis mouebitur antenna in <lb></lb>C, quàm in E, conſtituta. </s> </p> <p type="main"> <s id="s.000633">Plauſibilia ſunt hæc, at certè per veritatem ipſam, <lb></lb>non vera. </s> <s id="s.000634">Rogo, Si fulcimentum dum vectis mouetur, <expan abbr="cē-trum">cen<lb></lb>trum</expan> eſt, centrum vtique motus erit D. ſpirante igitur va<lb></lb>lidè vento inclinabitur malus, fietque; vbi FGD, quæ qui<lb></lb>dem inclinatio violentius fiet, vento pellente in F quàm <lb></lb>in G, vtpote puncto à fulcimento remotiore. </s> <s id="s.000635">Impulſo ma<lb></lb>lo, duo neceſſariò <expan abbr="cōſequentur">conſequentur</expan>, vel enim ad ipſam ſedem <lb></lb>D. frangetur vel puppis ipſa circa D punctum conuerſa, <lb></lb>vt mali ſequatur motum eleuabitur. </s> <s id="s.000636">Prora verò ſubmer<lb></lb>getur facta naui in HDI. </s> <s id="s.000637">Quod ſi quiſpiam funem ad ma<lb></lb>li ſummitatem annexam ad ipſam puppim alligauerit in <lb></lb>B, impedietur ſanè mali inclinatio ad partes F, & ideo nul<lb></lb>la vis prorſus fiet in D ex vectis ratione. </s> <s id="s.000638">Attamen nihilo <lb></lb>ſecius, quo ſublimior fuerit antenna, eo faciliùs à ſpirante <lb></lb>vento puppis eleuabitur. </s> <s id="s.000639">quatenus igitur malus vectis <lb></lb>eſt, hoc tantum quod dicimus operatur. </s> <s id="s.000640">Quod ſi contrà <lb></lb>obiectum fuerit, experientiam docere, quo ſublimior an<lb></lb>tenna fuerit, eo citiùs nauigium, ſpiritu flante moueri. <lb></lb></s> <s id="s.000641">Reſponſio facilis, nempe, mirum non eſſe, ſi mali pars ſub<lb></lb>limior validius à vento feriatur. </s> <s id="s.000642">Videmus enim, & turres <lb></lb>quo ſublimiores fuerint, eo magis à ventorum impetuoſis <lb></lb>flatibus infeſtari, quod ſanè ad vectis longitudinem refer<lb></lb>re, eſſet ridiculum. </s> <s id="s.000643">Cæterùm quod ad puppis faciliorem <lb></lb>eleuationem ex mali ipſius altitudine pertinet, ad vectis <pb xlink:href="007/01/072.jpg"></pb>contemplationem reducimus. </s> <s id="s.000644">eſt enim quæ dam vectium <lb></lb>ſpecies ab alijs non conſiderata, cuius brachia in angu<lb></lb>lum deſinunt, vt ipſe angulus in operatione ſit fulcimen<lb></lb>tum. </s> </p> <figure id="id.007.01.072.1.jpg" xlink:href="007/01/072/1.jpg"></figure> <p type="main"> <s id="s.000645">Eſto enim vectis, de quo agimus, <lb></lb>ABC, cuius brachia AB, BC. iuncta <lb></lb>ad angulum B, ſitqueue B in operatione <lb></lb>fulcimentum. </s> <s id="s.000646">Nec quicquam refert <lb></lb>quatenus ad vſum pertinet, vtrum an<lb></lb>gulus ipſe rectus ſit, acutus vel obtu<lb></lb>ſus. </s> <s id="s.000647">ſit autem modò rectus. </s> <s id="s.000648">Ponatur i<lb></lb>gitur pondus aliquod in C, tum po<lb></lb>tentia quædam applicetur in A, quae i<lb></lb>pſam vectis extremitatem A propel<lb></lb>lat in D. erit igitur AB in DB & an<lb></lb>gulo ſeruato BC in BE. </s> <s id="s.000649">Pondus igi<lb></lb>tur cum parte vectis BC eleuabitur in E. </s> <s id="s.000650">In hoc autem <lb></lb>vectis genere attenditur proportio quam habet AB ad <lb></lb>BC. </s> <s id="s.000651">Si enim potentia quæ applicatur in A ita ſe habet ad <lb></lb>pondus in C vt CB, ipſi BA, fiet æquilibrium. </s> <s id="s.000652">Si maior <lb></lb>autem fuerit proportio potentiæ in A, ad pondus in C, ea <lb></lb>quam habet AB ad BC, ſuperatâ ponderis reſiſtentiâ fiet <lb></lb>motus. </s> <s id="s.000653">Res autem haud aliter ſe habet, ac ſi producta in <lb></lb>F, fieret BF æqualis BC. </s> <s id="s.000654">Tunc enim vectis ad rectitudi<lb></lb>nem, ſeruatâ proportione, redigeretur, & ita potentia in <lb></lb>A, fulcimento B operaretur in F, vt operabatur in C. </s> </p> <p type="main"> <s id="s.000655">Ad huius vectis naturam referuntur fabrorum mal<lb></lb>lei, quibus clauos reuellunt, forcipes item quæ tenaci <lb></lb>morſu clauorum capita vmbellasue apprendentes, vio<lb></lb>lenter è tabulis extrahunt. </s> <s id="s.000656">In malleo itaque ſubtili, vt in <lb></lb>figura videre eſt, AB vectis eſt pars quæ à fulcimento ad <lb></lb>potentiam; ac verò quæ à fulcimento ad pondus, ponderi <pb xlink:href="007/01/073.jpg"></pb><figure id="id.007.01.073.1.jpg" xlink:href="007/01/073/1.jpg"></figure><lb></lb>ſiquidem æquiparatur reſi<lb></lb>ſtentia quae fit in C. </s> <s id="s.000657">I dem ob<lb></lb>ſeruamus in forcipe, in quo <lb></lb>duo quidem brachia AD, <lb></lb>CB, quatenus ad apprenſio<lb></lb>nem pertinet, fulcimentum, <lb></lb>habent in ipſo <expan abbr="cētro">centro</expan> ſeu ver<lb></lb>tebra, & ideo quo longiores <lb></lb>fuerint, eo tenaciùs appre<lb></lb>hendunt & retinent. </s> <s id="s.000658">quate<lb></lb>nus autem ad extractionem, <lb></lb>facit, pro vnico forceps totus habetur vecte, cuius <expan abbr="quidē">quidem</expan> <lb></lb>pars à potentia ad fulcimentum AB. quæ verò à <expan abbr="fulcimē-to">fulcimen<lb></lb>to</expan> ad hoc eſt clauum ipſum qui reuellitur AC. </s> <s id="s.000659">Violentiſ<lb></lb>ſimè autem extrahunt forcipes, propterea quod maxima <lb></lb>ſit proportio longitudinis brachij BA, ad eam quæ eſt ab <lb></lb>A ad C. </s> </p> <p type="main"> <s id="s.000660">His igitur hoc pacto examinatis, ad nauim & malum <lb></lb>reuertentes, dicimus, tunc facillimam fieri puppis eleua<lb></lb>tionem, proræ verò demerſionem, cum maxima fuerit <lb></lb>proportio, quam habet altitudo mali, ad eam nauis <expan abbr="partē">partem</expan> <lb></lb>quæ à malo ad ipſam puppis extremitatem, pertingit. <lb></lb></s> <s id="s.000661">Quamobrem prudentes nauium fabri, vt huic difficultati <lb></lb>occurrant, malum non in medio quidem nauis, ſed in ter<lb></lb>tia ferè parte longitudinis quæ à prora eſt, puppim verſus <lb></lb>conſtituunt. </s> </p> <figure id="id.007.01.073.2.jpg" xlink:href="007/01/073/2.jpg"></figure> <p type="main"> <s id="s.000662">Eſto enim nauis AB; cuius <lb></lb>malus CD: prora A: puppis B; <expan abbr="vē-to">ven<lb></lb>to</expan> igitur velum impellente, <expan abbr="malū">malum</expan> <lb></lb>ad partem contrariam vergit, pu<lb></lb>ta in FD. </s> <s id="s.000663">At <expan abbr="quoniã">quoniam</expan> catcheſium <lb></lb>funi ad puppim vnitur in B, nauim, <lb></lb>hoc eſt, ipſam puppim trahat ne<pb xlink:href="007/01/074.jpg"></pb>ceſſe eſt. </s> <s id="s.000664">non poteſt autem; quoniam ſuburræ grauitas & <lb></lb>onera, quæ naui impoſita inter D. & <emph type="italics"></emph>B.<emph.end type="italics"></emph.end> grauitatis centrum <lb></lb>circa punctum E conſtituunt, quod quidem vi ventorum <lb></lb>inclinante malo ab E, in G eleuaretur, quo igitur minor <lb></lb>fuerit proportio CD ad DE & maius pondus ipſum cu<lb></lb>ius grauitatis centrum in E minus præualebit potentia <lb></lb>pellens in C ad eleuationem partis nauigij, quæ à mali ſe<lb></lb>de ad puppim intercedit, An igitur malus ſit vectis, pes ve<lb></lb>rò fulcimentum, pondus autem quod vecte mouetur, <expan abbr="ipsū">ipsum</expan> <lb></lb>nauigium, vt placuit Ariſtoteli, & qua item ratione malus <lb></lb>in nauim vt vectis operetur, ex ijs quae dicta ſunt, facilè pa<lb></lb>tet. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000665">QVÆSTIO VII.</s> </p> <p type="head"> <s id="s.000666"><emph type="italics"></emph>Quæritur, Cur quando ex puppi nauigare voluerint, non flante ex <lb></lb>puppi vento, veli quidem partem, quæ ad gubernatorem vergit, <lb></lb>conſtringunt; illam verò quæ proram verſus eſt, pedem <lb></lb>facientes, relaxant?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000667">Mirabilis huius effectionis cauſſam explicat Ariſtote<lb></lb>les. </s> <s id="s.000668">inquit enim, An quia retrahere quidem multo <lb></lb>exiſtente vento gubernaculum non poteſt, pauco autem <lb></lb>poteſt, quem conſtringunt? </s> <s id="s.000669">propellit igitur quidem ipſe <lb></lb>ventus, in puppim verò illum conſtituit gubernaculum, <lb></lb>retrahens, & mare compellens: ſimul & nautæ ipſi cum <lb></lb>vento contendunt; in contrariam enim ſe reclinant par<lb></lb>tem. </s> <s id="s.000670">Hæc ille. </s> </p> <p type="main"> <s id="s.000671">Cuius ſenſum breuitate ſubobſcurum, mirâ facilita<lb></lb>te explicat Picolomineus. </s> <s id="s.000672">Nos autem vt rem lucidiorem <lb></lb>faciamus, ſchema, quod nec ipſe fecit, nec Philoſophus, <lb></lb>proponemus. </s> </p> <p type="main"> <s id="s.000673">Eſto nauis A <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, cuius prora A, puppis verò D, guber<lb></lb>naculum C<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, temonis ala <emph type="italics"></emph>B<emph.end type="italics"></emph.end>D, veli ſinus EF, velum vero <lb></lb>ita conſtitutum, vt directè ex puppi flantem ventum exci-<pb xlink:href="007/01/075.jpg"></pb><figure id="id.007.01.075.1.jpg" xlink:href="007/01/075/1.jpg"></figure><lb></lb>piat. </s> <s id="s.000674">Hoc vbi euenerit, naui<lb></lb>gium, rectâ è puppi mouetur <lb></lb>in proram; Si autem ventus la<lb></lb>teraliter ſpirat, puta à parte <lb></lb>G verſus H & nihilo ſecius na<lb></lb>uigium, ac ſi ventus ex pup<lb></lb>pi eſſet antrorſum propelle<lb></lb>re volunt, velum quidem obli<lb></lb>quant partem eius infimam, <lb></lb>pedem nempe, quæ eſt in F <lb></lb>contrahentes, Cornu verò <lb></lb>antennæ vbi E, proram verſus <lb></lb>laxantes ventumque; ipſum obliquè excipientes id <expan abbr="efficiūt">efficiunt</expan>, <lb></lb>vt ventus minus violenter feriat, & minori ſui parte <expan abbr="velū">velum</expan> <lb></lb>impleat, & quoniam ventus velum pellit in partem con<lb></lb>trariam, nempe in H, ipſi vt vento reſiſtant conuerſo gu<lb></lb>bernaculo ex C in L, & temone <emph type="italics"></emph>B<emph.end type="italics"></emph.end>D, in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>M compellunt <lb></lb>proram ad partem à qua ventus ipſe ſpirat. </s> <s id="s.000675">Sit igitur inter <lb></lb>ventum & temonem pugna, illo proram in dextram, hoc <lb></lb>verò eandem in ſiniſtram pellente, <expan abbr="itaq;">itaque</expan> cum neuter præ<lb></lb>ualeat, neceſſario nauis mediam viam, quæ inter <expan abbr="vtramq;">vtramque</expan> <lb></lb>eſt, ſuo curſu tenet. </s> <s id="s.000676">Nautæ autem ideo in partem nauis <lb></lb>AE<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, quæ verſus ventum eſt, ſe conferunt, vt vento æqui<lb></lb>librium faciant, ne ſcilicet naui in <expan abbr="cōtrariam">contrariam</expan> partem pel<lb></lb>lente ſpiritu, eam demergat. </s> <s id="s.000677">Cæterùm quod nec Ariſto<lb></lb>teles nec Picolomineus animaduerterunt, velum obli<lb></lb>què conſtitutum à vento in anteriora impellitur eandem <lb></lb>ob cauſſam, quam retulimus, vbi de temone & velis, qui<lb></lb>bus farinariæ molæ <expan abbr="cōuertuntur">conuertuntur</expan>, verba faceremus. </s> <s id="s.000678">Quod <lb></lb>autem addit Picolomineus rem ad vectem reduci poſſe, <lb></lb>non eſt cur ſub ſilentio prætereamus. </s> <s id="s.000679">Ventus, inquit, pon<lb></lb>deris gubernaculum mouentis vicem obtinet; centrum <lb></lb>verò (fulcimentum intelligit) in medio nauis eſt, quod ta-<pb xlink:href="007/01/076.jpg"></pb>men ad proram vergit, vt faciliùs ipſi vento reſiſtere poſ<lb></lb>ſit. </s> <s id="s.000680">Tunc enim in rectum mouebitur nauis, cum ſibi inui<lb></lb>cem æquatæ vires, quaſi libramentum conſtituerint. </s> <s id="s.000681">Hæc <lb></lb>ille, cuius ſenſum figurâ propoſitâ facilè aperiemus. </s> </p> <figure id="id.007.01.076.1.jpg" xlink:href="007/01/076/1.jpg"></figure> <p type="main"> <s id="s.000682">Eſto carina AB, cuius prora <lb></lb>A, puppis, B temo BC, ventus verò <lb></lb>obliquè feriens H. </s> <s id="s.000683">Conuerſus ita<lb></lb>que temo vt in BC vndarum vi cur<lb></lb>rente naui repulſus ſit in EF ten<lb></lb>dens verſus I, quo caſu prora con<lb></lb>uertitur in D, nempe contra <expan abbr="ventū">ventum</expan> <lb></lb>qui ſpirat ex H. fit autem conuer<lb></lb>ſio circa punctum G, quod fulcimenti locum obtinet. </s> <s id="s.000684"><expan abbr="Vē-tus">Ven<lb></lb>tus</expan> verò ad contrariam <expan abbr="partē">partem</expan> proram impellit, repugnans <lb></lb>Temonis violentiæ contra ipſam proram dirigentis. </s> <s id="s.000685">Eſt i<lb></lb>gitur AB, ſeu DE carina, inſtar vectis, cuius fulcimentum <lb></lb>G, vis mouens mare quo temo EF repellitur, pondus ve<lb></lb>ro, ventus premens in D; quo igitur remotior erit temo à <lb></lb>fulcimento G, D autem vbi pondus ei vicinius, eo magis <lb></lb>temo venti vim ſuperabit. </s> <s id="s.000686">Hæc Picolominei ratio, quam <lb></lb>explicauimus, ſanè ingenioſa eſt, verum enimuero, quo<lb></lb>niam fulcimentum ſui naturâ ſtare debet, hic verò <expan abbr="nullã">nullam</expan> <lb></lb>habeat ſtabilitatem, difficultatem patitur. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000687">QVÆSTIO VIII.</s> </p> <p type="head"> <s id="s.000688"><emph type="italics"></emph>Quæritur, Cur ex figuris omnibus rotundæ faciliùs <lb></lb>moueantur?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000689">Trifariam, inquit Ariſtoteles, circulum rotari contin<lb></lb>git; Aut ſecundum abſidem <expan abbr="cētro">centro</expan> ſimul moto, quem<lb></lb>admodum plauſtri vertitur rota; aut circa manens cen<lb></lb>trum, veluti trochleæ puteorum, ſtante centro: Aut in pa<lb></lb>uimento manente centro, ſicuti figuli rota conuertitur. <pb xlink:href="007/01/077.jpg"></pb>Cauſſam verò explicans, ait, celerrima eiuſmodi corpora <lb></lb>eſſe, eo quod paruâ ſui parte planum contingunt, vti cir<lb></lb>culus ſecundum punctum, item quoniam non offenſant: <lb></lb>Non offenſandi vero eſſe cauſſam, quod ſemotum à terra <lb></lb>habeant angulum. </s> <s id="s.000690">Item propterea quod corpus, cui fiunt <lb></lb>obuiam, ſecundum puſillum tangunt. </s> <s id="s.000691">Rectilineo autem <lb></lb>aliter euenire, quippe quod rectitudine ſuâ, multum pla<lb></lb>ni contingat. </s> <s id="s.000692">Ad hæc, quo nutat pondus eo mouentem <lb></lb>mouere. </s> </p> <p type="main"> <s id="s.000693">Hæc ferè Philoſophus, cuius rationes ad eum ſolum<lb></lb>modo circularem motum faciunt, qui fit ſecundum abſi<lb></lb>dem, vt in carrorum rotis vſu venit, nec aptantur rotis fi<lb></lb>gulorum trochleiſqueue, cuiuſmodi ſunt illæ, quæ ſupra <lb></lb>puteos appenduntur. </s> <s id="s.000694">Nos igitur, ad Ariſtotelis mentem, <lb></lb>primam rotationis ſpeciem, quæ eſt ſecundum abſidem, <lb></lb>examinabimus. </s> </p> <figure id="id.007.01.077.1.jpg" xlink:href="007/01/077/1.jpg"></figure> <p type="main"> <s id="s.000695">Eſto rota ſphæ<lb></lb>raue AB, cuius cen<lb></lb>trum C; Horizontis <lb></lb>planum DE; conta<lb></lb>ctus circuli in plano <lb></lb>B. <expan abbr="perpēdicularis">perpendicularis</expan> ho<lb></lb>rizonti à puncto <expan abbr="cō-tactus">con<lb></lb>tactus</expan> B ipſa <emph type="italics"></emph>B<emph.end type="italics"></emph.end>CA, <lb></lb>tranſiens per <expan abbr="centrū">centrum</expan> <lb></lb>C, partes rotæ circa <lb></lb>perpendicularem AF<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, AG<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, angulus contactus G<emph type="italics"></emph>B<emph.end type="italics"></emph.end>E. <lb></lb></s> <s id="s.000696">Primo itaque id conſtat, circulum in puncto planum, ſeu <lb></lb>lineam contingere. </s> <s id="s.000697">At quoniam, vt Mechanici, de circulis <lb></lb>rotiſqueue ſeu ſphæris agimus materialibus, rectè Philoſo<lb></lb>phus non in puncto planum præcisè tangere dixit, ſed ſe<lb></lb>cundum partem ſui minimam. </s> <s id="s.000698">Angulum porro, quem à <lb></lb>terra ſemotum dicit, ipſe angulus eſt contingentiae. </s> <s id="s.000699">eleua<pb xlink:href="007/01/078.jpg"></pb>tur enim ex <emph type="italics"></emph>B<emph.end type="italics"></emph.end> in G. </s> <s id="s.000700">Si autem corpus quodpiam in plano <lb></lb>fuerit, puta HI in puncto illud tanget ci culus ei occur<lb></lb>rens, exempli gratiâ in K. </s> <s id="s.000701">Hæc igitur accidunt circulari <lb></lb>figuræ. </s> <s id="s.000702">In lateratis autem ſecus fit, quippe quæ nec in <expan abbr="pū-cto">pun<lb></lb>cto</expan> ſeu ſecundum paruam ſui partem, planum tangunt, <lb></lb>nec ſemotum vt circulus à plano habent angulum, nec <lb></lb>impingentes offendiculum in puncto tangunt. </s> <s id="s.000703">Cæterùm <lb></lb>potiſſimam facilitatis motus in rotatione quæ fit ſecun<lb></lb>dum abſidem, eſſe cauſſam dixit, nempe quò nutat pon<lb></lb>dus eò à mouente impelli ac moueri. </s> <s id="s.000704">Primò igitur circu<lb></lb>laris ſphæricaue figura in æquilibrio ſtat; æquales enim <lb></lb>ſunt partes quæ circa perpendicularem: ceu ſunt AF<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <lb></lb>AG<emph type="italics"></emph>B.<emph.end type="italics"></emph.end> ſi enim impulſus fiat ex parte F, pars oppoſita nuta<lb></lb>bit, & propendet in partem G, & ſuo nutu motuque; ſecum <lb></lb>trahet partem AF<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, fietqueue progreſſus. </s> <s id="s.000705">Si enim ducatur <lb></lb>FCG diameter, ipſi horizonti æ que diſtans, erit veluti li<lb></lb>bra, cuius pondera vtrinque AF<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, AG<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, brachia verò <lb></lb>æqualia CF, CG. </s> <s id="s.000706">Potentia autem quâ trahitur pellitur<lb></lb>ue ad inſtar ponderis ſe habet, quo addito partium alteri, <lb></lb>facto queue receſſu ab æquilibrio, ſequetur motus. </s> <s id="s.000707">Putauêre <lb></lb>quidam, vt refert Philoſophus, <expan abbr="circularē">circularem</expan> lineam, ita per<lb></lb>peti motu verſatum iri, vt manentia, propter contrarium <lb></lb>nixum, manent, neque enim circulus in plano contrarium <lb></lb>nixum habet, cum ſit, veluti dicebamus, in æquilibrio & <lb></lb>facilis in vtramuis partem moueri. </s> <s id="s.000708">Veruntamen perpe<lb></lb>tuum eſſe non poſſe horum corporum motum, ea eſt cauſ<lb></lb>ſa, quod violentum accidat naturæ, & ideo non durabile. <lb></lb></s> <s id="s.000709">Ad hæc, addit Philoſophus, Maiores circulos ad minores <lb></lb>nutum habere <expan abbr="quēdam">quendam</expan>; & nutum maioris ad minoris nu<lb></lb>tum, ſe habere vt angulos ad angulos, & <expan abbr="diametrū">diametrum</expan> ad dia<lb></lb>metrum. </s> <s id="s.000710">Angulos autem hîc ſectores ipſos vocat; oportet <lb></lb>enim circulos tum maiores tum minores circa idem cen<lb></lb>trum eſſe conſtitutos. </s> <s id="s.000711">Hæc autem non abſimili ab eo <lb></lb>quod ſuprà poſuimus ſchemate explicantur. </s> </p> <pb xlink:href="007/01/079.jpg"></pb> <figure id="id.007.01.079.1.jpg" xlink:href="007/01/079/1.jpg"></figure> <p type="main"> <s id="s.000712">Eſto enim circulus <lb></lb>AB circa centrum, C, <lb></lb>Horizontis planum DE, <lb></lb>tangens circulum in B, <lb></lb>linea verò perpendicu<lb></lb>laris per centrum BCA. <lb></lb></s> <s id="s.000713">Sit autem circa idem <expan abbr="cē-trum">cen<lb></lb>trum</expan> C, minor circulus <lb></lb>FG, ducatur queue CH ſe<lb></lb>cus minorem circulum in I, tangens verò maiorem in H, <lb></lb>conſtituenſqueue cum AC linea angulum ACH, duos an<lb></lb>gulos, ex Ariſtotelis mente comprehendentem, hoc eſt, <lb></lb>duos ſectores ACH, FCI. quoniam igitur ſector ſeu an<lb></lb>gulus ACH, ſuo ſpatio ſuperat angulum ſeu ſectorem <lb></lb>FGI, facilè ex nutu quem maior ſupra minorem habet, <lb></lb>maior ipſe mìnorem mouet. </s> <s id="s.000714">Videtur autem tacitè Philo<lb></lb>ſophus hæc ad vectis naturam referre, cuius altera extre<lb></lb>mitatum in centro ſit, altera verò in ab ſide, & ita ſe habe<lb></lb>re nutum maioris ſupra minorem, vt vectis ad vectem, hoc <lb></lb>eſt, ſemidiameter ad ſemidiametrum, ſeu ſector ad ſecto<lb></lb>rem, quos quidem ſectores, vt vidimus, angulos appellat. <lb></lb></s> <s id="s.000715">Hæc autem quæ de nutu refert, licet ſubtilia ſint, vera eſ<lb></lb>ſe non videntur. </s> <s id="s.000716">Si enim in figura producatur ad oppoſi<lb></lb>tam partem ſemidiameter HC in K ſecans minorem cir<lb></lb>culum in L, duos alios ſectores angulosue habebimus, <expan abbr="nē-pe">nem<lb></lb>pe</expan> KCB, LCG, ipſis ACHFCI æquales. </s> <s id="s.000717"><expan abbr="Itaq;">Itaque</expan> quan<lb></lb>tum adiuuat motum anguli ACH maioris nutus, in de<lb></lb>ſcendendo ad partes B, tantundem retardat anguli item <lb></lb>maioris KCB, contra nutus (vt ita appellem) in <expan abbr="aſcendē-do">aſcenden<lb></lb>do</expan> ad partes A. & ſanè quatenus ad rei naturam pertinet <lb></lb>& ad ipſum æquilibrium, non differunt maiores circuli à <lb></lb>minoribus, nec ſunt maiores minoribus mobiliores, imo <lb></lb>ex aliqua ratione minores videntur fore ad motum faci<pb xlink:href="007/01/080.jpg"></pb>liores, tum quia data materiæ æqualitate ſunt leuiores, <lb></lb>tum etiam quod maior eſt angulus contactus ad planum <lb></lb>circumferentiae minoris quàm maioris circuli, vt in ſubie<lb></lb><figure id="id.007.01.080.1.jpg" xlink:href="007/01/080/1.jpg"></figure><lb></lb>cta figura angulus ABC maior <lb></lb>eſt angulo DBC, in materiali i<lb></lb>gitur circulo rotaue maiore ſui <lb></lb>parte tanget planum DB circu<lb></lb>lus, ipſo AB. quicquid tamen fit, <lb></lb>mobiliores ſunt maiores circuli <lb></lb>non quidem ex natura circuli, <lb></lb>quæ tam in maioribus quàm in <lb></lb>ipſis minoribus eſt par, ſed alijs de cauſſis, quas ſuo loco <lb></lb>examinabimus. </s> </p> <p type="main"> <s id="s.000718">Cæterùm vt aliquid de motu qui ſecundum abſidem <lb></lb>fit, ex noſtro penu promamus, Dicimus, Circulos, rotaſue, <lb></lb>quæ hoc pacto mouentur, vel per horizontis planum mo<lb></lb>ueri, vel per accliue, aut decliue. </s> <s id="s.000719">Si autem per horizontis <lb></lb>planum, ideo facilem eſſe motum, quòd nunquam, cæte<lb></lb>ris paribus, centrum grauitatis ipſius corporis à centro <lb></lb>mundi, in ipſa rotatione, fiat remotius. </s> </p> <figure id="id.007.01.080.2.jpg" xlink:href="007/01/080/2.jpg"></figure> <p type="main"> <s id="s.000720">Eſto enim planum, <lb></lb>horizontis AB, cui circu<lb></lb>lus inſiſtat AD, circa cen<lb></lb>trum C, diuiſus per <expan abbr="centrū">centrum</expan> <lb></lb>ipſum à perpendiculari <lb></lb>ACD; Ducatur autem per <lb></lb>centrum C recta linea ho<lb></lb>rizonti æquidiſtans, ECFG: dum diuidatur circulus vt<lb></lb>cunque in partes AH, HF, FI, ID, & CI, CH iungan<lb></lb>tur. </s> <s id="s.000721">Poſthæc intelligatur circulum ſecundum abſidem <lb></lb>moueri ad partes G, erit igitur aliquando punctum H, <lb></lb>tangens horizontis planum, tangat autem in K, tum F in <pb xlink:href="007/01/081.jpg"></pb>L, I in N. D verò in O. </s> <s id="s.000722">Ducanturqueue KP, LQ, NR, OS <lb></lb>ipſi AC parallelæ horizonti autem perpendiculares. <lb></lb></s> <s id="s.000723">Centrum ergo circuli, quod idem & grauitatis eſt <expan abbr="centrū">centrum</expan>, <lb></lb>feretur per rectam CPQRS, ſunt enim KP, LQ, NR, <lb></lb>OS ipſi AC ſemidiametro æquales, <expan abbr="nūquam">nunquam</expan> igitur cen<lb></lb>trum ipſum C in circuli rotatione ab horizontis plano e<lb></lb>leuabitur, nec à mundi centro fiet remotius. </s> </p> <p type="main"> <s id="s.000724">Hoc autem longè aliter cæteris figuris contingit, <lb></lb>quarum motus ideo in æqualis, quòd non ſemper in rota<lb></lb>tione centrum grauitatis eandem ſeruet à mundi centro <lb></lb>diſtantiam. </s> </p> <figure id="id.007.01.081.1.jpg" xlink:href="007/01/081/1.jpg"></figure> <p type="main"> <s id="s.000725">Eſto enim Ellipſis <lb></lb>ABCD, cuius <expan abbr="cētrum">centrum</expan> <lb></lb>E, diameter longior <lb></lb>BED, breuior AEC, <lb></lb>Horizontis planum, <lb></lb>FCG. locus contactus <lb></lb>C perpendicularis à <lb></lb>contactu per centrum i<lb></lb>pſa CEA diuidens El<lb></lb>lipſim in partes æquales, & æqueponderantes ABC, <lb></lb>ADC. </s> <s id="s.000726">Sumantur in quadrante CD, <expan abbr="pūcta">puncta</expan> HI, tum EH, <lb></lb>HI iungantur, erit autem EH longior ipſa EC, tum EI, <lb></lb>ipſa EH & ED, pſa EI. </s> <s id="s.000727">Rotetur ellipſis ſecun dum abſi<lb></lb>dem, fiet igitur punctum H in K, & à puncto K horizonti <lb></lb>perpendicularis erigatur KL, quæ fiat æqualis EH. </s> <s id="s.000728">Poſt <lb></lb>hæc punctum I erit in M, & ab M perpendicularis, æqua<lb></lb>lis EI. rurſus D fiat in O, & ipſi ED, æqualis perpendicu<lb></lb>laris OP. </s> <s id="s.000729">Mota igitur ellipſi à C in K, haud ita difficilis e<lb></lb>rit motus, quippe quod haud multum EH ſuperet EC, at <lb></lb>difficilior erit translatio in M, difficillima verò in O. Val<lb></lb>de enim à ſitu E, ibi attollitur grauitatis centrum, aſcen<lb></lb>dens nempe vbi P. </s> <s id="s.000730">Videmus igitur ex his eandem poten<pb xlink:href="007/01/082.jpg"></pb>tiam in mouendo ellipſim, haud pariter ſe habere, vt in <lb></lb>mouendo circulum. </s> <s id="s.000731">ibi enim centrum grauitatis fertur <lb></lb>per æquidiſtantem horizonti, hic verò modò attollitur, <lb></lb>modò deprimitur, quod ſanè moleſtiam & difficultatem <lb></lb>facit. </s> <s id="s.000732">Sed idem alijs figuris contingere, & maximè latera<lb></lb>tis, ita docebimus. </s> </p> <figure id="id.007.01.082.1.jpg" xlink:href="007/01/082/1.jpg"></figure> <p type="main"> <s id="s.000733">Eſto enim triangulum <lb></lb>æquilaterum ABC, cuius <lb></lb>grauitatis centrum E hori<lb></lb>zontis planum BD. </s> <s id="s.000734">Demit<lb></lb>tatur à vertice A perpendi<lb></lb>cularis horizonti AF tranſ<lb></lb>ibit autem per centrum E, <lb></lb>& bifariam diuidet baſim <lb></lb>BC in F. </s> <s id="s.000735">Sunt autem trianguli ABF, ACF, æquales & <lb></lb>æqueponderantes. </s> <s id="s.000736">angulus verò AFC rectus. </s> <s id="s.000737">lungatur <lb></lb>EC, erit igitur maior EC, ipſa EF. </s> <s id="s.000738">Rotetur iraque trian<lb></lb>gulum circa punctum C, fiatque; EC horizonti perpendi<lb></lb>cularis, ſitqueue GH, & per E horizonti parallela ducatur <lb></lb>EK, moto igitur triangulo, centrum grauitatis E transla<lb></lb>tum erit in H, ſed KC æqualis eſt EF, minor autem ipſa <lb></lb>CH, eleuatur ergo centrum grauitatis ab E in H, nempe <lb></lb>ſupra K, totum ſpatium KH. ex qua eleuatione fit in mo<lb></lb>tu difficultas. </s> <s id="s.000739">Idem prorſus eadem demonſtratione oſten<lb></lb>deretur fieri in quadrato & alijs lateratis figuris. </s> <s id="s.000740">Cur igi<lb></lb>tur in plano horizontis facillimè circularia, difficile <expan abbr="autē">autem</expan> <lb></lb>laterata & quæ inæquales habent ſemidiametros, mo<lb></lb>ueantur, ex dictis clarè patet. </s> </p> <p type="main"> <s id="s.000741">Ad hanc quæſtionem illud quoque facit, cur per de<lb></lb>cliue planum grauiora corpora, & rotunda maximè; ma<lb></lb>gno impetu dimiſſa, delabantur. </s> </p> <p type="main"> <s id="s.000742">Eſto enim rota ſphæraue aut Cylindrus CD, cuius <lb></lb>centrum E, tangens decliue planum AB in D, quæritur <pb xlink:href="007/01/083.jpg"></pb>cur dimiſſa hæc magno impetu deferantur ad partes B, <lb></lb>Ducatur per grauitatis centrum E ad horizontem, BK <lb></lb>perpendicularis FEL ſecans decliue planum in G, cir<lb></lb>cumferentiam verò in H. opponitur autem EG angulo <lb></lb>recto EDG, maior ergo EG ipſa ED, hoc eſt, EH, inter <lb></lb><figure id="id.007.01.083.1.jpg" xlink:href="007/01/083/1.jpg"></figure><lb></lb>circumferentiam igitur & pla<lb></lb>num decliue, ſpatium interce<lb></lb>dit HG. </s> <s id="s.000743">Ducatur item DI ipſi <lb></lb>FG æquidiſtans. </s> <s id="s.000744">non tranſibit <lb></lb>igitur per centrum E. minor e<lb></lb>rit igitur diametro CD, quare <lb></lb>circulum in partes inæquales <lb></lb>ſecabit, & non per grauitatis <lb></lb>centrum, quod idem cum ma<lb></lb>gnitudinis ſeu figuræ centro ſupponitur. </s> <s id="s.000745">Dimiſſa igitur <lb></lb>rota, contingit quidem planum decliue in puncto D. </s> <s id="s.000746">At <lb></lb>centrum grauitatis premit ſecundam per lineam perpen<lb></lb>dicularem FG, non ſuſtentatur autem in H, quippe quod <lb></lb>inter planum & circum <expan abbr="ferentiã">ferentiam</expan> intercedat ſpatium HG, <lb></lb>nec H locum habeat cui innitatur, corpus autem ita per <lb></lb>lineam DI eſt diuiſum, vt longè maior ſit pars IFCHD <lb></lb>ipſa DI, & centrum in ea parte cadat quæ non fulcitur. </s> <s id="s.000747">i<lb></lb>taque ſuopte nutu, cum extra fulcimentum ſit D & per<lb></lb>pendicularem DI ad inferiores partes rapidè rotans de<lb></lb>labitur. </s> <s id="s.000748">Ducatur autem perpendicularis GL, parallela <lb></lb>MN, & quoniam BN breuior eſt BL, erit MN ipſa GL <lb></lb>breuior. </s> <s id="s.000749">Eſt igitur punctum M mundi centro propius <lb></lb>quàm D & G, quare eò non impedita rota ipſa ſuo nutu <lb></lb>feretur, nec ſtabit donec in fimum <expan abbr="locū">locum</expan> vbi quieſcat nan<lb></lb>ciſcatur. </s> <s id="s.000750">Poſſumus etiam Rota ſphæraue in plano decliui <lb></lb>collocata, datam potentiam inuenire, quæ extremitati <lb></lb>diametri ad eam partem qua vergit applicata ipſam rotam <lb></lb>ſphæramue impediatne delabatur. </s> </p> <pb xlink:href="007/01/084.jpg"></pb> <figure id="id.007.01.084.1.jpg" xlink:href="007/01/084/1.jpg"></figure> <p type="main"> <s id="s.000751">Eſto planum in clinatum <lb></lb>AB, cui Rota ſphæraue inſi<lb></lb>ſtat tangatque; illud in C. </s> <s id="s.000752">Rota <lb></lb>verò ipſa ſphæraue DC, cu<lb></lb>ius centrum E, diameter ve<lb></lb>rò DEC ipſi BA ad <expan abbr="punctū">punctum</expan> <lb></lb>contactus C, perpendicula<lb></lb>ris. </s> <s id="s.000753">Ducatur per C ipſi hori<lb></lb>zonti perpendiculatis FCG <lb></lb>circulum <expan abbr="ſecãs">ſecans</expan> in G tum per <lb></lb>E ipſi CG perpendicularis, ipſi verò BF horizonti æqui<lb></lb>diſtans HEI ceu vectis, cuius fulcimentum I reſpondens <lb></lb>ipſi C, pondus verò in E, vbi grauitatis eſt centrum. </s> <s id="s.000754">Ap<lb></lb>plicata igitur potentia in H erit pondus inter fulcimen<lb></lb>tum & potentiam, quare vt IE ad IH ita potentia ſuſti<lb></lb>nens in H ad pondus in E, quod demonſtrandum fuerat. </s> </p> <p type="main"> <s id="s.000755">Quippiam ſimile oſtendit Pappus 1. 8. prop. 9. alijs <lb></lb>tamen ſuppoſitis & conſideratis. </s> <s id="s.000756">Dico præterea, ijſdem <lb></lb>ſtantibus angulum ECI æqualem eſſe angulo inclinatio<lb></lb>nis CBF. </s> <s id="s.000757">Producatur HI concurrens cum ipſa AB in K, <lb></lb>concurret autem propterea, quod CIK rectus ſit, ICA <lb></lb>minor recto, & quoniam HK parallela eſt horizonti BF <lb></lb>alterni anguli IKC, CBF, æquales erunt. </s> <s id="s.000758">Similes autem <lb></lb>ſunt ECI, ECK, trianguli, eſtqueue ECI angulus æqualis <lb></lb>angulo EKC, hoc eſt, ipſi CBF. vnde ſequitur, quo mi<lb></lb>nor fuerit inclinationis angulus, eo facilius rotam ſphæ<lb></lb>ramue in plano inclinato ſuſtineri. </s> <s id="s.000759">quo enim minor fuerit <lb></lb>angulus ECI, eo minus latus EI & minor proportio EI <lb></lb>ad IH, & ideo minor potentia ſuſtinens requiratur in H. <lb></lb></s> <s id="s.000760">Cæterùm accliue & decliue planum nihil differunt niſi <lb></lb>reſpectu. </s> </p> <p type="main"> <s id="s.000761">His ita conſideratis, admonet nos locus, vt pulcher<lb></lb>rimam dubitationem diluamus. </s> <s id="s.000762">Quæritur, Cur maiores <pb xlink:href="007/01/085.jpg"></pb>rotae impingentes, facilius offendicula ſuperent quàm mi<lb></lb>nores. </s> <s id="s.000763">Neque enim ſatisfacere videtur quod ait Ariſtote<lb></lb>les, ex contactu in puncto eo anguli à plano eleuatione id <lb></lb>fieri, alijs ergo principijs dubitatio ſoluitur. </s> </p> <figure id="id.007.01.085.1.jpg" xlink:href="007/01/085/1.jpg"></figure> <p type="main"> <s id="s.000764">Eſto rota quidem maior <lb></lb>AB, circa centrum C minor <lb></lb>vero DB circa centrum, E, <lb></lb><expan abbr="tãgentes">tangentes</expan> horizontis planum <lb></lb>in B. </s> <s id="s.000765">Diameter maioris AB, <lb></lb>minoris DB, offendiculum, <lb></lb>horizonti perpendiculare <lb></lb>FG. </s> <s id="s.000766">Ducatur per F horizonti <lb></lb>parallela FK ſecans minoris <lb></lb>rotæ peripheriam in H, dia<lb></lb>metrum verò AB in K, & à <lb></lb>puncto H ad <expan abbr="planū">planum</expan> horizon<lb></lb>tis perpendicularis demittatur HI: erit autem HI æqua<lb></lb>lis ipſi offendiculo FG, & iungantur BH, BF. <expan abbr="Itaq;">Itaque</expan> quo<lb></lb>niam BH ab extremo B cadit in triangulum KFB, erit <lb></lb>KHB angulus maior angulo KFB. </s> <s id="s.000767">Parallelæ autem ſunt <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>F, BG, pares ergo anguli <emph type="italics"></emph>K<emph.end type="italics"></emph.end>HB, HBG, pares item <emph type="italics"></emph>K<emph.end type="italics"></emph.end>FB, <lb></lb>FBG, Maior ergo HBI, ipſo FBC. </s> <s id="s.000768">At minoris rotæ gra<lb></lb>uitatis centrum mouetur ſecundum lineam BH, maius <lb></lb>verò ſecundum literam BF, difficilius ergo mouebitur, & <lb></lb>ſuperabit offendiculum minor rota, quàm maior: quod <lb></lb>fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.000769">Poſſumus idem oſtendere magis mechanicè, hoc <lb></lb>eſt, tem ad vectem reducendo. </s> <s id="s.000770">Eſto horizontis planum <lb></lb>AB, rota maior CD planum tangens in D. rotæ verò ma<lb></lb>ioris centrum E. </s> <s id="s.000771">Rota verò minor FD, tangens itidem <lb></lb>planum in D. rotæ autem centrum G, offendiculi verò re<lb></lb>ctitudo DH. </s> <s id="s.000772">Ducatur per H ipſi AB horizonti æquidi<lb></lb>ſtans HI ſecans minorem circulum in K, maiorem verò <pb xlink:href="007/01/086.jpg"></pb><figure id="id.007.01.086.1.jpg" xlink:href="007/01/086/1.jpg"></figure><lb></lb>in I. </s> <s id="s.000773">Ducantur etiam dia<lb></lb>metri maioris quidem <lb></lb>LEM, minoris NGO, <lb></lb>Tum à puncto K perpen<lb></lb>dicularis ducatur ad <lb></lb>GO, ipſa KP, item à pun<lb></lb>cto I ad EM perpendi<lb></lb>cularis <expan abbr="Iq.">Ique</expan> Dico EQ ad <lb></lb>QL, minorem habere <lb></lb>proportionem quam GP, <lb></lb>ad PN. </s> <s id="s.000774">Connectatur <lb></lb>GK, & ei per E parallela <lb></lb>ducatur ER, ſecans maiorem circulum in R, & ab R ipſi <lb></lb>EM perpendicularis ducatur RS. quoniam igitur ER <lb></lb>parallela eſt ipſi GK, erit GER angulus HGK angulo <lb></lb>æqualis. </s> <s id="s.000775">Recti autem ſunt HGP, GES reliqui ergo KGP, <lb></lb>RES ad inuicem ſunt æquales. </s> <s id="s.000776">Sed & ESR, GPK recti <lb></lb>ſunt, quare ERSGKP anguli æquales ſunt, & trianguli <lb></lb>GPKESR, per pr. diff. </s> <s id="s.000777">1.6. ſimiles. </s> <s id="s.000778">Vt ergo GK hoc eſt <lb></lb>GN ad GP, ita ER hoc eſt EL ad ES. </s> <s id="s.000779">Componendo igi<lb></lb>tur vt NP ad PG, ita LS ad SE. quamobrem ſi fulcimen<lb></lb>tum eſſet in S, pondus in E, <expan abbr="potētia">potentia</expan> in L, idem fieret ac fiat <lb></lb>fulcimento in P, pondere in G, potentia verò in N conſti<lb></lb>tuta. </s> <s id="s.000780">& id quidem ſi eiuſdem ponderis vtraque rota ſup<lb></lb>ponatur. </s> <s id="s.000781">Rurſus quoniam vt DK ad totum circulum DF, <lb></lb>ita DR ad totum DC. </s> <s id="s.000782">Minor eſt autem proportio DI ad <lb></lb>totum circulum DC, ergo minor eſt DI ipſa DR. </s> <s id="s.000783">Maior <lb></lb>ergo MI ipſa MR, maior ergo QI ipſa SR, propius ergo <lb></lb>centro E eſt Q ipſo puncto S, minor eſt igitur proportio <lb></lb>EG ad LQ quàm ES ad SL. </s> <s id="s.000784">Minor ergo potentia requi<lb></lb>ritur in L ad ſuſtinendum pondus E ex fulcimento Q hoc <lb></lb>eſt I, quàm requiratur in N ad ſuſtinendum pondus G ex <lb></lb>fulcimento P, hoc eſt K. </s> <s id="s.000785">Minor ergo potentia requiritur <pb xlink:href="007/01/087.jpg"></pb>ad transferendam maiorem retam CD vltra offendicu<lb></lb>lum IV, hoc eſt, DH, quàm requiratur ad trans ferendam <lb></lb>minorem vltra offendiculum KT, hoc eſt HD, quod fue<lb></lb>rat oſtendendum. </s> </p> <p type="main"> <s id="s.000786">Ad hæc, quæri poteſt, quo pacto plauſtrorum rotæ <lb></lb>in ipſa plauſtri conuerſione ſe habeant, nempe quæ ſit li<lb></lb>nea illa curua, quam in conuerſione deſcribunt. </s> </p> <figure id="id.007.01.087.1.jpg" xlink:href="007/01/087/1.jpg"></figure> <p type="main"> <s id="s.000787">Eſto rotarum in <lb></lb>plano orbita, <expan abbr="dū">dum</expan> plau<lb></lb>ſtrum rectâ procedit <lb></lb>AB, CD, Sunt autem i<lb></lb>pſæ lineæ, quod oſten<lb></lb>demus poſtea, æquedi<lb></lb>ſtantes. </s> <s id="s.000788">Sit itaque pun<lb></lb>ctum. </s> <s id="s.000789">B illud in quod <lb></lb>rota quæ per AB ſer<lb></lb>tur, eò delata planum <lb></lb>tangit. </s> <s id="s.000790">D verò alterius rotæ at que plani contactus. </s> <s id="s.000791">Igitur <lb></lb>dum plauſtri fit conuerſio, punctum D conuerſionis fit <lb></lb>centrum. </s> <s id="s.000792">Stat enim interim rota & circa lineam conuer<lb></lb>titur, quæ å puncto contactus D per rotæ centrum ducta <lb></lb>horizontis plano eſt perpendicularis. </s> <s id="s.000793">ea autem ſtante, ro<lb></lb>ta quæ in B circa centrum D <expan abbr="ſemicirculū">ſemicirculum</expan> pertranſit DEF, <lb></lb>vbi autem rota B, peruenerit in F, plauſtro iam in oppoſi<lb></lb>tam partem conuerſo, rota quæ eſt in D per lineam DC, <lb></lb>quæ verò in F per rectam FG mouetur, plauſtriqueue fit re<lb></lb>greſſus. </s> <s id="s.000794">Et quoniam vel D in ipſa conuerſione ſtat omnino <lb></lb>nec quicquam progreditur, vt in prima figura, vel non ſtat <lb></lb>vt in ſecunda, quo caſu portionem parui circuli deſcribit, <lb></lb>ipſi maiori circulo & exteriori concentricam. </s> <s id="s.000795">Vnde col<lb></lb>ligimus, Plauſtrorum conuerſiones flexioneſque ſemper <lb></lb>circa centrum, & concentricorum circulorum portiones <lb></lb>fieri, <emph type="italics"></emph>H<emph.end type="italics"></emph.end>inc etiam diſcimus, cur veteres, vt ex antiquis co<pb xlink:href="007/01/088.jpg"></pb>gnoſcimus veſtigijs, circos in quibus curſus quadrigarum <lb></lb>fiebant ea forma quæ apparet, efformauerint. </s> <s id="s.000796">Hoc etiam <lb></lb>theorema probamus. </s> </p> <p type="main"> <s id="s.000797">Cylindros, quorum baſes axi ſunt perpendiculares, <lb></lb>dum in æquato plano conuoluuntur, rectâ incedere & <lb></lb>per parallelas, quarum diſtantia axis ſeu latoris longitudi<lb></lb>ne præfinitur. </s> </p> <figure id="id.007.01.088.1.jpg" xlink:href="007/01/088/1.jpg"></figure> <p type="main"> <s id="s.000798">Eſto enim Cylin<lb></lb>drus ABCD, cuius a<lb></lb>xis GH, <expan abbr="horizōtis">horizontis</expan> pla<lb></lb>no inſiſtens ſecundum <lb></lb>latus AB, cui latus op<lb></lb>poſitum & aequale CD. <lb></lb></s> <s id="s.000799">Moueatur Cylindrus <lb></lb>rotans, donec latus <lb></lb>CD, in plano ſit vbi EF. </s> <s id="s.000800">Deſcribat autem circuli CB <expan abbr="lineã">lineam</expan> <lb></lb>BF. </s> <s id="s.000801">Circulo verò AD lineam AE. </s> <s id="s.000802">Dico eas rectas eſſe, & <lb></lb>parallelas. </s> <s id="s.000803">Si enim ſuperficies baſium DA, CB, extendan<lb></lb>tur ita vt horizontis planum ſecent, illud ſecabunt iuxta <lb></lb>lineas AE BF, recta ergo eſt vtraque. </s> <s id="s.000804">Sed & parallelas eſſe <lb></lb>ad inuicem ita oſtendimus. </s> <s id="s.000805">quoniam ſemicirculus AD, <lb></lb>æqualis eſt ſemicirculo BC, erit linea AE, æqualis lineæ <lb></lb>BF, ſed & AB, æqualis eſt ipſi DC, quare & ipſi EF. </s> <s id="s.000806">Oppo<lb></lb>ſita igitur quadrilateri figura ABFE latera æqualia ſunt, <lb></lb>quare EF æquediſtat ipſi AB, tum AE ipſi BF, quod fue<lb></lb>rat demonſtrandum. </s> </p> <p type="main"> <s id="s.000807">Probabimus etiam ſi cylindri baſes axi perpendicu<lb></lb>lares non fuerint, & ideo ellipſes in ipſa rotatione perpla<lb></lb>num, parallelas quidem deſcribere, ſed non rectas. </s> </p> <p type="main"> <s id="s.000808">Eſto enim Cylindrus ABCD, cuius baſes ellipſes <expan abbr="inuicē">inuicem</expan> <lb></lb><expan abbr="æquediſtãtes">æquediſtantes</expan>, quarum axes longiores AB, CD, Commu<lb></lb>nis autem ſectio cylindri & plani ad axem & horizontem <lb></lb>planum perpendicularis EHF. </s> <s id="s.000809">Diuidatur autem ſemicir-<pb xlink:href="007/01/089.jpg"></pb>culus EHF in partes æquales quatuor FI, IH, HG, GE. <lb></lb><figure id="id.007.01.089.1.jpg" xlink:href="007/01/089/1.jpg"></figure><lb></lb>Tum per diuiſionum puncta lateri parallelae, rectæ ducan<lb></lb>tur KGL, M<emph type="italics"></emph>H<emph.end type="italics"></emph.end>N, OIP, quæ quidem <expan abbr="cū">cum</expan> baſes AMB, DNC <lb></lb>parallelæ ſint, erunt inuicem æquales, cumqueue circum<lb></lb>ferentia E<emph type="italics"></emph>H<emph.end type="italics"></emph.end>F æquales, eosqueue rectos angulos <expan abbr="cōſtituent">conſtituent</expan>. <lb></lb></s> <s id="s.000810">Ducatur poſt hæc ſeorſum recta QR, & eidem perpendi<lb></lb>cularis ST eam ſecans in V. applicetur autem rectæ ST <lb></lb>æqualis Cylindri lateri BC, ipſa <foreign lang="grc">ηζ. </foreign></s> <s id="s.000811">ita tamen vt punctum <lb></lb>E congruat puncto V, ſitqueue V<foreign lang="grc">η</foreign> æqualis EB, V<foreign lang="grc">ζ</foreign> verò æ<lb></lb>qualis EC. </s> <s id="s.000812">Tum fiant VX, XY, YZ, Z<foreign lang="grc">α</foreign> æquales ipſis EG, <lb></lb>G<emph type="italics"></emph>H<emph.end type="italics"></emph.end>, <emph type="italics"></emph>H<emph.end type="italics"></emph.end>I, IF, & per puncta X, Y, Z, <foreign lang="grc">α</foreign> & paralleli ipſi ST du<lb></lb>cantur <foreign lang="grc">ο α π, ν Ζ ξ, λ γ μ, κ χ θ</foreign>, tum & his ex altera parte re<lb></lb>ſpondentes parallelæ per puncta <foreign lang="grc">β, γ, δ, ε. </foreign></s> <s id="s.000813">Sit autem <foreign lang="grc">ο α</foreign> æ<lb></lb>qualis AF, <foreign lang="grc">α</foreign> <11> æqualis FD, item <foreign lang="grc">ε</foreign> <10>, æqualis EC, <foreign lang="grc">ε σ</foreign> æqualis <lb></lb>EB, ſed & <foreign lang="grc">ν Ζ</foreign> aequalis OI, <foreign lang="grc">Ζ ξ</foreign> ipſi P, <foreign lang="grc">λ</foreign>y ipſi MH, y <foreign lang="grc">μ</foreign> verò ipſi <lb></lb>HN, <expan abbr="tū">tum</expan> <foreign lang="grc">κ χ</foreign> ipſi KG. & <foreign lang="grc">χ θ</foreign>, ipſi GL & ipſis æquales & aequa<lb></lb>liter poſitæ ad partes R, aliæ parallelæ <expan abbr="aptētur">aptentur</expan> per <foreign lang="grc">β, γ, δ, ξ</foreign>, <pb xlink:href="007/01/090.jpg"></pb>quibus ita diſpoſitis per puncta <foreign lang="grc">ο, ν, λ, κ, η</foreign>, item per <foreign lang="grc">π, ξ, μ, θ, ζ</foreign>. <lb></lb></s> <s id="s.000814">ducantur lineæ <foreign lang="grc">οη, πζ</foreign>, curuæ quidem & eodem pacto a<lb></lb>liæ curuæ illis reſpondentes <foreign lang="grc">η <10>, ζς</foreign>, Erunt igitur <foreign lang="grc">ο, η, <10>, <lb></lb>π, ζ, σ</foreign>, parallelæ quidem eo quod lineae quæ inter ipſas du<lb></lb>cuntur, parallelæ ſint & æquales, non tamen rectæ illæ, <lb></lb>ſed curuæ. </s> <s id="s.000815">Moto igitur Cylindro circulus EHF rectam <lb></lb>deſcribet<foreign lang="grc">αε</foreign>, ellipſis verò AMB, curuam <foreign lang="grc">ο η ρ</foreign>, ellipſis au<lb></lb>rem DNC, ipſam curuam <foreign lang="grc">πζς. </foreign></s> <s id="s.000816">In hoc <expan abbr="autē">autem</expan> Cylindri mo<lb></lb>tu illud mirabile, velociores nempe, in ipſa rotatione eſſe <lb></lb>ellipſes ipſo circulo EHF. </s> <s id="s.000817">Ducatur enim recta<foreign lang="grc">ο<10></foreign> quæ oc<lb></lb>currat ipſi VS in S, & <foreign lang="grc">οη</foreign> iungatur, fietqueue triangulum <lb></lb><foreign lang="grc">οη</foreign>S. eſt autem, angulus <foreign lang="grc">ο</foreign> S <foreign lang="grc">η</foreign> rectus, maior erg. <foreign lang="grc">οη</foreign> i<lb></lb>pſa <foreign lang="grc">ο</foreign> S, ſed recta <foreign lang="grc">ο</foreign> S æqualis eſt ipſi<foreign lang="grc">αν</foreign>, hoc eſt, ſemicircu<lb></lb>lo FHE. multo maior eſt autem curua, <foreign lang="grc">ο, ν, λ, κ, η</foreign>, ipſa recta <lb></lb><foreign lang="grc">οη</foreign>, ſed eodem tempore quo ſemicirculus EHF conficit <lb></lb>in rotatione <expan abbr="ſpatiū">ſpatium</expan> <foreign lang="grc">α</foreign> V, eodem dimidia ellipſis BMA me<lb></lb>titur curuam <foreign lang="grc">ονλκη. </foreign></s> <s id="s.000818">velocior igitur eſt ellipſis ipſo cir<lb></lb>culo. </s> </p> <p type="main"> <s id="s.000819">Hæc quoque ſpeculatio ad motum qui ſecundum <lb></lb>abſidem fit, manifeſtè pertinet. </s> <s id="s.000820">Coni, quorum baſes cir<lb></lb>culi ſunt, ſi in plano ſecundum latus rotentur, baſi circu<lb></lb>lum deſcribunt, cuius centrum immobile coni ipſius eſt <lb></lb>vertex, ſemidiameter verò ipſum latus. </s> </p> <figure id="id.007.01.090.1.jpg" xlink:href="007/01/090/1.jpg"></figure> <p type="main"> <s id="s.000821">Eſto conus ABC cu<lb></lb>ius vertex C baſis AB, axis <lb></lb>DC, baſis verò centrum, <lb></lb>D, latus quo planum tan<lb></lb>git BC, ſecatur itaque Co<lb></lb>nus per latus BC & axem <lb></lb>DE à plano horizonti per<lb></lb>pendiculari, cuius & coni <lb></lb>communis ſectio eſt ABC <lb></lb>triangulum, & quoniam coni grauitatis centrum eſt in <pb xlink:href="007/01/091.jpg"></pb>axe ipſo, conus in partes æque <expan abbr="pōderantes">ponderantes</expan> ſecatur AEBC, <lb></lb>AFBC, ſtat ergo conus ſibimet æquilibris. </s> <s id="s.000822">Si autem à po<lb></lb>tentia quadam moueatur, puta ab A verſus F, trahitur ſe<lb></lb>micirculus BEA, à ſemicirculo AFB, & ita fit rotatio. </s> <s id="s.000823">Ita<lb></lb>que ſi imaginemur, in finitos vſque ad verticem parallelos <lb></lb>baſi circulos, eorum ſemicirculi in ipſo motu & trahent & <lb></lb>trahentur; at cum ad verticem circuli deſinant, nec ibi ſe<lb></lb>micirculi ſint qui trahant & trahantur, motus rotationis <lb></lb>prorſus ceſſat & vertex ipſe immobilis fit rotationis cen<lb></lb>trum. </s> <s id="s.000824">Quoniam igitur lateris BC, punctum C ſtat, B verò <lb></lb>circa ipſum mouetur, in ipſo motu circulus deſcribitur <lb></lb>BHIK, cuius ſemidiameter BC, & eodem pacto alij cir<lb></lb>culi in cono, qui baſi HEBF ſunt æquediſtantes, circulos <lb></lb>in plano circa idem centrum deſcribent, vt facile videre <lb></lb>eſt in obiecto ſchemate. </s> <s id="s.000825">Huic ſimilem demonſtrationem <lb></lb>affert Heron in libello Automatum, quem nos Tyrones <lb></lb>adhuc vernacule è Græco translatum, Venetijs prælo <lb></lb>ſubiecimus. </s> </p> <p type="main"> <s id="s.000826">Porrò ſi conus rotundus pro baſi ellipſim habeat, <lb></lb>ſectionem videlicet per planum axi non perpendiculare, <lb></lb>in ipſa rotatione, ſtante vertice, ellipſis baſis, ellipſim de<lb></lb>ſcribit in plano, cuius maior diameter à puncto quod co<lb></lb>ni vertex eſt, ita diuiditur, vt diametri pars maior æqualis <lb></lb>ſit lateri maximo; minor verò æqualis lateri minimo. </s> <s id="s.000827">Sed <lb></lb>hæc ad aliam pertinent ſpeculationem. </s> </p> <p type="main"> <s id="s.000828">His itaque de motu rotundorum, qui circa abſidem <lb></lb>fit, conſideratis, reliquum eſſet de motu trochlearum, qui <lb></lb>circa centrum ſit, opportunè agere, ſed cùm in ſequenti <lb></lb>quæſtione de hoc ſermonem faciat Philoſophus, ad ea <lb></lb>quæ ibi diſputabuntur, lectorem ablegamus. </s> </p> <p type="main"> <s id="s.000829">Modò de tertia motus ſpecie nobis erit ſermo; in <lb></lb>qua quidem ſpecie nonnulla perpendemus, quæ omiſit A<lb></lb>riſtoteles. </s> <s id="s.000830">Agitur autem hîc de rotundorum corporum <pb xlink:href="007/01/092.jpg"></pb>motu, qui fit çirca axem horizonti perpendicularem, axis <lb></lb>altera extremitate in eodem horizontis plano manente, <lb></lb>vti videre eſt in ipſis figulorum rotis. </s> </p> <p type="main"> <s id="s.000831">Hanc motus ſpeciem in extrema quæſtionis parte <lb></lb>cum duabus alijs ſpeciebus comparans ait, eam quæ in <lb></lb>obliquo fit motionem (ita enim hanc, de qua agimus, ap<lb></lb>pellat) ipſam impellere mouentem, hoc eſt, nullum ex ſe <lb></lb>ad motum propenſionem habere, nutumue, & omnia illi <lb></lb>eſſe à motore, ſecundum verò eam motionem, quæ ſupra <lb></lb>diametrum eſt, ſe ipſum mouere circulum. </s> <s id="s.000832">Dixerat enim, <lb></lb>ea referens quæ ſuperiùs circa principium de circulo ver<lb></lb>ba faciens, examinauerat, circulum ex duabus fieri latio<lb></lb>nibus, altera præter, altera verò ſecundum naturam, & <lb></lb>ideo hanc ſemper nutum habere, & ceu continuo motam <lb></lb>ab eo moueri qui mouet. </s> <s id="s.000833">Videtur autem clarè profiteri, <lb></lb>ideo difficiliorem eſſe huius terræ ſpeciei motum, eo <lb></lb>quòd nutu careat proprio & tantum ab alieno, vt ita di<lb></lb>cam, motore, moueatur. </s> </p> <p type="main"> <s id="s.000834">Veruntamen motum hunc facilitate alijs illis duo<lb></lb>bus nequaquam cedere, facilè ex ſequentibus oſtende<lb></lb>mus. </s> </p> <p type="main"> <s id="s.000835">Primo, quia pondus totum rotati corporis, ex graui<lb></lb>tatis centro quod in ipſo axe eſt à plano cui nititur, ſuſti<lb></lb>netur: minima quidem ſui parte axe ipſo tangente <expan abbr="planū">planum</expan> <lb></lb>vnde fit, nullam ferè dum rotatur corpus, circa centrum <lb></lb>vbi nititur, frictionem partium fieri. </s> <s id="s.000836">Præterea grauitatis <lb></lb>centrum ſemper ſtat, nec minimum quidem in ipſa rota<lb></lb>tione attollitur, quod ſanè cum naturæ ſit repugnans, dif<lb></lb>ficultatem facit. </s> <s id="s.000837">Ad hæc circa axem ita libratur rota, vt <lb></lb>quantumuis exigua potentia alteri parti applicetur, alte<lb></lb>ra illico ſuperata moueatur. </s> <s id="s.000838">Licet enim propriè ea <expan abbr="tantū">tantum</expan> <lb></lb>corpora æquilibrare dicantur, quæ ob ponderis hinc in de <pb xlink:href="007/01/093.jpg"></pb>æqualitatem horizonti fiunt æquidiſtantes, nihilominus <lb></lb>& hic aliquam eſſe æquilibrij ſimilitudinem patebit. </s> </p> <figure id="id.007.01.093.1.jpg" xlink:href="007/01/093/1.jpg"></figure> <p type="main"> <s id="s.000839">Eſto enim rota ABCD, <lb></lb>cuius axis horizonti perpendi<lb></lb>cularis FEG tranſiens per cen<lb></lb>trum E, tangens autem planum <lb></lb>in puncto G. </s> <s id="s.000840">Ducatur diame<lb></lb>ter BED, Itaque ſi per diame<lb></lb>trum BED, & axem FEG cor<lb></lb>pus diuidatur, eo quòd <expan abbr="centrū">centrum</expan> <lb></lb>grauitatis in axe inueniatur, <lb></lb>corpus ipſum in duas partes <expan abbr="tū">tum</expan> <lb></lb>mole tum <expan abbr="pōdere">pondere</expan> æquales ſecabitur, nempe BAD, BCD. <lb></lb></s> <s id="s.000841">Nulla igitur adhibita vi extranea ſtabit corpus in <expan abbr="quodã">quodam</expan>, <lb></lb>vt diximus, æquilibrio. </s> <s id="s.000842">At alteri partium potentiâ quauis <lb></lb>licet exigua appoſitâ, puta in C, præualebit pars BCD, & <lb></lb>partem BAD vel impellet vel rapiet, alterâ interim eius <lb></lb>motui obſequente. </s> <s id="s.000843">Potentia igitur quæ in C, nullam rem <lb></lb>quæ impediat inueniens, velociſſimè rotam mouet, quod <lb></lb>eo faciliùs velocius queue fit, quo magis rota eſt in motu, e<lb></lb>ius verò diameter maior & potentia mouens à centro re<lb></lb>motior, & ſanè motus <expan abbr="facilitatē">facilitatem</expan> inde cognoſcimus, quòd <lb></lb>ipſo impulſore ab impulſu ceſſante, diutiſſimè rota im<lb></lb>preſſum motum ſeruet, nec niſi poſt longam rotationem <lb></lb>omnino quieſcat. </s> </p> <p type="main"> <s id="s.000844">Cæterùm quia ſicco, vt aiunt, pede Ariſtoteles quæ <lb></lb>ad hunc motum <expan abbr="pertinēt">pertinent</expan> pertranſijt, nos quædam quæ ad <lb></lb>hanc rem faciunt, diligentiùs expendemus. </s> </p> <p type="main"> <s id="s.000845">Quærimus igitur primò; Cur ea quæ hoc pacto <expan abbr="ro-tãtur">ro<lb></lb>tantur</expan>, in ipſa rotatione locum non mutent, niſi extrinſeca <lb></lb>aliqua id fiat ex cauſſa. </s> </p> <p type="main"> <s id="s.000846">Eſto enim rota aut aliud quippiam rotundum ceu <lb></lb>Turbines ſunt, quibus pueri ludunt, quod circa axem ho<pb xlink:href="007/01/094.jpg"></pb><figure id="id.007.01.094.1.jpg" xlink:href="007/01/094/1.jpg"></figure><lb></lb>rizonti perpendicularem mo<lb></lb>ueatur, ABCD, cuius centrum <lb></lb>E, Diameter AEC. </s> <s id="s.000847">Modò circa <lb></lb>centrum E in finiti imaginentur <lb></lb>circuli, alij alijs minores vſque <lb></lb>ad <expan abbr="centrū">centrum</expan> ipſum, vti ſunt FGH; <lb></lb>ibi enim circuli eſſe deſinunt, <lb></lb>vbi nullum amplius eſt ſpatium. <lb></lb></s> <s id="s.000848">Applicetur itaque potentia in <lb></lb>B, quæ rotam v. geat verſus A. <lb></lb>eodem igitur tempore & inſimul A verſus D, D verſus C, <lb></lb>& C verſus B mouebitur. </s> <s id="s.000849">quantum enim ſemicirculorum <lb></lb>à parte CBA tranſit vltra diametrum AEC, tantundem <lb></lb>ſemicirculorum, qui ſunt ad partem ADC, tranſibit ad <lb></lb>partes CBA. </s> <s id="s.000850">At vbi deſierit motus, ibi deſinit rotatio; vbi <lb></lb>autem deſinit ſpatium, deſinit motus, ſed vbi deſinunt cir<lb></lb>culi, deſinit ſpatium, quare in centro cum non ſint circuli, <lb></lb>nec ſpatium ibi deſinit motus. </s> <s id="s.000851">nulla enim adeſt ratio, cur <lb></lb>ipſum corpus alio à loco in quo eſt, ex rotatione transfe<lb></lb>ratur. </s> <s id="s.000852">Stat ergo rotans, quod fuerat demonſtrandum. </s> <s id="s.000853">Eſt <lb></lb>autem hæc demonſtratio ei ſimilis, quam ſuprà retuli<lb></lb>mus de coni in plano circa verticem rotatione, quam ab <lb></lb>Herone in Automatis excogitatam diximus. </s> </p> <p type="main"> <s id="s.000854">Addimus in hoc rotationis genere corpus in ipſo <lb></lb>motu fieri leuius, idqueue eo magis, quo rotatio velocior. <lb></lb></s> <s id="s.000855">Cauſſa eſt, quod lateralis motus eum motum aliqualiter <lb></lb>impedit, qui ex naturali grauitate fit ad centrum, idcirco <lb></lb>experientiâ docemur, leuiſſimos eſſe turbines, quibus pu<lb></lb>eri ludunt, ſi manus teneantur palmâ, dum citiſſima rota<lb></lb>tione mouentur. </s> </p> <p type="main"> <s id="s.000856">Ad hæc alia proponitur, & ſoluitur quæſtio, Cur ro<lb></lb>tunda corpora huic motionis generi ſint aptiora. </s> </p> <p type="main"> <s id="s.000857">Exploratiſſimum eſt, corporum, quæ ita mouentur, <pb xlink:href="007/01/095.jpg"></pb>partes eo eſſe velociores, quo magis à centro, circa quod <lb></lb>mouentur, fuerint remotiores. </s> <s id="s.000858">maius enim eodem tem<lb></lb>pore ſpatium pertranſeunt. </s> <s id="s.000859">quo igitur figura ijs partibus, <lb></lb>quæ longius à centro abſunt, abundauerit magis, eo faci<lb></lb>lius, & velocius in circulum rotata mouebitur. </s> <s id="s.000860">Modò o<lb></lb>ſtendemus, circularem cæteras omnes ea qua diximus <lb></lb>partium à centro remotiſſimarum copiâ abundare. </s> </p> <figure id="id.007.01.095.1.jpg" xlink:href="007/01/095/1.jpg"></figure> <p type="main"> <s id="s.000861">Eſto triangulum puta æqui<lb></lb>laterum, ABC circa centrum D. <lb></lb></s> <s id="s.000862">Ducantur Catheti per centrum ab <lb></lb>oppoſitis angulis ad oppoſita late<lb></lb>ra ADG, BDF, CDE, erunt autem <lb></lb>lateribus perpendiculares. </s> <s id="s.000863"><expan abbr="quoniã">quoniam</expan> <lb></lb>igitur latera AD, DB, DC, rectis <lb></lb>angulis ſubtenduntur, maiora <expan abbr="erūt">erunt</expan> <lb></lb>lateribus DE, DF, DG. tres igitur <lb></lb>lineæ in hoc triangulo ſunt longiſſimæ DA; DB, DC. tres <lb></lb>verò breuiſſimæ DE, DG, DF, quamobrem rotato ſuper <lb></lb>centrum D triangulo, tres tantum partes eius ABC velo<lb></lb>ciſſimæ erunt, tres verò tardiſſimæ E, G, F. </s> <s id="s.000864">Minus igitur a<lb></lb>pta eſt motui huic triangularis figura, quam quadrata, in <lb></lb>qua partes à centro remotiſſimè, & ideo velociſſimè ſunt <lb></lb>quatuor. </s> <s id="s.000865"><expan abbr="Itaq;">Itaque</expan> quo magis laterata figura angulis abunda<lb></lb>bit, eo magis erit ad hunc, & cæteros omnes circulares <lb></lb>motus aptior. </s> <s id="s.000866">At circulus infinitas, vt ita dicam, partes à <lb></lb>centro remotiſſimas habet, itaque nulla figura eſt circu<lb></lb>lari, in ipſa rotatione, commodior atque velocior. </s> <s id="s.000867">Alia <lb></lb>quoque de cauſſa id fit, quod dum circularis figura mo<lb></lb>uetur, nullis eminentibus angulis aërem verberet <expan abbr="circū-ſtãtem">circum<lb></lb>ſtantem</expan>, ex qua verberatione motus impeditus ſit tardior. <lb></lb></s> <s id="s.000868">Quæri etiam poteſt, Num axe in clinato, rotæ motus ali<lb></lb>qualiter impediatur? </s> <s id="s.000869">Nos negatiuam partem amplecti<lb></lb>mur. </s> </p> <pb xlink:href="007/01/096.jpg"></pb> <figure id="id.007.01.096.1.jpg" xlink:href="007/01/096/1.jpg"></figure> <p type="main"> <s id="s.000870">Eſto enim tota ABCD, cuius cen<lb></lb>trum E axis inclinatus, circa quem <lb></lb>conuertitur EGF. </s> <s id="s.000871">Duobus aute pun<lb></lb>ctis fulcitur GF. </s> <s id="s.000872">Sit autem tum gra<lb></lb>uius tum figuræ centrum E, Perpen<lb></lb>dicularis vero per inferius fulcimen<lb></lb>tum tranſiens HFI. </s> <s id="s.000873">Conuerſa igitur <lb></lb>rota, grauitatis centrum ſtabit nec à <lb></lb>ſuo ſitu ſurſum deorſumue mouebi<lb></lb>tur. </s> <s id="s.000874">Eſt autem axis FEG, ceu vectis in <lb></lb>quo pondus in E, potentiæ ſuſtinentes GF; non enim hic <lb></lb>vt in axe perpendiculari pondus totum ab inferiori fulci<lb></lb>mento ſuſtinetur. </s> <s id="s.000875">quo igitur minor erit proportio FE ad <lb></lb>FG, eo minori indigebit potentiâ is qui pondus ſuſtinet in <lb></lb>G. </s> <s id="s.000876">Et hæc ſanè ita ſe habent, grauitatis çentro in axe ipſo <lb></lb>conſtituto, ſi enim extra fuerit motus impeditur & moto<lb></lb>re ceſſante citò quieſcit. </s> <s id="s.000877">Eſto enim grauitatis centrum in <lb></lb>K. </s> <s id="s.000878">Dum igitur circa axem fit motus, centrum circulatum <lb></lb>aliquando erit in L; Secet autem rotæ diameter AC per<lb></lb>pendicularem Hl in M. </s> <s id="s.000879">Porrò à punctis LK ad ipſam <expan abbr="per-pēdicularem">per<lb></lb>pendicularem</expan> ducantur ad rectos angulos lineæ LN, KO. <lb></lb></s> <s id="s.000880">Maior eſt autem MK ipſa ML, maior ergo MO, ipſa MN. <lb></lb>magis igitur à mundi centro diſtat punctum N puncto O. <lb></lb></s> <s id="s.000881">Centrum ergo grauitatis K ſi liberè dimittatur, requieſcet <lb></lb>in K & contranaturam transferetur in L. </s> <s id="s.000882">Ceſſante igitur <lb></lb>violentiâ & præualente naturâ citò rota ſuâ ſponte quie<lb></lb>ſcet, quod fuerat oſtendendum. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000883">QVÆSTIO IX.</s> </p> <p type="head"> <s id="s.000884"><emph type="italics"></emph>Quæritur, Cur ea quæ per maiores circulos tolluntur, & trahuntur <lb></lb>faciliùs, & celeriùs moueri contingat, veluti maioribus tro<lb></lb>chleis, & ſcytalis ſimiliter?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000885">Reſpondet ad hæc Philoſophus, forte id euenire, quo-<pb xlink:href="007/01/097.jpg"></pb>niam quanto maior fuerit illa quæ à centro eſt, in æquali <lb></lb>tempore maius mouetur ſpatium. </s> <s id="s.000886">quamobrem æquali <lb></lb>exiſtente onere idem faciet. </s> <s id="s.000887">Ita enim dixerat de <expan abbr="librarū">librarum</expan> <lb></lb>natura, & differentijs agens, maiores minoribus exactio<lb></lb>res eſſe. </s> <s id="s.000888">Circulos verò libras, in quibus centrum ſpartum, <lb></lb>ſemidiametri hinc in de æqualia brachia. </s> </p> <p type="main"> <s id="s.000889">Quod vltimo loco affirmauit, trochleas eſſe inſtar <lb></lb>librarum, verum eſt. </s> <s id="s.000890">Quod autem dixit, faciliùs & cele<lb></lb>rius mouere maiores libras ijs quæ minores ſunt, ſi ſimpli<lb></lb>citer intelligatur, falſum, quippe quod facilitas motus, in <lb></lb>tractorijs machinis velocitati ſit contraria, quod demon<lb></lb>ſtrauit Guid. Vbald. in tractatu de Trochlea in 2. Corol<lb></lb>lario propoſitione vltima. </s> </p> <p type="main"> <s id="s.000891">Ad id autem quod dixit, quo <expan abbr="maiorēs">maiores</expan> fuerint tro<lb></lb>chleæ, eo faciliùs mouere, non eſt, vt dicebamus, ſimpli<lb></lb>citer verum, quod facilè oſtendemus. </s> </p> <figure id="id.007.01.097.1.jpg" xlink:href="007/01/097/1.jpg"></figure> <p type="main"> <s id="s.000892">Eſto enim trochlea AB circa centrum C, appenſa in <lb></lb>puncto D, perpendicularis quæ ad mundi centrum DCE, <lb></lb>pondera æqualia vtrinque appenſa FG. </s> <s id="s.000893">Eſto item alia <lb></lb>Trochlea, eaque; maior HI, circa centrum K appenſa in L, <lb></lb>perpendicularis, quæ ad mundi centrum LKM, æqualia <pb xlink:href="007/01/098.jpg"></pb>pondera vtrinque appenſa N, O. </s> <s id="s.000894">Dico maiorem Hl ipſa <lb></lb>minori DE facilius pondera non mouere, eo quòd ſit ma<lb></lb>ior, illa verò difficiliùs, propterea quòd ſit minor. </s> <s id="s.000895">Et enim, <lb></lb>quoniam vtraque trochlea per centrum grauitatis à per<lb></lb>pendiculari diuiditur, erunt partes DAE, DBE, æque <expan abbr="pō-derantes">pon<lb></lb>derantes</expan>. </s> <s id="s.000896">Eadem ratione ipſæ quoque LHM, LIM æquè <lb></lb>ponderabunt. </s> <s id="s.000897">Itaque ſi quantumuis puſilla pondera ad<lb></lb>das, <expan abbr="vtriq;">vtrique</expan> earum ad alteram partem tolletur <expan abbr="æquilibriū">æquilibrium</expan>, <lb></lb>nec minus requiritur pondus vt recedat ab æquilibrio <lb></lb>Trochlea minor, quàm maior. </s> <s id="s.000898">Vnico autem verbo con<lb></lb>cludi poteſt diſputatio, <expan abbr="tã">tam</expan> in minori quàm in maiori, bra<lb></lb>chia ſiqui dem bifariam diuiduntur, ergo in <expan abbr="vtriſq;">vtriſque</expan> eadem <lb></lb>brachiorum proportio, & eadem ponderum ratio. </s> </p> <p type="main"> <s id="s.000899">Exploratiſſima ſunt hæc. </s> <s id="s.000900">Veruntamen cùm res ipſa <lb></lb>doceat, verum eſſe quod ſcribit Ariſtoteles, huius effe<lb></lb>ctus cauſſa aliunde à nobis, nempe à mechanicis princi<lb></lb>pijs, eſt mutuanda. </s> <s id="s.000901">Dico igitur, Axium, circa quos tro<lb></lb>chleæ rotæue conuertuntur ad rotas ipſas, varias habere <lb></lb>proportiones. </s> <s id="s.000902">Oſtendemus autem <expan abbr="rotã">rotam</expan> illam, trochleam<lb></lb>ue faciliùs moueri, & mouere pondera, quo rotæ diame<lb></lb>ter ad axis diametrum maiorem habuerit proportionem, <lb></lb>& ideo fieri poſſe rotam maiorem ad ſuum axem <expan abbr="minorē">minorem</expan> <lb></lb>habere proportionem quam rotam minorem ad ſuum. </s> </p> <figure id="id.007.01.098.1.jpg" xlink:href="007/01/098/1.jpg"></figure> <p type="main"> <s id="s.000903">Eſto enim <lb></lb>trochlea AB cir<lb></lb>ca centrum C, <lb></lb>cuius diameter <lb></lb>DCE ſit in ipſa <lb></lb>quæ ad mundi <lb></lb>centrum <expan abbr="perpē-diculari">perpen<lb></lb>diculari</expan>: ſit au<lb></lb>tem appenſa in D. </s> <s id="s.000904">Alia ſimiliter ei æqualis ſit trochlea F <lb></lb>G circa centrum H, cuius diameter IHK, conueniens <pb xlink:href="007/01/099.jpg"></pb>cum perpendiculari quæ ad mundi centrum. </s> <s id="s.000905">appendatur <lb></lb>autem in I. </s> <s id="s.000906">Habeant autem & axes, circa quos conuertan<lb></lb>tur. </s> <s id="s.000907">Hi ſi æquales fuerint, proportione non mutatâ idem <lb></lb>operabuntur. </s> <s id="s.000908">Modò ponantur inæquales, ſitqueue axis ro<lb></lb>tæ AB, craſſior axe rotæ FG, ſitqueue craſſioris quidem ſemi<lb></lb>diameter CL, ſubtilioris autem HM. </s> <s id="s.000909">Dico per trochleam <lb></lb>FG facilius attolli pondera æqualia quàm per AB, licet <lb></lb>altera trochlearum alteri ſit æqualis. </s> <s id="s.000910">Quoniam enim me<lb></lb>chanica corpora ſine materia & pondere non ſunt, onera <lb></lb><expan abbr="appēſa">appenſa</expan> & trochlearum ipſarum grauitas ex ſuperiori par<lb></lb>te prement axes, vbi puncta L, M, quæres, ſecutâ in uicem <lb></lb>corporum ſolidorum fricatione, motum ipſum trochlea<lb></lb>rum difficiliorem & aſperiorem facit. </s> <s id="s.000911">Succedit igitur im<lb></lb>pedimentum loco ponderis. </s> <s id="s.000912">Duos igitur habemus vectes <lb></lb>DC, IH, quorum fulcimenta contra ipſa C, H. </s> <s id="s.000913">Pondera <lb></lb>verò inter fulcimenta & potentias in L, M. </s> <s id="s.000914">Intelligantur <lb></lb>autem potentiæ applicatæ punctis DI. </s> <s id="s.000915">Igitur ex natura e<lb></lb>iuſmodi vectis, in quo pondus inter fulcimentum eſt & <lb></lb>potentiam erit vt CL, ad CD, ita potentia in D ad <expan abbr="pōdus">pondus</expan>, <lb></lb>hoc eſt, reſiſtentiam fricationis, quæ fit in L. </s> <s id="s.000916">Sed maior <lb></lb>eſt proportio CL ad CD quàm HM ad HI. </s> <s id="s.000917">Maior igitur <lb></lb>ad ſuperandum idem ſeu æquale impedimentum poten<lb></lb>tia requiritur in D, quam in I. </s> <s id="s.000918">Itaque cum vis tota in rota<lb></lb>rum & axium, diametrorum proportione conſiſtat, fieri <lb></lb>poteſt, quod dicebamus, minorem trochleam dari, quæ <lb></lb>maiorem habeat proportionem ad ſuum axem, quàm, <lb></lb>maior ad ſuum, quo caſu minor rota facilius impedimen<lb></lb>tum, quod diximus, ipſa maiori rota ſeu trochlea ſupera<lb></lb>bit. </s> <s id="s.000919">Veruntamen quoniam ex materia fiunt tum axes tum <lb></lb>rotæ, nec rei natura patitur axes ſubtiles, & imbecilles <lb></lb>magna <expan abbr="pōdera">pondera</expan> ſuſtinere poſſe, idcirco craſſiores fiunt, quę <lb></lb>craſſitudo cum proportione magis à magnarum rotarum <lb></lb>diametris ſuperetur; fit hinc maiores rotas datâ axium pa-<pb xlink:href="007/01/100.jpg"></pb>ritate facilius impedimentum ſuperare quàm minores, & <lb></lb>hoc videtur ſenſiſſe Philoſophus in ipſa quæſtionis huius <lb></lb>propoſitione, Hinc aurigæ vulgo axungiâ (quæ inde no<lb></lb>men trahit) axium aſperitates mitigant, vt minor in rotan<lb></lb>do, ex fricatione fiat reſiſtentia. </s> <s id="s.000920">Concludimus igitur, fa<lb></lb>cillimè trochleam illam pondus trahere, quæ cum maxi<lb></lb>ma ſit, axem habet minimum, cumqueue axungiâ aliaue vn<lb></lb>ctuoſa materia perfuſum. </s> <s id="s.000921">De manubrijs, quæ rotarum a<lb></lb>xibus aptantur, nemo ferè verba fecit; nos igitur de his a<lb></lb>liquid; ſiquidem res ad ſpeculationem, qua de agimus, <expan abbr="nē-pe">nem<lb></lb>pe</expan> Mechanicam pertinet. </s> </p> <p type="main"> <s id="s.000922">Manubria vectes ſunt, & ad vectium naturam redu<lb></lb>cuntur, eorum ſcilicet, in quibus fulcimentum eſt inter <lb></lb>pondus & potentiam. </s> <s id="s.000923">In his autem attenditur proportio, <lb></lb>quam habet manubrij longitudo ad ipſum axis ſemidia<lb></lb>metrum, eo enim faciliùs mouent, quo eorum longitudo <lb></lb>ad axium ſemidiametros proportionem, habuerit ma<lb></lb>iorem. </s> <s id="s.000924">Duabus autem partibus conſtant, alterâ, quæ ab <lb></lb>axe ad angulum; quæ verè vectis eſt; alterâ, cui manus i<lb></lb>pſa admouetur, ex qua res tota manubrium dicitur. </s> <s id="s.000925">Fiunt <lb></lb>autem manubria hæc vt plurimum amouibilia, ſunt <expan abbr="tamē">tamen</expan> <lb></lb>ceu rotarum ipſarum partes, & rotis ipſis commodè affi<lb></lb>gerentur, niſi in rotatione à tranſuerſarijs, quibus rotæ ſu<lb></lb>ſtinentur, impedimentum fieret. </s> </p> <figure id="id.007.01.100.1.jpg" xlink:href="007/01/100/1.jpg"></figure> <p type="main"> <s id="s.000926">Eſto enim rota AB, cu<lb></lb>ius axis E, terebretur autem <lb></lb>in F, ibiqueue paxillus affigatur <lb></lb>FK. </s> <s id="s.000927">Sit & alia rota CD, cu<lb></lb>ius axis G, manubrium axi <lb></lb>appoſitum GHI. </s> <s id="s.000928">Sint autem <lb></lb>rotæ æquales & axes æqua<lb></lb>les. </s> <s id="s.000929">Sint etiam æqualia ipſa <lb></lb>ſpatia EF, GH, hoc eſt, ma-<pb xlink:href="007/01/101.jpg"></pb>nubrij GHI longitudo. </s> <s id="s.000930">Dico, eâdem facilitate moueri AB <lb></lb>rotam à potentia in FK, quâ mouetur CB, à potentia po<lb></lb>ſita in HI, datis ipſi nempe potentijs æqualibus. </s> <s id="s.000931">Produca<lb></lb>tur enim IH, vſque ad rotæ CD latus in L, & LG ducatur, <lb></lb>& FE in rota AB iungatur. </s> <s id="s.000932">Erunt igitur FE LG inter ſe æ<lb></lb>quales. </s> <s id="s.000933">Sunt autem eorum circulorum ſemidiametri, qui <lb></lb>à punctis FL, in ipſa rotatione deſcribuntur. </s> <s id="s.000934">Ita igitur ſe <lb></lb>habebit potentia applicata in L ad diametrum ſemidia<lb></lb>metrumue axis rotæ CD, vt ſe habet potentia applicata <lb></lb>in F, ad diametrum ſemidiametrumue axis E rotæ AB, ſed <lb></lb>ſpatia ſunt æqualia & potentiæ æquales, quare nihil re<lb></lb>fert, vtrum manubrium lateri affigatur, vel axi à latere ro<lb></lb>tæ ſeparatum applicetur. </s> </p> <figure id="id.007.01.101.1.jpg" xlink:href="007/01/101/1.jpg"></figure> <p type="main"> <s id="s.000935">Duplex autem eſt ma<lb></lb>nubriorum forma; altera e<lb></lb>nim rectis partibus conſtat, <lb></lb>altera verò curua eſt tota, <lb></lb>ſed rectis vtimur vt mani<lb></lb>bus apprendamus, curuis <lb></lb>verò vt locum illis appona<lb></lb>mus, & pedis preſſione ceu <lb></lb>in molis lapideis, quibus <lb></lb>gladij acuuntur, fieri aſſolet, conuertantur. </s> <s id="s.000936">Cur autem <lb></lb>manubria hæc curua fiant, ea videtur ratio, ne videlicet <lb></lb>manubrij capite ſupra centrum in linea quæ per centrum <lb></lb>tranſit, <expan abbr="cōſtituto">conſtituto</expan>, factâ interim preſſione motus à centro, <lb></lb>ad quod directè fieret preſſio, impediretur. </s> <s id="s.000937">Curuitas <expan abbr="autē">autem</expan> <lb></lb>facilitatem quandam habet, ex qua factâ modicâ flexione <lb></lb>axis caput, dum premitur ab ipſa perpendiculari linea le<lb></lb>niter abducitur. </s> <s id="s.000938">quæ cum ceſſent in manubrijs quæ manu <lb></lb>aguntur, ideo alia forma, nempe ex rectis partibus paſſim <lb></lb>fiunt. </s> <s id="s.000939">Eſto igitur illud quod ex rectis partibus AB, curuum <lb></lb>verò CD, linea verò, ſecundum quam pede fit preſſio <pb xlink:href="007/01/102.jpg"></pb>CDE. </s> <s id="s.000940">Hæc itaque de manubrijs ſeu vectibus nos conſi<lb></lb>deraſſe ſit ſatis. </s> </p> <p type="main"> <s id="s.000941">Quæri interim poſſet, Cur duabus datis rotis æqua<lb></lb>lis magnitudinis in æqualis ponderis, circa æquales axes <lb></lb>conſtitutis leuior faciliùs moueatur & citiùs quieſcat; <lb></lb>grauior verò difficilius moueatur & tardiùs ceſſet à mo<lb></lb>tu, ea videtur ratio, quod grauior reſiſtens magis cum ſu<lb></lb>peratur impreſſam vim ſuſcipit, & diutiùs retinet, quod <lb></lb>ceſſat in leuiore. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000942">QVÆSTIO X.</s> </p> <p type="head"> <s id="s.000943"><emph type="italics"></emph>Dubitat Ariſtoteles, Cur faciliùs, quando ſine pondere est, mouea<lb></lb>tur libra, quàm cum pondus habet. </s> <s id="s.000944">Simili modo rota, & eiuſmodi <lb></lb>quidpiam, quod grauius quidem est, item quod maius & <lb></lb>grauius minori, & leuiori?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000945">Breuiter autem ſoluit, ait enim, An quia non ſolum in <lb></lb>contrarium quod graue eſt, ſed in obliquam etiam dif<lb></lb>ficulter mouetur? </s> <s id="s.000946">In contrarium enim ei ad quod vergit <lb></lb>onus mouere difficile eſt, quo autem vergit, eſt facile. </s> <s id="s.000947">In <lb></lb>obliquum autem haudquaquam vergit. </s> <s id="s.000948">Nos quod ipſe <lb></lb>non fecit figurâ ipſa appoſitâ rem clariorem faciemus. </s> </p> <figure id="id.007.01.102.1.jpg" xlink:href="007/01/102/1.jpg"></figure> <p type="main"> <s id="s.000949">Eſto libra AB, cuius ful<lb></lb>cimentum C, pondera vtrin<lb></lb>que appenſa AB, quorum v<lb></lb>trumque ponderet 10. Item <lb></lb>libra DE, cuius fulcimentum <lb></lb>F pondere vero appenſa D, E, <lb></lb>ipſis A, B, dimidio leuiora, <expan abbr="nē-pe">nem<lb></lb>pe</expan> S. </s> <s id="s.000950">Addatur ponderi B pon<lb></lb>dus G, & ponderi E pondus <lb></lb>H, quorum ſimiliter <expan abbr="vtrumq;">vtrumque</expan> <lb></lb>ponderet S, nutabunt igitur <lb></lb>libræ ponderibus appoſitis, & <pb xlink:href="007/01/103.jpg"></pb>BG ſecetur in K, EH verò in N, grauius eſt autem GB, eſt <lb></lb>enim IS, ipſo EH, quod eſt 10. Difficiliùs autem deſcen<lb></lb>det BG, quàm EH. hoc autem ex doctrina Ariſtotelis, <lb></lb>quia non ſolum in contrarium quod graue eſt, ſed in obli<lb></lb>quum etiam difficulter mouetur, in contrarium enim ei <lb></lb>ad quod vergit onus mouere difficile eſt, quò autem ver<lb></lb>git facilè in obliquum autem puta per lineas BK, EN non <lb></lb>vergit onus. </s> <s id="s.000951">Difficiliùs ergo in obliquum mouebitur pon<lb></lb>dus BG ipſo pondere EH. vtrumque autem in deſcenſu <lb></lb>retrahitur nempe à perpendicularibus BI, EM & retra<lb></lb>ctionis quidem anguli ſunt æquales & æquales ipſæ retra<lb></lb>ctiones. </s> <s id="s.000952">Sed grauius eſt pondus GB. quod autem grauius <lb></lb>eſt, violentius <expan abbr="deſcēdit">deſcendit</expan> eo quod eſt leuius. </s> <s id="s.000953">maiori igitur ni<lb></lb>ſu atque impetu cum cætera paria ſint, deſcendet pondus <lb></lb>BG, ipſo EH, quod è diametro Ariſtotelis aſſertioni eſt <lb></lb>contrarium. </s> <s id="s.000954">ex alijs igitur principijs veritas ipſa eſt eruen<lb></lb>da. </s> <s id="s.000955">Dicimus autem id ex proportionum fieri inæqualita<lb></lb>te; quia enim is ad 10. proportionem habet ſeſquialteram, <lb></lb>10. verò ad 5. duplam, maiorem proportionem habet EH <lb></lb>ad oppoſitum pondus D, quàm BG ad pondus A, facilius <lb></lb>ergo trahet libra DE leuior pondus D, quàm ipſa AB, gra<lb></lb>uior pondus A, quod vtique fuerat oſtendendum. </s> <s id="s.000956">Alia <lb></lb>quoque cauſſa & hæc accidentalis ad hunc effectum pa<lb></lb>riendum concurrit, axium nempe ad fulcimenta, in qui<lb></lb>bus rotantur, fricatio. </s> <s id="s.000957">quo enim maius eſt pondus cæteris <lb></lb>paribus, quod nos in præ cedente quæſtione demonſtra<lb></lb>uimus, eò maiìor fit ipſa colliſio. </s> </p> <p type="main"> <s id="s.000958">Porrò huius <expan abbr="quoq;">quoque</expan> ſpeculationis eſt, Cur æqualia & <lb></lb>ſimilia corpora in æqualibus ſimilibuſqueue baſibus conſti<lb></lb>tuta eodem ſimiliqueue plano fulta, ponderibus tamen in<lb></lb>æqualia, non eâdem facilitate euertantur, ſed horum gra<lb></lb>uiora difficilius. </s> </p> <pb xlink:href="007/01/104.jpg"></pb> <figure id="id.007.01.104.1.jpg" xlink:href="007/01/104/1.jpg"></figure> <p type="main"> <s id="s.000959">Sit enim Priſma ſeu <lb></lb>Cylindrus ABCD, cuius <lb></lb>grauitatis centrum E in <lb></lb>plano Cl, baſi fultus CD. <lb></lb></s> <s id="s.000960">Sit & alter Cylindrus <lb></lb>FGHI, cuius grauitatis <lb></lb>centrum K fultus baſi HI <lb></lb>æqualis quidem & ſimilis <lb></lb>ipſi AD. </s> <s id="s.000961">Sit autem grauior FGHI, ipſo ABCD. Dico, pari <lb></lb>potentiâ vtrumque impellente, facilius euerſum iri Cy<lb></lb>lindrum AD, ipſo Fl. </s> <s id="s.000962">Ducantur EC, KH, & æquales po<lb></lb>tentiæ applicentur punctis BG, pellentes Cylindros ad <lb></lb>partes AF. </s> <s id="s.000963">Euerſio autem non fiet donec facta corporis <lb></lb>conuerſione circa puncta CH, grauitatis centra E, K <expan abbr="trãs-ferunturin">trans<lb></lb>feruntur in</expan> L, M, in ipſis ſcilicet <expan abbr="perpēdicularibus">perpendicularibus</expan> ACFH. <lb></lb></s> <s id="s.000964">Demittantur EN, KO, perpendiculares ipſis CD, HF. </s> <s id="s.000965">Et <lb></lb>quoniam CNE, HOK anguli recti ſunt, erunt EC KH i<lb></lb>pſis EN, KO, maiores, quare & LC, MH ipſis EN KO, ma<lb></lb>iores attolluntur ergo in ipſa euerſione, grauitatum cen<lb></lb>tra E in L, K in M. </s> <s id="s.000966">At quod grauius eſt, difficilius contra <lb></lb>ſui naturam mouetur, ideo difficilius euertetur corpus <lb></lb>FI, ipſo AD, quod fuerat demonſtrandum. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.000967">QVÆSTIO XI.</s> </p> <p type="head"> <s id="s.000968"><emph type="italics"></emph>Dubitat Philoſophus, Cur ſuper ſcytalas facilius portentur onera <lb></lb>quàm ſuper currus, cum tamen ij magnas habeant rotas, <lb></lb>illæ verò puſillas?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000969">Optimè reſpondet dubitationi. </s> <s id="s.000970">An, inquiens, quoniam <lb></lb>in ſcytalis nulla eſt offenſatio; in curribus verò axis <lb></lb>eſt, ad quem offenſant. </s> <s id="s.000971">Deſuper enim illum premunt, & <lb></lb>à lateribus. </s> <s id="s.000972">quod autem eſt in ſcytalis ad iſthæc duo mo<lb></lb>uetur & inferiori ſubſtrato ſpatio, & onere ſuperimpoſi-<pb xlink:href="007/01/105.jpg"></pb>to, in vtriſque enim ijs reuoluitur locis circulus, & motus <lb></lb>impellitur. </s> <s id="s.000973">Tam appoſitè paucis verbis veritatem expli<lb></lb>cauit, vt ferè quicquid inſuper ad datur, ſuperuacaneum <lb></lb>videri poſſit. </s> <s id="s.000974">quicquid tamen ſit, ad maiorem claritatem <lb></lb>aliquantulum in hac ipſa quæſtione immorabimur. </s> </p> <p type="main"> <s id="s.000975">Rotatas ſcytalas proponit hîc Ariſtoteles. </s> <s id="s.000976">Coniun<lb></lb>ctas autem eſſe rotas ipſis ſcytalis eſt intelligendum, nem<lb></lb>pe, vt ſimul rotæ cum ſcytalis conuertantur. </s> <s id="s.000977">Secus enim <lb></lb>axium & Rotarum fieret offenſatio, cuius offenſationis <lb></lb>vim & effectum cum nouerit Ariſtoteles, vel hoc ipſo lo<lb></lb>co teſte, mirum eſt, nihil de ea egiſſe quæſtione 9. vbi nos <lb></lb>hac de re fuſiſſimè tractauimus. </s> </p> <p type="main"> <s id="s.000978">Cæterùm quod de rotatis ſcytalis ſcribit Philoſo<lb></lb>phus, notandum, à Pappo quidem lib. 8. & à noſtris Me<lb></lb>chanicis paſſim abſque rotis Cylindrica ſimplici videli<lb></lb>cet, & tereti formâ ad vſum adhiberi. </s> <s id="s.000979">Eſto igitur Ari<lb></lb><figure id="id.007.01.105.1.jpg" xlink:href="007/01/105/1.jpg"></figure><lb></lb>ſtotelis quidem ſcytala <lb></lb>AB, Pappi verò ſeu vul<lb></lb>garis, & communis CD. <lb></lb></s> <s id="s.000980">His non modò lapicidæ <lb></lb>paſſim, ſed & nautæ na<lb></lb>uiumqueue fabri ſubdu<lb></lb>cendis & mari inducen<lb></lb>dis nauibus vtuntur, quod varare dicunt vernaculè, Hi<lb></lb>ſpanico, vt arbitror, vocabulo. </s> <s id="s.000981">ca enim natio teres lignum <lb></lb>baculumue appellat Varam. </s> </p> <p type="main"> <s id="s.000982">Quæri autem poſſet, vtra harum formatum ſit vti<lb></lb>lior atque commodior? </s> <s id="s.000983">Nos rotatas laudamus magis in <lb></lb>plano duroqueue ſolo, minus enim tangunt & minus offen<lb></lb>ſant; in molliori autem & minus duro proponimus non <lb></lb>rotatas, ſiquidem rotæ ſui naturâ pondere preſſæ ſolum, <lb></lb>facillimè ſcindunt & abſorbentur. </s> </p> <p type="main"> <s id="s.000984">Quatenus autem ad vſum pertinet. </s> <s id="s.000985">Eſto horizontis <pb xlink:href="007/01/106.jpg"></pb><figure id="id.007.01.106.1.jpg" xlink:href="007/01/106/1.jpg"></figure><lb></lb>planum AB, ſcytalae duae <lb></lb>CD, EF, Pondus verò <lb></lb>eis impoſitum G, tan<lb></lb>gens ipſas in <expan abbr="pūctis">punctis</expan> CE, <lb></lb>ſcytalæ autem planum <lb></lb>in punctis D, F, Pellatur <lb></lb>à potentia quapiam <expan abbr="pō-dus">pon<lb></lb>dus</expan> G ad anteriora, <expan abbr="nē-pe">nen<lb></lb>pe</expan> ad partes E. rotabuntur igitur ſcytalæ & pars quædam <lb></lb>ſcytalæ D, in qua ſit contactus aſcendet in I, C verò de<lb></lb>ſcendet in H, nulla re motum impediente, quippe quòd <lb></lb>nulla ponderis ſcytalarum, & plani ad inuicem fiat offen<lb></lb>ſatio. </s> <s id="s.000986">Præterea cum ſcytalarum centra ab horizontis pla<lb></lb>no æqualiter diſtent, pondus quidem horizonti æquidi<lb></lb>ſtanter mouetur, & ideo eius centrum grauitatis nequa<lb></lb>quam, in motu qui ſit, eleuatur. </s> </p> <p type="main"> <s id="s.000987">Cæterùm materiæ imperfectione remota nihil re<lb></lb>fert ad facilitatem, vtrum maioris minorisue diametri <lb></lb>ſint ſcytalæ, vt ea poſita eo quod maiores circuli faciliùs <lb></lb>offendicula ſuperent, quod demonſtratum eſt in quæſtio<lb></lb>ne 8. eo vtiliores ſunt ſcytalæ, quo craſſiores. </s> <s id="s.000988">Quatenus <lb></lb>autem ad plauſtri naturam ſpectat, cuius ad ſcytalas Phi<lb></lb>loſophus fecit comparationem, vt oſten damus difficilius <lb></lb>ex eo moueri pondera. </s> </p> <figure id="id.007.01.106.2.jpg" xlink:href="007/01/106/2.jpg"></figure> <p type="main"> <s id="s.000989">Eſto plauſtri rota <lb></lb>KL, cuius centrum M, a<lb></lb>xis verò NO circa quem <lb></lb>rota ipſa conuertitur KL. <lb></lb></s> <s id="s.000990">Funis quo rota ex axis <lb></lb>centro M trahitur MP, <lb></lb>pondus vero QR. </s> <s id="s.000991">Quo<lb></lb>niam igitur pondus axem <lb></lb>premit in N, axis autem rotæ modiolum in O, & eodem, <pb xlink:href="007/01/107.jpg"></pb>tempore potentia quæ trahit in P, axem admouet modio<lb></lb>lo in parte V. duplex itaque fit ex fricatione ſeu offenſa<lb></lb>tione impedimentum, infra nempe, vbi O, & ad latus vbi <lb></lb>V. quæ quidem offenſiones currus motum reddunt diffi<lb></lb>ciliorem, quæ quidem difficultas eo maior erit, quo ma<lb></lb>ior fuerit pondus axem premens, & minor proportio ſe<lb></lb>midiametri rotæ KM, ad axis ſemidiametrum MO. </s> <s id="s.000992">Cur <lb></lb>igitur ſcytalis facilius pondera transferantur quam plau<lb></lb>ſtris, apertè ex dictis ad Ariſto telis mentem demonſtra<lb></lb>uimus. </s> </p> <p type="main"> <s id="s.000993">Cæterùm quod ipſe reticuit, nos dicemus, nempe <lb></lb>validiſſimè enormia pondera per ſcytalas moueri, ſi ſcy<lb></lb>talis ipſis vectes adiungantur. </s> <s id="s.000994">Et ſanè motus erit tardiſſi<lb></lb>mus, veruntamen tarditas ipſa facilitate, quæ in de fit, v<lb></lb>berrimè compenſatur. </s> </p> <figure id="id.007.01.107.1.jpg" xlink:href="007/01/107/1.jpg"></figure> <p type="main"> <s id="s.000995">Eſto igitur horizontis planum AB, ſcytalæ CD, fo<lb></lb>ramina in ſcytalis EFGH, vectes foraminibus inſerti IE, <lb></lb>KF, LG, MH. </s> <s id="s.000996">Pondus vero ſcytalis impoſitum N. </s> <s id="s.000997">Appli<lb></lb>catis igìtur quatuor potentijs extremitatibus vectium I, <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>, L, M, ijſque in anteriora propulſis, fiet ſcytalarum rota-<pb xlink:href="007/01/108.jpg"></pb>tio, & ponderis N translatio ad anteriores partes B. </s> <s id="s.000998">Eſto <lb></lb>item ſeorſum ſcytala PR, cuius centrum Q, vectis eidem <lb></lb>per centrum inſertus O, P, Q, R. facto igitur vectis motu <lb></lb>O P Q R fiet ex O; centro <expan abbr="autē">autem</expan> Q circuli quadrans O T. <lb></lb>exiſtente igitur O in T erit P in S. facta quartæ partis ipſius <lb></lb>ſcytalæ rotatione. </s> <s id="s.000999">Et quoniam ex eodem centro ſunt qua<lb></lb>drantes PSOT. erit vt OQ ad QP. ita quadrans OT, ad <lb></lb>quadrantem PS. </s> <s id="s.001000">Maxima autem eſt proportio OQ, ad <lb></lb>QP. </s> <s id="s.001001">Maxima igitur proportio OT ad PS. </s> <s id="s.001002">Ex magno igitur <lb></lb>motu O ad T, paruus ſit ſcytalæ motus à P in S. tardius i<lb></lb>gitur progreditur ſcytala, quæ longioribus vectibus rota<lb></lb>tur, vis tamen maxima, quippe quod vt ſe habet QP, hoc <lb></lb>eſt, QR ad QO, ita potentia in O ad pondus quod premit <lb></lb>in P vel in V. </s> <s id="s.001003">Facillimè itaque pondera vectibus & ſcyta<lb></lb>lis per horizontis planum transferri, exiſtis patet. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001004">QVAESTIO XII.</s> </p> <p type="head"> <s id="s.001005"><emph type="italics"></emph>Quæritur, Cur Miſſilia longius funda mittantur quam manu, <lb></lb>præſertim cum proijcienti fundæ pondus addatur lapidis ſeu miſſi<lb></lb>lis ponderi: & minus miſſili, manu proiecto, com<lb></lb>prehendatur?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001006">Soluit Philoſophus, inquiens, fortè ita fieri, quòd fun<lb></lb>ditor miſſile proijciat iam ex funda commotum, ſiqui<lb></lb>dem fundam circulo ſubinde rotans, iaculatur, ex manu <lb></lb>autem à quiete eſt initium. </s> <s id="s.001007">Omnia autem cum in motu <lb></lb>ſunt, quàm cum quieſcunt, facilius mouentur. </s> <s id="s.001008">Addit præ<lb></lb>terea, An & ob eam cauſſam eſt, ſed nec minus etiam, quia <lb></lb>infunde vſu manus quidem fit centrum, funda verò quod <lb></lb>à centro exit? </s> <s id="s.001009">quantò igitur productius fuerit quod à cen<lb></lb>tro eſt, tanto citiùs mouetur; iactus autem, qui manu fit, <lb></lb>fundæ reſpectu breuior eſt. </s> </p> <p type="main"> <s id="s.001010">Hæc Philoſophus. </s> <s id="s.001011">Et ſanè perquàm appoſitè, <expan abbr="itaq;">itaque</expan> <pb xlink:href="007/01/109.jpg"></pb>illi prorſus aſſentirer, niſi pro comperto haberem, in iactu <lb></lb>qui fundâ fit, non eſſe manum ipſam motus centium, ſed <lb></lb>potius partem illam brachij, quæ humero iungitur, & id<lb></lb>eo motum eo fieri velociorem, quo longior eſt linea quæ <lb></lb>ab humero ad ſummitatem fundæ eſt, ea quæ ab humero <lb></lb>ad manum ipſam. </s> <s id="s.001012">Illud quoque mirabile eſt, quod non <lb></lb>obſeruat Ariſtoteles, nempe à funditoribus in ipſo eiacu<lb></lb>landi actu, tardam fieri circa caput fundæ rotationem. <lb></lb></s> <s id="s.001013">Quamobrem conſiderandum eſt, quo pacto fiat à tardi<lb></lb>tate velocitas. </s> <s id="s.001014">Reſpondemus, velocitatem acquiri non ex <lb></lb>ſimplici, quæ circa funditoris caput ſit, rotatione, ſed ex <lb></lb>eo impetu qui fit in ipſa lapidis emiſſione, qui quidem im<lb></lb>petus ſi ante vel poſt illud tempus fiat, quod à funditore <lb></lb>captatur, caſſa prorſus & inualida fit ipſa iaculatio. </s> </p> <figure id="id.007.01.109.1.jpg" xlink:href="007/01/109/1.jpg"></figure> <p type="main"> <s id="s.001015">Eſto funda AB, manus <lb></lb>B, brachium BC. </s> <s id="s.001016">Vt igitur ſe <lb></lb>habet CH, ad CB, ita veloci<lb></lb>tas AD ad velocitatem, BE; <lb></lb>Vidimus nos pueros, arundi<lb></lb>ni ad caput ſciſſæ, paruos la<lb></lb>pides inſerentes, arundinem<lb></lb>queue manu rotantes longiſſi<lb></lb>mè lapides ipſos proijcere; A<lb></lb>rundo FG, lapis F, manus G, <lb></lb>brachium GH. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001017">QVÆSTIO XIII.</s> </p> <p type="head"> <s id="s.001018"><emph type="italics"></emph>Quæritur, Cur circa idem iugum, maiores collopes (vectes ſunt, <lb></lb>quos alij ſcytalas appellant, vt Pappus & Heron) faciliùs quàm mi<lb></lb>nores mouentur: & item ſuculæ, quæ graciliores ſunt eadem <lb></lb>vi quam craſſiores?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001019">Ideo hoc fieri poſſe docet Philoſophus, quòd <expan abbr="tamiugū">tam iugum</expan> <lb></lb>quam ſucula <expan abbr="cētrum">centrum</expan> ſit, prominentes autem collopum <pb xlink:href="007/01/110.jpg"></pb>longitudines eæ lineæ quæ ſunt à centro. </s> <s id="s.001020">Celeriùs autem <lb></lb>moueri & plus ab eadem vi quæ maiorum ſunt <expan abbr="circulorū">circulorum</expan> <lb></lb>quàm quæ minorum. </s> <s id="s.001021">quippe quod ab ea dem vi plus <expan abbr="trãſ-feratur">tranſ<lb></lb>feratur</expan> illud extremum quod longius à centro diſtat. </s> <s id="s.001022">In <lb></lb>gracilioribus verò ſuculis datâ collopum paritate plus eſ<lb></lb>ſe id quod à ligno diſtat. </s> </p> <figure id="id.007.01.110.1.jpg" xlink:href="007/01/110/1.jpg"></figure> <p type="main"> <s id="s.001023">Eſto iugum ſucu<lb></lb>laue maior, AB circa <lb></lb>centrum C, minor verò <lb></lb>circa idem <expan abbr="centrū">centrum</expan> DE. <lb></lb></s> <s id="s.001024">Collops <expan abbr="autē">autem</expan> AF, pon<lb></lb>dus quod per iugum at<lb></lb>tollitur G. </s> <s id="s.001025">A it igitur A<lb></lb>riſtoteles, ſuculas, iu<lb></lb>gaue AB, DE ceu cen<lb></lb>tra eſſe, à quibus extat colops AB, ex maiori quidem, totâ <lb></lb>ſui parte BF, ex minori autem EF. quo igitur, ait, longior <lb></lb>fuerit collops extans, eo maior, & deo velocior ad <expan abbr="partē">partem</expan> <lb></lb>F per maiorem circulum FH, fiet collopis motus & pon<lb></lb>deris eleuatio, at maior eſt collops EF ipſo BF, facil. </s> <s id="s.001026">us er<lb></lb>go mouebitur pondus per ſuculam DE, ex collope EF, ab <lb></lb>eadem vi, quam per ſuculam AB, & collopem BF. </s> </p> <p type="main"> <s id="s.001027">Hæc ſenſiſſe videtur Ariſto teles, qui craſſa, vt aiunt, <lb></lb>Minerua rem pulchram & ſubtilem eſt proſequutus. </s> <s id="s.001028">Di<lb></lb>cimus igitur primò, inſtrumentum illud quod Latini ſu<lb></lb>culam, id eſt, ſeroſulam, à ſtridore arbitror qui in conuer<lb></lb>ſione fit, appellauere, Græci verò <foreign lang="grc">ὄνον</foreign>, id eſt, A ſinum, quip<lb></lb>pe quod ceu A ſinus pondera ſuſtineat portetque. </s> <s id="s.001029">Hanc <lb></lb>eandem Machinam veteres Mechanici vocauere Axem <lb></lb>in Peritrochio, cuius nos imaginem, è Pàppo in 8. Col<lb></lb>lect. Mathematicarum deſumptam in ipſo huius noſtri o<lb></lb>peris initio, inter quinque Potentias propoſuimus. </s> <s id="s.001030">Huius <lb></lb>vim inter antiquos diligentiſſime examinauêre Heron, & <pb xlink:href="007/01/111.jpg"></pb>ipſemet Pappus, inter iuniores verò Guilibaldus eo Tra<lb></lb>ctatu quem hac de Potentia Mechanicis ſuis inſeruit. <lb></lb></s> <s id="s.001031">Summa eſt, hanc Machinam ad vectem reduci. </s> <s id="s.001032">Nec ve<lb></lb>rum eſt quod ſcribit Ariſto teles, iugum ſuculamue cen<lb></lb>tra eſſe, hæc enim centrum habent, quod in figura ſupe<lb></lb>rius poſita notatur ſigno C. igitur vt ſe habet FC, ad CA, <lb></lb>ita pondus G ad potentiam in F; eſt autem maior propor<lb></lb>tio FC ad CD, quàm FC, ad CA. faciliùs ergo mouebit <lb></lb>potentia quæ in F, pondus in D, quàm eadem potentia F, <lb></lb>pondus in A, hoc eſt, G. </s> <s id="s.001033">Huius naturæ ſunt quo que Erga<lb></lb>tæ, quas machinas noſtri, Græco luxato vocabulo Arga<lb></lb>nos appellant. </s> <s id="s.001034">Suculæ enim reuera ſunt, poſitione <expan abbr="tantū">tantum</expan> <lb></lb>ab eis differentes, non enim plano horizontis ergatæ æ<lb></lb>quidiſtant, ceu ſuculæ & Axis in Peritrochio, ſed eidem <lb></lb>fiunt perpendiculares. </s> <s id="s.001035">Cæterùm facilitatem à velocitate <lb></lb>non oriri ſuperius demonſtrauimus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001036">QVAESTIO XIV.</s> </p> <p type="head"> <s id="s.001037"><emph type="italics"></emph>Proponitur dubitatio, Cur eiuſdem magnitudinis lignum facilius <lb></lb>genu frangatur ſi quiſpiam æque diductis manibus extrema com<lb></lb>prehendens fregerit, quàm ſi iuxta genu. </s> <s id="s.001038">Et ſi terræ applicans pede <lb></lb>ſuperpoſito manu hinc inde diducta confregerit <lb></lb>quàm propè.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001039">Soluitur à Philoſopho paucis verbis, An quia ibi genu <lb></lb>centrum eſt, hic verò ipſe pes? </s> <s id="s.001040">quanto autem remotius <lb></lb>à centro fuerit, facilius mouetur quodcunque: Moueri <lb></lb>autem quod frangitur neceſſe eſt. </s> </p> <p type="main"> <s id="s.001041">Eſto lignum quod frangi debet AB, genu vel pedis <lb></lb>locus C, manuum latè diductarum ſitus DE, minus didu<lb></lb>ctarum FG; ltaque quoniam DE magis à centro C diſtant <lb></lb>quàm FG, velocius mouebuntur puncta DE ipſis FG, er<lb></lb>go inde facilius fiet fractio quam ex FG. </s> <s id="s.001042">Hæc ille ex ſuis <pb xlink:href="007/01/112.jpg"></pb><figure id="id.007.01.112.1.jpg" xlink:href="007/01/112/1.jpg"></figure><lb></lb>principijs. </s> <s id="s.001043">Nos dili<lb></lb>gentius, ſi fieri poterit, <lb></lb>effectus huius cauſſam <lb></lb>perſcrutemur. </s> <s id="s.001044">Eſto igi<lb></lb>tur in ſecunda figura <lb></lb>lignum oblongum AB, <lb></lb>cuius medium C, linea <lb></lb>ducatur CD perpen<lb></lb>dicularis ipſi AB. </s> <s id="s.001045">Ad<lb></lb>moueatur genu <expan abbr="pūcto">puncto</expan> <lb></lb>C, manus verò diuari<lb></lb>centur in AB, facta i<lb></lb>gitur vtrinque impreſ<lb></lb>ſione, lignum non <expan abbr="frã-getur">fran<lb></lb>getur</expan>, niſi partium in CD coniunctarum ſeparatio fiat, <lb></lb>ſitqueue altera in E, altera verò in F, fractum ergo erit <expan abbr="lignū">lignum</expan>, <lb></lb>& centro C immobili permanente, partes facto angulo <lb></lb>GCH erunt in GC, HC: Modò lignum ſuæ integritati re<lb></lb>ſtituetur, & denuò admoto genu puncto C, manus didu<lb></lb>cantur in I, K, quæ loca viciniora ſint ipſi C, quam AB, Di<lb></lb>co hinc difficilius fractionem fieri quam ex AB. </s> <s id="s.001046">Conſide<lb></lb>ramus enim in integro ligno AB, duos vectes ACD, BCD, <lb></lb>quorum anguli concurrunt in commune fulcimentum C, <lb></lb>Sunt autem vectes angulati, & eius naturæ, quam exami<lb></lb>nauimus in quæſtiones. </s> <s id="s.001047">Eſt igitur reſiſtentia, qua ligni <lb></lb>partes vniuntur in D, loco ponderis: ſuperanda hæc eſt, vt <lb></lb>ligni fiat fractio. </s> <s id="s.001048">Dico id facilius ceſſurum, ſi fiat ex pun<lb></lb>ctis A, B, remotioribus quam ex IK, ipſi puncto C propio<lb></lb>ribus: etenim vt AC, ad CD, ita reſiſtentia quæ fit in D ad <lb></lb>potentiam in A, item vt ſe habet IC ad CD, ita reſiſtentia <lb></lb>in Dad potentiam in I, ſed minor eſt proportio IC ad CD, <lb></lb>quam AC ad CD. ergo facilius potentia quæ eſt in A, re<lb></lb>ſiſtentiam ſuperabit, quæ eſt in D, quam ea quæ eſt in I, <pb xlink:href="007/01/113.jpg"></pb>quod fuerat demonſtrandum. </s> <s id="s.001049">Idem autem <expan abbr="intelligendū">intelligendum</expan> <lb></lb>eſt de parte CB; eadem enim eſt ratio. </s> <s id="s.001050">Cur igitur longio<lb></lb>ra & graciliora ligna facilè frangantur, ex iſtis clare patet: <lb></lb>nempe quia maxima eſt proportio longitudinis ad craſſi<lb></lb>tudinem, cuius quidem craſſitudinis ſpatium loco partis <lb></lb>illius in vecte ſuccedit, quæ pertingit à fulcimento ad <expan abbr="pō-dus">pon<lb></lb>dus</expan>, hoc eſt, ad ipſam reſiſtentiam. </s> <s id="s.001051">Sed nos hac eadem de <lb></lb>re nonnulla in declaranda quæſtione 16. perpendemus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001052">QVAESTIO XV.</s> </p> <p type="head"> <s id="s.001053"><emph type="italics"></emph>Proponitur inuestigandum, Cur litterales crocæ (glareas dicunt <lb></lb>Latini, vel calculos, quos vmbilicos appellat Cicero lib. 2. de Orat.) <lb></lb>rotundâ ſint figurâ, cum aliquando ex magnis ſint la<lb></lb>pidibus teſtisue?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001054">A It Philoſophus, ideo fortaſſe fieri, quòd ca quæ à me<lb></lb>dio magis recedunt, in motionibus, celerius feran<lb></lb>tur; me dium eſſe centrum, interuallum vero quæ à cen<lb></lb>tro, ſemper autem maiorem ab æquali motione maiorem <lb></lb>deſcribere circulum; quod autem maius in æquali tem<lb></lb>pore ſpatium tranſit, celerius ferri; quæ autem celerius ex <lb></lb>æquali feruntur ſpatio vehementius impetere, quæ <expan abbr="autē">autem</expan> <lb></lb>impetunt, impeti magis, & ideo quæ magis à centro di<lb></lb>ſtant, neceſſe eſſe confringi, quod cum glareæ ſeu crocæ <lb></lb>patiantur, neceſſariò rotundas fieri. </s> <s id="s.001055">Hactenus ille, & ſanè <lb></lb>probabiliter. </s> <s id="s.001056">Verum enimuerò aliter ſeres habere vide<lb></lb>tur: ſiquidem enim à rotatione ex maiori à centro diſtan<lb></lb>tia id fieret, maiores quidem glareæ crocæue eſſent ro<lb></lb>tundiores, at nos non maximas modò, ſed & minimas, <lb></lb>eaſque magis angulis carere, & ad rotunditatem accede<lb></lb>re videmus. </s> <s id="s.001057">Præterea non moueri eas circa centrum pa<lb></lb>lam eſt, imò vt varia ſunt figura, ita varijs quoque motio<lb></lb>nibus, ex agitatione moueri. </s> <s id="s.001058">Id ſanè exploratiſſimum eſt, <pb xlink:href="007/01/114.jpg"></pb>angulos omnes, & eminentias quaslibet in corporibus eſ<lb></lb>ſe infirmiores, offenſionibus enim expoſitæ ſunt, nec reſi<lb></lb>ſtendi habent facultatem. </s> <s id="s.001059">Itaque in attritione quæ fit in <lb></lb>eorum agitatione perpetua, eminentiæ contunduntur, & <lb></lb>ſuperficies ipſa paullatim leuigatur. </s> </p> <figure id="id.007.01.114.1.jpg" xlink:href="007/01/114/1.jpg"></figure> <p type="main"> <s id="s.001060">Eſto angulatus lapis ABCD. <lb></lb></s> <s id="s.001061">Dum igitur perpeti motione <expan abbr="atq;">atque</expan> <lb></lb>aſſiduâ verſatione agitatur, fer<lb></lb>turqueue, eminentiæ anguliqueue, vt<lb></lb>pote debiles & imbecilli, conte<lb></lb>runtur, & inde figura fit quædam <lb></lb>irregularis, ad primam quidem la<lb></lb>pidis <expan abbr="formã">formam</expan> accedens, leuis tamen <lb></lb>& quouis angulo carens, qualis eſt E remotis ABCD, an<lb></lb>gularibus eminentijs. </s> </p> <p type="main"> <s id="s.001062">Hanc eandem ob cauſſam, ſculptores antequam mar<lb></lb>moribus vltimum læuorem inducant, dentato malleo pri<lb></lb>mum quidem vtuntur, tum demum eminentiores parti<lb></lb>culas radula facilè amouentes ſuperficiem ipſam læuem <lb></lb>& adæquatam reddunt. </s> </p> <p type="main"> <s id="s.001063">Hinc etiam noſtrates Architecti, in arcium propu<lb></lb>gnaculis efformandis acutos angulos <expan abbr="deuitãt">deuitant</expan>, vtpote de<lb></lb>biliores, & magis offenſionibus obnoxios. </s> <s id="s.001064">quod nec Vi<lb></lb>truuium latuit, qui ideo lib. 1. cap. 5. ita ſcribit: <emph type="italics"></emph>Turres itaque <lb></lb>rotundæ aut polygoniæ ſunt faciendæ, quadratas enim machinæ <lb></lb>celerius diſſipant; & angulos, Arietes tundendo frangunt, in ro<lb></lb>tundationibus autem, vti cuneos ad centrum adigendo lædere non <lb></lb>poſſunt.<emph.end type="italics"></emph.end> Hæc ille. </s> <s id="s.001065">Cur autem noſtri rotundas figuras alias <lb></lb>vtiles reijciant, ab ijs petendum qui in ea facultate ver<lb></lb>ſantur. </s> <s id="s.001066">Porrò quod ad hanc eandem ſpeculationem facit, <lb></lb>videmus, antiquas ſtatuas, vt ſæpius auribus, naſo, digitis, <lb></lb>manibuſue atque pedibus carere, quippe quod imbecillæ <lb></lb>ſint partes, & facilè quouis occurſu mutilentur. </s> <s id="s.001067">Quæ o-<pb xlink:href="007/01/115.jpg"></pb>mnia cùm vera ſint, nemo, vt arbitror, dixerit, abſolutè, <lb></lb>quod voluit Ariſtoteles, id ex rotatione velociori & par<lb></lb>tium à centro remotione, fieri. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001068">QVAESTIO XVI.</s> </p> <p type="head"> <s id="s.001069"><emph type="italics"></emph>Dubitatur, quare, quò longiora ſunt ligna, <expan abbr="tãto">tanto</expan> imbecilliora fiant, <lb></lb>& ſi tolluntur, inflectuntur magis: tametſi quod breue est ceu bi<lb></lb>cubitum fuerit, tenue, quod verò cubitorum cen<lb></lb>tum craſſum?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001070">Ex ſuis principijs ſoluit Ariſtoteles. </s> <s id="s.001071">Inquit enim: An <lb></lb>quia & vectis & onus & hypomochlium, id eſt, fulci<lb></lb>mentum in leuando, fit ipſa ligni proceritas? </s> <s id="s.001072">Prior <expan abbr="namq;">namque</expan> <lb></lb>illius pars ceu hypomochlium fit, quod verò in extremo <lb></lb>eſt, pondus: quamobrem quanto extenſius fuerit id quod <lb></lb>à fulcimento eſt, in flecti neceſſe eſt magis; quo enim plus <lb></lb>à fulcimento diſtat, eo magis incuruari neceſſe eſt. </s> <s id="s.001073">Ne<lb></lb>ceſſariò igitur extrema vectis eleuantur. </s> <s id="s.001074">Si igitur flexilis <lb></lb>fuerit vectis, ipſum inflecti magis cum extollitur neceſſe <lb></lb>eſt, quod longis accidit lignis, in breuibus autem quod vl<lb></lb>timum eſt, quieſcenti hypomochlio depropè fit. </s> <s id="s.001075">Hæc <lb></lb>ſubiectâ figurâ ob oculos ponimus. </s> </p> <figure id="id.007.01.115.1.jpg" xlink:href="007/01/115/1.jpg"></figure> <p type="main"> <s id="s.001076">Eſto longum ac fle<lb></lb>xile lignum AB, manu ele<lb></lb>uetur in A, flectetur <expan abbr="itaq;">itaque</expan> <lb></lb>in B, & declinabit in C. et<lb></lb>enim manus quæ ſuſtinet <lb></lb>in A, fulcimenti loco ſuccedit: longitudo vero AB ponde<lb></lb>ris vices refert, at que vectis, quare quo longius abfuerit à <lb></lb>fulcimento, id eſt, manu extremum B, eo magis flectetur; <lb></lb>ſi autem lignum breuius fuerit, nempe terminatum in D, <lb></lb>nequaquam flectetur, eò quòd eius extremum D minus à <lb></lb>fulcimento quod eſt in A ſit remotum. </s> <s id="s.001077">Hæc igitur eſt <expan abbr="mēs">mens</expan> <pb xlink:href="007/01/116.jpg"></pb>Ariſtotelis, cuius quidem ſententiam non damnamus; <lb></lb>quippiam tamen addimus. </s> <s id="s.001078">Dicimus autem materiam, <lb></lb>quatenus ad hanc contemplationem ſpectat, in duplici <lb></lb>eſſe differentia. </s> <s id="s.001079">aut enim rarefactionis & conſtipationis <lb></lb>eſt incapax, vt in chalybe videmus, nitro, metallo, mar<lb></lb>more, aut capax quidem, & hæc duplex: Vel enim natura <lb></lb>nata eſt ad rectitudinem quandam, vt arborum flagella <lb></lb>virgæque, aut non item, ceu ſtannum, plumbum, & cæte<lb></lb>ra eiuſmodi. </s> </p> <figure id="id.007.01.116.1.jpg" xlink:href="007/01/116/1.jpg"></figure> <p type="main"> <s id="s.001080">Eſto primò vitreum <lb></lb>corpus gracile, procerum, <lb></lb>teres AB, manu capiatur in <lb></lb>A, <expan abbr="itaq.">itaque</expan> pondere ipſius cor<lb></lb>poris præualente ad partes <lb></lb>B, quia in C puncto, quod <lb></lb>circa medium eſt, ex parte <lb></lb>ſuperiori non fit rarefactio, <lb></lb>nec in in feriori conſtipatio, <lb></lb>nec interim datur penetra<lb></lb>tio corporum, fit fractio à <lb></lb>ſuperiori parte, & pars CB à <lb></lb>reliqua parte AC, auulſa & <lb></lb>ſeparata cadit in D, ſuccedit autem ipſa ſeparatio rarefa<lb></lb>ctioni. </s> <s id="s.001081">Porrò quod materias haſce non flexibiles diximus, <lb></lb>ſed frangibiles, non ideo negamus vel ſenſu docente, ali<lb></lb>quam in ijs fieri flexionem. </s> <s id="s.001082">Si autem lignea fuerit mate<lb></lb>ria, caque; flexibilis, vt EF, ſi manu eleuetur in E, præualen<lb></lb>te pondere in F flectetur vbi G. ibi enim à parte ſuperiori <lb></lb>fit rarefactio, ab in feriori verò conſtipatio, & pars GF de<lb></lb>clinabit in H, quæ declinatio eò vſque procedet, quo ra<lb></lb>refactio & conſtipatio competens naturæ illius materiæ, <lb></lb>quæ flectitur ad ſummam intenſionem deuenerint; tunc <lb></lb>ſi vis maior ingruerit, frangetur omnino: ſi ſecus facta ibi <pb xlink:href="007/01/117.jpg"></pb>reſiſtentia, vbi rarefactio fit & conſtipatio per inclina<lb></lb>tionem ſurſum feretur pars in clinata & nutans, tum in <lb></lb>contrariam partem tendens reflectetur, vt videre eſt in <lb></lb>virga IN. </s> <s id="s.001083">Declinans enim in KL, repellente ea quæ infra <lb></lb>K fit materiæ condenſatione, impetu ex deſcenſu acqui<lb></lb>ſito facta reflexione aſcendit in KM, donec paullatim cir<lb></lb>ca priſtinam rectitudinem reuertatur, & hic quidem mo<lb></lb>tus vibratio dicitur, agitatioue. </s> <s id="s.001084">Si autem virga plumbea <lb></lb>fuerit, naturâ non factâ ad rectitudinem, puta OP, pro<lb></lb>prio vincente pondere, ad partes declinabit QS, fietque; in <lb></lb>QR rarefacta, nempe ſuperiori parte ea conſtipata infe<lb></lb>riori in Q, nec reflectetur, quippe quòd eius natura con<lb></lb>denſationem & rarefactionem commodè patiatur, nec <lb></lb>facta ſit ad rectitudinem. </s> </p> <p type="main"> <s id="s.001085">Porrò tripliciter fieri poteſt horum oblongorum <lb></lb>corporum eleuatio, nempe vel extremorum alteio, aut ſi <lb></lb>ambobus, ſi vtrinque ſuſpendatur, vel alicubi inter extre<lb></lb>ma. </s> <s id="s.001086">De priori modo iam egimus. </s> <s id="s.001087">Modò ſuſpendatur in <lb></lb>medio vt AB, in C. eo igitur caſu cum fulcimentum ſit in <lb></lb>C, <expan abbr="vtrinq;">vtrinque</expan> fit flexio in D, & E, & id quidem ſi materia fle<lb></lb>xionem patitur: ſin minus, fractio fit in C. </s> <s id="s.001088">Si autem ab ex<lb></lb><figure id="id.007.01.117.1.jpg" xlink:href="007/01/117/1.jpg"></figure><lb></lb>tremis fiat ſuſpenſio, vt in <lb></lb>AB, tunc ceu duo vectes <lb></lb>fient, quorum fulcimenta in <lb></lb>extremis AB. </s> <s id="s.001089">Pondera au<lb></lb>tem communia in medio vbi <lb></lb>C remotiſſima enim ea pars eſt ab extremis AB. </s> <s id="s.001090">Cedente <lb></lb><figure id="id.007.01.117.2.jpg" xlink:href="007/01/117/2.jpg"></figure><lb></lb>igitur materia ſuomet pon<lb></lb>deri, ſiquidem in flexibilis fu<lb></lb>erit, frangetur, & fiet <expan abbr="partiū">partium</expan> <lb></lb>ſeparatio in C, duoque in de <lb></lb>corpora AD, BE. </s> <s id="s.001091">Si autem fle<lb></lb>xionis capax, vt AB in poſtre<pb xlink:href="007/01/118.jpg"></pb>ma figura, facta ex contrario, nempe in in feriori parte cir<lb></lb>ca C rarefactione, in ſuperiori verò condenſatione, pon<lb></lb>dere præualente curuabitur, fietque; lignum quidue aliud <lb></lb>huiuſmodi, vt ADB, nec amplius pondere ſuapte naturâ <lb></lb>inferiùs vergente ad rectitudinem reuertetur. </s> </p> <p type="main"> <s id="s.001092">Cæterùm cur oblonga & graciliora corpora facilius <lb></lb>illis, quæ contrario ſe habent modo, frangantur, ex me<lb></lb>chanicis principijs in quæſtione 14. apertè demonſtraui<lb></lb>mus. </s> <s id="s.001093">Modò vt ex hac contemplatione, quæ aliàs inutilis <lb></lb>videtur, aliquam vtilitatem capiamus, & ex his quæ con<lb></lb>templabimur, Architecti prudentiores fiant, iſthæc ipſa, <lb></lb>de quibus agimus, ad rem ædificatoriam commodè apta<lb></lb>bimus. </s> <s id="s.001094">Transferamus igitur cogitationem ad eam <expan abbr="trabiū">trabium</expan> <lb></lb>compagem, quæ ad tecta ſuſtinenda ex tranſuerſario ar<lb></lb>rectarioque; ſit, & duobus cauterijs, quam noſtri à Latinis <lb></lb>detorto vocabulo Biſcauterium dicunt. </s> <s id="s.001095">Perſcrutabimur <lb></lb>enim, vnde illi tanta ad ſuſtinendum vis, & quæ compa<lb></lb>gem hanc conſequantur paſſiones. </s> <s id="s.001096">quamuis enim fabri <lb></lb>meræ praxi, quod vtile eſt efficiant, nos meliorum inge<lb></lb>niorum gratiâ, rei ipſius cauſſas diligenter examinatas in <lb></lb>medium proferemus; nec de hac re tantùm agemus, ſed <lb></lb>de Cameris quoque, fornicibus eorumqueue vitijs & virtu<lb></lb>tibus quatenus ad Mechanicum pertinet, ſermonem ha<lb></lb>bebimus. </s> <s id="s.001097">Quærimus primo, cur perpendiculariter erectae <lb></lb>trabes ſuperimpoſita pondera validiſſime ſuſtineant? </s> <s id="s.001098">Et <lb></lb>ſane hoc omnes norunt, ſed non per cauſſas. </s> </p> <p type="main"> <s id="s.001099">Eſto horizontis planum, illudqueue ſolidiſſimum, & <lb></lb>impenetrabile AB, trabs eidem ad perpendiculum erecta <lb></lb>CD fulta baſi vbi C grauitatis centrum F. pondus ſuper<lb></lb>impoſitum FG, cuius grauitatis centrum H: Sint autem <lb></lb>H & E in eadem perpendiculari, quæ ad mundi centrum <lb></lb>HEC. </s> <s id="s.001100">Itaque eo quod tum penderis tum trabis centra <lb></lb>grauitent in perpendiculari, illa verò fulciatur in C, to-<pb xlink:href="007/01/119.jpg"></pb><figure id="id.007.01.119.1.jpg" xlink:href="007/01/119/1.jpg"></figure><lb></lb>tius ponderis moles recumbet <lb></lb>in C: non deſcendet autem in I, <lb></lb>propterea quod ſupponatur i<lb></lb>pſum planum AB, impenetrabi<lb></lb>le. </s> <s id="s.001101">Igitur vt pondus H deſcen<lb></lb>dat in C, alterum duorum eſt <lb></lb>neceſſarium, nempe vel trabem <lb></lb>ſubiectam comminui, aut eius <lb></lb>partes ſeſe penetrare, & plura <lb></lb>corpora eſſe in eodem loco, pu<lb></lb>ta KC, quorum hoc ſecundum <lb></lb>naturæ penitus repugnat, illud <lb></lb>vero primum, penè impoſſibile. </s> <s id="s.001102">Diuidatur enim trabs in <lb></lb>partes æquales tres, lineis KL, ipſa igitur KC infima ſuſti<lb></lb>net mediam KL, hæc verò ſupremam LD, hæc autem <expan abbr="pō-dus">pon<lb></lb>dus</expan>, ipſum ſuperpoſitum in H. </s> <s id="s.001103">Seigitur ſuſtinent partes. <lb></lb></s> <s id="s.001104">Sed illud totum partibus conſtat. </s> <s id="s.001105">ergo pondus totum à <lb></lb>trabe tota, hoc eſt, à ſe toto ſuſtinetur. </s> </p> <p type="main"> <s id="s.001106">Præterea in præcedenti quæſtione monſtrauimus <lb></lb>tunc facilem eſſe gracilis & oblongi ligni fractionem, <expan abbr="cū">cum</expan> <lb></lb>maxima eſt longitudinis ad craſſitudinem proportio. </s> <s id="s.001107">Hîc <lb></lb>verò contrà accidit, etenim MD pars vectis quæ à fulci<lb></lb>mento eſt ad potentiam minimam habet proportionem <lb></lb>ad rectam DC, quæ à fulcimento ad locum fractionis ex<lb></lb>tenditur, vbi C, quod vt euidentius pateat, </s> </p> <figure id="id.007.01.119.2.jpg" xlink:href="007/01/119/2.jpg"></figure> <p type="main"> <s id="s.001108">Eſto ſeorſum trabs AB, <lb></lb>cuius medium C. </s> <s id="s.001109">Sit autem <lb></lb>pondus D impoſitum pun<lb></lb>cto C. facilè igitur frange<lb></lb>tur lignum AB, propterea <lb></lb>quòd maxima ſit proportio <lb></lb>AC ad CE; reſiſtentia verò <lb></lb>fiat in E, addatur vniaturque; <pb xlink:href="007/01/120.jpg"></pb>ligno AB lignum FH. </s> <s id="s.001110">Craſſius igitur eſt totum AL, ipſo <lb></lb>AH, & ideo minor proportio AC ad CG quàm AC, ad <lb></lb>CE. </s> <s id="s.001111">Addatur adhuc & IM. </s> <s id="s.001112">Longè itaque difficilius fran<lb></lb>getur in K propterea quòd longè minor ſit proportio AC <lb></lb>ad CK quàm eiuſdem ad CE & CG. </s> <s id="s.001113">His igitur conſide<lb></lb>ratis, & demonſtratis concludimus, impoſſibile eſſe ere<lb></lb>ctam trabem ponderi cedere, & frangi. </s> </p> <p type="main"> <s id="s.001114">Dicet autem quiſpiam, haec ſi vera ſunt, quo gracilius <lb></lb>fuerit fulcrum, eo validiùs ſuſtinebit, & frangetur minus, <lb></lb>quod oppido falſum eſt. </s> <s id="s.001115">Reſpondemus, id non ex propor<lb></lb>tionum naturâ, ſed ex materiæ ipſius infirmitate fieri. </s> <s id="s.001116">Ita <lb></lb>quoque invecte non materiam, quatenus ad vim pertinet, <lb></lb>ſed proportiones partium conſideramus. </s> <s id="s.001117">Vtrumque igi<lb></lb>tur requiritur ad fulcri validitatem proportio longitudi<lb></lb>nis ad craſſitudinem debita, & materiæ ipſius robur & <lb></lb>fortitudo. </s> <s id="s.001118">Præterea, quoniam pondus, cui fulcrum reſi<lb></lb>ſtit, vel ex natura premit, vel ex violentia, illud quidem <lb></lb>per lineam perpendicularem, quæ ad mundi <expan abbr="cētrum">centrum</expan>, hoc <lb></lb>autem lateraliter & diuerſi modè, varia fit fulcrorum diſ<lb></lb>poſitio. </s> <s id="s.001119">Cuius rei ſumma hæc eſt, vt ſemper contra impe<lb></lb>tum ſupponantur. </s> </p> <figure id="id.007.01.120.1.jpg" xlink:href="007/01/120/1.jpg"></figure> <p type="main"> <s id="s.001120">Eſto enim horizontis planum <lb></lb>AB, <expan abbr="eidē">eidem</expan> perpendiculares CADB, <lb></lb>ítaque ſi naturaliter pondus pre<lb></lb>mat ex C, fulcrum ſupponetur AE. <lb></lb></s> <s id="s.001121">Si autem ex F ipſum GE, ſi verò ex <lb></lb>H, ſupponatur iuxta BE. </s> <s id="s.001122">Si verò ſe<lb></lb>cundum I ponderi opponatur KE. <lb></lb></s> <s id="s.001123">Hæc nos de arrectarijs fulcrisue; <lb></lb>nunc de tranſuerſarijs, & inclinatis agemus, & primum <lb></lb>de tranſuerſarijs, quatenus ad tectorum trabeationes ſpe<lb></lb>ctat. </s> </p> <p type="main"> <s id="s.001124">Eſto tranſuerſaria trabs AB, muris <expan abbr="vtrinq;">vtrinque</expan> fulta CD, <pb xlink:href="007/01/121.jpg"></pb><figure id="id.007.01.121.1.jpg" xlink:href="007/01/121/1.jpg"></figure><lb></lb>cuius grauitatis centrum <lb></lb>E, in <expan abbr="perpēdiculari">perpendiculari</expan> FEG, <lb></lb>quæ quidem ad mundi <lb></lb>centrum vergit. </s> <s id="s.001125"><expan abbr="Itaq;">Itaque</expan> eo<lb></lb>dem tendente grauitatis <lb></lb>centro, ſi pondus quod <lb></lb>premit in E, non præua<lb></lb>leat vnioni <expan abbr="partiū">partium</expan> ipſius <lb></lb>materiæ quæ eſt in E, reſiſtet trabs ſuomet ponderi, nec <lb></lb>frangetur. </s> <s id="s.001126">Si autem vel in firmitate materiæ, aut vitio, vel <lb></lb>maxima existente proportione AF ad FE, fractio fiet in E, <lb></lb>& ſecutâ partium ſeparatione duæ fient vtrin que trabes <lb></lb>AH, Bl, quorum grauitatis centra KL. </s> <s id="s.001127">Erunt igitur duo <lb></lb>vectes AE, BE, quorum fulcimenta MN, quamobrem ſi <lb></lb>proportio EM ad MH ita præualeat, vt pondus quod e ſt <lb></lb>in E, ſuperet pondus muri O ſuperimpoſiti, & item muri <lb></lb>P, corruent quidem trabes, & murorum fiet hinc inde diſ<lb></lb>ſipatio. </s> <s id="s.001128">Si autem non præualuerit ea, quam diximus, pro<lb></lb>portio, ſuſpenſæ remanebunt vtrinque trabes vt AHBI. </s> </p> <p type="main"> <s id="s.001129">Huic difficultati egregiè occurrunt Architecti, ali<lb></lb>quando autem hoc modo: </s> </p> <figure id="id.007.01.121.2.jpg" xlink:href="007/01/121/2.jpg"></figure> <p type="main"> <s id="s.001130">Eſto tranſuerſaria <lb></lb>trabs ſuâ gracilitate, alia<lb></lb>ue de cauſſa imbecilla <lb></lb>AB, muri quibus <expan abbr="vtrinq;">vtrinque</expan> <lb></lb>ſuſtinetur CD, Trabis i<lb></lb>pſius grauitatis centrum <lb></lb>G. </s> <s id="s.001131">Itaque adpactis trabi <lb></lb>lignis EF, capreolos ad<lb></lb>dunt muro vtrinque ful<lb></lb>tos CE, DF, eorum capita adpactis lignis admouentes EF, <lb></lb>ſed & tunc validiſſima fit colligatio, ſi inter E & F capreo<lb></lb>lorum capita integrum lignum trabi ſupponatur EF. </s> <s id="s.001132">Ra<pb xlink:href="007/01/122.jpg"></pb>tio autem validitatis patet; premente enim grauitatis <expan abbr="cē-tro">cen<lb></lb>tro</expan> in G, fulcra hinc inde ſuccurrunt CE, DF, quæ cum ſe<lb></lb>ipſis fieri non valeant breuiora, ne corpori detur penetra<lb></lb>tio, reſiſtunt & robuſtiſſimè ipſi ponderi ſuperimpoſito <lb></lb>contra nituntur. </s> <s id="s.001133">Videntur autem in hoc opere duo con<lb></lb>ſiderari vectes, GH, GB, quorum fulcimenta EF, potentia <lb></lb>premens vtrinque G. </s> <s id="s.001134">Pondera autem parietum partes ca<lb></lb>pitibus trabis impoſitæ in A & B. </s> <s id="s.001135">Quoniam igitur parua <lb></lb>eſt proportio GE ad EH, parua potentia premens in G, <lb></lb>maximè autem pondus in A, fieri non poteſt trabem fran<lb></lb>gi aut muros vtrinque diſſipare in AB. </s> <s id="s.001136">Poſſunt etiam to<lb></lb>tius trabis tres partes conſiderari AE, EF, FB, quarum ful<lb></lb>cimenta quatuor A, E, F, B, Diuiſo igitur pondere & mul<lb></lb>tiplicatis fulcimentis impoſſibile eſt trabem conuelli & <lb></lb>vitium facere. </s> </p> <p type="main"> <s id="s.001137">Sed & tectorum contignationes imbecillaque; tranſ<lb></lb>uerſaria Mechanici corroborare ſolent, additis nempe <lb></lb>arrectaria trabe atque cauterijs. </s> </p> <figure id="id.007.01.122.1.jpg" xlink:href="007/01/122/1.jpg"></figure> <p type="main"> <s id="s.001138">Eſto enim tranſ<lb></lb>uerſaria trabs AB <lb></lb>parietibus vtrinque <lb></lb>fulta I, K, <expan abbr="arrectariū">arrectarium</expan> <lb></lb>CD. </s> <s id="s.001139">Cauterij vtrin<lb></lb>que AD, BD, ita <lb></lb>tranſuerſariæ trabi <lb></lb>in AB, & arrectario <lb></lb>in D inſerti, vt ne<lb></lb>quaquam inde ela<lb></lb>bi valeant. </s> <s id="s.001140">Tum ferrea faſcia EF mediam tranſuerſariam <lb></lb>trabem AB, à parte inferiori ipſi arrectario connectens, <lb></lb>Debet autem arrectarij pes vbi C, aliquantulum à tranſ<lb></lb>uerſaria trabe diſtare, ne deorſum ex pondere vergente <lb></lb>paululum arrectario ipſam tranſuerſariam premat. </s> <s id="s.001141">His i-<pb xlink:href="007/01/123.jpg"></pb>gitur ita conſtitutis pondus quidem tranſuerſariæ trabis, <lb></lb>quod ſuapte naturâ premit in medio vbi C, ferrea faſcia, <lb></lb>arrectariæ trabi affixa diſtinetur, Arrectariam cauterij ſu<lb></lb>ſtinent, hos verò tranſuerſariæ capita AB, quibus indun<lb></lb>tur. </s> <s id="s.001142">Tota igitur eiuſcemodi operis vis in eo conſiſtit, vt <lb></lb>probè cauterij tranſuersariæ & arrectariæ trabi inſeran<lb></lb>tur. </s> <s id="s.001143">fixis enim cauteriorum pedibus in AB, non <expan abbr="deſcendēt">deſcendent</expan> <lb></lb>à partibus ſeu capitibus D, ijs verò ſtantibus ſtabit & arre<lb></lb>ctarium, quo inde ſuſpenſo tranſuerſaria trabs ei ex ferrea <lb></lb>faſcia alligata nequaquam pendebit. </s> <s id="s.001144">Stabit ergo compa<lb></lb>ges tota & ſuapte vi robuſtiſſimè connexa totius tecti <lb></lb>pondus ſuſtinebit. </s> </p> <p type="main"> <s id="s.001145">Quoniam autem vſu venire ſolet, cauterios nimia <lb></lb>longitudine debiles, aliquando tum proprio tum extra<lb></lb>neo cedentes ponderi deorſum vergentes pandare, Ar<lb></lb>chitecti capreolis hinc inde ſuppoſitis, ceu fulcris, huic <lb></lb>medentur infirmitati. </s> </p> <figure id="id.007.01.123.1.jpg" xlink:href="007/01/123/1.jpg"></figure> <p type="main"> <s id="s.001146">Sint enim cauterij <lb></lb>debiles hinc inde AB, <lb></lb>AC, media trabs arre<lb></lb>ctaria, quam <expan abbr="Monachū">Monachum</expan> <lb></lb>dicimus AD. </s> <s id="s.001147">Cauterio<lb></lb>rum mediæ partes E, F, <lb></lb>in punctis igitur EF, vtpote maximè ab extremis diſtanti<lb></lb>bus debiles cauterij valde laborant. </s> <s id="s.001148">Itaque ſuppoſitis v<lb></lb>trinque arrectariolis EH, Fl, eorum capitibus E, F, duos <lb></lb>cauteriolos ſibi ipſis ad pedem arrectarij in D, reſiſtentes <lb></lb>apponunt. </s> <s id="s.001149">quibus ita conſtitutis nec E, nec F ad partes H, <lb></lb>I, deſcendere valent. </s> <s id="s.001150">Capiatur enim inter EH, quoduis <lb></lb>punctum G, & BG, DG, connectantur, erunt autem BG, <lb></lb>DG ipſis BE ED breuiores ex 21. primi elem. </s> <s id="s.001151">Tunc igitur <lb></lb>punctum E fiet in G cum BE, ED fient in BG, DG, quod <lb></lb>non cedentibus B, D, & ſibi ipſis breuioribus factis parti-<pb xlink:href="007/01/124.jpg"></pb>bus BE, ED, prorſus eſt impoſſibile. </s> <s id="s.001152">ſtabunt igitur in eo<lb></lb>rum rectitudine cauterij AB, AC, nec pandabunt, quod <lb></lb>fieri querebatur. </s> </p> <p type="main"> <s id="s.001153">Hîc autem damnandi veniunt ij, qui tranſuerſariæ <lb></lb>quidem trabis capitibus cauteriorum pedes non <expan abbr="inſerūt">inſerunt</expan>, <lb></lb>ſed ea vice tranſuerſariolo quodam medios cauterios v<lb></lb>trinque connectunt ad inſtar elementi A, quam compa<lb></lb>gem, capram, appellant. </s> <s id="s.001154">Sint enim cauterij hinc inde AB, <lb></lb>AC, quorum medias partes connectit tranſuerſariolum, <lb></lb>DE. </s> <s id="s.001155">Dico igitur colligationem iſtam magnopere impro<lb></lb>bandam. </s> <s id="s.001156">Sunt enim AB, AC vectes, quorum commune <lb></lb>fulcimentum A, potentiæ hinc inde diuaricantes B, C, <lb></lb>pondera inter fulcimentum & potentias DE. quoniam i<lb></lb>gitur vt DH ad AB, ita potentia in B, ad pondus in D, par<lb></lb>ua quidem potentia, pondus in D diſtrahet & ſuperabit: <lb></lb>facillimaque; in de fiet tranſuerſariolì à capreolis ipſis vtrin<lb></lb>que reuulſio: Et quoniam centrum quidem eſt A, facta in <lb></lb>D, E, parua diuaricatione, maxima fit in BC, vtpote parti<lb></lb>bus ab ipſo centro A quam remotis. </s> <s id="s.001157">Calcitrant igitur li<lb></lb>beri prope cauteriorum pedes, & muros ipſos ſummos, <lb></lb>non ſine magno operis totius vitio, ſua calcitratione pro<lb></lb>pellunt. </s> </p> <p type="main"> <s id="s.001158">Hæc nos de trabeationibus, modò ad fornicum ca<lb></lb>merarumque; naturam ſtilum transferemus; id enim ſuadet <lb></lb>vtilitas, imo & neceſſitas ipſa. </s> <s id="s.001159">Pauci enim ante nos hæc <lb></lb>tractarunt, & ſanè his probè non cognitis aut neglectis, <lb></lb>Architecti fabriqueue ingentes perſæpe incurrunt, & inex<lb></lb>plicabiles difficultates. </s> <s id="s.001160">Dicimus igitur primò, coctiles la<lb></lb>teres, & non cuneatos lapides ad rectam lineam diſpoſi<lb></lb>tos, non ſtare. </s> </p> <p type="main"> <s id="s.001161">Sint enim muri vtrinque AC, BD. </s> <s id="s.001162">Ducatur hori<lb></lb>zonti æquidiſtans CD, iuxta quam lateres lapideſue non <lb></lb>cuneati, ſeriatim collocentur EF. </s> <s id="s.001163">Dicimus amoto arma<pb xlink:href="007/01/125.jpg"></pb><figure id="id.007.01.125.1.jpg" xlink:href="007/01/125/1.jpg"></figure><lb></lb>mento, hoc eſt, pro<lb></lb>hibente ipſo lateres <lb></lb>ruere. </s> <s id="s.001164">Producantur <lb></lb>enim AC in G, BD <lb></lb>verò in H, cum ipſis <lb></lb>CG, DH, æquales <lb></lb>fiant CI, DK, & recta <lb></lb>IK iungatur, erit igi<lb></lb>tur GD ſpatium ipſi <lb></lb>CK ſpatio ſimile qui<lb></lb>dem & æquale, quod <lb></lb>cùm ita ſit, nihil prohibet quin tota laterum GD moles in <lb></lb>ſpatium CK transferatur, & corruat. </s> </p> <p type="main"> <s id="s.001165">Si autem cunei ipſi latereſue, cuneatim diſpoſiti, ita <lb></lb>ſint vt ad vnum centrum tendant, licet ad rectam lineam <lb></lb>collocentur, non delabentur, ſed ſtabunt; quod ita oſten<lb></lb>demus. </s> </p> <figure id="id.007.01.125.2.jpg" xlink:href="007/01/125/2.jpg"></figure> <p type="main"> <s id="s.001166">Sint cunei latereſue <lb></lb>cuneatim diſpoſiti ABCD, <lb></lb>tendentes ad centrum, ſeu <lb></lb>commune punctum E, Du<lb></lb>cantur CAE, DBE, ſintqueue <lb></lb>muri vtrinque ponderi reſi<lb></lb>ſtentes CL, DM, Demitta<lb></lb>tur perpendicularis, quæ ad <lb></lb>mundi centrum FGE ſecans AB, in G. </s> <s id="s.001167">Tum fiat GK aequa<lb></lb>lis GF & per K ipſi AGB parallela ducatur, HKI claudens <lb></lb>ſpatium AHIB. </s> <s id="s.001168">Quoniam igitur vt EC, ad EA, ita CD ad <lb></lb>AB per 4. propoſ. lib. 6. maior erit CD ipſa AB, & eâdem <lb></lb>de cauſſa maior AB, ipſa HI, & idcirco maius ABDC ſpa<lb></lb>tium, ſpatio AHIB. </s> <s id="s.001169">Non igitur poteſt linea CD, fieri in <lb></lb>AB, neque AB, in HI, neque ſpatium totum CABD, tranſ<lb></lb>ferri in ſpatium AHIB non data (quod naturæ ipſi repu<pb xlink:href="007/01/126.jpg"></pb>gnat) corporum penetratione. </s> <s id="s.001170">Stabunt ergo cunei, quod <lb></lb>fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.001171">Verum enimuero, debilis hæc ſtructura eſt, & eo de<lb></lb>bilior, quo vani latitudo fuerit maior, cuneorum verò al<lb></lb>titudo minor. </s> <s id="s.001172">Idem enim patitur quod epiſtylia in ſpecie <lb></lb>Aræosſtyla, quæ, vt ſcribit Vitruuius lib. 3. c. 2. propter in<lb></lb>teruallorum magnitudinem franguntur. </s> <s id="s.001173">Id quoque ha<lb></lb>bet vitij, quod cunei ita diſpoſiti ſuo pondere incumbas <lb></lb>vtrinque violentiſſimè pellant. </s> <s id="s.001174">Vtilis tamen eſſe poteſt <lb></lb>ad portarum & feneſtrarum, quæ in medijs muris ſunt, & <lb></lb>mediocri vano aperiuntur, ſuperliminaria. </s> </p> <p type="main"> <s id="s.001175">Si verò ad minorem circuli portionem curuetur Ca<lb></lb>mera, vtilior quidem erit ſtructura ea ipſa, de qua locuti <lb></lb>ſumus; non tamen omninò ſine vitio. </s> </p> <figure id="id.007.01.126.1.jpg" xlink:href="007/01/126/1.jpg"></figure> <p type="main"> <s id="s.001176">Eſto fornix ex minori <lb></lb>circuli portione AB, cuius in<lb></lb>cumbæ AF, BH muris fultæ <lb></lb>AC, BD. </s> <s id="s.001177">Conſtet autem vel <lb></lb>ex lapidibus cuneatis, vel ex <lb></lb>coctilibus lateribus ad E <expan abbr="cē-trum">cen<lb></lb>trum</expan> tendentibus. </s> <s id="s.001178">Sitque; for<lb></lb>nicis linea exterior FGH, in<lb></lb>terior AIB. </s> <s id="s.001179">Ducantur EA, <lb></lb>ED, & producantur in M, N. <lb></lb></s> <s id="s.001180">Quoniam igitur vt EM ad EA, ita MGN ad AIB, maior e<lb></lb>rit MGN linea ipſa AIB, quamobrem fieri non poteſt vt <lb></lb>aptetur lineæ AIB, & in eius locum deſcendat. </s> <s id="s.001181">Stabit igi<lb></lb>tur, incumbis vtrinque non cedentibus. </s> <s id="s.001182">Validè autem <lb></lb>ſpeciem hanc, loca quibus incumbit, propellere, ita o<lb></lb>ſtendemus. </s> </p> <p type="main"> <s id="s.001183">Producatur in eadem figura CA in K, & DB in L. <lb></lb></s> <s id="s.001184">Partes igitur quæ muris ad perpendiculum fulciuntur, <lb></lb>ſunt AKF, BLH, minimæ illæ quidem, maxima verò pars <pb xlink:href="007/01/127.jpg"></pb>eſt extra fulcimenta, nempe tota AKLB quæ idcircó ſuo<lb></lb>pte pondere deorſum vergens & in incumbas <expan abbr="vtrinq;">vtrinque</expan> pel<lb></lb>lens aperitur, & facillimè vitium facit. </s> <s id="s.001185">Eiuſdem ferè na<lb></lb>turæ ea ſpecies eſt, quæ vel ex media, vel ex minori ellipſis <lb></lb>ſecundum maiorem diametrum fit ſegmento. </s> <s id="s.001186">Vtilior ta<lb></lb>men hæc eſt, præcipuè circa incumbas, propterea quod <lb></lb>partes habeat erectiores, & circulari illa de qua egimus, <lb></lb>magis fultas. </s> <s id="s.001187">circa medium autem poteſt videri debilior, <lb></lb>quippe quod ellipſis ibi circulo curuetur minus. </s> </p> <p type="main"> <s id="s.001188">Ea verò forma, qua mirum in modum delectati ſunt <lb></lb>Barbari, qui declinante imperio Italiam inuaſerunt, & <lb></lb>bonam emendatiſſimamqueue antiquorum ædificandi ra<lb></lb>tionem deturparunt, ex duobus conſtat circuli portioni<lb></lb>bus, quamobrem Albertus lib. 3. hoſce arcus, compoſitos, <lb></lb>appellat. </s> <s id="s.001189">Circinantur autem hoc pacto, diuiſa nempe <lb></lb>ſubtenſa, in partes tres, eaſque æquales, ponitur circini <lb></lb>pes in altero diuiſionum puncto & pars circuli deſcribi<lb></lb>tur, mox in altero puncto circini pede collocato alia cir<lb></lb>culi portio lineatur, quibus arcus ipſe integratur. </s> <s id="s.001190">Appel<lb></lb>lant autem tertium acutum, eo quod ex ſubtenſa in tres <lb></lb>partes diuiſa, arcus non fiat rotundus, ſed in acutum an<lb></lb>gulum ex duabus circuli portionibus deſinens. </s> </p> <figure id="id.007.01.127.1.jpg" xlink:href="007/01/127/1.jpg"></figure> <p type="main"> <s id="s.001191">Sint igitur muri <lb></lb>AC, BD, in quibus v<lb></lb>trinque incumbæ KA, <lb></lb>BI. </s> <s id="s.001192">Ducatur itaque ſub<lb></lb>tenſa horizonti æquidi<lb></lb>ſtans AP, quæ in tres æ<lb></lb>quales partes diuidatur <lb></lb>punctis E, F, tum centris <lb></lb>EF, circulorum portio<lb></lb>nes deſcribantur hinc <lb></lb>AG, HK, inde verò BG, <pb xlink:href="007/01/128.jpg"></pb>IH, ex quibus arcus totus integratur. </s> <s id="s.001193">Vtilis hæc quidem <lb></lb>ſpecies eſt, licet inuenuſta, propterea quod haud violen<lb></lb>ter incumbas vtrinque repellat, & in ſummo magnis ſuſti<lb></lb>nendis oneribus ſit apta. </s> <s id="s.001194">Producantur CH in N, DB verò <lb></lb>in O, ſitqueue centrum grauitatis AG in L, partis vero BG <lb></lb>in M. </s> <s id="s.001195">Quoniam igitur centra hæc ob elatam portionum <lb></lb>conſtitutionem quam proxima lineis AN, BO, fulcimen<lb></lb>torum fiunt, maximè <expan abbr="ſuſtinētur">ſuſtinentur</expan>, & deorſum potius quam <lb></lb>lateraliter incumbas ipſas premunt. </s> <s id="s.001196">Si quid tamen <expan abbr="habēt">habent</expan> <lb></lb>vitij, illud eſt quod grauitatis centra momentum haben<lb></lb>tia ad interiorem partem verſus PQ vim faciant, & niſi <lb></lb>partes magno ſuperimpoſito pondere comprimantur, <lb></lb>partes quæ ſunt circa HG, ſurſum pellentes aliquali ſibi <lb></lb>rectitudine comparata corruunt, facta nempe circa L, M, <lb></lb>coniunctarum partium ſeparatione. </s> </p> <p type="main"> <s id="s.001197">His hoc pacto explicatis de ſemicirculari fornice a<lb></lb>gemus, quæ cæteris omnibus vtilior eſt, & longè pulcher<lb></lb>rima, quamobrem Antiquis Architectis omnibus inpri<lb></lb>mis admodum familiaris: </s> </p> <figure id="id.007.01.128.1.jpg" xlink:href="007/01/128/1.jpg"></figure> <p type="main"> <s id="s.001198">Eſto vanum <lb></lb>ABCD, muris v<lb></lb>trinque clauſum. <lb></lb></s> <s id="s.001199">Ducatur per <expan abbr="sū-mitates">sum<lb></lb>mitates</expan> <expan abbr="murorū">murorum</expan> <lb></lb>horizonti æqui<lb></lb>diſtans recta AD, <lb></lb>hac bifariam ſe<lb></lb>cta in E, eodem <lb></lb>centro E, ſpatio <lb></lb>verò EA ſemicir<lb></lb>culus deſcribatur <lb></lb>AFD, concaua <lb></lb>nempe ipſius for-<pb xlink:href="007/01/129.jpg"></pb>nicis pars; tum eodem centro, ſpatio verò EG, circinetur <lb></lb>GHI eiuſdem fornicis pars conuexa. </s> <s id="s.001200">Poſt hæc productis <lb></lb>lineis BH, CD, in OP, ſecetur fornix tota in tres æquales <lb></lb>partes AGKM, MNLK, NDIL, & KME, LNE iungantur, <lb></lb>ſint autem partium ipſarum grauitatis centra QRS. </s> <s id="s.001201">Eſt <lb></lb>autem R in ipſa perpendiculari HE. </s> <s id="s.001202">Quoniam igitur <lb></lb>partium AGKM, DILN, quæ <expan abbr="vtrinq;">vtrinque</expan> ſunt grauitatis cen<lb></lb>tra QS, in ipſis ſunt fulcimentorum lineis OH PD, ſuâ <lb></lb>ſponte fulcimentis eas ſuſtinentibus partes ipſæ ſtabunt. <lb></lb></s> <s id="s.001203">Pars autem media KMNL deorſum vergente per ipſam <lb></lb>HE lineam grauitatis centro, ſi parumper vel incumbæ <lb></lb>vel partes vtrinque AG<emph type="italics"></emph>K<emph.end type="italics"></emph.end>M, DILN cedant, vtpote quæ à <lb></lb>fulcimentis eſt remotiſſima, magno impetu ſuopte pon<lb></lb>dere deorſum feretur. </s> <s id="s.001204">quæ igitur in his ſemicircularibus <lb></lb>fornicibus partes ſtabiliores ſint, quæ verò caſibus obno<lb></lb>xiæ, ex his quæ diximus, clarè patet. </s> </p> <p type="main"> <s id="s.001205">Cæterùm cur incumbis manentibus fornix ſtet, ea <lb></lb>cauſſa eſt, quod partes exteriores G<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, <emph type="italics"></emph>K<emph.end type="italics"></emph.end>L, LI, maiores ſint <lb></lb>in ferioribus & oppoſitis AM, MN, NG; quod ſuprà de<lb></lb>monſtrauimus. </s> </p> <p type="main"> <s id="s.001206">Si quid autem vitij in hac ſpecie eſt, illud quidem <lb></lb>eſt, quod ſumma pars <emph type="italics"></emph>K<emph.end type="italics"></emph.end>MNL deorſum vergens magnâ vi <lb></lb>partes, quæ vtrinque ſunt, repellat, ex qua re ſolidarum <lb></lb>partium fit ſolutio, & inde ruina. </s> </p> <p type="main"> <s id="s.001207">Huic difficultati vt occurrerent peritiores Archite<lb></lb>cti, plura excogitârunt remedia. </s> <s id="s.001208">Primum enim parietes <lb></lb>hinc inde ita ſolidos, craſſos & firmos faciunt, vt ſuapte vi <lb></lb>reſiſtentes dimoueri loco nequeant, vel paraſtatas <expan abbr="addūt">addunt</expan> <lb></lb>vt in figura TX, VY. </s> <s id="s.001209">Præterea & ferrea claui ex incumba <lb></lb>in incumbam ducta & vtrinque firmata contrarias partes <lb></lb>validiſſimè connectunt, quæ calcitrantes (ita enim lo<lb></lb>quuntur noſtrates <emph type="italics"></emph>A<emph.end type="italics"></emph.end>rchitecti,) fornicis pedes cohibent, & <lb></lb>ſolidum ne ſoluatur impediunt. </s> <s id="s.001210">qua in ſpecie dubitan<expan abbr="dū">dum</expan> <pb xlink:href="007/01/130.jpg"></pb>eſſet, an optimo loco ſit a ſit clauis, quæ per centrum? </s> <s id="s.001211">Et <lb></lb>ſanè videtur, quippe quod circa incumbas impetus fiat <lb></lb>maior. </s> <s id="s.001212">Ego autem vtilius ibi poni arbitror, vbi <expan abbr="punctaq.">punctaque</expan> <lb></lb>5. hoc eſt, in medio tertiarum illarum partium, quæ vtrin<lb></lb>que incumbis inſiſtunt, propterea quod primus impulſus <lb></lb>ex media parte quæ impendet, ibi fiat. </s> <s id="s.001213">Rarò tamen boni <lb></lb>Architecti eo loco aptare ſolent, eo quòd eiuſmodi cla<lb></lb>ues vel pulcherrimis ædificijs minuant gratiam. </s> <s id="s.001214">Vnde fit <lb></lb>vt nunquam ſatis laudetur Lucianus ille Benuerardus <lb></lb>Lauranenſis Dalmata, qui nullibi apparentes eas poſuit <lb></lb>in admirabili illa Vrbini Aula, quam Federico Feltrio, fe<lb></lb>liciſſimo æquè & inuictiſſimo Duci, ædificauit. </s> </p> <p type="main"> <s id="s.001215">Tertio denique modo huic infirmitati me dentur, <lb></lb>vt videre eſt in ſequenti figura, in qua vanum ADBC, mu<lb></lb>ri vtrinque AF, BH, fornix verò FGH. </s> <s id="s.001216">Itaque dum muros <lb></lb><figure id="id.007.01.130.1.jpg" xlink:href="007/01/130/1.jpg"></figure><lb></lb>exſtruunt, arre<lb></lb>ctarias trabes, ro<lb></lb>bore aliaue mate<lb></lb>ria firmiſſima, illis <lb></lb>inſerunt, quales <lb></lb>ſunt IF<emph type="italics"></emph>K<emph.end type="italics"></emph.end> LHM, <lb></lb>ea proceritate vt <lb></lb>futuri fornicis ſu<lb></lb>perent ſummita<lb></lb>tem. </s> <s id="s.001217">Conſumma<lb></lb>to enim fornice, <lb></lb>nondum tamen, <lb></lb>exarmato, tranſ<lb></lb>uerſariam <expan abbr="trabē">trabem</expan> à <lb></lb>ſummo fornicis <lb></lb>dorſo parumper <lb></lb>eminentem in punctis I, L, arrectarijs trabibus validiſſi<lb></lb>mis clauibus connectunt, tum punctis NP, Oq, capreolos <pb xlink:href="007/01/131.jpg"></pb>tranſuerſario, & arrectarijs ferreis, clauis affigunt. </s> <s id="s.001218">Qui<lb></lb>bus ita concinnatis, facta fornicis validâ preſſione in G, <lb></lb>incumbiſque F, H, ad exteriora repulſis, AB ſpatium non <lb></lb>fit maius. </s> <s id="s.001219">Repulſis enim incumbis & muros propelli ne<lb></lb>ceſſe eſt, & cum muris ipſas inſertas trabes, I<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, LM. </s> <s id="s.001220">At va<lb></lb>ricari non poſſunt, nî ſecum trahant puncta PQ, quod fie<lb></lb>ri non poteſt, propterea quod in punctis N, O, validè diſ<lb></lb>tineantur. </s> <s id="s.001221">Itaque ſpatio AB non dilatato nulla fit ipſius <lb></lb>fornicis diſſolutio, quod vtique à principio ceu propoſi<lb></lb>tus finis quærebatur. </s> <s id="s.001222">Sed dicet quiſpiam, Nonne pende<lb></lb>bit tranſuerſaria trabs in ipſa diſtractione arrectariorum, <lb></lb>preſſa in punctis N, O? aut parum dicimus, aut nihil. </s> <s id="s.001223">Cum <lb></lb>enim PQ proxima ſint punctis FH, quæ cum arrectarijs à <lb></lb>muro diſtinentur, magna in ijs fit vtrobique reſiſtentia. </s> </p> <p type="main"> <s id="s.001224">Rebus igitur ita ſe habentibus cum obſeruaſſent Ar<lb></lb>chitecti, ob enormitatem ponderis fornices in tertia illa <lb></lb><figure id="id.007.01.131.1.jpg" xlink:href="007/01/131/1.jpg"></figure><lb></lb>parte quæ ſumma eſt <lb></lb>laborare, <expan abbr="quãtum">quantum</expan> ter<lb></lb>tijs vtrinque partibus <lb></lb>ſoliditatis addunt, tan<lb></lb>tundem ex illa parte <lb></lb>ſuprema demere <expan abbr="ſolēt">ſolent</expan>, <lb></lb>vt videre eſt in ſubie<lb></lb>cta figura, in qua par<lb></lb>tes A, B, ſolidæ & craſ<lb></lb>ſiores, quibus hærent <lb></lb>partes, quæ CE, DG <lb></lb>craſſæ quidem & illæ, <lb></lb>tum vero ſumma EFG, <lb></lb>alijs ſubtilior. </s> <s id="s.001225">Minus <lb></lb>igitur grauante ponde<lb></lb>re in F, minor fit ad incumbas preſſio, aut ſi qua fit, à <expan abbr="partiū">partium</expan> <lb></lb>ACE, BDG ſoliditate haud inualidè ſuſtinetur. </s> </p> <pb xlink:href="007/01/132.jpg"></pb> <p type="main"> <s id="s.001226">Cæterùm admonet nos locus, vt aliquid de forni<lb></lb>cum diſſolutionibus in medium afferamus: cauſſis enim <lb></lb>morborum cognitis, facilius periti medici adhibere ſo<lb></lb>lent remedia. </s> </p> <figure id="id.007.01.132.1.jpg" xlink:href="007/01/132/1.jpg"></figure> <p type="main"> <s id="s.001227">Eſto enim ſemicircula<lb></lb>ris fornix ABC, cuius cen<lb></lb>trum E, perpendicularis ve<lb></lb>rò quæ per centrum DBE, ſe<lb></lb>micirculi ABC, diameter <lb></lb>AEC, incumbæ <expan abbr="vtrinq;">vtrinque</expan> A, C. <lb></lb></s> <s id="s.001228">Itaque ſi nulla fiat incumba<lb></lb>rum repulſio, ſtabit fornix; ſi verò fiat, ruinam faciet. </s> </p> <p type="main"> <s id="s.001229">Pellantur itaque ad exteriores partes, vt in ſecunda <lb></lb><figure id="id.007.01.132.2.jpg" xlink:href="007/01/132/2.jpg"></figure><lb></lb>figura, H in F, & C in G, <lb></lb>ex qua pulſione cum ma<lb></lb>ius fiat ſpatium quod in<lb></lb>tegro fornice impleba<lb></lb>tur, iam diſtractis <expan abbr="vtrinq;">vtrinque</expan> <lb></lb>fornicis partibus <expan abbr="nō">non</expan> im<lb></lb>pletur, Diuiditur igitur <lb></lb>locus maior factus in tres partes, quarum hinc inde duas <lb></lb>replent fornicis partes, tertiam verò quæ media eſt, re<lb></lb>plet inſertus, ne vacuum detur, aër, vt in figura videre eſt, <lb></lb>in qua ſolutæ vtrinque fornicis partes HIKF, PMNG, aër <lb></lb>autem medius ſpatium replens IKMN. </s> <s id="s.001230">Diuidantur ſin<lb></lb>guli quadrantes FK, GN, in partes tres, quarum duæ ſint <lb></lb>hinc inde FQ, GR, & à centris, quæ ſeparatis quadranti<lb></lb>bus facta ſunt in ST, rectæ ducantur SQV. TRX. </s> <s id="s.001231">Quo<lb></lb>niam igitur tertiæ partes vtrinque VIKQ MNRX pro<lb></lb>pria grauitate depreſſæ, nullum quo ſuſtineantur fulci<lb></lb>mentum habent, corruent quidem. </s> <s id="s.001232">Ducantur autem re<lb></lb>ctæ QI, RM, conſtituentes cum ipſis QV, RX pares an<lb></lb>gulos VQI MRX. </s> <s id="s.001233">Itaque centris QR partes QIRM ad <pb xlink:href="007/01/133.jpg"></pb>inferiores partes deuoluentur, fientqueue QI, RM, vbi QZ, <lb></lb>RZ. </s> <s id="s.001234">Si autem QI, RM perpendicularibus quæ à punctis <lb></lb>QR ad perpendicularem DE ducuntur, fuerint maiores <lb></lb>conuenient alicubi in ipſa perpendiculari, & altera alte<lb></lb>ram ſuſtinebit; ſi autem æquales tangent ſe & nihilomi<lb></lb>nus fiet ruina, ſi minores nec ſe inuicem tangent, & nullà <lb></lb>re prohibente deorſum corruent. </s> <s id="s.001235">tangant autem ſe in <expan abbr="pū-cto">pun<lb></lb>cto</expan> Z. quo pacto igitur fornices incumbis cedentibus in <lb></lb>medio aperti, <expan abbr="diſſoluãtur">diſſoluantur</expan> & ruinam faciant, ex iſtis patet. </s> </p> <p type="main"> <s id="s.001236">Ex demonſtratis quaſi ex conſectario habemus for<lb></lb>nices quo fuerint craſſiores dato pari incumbarum ſeceſ<lb></lb>ſu, ruinæ minus eſſe obnoxios quàm tenuiores, hoc eſt, <lb></lb>maiori aperitione indigere ad ruinam craſſiores quam te<lb></lb>nuiores, quod licet ex iam dictis reſultet, nos tamen cla<lb></lb>rius ex ſubiecto ſchemate demonſtrabimus. </s> </p> <figure id="id.007.01.133.1.jpg" xlink:href="007/01/133/1.jpg"></figure> <p type="main"> <s id="s.001237">Eſto enim craſſioris <lb></lb>fornicis pars <expan abbr="quidē">quidem</expan> ABCD, <lb></lb>tenuioris EFCD circa <expan abbr="idē">idem</expan> <lb></lb>centrum R. </s> <s id="s.001238">Ducatur au<lb></lb>tem RM, ſecans CD in G. <lb></lb>EF in H AB, in M. </s> <s id="s.001239">Centro <lb></lb>igitur G fiet euerſio portio<lb></lb>num fornicum, MD, HD, <lb></lb>Ducantur GA, GE & producta AD in N ipſi AN perpen<lb></lb>dicularis ducatur GN. quoniam igitur GE cadit in trian<lb></lb>gulo AGN erit ex 21. propoſ. lib. 1. elem. GA, maior GE. <lb></lb></s> <s id="s.001240">Corruente igitur maioris fornicis portione MD, recta <lb></lb>GA centro G punctum A deſcribet portionem AI, mino<lb></lb>ris interim ex GE, deſcribente EL, at cadenti angulo A <lb></lb>occurrit in perpendiculari IK in puncto I angulus oppo<lb></lb>ſitæ portionis, O, ipſi autem E cadenti per EL non occur<lb></lb>ret punctum P, cadens per Pq eo quod neutrum eorum <lb></lb>pertingat ad perpendicularem IK. Tenuioris ergo forni<pb xlink:href="007/01/134.jpg"></pb>cis partes è ſuis locis auulſæ ex eadem aperitione ruinam <lb></lb>facient, quod non contingit partibus craſſioris. </s> <s id="s.001241">quod ſa<lb></lb>nè fuerat de clarandum. </s> </p> <p type="main"> <s id="s.001242">Quæritur adhuc, quare grauiores fornices in ſum<lb></lb>mis ædificijs non ſine vitio fiant? </s> </p> <p type="main"> <s id="s.001243">Eſto ædificium ABGH, cuius <expan abbr="vtrinq;">vtrinque</expan> muri ABCD, <lb></lb>EFGH, maiorum ſummitates AD, EH, mediæ murorum <lb></lb>partes KL, fornicum ſummus quidem DIE, medius verò <lb></lb><figure id="id.007.01.134.1.jpg" xlink:href="007/01/134/1.jpg"></figure><lb></lb>KML. Dico, magis cedere pul<lb></lb>ſos muros ſummos circa DE, <lb></lb>quam in medio circa KL. </s> <s id="s.001244">Sunt <lb></lb>enim muri BA, GH ceu vectes <lb></lb>quidam, <expan abbr="quorū">quorum</expan> extremis par<lb></lb>tibus à fulcimentis BG remo<lb></lb>tiſſimis potentia admouetur, <lb></lb>hoc eſt, ipſius fornicis DIE ad <lb></lb>DE incumbans repulſio; lon<lb></lb>gior eſt autem pars à <expan abbr="fulcimē-to">fulcimen<lb></lb>to</expan> ad potentiam AB, ipſa BK. <lb></lb></s> <s id="s.001245">Data igitur paritate potentia<lb></lb>rum plus operabitur ea quæ in <lb></lb>D, illa quæ K. facilius ergo re<lb></lb>pellentur muri in DE quàm in <lb></lb>KL. </s> <s id="s.001246">Alia quoque ratio intercedit, ſiquidem pondus muri <lb></lb>ſuperioris ADK, premens inferiorem murum KBC, cum <lb></lb>ſua grauitate firmiorem, & pulſionibus minus obnoxium <lb></lb>reddit. </s> <s id="s.001247">Difficilius enim propellitur id quod graue eſt <expan abbr="quã">quam</expan> <lb></lb>quod leue, vt nos quæſtione 10. demonſtrauimus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001248">QVÆSTIO XVII.</s> </p> <p type="head"> <s id="s.001249"><emph type="italics"></emph>Quærit Ariſtoteles, Cur paruo exiſtente cuneo magna ſcindantur <lb></lb>pondera & corporum moles, validaque, fiat impreſſio?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001250">In parua re magnum negotium. </s> <s id="s.001251">Etenim quæſtio hæc <pb xlink:href="007/01/135.jpg"></pb>clariſſimorum virorum ingenia magnopere fatigauit. </s> <s id="s.001252">Ex <lb></lb>quibus Ariſtoteles inter veteres, Guid. Vbald. inter re<lb></lb>centiores ad vectis naturam (ne quid in Mechanicis ad <lb></lb>vectem non reduci putaretur) cuneum ipſum trahere co<lb></lb><figure id="id.007.01.135.1.jpg" xlink:href="007/01/135/1.jpg"></figure><lb></lb>nati ſunt. </s> <s id="s.001253">Nos autem pro <lb></lb>veritate certantes, ſi in <lb></lb>horum ſententiam vltrò <lb></lb>non tranſierimus, multa <lb></lb>venia digni à non iniquo <lb></lb>iudice exiſtimabimur. </s> <s id="s.001254">A<lb></lb>riſtotelis mentem clarè <lb></lb>& fusè explicat G. V<lb></lb>bald. in Mechan. vbi de <lb></lb>Cuneo peculiariter a<lb></lb>git. </s> </p> <p type="main"> <s id="s.001255">Eſto igitur ſcindendum quippiam ABCD, Cuneus <lb></lb>EFG, cuius pars HFI ſciſſuræ inſerta HI, facta igitur vali<lb></lb>da percuſſione in EG, fiet vt cum EG fuerit in NO, H ſit v<lb></lb>bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit <lb></lb>ſciſſio NSO, toti nempe cuneo EFG, æqualis. </s> <s id="s.001256">Vult igitur <lb></lb>Ariſtoteles, duos in cuneo vectes conſiderari EF, GF, quo<lb></lb>rum alterius, nempe EF, fulcimentum ſit in H, pondus ve<lb></lb>ro in F; alterius autem, hoc eſt, GF fulcimentum quidem <lb></lb>ſit in I, pondus verò itidem ſit in F. </s> <s id="s.001257">His nequaquam con<lb></lb>ſentiens G. Vbald. aliam viam ingreditur. </s> <s id="s.001258">Ait enim EHF <lb></lb>vectes quidem eſſe, quorum commune fulcimentum F, <lb></lb>potentias verò mouentes in EG. </s> <s id="s.001259">Pondera vtrinque inter <lb></lb>fulcimenta & potentias, vbi HI, idemque; eſſe ac ſi EF, GF, <lb></lb>ſeorſum à cuneo conſiderati in puncto F, adinuicem fulti <lb></lb>atque diſtracti pondera pellerent H in NP, I verò in O, <lb></lb>Q.</s> <s id="s.001260"> Verum enimuerò quoniam cunei angulus non muta<lb></lb>tur, nec vertex ipſe centri vllum prorſus præbet vſum, nec <lb></lb>eius latera vtrinque diſtracta ad contrarias partes didu<pb xlink:href="007/01/136.jpg"></pb>cuntur, vectes in cuneo hoc pacto conſiderare videtur à <lb></lb>veritate alienum. </s> <s id="s.001261">Ariſtotelis autem ſolutionem falſam eſ<lb></lb>ſe, clarè patet. </s> <s id="s.001262">quo pacto enim F pellet ex fulcimento H i<lb></lb>pſam ligni partem OS, & idem F ex fulcimento I pellet <lb></lb>oppoſitam partem NS, ſi inuicem contendentes extremæ <lb></lb>vectium partes in F, altera alteri ne quicquam operentur, <lb></lb>eſt impedimento? </s> <s id="s.001263">Et ſanè opinionis falſitas inde patet, <lb></lb>quòd videamus materiæ partes ſciſſas, in ipſo ſciſſionis a<lb></lb>ctu facta diſtractione à cunei vertice nequaquam tangi. <lb></lb></s> <s id="s.001264">At eiuſmodi operationes per contactum fieri nulli eſt i<lb></lb>gnotum. </s> <s id="s.001265">Solutio igitur iſta meo iudicio, tanto Philoſo<lb></lb>pho prorſus videtur indigna. </s> </p> <p type="main"> <s id="s.001266">Porrò G. Vbald. ijs quæ de diuaricatis vectibus in <lb></lb>medium adduxerat non acquieſcens alias quærit cauſſas, <lb></lb>cur cuneus minoris anguli validiùs ſcindat. </s> <s id="s.001267">Idque; ex quo<lb></lb>dam lemmate demonſtrare conatur, figura autem eius ita <lb></lb>ferè ſe habet. </s> </p> <figure id="id.007.01.136.1.jpg" xlink:href="007/01/136/1.jpg"></figure> <p type="main"> <s id="s.001268">Eſto cuneus ABC, <lb></lb>item alius DEF. <expan abbr="Demō-ſtrauit">Demon<lb></lb>ſtrauit</expan> igitur ex aſſum<lb></lb>pto, quo acutior fuerit <lb></lb>angulus BIM, eo facilius <lb></lb>pondera moueri, & ideo <lb></lb>facilius ceu vecte AB <lb></lb>moueri pondus I quàm <lb></lb>vecte DE pondus Q.</s> <s id="s.001269"> In<lb></lb>geniosè quidem. </s> <s id="s.001270">At ma<lb></lb>gnam hæc apud me ha<lb></lb>bent difficultatem. </s> <s id="s.001271">Si e<lb></lb>nim ita ſe habet AB, ad BI, vt DE, ad EQ (ipſæ enim DE, <lb></lb>EQ ſupponuntur æquales) ergo eadem æqualiſue poten<lb></lb>tia æqualiter mouebit pondera I & Q.</s> <s id="s.001272"> quod ipſi eiuſdem <lb></lb>demonſtrationi prorſus concludit contrarium. </s> <s id="s.001273">Nec meo <pb xlink:href="007/01/137.jpg"></pb>quidem iudicio id ſequi videtur, propterea quod ex Pap<lb></lb>po ea quæ in planis inclinatis mouentur, redigantur ad li<lb></lb>bram. </s> <s id="s.001274">Ratio enim valde eſt diuerſa, ſiquidem pondera <lb></lb>quæ in planis inclinatis mouentur, certa habent fulci<lb></lb>menta & determinatas tum brachiorum tum ponderum <lb></lb>proportiones, quæ omnia in cuneo, nec quidem mente <lb></lb>concipi poſſe, clarè patet. </s> </p> <p type="main"> <s id="s.001275">His igitur difficultatibus conſideratis, Nos cunei <lb></lb>vim, ad alia eſſe principia referendam pro comperto ha<lb></lb>bemus. </s> <s id="s.001276">Ordimur igitur hoc pacto. </s> <s id="s.001277">Cuneo quidem res di<lb></lb>uidi certum eſt. </s> <s id="s.001278">Cæterùm quæ natura diuidere apta ſunt, <lb></lb>tria ſunt, punctum, linea, ſuperficies, Puncto enim linea, <lb></lb>lineâ ſuperficies, ſuperficie autem corpus ipſum diuidi<lb></lb>tur. </s> <s id="s.001279">quæ omnia à Mathematico abſque materia conſide<lb></lb>rantur. </s> <s id="s.001280">De diuiſione autem quæ fit ex puncto, nihil agit <lb></lb>Mechanicus, qui corporibus quidem vtitur, ad cuius na<lb></lb>turam non trahitur punctum, cuius partes ſunt nullæ. </s> <s id="s.001281">At <lb></lb>non lineis & ſuperficiebus modò corpora diuiduntur, ſed <lb></lb>etiam corporibus, quod verum eſt, at ea corpora ad linea<lb></lb>rum & ſuperficierum naturam quodammodo aptari faci<lb></lb>lè docebimus. </s> <s id="s.001282">Dicimus igitur, duplicem eſſe Cuneorum <lb></lb>ſpeciem, linearem vnam, ſuperficialem alteram. </s> <s id="s.001283">linearem <lb></lb>appello, quæ ad lineæ naturam magnopere accedit. </s> <s id="s.001284">Tales <lb></lb>ſunt orbiculares illæ cuſpides, quibus ad perforandum v<lb></lb>timur, & ideo vernaculè Pantirolos vocamus. </s> <s id="s.001285">Acus item <lb></lb>ſutorij, & cætera quæ non ſecus ac linea in punctum deſi<lb></lb>nunt, & imaginariam quandam lineam ceu axem in eo <lb></lb>puncto deſinentem continent. </s> <s id="s.001286">Ad lineam quoque refe<lb></lb>runtur lateratæ cuſpides oblongæ, & ſubtiles ceu ſubulæ, <lb></lb>claui, enſes, pugiones, & his ſimilia, quæ cum adacta vali<lb></lb>dam faciant partium ſeparationem ad cunei naturam <expan abbr="nō">non</expan> <lb></lb>referre magnæ videretur dementiæ. </s> <s id="s.001287">Et tunc quanto ma<lb></lb>gis corpora hæc ad linearem naturam accedunt, eo ma<pb xlink:href="007/01/138.jpg"></pb>gis penetrant. </s> <s id="s.001288">Sed & hoc idem in rebus non ab arte, ſed <lb></lb>ab ipſa natura productis facile eſt cognoſcere. </s> <s id="s.001289">Quis enim <lb></lb>non experitur, quàm validè culex, infirmiſſimum animal, <lb></lb>& ea paruitate qua eſt, hominum & cæterorum <expan abbr="animaliū">animalium</expan>, <lb></lb>cutes aculeata proboſcide penetret? </s> <s id="s.001290">Id vtique non alia de <lb></lb>cauſſa fit, quod ad imaginariæ lineæ ſubtilitatem quam, <lb></lb>proximè accedat. </s> <s id="s.001291">Veſpæ quoque, Apes, Scorpiones a<lb></lb>culeis iſtis ceu linearibus cuneis vtuntur. </s> <s id="s.001292">Nec refert, vt <lb></lb>diximus, vtrum laterati ſint, ceu ſubulæ, & claui, vel ro<lb></lb>tundi & vtrum plura paucioraue latera habeant, dummo<lb></lb>do in punctum & aculeatam aciem deſinant. </s> <s id="s.001293">Altera por<lb></lb>ro cuneorum ſpecies ſuperficiei naturam ſapit, acie ſiqui<lb></lb>dem in lineam deſinit, quæ ſuperficiei eſt terminus, <expan abbr="quã">quam</expan>obrem huc ea omnia referuntur, quæ acie ipsâ ſcindunt, <lb></lb>ceu ſunt cunei propriè dicti, de quibus hoc loco eſt ſer<lb></lb>mo, cultra, enſes, aſciæ, ſecures, ſcalpra lata, & cætera e<lb></lb>iuſmodi, quibus corpora acie ſcinduntur. </s> <s id="s.001294">Quidam his ad<lb></lb>dunt ſerras, quibus haud prorſus aſſentimur. </s> <s id="s.001295">Etenim alia <lb></lb>ratione diuidunt, ſicut & limæ ſolent, deterendo enim, <expan abbr="nō">non</expan> <lb></lb>ſcindendo ferri, ligni, & marmorum duritiem diuidunt & <lb></lb>domant. </s> <s id="s.001296">His igitur <expan abbr="cōſideratis">conſideratis</expan>, ſi daretur ex materia qua<lb></lb>piam in frangibili cuneus, qui maximè ad ſuperficiei natu<lb></lb>ram accederet, vel paruo labore tenaciſſima ligna validiſ<lb></lb>ſimè ſcinderet, & ideo optimè res gladijs illis diuiditur, <lb></lb>qui magis ad ſuperficiei naturam accedunt. </s> <s id="s.001297">Ex quibus o<lb></lb>mnibus, nî fallimur, clarè patet, cur acutiores angulo cu<lb></lb>nei obtuſioribus facilius ſcindant, quæ quidem ratio lon<lb></lb>gè ab ea diſtat, ex qua cæteri ferè omnes Cuneum ad ve<lb></lb>ctis naturam referre hactenus contenderunt. </s> </p> <p type="main"> <s id="s.001298">Cæterùm vtramque eorum quos diximus, <expan abbr="cuneorū">cuneorum</expan> <lb></lb>ſpeciem ſolertiſſima cognouit Natura, & ideo quoniam <lb></lb>res vel contuſione vel perforatione, vel ſecatione confi<lb></lb>ciuntur, triplicem dentium qualitatem dentatis animali-<pb xlink:href="007/01/139.jpg"></pb><figure id="id.007.01.139.1.jpg" xlink:href="007/01/139/1.jpg"></figure><lb></lb>bus dedit, Molares, <lb></lb>qui & Maxillares ap<lb></lb>pellantur, quibus <lb></lb>cibus contunditur, <lb></lb>Canini, quibus fit <lb></lb>perforatio, Anterio<lb></lb>res, quibus cibus <lb></lb>ſcinditur, quos ideo <lb></lb><foreign lang="grc">τεμνικοὺς</foreign>, id eſt, ſecan<lb></lb>tes appellant Graeci. </s> </p> <p type="main"> <s id="s.001299">Molares KK, <lb></lb>Canini L, L, Temni<lb></lb>ci ſeu ſecantes M. </s> <s id="s.001300">Cuneus orbicularis lineariſqueue AB, in <lb></lb>quo axis linea eſt, ad cuius naturam accedit AB cuneus <lb></lb>ſuperficialis CD, accedens ad ſuperficiei naturam, quam <lb></lb>vitro imaginamur EFGD, in aciem cunei deſinentem, <lb></lb>GD, Lateratus lineariſque cuneus, clauus HI. </s> </p> <p type="main"> <s id="s.001301">Cunei autem omnes dupliciter ſunt efficaces, vel e<lb></lb>nim malleo, vt in ijs fit, quibus lìgna ſcinduntur & ſcalpris <lb></lb>fieri ſolet, adiguntur, vel impulſu & preſſione, vt in gla<lb></lb>dijs fit, pugionibus, cælatorum ſcalpris, ſubulis, & cæteris <lb></lb>eiuſmodi. </s> <s id="s.001302">Quidam etiam ſunt, qui licet mallei ictu non <lb></lb>adigantur, malleum coniunctum habent, ceu ſunt ſecu<lb></lb>res, ligones, Aſciæ, & his ſimilia, quæ ex percuſſione ſe<lb></lb>metipſa ſcindendis rebus inſerunt & validè penetrant. <lb></lb></s> <s id="s.001303">De vi autem & efficacia ictus ſeu percuſſionis hic ſuper<lb></lb>ſedemus aliquid, ea de re, in ſequenti quæſtione verba fa<lb></lb>cturi. </s> </p> <p type="main"> <s id="s.001304">Multa hîc addere potuiſſemus ad Cochleam ſpe<lb></lb>ctantia, quippe quòd Cochlea cuneus ſit Cylindro inuo<lb></lb>lutus, qui quidem ad mallei, ſed vectis virtute ſibi adiun<lb></lb>ctâ, validiſſimè operatur, & ſexcentis inſeruit vſibus. </s> <s id="s.001305">Ve<lb></lb>runtamen cùm de hac ſpecie egregiè diſſerat G. Vbaldus, <pb xlink:href="007/01/140.jpg"></pb>conſultò hanc diſputationem omittimus; idque hac quo<lb></lb>que de cauſſa, quod nihil de cochlea, ac ſi eam non nouiſ<lb></lb>ſet, locutus ſit Ariſtoteles. </s> </p> <p type="main"> <s id="s.001306">Poſſumus autem in actu ſciſſionis, quæ cuneo fit, a<lb></lb>liâ tamen ratione vectem conſiderare, nempe non in cu<lb></lb>neo quidem, ſed in ipſa re quæ ſcinditur. </s> </p> <figure id="id.007.01.140.1.jpg" xlink:href="007/01/140/1.jpg"></figure> <p type="main"> <s id="s.001307">Eſto enim quip<lb></lb>piam ſciſſile ABCD, <lb></lb>cui alteri extremita<lb></lb>tum, puta BD, cuneus <lb></lb>adigatur EFG, <expan abbr="fiatq;">fiatque</expan> <lb></lb>ſciſſio per longitudi<lb></lb>nem ſecundum <expan abbr="lineã">lineam</expan> <lb></lb>EH. facta igitur ex <lb></lb>cunei ingreſſu <expan abbr="partiū">partium</expan> ſeparatione B, expelletur in I, D ve<lb></lb>rò in K. fient igitur materiæ ſciſſæ partes AIBH, CKDH, <lb></lb>ceu duo vectes, quorum hinc inde in corpore ipſo fulci<lb></lb>menta L, M potentiæ vtrinque dilatantes BD, pondus ve<lb></lb>rò materiæ reſiſtentia, in ſeparationis loco vbi N. </s> <s id="s.001308">Duca<lb></lb>tur NL, quanto itaque BN maiorem habebit proportio<lb></lb>nem ad LN, eo faciliùs reſiſtentia quæ in N, ſuperabitur. <lb></lb></s> <s id="s.001309">Mutatur <expan abbr="autē">autem</expan> aſſiduè in ipſa ſciſſione fulcimentum, & <expan abbr="cū">cum</expan> <lb></lb>fulcimento ipſa proportio. </s> <s id="s.001310">Pertingente enim ſciſſione in <lb></lb>O, <expan abbr="fulcimētum">fulcimentum</expan> fit in P. quo caſu ſciſſura eſt facilior, quip<lb></lb>pe quod maiorem habeat proportionem BO ad OP, <expan abbr="quã">quam</expan> <lb></lb>BN ad NL. </s> <s id="s.001311">Hoc autem experiuntur materiarij, qui primis <lb></lb>ictibus, ſecuriculâ nondum probè adactâ, & nondum fa<lb></lb>ctâ notabili ſciſſione difficultatem ſentiunt, mox <expan abbr="factaiã">facta iam</expan> <lb></lb>ſeparatione facillima paullatim fit materiæ totius ſepara<lb></lb>tio. </s> <s id="s.001312">Hoc idem & nos abſque cunei vſu experimur, cum ba<lb></lb>culum aut quippiam tale manibus diductis ſcindimus. </s> <s id="s.001313">à <lb></lb>principio enim difficultatem ſentimus, deinde ex ea <expan abbr="quã">quam</expan> <lb></lb>diximus proportione ſciſſio ipſa fit apprime facilis. </s> <s id="s.001314">Vti-<pb xlink:href="007/01/141.jpg"></pb>mur etiam vecte cuneato ad ſcindendum & aperiendum: <lb></lb>adacto enim ſciſſuræ cuneo, idqueue manu malleoue, tum <lb></lb>ab altera extremitate preſſo, valida fit ex vectis vi <expan abbr="cōtinui">continui</expan> <lb></lb><figure id="id.007.01.141.1.jpg" xlink:href="007/01/141/1.jpg"></figure><lb></lb>corporis ſeparatio. </s> <s id="s.001315">Ma<lb></lb>teria ſciſſilis AB <expan abbr="ſcalprū">ſcalprum</expan> <lb></lb>ceu vectis cuneatus CD, <lb></lb>cuius fulcimentum, E, <lb></lb>pondus verò vbi C, po<lb></lb>tentia vbi D, quo caſu <lb></lb>quo maior eſt proportio <lb></lb>DE ad EC, eo eſt ipſa ſciſſio leuior & facilior. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001316">QVAESTIO XVIII.</s> </p> <p type="head"> <s id="s.001317"><emph type="italics"></emph>Quærit hic Ariſtoteles, Cur per Trochleas ab exigua potentia in<lb></lb>gentia moueantur pondera?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001318">De Trochlea Pappus, & veteres: inter recentiores e<lb></lb>gregiè admodum, vt omnia examinauit in Mechani<lb></lb>cis G. Vbaldus. </s> <s id="s.001319">Nos tamen interim poſt clariſſimos illos <lb></lb>viros aliquid quod nouitatem & ſubtilitatem ſapiat, de <lb></lb>noſtro penu promemus. </s> <s id="s.001320">Et ſanè inuentis quidem addere <lb></lb>res eſt facilis, at quod inuentis addas inuenire haud adeo <lb></lb>facile. </s> <s id="s.001321">Sed nos primum Philoſophi ipſius dicta ad <expan abbr="trutinã">trutinam</expan> <lb></lb>reuocemus. </s> <s id="s.001322">Ita autem quæſtionem proponit; Cur ſi quiſ<lb></lb>piam Trochleas componens duas, in ſignis duobus, ad ſe <lb></lb>inuicem iunctis contrario ad Trochleas modo circulo fu<lb></lb>nem circumduxerit, cuius alterum quidem caput tigno<lb></lb>rum appendatur alteri, alterum verò Trochleis ſit <expan abbr="innixū">innixum</expan> <lb></lb>& à funis initio trahere cœperit, magna trahit pondera, li<lb></lb>cet imbecillium fuerit virium? </s> </p> <p type="main"> <s id="s.001323">Obſeueriſſima expoſitio, & nî res eſſet vulgò per ſe <lb></lb>nota, dequeue ea Vitruuius & Mechanici non egiſſent, diffi<lb></lb>cile vtique eſſet ex eius verbis ſenſum aſſequi. </s> </p> <pb xlink:href="007/01/142.jpg"></pb> <p type="main"> <s id="s.001324">Tigna ſanè vocaſſe videtur ea ligna, quæ à Vitruuio <lb></lb>Rechami dicuntur, in quibus nempe ipſi inſeruntur orbi<lb></lb>culi. </s> <s id="s.001325">Etſi de tignis eiuſmodi aliud quippiam ſentire videa<lb></lb>tur Picolomineus. </s> <s id="s.001326">Græca lectio pro tignis habet <foreign lang="grc">ξύλα</foreign>, id <lb></lb>eſt, ligna; item vbi Leoniceni verſio legit, ad ſe inuicem <lb></lb>iunctis, textus habet <foreign lang="grc">συμβαίνουσιν ἑαυτοῖς ἐναντίως</foreign>, hoc eſt, in<lb></lb>uicem ex oppoſito concurrunt. </s> <s id="s.001327">Certè locum totum ita <lb></lb>redderem: Cur ſi quis duas Trochleas fecerit, in duobus <lb></lb>lignis ſibi ex oppoſito concurrentibus, eiſqueue Trochleis <lb></lb>circumpoſuerit funem, cuius alterum caput alteri ligno<lb></lb>rum ſit annexum, alterum verò Trochleis cohæreat, vel <lb></lb>apponatur. </s> <s id="s.001328">Si quis alterum funis principium trahat, ma<lb></lb>gna trahat pondera, etſi trahens potentia ſit exigua? </s> <s id="s.001329">Nos <lb></lb>verbis figuram, & figurâ verba ipſa elucidabimus. </s> </p> <figure id="id.007.01.142.1.jpg" xlink:href="007/01/142/1.jpg"></figure> <p type="main"> <s id="s.001330">Sint duo ligna ex oppoſito concurrentia, <lb></lb>in quibus Trochleæ, hoc eſt, orbiculi AB, fu<lb></lb>nis ductarius DABC, cuius alterum caput re<lb></lb>ligatum eſt ligno trochleæ A, vbi eſt C. </s> <s id="s.001331">Tro<lb></lb>chlea A loco ſtabili commendata, vbi E. </s> <s id="s.001332">Pon<lb></lb>dus alteri ligno Trochleæ appenſum F. </s> <s id="s.001333">Tra<lb></lb>cto itaque fune DABC, eleuatur & trahitur <lb></lb>pondus F. </s> <s id="s.001334">Ex quibus clarè patet, <expan abbr="Philoſophū">Philoſophum</expan> <lb></lb>propoſuiſſe Trochleam duobus tantum orbi<lb></lb>culis munitam, quod vtique ſatis erat ad ex<lb></lb>plicationem. </s> <s id="s.001335">Inquit autem, faciliùs vecte <expan abbr="quã">quam</expan> <lb></lb>manu pondus moueri. </s> <s id="s.001336">Trochleam vero (id <lb></lb>eſt, orbiculum; ita enim eſt intelligendum) eſ<lb></lb>ſe vectem, aut vectis virtute operari. </s> <s id="s.001337">Ita autem <lb></lb>videtur argumentari. </s> <s id="s.001338">Si vnicâ Trochleâ plus trahitur <lb></lb>quàm manu, multo faci ius & velocius id fiet duobus, <lb></lb>quibus plus, vt ipſe ait, quàm in duplici velocitate pon<lb></lb>dus leuabitur. </s> <s id="s.001339">Summa dictorum eſt, ex multiplicatione <lb></lb>orbiculorum pondus ipſum imminui, & minori difficul-<pb xlink:href="007/01/143.jpg"></pb>tate leuari, quod ſanè verum eſt. </s> <s id="s.001340">Nos tamen nonnulla <expan abbr="cō-ſiderabimus">con<lb></lb>ſiderabimus</expan>. </s> <s id="s.001341">quod ait, vecte facilius moueri pondera <lb></lb>quam manu, ſemper non eſt verum. </s> <s id="s.001342">Si enim vectis pars <lb></lb>quæ à fulcimento ad manum breuior fuerit illâ, quæ à <lb></lb>fulcimento ad pondus difficilius vecte pondus mouebi<lb></lb>tur quam manu. </s> <s id="s.001343">Idem quoque accidet, ſi eo modo vecte <lb></lb>vtamur, quem obſeruat Guidus Vbald. Tract. </s> <s id="s.001344">de Vecte <lb></lb>prop. 3. Poſita nempe inter fulcimentum & pondus ſuſti<lb></lb>nente potentiâ. </s> <s id="s.001345">Præterea quod aſſeruit Ariſtoteles, Tro<lb></lb>chleas ad vectem reduci, verum quidem eſt, ſed aptius di<lb></lb>xiſſet ad libram, etenim vectis vtcunque à fulcimento di<lb></lb>uiditur. </s> <s id="s.001346">Libra verò quod & orbiculis ex centro accidit, <lb></lb>ſemper bifariam. </s> <s id="s.001347">Ad hæc videtur ille ad orbiculorum <lb></lb>multiplicitatem Trochlearum vim referre. </s> <s id="s.001348">Si enim, ait, <lb></lb>vnicâ Trochleâ pondus facile trahitur, id multo validius <lb></lb>pluribus fiet. </s> <s id="s.001349">Veruntamen non abſolutè ex orbiculorum <lb></lb>multiplicatione id fieri ita oſtendemus. </s> </p> <figure id="id.007.01.143.1.jpg" xlink:href="007/01/143/1.jpg"></figure> <p type="main"> <s id="s.001350">Sint duæ op<lb></lb>poſitæ lineae rectae, <lb></lb>vtpote trabes AB, <lb></lb>CD, <expan abbr="inuicē">inuicem</expan> æqui<lb></lb>diſtantes & ipſæ <lb></lb>ſtabiles: ſuperiori <lb></lb>tres appendantur <lb></lb>orbiculi ex <expan abbr="pūctis">punctis</expan> <lb></lb>E, F, G, <expan abbr="nēpe">nempe</expan> ML, <lb></lb>PQ, TV. inferiori <lb></lb><expan abbr="autē">autem</expan> duobus pun<lb></lb>ctis IH, nempe <lb></lb>NO, RS. </s> <s id="s.001351">Erunt i<lb></lb>gitur in vniuerſum <lb></lb>quinque, indatur per eos funis ductarius KLMNOP <lb></lb>QRSTVX, ex cuius extremitate pendeat pondus X, <pb xlink:href="007/01/144.jpg"></pb>Trahatur funis in K. </s> <s id="s.001352">Dico ex multiplicatione <expan abbr="orbiculorū">orbiculorum</expan>, <lb></lb>trahenti pondus nequaquam minui. </s> <s id="s.001353">Sint autem orbicu<lb></lb>lorum diametri, LM, NO, PQ, RS, TV, applicetur poten<lb></lb>tîa in S. </s> <s id="s.001354">Erit igitur ad hoc vt ſuſtineat æqualis ponderi X, <lb></lb>orbiculi enim TV ſemidiametri ſunt æquales. </s> <s id="s.001355">Transfe<lb></lb>ratur <expan abbr="potētia">potentia</expan> in q, & ita deinceps donec perueniatur in K, <lb></lb>vbi funis ipſius eſt principium, Idem eſt igitur ſeruata ſem<lb></lb>per ſemidiametrorum æqualitate ac ſi potentia quæ eſt in <lb></lb>K, applicata intelligatur in T vel in V. vbicunque enim <lb></lb>collocetur, ponderi erit æqualis. </s> <s id="s.001356">Nihil igitur rebus ita <lb></lb>diſpoſitis, orbiculorum multiplicatio ad facilitatem ope<lb></lb>ratur. </s> <s id="s.001357">Alia itaque ratio quærenda eſt, quam non ſatis ex<lb></lb>plicaſſe videtur Ariſtoteles. </s> <s id="s.001358">Probabimus autem, nullam <lb></lb>ex ſuperioribus orbiculis fieri ponderum imminutionem, <lb></lb>ſed totam vim in inferioribus conſiſtere. </s> <s id="s.001359">At nos interim <lb></lb>quippiam quod ad rem faciat, proponamus. </s> </p> <figure id="id.007.01.144.1.jpg" xlink:href="007/01/144/1.jpg"></figure> <p type="main"> <s id="s.001360">Eſto punctum A, cui rectæ ap<lb></lb>pendantur lineæ BAC, diuiſæ qui<lb></lb>dem in A, ſit autem lineæ BA caput <lb></lb>B, ipſius verò CA caput C. </s> <s id="s.001361">Modò <lb></lb>intelligantur vnitæ in A, ſitqueue vni<lb></lb>ca linea à puncto A ceu funiculus <lb></lb>dependens BAC; Appendatur capi<lb></lb>ti B pondus B. </s> <s id="s.001362">Capiti vero C, <expan abbr="pōdus">pondus</expan> <lb></lb>C, inter ſe æqualia. </s> <s id="s.001363">Potentia igitur <lb></lb>in A, duo ſuſtinebit pondera BC. <lb></lb></s> <s id="s.001364">Pondera verò ex æqualitate æque<lb></lb>ponderabunt. </s> <s id="s.001365">Quod ſi B potentia <lb></lb>dicatur ſuſtinens pondus C, aut C <lb></lb>potentia ſuſtinens pondus D, vel <lb></lb>duæ potentiæ inter ſe æquales, nihil <lb></lb>refert. </s> <s id="s.001366">Vtcunque enim id ſit, fiet æquilibrium. </s> <s id="s.001367">Habemus <lb></lb>igitur ex iſtis ad ſuſtinendum pondus ex ſuperiori parte <pb xlink:href="007/01/145.jpg"></pb>appenſum potentiam requiri ipſi ponderi æqualem. </s> <s id="s.001368">Ani<lb></lb>mo poſthæc concipiatur alia recta linea DEF, cuius inte<lb></lb>gra longitudo ſi extenderetur, eſſet DE, EF. </s> <s id="s.001369">Appendatur <lb></lb>in E pondus E æquale alteri ponderum B vel, C, ſint autem <lb></lb>duæ potentiæ pondus E ſuſtinentes D, F. </s> <s id="s.001370">Vtraque igitur <lb></lb>dimidium ſuſtinebit ponderis E, ſed potentia quæ ſuſti<lb></lb>nebat pondus B, in C erat ipſi B æqualis, vbi appenſio pon<lb></lb>deris erat in ſuperiori parte in A, hîc autem, vbi appenſio <lb></lb>eſt in parte in feriori, vtraque potentia dimidium ſuſtinet <lb></lb>appenſi ponderis. </s> <s id="s.001371">Videmus igitur illam appenſionem <lb></lb>quidem pondus nullatenus imminuere, hanc verò pon<lb></lb>dus ipſum, bifariam diuiſum, ſuſtinentibus potentijs im<lb></lb>partiri. </s> <s id="s.001372">Hæc in lineis, Mathematicâ vſi abſtractione, con<lb></lb>ſiderauimus, nunc verò eadem mechanicè perpenda<lb></lb>mus. </s> </p> <figure id="id.007.01.145.1.jpg" xlink:href="007/01/145/1.jpg"></figure> <p type="main"> <s id="s.001373">Sit igitur <lb></lb>punctum A, vt <lb></lb>in ſequenti figu<lb></lb>ra clauus paxil<lb></lb>luſue, cui appen<lb></lb>ſus funiculus <lb></lb>BAC, & funicu<lb></lb>li capitibus pon<lb></lb>dera BC, ſit quo<lb></lb>que anulus D, <lb></lb>per quem traìe<lb></lb>ctus funiculus <lb></lb>EDF. </s> <s id="s.001374">Anulo au<lb></lb>tem <expan abbr="cōiunctum">coniunctum</expan> <lb></lb>pondus G. </s> <s id="s.001375">His igitur ita conſtitutis, eadem demonſtra<lb></lb>buntur quæ ſuperius, nempe oportere vt fiat æquilibrium <lb></lb>B, C, eſſe æqualia, tum potentias, quæ ſunt in EF pondus <lb></lb>G inter eas diuiſum ſuſtinere. </s> <s id="s.001376">Porrò volentes Mechanici <pb xlink:href="007/01/146.jpg"></pb>funiculos circa paxillum, & anulum ad attollenda & de<lb></lb>primenda pondera mouere incommodè illis vtique ſuc<lb></lb>cedebat, clauo & anulo motum difficilem facientibus. <lb></lb></s> <s id="s.001377">Quamobrem vt difficultati occurrerent, ad locum claui <lb></lb>clauo ipſi orbiculum circumpoſuerunt, & anuli itidem <lb></lb>loco orbiculum aptauerunt. </s> <s id="s.001378">Hæc autem agentes rei i<lb></lb>pſius naturam non mutauerunt, ſed ſibi, vt diximus, ex or<lb></lb>biculis maximam commoditatem <expan abbr="atq;">atque</expan> facilitatem com<lb></lb>parârunt. </s> </p> <p type="main"> <s id="s.001379">Ex his principîjs tota Trochlearum ratio pendet, <lb></lb>quæ tamen alia quoque conſideratione in idem tenden<lb></lb>te examinari poteſt, quod quidem fecere veteres, & ipſe, <lb></lb>qui veteres optimè imitatus eſt, Guid. </s> <s id="s.001380">Vbaldus. </s> </p> <p type="main"> <s id="s.001381">Vidimus vtique nos, à potentia quæ eſt in B, pondus <lb></lb>par ſuſtineri in C, Potentiam autem quæ eſt in E <expan abbr="dimidiū">dimidium</expan> <lb></lb>ſuſtinere ponderis quod eſt in G. </s> <s id="s.001382">Nos igitur ijſdem inſi<lb></lb>ſtentes adiecta libra, vecteue, bifariam diuiſo rem ipſam <lb></lb>ex ſubiecto diagrammate lucidiorem faciemus. </s> </p> <p type="main"> <s id="s.001383">Eſto linea quædam ſtabilis ceu trabs horizonti æ<lb></lb>quediſtans AB, cui in A funiculus annectatur AC, cuius <lb></lb>extremum C vecti cuidam alligetur CD, in medio diuiſo <lb></lb>vbi E, tum alteri vectis eiuſdem extremitati D, funiculus <lb></lb>nectatur DG, & à puncto E pondus appendatur F. puta li<lb></lb>brarum mille, Tum puncto G in medio vectis HI, funis re<lb></lb>ligetur DG, & ex altero vectis extremo alligato fune HK <lb></lb>commendetur loco ſtabili in K, & ab alio capite vectis vbi <lb></lb>I ad medium vectis MN, vbi L, funis annectatur lL, tum <lb></lb>ex vectis capite M, funis commendetur MO, loco ſtabili <lb></lb>in O, & alteri capiti N, funis, NP, qui alligetur medio ve<lb></lb>cti QR in P, & ex Q, funis QS. </s> <s id="s.001384">Commendetur loco ſtabili <lb></lb>in S, & alteri vectis extremo R funis alligetur RT, cui <lb></lb>quidem potentia ſuſtinens applicetur in T. </s> <s id="s.001385">Dico igitur, <pb xlink:href="007/01/147.jpg"></pb><figure id="id.007.01.147.1.jpg" xlink:href="007/01/147/1.jpg"></figure><lb></lb>rebus ita diſpoſitis, <lb></lb>potentiam in T ita <lb></lb>ſe habere ad pondus <lb></lb>F, vt vnum ad ſexde<lb></lb>cim, hoc eſt, in pro<lb></lb>portione eſſe ſub<lb></lb>ſexdecupla. </s> <s id="s.001386">Sunt <lb></lb>autem, hic vectes <lb></lb>quatuor in feriorum <lb></lb>cubiculorum, loco, <lb></lb>CD, HI, MN, QR, <lb></lb>quorum, centra E, <lb></lb>G, L, P. quoniam e<lb></lb>nim A hoc eſt, C, v<lb></lb>nà cum potentia G, <lb></lb>hoc eſt, D, ſuſtinet <lb></lb>pondus F alterum, <lb></lb>ponderis dimidium <lb></lb>ſuſtinebit C, <expan abbr="alterū">alterum</expan> <lb></lb>vero D. erunt igitur <lb></lb>vtrinque librae quin<lb></lb>gentæ. </s> <s id="s.001387">Tum potentia in K, hoc eſt, in H, vna cum poten<lb></lb>tia in L, hoc eſt, in I ſuſtinebunt quingenta. </s> <s id="s.001388">Quare <expan abbr="vtraq;">vtraque</expan> <lb></lb>ducenta quinquaginta, ſed hoc totum bifariam diuiditur <lb></lb>inter potentias, O, id eſt, M, & P, id eſt H. erunt igitur v<lb></lb>trinque centum viginti quinque. </s> <s id="s.001389">Ea autem ſumma <expan abbr="iterū">iterum</expan> <lb></lb>bifariam diuìditur, hoc eſt, inter potentias S, id eſt, Q & <lb></lb>T, id eſt, R, quare vtraque ſuſtinet ſexaginta duo cum di<lb></lb>midio. </s> <s id="s.001390">Sed numerus iſte ad Millenarium ita ſe habet vt v<lb></lb>num ad ſexdecim. </s> <s id="s.001391">Hinc colligimus, pondus totum inter <lb></lb>loca ſtabilia diuidi, nempe A, K, O, S, & ipſam potentiam <lb></lb>quæ ſuſtinet in T, & locis ipſis ſtabilibus quindecim par<lb></lb>tes integri ponderis, potentia verò T ſextam decimam <pb xlink:href="007/01/148.jpg"></pb>tantùm commendari. </s> <s id="s.001392">Itaque ſi ex puncto V appendere<lb></lb>tur AB, in X potentia, quæ in X ſuſtineret mille, minus <lb></lb>ſexaginta duo cum dimidio, quod quidem à potentia in <lb></lb>T ſuſtinetur; quod ſi alius adderetur orbiculus, & fierent <lb></lb>quinque, potentia in T ſuſtineret trigeſimam ſecundam <lb></lb>partem integri ponderis, hoc eſt, dimidium librarum ſe<lb></lb>xaginta duarum cum dimidio, nempe triginta & vnam <lb></lb>cum quarta parte, ſi item textus adderetur, potentia in T <lb></lb>ſexageſimam partem ſuſtineret integri ponderis, hoc eſt, <lb></lb>libras quindecim & 5/8 libræ vnius. </s> <s id="s.001393">Vnde patet clarè pon<lb></lb>deris diminutionem fieri ex orbiculis inferioribus, non <lb></lb>autem ex ſuperioribus, ſuperiores autem addi non neceſ<lb></lb>ſitatis quidem, ſed commoditatis gratiâ: neque enim abſ<lb></lb>que ſuperioribus vnico ductario fune fieri poſſet attractio <lb></lb>& ponderis ipſius eleuatio. </s> <s id="s.001394">Hactenus igitur nobis iſthæc <lb></lb>de Trochleæ natura & vi poſt alios, conſideraſſe ſit ſatis. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001395">QVÆSTIO XIX.</s> </p> <p type="head"> <s id="s.001396"><emph type="italics"></emph>Dubitat Philoſophus, Cur ſi quis ſuper lignum magnam imponat <lb></lb>ſecurim, deſuperque magnum adijciat pondus, ligni quippiam quod <lb></lb>curandum ſit, non diuidit; ſi verò ſecurim extollens percutiat, illud <lb></lb>ſcindit, cum alioquin multo minus habeat ponderis id quod <lb></lb>percutit, quam illud quod ſuperiacet <lb></lb>& premit?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001397">Poterat Ariſtoteles, nî fallimur, rem breuius & vniuer<lb></lb>ſalius proponere. </s> <s id="s.001398">Scilicet cur motus ponderi addat <lb></lb>pondus & efficacius ex motu quam ex immoto pondere <lb></lb>mota res operetur. </s> <s id="s.001399">Soluit autem. </s> <s id="s.001400">An, inquiens, ideo fit, <lb></lb>quia omnia cum motu fiunt, & graue ipſum grauitatis ma<lb></lb>gis aſſumit motum, dum mouetur quam dum quieſcit? <lb></lb></s> <s id="s.001401">Incumbens igitur connatam graui motionem non moue<lb></lb>tur, motum verò & ſecundum hanc mouetur & ſecun-<pb xlink:href="007/01/149.jpg"></pb>dum eam quæ eſt <expan abbr="percutiētis">percutientis</expan>? </s> <s id="s.001402">Hæc præclarè quidem, cæ<lb></lb>tera autem, quæ de cuneo iterat, nempe ad vectem eius o<lb></lb>perationem referri ſuperius confutauimus. </s> <s id="s.001403">Porrò effe<lb></lb>ctus huius, de quo agitur, diſputatio illuc ſpectat, videli<lb></lb>cet ad cadentium atque proiectorum naturam. </s> <s id="s.001404">Ad maio<lb></lb>rem autem rei euidentiam hæc addimus. </s> </p> <figure id="id.007.01.149.1.jpg" xlink:href="007/01/149/1.jpg"></figure> <p type="main"> <s id="s.001405">Eſto libra AB, cu<lb></lb>ius centrum C, libra<lb></lb>ta æqualibus ponde<lb></lb>ribus DE, apponatur <lb></lb>ponderi E pondus F, <lb></lb>item ponderi D pon<lb></lb>dus G ipſi ponderi F <lb></lb>æquale, æquilibrabit <lb></lb>itidem, Modò non apponatur ſimpliciter pondus G ſex <lb></lb>ex H in lancem A dimittatur, tunc ſanè non æquilibrabit, <lb></lb>ſed libram deprimet. </s> <s id="s.001406">Duo enim in pondere dimiſſo con<lb></lb>ſiderantur pondera; naturale ſcilicet, & quod motu ipſi <lb></lb>moto, ponderi eſt acquiſitum. </s> <s id="s.001407">Itaque quo motus fuerit <lb></lb>maior, puta ſi cadat ex I, grauitas ex maiori motu fiet ma<lb></lb>ior. </s> <s id="s.001408">quod vtique efficacius fieret ſi pondus G non dimit<lb></lb>tetur modo remoto prohibente, ſed proijceretur. </s> <s id="s.001409">Tunc <lb></lb>enim tria concurrerent, grauitas naturalis, grauitas ac<lb></lb>quiſita ex naturali motu, & ea quæ naturali adijcitur ex <lb></lb>violentia. </s> <s id="s.001410">Pondus igitur ſecuri impoſitum & ſecuris ipſius <lb></lb>naturalis grauitas naturali tantum grauitate operantur, <lb></lb>& ideo minus efficaciter. </s> <s id="s.001411">Huc autem ea ferè pertinent <lb></lb>quæ nos à principio de duobus centris retulimus, natura<lb></lb>lis nempe grauitatis, & acquiſitæ. </s> </p> <p type="main"> <s id="s.001412">Cæterùm cur mallei & ſecuris ictus ſit violentiſſi<lb></lb>mus, ideo fit quod non ex vnico neque duplici, ſed ex tri<lb></lb>plici grauitate operetur. </s> <s id="s.001413">Eſto enim ſecuris A, cuius manu<lb></lb>brium AB, brachium vero ſecuri vtentis BC, erit igitur C <pb xlink:href="007/01/150.jpg"></pb><figure id="id.007.01.150.1.jpg" xlink:href="007/01/150/1.jpg"></figure><lb></lb>locus vbi humero <lb></lb>brachium iungi<lb></lb>tur, motus ipſius <lb></lb>centrum, attollit <lb></lb>autem ſecurim is <lb></lb>qui percutit, & re<lb></lb>tro ad ſcapulas re<lb></lb>ducens totis viri<lb></lb>bus ex centro C <lb></lb>ſecurim vibrat, <lb></lb>portionem circuli <lb></lb>deſcribens ADE <lb></lb>ictumqueue faciens <lb></lb>in E. </s> <s id="s.001414">Vires igitur acquirit ſecuris, tum ex naturali grauita<lb></lb>te, cadens ex D, in E, tum ex proprio pondere, tum etiam <lb></lb>ex violentia eidem à percutiente impreſſa. </s> <s id="s.001415">Fiunt autem <lb></lb>motus tam naturalis quàm violentus eo validiores, quo <lb></lb>maius eſt ſpatium, quo res mota mouetur, idqueue praecipuè <lb></lb>cum violentia ipſam ſecundat naturam. </s> <s id="s.001416">Itaque maior fit <lb></lb>ictus in E quàm in F, & in F maior quàm in D. </s> <s id="s.001417">Item violen<lb></lb>tius feriret percutiens, ſi manubrium eſſet longius, puta <lb></lb>BG. </s> <s id="s.001418">Tunc enim maior eſſet circulus GH, & motus tum <lb></lb>prolixior, tum velocior. </s> <s id="s.001419">quo igitur longiora habet bra<lb></lb>chia is qui ſecuri malleoue vtitur, data virium paritate, ex <lb></lb>eadem ratione validius percellit. </s> <s id="s.001420">Eſt autem ſecuris, vel <lb></lb>malleus cuneatus, vel cuneus malleatus manubrio inſer<lb></lb>tus. </s> <s id="s.001421">An autem operetur efficacius cuneus malleo percuſ<lb></lb>ſus, aut cum manubrio motus, vt fit in ſecuri, data aciei & <lb></lb>ponderis æqualitate, difficile eſt determinare. </s> <s id="s.001422">Certè va<lb></lb>lidius, & certius fieri ſciſſionem ex cuneo & malleo, ea ra<lb></lb>tio eſt, quod cuneus adactus, nec inde remotus eam inte<lb></lb>rim ſeruat, quam antea fecerat partium ſeparationem, <pb xlink:href="007/01/151.jpg"></pb>quod quidem ſecuri non accidit, quæ adacta ad nouam <lb></lb>percuſſionem faciendam extrahitur. </s> </p> <p type="main"> <s id="s.001423">Hoc etiam conſideramus, ſecuris in circulo motum, <lb></lb>ex A in D, eſſe videndum, id eſt, non ſecundum naturam, <lb></lb>ſurſum enim fertur quod eſt graue, ex D verò in F <expan abbr="mixtū">mixtum</expan>: <lb></lb>magis autem ad naturalem accedere qui fit ex F in E. </s> <s id="s.001424">Tar<lb></lb>dior ergo ex A in D, velocior ex D, in F, velociſſimus ex F <lb></lb>in E; quædam quæ ad hanc rem faciunt, egregiè conſide<lb></lb>rat Guid, Vbald. in calce Tractatus, De Cuneo; ipſum <lb></lb>conſule. </s> </p> <p type="main"> <s id="s.001425">Ad hæc ſuccurrit nobis pulcherrima quæſtio. </s> <s id="s.001426">Du<lb></lb>bitari enim poteſt, vtrum ictus ex enſe efficacior ſit à par<lb></lb>te quæ eſt circa aciem, aut circa medium enſem, vel pro<lb></lb>pe manubrium capulumue; etenim hinc inde ſunt ra<lb></lb>tiones. </s> </p> <p type="main"> <s id="s.001427">Eſto quidem enſis AB, cuius capulus A, ſpiculum ve <lb></lb>rò B, centrum grauitatis C, pars capulo proxima D. </s> <s id="s.001428">Libra<lb></lb>to itaque gladio tres fiunt circulorum portiones BE, CF, <lb></lb>DG, quæritur quo loco ictus ſit validior, nempe in E, in F, <lb></lb>velin G. </s> <s id="s.001429">Videtur validiorem futurum in E, quippe quod <lb></lb>ex maiori ſemidiametro AB, maioris ſit circuli portio BE, <lb></lb>& ideo velocior motus ex B in E. </s> <s id="s.001430">Contra efficaciorem <lb></lb>futurum apparet in F, propterea quod ibi ex centro C to<lb></lb>tius fiat grauitatis impreſſio, fieri autem validiſſimum in <lb></lb>G, licet ibi motus ſit tardior inde videtur, quod ſi conſide<lb></lb>retur enſis vt vectis, cuius fulcimentum eſt A, potentia <lb></lb>premens in B, ponderis vero loco reſiſtentia rei quæ per<lb></lb>cutitur in D. </s> <s id="s.001431">Maior eſt autem proportio BA, ad AD, quam <lb></lb>BA ad AC, & ideo violentior fiet preſſio ex ictu in D, <expan abbr="quã">quam</expan> <lb></lb>in C. </s> <s id="s.001432">Hiſce hoc pacto conſideratis, putarem ictum effica<lb></lb>ciorem fieri in F ex medio C, quam ex extremis & oppo<lb></lb>ſitis partibus EG. </s> <s id="s.001433">Licet enim in B velocitas ſit maior, deeſt <lb></lb>ibi pondus. </s> <s id="s.001434">Si enim enſis iterum vt vectis conſideretur, e<pb xlink:href="007/01/152.jpg"></pb>runt AB. duo fulcimenta ſuſtinentía pondus in C, vbi gra<lb></lb>uitatis eſt centrum. </s> <s id="s.001435">Si igitur paria fuerint ſpatia BC, CA, <lb></lb><figure id="id.007.01.152.1.jpg" xlink:href="007/01/152/1.jpg"></figure><lb></lb>in B erit dìmidium <lb></lb>ponderis C, quantum <lb></lb>ergo velocitate præ<lb></lb>ualet ictus in B, <expan abbr="tantū">tantum</expan> <lb></lb>ponderis amittit. </s> <s id="s.001436">D <lb></lb>verò plus quidem de <lb></lb>pondere participat, <lb></lb>ſed velocitatis habet <lb></lb>minimum, in C verò <lb></lb>velocitas eſt medio<lb></lb>cris, tota tamen ipſius <lb></lb>ex grauitatis centro <lb></lb>ponderis fit impreſ<lb></lb>ſio. </s> </p> <p type="main"> <s id="s.001437">Quidam, quod huc pertinet, vt ex acie ipſa quæ lon<lb></lb>gius à capulo abeſt, violentiſſimum facerent ictum, Ar<lb></lb>gentum viuum, quod ſui naturâ grauiſſimum quidem eſt <lb></lb>& mobiliſſimum in canali à manubrio ad verticem exca<lb></lb>uato infundunt, quo in gladij deſcenſu ad verticem velo<lb></lb>ciſſimè delato illuc transfert grauitatem totam, quare <lb></lb>tum velocitate tum grauitate concurrentibus ictus fit <lb></lb>violentiſſimus & longè validiſſimus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001438">QVAESTIO XX.</s> </p> <p type="head"> <s id="s.001439"><emph type="italics"></emph>Dubitatur, Cur ſtatera qua carnes ponderantur, paruo appendicu<lb></lb>lo, magna trutinet onera, cum alioqui tota, dimidiata exiſtat <lb></lb>libra, altera vero parte ſola ſit <lb></lb>ſtatera?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001440">Soluit Philoſophus, inquiens, ſtateram ſimul, & vectem <lb></lb>eſſe & libram, ipſius verò libræ centra ſeu fulcimenta <pb xlink:href="007/01/153.jpg"></pb>eſſe ibi vbi fit ſuſpenſio. </s> <s id="s.001441">Pondera verò hinc in de in lance <lb></lb>& appendiculo, loco ſcilicet æquipondij, appendiculo <lb></lb>ſuccedente. </s> <s id="s.001442">Reducit autem demonſtrationem ad ea quæ <lb></lb>ſtatuit ipſe Mechanica principia; nempe ad circulum & <lb></lb>circuli virtutem. </s> <s id="s.001443">Ait igitur, appendiculum licet parui <expan abbr="pō-deris">pon<lb></lb>deris</expan> ſit, ideo maiori ponderi virtute æquari, quod lon<lb></lb>gius à centro, hoc eſt, ab ipſo fulcimento ſiſtatur. </s> <s id="s.001444">quic<lb></lb>quid tamen ſit, ſtateram eſſe vectem, res eſt exploratiſ<lb></lb>ſima. </s> </p> <figure id="id.007.01.153.1.jpg" xlink:href="007/01/153/1.jpg"></figure> <p type="main"> <s id="s.001445">Eſto igitur ſtatera AB, <lb></lb>cuius appendiculum cur<lb></lb>rens F, fulcimentum cen<lb></lb>trumue C, lanx quæ cate<lb></lb>na ſuſpenditur E ſpatium <lb></lb>à loco fulcimenti ad ap<lb></lb>pendiculum CF. quod ve<lb></lb>rò à fulcimento ad cate<lb></lb>nam, ex qua lanx appen<lb></lb>ditur AC. </s> <s id="s.001446">Intelligatur autem & aliud fulcimentum D, ſit<lb></lb>queue maius ſpatium AD, quam AC. </s> <s id="s.001447">Porrò ita ſe habeat <lb></lb>pondus in E ad appendiculi F pondus, vt CF ſpatium, ad <lb></lb>ſpatium AC, quo caſu ſeruata, permutatim, ponderum & <lb></lb>brachiorum proportione, fiet aequilibrium. </s> <s id="s.001448">Si autem pon<lb></lb>deribus ita conſtitutis iterum ſuſpendatur in D, non fiet <lb></lb>æquilibrium, propterea quod minor ſit proportio DF ad <lb></lb>DA, ea quæ eſt FC ad CA. </s> <s id="s.001449">Minor ergo eſt proportio FD <lb></lb>ad DA, quam ponderis E ad pondus F, & idcirco facta <lb></lb>ſuſpenſione præualebit pondus E ponderi F. </s> <s id="s.001450">Ita que vt it e<lb></lb>rum fiat æquilibrium, neceſſe eſt <expan abbr="iterū">iterum</expan> proportiones bra<lb></lb>chiorum ſeu ſpatiorum proportionibus ponderum æqua<lb></lb>re. </s> <s id="s.001451">Transferatur igitur (lancis interim immoto pondere) <lb></lb>ipſum appendiculum in B, fiatque vt FC ad CA, ita BD ad <lb></lb>DA. </s> <s id="s.001452">Stabit autem iterum ſtatera ad eam redacta quam <pb xlink:href="007/01/154.jpg"></pb>diximus brachiorum & ponderum permutatam propor<lb></lb>tionem. </s> </p> <p type="main"> <s id="s.001453">Nos ſtateris vtimur ex duplici fulcimento, altero <lb></lb>propiori, altero à lance ſeu loco, vbi lanx appenditur, re<lb></lb>motiori, illa grauiora appendimus pondera, & non per <lb></lb>vncias & libras, ſed per libras tantum & ſelibra ponde<lb></lb>ramus; & hoc ſtateræ latus eo quod minus minutè ſit di<lb></lb>uiſum; vulgo noſtrates Groſſum, hoc eſt, rude & craſſum <lb></lb>appellant. </s> <s id="s.001454">Aliud verò, cum fulcimentum eſt loco appen<lb></lb>ſionis lancis vicinius, & per libras, ſelibras & vncias diui<lb></lb>ditur, quo quidem minora appendimus pondera, eò quod <lb></lb><expan abbr="exquiſitiorē">exquiſitiorem</expan> contineat diuiſionem, ſubtile dicunt. </s> <s id="s.001455">Rectè <lb></lb>igitur dicebat Philoſophus, in ſtatera plures eſſe libras, <lb></lb>quamquam & ea quoque de cauſſa dici poſſit, quod, quot <lb></lb>ſunt appendiculi, è loco in locum translationes, totidem <lb></lb>ex proportionum variatione fiant libræ. </s> <s id="s.001456">Et hoc quidem <lb></lb>ſenſiſſe videtur Ariſtoteles. </s> </p> <figure id="id.007.01.154.1.jpg" xlink:href="007/01/154/1.jpg"></figure> <p type="main"> <s id="s.001457">Poſſemus & alio <lb></lb>modo ſtatera vti, nempe <lb></lb>ſtabili appendiculo, mo<lb></lb>bili autem fulcimento. <lb></lb></s> <s id="s.001458">Eſto enim ſtatera AB, <lb></lb>cuius lanx C appenſa in <lb></lb>A, appendiculum verò <lb></lb>ſtabile D, appenſum in <lb></lb>B, Apponatur ipſi ląnci <lb></lb>C, pondus E. </s> <s id="s.001459">Vnicum ergo fiet corpus CEABD conſtans <lb></lb>ex lance, libra & ponderibus. </s> <s id="s.001460">Habet ergo hoc totum gra<lb></lb>uitatis ſuæ centrum, quod quidem vbi ſit eſt ignotum. </s> <s id="s.001461">Ex <lb></lb>illo autem inuento ſi corpus totum appendatur, partes æ<lb></lb>queponderabunt. </s> <s id="s.001462">Appendatur autem, puta in G, ſit <expan abbr="autē">autem</expan> <lb></lb>grauitatis centrum in H. </s> <s id="s.001463">Quoniam igitur H eſt extra ful<lb></lb>cimentum G, declinabit ſtateræ pars GA, centro G per <pb xlink:href="007/01/155.jpg"></pb>circuli portionem Hl, à centro grauitatis in ipſa deſcen<lb></lb>ſione deſcriptam. </s> <s id="s.001464">Si autem grauitatis centrum fuerit vbi <lb></lb>K, eo quod ibi quoque ſit extra fulcimentum G, deſcen<lb></lb>det pars GB, deſcribente interim grauitatis centro K, cir<lb></lb>culi portionem KL. ltaque ſi ſtateram totam eum ponde<lb></lb>ribus trahamus <expan abbr="pellamuſq;">pellamuſque</expan> vltro citroque;, immoto appen<lb></lb>diculo erit aliquando fulcimentum in ea linea perpendi<lb></lb>culari vel loco ipſo, vbi eſt grauitatis centrum, quo caſu <lb></lb>ſtatera ſtabit, & tunc ita erit diuiſa, vt fiat brachiorum & <lb></lb>ponderum eadem ratio, ordine permutato. </s> <s id="s.001465">Hic autem <lb></lb>modus ideo non eſt in vſu, quod moleſtum ſit libram ſeu <lb></lb>ſtateram cum ponderibus vltro citroqueue transferre, quæ <lb></lb>difficultas commodè appendiculi mobilitate vitatur. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001466">QVAESTIO XXI.</s> </p> <p type="head"> <s id="s.001467"><emph type="italics"></emph>Quæritur, Cur facilius dentes extrahunt Chirurgi, denti forcipis <lb></lb>onere adiecto, quam ſi ſola manu vtantur?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001468">Reſpondet Philoſophus, An quia ex manu, magis quam <lb></lb>ex dentiforcipe lubrius elabitur dens? </s> <s id="s.001469">An ferro id po<lb></lb>tius accidit quam digitis, quoniam vndique dentem non <lb></lb>comprehendunt, quod mollis facit digitorum caro; ad<lb></lb>hæret enim & complectitur magis. </s> <s id="s.001470">Hæc ſecunda ratio <lb></lb>videtur primam deſtruere, & contrarium prorſus ſenten<lb></lb>tiæ, quæ in problemate proponitur, aſſerere. </s> <s id="s.001471">Si Græca ad <lb></lb>verbum reddas ita habent: An magis ipſa manu labile eſt <lb></lb>ferrum, & ipſum vndique (dentem nempe) non comple<lb></lb>ctitur, caro autem digitorum cum mollis ſit, adhæret ma<lb></lb>gis, & vndique congruit. </s> <s id="s.001472">Certè vt ſententia non ſit con<lb></lb>traria propoſitioni, Græca verſio ita videtur concinnan<lb></lb>da: Vel magis è manu elabitur, mollis enim eſt digitorum <lb></lb>caro, ferrum autem circumplectitur, & haeret magis. </s> <s id="s.001473">quic<lb></lb>quid ſit, Græcam lectionem contrarium ei quod quæri-<pb xlink:href="007/01/156.jpg"></pb>tur, affirmare certum eſt. </s> <s id="s.001474">Picolomineus, Ideo, inquit, di<lb></lb>gitorum caro mollis minus aptè extrahit, quod dentem <lb></lb>totum comprehendere non poteſt, quod ferrum ob ſuam <lb></lb>durítiem & conſtantiam commodiſſimè facit. </s> <s id="s.001475">Senſum ex <lb></lb>mente reddidit, quod ex verbis non poterat. </s> <s id="s.001476">Subiungit <lb></lb>denique Ariſtoteles, An quia dentiforcipes ſint duo con<lb></lb>trarij vectes vnicum habentes fulcimentum, ipſam ſcili<lb></lb>cet in ſtrumenti partium connexionem. </s> <s id="s.001477">Hoc igitur ad ex<lb></lb>tractionem vtuntur^{**}, vt facilius moueant. </s> <s id="s.001478">Figuram hoc <lb></lb>pacto proponit Philoſophus. </s> </p> <figure id="id.007.01.156.1.jpg" xlink:href="007/01/156/1.jpg"></figure> <p type="main"> <s id="s.001479">Eſto dentiforcipis alterum <lb></lb>quidem extremum vbi A, alte<lb></lb>rum autem quod extrahit B, ve<lb></lb>ctis vbi ADF, alter vectis, vbi <lb></lb>BCE, fulcimentum verò CGD <lb></lb>connexio vbi G. </s> <s id="s.001480">Dens autem pondus: vtroque igitur ve<lb></lb>cte B, & F ſimul comprehendentes mouent, Hæc ille. </s> <s id="s.001481">At<lb></lb>tamen rem ipſam ſubtilius conſiderantibus aliter videtur <lb></lb>habere, ac ipſe aſſerat. </s> <s id="s.001482">Et ſanè dentisforcipis brachia ve<lb></lb>ctes eſſe, quorum commune fulcimentum eſt in ipſo cen<lb></lb>tro vbi vertebra, nemo negauerit. </s> <s id="s.001483">Dentem autem eſſe <lb></lb>pondus, ego quidem abſolute non dixerim. </s> <s id="s.001484">Pondus <expan abbr="autē">autem</expan> <lb></lb>hîc proprie eſt ipſa dentis durities, cuius reſiſtentia eo fa<lb></lb>cilius ſuperatur, quo maior eſt proportio brachiorum à <lb></lb>manu ad vertebram, ad partem illam quæ à vertebra eſt <lb></lb>ad dentem. </s> <s id="s.001485">At dentis ex conſtrictione fractio nihil facit <lb></lb>prorſus ad extractionem: id tamen operatur brachio<lb></lb>rum longitudine dentiforceps, quod valide ex vectium <lb></lb>oppoſitorum vi dentes conſtringit & extractioni commo<lb></lb>dum reddit & facilem. </s> <s id="s.001486">Neque enim totus Dentiforceps <lb></lb>hic ceu vectis vnicus operatur, quod fit in forcipibus quas <lb></lb>Tenaleas vocamus, quibus è tabulis claui reuelluntur, <lb></lb>qua de re nos quaeſtione 6. verba fecimus. </s> <s id="s.001487">Quo pacto <expan abbr="autē">autem</expan> <pb xlink:href="007/01/157.jpg"></pb>dentis ex Dentiforcipe extractio ad vectem reducatur, <lb></lb>ſubtilius eſt perpendendum, neque enim res eſt in propa<lb></lb>tulo. </s> </p> <p type="main"> <s id="s.001488">Dicimus igitur, tum dentem ipſum, tum dentifor<lb></lb>cipem vectes eſſe, varia tamen ratione & ſatis ſane diuer<lb></lb>ſa. </s> <s id="s.001489">Dens enim fit vectis eius nempe naturæ quæ fulcimen<lb></lb>tum habet in angulo, quo caſu ipſius Dentiforcipis <expan abbr="partiū">partium</expan>, <lb></lb>quibus Dens apprehenditur, ea quæ longior eſt poten<lb></lb>tiæ mouentis loco ſuccedit, breuior vero fulcimentum <lb></lb>facit, Dentis vero reſiſtentia ponderis vices refert. </s> </p> <figure id="id.007.01.157.1.jpg" xlink:href="007/01/157/1.jpg"></figure> <p type="main"> <s id="s.001490">Eſto enim dens qui<lb></lb>dem A, cuius diameter <lb></lb>BC, longitudo vſque ad <lb></lb>extremas radices CD, <lb></lb>pars dentiforcipis breui<lb></lb>or CG, longior BG. </s> <s id="s.001491">Fit <lb></lb>ergo vectis BCD, habens <lb></lb>fulcimentum in C. </s> <s id="s.001492">Den<lb></lb>te igitur apprehenſo in BC, & manu dentiforcipe ceu ve<lb></lb>cte ad inferiora compreſſo C, fit fulcimentum centrum<lb></lb>ue. </s> <s id="s.001493">Stante enim puncto C, trahente autem potentia quæ <lb></lb>eſt in B, fit motus ipſius B, per circuli portionem BE, radi<lb></lb>cis vero D, fit motus per DF, & inde ipſius dentis extra<lb></lb>ctio facilis. </s> <s id="s.001494">Quibus conſideratis vt rem ad proportiones <lb></lb>quatenus fieri poteſt reducamus, dicimus, quo maior fu<lb></lb>erit proportio BC, ad CD, hoc eſt, partis vectis, quæ à ful<lb></lb>cimento ad potentiam ad eam quæ à fulcimento eſt ad <lb></lb>pondus, eo facilius fieri dentis auulſionem, quod vtique <lb></lb>demonſtrandum fuerat. </s> </p> <p type="main"> <s id="s.001495">Porro quod in calce quæſtionis addit Philoſophus, <lb></lb>Dentes commotos facilius manu extrahi quam inſtru<lb></lb>mento, nulla ratione probat. </s> <s id="s.001496">Ego autem arbitror, huc <lb></lb>pertinere ea verba, quæ ſuperius habentur, videlicet fer<pb xlink:href="007/01/158.jpg"></pb>rum quidem non vndique dentem <expan abbr="comprehēdere">comprehendere</expan>, quod <lb></lb>mollis facit digitorum caro, quæ id circo adhæret & com<lb></lb>plectitur magis. </s> <s id="s.001497">An autem ita ſit, alij videant, nobis enim <lb></lb>digito rem oſtendiſſe fuerit ſatis. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001498">QVÆSTIO XXII.</s> </p> <p type="head"> <s id="s.001499"><emph type="italics"></emph>Hîc quærit Ariſtoteles, Cur nuces abſque ictu facile confringuntur <lb></lb>inſtrumentis quæ ad eum faciunt vſum, & hoc licet multum aufe<lb></lb>ratur virium, ceſſante motu & violentia, quod accidit dum mal<lb></lb>leo confringuntur. </s> <s id="s.001500">Addit præterea, citius fieri confractionem <lb></lb>graui, & duro inſtrumento ferreo vide<lb></lb>licet quàm ligneo.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001501">Soluit, inquiens, id fieri quod inſtrumentum duobus <lb></lb>vectibus conſtet, coëuntibus in connexione ſeu verte<lb></lb>bra, & idcirco eo violentius fieri confractionem, quo mi<lb></lb>nus eſt ſpatium à nuce, quæ frangitur, ad vertebram. </s> <s id="s.001502">ma<lb></lb>ius verò quod à vertebra ad extremitates, quæ confrin<lb></lb>gentis manu comprimuntur. </s> <s id="s.001503">Ait igitur, & id quam oppo<lb></lb>ſite, vim ex vectibus ictus loco ſuccedere & idem operari. </s> </p> <figure id="id.007.01.158.1.jpg" xlink:href="007/01/158/1.jpg"></figure> <p type="main"> <s id="s.001504">Eſto igitur in ſtrumentum, <lb></lb>de quo agimus CDBF, ex duo<lb></lb>bus vectibus conſtans, quorum <lb></lb>alter CAF, alter vero DAB ver<lb></lb>tebra ſeu connexio A locus v<lb></lb>bi nux frangitur K, manubria <lb></lb>vero BF. quo igitur prolixiores <lb></lb>erunt AB, AF, breuiores vero ACAD, violentius fiet <expan abbr="cō-fractio">con<lb></lb>fractio</expan>. </s> <s id="s.001505">Erit autem nucis reſiſtentia loco ponderis A, ful<lb></lb>cimentum BF loco potentiæ. </s> <s id="s.001506">Itaque nî maior ſit propor<lb></lb>tio potentiæ ad reſiſtentiam, quam brachij à potentia ad <lb></lb>fulcimentum ad eam partem quæ à fulcimento eſt ad nu<lb></lb>cem, non fiet confractio. </s> <s id="s.001507">eo autem magis ſuperabit, quo <pb xlink:href="007/01/159.jpg"></pb>maior fuerit pars vectis quæ à potentia ad fulcimentum. </s> </p> <p type="main"> <s id="s.001508">Quod autem addit Ariſtoteles, eo maiorem fieri <lb></lb>vectium eleuationem, hoc eſt, inſtrumenti aperitionem, <lb></lb>quo magis nux quæ frangitur, fuerit propior fulcimento, <lb></lb>hoc eſt, ipſi vertebræ, facile oſtenditur ex conuerſa 21. <lb></lb>propoſ. lib. 1. Elem. </s> <s id="s.001509">ſi enim ab extremitatibus vnius lineæ <lb></lb>ad eaſdem partes conſtituantur duæ lineæ maiores con<lb></lb>currentes in angulo, & ab ijſdem extremitatibus duæ a<lb></lb>liæ minores, quæ intra triangulum à maioribus conſtitu<lb></lb>tum cadant, maiorem angulum continebunt. </s> <s id="s.001510">At talis eſt <lb></lb>angulus qui fit in in ſtrumento, cum partes vectis à verte<lb></lb>bra ad nucem fuerint breuiores. </s> <s id="s.001511">magìs ergo dilatantur <lb></lb>vectes, & magis dilatati magis comprimuntur, magis au<lb></lb>tem compreſſi validius frangunt, quod dixerat Ariſto<lb></lb>teles. </s> </p> <p type="main"> <s id="s.001512">Cæterum & illud quod ſcribit, ex grauiori & durio<lb></lb>ri materia inſtrumentum citius fractionem facere, quam <lb></lb>ex leuiori & minus dura, ex parte quidem materiæ verum <lb></lb>eſt, nec pertinet ad proportionem, quæ ſane in <expan abbr="huiuſmodī">huiuſmodi</expan> <lb></lb>inſtrumentis formæ ferè habent rationem. </s> <s id="s.001513">Nos hiſce in<lb></lb>ſtrumentis non vtimur. </s> <s id="s.001514">Sunt autem ſimilia inſtrumentis <lb></lb>illis, quibus figuli cretaceas pilas ad chirobaliſtarum vſum <lb></lb>facere & efformare conſueuerunt. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001515">QVÆSTIO XXIII.</s> </p> <p type="main"> <s id="s.001516">Pvlcherrimam proponit hoc loco Philoſophus con<lb></lb>templationem, eamque ad mixtos motus <expan abbr="pertinētem">pertinentem</expan>. <lb></lb></s> <s id="s.001517">Mixtorum autem motuum ſpeculationem antiquis Me<lb></lb>chanicis fuiſſe tum vtilem tum etiam familiarem, norunt <lb></lb>ij qui norunt quæ de lineis ſpiralibus Heliciſue, cyſſoidi<lb></lb>bus, conchoidibus & alijs eiuſcemodi ſcripta & contem<lb></lb>plata reperiuntur, quibus tum ad duarum mediarum pro<pb xlink:href="007/01/160.jpg"></pb>portionalium inuentionem, tum ad circuli quadratio<lb></lb>nem vti ſolent. </s> <s id="s.001518">Quod autem hîc quærit Ariſtoteles, ita ſe <lb></lb>habet. </s> </p> <p type="head"> <s id="s.001519"><emph type="italics"></emph>Cur ſi duo extrema in Rhombo puncta duabus ferantur lationibus, <lb></lb>haudquaquam æqualem vtrumque eorum pertranſit rectam, ſed <lb></lb>multo plus alteram? </s> <s id="s.001520">Item cur quod ſuper latus fertur, minus per<lb></lb>tranſeat quam ipſum latus. </s> <s id="s.001521">Illud enim diametrum pertranſire <lb></lb>certum est, hoc vero maius latus, licet hoc vnica, illud au<lb></lb>tem duabus feratur lationibus?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001522">Difficile hoc intellectu prima fronte, & ſane admi<lb></lb>rabile, itaque in tentam contemplationem requirit. </s> <s id="s.001523">Nos <lb></lb>primo cum Ariſtotele, rem totam explicabimus, tum ali<lb></lb>quid fortaſſe non pœnitendum noſtro de promptuario <lb></lb>proferemus. </s> </p> <figure id="id.007.01.160.1.jpg" xlink:href="007/01/160/1.jpg"></figure> <p type="main"> <s id="s.001524">Eſto itaque Rhombus ABCD, <lb></lb>cuius latera AB, BD, DC, CA, diame<lb></lb>trorum maior AD, minor BC, ſecan<lb></lb>tes ſe inuicem in puncto ſeu figuræ <lb></lb>centro K. </s> <s id="s.001525">Sunt <expan abbr="autē">autem</expan> ex ipſius Rhom<lb></lb>bi natura latera æqualia & parallela, <lb></lb>Angulorum vero qui maiori diame<lb></lb>tro opponuntur, recto maiores, qui <lb></lb>vero minori minores. </s> <s id="s.001526">His igitur con<lb></lb>ſideratis, intelligatur punctum A mo<lb></lb>ueri peculiari & ſimplici motu, per li<lb></lb>neam AB, ab A verſus B, & eodem <expan abbr="tē-pore">tem<lb></lb>pore</expan> moueri totam lineam AB, verſus lineam DC, hac ta<lb></lb>men lege, vt ſemper eidem DC feratur parallela, & eius <lb></lb>alterum extremorum feratur per AC, alterum vero per <lb></lb>BD, Intelligatur etiam punctum B moueri eodem tem<lb></lb>pore proprio motu, eoque ſimplici, per eandem rectam <lb></lb>BA, verſus A, & cum eadem, vt dictum eſt, mota; ferri ver-<pb xlink:href="007/01/161.jpg"></pb>ſus CD. </s> <s id="s.001527">Erunt autem ſemper AB puncta in eadem linea <lb></lb>quæ mouetur, ſibi inuicem ex contrarijs partibus occur<lb></lb>rentia. </s> <s id="s.001528">Itaque cum ex duobus motibus ſemper propor<lb></lb>tionalibus, hoc eſt, laterum proportione ſeruata, recta <lb></lb>producatur, vt demonſtratum eſt à principio, vbi produ<lb></lb>ctio circuli ex Philoſophi mente eſt declarata, <expan abbr="vtraq;">vtraque</expan> pun<lb></lb>cta quæ eandem laterum proportionem ſeruantia <expan abbr="mouē-tur">mouen<lb></lb>tur</expan>, rectas lineas <expan abbr="producēt">producent</expan> A quidem AD, B autem ipſam <lb></lb>BC. </s> <s id="s.001529">Feratur igitur A, tum mixto tum ſimplici motu per <lb></lb>diametrum AD. </s> <s id="s.001530">B vero quoque tum mixto, tum proprio <lb></lb>per diametrum BC, ſupponitur autem motus omnes ſim<lb></lb>plices, tum punctorum, tum etiam lineae, à qua puncta ipſa <lb></lb>feruntur, æquali velocitate fieri. </s> <s id="s.001531">Illud igitur mirabile eſt, <lb></lb>cuius etiam ratio quæritur, quo pacto eodem tempore ea<lb></lb>dem que velocitate latum A quidem totam percurrat AD <lb></lb>maiorem, B vero totam BC, eamque longe minorem? <lb></lb></s> <s id="s.001532">Porro neceſſe fuit rem in Rhombo ſpeculari, non autem <lb></lb>in quadrato & altera parte longiori rectangulo, in quibus <lb></lb>diametri (quod Rhombo non accidit) ſunt æquales. </s> <s id="s.001533">Ima<lb></lb>ginemur igitur A, proprio motu percurriſſe ſpatium AE, <lb></lb>nempe ipſius AB lineæ dimidium. </s> <s id="s.001534">Erit igitur in E, item li<lb></lb>neam totam AB eodem tempore pertranſiſſe dimidia op<lb></lb>poſitarum linearum, ACBD, & eſſe translatam, vbi FKG. <lb></lb></s> <s id="s.001535">Quoniam igitur æquali celeritate lineæ AB extremitas <lb></lb>A, translata eſt in F & A, punctum per eam motum in E, e<lb></lb>rit ſpatium AE, æquale ſpatio AF. </s> <s id="s.001536">Ductis igitur lineis <lb></lb>FKG, EKH lateribus AB, AC æquidiſtantibus, erit figura <lb></lb>AEKF. </s> <s id="s.001537">Rhombus ſimilis quidem Rhombo ABCD, recta <lb></lb>igitur FK æqualis erit oppoſitæ AE. quare A punctum <lb></lb>translatum erit ex mixto motu in K. </s> <s id="s.001538">Eodem pacto <expan abbr="quoniã">quoniam</expan> <lb></lb>punctum B. eadem velocitate mouetur verſus A, & linea <lb></lb>AB verſus CD, cum B fuerit in E extremum lineæ motæ <lb></lb>BA, <expan abbr="nēpe">nempe</expan> B erit in G. æquales ergo ſunt BE, BG & Rhom<pb xlink:href="007/01/162.jpg"></pb>bus EBGK, circa diametrum BKC ipſi Rhombo ABCD <lb></lb>ſimilis, & ideo GK æqualis oppoſitæ BE & BG æqualis <lb></lb>EK. </s> <s id="s.001539">Cum ergo B confecerit ſpatium BE, erit ex mixto <lb></lb>motu in K, ſuperato nempe ſpatio BK, idque eodem tem<lb></lb>pore quo A percurrerat totum ſpatium AK. </s> <s id="s.001540">Ex æquali i<lb></lb>gitur ſimplicium motuum velocitate, in æqualia ſpatia <lb></lb>AB puncta pertranſierunt, quæ res miraculo, cuius dilu<lb></lb>tio quæritur, præbet occaſionem. </s> </p> <p type="main"> <s id="s.001541">Porro quod de dimidijs diametris demonſtratum <lb></lb>eſt, poſſumus & de totis eadem ratione concludere, quip<lb></lb>pe quod eadem ſit proportio partium ad partes, quæ to<lb></lb>tius ad totum. </s> <s id="s.001542">Hæc igitur prima eſt pars propoſitæ quæ<lb></lb>ſtionis. </s> <s id="s.001543">Secunda vero dubitatio ita habet; Nempe mirum <lb></lb>videri punctum B, cum peruenerit in C, extremum lineæ <lb></lb>BA, videlicet ipſum B, translatum eſſe in D, licet æquali<lb></lb>ter moueantur linea BA, per lineam BD, & punctum B per <lb></lb>lineam BA. ſitque BC ipſa BD maior. </s> <s id="s.001544">Primam dubitatio<lb></lb>nem hoc pacto ſoluit Philoſophus; A fertur tum proprio, <lb></lb>tum alieno motu, hoc eſt, lineæ AB verſus oppoſitam par<lb></lb>tem CD, Itaque cum vterque motus deorſum vergat, mo<lb></lb>tus fit velocior. </s> <s id="s.001545">Contra vero B proprio quidem motu fer<lb></lb>tur verſus A, hoc eſt, ſurſum, alieno vero, hoc eſt, lineæ BA <lb></lb>verſus D, hoc eſt, deorſum, qui motus cum inuicem aduer<lb></lb>ſentur, motus ipſe fit tardior, non igitur eſt mirum, A eo<lb></lb>dem tempore maius ſpatium pertranſire quam B. </s> </p> <p type="main"> <s id="s.001546">Hæc ſolutio non modo vera videtur, ſed mirabilis <lb></lb>& ipſomet Philoſopho digniſſima, cui quidem <expan abbr="temerariū">temerarium</expan> <lb></lb>iudicaremus contradicere, nî in genere verſaremur, in <lb></lb>quo non probabilia quæruntur, ſed demonſtrata, ſed ve<lb></lb>ra. </s> <s id="s.001547">Futilem igitur eſſe rationem hanc ipſius Ariſtotelis <lb></lb>pace, hoc pacto oſtendemus. </s> </p> <p type="main"> <s id="s.001548">Eſto quadratum ABCD, cuius diametri ACBD ſe<lb></lb>cantes ſeſe in E, moueatur eodem pacto BA, verſus CD, <pb xlink:href="007/01/163.jpg"></pb><figure id="id.007.01.163.1.jpg" xlink:href="007/01/163/1.jpg"></figure><lb></lb>item A, verſus B, & B verſus A, ita<lb></lb>que punctum A tum proprio tum <lb></lb>alieno, hoc eſt lineæ illud <expan abbr="deferē-tis">deferen<lb></lb>tis</expan> motu deorſum trudet, hoc eſt, <lb></lb>verſus CD. </s> <s id="s.001549">Motus ergo velocior <lb></lb>erit motu puncti B, quod lationi<lb></lb>bus fertur ferè contrarijs, hoc eſt, <lb></lb>ex B verſus A ſurſum, cum linea <lb></lb>autem BA verſus C deorſum. </s> <s id="s.001550">Ve<lb></lb>locius tamen non mouetur, quip<lb></lb>pe quod æquali tempore æquale <lb></lb>ſpatium vtrum que punctum conficiat. </s> <s id="s.001551">Stante igitur cauſ<lb></lb>ſa ſequi debuiſſet effectus; non ſequitur autem, Ariſtote<lb></lb>lis igitur cauſſa non eſt cauſſa. </s> <s id="s.001552">Rhombo quoque inuerſo <lb></lb>idem clarius oſtendemus hoc pacto: Sit Rhombus ABCD, <lb></lb><figure id="id.007.01.163.2.jpg" xlink:href="007/01/163/2.jpg"></figure><lb></lb>cuius diametri AC, BD ſecan<lb></lb>tes ſeſe in E. </s> <s id="s.001553">Mota igitur linea <lb></lb>AB verſus CD, nempe deorſum <lb></lb>& A quoque deorſum verſus B, <lb></lb>contra vero B quidem ſur<lb></lb>ſum verſus A, deorſum vero <lb></lb>verſus C, erit B tardior A, ſed <lb></lb>contrarium fit, quippe quod <lb></lb>longior ſit BD, per quam mouetur B ipſa AC, per quam <lb></lb>mouetur A. </s> </p> <p type="main"> <s id="s.001554">His igitur non ſatisfacientibus veriorem ſi per im<lb></lb>becillitatem noſtram licuerit, huius effectus cauſſam in<lb></lb>ueſtigabimus. </s> <s id="s.001555">Rationibus igitur & veritate contra aucto<lb></lb>ritatem & probabilitatem eſt nobis pugnandum: quod & <lb></lb>intrepide faciemus. </s> </p> <p type="main"> <s id="s.001556">Dicimus igitur, in quouis parallelogrammo ſit illud <lb></lb>quadratum aut altera parte longius, vel idem Rhombus <lb></lb>Rhomboiſue ſemper mixtos motus proportione ſeruata <pb xlink:href="007/01/164.jpg"></pb>fieri per diametros. </s> <s id="s.001557">Cæterum díametrorum ad latera <lb></lb>proportiones eſſe varias (quadratis exceptis, in quibus ea<lb></lb>dem eſt ſemper) exploratiſſimum. </s> <s id="s.001558">Illud quoque certum <lb></lb>eſt, in rectangulis nunquam dari poſſe diametros lateri<lb></lb>bus vtcunque captis æquales, ſemper enim diametri re<lb></lb>ctis angulis ſubtruduntur. </s> <s id="s.001559">In Rhombis vero & Rhombo<lb></lb>idibus diametrorum ad latera proportiones variant. </s> <s id="s.001560">Dari <lb></lb>enim poſſunt diametri lateribus longiores item æquales, <lb></lb>& lateribus quoque ipſis breuiores. </s> </p> <p type="main"> <s id="s.001561">Itaque diametrorum & laterum varia adinuicem <lb></lb>ratione ſe habentibus, attentis proportionibus, <expan abbr="mixtorū">mixtorum</expan> <lb></lb>& ſimplicium motuum diuerſa fiet, & varia comparatio. <lb></lb></s> <s id="s.001562">in quadratis motus mixtus, qui per diametros ſemper ve<lb></lb>locior erit ſimplici qui per latera, Idem quoque in altera <lb></lb>parte longiori, in quo mixti quidem motus per diametros <lb></lb>erunt velociores, ſimplices vero qui per latera, tardiores <lb></lb><expan abbr="quidē">quidem</expan>, ſed ex illis tardior qui per latus breuius. </s> <s id="s.001563">In Rhom<lb></lb>bis autem mixtus motus qui fit per diametros inæqualis. <lb></lb></s> <s id="s.001564">Velocior enim qui per longiorem diametrum, tardior <lb></lb>qui per breuiorem. </s> <s id="s.001565">Itaque ſimplices motus punctorum <lb></lb>per latera ad eum qui fit per diametros non eodem pacto <lb></lb>ſe habent. </s> <s id="s.001566">Porro cum Rhomboides variæ ſint <expan abbr="diametrorū">diametrorum</expan> <lb></lb>ad latera habitudines, varia quoque dari poteſt propor<lb></lb>tio. </s> <s id="s.001567">aliquando enim diametri dari poſſunt lateribus maio<lb></lb>res quando que, alter eorum minor. </s> <s id="s.001568">Si autem Rhombus in <lb></lb>duos ſoluatur triangulos, alter diametrorum datur æqua<lb></lb>lis æqualibus lateribus æquicrurium triangulorum; <expan abbr="itaq;">itaque</expan> <lb></lb>in iſtis mixti motus per diametros aequeveloces erunt ſim<lb></lb>plicibus, qui per latera longiora, velociores autem illis <lb></lb>qui per latera breuiora. </s> <s id="s.001569">His igitur hoc pacto non perfun<lb></lb>ctoriè conſideratis, facile ex proprijs cauſſis, nî fallimur, <lb></lb>hocce Ariſtotelicum & mirabile Problema ſoluitur. </s> </p> <pb xlink:href="007/01/165.jpg"></pb> <figure id="id.007.01.165.1.jpg" xlink:href="007/01/165/1.jpg"></figure> <p type="main"> <s id="s.001570">Eſto enim Rhombus ABDC, <lb></lb>cuius diameter longior AD maior ſit <lb></lb>tum lateribus, tum etiam altera dia<lb></lb>metro BC. ſecent autem ſe inuicem <lb></lb>diametri in E. </s> <s id="s.001571">Ducatur queue ipſis AB, <lb></lb>CD, parallela FG ſecans longiorem <lb></lb>diametrum AD, in H, breuiorem ve<lb></lb>ro BC in I. & per I ipſis BD AC paral<lb></lb>lela ducatur KIL, Cum ergo B mixto <lb></lb>motu per diametrum BC erit in I & <lb></lb>A per diametrum AD, mixto ſimili<lb></lb>ter motu erit in H, & quia motus mi<lb></lb>xti fiunt per diametros, vt dictum eſt, <lb></lb>vt ſe habet AD ad BC, ita AE ad EB, per 15. propos. 5. elem. <lb></lb></s> <s id="s.001572">item vt AE ad EB, ita per 4. propoſ. 6. AH ad BI. eſt enim <lb></lb>IH ipſi AB parallela. </s> <s id="s.001573">Longior eſt autem AH ipſa BI, quip<lb></lb>pe quod AE longior ſit ipſa EB. motus igitur mixtus pun<lb></lb>cti A per diametrum AD vſque ad H velocior eſt motu B, <lb></lb>per diametrum BC vſque ad I. </s> <s id="s.001574">Mota igitur linea AB mo<lb></lb>uebuntur communia eius & diametrorum BC, AD pun<lb></lb>cta, quibus ſecantur ſemper diametrorum proportione <lb></lb>ſeruata. </s> <s id="s.001575">Quibus ita ſe habentibus, nil mirum eſt punctum <lb></lb>A motum per AD velociorem eſſe mixto motu puncti B, <lb></lb>quod per minorem diametrum fertur BC. quod fuerat <lb></lb>demonſtrandum. </s> <s id="s.001576">quatenus vero ad ſecundam problema<lb></lb>tis partem pertinet, dicimus Propoſitionem non eſſe vni<lb></lb>uerſalem. </s> <s id="s.001577">Si enim Rhombus detur, ex duobus æquilateris <lb></lb>triangulis conſtans, breuior diameter lateribus erit aequa<lb></lb>lis, quare non mouebitur citius motu ſimplici punctum <lb></lb>per latus ac faciat mixto per minorem diametrum, quod <lb></lb>vt mirum propoſuerat Ariſtoteles. </s> <s id="s.001578">Si autem latus ipſum <lb></lb>breuiori diametro ſit <expan abbr="lōgius">longius</expan>, nec mirum quoque erit ſim<lb></lb>plici motu moueri velocius quam mixto, quippe quod, vt <pb xlink:href="007/01/166.jpg"></pb>dictum eſt, motus iſti à proportionibus linearum, per quas <lb></lb>mouentur, legem velocitatis atque tarditatis accipiant. <lb></lb></s> <s id="s.001579">Hæc igitur nos circa hoc mirabile Ariſtotelicum proble<lb></lb>ma conſiderare ſit ſatis. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001580">QVÆSTIO XXIV.</s> </p> <p type="main"> <s id="s.001581">Mirabilem aliam quæſtionem proponit Ariſtoteles, <lb></lb>quæ itidem ad mixtos motus pertinet. </s> </p> <p type="main"> <s id="s.001582"><emph type="italics"></emph>Dubitatio est, quam ob cauſſam maior circulus æqualem minori <lb></lb>circulo circumuoluitur lineam, quando circa idem centrum fue<lb></lb>rint poſiti. </s> <s id="s.001583">Seorſum autem reuoluti quemadmodum alterius ma<lb></lb>gnitudo ad alterius magnitudinem ſe habet, ita & illorum adin<lb></lb>uicem fiunt lineæ? </s> <s id="s.001584">Præterea vno etiam & eodem vtriſque exiſten<lb></lb>te centro. </s> <s id="s.001585">Aliquando quidem tanta ſit linea, quam conuoluuntur, <lb></lb>quantum minor per ſe conuoluitur circulus, quandoque vero quan<lb></lb>tum maior.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001586">Hæc ille, qui vt prober maiorem circulum in ſua ro<lb></lb>tatione maiorem lineam pertranſire, minorem vero mi<lb></lb>norem; ait ſenſu cognoſci angulum maioris circuli, id eſt, <lb></lb>eius qui maiorem habet circumferentiam, eſſe maiorem, <lb></lb>eius vero qui minorem, minorem. </s> <s id="s.001587">Ita autem ſe habere cir<lb></lb>cumferentias vt ſe habent anguli, & eandem <expan abbr="proportionē">proportionem</expan> <lb></lb>habere per quas tum maior, tum minor circulus circum<lb></lb>uoluuntur. </s> <s id="s.001588">Ad quorum clariorem intelligentiam ea re<lb></lb>uocare oportet in memoriam, quæ dixit de maiorum cir<lb></lb>culorum ad minores circulos nutu. </s> <s id="s.001589">Hic enim, quod ibi <lb></lb>quoque fecerat, ſectorem ipſum angulum appellauit, an<lb></lb>gulum vero maiorem maioris circuli ſectorem, & mino<lb></lb>rem angulum minoris ipſius circuli ſectorem dixit. </s> <s id="s.001590">Clau<lb></lb>dit igitur dicens: quoniam circumferentiæ ſe habent vt <lb></lb>anguli, hoc eſt, vt ſectores, maior erit circumferentia ma<lb></lb>ioris circuli, & ex conſequenti maior linea, per quam cir-<pb xlink:href="007/01/167.jpg"></pb>cumuoluitur, ea per quam minor. </s> <s id="s.001591">Demonſtrationem ve<lb></lb>ro ex ſenſu petijt. </s> <s id="s.001592">Sat autem erat ſi dixiſſet, ita ſe habere <lb></lb>circumferentias vt ſe habent diametri ſeu ſemidiametri, <lb></lb>& ideo lineas in rotatione deſcriptas inuicem ſe habere vt <lb></lb>diametros. </s> <s id="s.001593">Obſcuriuſculè, hæc ſua figura oſtendit Ariſto<lb></lb>teles. </s> <s id="s.001594">Nos igitur claritatem amantibus, noſtram aliquan<lb></lb>to, nî fallimur, clariorem, proponemus. </s> </p> <figure id="id.007.01.167.1.jpg" xlink:href="007/01/167/1.jpg"></figure> <p type="main"> <s id="s.001595">Eſto circulus <lb></lb>maior ABCD, mi<lb></lb>nor FGHI, circa i<lb></lb>dem, & commune <lb></lb><expan abbr="cētrum">centrum</expan> E. </s> <s id="s.001596">Circum<lb></lb>uoluatur maior ad <lb></lb>partes D. </s> <s id="s.001597">Sint <expan abbr="autē">autem</expan> <lb></lb>diametri, maioris <lb></lb><expan abbr="quidē">quidem</expan> AEC, BED, <lb></lb>minoris verò FEH, <lb></lb>GEI, fitque CD, <lb></lb>quadrans maioris, <lb></lb>HI vero minoris circuli. </s> <s id="s.001598">Moto igitur maiori circulo <expan abbr="ſecū-dum">ſecun<lb></lb>dum</expan> abſidem, cum D fuerit in K erit CK ipſi CD æqualis, <lb></lb>fietque; DE ex puncto K perpendicularis ipſi CK, eritque vbi <lb></lb>KO, & quia punctum I eſt in linea DE, erit I facta <expan abbr="quadrã-tis">quadran<lb></lb>tis</expan> rotatione in linea KO vbi L, centrum vero E in ipſa <lb></lb>KO, vbi O. </s> <s id="s.001599">Reuoluto igitur quadrante maioris, & confe<lb></lb>cto ſpatio CK minoris circuli quadrans HI conficiet ſpa<lb></lb>tium HL, quod ipſi CK ſpatio eſt æquale. </s> <s id="s.001600">quod autem in <lb></lb>quadrantibus fit, in totis etiam fit circulis. </s> <s id="s.001601">Motus igitur <lb></lb>minor circulus circa centrum E, vnica rotatione æquauit <lb></lb>ſpatium rotationis maioris circuli. </s> <s id="s.001602">Mirabile itaque eſt mi<lb></lb>norem circulum eodem tempore & circa idem centrum <lb></lb>circumuolutum, lineam pertranſiſſe æqualem circumfe<lb></lb>rentiæ maioris circuli. </s> <s id="s.001603">Nec ſecius admirationem facit ro<pb xlink:href="007/01/168.jpg"></pb>tato minori circulo, maiorem vna <expan abbr="circumuolutū">circumuolutum</expan> lineam <lb></lb>metiri circumferentiæ minoris circuli æqualem. </s> <s id="s.001604">Rotetur <lb></lb>enim minoris circuli quadrans HI per rectam HL. erit i<lb></lb>gitur punctum I vbi M, æquali exiſtente recta HM, ipſi <lb></lb>curuæ HI. </s> <s id="s.001605">Tunc autem facto motu centrum E erit vbi P, <lb></lb>exiſtente EP, ipſi HM æquali, demittatur autem ex P per <lb></lb>M, ipſis HL CK perpendicularis PMN. </s> <s id="s.001606">Et quoniam in <lb></lb>eadem linea ſunt DIE, vbi E fuerit in PI erit in M, & D in <lb></lb>N. quamobrem rotata quarta minoris circuli parte, ma<lb></lb>ioris interim circuli quadrans confecit ſpatium CN æ<lb></lb>quale ipſi HM, hoc minus circuli quadranti HI, quod vti<lb></lb>que eſt admirabile. </s> </p> <p type="main"> <s id="s.001607">Porro cauſſam effectus huius mirifici diligenter quæ<lb></lb>rit Philoſophus, & inueneram accurate explicat. </s> <s id="s.001608">Occur<lb></lb>rit autem primo abſurdæ cuidam opinioni. </s> <s id="s.001609">Diceret enim <lb></lb>quiſpiam, ideo tardius moueri maiorem circulum, ad mo<lb></lb>tum minoris, quod interim <expan abbr="dū">dum</expan> minor moueretur, aliquas <lb></lb>inter rotandum moras interponeret, minor vero ad mo<lb></lb>tum maioris ſpatia aliqua tranſiliret, & ita ſpatiorum fieri <lb></lb>ad æquationem. </s> <s id="s.001610">Porro demonſtrationem aggreſſurus haec <lb></lb>aſſumit principia. </s> <s id="s.001611">Eandem aequalemue potentiam, <expan abbr="aliquã">aliquam</expan> <lb></lb>magnitudinem tardius quidem mouere, aliquam vero <lb></lb>celerius. </s> <s id="s.001612">quod autem natum eſt aptum moueri, tardius <lb></lb>moueri, ſi ſimul cum non apto nato moueri, moueatur, <lb></lb>quam ſi ſeparatim moueretur, celerius autem ſi non ſimul <lb></lb><figure id="id.007.01.168.1.jpg" xlink:href="007/01/168/1.jpg"></figure><lb></lb>cum eo moueatur. </s> <s id="s.001613">Eſto enim corpus A leue <lb></lb>quidem & aptum natum moueri ſurſum, cui <lb></lb>connectatur B, aptum natum moueri deor<lb></lb>ſum, Si quis igitur mouere conetur corpus A <lb></lb>ſurſum difficilius mouebit, & tardius <expan abbr="iunctū">iunctum</expan> <lb></lb>nempe ipſi B, quam ſi ab ipſo eſſet <expan abbr="ſeiūctum">ſeiunctum</expan>. <lb></lb></s> <s id="s.001614">Praeterea quod non ſuo, ſed alieno motu mo<lb></lb>uetur, impoſſibile eſſe plus eo moueri qui <pb xlink:href="007/01/169.jpg"></pb>mouet, ſiquidem non ſuo, ſed alieno motu mouetur. </s> <s id="s.001615">Mo<lb></lb>to igitur ſuo motu maiori circulo, minor non ſuo moue<lb></lb>tur, ſed motu maioris circuli, & ideo non plus mouetur <lb></lb>quam ille moueatur, mouetur autem maiori ſpatio quam <lb></lb>ex ſe moueretur, propterea quod maior ſit maioris circu<lb></lb>li, à quo ſimul defertur, circumferentia. </s> <s id="s.001616">Item ſi minor ſuo <lb></lb>motu circumuoluatur, maiorem feret ſecum, & ideo non <lb></lb>plus in ſua rotatione mouebitur maior, quam ipſe minor <lb></lb>circulus moueatur. </s> <s id="s.001617">Summa rei haec eſt, alterum ferri ab al<lb></lb>tero & latum ad ferentis ſpatium moueri. </s> <s id="s.001618">Licet enim al<lb></lb>tero moto, alter interim moueatur, nihil refert. </s> <s id="s.001619">Eſt enim <lb></lb>ac ſi is qui fertur, nullam habeat motionem, aut ſi eam ha<lb></lb>beat, ipſa nequaquam vtatur. </s> <s id="s.001620">quod non fit ſi vterque ſe<lb></lb>paratim circa proprium centrum moueatur, tunc enim <lb></lb>magnus magnum, paruus vero paruum ſpatium conficit. <lb></lb></s> <s id="s.001621">Hinc decipi ait Ariſtoteles illum, qui putat vtrum que cir<lb></lb>culum per ſe ſuper idem centrum in rotatione moueri, li<lb></lb>cet enim videatur, re vera non eſt. </s> <s id="s.001622">Id enim vtique certum <lb></lb>eſt, cum à maiori circulo minor fertur, circa maioris cen<lb></lb>trum motum fieri. </s> <s id="s.001623">Si vero maior à minori feratur circa mi<lb></lb>noris circuli centrum motum fieri. </s> <s id="s.001624">Hæc ferè Philoſophi <lb></lb>eſt mens, cuius ſolutionem eſſe certiſſimam, & ex veris <lb></lb>cauſſis non dubitamus. </s> </p> <p type="main"> <s id="s.001625">Hinc ad aliam eamqueue certam aſſertionem tranſi<lb></lb>mus. </s> <s id="s.001626">Dicimus enim, nullam materialem <expan abbr="rotã">rotam</expan> circa axem <lb></lb>eidem affixum, dum rotatur, poſſe eundem locum ſeruare, <lb></lb>niſi cauum fiat, quod axem ipſum recipiat, in tranſuerſa<lb></lb>rijs quibus rota ſuſtinetur & progreſſiuum axis motum <lb></lb>impediat. </s> </p> <p type="main"> <s id="s.001627">Eſto enim rota ABCD, cuius centrum E, diametri <lb></lb>AEC, BED, eſto alia minor rota GH, item minor KL, tum <lb></lb>minor NO, & adhuc minor QR, circa idem centrum E. <lb></lb></s> <s id="s.001628">Rotetur itaque ſecundum abſidem integri quadrantis <pb xlink:href="007/01/170.jpg"></pb><figure id="id.007.01.170.1.jpg" xlink:href="007/01/170/1.jpg"></figure><lb></lb>ſpatium CD, eritque <lb></lb>D, in F, item ſi ex rota <lb></lb>GH, ex quadrante <lb></lb>HT, erit T in I. </s> <s id="s.001629">Ex a<lb></lb>lijs item minoribus in <lb></lb>M, P, S. erit <expan abbr="itaq;">itaque</expan> <expan abbr="lon-giſſimū">lon<lb></lb>giſſimum</expan> ſpatium CF, <lb></lb><expan abbr="breuiſſimū">breuiſſimum</expan> vero RS, <lb></lb>Mota igitur rota cir<lb></lb>ca <expan abbr="circulū">circulum</expan> ſeu axem, <lb></lb>QR, maior rota ſpa<lb></lb>tio mouebitur RS, <lb></lb>quod ſi intra QR, circa centrum E alij infiniti imaginen<lb></lb>tur circuli, quo propio es centro fuerint, eo maioris rotæ <lb></lb>progreſſus erit minor, donec ad centrum deueniatur, vbi <lb></lb>cum non ſit circulus, nullus fiet progreſſiuus motus, ſed <lb></lb>circa ipſum centrum nulla facta loci mutatione rotabi<lb></lb>tur. </s> <s id="s.001630">At cum nulla materialis rota circa lineam punctumue <lb></lb>imaginarium conuerti poſſit, ideo axi ferreo alteriuſue <lb></lb>materiæ circa quem & cum quo circumuoluatur rota, ca<lb></lb>uum ſemirotundum incidere oportet, in quo inſertus axis <lb></lb>dum conuertitur à loco in quo conuertitur, non recedat. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001631">QVÆSTIO XXV.</s> </p> <p type="head"> <s id="s.001632"><emph type="italics"></emph>Quæritur, Cur lectulorum ſpondas ſecundum duplam faciant pro<lb></lb>portionem, hanc quidem ſex pedum, vel paulo ampliorem, illam <lb></lb>vero trium. </s> <s id="s.001633">Item cur vectes funesue non ſecundum <lb></lb>diametrum extendantur?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001634">Primam quæſtionis partem ita diluit Philoſophus, for<lb></lb>taſſe tantæ fieri ſolitos magnitudinis lectulos vt corpo<lb></lb>ribus ſint proportionem habentes, & ideo fieri ſecundum <lb></lb>ſpondas dupli longitudine nempe cubitorum quatuor, <lb></lb>latitudine vero duorum. </s> </p> <pb xlink:href="007/01/171.jpg"></pb> <p type="main"> <s id="s.001635">Noſtrates alia vtuntur proportione, ſeſquialtera, vi<lb></lb>delicet, quam Græci Hemioliam dicunt, communiter e<lb></lb>nim pedes quatuor latos faciunt plus minuſue, longos ve<lb></lb>ro circiter ſex. </s> <s id="s.001636">quod ideo fit vt in eis duo corpora commo<lb></lb>dius cubare poſſint. </s> <s id="s.001637">Lecturi autem, de quibus loquitur <lb></lb>Philoſophus, ad vnum tantummodo ſuſtinendum facti <lb></lb>videntur, quicquid tamen ſit, nullam ferè habet res ex <lb></lb>hac parte dubitationem. </s> </p> <p type="main"> <s id="s.001638">Secunda quæſtionis ſectio ea erat, Cur non <expan abbr="ſecundū">ſecundum</expan> <lb></lb>diametros funes extendantur? </s> <s id="s.001639">Reſtium funiumue in le<lb></lb>ctulis muniendis vſus non eſt apud nos. </s> <s id="s.001640">etenim feretra <lb></lb>tantum, ſeu ſandapilas, quibus defunctorum corpora ef<lb></lb>feruntur, funibus ad ea ſuſtinenda inteximus. </s> </p> <p type="main"> <s id="s.001641">Cæterum lectos tabulis ſeu aſſeribus ſternimus, qui<lb></lb>bus ſaccos paleis plenos imponimus, ſaccis vero culcitras, <lb></lb>& tormenta, ne tabularum durities cubantes offendat. <lb></lb></s> <s id="s.001642">Atqui in re facili multum laboraſſe videtur Ariſtoteles, <lb></lb>tum etiam obſcure & inuolute nimis quæſtionem tractaſ<lb></lb>ſe. </s> <s id="s.001643">Difficilem enim apud eum habet hæc explicationem, <lb></lb>tum ea quam diximus de cauſſa, tum etiam quod Græca <lb></lb>lectio & Latina verſio corrupta, vt apparet, præ manibus <lb></lb>habeantur. </s> <s id="s.001644">Sane vt veritatem hoc loco vindicaret in lu<lb></lb>cem, egregie laborauit Picolomineus nec parum profe<lb></lb>cit. </s> <s id="s.001645">Cæterum currentes non ſecundum diametrum extru<lb></lb>dantur, triplicem affert Philoſophus rationem. </s> <s id="s.001646">Prima eſt <lb></lb>vt ſpondarum ligna, minus diſtrahantur. </s> <s id="s.001647">Secunda, vt <expan abbr="pō-dus">pon<lb></lb>dus</expan> inde commodius ſuſtineatur. </s> <s id="s.001648">Tertia, vt in ipſa textura <lb></lb>minus reſtium funiumue abſumatur. </s> </p> <p type="main"> <s id="s.001649">Ad primam, cur extenſis diametraliter funibus <expan abbr="ſpō-dæ">ſpon<lb></lb>dæ</expan> ipſæ diſtrahantur diſcindanturue, nec ille nec alij do<lb></lb>cent. </s> <s id="s.001650">Ego autem demonſtrarem hoc pacto. </s> </p> <p type="main"> <s id="s.001651">Eſto ſponda ABCD, cuius longitudo AB, craſſitudo <lb></lb>AC, in ea foramen vtrinque pertinens EF, reſtis per fora-<pb xlink:href="007/01/172.jpg"></pb><figure id="id.007.01.172.1.jpg" xlink:href="007/01/172/1.jpg"></figure><lb></lb>men inditus GFE, ſitque E pars ſeu ca<lb></lb>put exterius, quod nodo in E diſtine<lb></lb>tur. </s> <s id="s.001652">Sit autem ſpondæ lignum iuxta <lb></lb>longitudinem vt natura aſſolet ſciſſile. <lb></lb></s> <s id="s.001653">Vis quædam, fune ita extento applice<lb></lb>tur in G, quae funem ipſum ad ſe violen<lb></lb>ter trahat. </s> <s id="s.001654">non diſcindetur idcirco <lb></lb>ſponda eo quod non diametraliter fu<lb></lb>nis extendatur. </s> <s id="s.001655">Modo facta capitis G <lb></lb>translatione in H, trahatur valide fu<lb></lb>nis, fiet autem preſſio valida in F. ibi e<lb></lb>nìm impedimentum facit angulus, ne funis ipſa dum tra<lb></lb>hitur, rectitudinem aſſequatur. </s> <s id="s.001656">Itaque vi præualente, li<lb></lb>gno vero ſciſſili, minus reſiſtente, funis, aſſecuta rectitudi<lb></lb>ne, fiet in HIE ſciſſa ſponda ad <expan abbr="quãtitatem">quantitatem</expan> trianguli FIE, <lb></lb>quod fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.001657">Cur autem funes ab angulo in angulum extenſæ mi<lb></lb>nus commode pondus ſuſtineant, ſatis patet. </s> <s id="s.001658">quo enim fu<lb></lb>nis <expan abbr="lōgior">longior</expan>, eo debilior, & preſſio quæ in medio fit, ea vide<lb></lb>licet parte quæ ab extremis eſt remotiſſima, magis funem <lb></lb>fatigat. </s> <s id="s.001659">Longiores autem funes ſunt quæ diametraliter <lb></lb>extenduntur. </s> </p> <figure id="id.007.01.172.2.jpg" xlink:href="007/01/172/2.jpg"></figure> <p type="main"> <s id="s.001660">Quatenus ad <expan abbr="tertiã">tertiam</expan> <lb></lb>rationem pertinet, hoc <lb></lb>pacto funes intexit <lb></lb>Philoſoph^{9}. Eſto lectu<lb></lb>lus cum ſuis <expan abbr="ſpōdis">ſpondis</expan> AB <lb></lb>CD, cuius ſponda AD, <lb></lb>ſit pedum ſex, AB vero <lb></lb><expan abbr="triū">trium</expan>, Diuidatur AD bi<lb></lb>faríam in E & BC in F. item AE in tres AG, GH, HE & in <lb></lb>totidem ED, nempe EL, LM, MD. </s> <s id="s.001661">Similiter medietas al<lb></lb>terius <expan abbr="ſpōdæ">ſpondæ</expan> BF in tres partes diſtinguatur BN, NO, OF, <pb xlink:href="007/01/173.jpg"></pb>& FC ſimiliter in tres FI, IK, KC, tum altero funis capite <lb></lb>in ducto per foramen A, ibique probe firmato, indatur per <lb></lb>F, inde per I, poſtea per GHK CE, & in E probe alligetur: <lb></lb>Erunt igitur funis quatuor partes æquales AF, IG, HK, <lb></lb>EC, quibus adijciuntur particulæ cadentes extra, quæ <lb></lb>ſunt FI, GH, KC. </s> <s id="s.001662">Poſt hæc alterius funis principium per <lb></lb>foramen traijcitur, quod eſt in angulo B. </s> <s id="s.001663">Deinde per E, in<lb></lb>de per L, N, O, M, D, F & in F probe vincitur, & nodo fa<lb></lb>cto obfirmatur. </s> <s id="s.001664">Erunt igitur aliæ quatuor alterius funis <lb></lb>partes, tum inter ſe, tum etiam ſupra dictis æquales, nem<lb></lb>pe BE, NL, OM, FD, quibus illæ pariter adijciuntur par<lb></lb>ticulæ, quæ cadat extra, videlicet EL, NO, MD. <expan abbr="quoniã">quoniam</expan> <lb></lb>igitur quadratis ex BA, AE æquale eſt quadratum BE, erit <lb></lb>BE quadratum 18. cuius latus radixue 4 1/3 quam proxime. <lb></lb></s> <s id="s.001665">Sunt autem huius longitudinis funes æquales octo. </s> <s id="s.001666">Ea<lb></lb>rum igitur ſimul ſumptarum longitudo erit pedum 34 2/3 vel <lb></lb>circiter, quibus ſi ad dantur pedes ſex funium qui cadunt <lb></lb>extra, erit reſtis totius longitudo expanſa pedum 40 2/3 plus <lb></lb>minuſue. </s> <s id="s.001667">Picolomineus vero ait 34 2/3, omiſit enim particu<lb></lb>las illas ſex, quæ, vt diximus, cadunt extra. </s> <s id="s.001668">Idem rationem <lb></lb>funium diametraliter extenſarum in idem, ait eſſe longi<lb></lb>tudinis pedum 40 1/2. Hic autem eas <expan abbr="quoq;">quoque</expan> particulas præ<lb></lb>termittit, quæ extra cadunt. </s> <s id="s.001669">Itaque his additis clare pa<lb></lb>tet, plus reſtium inſumi diametraliter ipſis, quam latera<lb></lb>liter extenſis. </s> <s id="s.001670">Cæterum ratio, qua Philoſophus hæc pro<lb></lb>bare conatur, adeo eſt mutila, inuoluta, obſcura, vt Delio <lb></lb>prorſus, vt aiunt, indigeat natatore. </s> <s id="s.001671">Huius loci in explica<lb></lb>bilem difficultatem, vidit Picolomineus, qui idcirco at<lb></lb>teſtatus eſt, interpretes in hac exponenda fuiſſe halluci<lb></lb>natos. </s> <s id="s.001672">Certe Græca lectio verſione ipſa Latina non eſt <lb></lb>clarior. </s> <s id="s.001673">Nos interim ne inutilem ferè ſpeculationem ni<lb></lb>mia diligentia, eaque fortaſſe fruſtranea proſequamur, a<lb></lb>lijs difficultatem hanc diſſoluendam aut ceu Gordij no<pb xlink:href="007/01/174.jpg"></pb>dum gladio ſcindendo relinquemus. </s> <s id="s.001674">Sed interim ſubit <lb></lb>mirari, cur veteres vtiliori modo prætermiſſo, <expan abbr="inutilioiē">inutiliorem</expan> <lb></lb>fuerint amplexati. </s> <s id="s.001675">Poterant enim reticulatim hoc per li<lb></lb>neas lateribus æquidiſtantes intexere. </s> </p> <figure id="id.007.01.174.1.jpg" xlink:href="007/01/174/1.jpg"></figure> <p type="main"> <s id="s.001676">Eſto enim lectulus <lb></lb>eiuſdem dimenſionis <lb></lb>ABCD, in cuius latere <lb></lb>AD ſint foramina quin<lb></lb>que E, F, G, H, I, totidem <lb></lb>in latere oppoſito QP, <lb></lb>ONM. </s> <s id="s.001677">Duo vero in la<lb></lb>tere breuiori AB, nempe <lb></lb>RS, & toti dem in oppoſito KL incipiatur extenſio à fora<lb></lb>mine E, per QP, F, GON, HIM & in M funis obfirmetur, <lb></lb>tum alterius funis caput in datur ſi lib et per K, & inde per <lb></lb>S, R, L & in L conſtringatur. </s> <s id="s.001678">Sunt autem omnes EQ, FP, <lb></lb>GO, NN, IM, pedum quindecim, quibus ſi addantur KS, <lb></lb>RL, ſinguli pedum ſex erunt pedum xxvii. </s> <s id="s.001679">quibus adiectis <lb></lb>particulis extra cadentibus QP, FG, ON, HI, & RS, erit <lb></lb>integra ſumma pedum xxxii. </s> <s id="s.001680">Vide igitur quantum hinc <lb></lb>minus inſumatur reſtium quam eo modo, quem proba<lb></lb>uit, & ceu vtiliorem propoſuit Ariſtoteles. </s> <s id="s.001681">Præterea vali<lb></lb>diſſimum eſt hoc texturæ opus nec ex eo fit vera ſponda<lb></lb>rum diſtractio ſciſſioue, quibus haud parum obnoxia eſt <lb></lb>ea ratio, quam præfert ipſe Philoſophus. </s> <s id="s.001682">Concludimus i<lb></lb>gitur, aut nos eius verba & ſenſum non intellexiſſe, aut <lb></lb>veteres ipſos, quorum vſum ipſe explicat, rei, quam nos <lb></lb>proponimus, naturam & commoditatem (quod ta<lb></lb>men vix credibile eſt) igno<lb></lb>rare. </s> </p> <pb xlink:href="007/01/175.jpg"></pb> </subchap1> <subchap1> <p type="head"> <s id="s.001683">QVÆSTIO XXVI.</s> </p> <p type="head"> <s id="s.001684"><emph type="italics"></emph>Proponitur à Philo ſopho examinandum, Cur difficilius ſit, langa <lb></lb>ligna ab extremo ſuper humeros ferre, quam ſecundum me<lb></lb>dium, æquali exiſtente pondere?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001685">Dvo hîc conſiderat, vibrationem, & pondus. </s> <s id="s.001686">Ait enim <lb></lb>primo fieri poſſe, procera ligna vibratione impedien<lb></lb>te, difficilius ferri. </s> <s id="s.001687">Quærerer autem quiſpiam, (ipſe enim <lb></lb>id reticet) cur vibratio hæc ferenti ſit nocua. </s> <s id="s.001688">Nos itaque <lb></lb>id expliçare conabimur. </s> </p> <figure id="id.007.01.175.1.jpg" xlink:href="007/01/175/1.jpg"></figure> <p type="main"> <s id="s.001689">Eſto igitur lignum <lb></lb>oblongum, flexile, & vt <lb></lb>ita dicam, vibrabile <lb></lb>AB, imponatur hume<lb></lb>ro, eique hæreat in C, <lb></lb>manu vero ſuſtineatur facta compreſſione in B. </s> <s id="s.001690">Nutet i<lb></lb>gitur & vibretur, in ipſa vibratione, ad partem A. </s> <s id="s.001691">Sit au<lb></lb>tem centrum grauitatis eius D, Lignum igitur in ipſa vi<lb></lb>bratione deſcendet ſua preſſus grauitate in E, tum facta <lb></lb>ligni conſtipatione in ea parte quæ eſt inferius inter C & <lb></lb>D, & inde reſiſtentia, eodem fere impetu quo deſcende<lb></lb>rat, repulſum per D, nec enim in ſua rectitudine ſtabit, a<lb></lb>ſcendet in F, facta iterum materiæ conſtipatione inter C <lb></lb>& F. </s> <s id="s.001692">Mouebitur igitur lignum ſua grauitate, motu fre<lb></lb>quentiſſimo, ſurſum deorſum, & is interim qui lignum hu<lb></lb>mero fert, procedit antrorſum, impedit igitur motus iſte, <lb></lb>qui fit ſurſum deorſum lationem, quæ fit ad anteriora; La<lb></lb>torem ipſum quodammodo retrahens. </s> <s id="s.001693">Si autem medio <lb></lb>ligno ſupponatur humerus, eo quod vibratio ſit minor. <lb></lb></s> <s id="s.001694">breuiores enim partes ſunt, quæ à medío ad extrema mi<lb></lb>nus à vibratione remorabitur ferens. </s> </p> <p type="main"> <s id="s.001695">Quoniam autem non ſola vibratio in hoc lationis <lb></lb>modo, nempe ex ligni extremitate difficultatem facit, ait <pb xlink:href="007/01/176.jpg"></pb>Philoſophus, forte id fieri, quoniam licet nihil inflecta<lb></lb>tur, neque multam habeat longitudinem, difficilius <expan abbr="tamē">tamen</expan> <lb></lb>ſit ad ferendum ab extremo, eo quod facilius eleuetur ex <lb></lb>medio quam ab extremis, & ideo ſic ferre ſit facilius. <lb></lb></s> <s id="s.001696">Cur autem ex medio facilius eleuetur, cauſſam eſſe ait, <lb></lb>quod eleuato medio ligno extrema ſeſe inuicem ſuſpen<lb></lb>dant, & altera pars alteram bene ſubleuet. </s> <s id="s.001697">Medium enim <lb></lb>fieri velut centrum, vbi is ſupponit humerum qui eleuat <lb></lb>aut fert. </s> <s id="s.001698">Extremorum autem interim altero depreſſo al<lb></lb>terum ſuſtolli. </s> <s id="s.001699">Nos interim Mechanicis principijs, quod <lb></lb>ipſe non fecit, rem clariorem efficiemus. </s> </p> <p type="main"> <s id="s.001700">Eſto enim oblongum lignum AB, cui humerus ſup<lb></lb>ponatur in B, manus vero premendo ſuſtinens in B. ſit au<lb></lb>tem ligni pars maxima AC, minima CB, inaioris autem ad <lb></lb>minorem proportio exempli gratia ſit ſexcupla. </s> <s id="s.001701">Ad hoc i<lb></lb>gitur vt fiat æquilibrium inter potentiam ſuſtinentem in <lb></lb>B, & pondus comprimens in A, ita ſe habere oportet po<lb></lb>tentiam in B, ad pondus in A, vt ſe habet pars ligni AC ad <lb></lb><figure id="id.007.01.176.1.jpg" xlink:href="007/01/176/1.jpg"></figure><lb></lb>partem CD. </s> <s id="s.001702">Eſto igitur pon<lb></lb>dus in A, puta librarum ſex. <lb></lb></s> <s id="s.001703">Erit igitur potentia quæ in B <lb></lb>ad hoc vt ſuſtineat librarum <lb></lb>triginta ſex, quas ſi addas <expan abbr="pō-deri">pon<lb></lb>deri</expan> in A, fiet humerus in C <lb></lb>ſuſtinens pondus librarum quadraginta duo. </s> <s id="s.001704">Si autem <lb></lb>humerus medio ligno, hoc eſt, in D ſupponatur, ad hoc vt <lb></lb>fiat æquilibrium, neceſſe erit potentiam in B eſſe æqua<lb></lb>lem ponderi in A, quod eſt ſex, quare humerus ſuſtinebit <lb></lb>duodecim. </s> <s id="s.001705">Vnde patet, longe difficilius portari lignum <lb></lb>ex C extremo, quam ex D medio; quod Mechanice fue<lb></lb>rat demonſtrandum. </s> </p> <p type="main"> <s id="s.001706">Poſſumus & aliter idem oſtendere. </s> <s id="s.001707">Intelligatur e<lb></lb>nim ijſdem ſuppoſitis, vectem quidem eſſe AB, cuius ful-<pb xlink:href="007/01/177.jpg"></pb>cimentum quidem B, pondus A, potentia ſuſtinens in C, <lb></lb>nempe inter fulcimentum & pondus. </s> <s id="s.001708">Res igitur ad eum <lb></lb>vectis vſum reducitur, de quo G. Vbaldus tractatu de Ve<lb></lb>cte, propoſ. 3.</s> <s id="s.001709">Quare vtile oſtendit, ita ſe habere oportet <lb></lb>potentiam ſuſtinentem ad pondus, vt totus vectis ad par<lb></lb>tem eius quæ à potentia ad fulcimentum. </s> <s id="s.001710">Ita igitur ſe ha<lb></lb>bebit preſſio, quæ fit in C ad pondus in A, vt totus vectis <lb></lb>AB ad partem eius CB, quæ à potentia ad fulcimentum. <lb></lb></s> <s id="s.001711">Erit igitur potentia ſeptupla ponderi, & ideo ſuſtinebit <lb></lb>pondus librarum quadraginta duarum. </s> <s id="s.001712">quod fuerat o<lb></lb>ſtendendum. </s> </p> <p type="main"> <s id="s.001713">Hinc alia quæſtio huic affinis ſoluitur, Cur haſta ſa<lb></lb>riſſaue ſolo iacens manu ad alteram extremitatum ap<lb></lb>prenſa difficillime extollatur? </s> </p> <figure id="id.007.01.177.1.jpg" xlink:href="007/01/177/1.jpg"></figure> <p type="main"> <s id="s.001714">Eſto igitur ſariſſa ha<lb></lb>ſtaue iacens AB, cuius ex<lb></lb>tremitati A manus ad ſu<lb></lb>ſtollendum applicetur, ſit <lb></lb>autem pars quæ digitis capitur AC, quæritur cur pars re<lb></lb>liqua CB difficillime ſuſtollatur? </s> <s id="s.001715">Facile dubitatio ex præ<lb></lb>demonſtratis ſoluitur. </s> <s id="s.001716">Eſt enim C fulcimentum, ſupponi<lb></lb>tur enim loco, pugno ad ſuſtollendum clauſo, digitus in<lb></lb>dex, potentia autem premens in A, vt ſuperet grauitatem <lb></lb>CB, eſt manus ipſius corpus, hoc eſt illa manus ipſius pars, <lb></lb>qua pondus facta ſuppreſſione ſuſtollitur. </s> <s id="s.001717">Eſt igitur AB <lb></lb>vectis, cuius fulcimentum C, pondus B, potentia A, <expan abbr="Itaq;">Itaque</expan> <lb></lb>quoniam maxima eſt proportio BA ad AC, maximam eſ<lb></lb>ſe oportet potentiam pondus ſuſtollentem in C. </s> </p> <p type="main"> <s id="s.001718">Huc etiam illud pertinet, Cur haſta ſolo iacente, ſi <lb></lb>alterum extremorum manu ſuſtollatur, alterum vero ve<lb></lb>lociſſime ſurſum vibretur, & eodem tempore manus ha<lb></lb>ſtæ ſic vibratæ ſupponatur, haud magna difficultate haſtæ <lb></lb>ad perpendiculum fit erectio. </s> </p> <pb xlink:href="007/01/178.jpg"></pb> <figure id="id.007.01.178.1.jpg" xlink:href="007/01/178/1.jpg"></figure> <p type="main"> <s id="s.001719">Sit enim haſta AB, quæ <lb></lb>manu ex B capta eleuetur in <lb></lb>C, & fiat in AC, tum facta ex <lb></lb>C partis A veloci vibratione, <lb></lb>ipſa extremitas A transferatur <lb></lb>in D, ſitque vbi CD, tum velo<lb></lb>ci manus depreſſione extremi<lb></lb>tas C transferatur in E, <expan abbr="fiatq;">fiatque</expan> <lb></lb>EF horizonti perpendicularis; <lb></lb>quod vbi factum fuerit, erunt <lb></lb>in eadem linea quæ ad centrum mundi, manus ipſa quæ <lb></lb>ſuſtinet, & grauitatis ipſius centrum G, quare manus ipſa <lb></lb>facta vibratione tantum portat, quantum præciſe ipſius <lb></lb>eſt haſtæ pondus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001720">QVAESTIO XXVII.</s> </p> <p type="head"> <s id="s.001721"><emph type="italics"></emph>Dubitatur, Cur ſi valde procerum fuerit idem pondus, difficilius <lb></lb>ſuper humeros geſtatur, etiamſi medium quiſpiam illud fe<lb></lb>rat quam ſi breuius ſit?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001722">Qvæſtio hæc ſuperiori eſt affinis. </s> <s id="s.001723">Ait autem Philoſo<lb></lb>phus, cauſſam non eſſe id, quod in præcedenti quæ<lb></lb>ſtione dixerat, ſed vibrationem: quo enim longiora ſunt <lb></lb>ligna, eo magis eorum extrema vibrantur, debiliora enim <lb></lb>ſunt & à medio remotiora, quare ſuopte pondere facilius <lb></lb>nutant. </s> <s id="s.001724">Si autem breuiora ſint ea cauſſa ceſſante minor <lb></lb>fit aut nulla vibratio, quare breuiora feruntur facilius. <lb></lb></s> <s id="s.001725">Dupliciter autem vibratione ipſa, portans offenditur, <lb></lb>tum ex cauſſa quam in ſuperiori quæſtione conſideraui<lb></lb>mus, nempe quod motus ſurſum deorſum aſſiduus, pro<lb></lb>gredientis motum impediat, tum etiam quod duplici <lb></lb>preſſione grauetur ferentis humerus, quod Philoſophus <lb></lb>non animaduertit. </s> </p> <p type="main"> <s id="s.001726">Sit enim oblongum lignum AB, quod humero me-<pb xlink:href="007/01/179.jpg"></pb><figure id="id.007.01.179.1.jpg" xlink:href="007/01/179/1.jpg"></figure><lb></lb>dio loco ſuſtineatur in C. <lb></lb>nutabunt ergo extrema AB, <lb></lb>à centro C, valde remota, <lb></lb>cadent autem ſimul A m D, <lb></lb>& B in E trahere ſecum conantes medium C, quare is qui <lb></lb>in C ſuſtinet, non modo ligni ſuſtinet pondus ex grauita<lb></lb>tis centro quod eſt in C, ſed impetum quoque in ipſa ex<lb></lb>tremorum depreſſione acquiſitum ex ipsa violentia. </s> <s id="s.001727">Illud <lb></lb>autem ſubtiliter conſideramus, portantem ex vibratione <lb></lb>per inter ualla deprimi & ſubleuari. </s> <s id="s.001728">fiat enim vibratum li<lb></lb>gnum ex contrario motu, vbi FCG. alleuiabit igitur eo <lb></lb>caſu portantem, ſiquidem impetus ex motu ipſo acquiſi<lb></lb>tus, medium C trahat ad ſuperiora. </s> <s id="s.001729"><expan abbr="Itaq;">Itaque</expan> cum eſt in DCE <lb></lb>portans plus ſuſtinet in ACD, æquale, in FCG minus, <lb></lb>quod vtique demonſtrandum fuerat. </s> <s id="s.001730">Eſt autem quæſtio <lb></lb>hæc illi familiaris, quam 16. loco explicauimus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001731">QVAESTIO XXVIII.</s> </p> <p type="head"> <s id="s.001732"><emph type="italics"></emph>Quæritur, Cur iuxta puteos celonia faciunt eo quo viſuntur mo<lb></lb>do? </s> <s id="s.001733">Ligno enim plumbi adiungunt pondus, cum alioquin vas <lb></lb>ipſum & plenum & vacuum pon<lb></lb>dus habeat.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001734">Reſpondet optime Philoſophus, hauriendi opus duo<lb></lb>bus temporibus diuidi, nempe dum vas ipſum vacuum <lb></lb>demittitur, dum que extrahitur plenum: Contingere au<lb></lb>tem, vacuum facile demitti, plenum autem difficulter ex<lb></lb>trahi. </s> <s id="s.001735">Expedire nihilominus tardius, hoc eſt difficilius di<lb></lb>mitti vt facilius extrahatur, plumbo nempe coadiuuante, <lb></lb>& ſane Philoſophi ſolutio eſt lucidiſſima. </s> <s id="s.001736">Nos autem luci <lb></lb>ipſi lucem aliquam adhuc afferre conabimur. </s> </p> <p type="main"> <s id="s.001737">Eſto Celomum (Latine Tolenonem appellant) ABC, <lb></lb>cuius arrectarium BD, tranſuerſum lignum AC, quod <pb xlink:href="007/01/180.jpg"></pb><figure id="id.007.01.180.1.jpg" xlink:href="007/01/180/1.jpg"></figure><lb></lb>conuertitur, circa <expan abbr="pūctum">punctum</expan> ſeu <lb></lb>fulcimentum B, pondus, plum<lb></lb>bumue, vbi A, ſitula E, funi ap<lb></lb>penſa CE. </s> <s id="s.001738">Dico rebus ita con<lb></lb>ſtitutis difficilem quidem eſſe <lb></lb>vacuæ ſitulæ demiſſionem, fa<lb></lb>cile vero eiuſdem extractio<lb></lb>nem. </s> <s id="s.001739">Vectis diuiſi, ſitulæ, ac <lb></lb>ponderis, ad hoc vt fiat æquili<lb></lb>brium, ca debet eſſe propor<lb></lb>tio, vt quemadmodum ſe habet AB ad BC, ita ſe habeat <lb></lb>plenæ ſitulæ pondus E ad ipſum pondus A, ſuperabit ergo <lb></lb>pondus in A ſitulam vacuam in E nec fiet æquilibrium, i<lb></lb>taque vt vacua ſitula demittatur, tanta vis adhibenda eſt <lb></lb>quantum eſt ipſius aquæ, qua ſitula impletur pondus, quæ <lb></lb>vis dum apponitur difficilem, vt dicebamus, efficit ſitulæ <lb></lb>vacuæ demiſſionem. </s> <s id="s.001740">Plena vero ſitula ſit æquilibrium, vn<lb></lb>de quantumuis puſilla vi adhibita, ſitula extrahitur, quaſi <lb></lb>ex ſemetipſa ponderis appenſi virtute aſcendens. </s> <s id="s.001741">Quan<lb></lb>tum igitur pondus dum vacua demittitur impedit, tan<lb></lb>tundem plena dum extrahitur, adiuuat. </s> <s id="s.001742">Quae cum ita ſint, <lb></lb>ſi paria ſunt difficultas in demittendo, & facilitas in ex<lb></lb>trahendo, quæ ratio hoc in negotio vtilitatis? </s> <s id="s.001743">Sane ſitula <lb></lb>vacua, manu per funem facile demittitur, plena vero dif<lb></lb>ficile extrahitur, vſu autem Celonij res <expan abbr="permutãtur">permutantur</expan>. </s> <s id="s.001744">Cor<lb></lb>poris enim proprij pondere, dum premit, adiuuatur de<lb></lb>mittens, qui per funem ſimplicem extrahendo, ab eodem <lb></lb>proprij corporis pondere impediebatur. </s> <s id="s.001745">quod quidem ex <lb></lb>corporis pondere, auxilium, ingentem parit in extrahen<lb></lb>do commoditatem. </s> </p> <p type="main"> <s id="s.001746">Quippiam ſimile accidit, aquas è puteis extrahen<lb></lb>tibus vſu trochleæ. </s> <s id="s.001747">Sit enim trochlea puteo imminens <lb></lb>ABCD, cuius centrum E ſuſpenſa quidem in A, funis, cui <pb xlink:href="007/01/181.jpg"></pb>ſitula ſuſpenditur FCABG, ſitula vero G. </s> <s id="s.001748">Eſt igitur dia<lb></lb>meter CED, inſtar libræ, quare vt fiat æquilibrium neceſ<lb></lb>ſe eſt capiti funis F, potentiam applicare, quæ ſit æqualis <lb></lb><figure id="id.007.01.181.1.jpg" xlink:href="007/01/181/1.jpg"></figure><lb></lb>pondere ſitulæ aqua plenæ, itaque extra<lb></lb>hens proprijs viribus <expan abbr="corporīs">corporis</expan> pondus ad<lb></lb>ijciens facile ſitulam aqua plenam extra<lb></lb>hit, ex qua re magna extrahentibus fit <lb></lb>commoditas. </s> <s id="s.001749">Patet autem diuerſo modo <lb></lb>extrahentes iuuare Celonium. </s> <s id="s.001750">& Tro<lb></lb>chleam, ibi enim corporis mole adiuuatur <lb></lb>demittens vacuam, hic vero qui extrahit <lb></lb>plenam aqua ſitulam. </s> </p> <p type="main"> <s id="s.001751">Cæterum Celonij partem BC, qui à <lb></lb>fulcimento ad funem longe maiorem eſ<lb></lb>ſe oportet, ipſa AB, vt ſitula in profundum poſſit demitti, <lb></lb>quamobrem ita ſe debet habere pondus in A, ad pondus <lb></lb>ſitulæ plenæ, vt ſe habet brachium ſeu pars BC, ad par<lb></lb>tem BA. </s> <s id="s.001752">Tunc enim ex permutata proportione efficitur <lb></lb>æquilibrium. </s> </p> <p type="main"> <s id="s.001753">Illud addimus, nouum non aeſſe Architectis Mecha<lb></lb>niciſque, tum hominum tum animalium vt commodius <lb></lb>machinas moueant, adhibere pondera corporum. </s> <s id="s.001754">Nec e<lb></lb>nim alia ratione mouentur Rotæ illæ, quas ob hanc cauſ<lb></lb>ſam ambulatorias vocant; quarum vſus ad Mangana, ad <lb></lb>extrahendas è puteis aquas, & ad farinarias quoque mo<lb></lb>las agitandas adhibetur. </s> </p> <p type="main"> <s id="s.001755">Porro Tollenonem bellicam Machinam à Celonio <lb></lb>tum forma tum poteſtate nihil differre, videre eſt apud <lb></lb>veteres Mechanicos, Heronem Byzantium, & alios. </s> <s id="s.001756">apud <lb></lb>neotericos vero hac de re agunt Daniel Barbarus in Vi<lb></lb>truuium, & Iuſtus Lipſius in librum quem de bellicis <lb></lb>machinis edidit, elegantiſſi<lb></lb>mum. </s> </p> <pb xlink:href="007/01/182.jpg"></pb> </subchap1> <subchap1> <p type="head"> <s id="s.001757">QVAESTIO XXIX.</s> </p> <p type="head"> <s id="s.001758"><emph type="italics"></emph>Dubitatur, Cur quando ſuper ligno, aut huiuſmodi quopiam, duo <lb></lb>portauerint homines, idem pondus non æqualiter premun<lb></lb>tur, ſed ille magis cui vicinius fuerit <lb></lb>pondus?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001759">Soluit Ariſtoteles, inquiens, lignum eſſe vectem, pon<lb></lb>dus vero fulcimentum; res quæ mouetur is qui ponde<lb></lb>ri eſt proximior: mouens vero qui remotior. </s> <s id="s.001760">Itaque quo <lb></lb>magis remotus eſt à pondere, hoc eſt, à fulcimento is qui <lb></lb>mouet, eo violentius is premitur qui altera vectis parte <lb></lb>eaque breuiori, mouetur. </s> </p> <figure id="id.007.01.182.1.jpg" xlink:href="007/01/182/1.jpg"></figure> <p type="main"> <s id="s.001761">Eſto lignum AB, pondus <lb></lb>C appenſum in E, vicinius ex<lb></lb>tremo B quam ipſi A, ſit <expan abbr="autē">autem</expan> <lb></lb><expan abbr="portãtium">portantium</expan> alter quidem AF, <lb></lb>alter vero BG, Imaginemur <lb></lb>itaque locum E à pondere ita <lb></lb>figi & deprimi, vt ſurſum qui<lb></lb>dem ferri nequaquam poſſit, <lb></lb>circa vero punctum E, ceu <lb></lb>circa centrum fulcimentum<lb></lb>ne ipſum vectem conuerti. </s> <s id="s.001762">Lignum ergo AB vectis: mo<lb></lb>uens potentia A, pars vectis à potentia ad fulcimentum <lb></lb>AE pars eiuſdem quæ à fulcimento ad rem motam EB, & <lb></lb>quoniam quanto longior eſt pars vectis EA ipſa EB, eo fa<lb></lb>cilius potentia quæ eſt in A, operatur in id quod eſt in B, ſi <lb></lb>res ad proportiones redigatur, erit potentia in A, ad id <lb></lb>quod mouetur ſeu premitur in B, vt pars vectis EB ad par<lb></lb>tem EA, ſed maior eſt AE ipſa EB, ergo maiorem partem <lb></lb>ſuſtinet ponderis, & plus premitur is qui in E, & qui mo<lb></lb>uet in A. </s> <s id="s.001763">Hæc fere Philoſophi eſt ſententia: Picolomi<lb></lb>neus vero Paraphraſtes appoſite duos vectes in vnico li-<pb xlink:href="007/01/183.jpg"></pb>gno conſiderat, alterum AB, alterum BA, in primo A eſt <lb></lb>mouens B, motum in ſecundo B, mouens A vero motum <lb></lb>in quibus vectibus ſemper idem & commune fulcimen<lb></lb>tum E. </s> <s id="s.001764">Et quoniam in propoſito diagrammate breuior eſt <lb></lb>pars vectis EB, quæque à mouente ad fulcimentum, parte <lb></lb>illa quæ ab eodem fulcìmento ad rem motam, minus o<lb></lb>peratur B in A, quam A in B, & ideo qui in B mouetur plus <lb></lb>premitur, contra vero quia maior eſt pars EA ipſa parte <lb></lb>EB, magis operatur qui in A in ipſum B, quam econtra. </s> <s id="s.001765">Et <lb></lb>ſane conſideratio hæc ſubtilis eſt & ingenioſa, & quæ ſi <lb></lb>recte intelligatur, quatenus ad proportiones & effectum <lb></lb>ipſum demonſtrandum pertinet, à veritate ipſa non ab<lb></lb>horret, Quicquid tamen ſit, Mechanice magis hoc pacto <lb></lb>quæſtio diluetur. </s> <s id="s.001766">Dicimus enim, pondus quidem vere eſ<lb></lb>ſe pondus, non autem fulcimentum, vt ſibi fingebat Ari<lb></lb>ſtoteles: lignum vero vectem, duo autem qui pondus ſu<lb></lb>ſtinent pro duplici fulcimento haberi, vtriſque enim ve<lb></lb>ctis cum appenſo pondere innititur. </s> <s id="s.001767">Poteſt etiam alter <lb></lb>eorum pro potentia mouente, alter vero pro fulcimen<lb></lb>to, & ſic viciſſim. </s> <s id="s.001768">Eſt autem, quomodocunque res accipia<lb></lb>tur, pondus inter fulcimentum. </s> <s id="s.001769">& potentiam. </s> <s id="s.001770">Quare ex <lb></lb>ijs quæ demonſtrauit G. Vbald. de hoc vectis genere lo<lb></lb>quens, vt ſe habet AE pars ad AB vectem totum, ita po<lb></lb>tentia quæ ſuſtinet in B, ad pondus appenſum in E, & vt <lb></lb>BE ad BA ita potentia quæ ſuſtinet in A ad pondus quod <lb></lb>in E. </s> <s id="s.001771">At minor eſt proportio BE, ad BA, quam AE ad AB, <lb></lb>quare magis ſuperatur pondus in E à potentia quæ in A, <lb></lb>quam à potentia quæ in B, & ideo plus ponderis ſuſtinet <lb></lb>ferens in B, quam ferens in A, quod fuerat demonſtran<lb></lb>dum. </s> </p> <p type="main"> <s id="s.001772">Hinc colligimus, pondere in medio vecte appenſo <lb></lb>ferentes æqualiter ſuſtinere, propterea quod totius vectis <lb></lb>ad partes ipſas proportio ſit eadem, vel æqualis. </s> </p> <pb xlink:href="007/01/184.jpg"></pb> <p type="main"> <s id="s.001773">Pulchre autem dubitari poteſt, an idem prorſus con<lb></lb>tingat, ſi alterum eorum qui ſuſtinent, ſit ſtatura quidem <lb></lb>procerior, alter vero humilior. </s> </p> <figure id="id.007.01.184.1.jpg" xlink:href="007/01/184/1.jpg"></figure> <p type="main"> <s id="s.001774">Sit enim vectis AB, in cuius <lb></lb>medio pondus H libere appen<lb></lb>ſum ex C, alter portantium pro<lb></lb>cerior AD, humilior vero BE. ſit <lb></lb>autem horizontis planum DE, <lb></lb>demittatur à puncto Cad <expan abbr="horizō-tem">horizon<lb></lb>tem</expan> perpendicularis, ipſis vero <lb></lb>AD, BE, æquidiſtans CF. </s> <s id="s.001775">Tranſi<lb></lb>bit autem per ipſius ponderis, <lb></lb>grauitatis centrum H. Dico igi<lb></lb>tur, nil referre quatenus ad pondus ſuſtinendum perti<lb></lb>net, vtrum portantes ſint ſtatura pares velne. </s> <s id="s.001776">Ducatur e<lb></lb>nim horizonti æquidiſtans GB, ſecans perpendicularem <lb></lb>CF in I. </s> <s id="s.001777">Quoniam igitur AG æquidiſtans eſt ipſi CI erit <lb></lb>vt AC ad CB per 4. ſexti elem, ita GI ad IB. </s> <s id="s.001778">Sunt ergo GI, <lb></lb>IB inter ſe æquales. </s> <s id="s.001779">Intelligatur itaque pondus H, <expan abbr="ſolutū">ſolutum</expan> <lb></lb>à puncto C appenſum eſſe libere ex puncto I, hoc eſt, ex <lb></lb>medio vectis GB, æqualiter ergo diuiſum erit pondus in<lb></lb>ter portantes, licet alter procerior, alter vero ſtatura pu<lb></lb>milior, quod fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.001780">Si autem pondus ita vecti alligatum ſit vt libere non <lb></lb>pendeat, vecte ex vna parte eleuato, ex altera vero de<lb></lb>preſſo, grauitatis centrum ad eam partem verget quæ <lb></lb>magis ab horizonte attollitur, & ad eam ipſam partem <lb></lb>vectis à pondere ad ſuſtinentem fit breuior. </s> </p> <p type="main"> <s id="s.001781">Eſto enim vectis AB, cuius medium C, pondus vecti <lb></lb>in C alligatum CFG, cuius grauitatis centrum H eorum <lb></lb>qui portant procerior AB, humilior BE, horizontis <expan abbr="planū">planum</expan> <lb></lb>DE. </s> <s id="s.001782">Demittatur per centrum H horizonti perpendicu<lb></lb>laris IHK, ſecans vectem quidem in I, horizontis vero pla-<pb xlink:href="007/01/185.jpg"></pb><figure id="id.007.01.185.1.jpg" xlink:href="007/01/185/1.jpg"></figure><lb></lb>num in K. </s> <s id="s.001783">Poſt hæc intelligatur pon<lb></lb>dus ſolutum quidem à puncto C, ap<lb></lb>penſum vero ex puncto I. </s> <s id="s.001784">Stabit igitur <lb></lb>ex definitione centri grauitatis nec ſi<lb></lb>tu ſuo mouebitur. </s> <s id="s.001785">Dico autem par<lb></lb>tem AI ipſa IB eſſe breuiorem, hoc eſt, <lb></lb>punctum I cadere inter C & A. </s> <s id="s.001786">Si e<lb></lb>nim non cadat, vel cadet in C, aut in<lb></lb>ter C & B, cadat autem ſi fieri poteſt <lb></lb>in C. </s> <s id="s.001787">Erit igitur CHK horizonti perpendicularis, ſed ei<lb></lb>dem perpendicularis AD. </s> <s id="s.001788">Erunt igitur BCK BAD anguli <lb></lb>inter ſe æquales, ſed ipſi BAD angulo æqualis eſt CIH, <lb></lb>quare & BCH ipſi CIH æqualis erit. </s> <s id="s.001789">Producto igitur la<lb></lb>tere IC trianguli ICH erit exterior angulus æqualis inte<lb></lb>riori ex oppoſito, quod eſt abſurdum. </s> <s id="s.001790">non ergo I cadet in <lb></lb>C. </s> <s id="s.001791">Eadem autem ratione monſtrabitur non cadere inter <lb></lb>CB, cadet ergo inter CA, & ideo minor AI ipſa IB. </s> <s id="s.001792">Itaque <lb></lb>vt ſe habet BI ad BA, ita potentia in A ad pondus in I, ſed <lb></lb>maiorem proportionem habet BI ad BA, quam IA ad AB. <lb></lb></s> <s id="s.001793">Ergo minor potentia requiretur in B quam in A, & ſane <lb></lb>pars IB reſpondet potentiæ ſuſtinenti in A, at IA potentiæ <lb></lb>ſuſtinenti in B, minor eſt autem AI ipſa IB, ergo maior po<lb></lb>tentia requiritur in B, quam in A, quod fuerat demon<lb></lb>ſtrandum. </s> </p> <p type="main"> <s id="s.001794">Hoc item concludetur, ſi portantes ſtatura quidem <lb></lb>pares fuerint, ſed per planum ambulent horizonti accliue <lb></lb>aut decliue. </s> <s id="s.001795">Si enim pondus libere pendeat, vectis <expan abbr="partiū">partium</expan> <lb></lb>proportio non mutabitur; ſr autem libere non pendeat, <lb></lb>is magis laborabit qui in aſcenſu præibit, minus vero qui <lb></lb>in deſcenſu. </s> </p> <p type="main"> <s id="s.001796">Hinc quoque Carrucarum ratio pendet, quæ dupli<lb></lb>ci manubrio vnica rota vulgo ſunt in vſu, pro vecte enim <lb></lb>habentur, cuius fulcimentum ad contactum plani & ro<pb xlink:href="007/01/186.jpg"></pb>tæ; potentiæ vero ad extremitatem duplicis manubrij. <lb></lb></s> <s id="s.001797">Reducitur enim ad idem genus vectis, in quo pondus in<lb></lb>ter fulcimentum eſt & potentiam. </s> <s id="s.001798">quo igitur minor fue<lb></lb>rit proportio partis vectis quæ à centro grauitatis ad i<lb></lb>pſum fulcimentum, ad totum vectem eo facilius pondus <lb></lb>eleuabitur. </s> </p> <p type="main"> <s id="s.001799">Cur autem difficilime hæ per accliue horizonti pla<lb></lb>num pellantur, duplici fit de cauſſa, tum quia grauitatis <lb></lb>centrum ad ipſum portantem ſeu pellentem vergit, & id<lb></lb>eo pars quæ a fulcimento ad centrum grauitatis ponderis <lb></lb>fit maior, tum etiam quoniam ipſum graue contra ſui na<lb></lb>turam ſurſus pellitur ferturque. </s> </p> <p type="main"> <s id="s.001800">Quærere ad hæc quiſpiam poſſet, Cur Baiuli ma<lb></lb>gna ferentes pondera, curui in cedant? </s> <s id="s.001801">Dixerit autem ali<lb></lb>quis, ponderis grauitate eos deprimentis id fieri. </s> <s id="s.001802">Nos au<lb></lb>tem duplici item de cauſſa id fieri putamus, tum ea quam <lb></lb>conſiderauimus, tum etiam alia, nempe vt grauitatis cen<lb></lb>trum ipſius ponderis quod ſuſtinent, in perpendiculari <lb></lb>collocent, ne ſi extra ponatur is qui fert à centro extra <lb></lb>fulcimentum poſito, ad eam partem ad quam vergit tra<lb></lb>hatur, & pondere ipſo opprimatur. </s> </p> <p type="main"> <s id="s.001803">Eadem de cauſſa fit quoque vt ij qui magna ponde<lb></lb>ra ſiniſtro ferunt humero, in dextram partem inclinentur, <lb></lb>qui vero dextro, contrario modo ſe habeant, æquatur e<lb></lb>nim pondus eo pacto, & grauitatis centrum in ipſa per<lb></lb>pendiculari collocatur. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001804">QVÆSTIO XXX.</s> </p> <p type="head"> <s id="s.001805"><emph type="italics"></emph>Cur aſſurgentes omnes fœmori tibiam ad acutum angulum conſti<lb></lb>tuamus & pectori thoraciue ſimiliter fœmur, quod nî fiat <lb></lb>haudquaquam ſurgere poterunt?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001806">Ait Philoſophus, forte id fieri, quod æqualitas ſit o<lb></lb>mnino quietis cauſſa, rectum vero angulum quietis <pb xlink:href="007/01/187.jpg"></pb>angulum eſſe, & ſtationem facere, nec alia de cauſſa ſtan<lb></lb>tem ipſi terræ eſſe perpendicularem, & ideo caput & pe<lb></lb>des in eadem linea habere, ſedentem vero non item. </s> <s id="s.001807"><expan abbr="Tūc">Tunc</expan> <lb></lb>autem à ſeſſione ſurrectionem fieri, cum caput & pedes in <lb></lb>vna linea collocantur, quod ſane fit cum pectus & crura <lb></lb>acutum cum ipſo fœmore angulum faciunt. </s> </p> <figure id="id.007.01.187.1.jpg" xlink:href="007/01/187/1.jpg"></figure> <p type="main"> <s id="s.001808">Eſto enim ſtans AB hori<lb></lb>zonti IBK perpendicularis, cù<lb></lb>ius caput A, pedes vero B, ſedeat <lb></lb>modo ſitque eius cum capite <lb></lb>Thorax CD, fœmur DE, crura <lb></lb>EF, ſintque CDE, DEF anguli <lb></lb>recti, quibus ita conſtitutis non <lb></lb>ſunt in eadem linea caput C & <lb></lb>pedes F. </s> <s id="s.001809">Surgere itaque non po<lb></lb>terit ſedens, propterea quod <lb></lb>partes omnes corporìs non ſint <lb></lb>horizonti perpendiculares. </s> <s id="s.001810">Ad <lb></lb>hoc autem vt ſurrectio fiat, neceſſe eſt vt ſedens retrahat <lb></lb>quidem pedes in H, & pectore in clinato acutum cum fœ<lb></lb>more angulum conſtituat GDE, quo caſu fient in eadem <lb></lb>recta linea, eaque horizonti perpendiculari caput in G, <lb></lb>& pedes in H, ex cuius ſitus natura commoda fiet ab ipſo <lb></lb>ſedente ſurrectio. </s> <s id="s.001811">Hæc fere, licet alijs ab eo verbis expli<lb></lb>cata, ipſius eſt Philoſophi ſententia; quæ licet vera ſit, non <lb></lb>tamen ex proprijs, hoc eſt, Mechanicis principijs eſt peti<lb></lb>ta. </s> <s id="s.001812">quod quidem nos facere conabimur. </s> </p> <p type="main"> <s id="s.001813">Dicimus autem primo, ſedentem non ideo quieſce<lb></lb>re, vt ſentit Ariſtoteles, quod rectus angulus quietis ſit <lb></lb>cauſſa, ſed propterea quod eius thoracis tum etiam fœ<lb></lb>morum pondus ab ipſa ſede ſuſtineantur; crura vero & <lb></lb>pedes ideo non laborent, quod partim ſuſpenſa ſint, par<lb></lb>tim ſolo ipſi innitantur. </s> <s id="s.001814">Quare cum corpus totum nec ſe <pb xlink:href="007/01/188.jpg"></pb>ſuſtineat, nec à pedibus ſuſtineatur, fit quies & laſſitudi<lb></lb>nis alleuatio. </s> <s id="s.001815">Natura autem ideo commodam hominibus <lb></lb>ſeſſionem facere voluiſſe inde apparet, quod clunes, qui<lb></lb>bus tota ſuperior pars, & grauior nititur, carnoſam fece<lb></lb>rit, & ceruicalis cuiuſdam inſtar mollem & facilem. </s> <s id="s.001816">Sed <lb></lb>nos ad quæſtionem. </s> </p> <figure id="id.007.01.188.1.jpg" xlink:href="007/01/188/1.jpg"></figure> <p type="main"> <s id="s.001817">Eſto enim ſtans AB, cuius caput A, <lb></lb>Thorax AC, fœmora CD, crura DB, pe<lb></lb>des vero B, centrum vero grauitatis in i<lb></lb>pſo Thorace E. </s> <s id="s.001818">Modo ſedeat, ſitque ca<lb></lb>put in F, Thorax FG, fœmora GH, crura <lb></lb>HI, pedes I, grauitatis vero centrum vbi <lb></lb>K. </s> <s id="s.001819">Producatur recta FG in L, ſitque FL <lb></lb>horizonti perpendicularis. </s> <s id="s.001820">Centrum er<lb></lb>go grauitatis K fulcitur puncto G, hoc eſt, <lb></lb>puncto L, in quo poſteriores pedes ipſius <lb></lb>ſed is ſolo hærent. </s> <s id="s.001821">efficit autem ſedens <lb></lb>duos rectos angulos FGH, GHI. </s> <s id="s.001822">Rebus <lb></lb>igitur ita diſpoſitis ſeruatis rectis angulis, non fiet ſurre<lb></lb>ctio, & id quidem non ideo quod, vt ait Philoſophus, æ<lb></lb>qualitas & rectitudo angulorum quietis ſit cauſſa, ſed <lb></lb>propterea quod centro grauitatis extra pedum <expan abbr="fulcimē-tum">fulcimen<lb></lb>tum</expan> conſtituto, non habet centrum ſtabilem locum cui in <lb></lb>actu ſurrectionis hæreat, & fulciatur, vnde fit vt ſi ſedenti <lb></lb>ſubtrahatur ſedes remoto prohibente, ſedens prorſus cor<lb></lb>ruat. </s> <s id="s.001823">Modo retrahat qui ſedet crura, & pedes ponat in M, <lb></lb>à puncto autem M, horizonti perpendicularis erigatur <lb></lb>MN. erit ergo fulcimentum in M, ſed adhuc ſurgere non <lb></lb>poterit, centro grauitatis adhuc extra lineam MN, quæ <lb></lb>per fulcimentum eſt, conſtituto. </s> <s id="s.001824">Reclinetur autem pe<lb></lb>ctus ad anteriora, & cum fœmore acutum angulum faciat <lb></lb>ſitque vbi GO, erit igitur grauitatis centrum vbi P, hoc <lb></lb>eſt, in ipſa perpendiculari NM, fiet igitur inde commoda <pb xlink:href="007/01/189.jpg"></pb>ſurrectio, propterea quod in eadem linea facta ſint, graui<lb></lb>tatis centrum P, & fulcimentum ipſum M. </s> <s id="s.001825">Acutum vero <lb></lb>angulum in ſurrectione neceſſarium eſſe clare patet, non <lb></lb>autem effectus ipſius eſſe cauſſam, vt videtur ſenſiſſe Ari<lb></lb>ſtoteles; nisi dicamus, cauſſam eſſe cauſſæ, ſiquidem acuti <lb></lb>qui fiunt anguli centrum & pedes in eadem linea collo<lb></lb>cant, quicquid tamen ſit, nos ideo ſurrectionem fieri dici<lb></lb>mus, quod immutatis angulis centrum grauitatis ſupra <lb></lb>fulcimentum, fulcimento vero ſub ipſo grauitatis centro <lb></lb>collocetur, & hæc eſt cauſſa proxima. </s> <s id="s.001826">Hæc nos ad Ariſto<lb></lb>telem. </s> <s id="s.001827">Modo quaſdam alias quæſtiones, nec inutiles ſed <lb></lb>& eas non iniucundas quoque proponemus. </s> </p> <p type="main"> <s id="s.001828">Primum igitur quærimus, Cur hominum & cætero<lb></lb>rum animalium, quæ aliquando erecto corpore incedunt, <lb></lb>pedes non quidem breues ſint & rotundi, ſed longiores <lb></lb>potius, & in inferiorem partem porrecti? </s> <s id="s.001829">Item cur magis <lb></lb>ad digitos quam ad calcaneum porrigantur? </s> </p> <figure id="id.007.01.189.1.jpg" xlink:href="007/01/189/1.jpg"></figure> <p type="main"> <s id="s.001830">Eſto homo animalue quodpiam ſtans <lb></lb>AB, cuius pes CD, pedis pars quæ ad digitos <lb></lb>BC. quae vero ad calcaneum BD fœmoris ver<lb></lb>tebra E, centrum vero grauitatis ipſius cor<lb></lb>poris F. </s> <s id="s.001831">Primum igitur ſtatuendum eſt, ho<lb></lb>minem & cætera fere animalia à Natura fa<lb></lb>cta eſſe vt ad anteriora moueantur, & ideo o<lb></lb>mnes fere quod in ſenioribus manifeſte ap<lb></lb>paret, ad anteriora ex ipſa corporis diſpoſi<lb></lb>tione vergant. </s> <s id="s.001832">Itaque dum qui ſtat horizon<lb></lb>ti prorſus eſt perpendicularis, grauitatis centrum F in ipſa <lb></lb>perpendiculari conſtituitur quæ ad mundi centrum AB, <lb></lb>& ideo corporis moles ponduſque fulcitur puncto B. </s> <s id="s.001833">Mo<lb></lb>do fiat ex vertebra E thoracis AE, inclinatio in anteriora, <lb></lb>in GE & grauitatis centrum D diluetur in H, & per H per<lb></lb>pendicularis demittatur HI, non erit ** extra pedis ful<pb xlink:href="007/01/190.jpg"></pb>cimentum BC. </s> <s id="s.001834">Stabit ergo qui ita inclinatur, nec corruet: <lb></lb>ſi autem adhuc propendeat magis, fiatque in KE, centro <lb></lb>grauitatis conſtituto in M, ducatur per M perpendicula<lb></lb>ris ML, quare quoniam linea ML extra pedis fulcimen<lb></lb>tum cadit, corruet qui eo pacto inclinatur nec ſuſtinebi<lb></lb>tur. </s> <s id="s.001835">Cur igitur natura animalibus quae erecto corpore am<lb></lb>balant, pedes in anteriora porrectos fecerit, hinc clare <lb></lb>patet. </s> </p> <p type="main"> <s id="s.001836">Hinc etiam ceu conſectarium habemus, cur homi<lb></lb>nes ſi impellantur, magis ad caſum in poſteriora quam in <lb></lb>anteriora ſint proni. </s> <s id="s.001837">Nec non etiam cur ſimiæ, vrſi, & ſi <lb></lb>quæ cætera eiuſmodi animalia diutius erecto corpore <lb></lb>ambulare nequeant, nempe ideo quod eorum corporum <lb></lb>moles valde in anteriora propendeat, nec ita commodo, <lb></lb>vt humanis euenit corporibus, pedum ipſorum baſibus <lb></lb>fulciantur. </s> </p> <p type="main"> <s id="s.001838">Quærere item haud importune poſſumus, Cur gral<lb></lb>latores non ſtent erecti, niſi aſſidue moueantur? </s> <s id="s.001839">Solutio <lb></lb>facilis. </s> <s id="s.001840">grallæ etenim duobus tantum punctis ſolum tan<lb></lb>gunt, nec porrecti beneficio, quod ambulantibus accidit, <lb></lb>vti poſſunt. </s> <s id="s.001841">quamobrem grauitatis centrum fit extra ful<lb></lb>cimentum, & ideo coguntur grallatores aſſiduo motu <lb></lb>grauitatis centro fulcimentum ſupponere, quod dum fit, <lb></lb>à caſu prohibentur. </s> </p> <p type="main"> <s id="s.001842">Poteſt autem id quod fulcitur, tripliciter fulciri, <expan abbr="nē-peaut">nem<lb></lb>pe aut</expan> puncto, aut linea, aut ſuperficie. </s> </p> <p type="main"> <s id="s.001843">Quod puncto fulcitur, nulla reimpediente ad quam<lb></lb>uis partem cadere poteſt, centrum ſiquidem, motus, pun<lb></lb>ctum eſt. </s> </p> <p type="main"> <s id="s.001844">Quod linea fulcitur ad duas tantum partes, eaſque <lb></lb>oppoſitas, habet caſum. </s> <s id="s.001845">ſit illud ſuperficies, corpuſue in <lb></lb>latus conſtitutum. </s> </p> <pb xlink:href="007/01/191.jpg"></pb> <figure id="id.007.01.191.1.jpg" xlink:href="007/01/191/1.jpg"></figure> <p type="main"> <s id="s.001846">Eſto horizontis pla<lb></lb>num ABCD, cui ad re<lb></lb>ctos angulos inſiſtat ſu<lb></lb>perficies EFGH, ſecun<lb></lb>dum latus FG. </s> <s id="s.001847">Sit autem <lb></lb>ipſius ſuperficiei grauita<lb></lb>tis centrum I. à quo ad <lb></lb>horizontis planum per<lb></lb>pendicularis demittatur IK. Cadet autem in lineam FG. <lb></lb>per propoſ. </s> <s id="s.001848">38. vndecimi elem. </s> <s id="s.001849">& anguli IKG IKF recti e<lb></lb>runt. </s> <s id="s.001850">Itaque ſuperficie EFGH circa lineam FKG ceu cir<lb></lb>ca axem mota punctum I peripheriam deſcribet LIM, & <lb></lb>ſiquidem cadat ad partes CD, grauitatis centrum erit vbi <lb></lb>M. </s> <s id="s.001851">Si vero ad partes AB, fiet vbi L. </s> <s id="s.001852">Sunt autem LKM <expan abbr="pū-cta">pun<lb></lb>cta</expan> in recta LKM, quæ quidem communis ſectio eſt plani <lb></lb>horizontis, & plani per IKLM, tranſeuntis. </s> </p> <figure id="id.007.01.191.2.jpg" xlink:href="007/01/191/2.jpg"></figure> <p type="main"> <s id="s.001853">Idem quoque de cor<lb></lb>pore dicimus in latus col<lb></lb>locato. </s> <s id="s.001854">Eſto enim cubus <lb></lb>LO, cuius grauitatis cen<lb></lb>trum R, latus vero quo ful<lb></lb>citur, NO, Si enim ita col<lb></lb>locetur, vt interna ſuperfi<lb></lb>cies LNOQ ad rectos an<lb></lb>gulos horizonti ſit conſti<lb></lb>tuta, demiſſa perpendicu<lb></lb>laris à puncto R, ea det in S, in ipſa linea NSO. </s> <s id="s.001855">Cadente i<lb></lb>gitur corpore fiet motus circa lineam NO, centro graui<lb></lb>tatis interim peripheriam TRV. deſcribente. </s> </p> <p type="main"> <s id="s.001856">Hinc animaduertere licet, Cur prouidiſſima Natu<lb></lb>ra nulli animantium vnicum dederit pedem, ſed aut qua<lb></lb>ternos, aut ſaltem binos, & binos quidem ipſos virtute <lb></lb>quaternos, ſiquidem in quolibet animantium bipedum <pb xlink:href="007/01/192.jpg"></pb>pede duo ſaltem puncta conſiderantur, quibus ipſum ani-<lb></lb>mal fulcitur. </s> </p> <figure id="id.007.01.192.1.jpg" xlink:href="007/01/192/1.jpg"></figure> <p type="main"> <s id="s.001857">Sint enim humani pedis ve<lb></lb>ſtigia A, B, C, D, in vtroque igitur <lb></lb>duo puncta conſiderantur, A, B, <lb></lb>C, D, illa quidem ad digitos, hæc <lb></lb>autem ad calcaneum. </s> <s id="s.001858">Idem quo<lb></lb>que in auium pedibus obſerua<lb></lb>tur, ex quibus concludimus, bi<lb></lb>pedum omnium fulcimentum eſ<lb></lb>ſe quadruplex. </s> <s id="s.001859">Porro quadrupe<lb></lb>dia eo quod tota corporis mole <lb></lb>ad in feriora vergant, quatuor ful<lb></lb>cimenta, eaque diſtincta, & commode ab inuicem remo<lb></lb>ta eademmet Natura præparauit. </s> </p> <p type="main"> <s id="s.001860">Eadem quoque in artificialibus conſideramus. </s> <s id="s.001861">Sit <lb></lb>enim vas quodpiam ABC, cuius pes vnicus, iſque rotun<lb></lb>dus BC, grauitatis vero centrum D. </s> <s id="s.001862">Quoniam igitur in <lb></lb>pedis ipſius peripheria, infinita puncta intelligantur, dici <lb></lb>quodammodo poteſt vas ipſum infinitis fere punctis, licet <lb></lb><figure id="id.007.01.192.2.jpg" xlink:href="007/01/192/2.jpg"></figure><lb></lb>pes vnicus ſit, ſuſtineri. </s> <s id="s.001863">Non<lb></lb>nulla autem corpora artifi<lb></lb>cialia. </s> <s id="s.001864">quatuor pedibus ſu<lb></lb>ſtinentur, vt menſæ <expan abbr="quædã">quædam</expan>, <lb></lb>nonnulla etiam tribus, vt <lb></lb>tripodes, qui nomen ab ipſo <lb></lb>pedum numero ſortiuntur. <lb></lb></s> <s id="s.001865">Sit enim triangulum EFG, <lb></lb>cuius centrum grauitatis H, <lb></lb>nitatur autem tribus pun<lb></lb>ctis I, K, L, ſtabit igitur. </s> <s id="s.001866">Si <lb></lb>autem duobus tantum; non ſtabit. </s> <s id="s.001867">ducta enim IK ſi pun<lb></lb>ctis tantum IK innitatur, conſtituto grauitatis centro <pb xlink:href="007/01/193.jpg"></pb>extra fulcimentum IK, verget cedens verſis partes, L, Si <lb></lb>autem innitatur punctis IL, cadet ad partes K. </s> <s id="s.001868">Sivero ipſis <lb></lb>KL, cadet ad partes I.Ex quibus apparet, inanimata cor<lb></lb>pora aut vnico pede plurium virtutem habente, aut ſal<lb></lb>tem tribus actu, vt ſuſtineantur, indigere. </s> </p> <p type="main"> <s id="s.001869">Hinc etiam patet, cur ſenes, imbecilles, curui, & pe<lb></lb>dibus capti, baculi baculorumue fulcimento egeant, ete<lb></lb>nim cum hi debiles ſint, & in anteriorem partem magno<lb></lb>pere propendeant, ne grauitatis centrum extra fulcimen<lb></lb>tum fiat, baculo vel baculis indigent, quibus centrum i<lb></lb>pſum fulciatur. </s> </p> <p type="main"> <s id="s.001870">Cæterum cur duplici genu ingeniculati difficile in <lb></lb>eo ſitu permaneant, ea cauſſa eſt, quod grauitatis centrum <lb></lb>in thorace conſtitutum, duobus genibus fulciatur, eoſ<lb></lb>que premat. </s> <s id="s.001871">quæ quidem genua eo quod natura apta na<lb></lb>ta non ſint, veluti pedes, ad ſuſtinendam corporis molem <lb></lb>laborant, idque eo magis, quod cum oſſea ſint, cutem in<lb></lb>ter oſſium & plani duritiem conſtitutam, accidit arctari, <lb></lb>& ideo dolorem & moleſtiam ingeniculatis facere. </s> </p> <p type="main"> <s id="s.001872">Si autem vnico tantum genu quiſpiam nitatur, dif<lb></lb>ficultatem ſentiet longe minorem. </s> <s id="s.001873">Triplici enim fulci<lb></lb><figure id="id.007.01.193.1.jpg" xlink:href="007/01/193/1.jpg"></figure><lb></lb>mento eo caſu ingeniculatus <lb></lb>fulcitur. </s> <s id="s.001874">Sit enim ingenicula<lb></lb>tus ABCDE, cuius grauitatis <lb></lb>centrum F. dextrum vero ge<lb></lb>nu, cui nititur D, ſiniſtrum ve<lb></lb>ro, quod eleuatur B. </s> <s id="s.001875">Tribus ergo fulcimentis ingenicula<lb></lb>tus vt diximus, ſuſtinetur, CDE. </s> <s id="s.001876">Diuiditur itaque pondus <lb></lb>in tres partes, & ideo ſingulæ minus fatigantur. </s> <s id="s.001877">Magis ta<lb></lb>men laborat punctum D, vtpote illud, cui ad perpendicu<lb></lb>lum F grauitatis centrum innititur. </s> </p> <p type="main"> <s id="s.001878">Vtique illud quoque mirabile eſt, Aues dormientes <lb></lb>vnico tantum pede fulciri, & quod magis mirum eſt, dor<pb xlink:href="007/01/194.jpg"></pb>mientes poſſe, quod vel ipſis vigilantibus eſt difficile. </s> <s id="s.001879">Cur <lb></lb>id Natura docente faciant, eam puto eſſe cauſſam, quod <lb></lb>dum dormiunt, caput ſiniſtræ alæ, vt naturali calore iu<lb></lb>uentur, ſupponunt, quapropter ad eam partem declinan<lb></lb>tes, vt interim æquilibrium faciant, pedem ſubleuant, & <lb></lb>eo caſu ceu inutilem retrahunt atque ſuſpendunt: addita <lb></lb>item alia cauſſa, nempe vt pedem ipſum dormientes nati<lb></lb>uo calore confoueant. </s> </p> <p type="main"> <s id="s.001880">Quæritur etiam, Cur ij qui inclinantur, vt <expan abbr="rē">rem</expan> quam<lb></lb>piam à ſolo ſuſtollant, alterum crurium ad anteriora, <expan abbr="nē-peverſus">nem<lb></lb>pe verſus</expan> manum ipſam, quam porrigunt, extendant? </s> </p> <figure id="id.007.01.194.1.jpg" xlink:href="007/01/194/1.jpg"></figure> <p type="main"> <s id="s.001881">Eſto enim quiſpiam ABCD, <lb></lb>cuius crura BC, BD, grauitatis <lb></lb>centrum E, velit autem quippiam <lb></lb>à ſolo tollere quod ſit in F. ſit per<lb></lb>pendicularis, quæ per grauitatis <lb></lb>centrum GEH. </s> <s id="s.001882">Dum igitur ad <lb></lb>anteriora ínclinatur, centrum a<lb></lb>mouet à perpendiculari, quam<lb></lb>obrem docente Natura, crus BC <lb></lb>ad centrum ipſum fulciendum. <lb></lb></s> <s id="s.001883">ad anteriora, hoc eſt, verſus rem <lb></lb>ſuſtollendam porrigitur. </s> </p> <p type="main"> <s id="s.001884">Huius quoque ſpeculationis eſt inueſtigare, Cur <lb></lb>quadrupedia dum gradiuntur, pedes diametraliter mo<lb></lb>ueant. </s> <s id="s.001885">Cuius rei verba fecit ipſe quoque Philoſophus lib. <lb></lb> de animalium inceſſu cap. 12. </s> <s id="s.001886">Nos autem ad maiorem de<lb></lb>clarationem, quod ipſe Phyſicis principijs fecit, mecha<lb></lb>nicis demonſtrabimus. </s> </p> <p type="main"> <s id="s.001887">Sint duæ in plano parallelæ AB, CD, in quibus qua<lb></lb>drupedis pedes E, F, B, D, quorum EF, anteriores, BD vero <lb></lb>poſteriores. </s> <s id="s.001888">iungantur BDEF, eritque EBDF parallelo<lb></lb>grammum altera parte longius, cuius diametri ducantur <pb xlink:href="007/01/195.jpg"></pb><figure id="id.007.01.195.1.jpg" xlink:href="007/01/195/1.jpg"></figure><lb></lb>ED, BF, ſecantes ſeſe in G, vbi & grauitatis <lb></lb>centrum. </s> <s id="s.001889">Moto igitur poſteriori ſiniſtro pe<lb></lb>de B in K, ſi anteriorem E, eodem tempore <lb></lb>moueret in I, ſtantibus interim DF, ceu ful<lb></lb>cimentis, centrum G extra fulcimenta fieret <lb></lb>ad partes BE. </s> <s id="s.001890">Caderet igitur ad partes BE. </s> <s id="s.001891">Si <lb></lb>autem eodem tempore moueret dextros eo<lb></lb>dem pacto centrum extra fulcimenta poſi<lb></lb>tum caderet ad partes ipſas DF. </s> <s id="s.001892">Si autem <lb></lb>moto pede B in K, & eodem tempore F in L, <lb></lb>& D in H, E, in I, centrum erit in diametris HI, KL, hoc <lb></lb>eſt, vbi M, fultum quidem ab ipſis pedibus K, L, H, I. </s> <s id="s.001893">Hoc <lb></lb>igitur pacto transfertur viciſſim cum grauitatis centro ſi<lb></lb>mul translatis fulcimentis ſeſe diametraliter reſponden<lb></lb>tibus; quod vtique demonſtrandum fuerat. </s> </p> <p type="main"> <s id="s.001894">Sane & bipedia quoque alternatim gradiendo gra<lb></lb>uitatis centrum transferunt. </s> <s id="s.001895">Dum enim dextrum crus e<lb></lb>leuatur, centrum ſiniſtro fulcitur, & econtra. </s> </p> <p type="main"> <s id="s.001896">Naturalia iſthæc ſunt; in artificialibus autem quæri <lb></lb>poſſet, Cur Architecti, Arcium muros non ad perpendi<lb></lb>culum erectos, ſed introrſum inclinatos conſtituant? </s> </p> <figure id="id.007.01.195.2.jpg" xlink:href="007/01/195/2.jpg"></figure> <p type="main"> <s id="s.001897">Vtique hoc faciunt, vt minus <lb></lb>ſint ad ruinam proni. </s> <s id="s.001898">Eſto enim <lb></lb>murus ad interiorem partem ver<lb></lb>gens ABCD, Cuius grauitatis cen<lb></lb>trum E baſis BC erigatur à puncto <lb></lb>B horizonti perpendicularis BF, & <lb></lb>ad eundem à centro grauitatis E <lb></lb>demittatur EM, tum BE iungatur. <lb></lb></s> <s id="s.001899">Poſt hæc à puncto BG angulum. <lb></lb></s> <s id="s.001900">cum linea horizontis BK faciens recto maiorem. </s> <s id="s.001901">Ita que <lb></lb>murus hoc pacto conſtitutus ad interiorem partem ſuo <lb></lb>pondere vergit, cadere autem non poteſt, vel quod viuæ <pb xlink:href="007/01/196.jpg"></pb>rupi, cui forte hæret, fulciatur, vel antiſtatis, quos no<lb></lb>ſtrates ſperones & contra fortes appellant, innitatur. </s> <s id="s.001902">Sed <lb></lb>nec in anteriora corruet, quandoquidem ruinam factu<lb></lb>ras, neceſſe eſt vt grauitatis centrum ſecum trahat in per<lb></lb>pendiculari BF, & demum in eam quæ vltra perpendicu<lb></lb>larem eſt BG, facta nempe circa B, ceu circa centrum, <expan abbr="cō-uerſione">con<lb></lb>uerſione</expan>. </s> <s id="s.001903">Moueatur autem & ex ſemidiametro BE cen<lb></lb>tro B portio circuli deſcribatur EH, quæ ſecet BG in H, <lb></lb>& BF in I; Et quia EM ſemidiametro BK perpendicularis <lb></lb>per B, centrum non tranſit, erit EM ipſa BK, hoc eſt, BI <lb></lb>brevior. </s> <s id="s.001904">Abſcindatur ex BI, ipſi EM æqualis LB. </s> <s id="s.001905">Erit igi<lb></lb>tur punctum L infra punctum I, hoc eſt, ipſo I, mundi cen<lb></lb>tro propius. </s> <s id="s.001906">Neceſſe igitur erit ad hoc vt murus corruat, <lb></lb>centrum grauitatis E facta circa B, conuerſione aliquan<lb></lb>do fieri in I, vt demum transferri poſſit in H, ſed I remo<lb></lb>tius eſt à mundi centro ipſis E, L, aſcendet igitur graue <lb></lb>contra ſui naturam ex E in I, at hoc eſt impoſſibile; quod <lb></lb>fuerat demonſtrandum. </s> </p> <p type="main"> <s id="s.001907">Ex his ijſdem principijs alia ſoluitur quæſtio, Cur <lb></lb>ſcilicet Campanaria turris quæ Piſis viſitur, nec non alia <lb></lb>Bononiæ in foro prope Aſellorum turrim, quam à nobili <lb></lb>olim Cariſendorum familia exſtructam, Cariſendam vo<lb></lb>cant, cuius meminit & Dantes Poeta ſummus in ſua Co<lb></lb>mœdia. </s> <s id="s.001908">Propendet autem hæc in latus, & ita propendet <lb></lb>vt perpendicularis, quæ à ſummo inclinatæ partis in ſo<lb></lb>lum demittitur, longe cadat ab ipſa, cui nititur, baſi, quod <lb></lb>ſane mirabile videtur, muros nempe, in ruinam pronos, <lb></lb>ruinam non facere. </s> </p> <p type="main"> <s id="s.001909">Eſto enim turris ABCD, baſi fulta BC, horizontis <lb></lb>planum BCF latera AB, DC, centrum vero grauitatis to<lb></lb>tius molis E. </s> <s id="s.001910">Propendeat autem ad partes DC ex angulo <lb></lb>DCF. </s> <s id="s.001911">Ita autem conſtituta intelligatur vt perpendicula<lb></lb>ris ab A, in planum horizontis demiſſa per grauitatis cen-<pb xlink:href="007/01/197.jpg"></pb><figure id="id.007.01.197.1.jpg" xlink:href="007/01/197/1.jpg"></figure><lb></lb>trum E extra baſim BC, non cadat, <lb></lb>cadat autem in C. </s> <s id="s.001912">Quoniam igitur <lb></lb>ABCD moles per E grauitatis cen<lb></lb>trum diuiditur, in partes ſecatur æ<lb></lb>queponderantes, ſed & centrum. <lb></lb></s> <s id="s.001913">grauitatis extra fulcimentum non <lb></lb>cadit, quare nec pars ACD, trahet <lb></lb>partem ABC, nec centrum extra <lb></lb>fulcimentum poſitum locum petet <lb></lb>centro mundi viciniorem. </s> <s id="s.001914">Cur igitur Cariſenda ſtet, & e<lb></lb>gregia illa turris campanaria quæ Piſis prope ſummum <lb></lb>Templum marmoribus præclare exſtructa videtur, licet <lb></lb>ruinam minentur, ſtent æternum, nec cadant, ex his quæ <lb></lb>conſiderauimus, liquido patet. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001915">QVAESTIO XXXI</s> </p> <p type="head"> <s id="s.001916"><emph type="italics"></emph>Cur facilius moueatur commotum quam manens, veluti currus <lb></lb>commotos citius agitant, quam moueri incipientes?<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.001917"><emph type="italics"></emph>Hoc quæritur.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001918">Problema hoc eſt mere Phyſicum; verumtamen quo<lb></lb>niam ad localem motum pertinet, de quo ipſe quoque <lb></lb>Mechanicus agit, Hiſce quæſtionibus contemplatio hæc <lb></lb>interſeritur. </s> <s id="s.001919">Soluit autem Ariſtoteles inquiens, id fortaſ<lb></lb>ſe ea de cauſſa fieri, quod difficillimum ſit pondus moue<lb></lb>re, quod in contrarium mouetur. </s> <s id="s.001920">Demit enim quippiam <lb></lb>de motoris potentia reſiſtens, licet mouens ipſo moto ſit <lb></lb>longe potentius atque velocius. </s> <s id="s.001921">neceſſe enim eſſe id tar<lb></lb>dius moueri quod repellitur. </s> <s id="s.001922">Hæc verba licet de ea po<lb></lb>tentia dicta videantur, quæ rem motam in contrariam. <lb></lb></s> <s id="s.001923">partem repellit, nihilominus illi quoque aptantur quæ <lb></lb>rem immobilem à principio mouere conatur. </s> <s id="s.001924">eſt enim re<lb></lb>ſiſtentia rei quæ à ſtatu ad motum transfertur ceu <expan abbr="quidã">quidam</expan> <pb xlink:href="007/01/198.jpg"></pb>contrarius motus. </s> <s id="s.001925">Contra autem accidit illi qui rem mo<lb></lb>tam mouet in ipſo motu: eo enim caſu mouens ab ipſo rei <lb></lb>motu magnopere iuuatur, cooperatur enim motus moto<lb></lb>ri, in ipſam rem motam operanti. </s> <s id="s.001926">Auget autem res mota <lb></lb>quodammodo mouentis potentiam. </s> <s id="s.001927">quod enim à mouen<lb></lb>te pateretur, ex ſe ipſa agit res quæ mouetur. </s> </p> <figure id="id.007.01.198.1.jpg" xlink:href="007/01/198/1.jpg"></figure> <p type="main"> <s id="s.001928">Eſto horizontis pla<lb></lb>num AB, cui moles quæ<lb></lb>dam inſiſtat, CD. </s> <s id="s.001929">Modo <lb></lb>potentia quædam appli<lb></lb>cetur vbi E, quæ molem in <lb></lb>anteriora propellat, id <lb></lb>eſt, verſus B. Primum igitur, quoniam à quiete ad motum <lb></lb>fit tranſitus, reſiſtit ſua quiere corpus graue, potentiæ im<lb></lb>pellenti, ſuperata demum reſiſtentia moles quæ moueri <lb></lb>cœpit, fertur in F & mouetur, quare potentia quæ à prin<lb></lb>cipio reſiſtentiam rei non motæ ſuperauerat, pellendo <lb></lb>rem motam pergens facilius pellit: Duo enim ſunt quo<lb></lb>dammodo motores, mouens videlicet ipſe, & motus quo <lb></lb>res mota mouetur. </s> <s id="s.001930">facilius ergo pelletur ex F in G, quam <lb></lb>ex D in F, & ex G in B, quam ex F in G, & eo motus fiet in <lb></lb>progreſſu facilior atque in ipſa velocitate velocior, quo <lb></lb>magis in ipſa motione mouetur. </s> </p> <p type="main"> <s id="s.001931">Hinc ſoluitur ea quæſtio apud Phyſicos difficillima, <lb></lb>Cur nempe in motu naturali velocitas vſque augeatur; <lb></lb>etenim ibi Natura mouens eſt, atque eadem inſeparabilis <lb></lb>à remota, vrget igitur aſſidue, à principio quidem tar dius, <lb></lb>poſt hæc autem ea quam diximus, de cauſſa vſque & vſque <lb></lb>velocius. </s> <s id="s.001932">Motus ergo fit in motu, qui motus cum ſemper à <lb></lb>motore, & motu ipſo augeatur, creſcit ex progreſſu in im<lb></lb>menſum. </s> <s id="s.001933">Certe cauſſam velocitatis auctæ eam eſſe, quod <lb></lb>potentia mouens rem motam in motu ipſo moueat, nemo <lb></lb>vt arbitror, inficias ibit, acquirit enim corpus motum <expan abbr="pō-deroſitatem">pon-<pb xlink:href="007/01/199.jpg"></pb>deroſitatem</expan> quandam accidentalem, quæ cum ex motu <lb></lb>perinde augeatur, ipſum motum faciliorem, eoque velo<lb></lb>ciorem facit. </s> <s id="s.001934">Diſputat hæc & Simplicius lib. 7. Phyſic. c. <lb></lb>11. Ariſtotelis de Natura libros exponens. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001935">QVAESTIO XXXII.</s> </p> <p type="head"> <s id="s.001936"><emph type="italics"></emph>Quæritur hic, Cur ea quæ proijciuntur, ceſſent <lb></lb>à latione?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001937">Hoc itidem problema eſt mere Phyſicum. </s> <s id="s.001938">Ad quod ea <lb></lb>pertinent quæ à Philoſopho tractantur libro Natu<lb></lb>ralium 8. & lib. 1. de Cœlo. </s> <s id="s.001939">Tres autem affert ſubdubitan<lb></lb>do rationes, An quia impellens deſinit potentia, vel pro<lb></lb>pter retractionem, vel propter rei proiectæ in<expan abbr="clinationē">clinationem</expan>, <lb></lb>quando ea valentior fuerit quam proijcientis vires? </s> </p> <p type="main"> <s id="s.001940">Quicquid dicat Philoſophus, id vtique exploratiſ<lb></lb>ſimum eſt. </s> <s id="s.001941">Proiecta ideo à motu ceſſare, propterea quod <lb></lb>impreſſio, cuius impetu & virtute feruntur, non ſit proie<lb></lb>ctus quidem naturalis, ſed mere accidentalis & violenta, <lb></lb>at nullum accidentale & violentum quodque, non natu<lb></lb>rale eſt, perpetuum eſt. </s> <s id="s.001942">Ceſſat ergo accidentalis illa im<lb></lb>preſſio, eaque paullatim ceſſante proiecti motus elan<lb></lb>gueſcit, donec quietem prorſus adipiſcatur. </s> <s id="s.001943">Illud quoque <lb></lb>notamus, quod à multis vidimus non obſeruatum, nempe <lb></lb>violentum motum violentia præualente non differre à <lb></lb>naturali, & ideo tardiorem eſſe à principio poſt hæc, in i<lb></lb>pſo motu fieri velociorem, remittente demum paullatim <lb></lb>impreſſa violentia, tardiorem, donec impetus, & cum im<lb></lb>petu motus euaneſcat, & res ipſa mota quietem adipiſca<lb></lb>tur. </s> <s id="s.001944">Vnde etiam experientia docemur, ictum ex proiectis <lb></lb>violentius fieri, ſi fiat paullo remotior à principio, & tunc <lb></lb>demum eſſe innocentiſſimum, cum ibi fit, vbi proiectum <lb></lb>ex motu plene acquiſito, ſummam adeptum eſt velocita<pb xlink:href="007/01/200.jpg"></pb>tem. </s> <s id="s.001945">Hinc videmus, vel pueros ipſos, docente Natura <expan abbr="cū">cum</expan> <lb></lb>nuces, vel aliud quippiam, parieti alliſum frangere <expan abbr="conã-tur">conan<lb></lb>tur</expan>, à pariete moderato aliquo ſpatio recedere. </s> <s id="s.001946">Si autem <lb></lb>eos interroges, cur id faciant, reſpondebunt, vt inde ictus <lb></lb>valentius fiat atque efficacius. </s> <s id="s.001947">Eleganter ex Simplicij & <lb></lb>Alexandri Aphrodiſienſis doctrina, quæ lucidiſſima eſt, <lb></lb>quæſtionem hanc in ſua Paraphraſi explicat Picolomi<lb></lb>neus. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.001948">QVAESTIO XXXIII.</s> </p> <p type="head"> <s id="s.001949"><emph type="italics"></emph>Dubitatur, Cur proiecta moueantur, licet impellens à proiectis ſe<lb></lb>paretur; vel vt verbis Philoſophi vtar, Cur quippiam non pecu<lb></lb>liarem ſibi fertur lationem impulſore alioquin <lb></lb>non conſequente?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001950">Soluit, inquiens, an videlicet, quoniam primum, id eſt, <lb></lb>impellens ipſe, id efficit vt alterum, nempe proiectum <lb></lb>ipſum impellat, illud vero (hoc eſt proiectum) alterum <lb></lb>impellat, hoc eſt, aërem ipſum mediumue, quod à proie<lb></lb>cto repelletur. </s> <s id="s.001951">Ceſſare autem motum, cum res eo deue<lb></lb>nit, vt motus eidem à proijciente impreſſus, non poſſit <lb></lb>amplius rem proiectam mouere, & itidem rem ipſam, aë<lb></lb>rem videlicet non poſſit amplius repellere. </s> <s id="s.001952">Vel etiam <lb></lb>quando ipſius lati grauitas nutu ſuo declinat magis quam <lb></lb>impellentis in ante ſit potentia. </s> <s id="s.001953">Vtique res per ſe ſatis cla<lb></lb>ra. </s> <s id="s.001954">etenim motus impreſſus accidentalis eſt, quod vero la<lb></lb>tioni violentæ reſiſtit principium, naturale, & ab ipſo mo<lb></lb>to inſeparabile, vincente igitur quod natura eſt, paulla<lb></lb>tim remittitur quod ex accidenti eſt, & inde proiecti fit <lb></lb>quies. </s> <s id="s.001955">Eſt autem & hoc quoque Problema pure phyſicum, <lb></lb>& ſuperiori, de quo immediate egimus, perquam familia<lb></lb>re, quamobrem ex ijſdem prorſus ſoluitur <lb></lb>principijs. </s> </p> <pb xlink:href="007/01/201.jpg"></pb> </subchap1> <subchap1> <p type="head"> <s id="s.001956">QVÆSTIO XXXIV.</s> </p> <p type="head"> <s id="s.001957"><emph type="italics"></emph>Cur neque parua multum, neque magna nimis longe proijci queunt, <lb></lb>ſed proportionem quandam habere oportet proiecta ipſa ad <lb></lb>eius vires qui proijcit?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001958">Pvlchre dubitationem diluit, inquiens, An quia neceſ<lb></lb>ſe eſt quod proijcitur, & impellitur contraniti ei vnde <lb></lb>impellitur. </s> <s id="s.001959">Quod autem magnitudine ſua nihil cedit, aut <lb></lb>imbecillitate nihil contra nititur, non efficit <expan abbr="proiectionē">proiectionem</expan> <lb></lb>neque impulſionem. </s> <s id="s.001960">quod enim multo impellentis exce<lb></lb>dit vires, haud quaquam cedit. </s> <s id="s.001961">Quod vero eſt multo im<lb></lb>becillius, nihil contranititur, & impreſſionem non ſuſci<lb></lb>pit. </s> <s id="s.001962">Aliam quoque adiungit rationem, videlicet, Tantum <lb></lb>ferri id quod fertur quantum aëris mouerit ad <expan abbr="profundū">profundum</expan> <lb></lb>(hoc eſt, ad eam partem aëris remotiorem, ad quam fer<lb></lb>tur) etenim proiectum à principio dum fertur aërem pel<lb></lb>lit, non pellit autem ſi nihil mouetur. </s> <s id="s.001963">Accidit igitur vt <lb></lb>concludit Philoſophus, proiecta iſthæc contrarijs ex cau<lb></lb>ſis minus moueri. </s> <s id="s.001964">quod enim valde paruum eſt nihil mo<lb></lb>uet imbecillitate ſua impediente. </s> <s id="s.001965">quod vero valde ma<lb></lb>gnum eſt, ex contraria cauſſa nihil mouet, nempe quod <lb></lb>ob magnitudinem ſuam nihil moueatur. </s> <s id="s.001966">Vnde fit pro<lb></lb>portionem inter proiectum & proijcientem eſſe inprimis <lb></lb>ad motum, neceſſariam. </s> <s id="s.001967">Hæc eadem præclare in ſua Pa<lb></lb>raphraſi explicat Picolomineus. </s> </p> <p type="main"> <s id="s.001968">Huic nos, de proiectis quæſtioni, hæc addimus. </s> </p> <p type="main"> <s id="s.001969">Cur proiecta corpora non ſibimet ipſis ſecundum, <lb></lb>partes æquegrauia, ſi fuerint irregularis figuræ in ipſo mo<lb></lb>tu, ſecundum grauiorem partem antrorſus inuiolento, & <lb></lb>deorſum in naturali ferantur, & dum in latione conuer<lb></lb>tuntur, ſonitum edant. </s> </p> <p type="main"> <s id="s.001970">Eſto pila ABCD, cuius centrum E concinnata ex <lb></lb>diſpari materia leui, nempe BCD, & graui ABD. non ergo <pb xlink:href="007/01/202.jpg"></pb><figure id="id.007.01.202.1.jpg" xlink:href="007/01/202/1.jpg"></figure><lb></lb>erit <expan abbr="centrū">centrum</expan> grauitatis & cen<lb></lb>trum molis, ſit autem grauita<lb></lb>tis centrum F. </s> <s id="s.001971">Deſcendat cor<lb></lb>pus prohibente remoto per <lb></lb>rectam AG. </s> <s id="s.001972">Et quoniam gra<lb></lb>uiora deorſum tendunt ma<lb></lb>gis, ſi à principio motus gra<lb></lb>uior pars fuerit ſupra in ipſo <lb></lb>deſcenſu conuertet ir pila, & <lb></lb>ſitum non ſeruabit donec ſu<lb></lb>perior pars ea quæ grauior, <lb></lb>deorſum fiat, vt videre eſt in <lb></lb>pila HIK, cuius centrum eſt G. pars grauior HIK. </s> <s id="s.001973">Si au<lb></lb>tem eadem pila, laterali motu violenter feratur verſus <lb></lb>N, ad eam quoque partem conuertetur pars grauior. </s> <s id="s.001974">fa<lb></lb>cto enim molis ſeu magnitudinis centro vbi L, grauior <lb></lb>pars fiet in MNO; quæcunque igitur ſunt corpora ita <expan abbr="cō-ſtituta">con<lb></lb>ſtituta</expan>, vt in illis non ſit idem molis & grauitatis centrum <lb></lb>in ipſa latione conuertentur, & eorum pars grauior an<lb></lb>trorſus fiet. </s> <s id="s.001975">Sonitus porro in ipſo motu editi ea eſt cauſſa, <lb></lb>quod irregulare corpus à principio incipit conuerti, & in <lb></lb>ipſa conuerſione dum fertur aërem verberat, & ab eodem <lb></lb>viciſſim reuerberatur, ex qua reuerberatione fit corporis <lb></lb>rotatio dum fertur, & ipſe ſonitus, quem Græci <foreign lang="grc">ροίζον</foreign><lb></lb>Rhœzum appellant. </s> </p> <p type="main"> <s id="s.001976">Ad hanc quoque ſpeculationem pertinet, Cur lapi<lb></lb>des ad ſuperficiem aquæ proiecti non ſtatim demergan<lb></lb>tur, ſed aliquot vicibus a quæ ſuperficiem radentes, abea, <lb></lb>dem reſiliant. </s> </p> <p type="main"> <s id="s.001977">Eſto aquæ ſuperficies AB, lapis proiectus C, tangens <lb></lb>aquæ ſuperficiem in D, & inde reſiliens in E, mox iterum <lb></lb>eandem tangens in F, & reſiliens in G, donec <expan abbr="violēto">violento</expan> mo<lb></lb>tu ceſſante demergatur. </s> <s id="s.001978">Vtique lapis C, proiectus in D, <pb xlink:href="007/01/203.jpg"></pb><figure id="id.007.01.203.1.jpg" xlink:href="007/01/203/1.jpg"></figure><lb></lb>niſi medio denſiori, aqua vi<lb></lb>delicet, repelleretur, pene<lb></lb>traret per D, in H. </s> <s id="s.001979">At eo reſi<lb></lb>ſtente, & adhuc vigente im<lb></lb>petu, fertur in E ad angulos <lb></lb>fere pares. </s> <s id="s.001980">Dico autem fere, <lb></lb>ſiquidem maior eſt ADC ipſo EDF, propterea quod vis <lb></lb>non ſit eadem, ſed minor ea quæ ex D pellit in E. </s> <s id="s.001981">Durante <lb></lb>igitur impetu quo pellitur antrorſum, fiunt ipſæ reſilitio<lb></lb>nes, & eo ceſſante, reſilitiones ceſſant, & lapis ſuapte gra<lb></lb>uitate demergitur. </s> </p> <p type="main"> <s id="s.001982">Huc quoque ſpectat, Cur pila luſoria in horizontis <lb></lb>planum proiecta ad pares reſiliat, angulos nempe rectos? </s> </p> <figure id="id.007.01.203.2.jpg" xlink:href="007/01/203/2.jpg"></figure> <p type="main"> <s id="s.001983">Eſto horizontis planum <lb></lb>AB, in quod à puncto C per <lb></lb>lineam perpendicularem CE <lb></lb>cadat proijciaturue pila DE, <lb></lb>cuius grauitatis centrum F. <lb></lb></s> <s id="s.001984">Tangit autem planum in <expan abbr="pū-cto">pun<lb></lb>cto</expan> E. </s> <s id="s.001985">Perpendicularis ergo <lb></lb>EC, circulum DE per <expan abbr="centrū">centrum</expan> <lb></lb>ſecat, hoc eſt, in partes æ qua<lb></lb>les & æqueponderantes, ſed <lb></lb>dum pila cadit proijciturue, <lb></lb>agit in planum horizontis, vbi E, & in eodem puncto re. <lb></lb></s> <s id="s.001986">petitur, quare cum cadens & agens diuidatur in partes æ<lb></lb>quales & æqueponderantes & item repatiens & reſiliens <lb></lb>diuidatur item in partes æquales & æqueponderantes, ita <lb></lb>reſilit repatiendo, vti egerat in cadendo, hoc eſt; ad angu<lb></lb>los pares; quod fuerat demonſtrandum. </s> <s id="s.001987">Modo ſit <expan abbr="planū">planum</expan> <lb></lb>aliquod ita ad horizontem inclinatum, vt GH, & in illud <lb></lb>cadat proijciaturue eadem pila. </s> <s id="s.001988">Dico eam ab eodem in<lb></lb>clinato plano ad pares angulos reſilire non tamen rectos. <pb xlink:href="007/01/204.jpg"></pb>Vtique pila cadens, planum non tanget in E. eſſet enim <lb></lb>GH, vbi AB, Tangat autem in I, & à centro F ad contin<lb></lb>gentiæ punctum I, recta ducatur FI. </s> <s id="s.001989">Erit igitur FI (prop. <lb></lb>18. lib. 3. elem.) ipſi GH plano perpendicularis. </s> <s id="s.001990">Ducatur <lb></lb>item peri, ipſi EC, parallela IK, ſecans pilæ circumferen<lb></lb>tiam in K. </s> <s id="s.001991">Agit ergo & repatitur pila in puncto Inon æ. <lb></lb></s> <s id="s.001992">qualiter inæquales. </s> <s id="s.001993">etenim ſunt partes KDLEI, & IK, eo <lb></lb>quod IK ſecet circulum non per centrum. </s> <s id="s.001994">repellitur ergo <lb></lb>in repatiendo non æqualiter, ſed iuxta inæqualitatem ea<lb></lb>rundem partium. </s> <s id="s.001995">Ducatur autem recta in circulo LI æ<lb></lb>qualis ipſi IK. Erit igitur LEI, æqualis IK, & tota KDLI æ<lb></lb>qualis toti IKDL. </s> <s id="s.001996">Vt igitur actio eſt per deſcenſum iuxta <lb></lb>rectam KI, ita eſt repaſſio per aſcenſum ex IL. </s> <s id="s.001997">Dico autem <lb></lb>angulos KIH, LIG eſſe æquales & ſingulos recto minores. <lb></lb></s> <s id="s.001998">Connectantur FL, FK. </s> <s id="s.001999">Quoniam igitur IK portio æqualis <lb></lb>eſt portioni IEL, & recta LI æqualis rectæ IK, & LF æqua<lb></lb>lis ipſi FK, & FI communis, triangulum LFI, æquale eſt <lb></lb>triangulo IFK. </s> <s id="s.002000">Quare & angulus FIL aequalis angulo FIK, <lb></lb>ſed GIF, HIF recti ſunt, ergo reſidui LIG, KIH æquales <lb></lb>ſunt inter ſe comparati, & recto minores; quod fuerat o<lb></lb>ſtendendum. </s> </p> <p type="main"> <s id="s.002001">Hinc colligimus, quo magis planum ab æquidiſtan<lb></lb>tia horizontis receſſerit, eo pilam in eo proiectam in par<lb></lb>tes in æqualiores diuidi & ad minores ipſi plano angulos <lb></lb>reſilire. </s> <s id="s.002002">Nihil autem refert, vtrum planum, in quod pila <lb></lb>cadit, ad horizontem ſit inclinatum, vel eodem horizonti <lb></lb>æquediſtante pila non ad perpendiculas, ſed iuxta <expan abbr="aliquē">aliquem</expan> <lb></lb>angulum in illud proijciatur. </s> <s id="s.002003">Hæc ſane ita ex demonſtra<lb></lb>tione fieri oſtenduntur. </s> <s id="s.002004">Veruntamen quoniam proiecta <lb></lb>pila materialis eſt, & ideo nec æqualis, nec æqueponde<lb></lb>rans & ſua grauitate reſiſtens, non ad pares ex amuſſi reſi<lb></lb>lit angulos, ſed minores aliquantulum in reſilitione, re. <lb></lb></s> <s id="s.002005">mittente nimirum vi in ipſa reactione. </s> <s id="s.002006">Et ſane fieri non <pb xlink:href="007/01/205.jpg"></pb>poteſt, pilam à plano reſilientem eo peruenire vnde à <lb></lb>principio diſceſſerat; Id enim ſi daretur, æterna quoque <lb></lb>pilæ ipſius daretur reſilitio, & paullatim vi & impetu re<lb></lb>mittente per parua interualla motus eſſet, donec res quæ <lb></lb>mouebatur, omnino quieſcat. </s> </p> </subchap1> <subchap1> <p type="head"> <s id="s.002007">QVÆSTIO XXXV.</s> </p> <p type="head"> <s id="s.002008"><emph type="italics"></emph>Quærit hoc vltimo Problemate Ariſtoteles, Cur ea quæ in vorti<lb></lb>coſis feruntur aquis, ad medium tandem agan<lb></lb>tur omnia?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002009">Tribus rationibus ſoluit; quarum prima eſt: Quicquid <lb></lb>fertur, magnitudinem habet, cuius extrema in duo<lb></lb>bus ſunt circulis, hoc in minori, illud in maiori. </s> <s id="s.002010">Et quo<lb></lb>niam maior velocior eſt, magnitudo media, non æquali<lb></lb>ter fertur, ſed à maiori quidem pellitur, à minori vero re<lb></lb>trahitur, vnde transuerſus fit magnitudinis motus, & ipſa <lb></lb>magnitudo ad interiorem propellitur circulum, itaque <lb></lb>eodem pacto, è maiori in minorem propulſa in centrum. <lb></lb></s> <s id="s.002011">tantum fertur, & ibi quieſcit. </s> </p> <figure id="id.007.01.205.1.jpg" xlink:href="007/01/205/1.jpg"></figure> <p type="main"> <s id="s.002012">Eſto vortex AB, cuius cen<lb></lb>trum C, magnitudo quæ fer<lb></lb>tur AD, maior circulus AFB, <lb></lb>minor DHEG. </s> <s id="s.002013">Velocitas igi<lb></lb>tur in A maior eſt velocitate <lb></lb>quæ in D, magnitudinis ergo <lb></lb>extremum A, velocius rapitur <lb></lb>in A quam eiuſdem extremum <lb></lb>inferius D, in D. </s> <s id="s.002014">Velocitas igi<lb></lb>tur maioris circuli pellit Aver<lb></lb>ſus F. tarditas vero minoris cir<lb></lb>culi D retrahit ad partes G. conuertitur itaque magnitu<lb></lb>do inter pellentem & retrahentem circulum, donec ex<pb xlink:href="007/01/206.jpg"></pb>tremitas A in circulo minori fuerit vbi H, D vero vbi I, & <lb></lb>ita deinceps eadem ratione vbi KL, donec paullatim fe<lb></lb>ratur in centrum C, facto nempe à maiori in minorem cir<lb></lb>culum tranſitu. </s> </p> <p type="main"> <s id="s.002015">Secunda ratio ita habet, quia quod fertur, ſimili ſe <lb></lb>habet modo ad omnes circulos propter centrum, hoc eſt, <lb></lb>in quouis circulo, qui circa idem centrum fertur. </s> <s id="s.002016">Omnes <lb></lb>autem circuli mouentur, centrum vero ſtat, neceſſe eſt à <lb></lb>motu tandem id quod mouetur ad quietis locum, hoc eſt, <lb></lb>in centrum ipſum peruenire. </s> </p> <p type="main"> <s id="s.002017">Tertia, quoniam circulorum, qui in vorticibus fiunt, <lb></lb>velocitas, & ideo impetus non eſt æqualis, ſed ſemper ex<lb></lb>terior eſt interiore velocior & violentior, Æqualis autem <lb></lb>ſemper in mota magnitudine, grauitas, diuerſi mode ſe <lb></lb>habet ad circulos, à quibus mouetur, & ideo modo vin<lb></lb>citur, modo vincit: vincitur autem à velocioribus circulis, <lb></lb>vincit autem tardiores. </s> <s id="s.002018">Ita que quoniam ſua grauitate re<lb></lb>ſiſtens, maioris circuli motum prorſus non ſequitur, ad <lb></lb>tardiorem reijcitur, hoc eſt, interiorem, & ſic deinceps, <lb></lb>donec tandem centrum ipſum nanciſcatur, in quo nec ſu<lb></lb>perans, nec ſuperata quieſcit. </s> </p> <p type="main"> <s id="s.002019">Hæ ſunt rationes, licet obſcuriſſime propoſitæ, qui<lb></lb>bus, vt diximus, vtitur Ariſtoteles. </s> <s id="s.002020">acutæ ſane illæ <expan abbr="quidē">quidem</expan>, <lb></lb>attamen haudquaquam vltro admittendæ. </s> </p> <p type="main"> <s id="s.002021">Primo enim falſum videtur, quod aſſerit, vortices <lb></lb>circulos eſſe, & circa idem centrum fieri atque rotari. </s> <s id="s.002022">Spi<lb></lb>ræ enim potius ſunt, quæ ab exteriori parte <expan abbr="remotioreq;">remotioreque</expan> <lb></lb>incipientes ſpiraliter circumuolutæ, ad intimam tandem <lb></lb>partem, quæ media eſt & centri vices gerit, deueniunt. <lb></lb></s> <s id="s.002023">qua veritate cognita, omnis prorſus difficultas tollitur, <lb></lb>Cum enim ea quæ feruntur, ab aqua ferantur, aqua vero <lb></lb>feratur ſpiraliter, ea quoque ſpiraliter ferri, eſt neceſſa-<pb xlink:href="007/01/207.jpg"></pb>rium. </s> <s id="s.002024">Hæc autem clariora erunt ſi quo pacto vortices <lb></lb>fiant, quiſpiam conſiderauerit. </s> </p> <figure id="id.007.01.207.1.jpg" xlink:href="007/01/207/1.jpg"></figure> <p type="main"> <s id="s.002025">Eſto fluminis cuiuſpiam curua <lb></lb>eademque profunda ripa ABCD. <lb></lb></s> <s id="s.002026">Aquæ vero moles rapida EFDC, <lb></lb>quæ quidem eo quod magno impe<lb></lb>tu deferatur in C, ripæ ipſius <expan abbr="naturã">naturam</expan> <lb></lb>ſequens turbinatim circum uoluitur, <lb></lb>egreſſa autem extra locum ſeu ripam <lb></lb>B rotationis principium ſecundans, <lb></lb>in ſeipſam ſpiraliter contorquetur, <lb></lb>& vorticem efficit GHFIK, cuius <lb></lb>quidem centrum eſt vbi K. </s> </p> <p type="main"> <s id="s.002027">Alia quoque de cauſſa, ex quieſcente nimirum, & <lb></lb>mota aqua fiunt ſpiræ vorticesue. </s> <s id="s.002028">Eſto enim fluminis ripa <lb></lb><figure id="id.007.01.207.2.jpg" xlink:href="007/01/207/2.jpg"></figure><lb></lb>ABC, ſinum efficiens, qui a quam ex <lb></lb>ripæ ipſius obiectu contineat quie<lb></lb>ſcentem, Curſus vero fluminis liber & <lb></lb>rectus, ſit inter lineas AC, DE. </s> <s id="s.002029">Itaque <lb></lb>dum aqua AC rapide fertur ad partes <lb></lb>A, quieſcentem ABC iuxta lineam. <lb></lb></s> <s id="s.002030">CA lateraliter propellit, & eius qui<lb></lb>dem partem quam tangit, ſecum ra<lb></lb>pit, puta ex F in G. </s> <s id="s.002031">Delata igitur aqua <lb></lb>& currente ex F verſus G quieſcens <lb></lb>lateraliter eidem ſeſe aliqualiter op<lb></lb>ponit, & currentem repellit ex G in H. </s> <s id="s.002032">Cœpto <expan abbr="itaq;">itaque</expan> ſpirali <lb></lb>motu aqua circumuoluitur ſecundum lineam GHK, do<lb></lb>nec perueniat ad centrum I, vbi circumuolutæ aquæ par<lb></lb>tes ſeſe inuicem tangunt. </s> <s id="s.002033">Porro vortices iſti ſpiræue, quod <lb></lb>nos per Padum, Abduam, & magna flumina nauigantes <lb></lb>obſeruauimus, non eodem permanent loco, ſed rapientis <lb></lb>aquæ motum ſecundantes, paullatim in currentem <expan abbr="aquã">aquam</expan> <pb xlink:href="007/01/208.jpg"></pb>delati euaneſcunt, fiunt etiam eiuſcemodi vortices nau<lb></lb>tis quidem valde formidabiles etiam in mari, de quibus <lb></lb>Poëta libro Æneidos primo. </s> </p> <p type="main"> <s id="s.002034">— <emph type="italics"></emph>aſt illam ter fluctus ibidem <lb></lb>Torquet agens circum, & rapidus vorat æquore vortex.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002035">Sed & idem quoque de vorticibus, qui in fluminibus <lb></lb>fiunt libro 7. </s> </p> <p type="main"> <s id="s.002036">— <emph type="italics"></emph>hunc inter fluuio Tiberinus amœno <lb></lb>Vorticibus rapidis, & multa flauus arena <lb></lb>In mare prorumpit.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002037">Fiunt autem in mari partim occultis de cauſſis, partim <lb></lb>etiam ex violentia aquarum ſibi inuicem obuiantium a<lb></lb>gitatione. </s> <s id="s.002038">Sed nos hiſce explicatis commode ad ea quæ <lb></lb>dixerat Ariſtoteles, reuertemur. </s> </p> <p type="main"> <s id="s.002039">Dicimus igitur, primam eius rationem haud magni <lb></lb>videri ponderis, ſiquidem non per circulos actu diſtinctos <lb></lb>aqua circumfertur, ſed ipſamet ſua mole tota ſimul. </s> </p> <figure id="id.007.01.208.1.jpg" xlink:href="007/01/208/1.jpg"></figure> <p type="main"> <s id="s.002040">Eſto enim vortex AB, cu<lb></lb>ius centrum C, ſemidiameter <lb></lb>CA, fiat autem rotatio totius a<lb></lb>quæ CA ad partes D, in linea <lb></lb>autem AC, ſit corpus aliquod a<lb></lb>quæ rotatione <expan abbr="circumlatū">circumlatum</expan> AE, <lb></lb>inter circulos maiorem ADB, <lb></lb>minorem EFG. velocius autem <lb></lb>mouetur ADB, ipſo EFG, citius <lb></lb>ergo fertur pars ſuperior ipſius <lb></lb>corporis vbi A, quam inferior <lb></lb>vbi E. </s> <s id="s.002041">At id nec A repellit, nec E retrahit, ſiquidem eodem <lb></lb>tempore quo A permeauit <expan abbr="circulū">circulum</expan> ADB, eodem & E per<lb></lb>currit circulum EFG. <expan abbr="Itaq;">Itaque</expan> A reuerſo in A & E, punctum <lb></lb>reuerſum erit in E, nulla facta corporis E quoad ſitum, <lb></lb>muratione quod voluit Ariſtoteles. </s> </p> <pb xlink:href="007/01/209.jpg"></pb> <p type="main"> <s id="s.002042">Ad ſecundam vero dicimus, non ideo quod omnes <lb></lb>circuli æqualiter circa centrum ſerantur, niſi alia <expan abbr="quæpiã">quæpiam</expan> <lb></lb>extranea vis interceſſerit, quæ ea ab exterioribus circulis <lb></lb>pellens agat in medium. </s> </p> <figure id="id.007.01.209.1.jpg" xlink:href="007/01/209/1.jpg"></figure> <p type="main"> <s id="s.002043">Tertia quoque ratio la<lb></lb>borare videtur. </s> </p> <p type="main"> <s id="s.002044">Eſto enim vortex AB, <lb></lb>cuius centrum C, ſit autem <lb></lb>corpus aliquod E, cuius na<lb></lb>tura apta ſit rotationi aliqua<lb></lb>tenus reſiſtere. </s> <s id="s.002045">Quoniam i<lb></lb>gitur eius reſiſtentia <expan abbr="aliquã-tulum">aliquan<lb></lb>tulum</expan> ab aqua rapiente ſu<lb></lb>peratur in ipſa rotatione, par<lb></lb>tim aquae impetum ſequetur, <lb></lb>partim ſuapte natura retardabitur. </s> <s id="s.002046">Quamobrem aqua <lb></lb>quæ eſt in A, translata in H, corpus ipſum non erit in H, <lb></lb>ſed in G. </s> <s id="s.002047">Tardius igitur corpus quam aqua ipſa, rotatio<lb></lb>nem complebit, non tamen propterea, niſi alia quæ piam <lb></lb>adſit cauſſa, feretur in medium. </s> </p> <p type="main"> <s id="s.002048">Cæterum horum vorticum effectum & cauſſam ob<lb></lb>ſeruare licet, ſi vaſe quopiam aqua pleno aquam ipſam <lb></lb>baculo manuue circulariter agitauerimus, fiet enim vor<lb></lb>tex, & ſi quippiam quod leue ſit, in aquam motam proie<lb></lb>cerimus, ea quam diximus de cauſſa in motum ipſum, hoc <lb></lb>eſt, vorticis ſpiræue, centrum feretur. </s> </p> <p type="main"> <s id="s.002049">Hæc nos, vt vera proponimus, & fortaſſe decipimur. <lb></lb></s> <s id="s.002050">Certe Philoſopho tantæ auctoritatis contradicere, ma<lb></lb>gnæ videtur audaciæ, aut potius inſaniæ. </s> <s id="s.002051">Quicquid ta<lb></lb>men ſit, pro pulcherrima veritate laboraſſe, à parte <lb></lb>aliqua laudis non fuerit prorſus, vt <lb></lb>arbitror, alienum. </s> </p> <pb xlink:href="007/01/210.jpg"></pb> </subchap1> </chap> <chap> <p type="head"> <s id="s.002052">APPENDIX.</s> </p> <p type="main"> <s id="s.002053">Modum inueniendarum duarum mediarum propor<lb></lb>tionalium non tantum vtilem eſſe, ſed prorſus neceſ<lb></lb>ſarium, illi norunt, qui in Mechanicis diſciplinis vel <expan abbr="parū">parum</expan> <lb></lb>fuerint verſati. </s> <s id="s.002054">Nulla enim alia ratio eſt, qua corporeae ma<lb></lb>gnitudines ſeruata figura & ſimilitudine augeri propor<lb></lb>tionaliter imminuiue poſſint. </s> <s id="s.002055">Quamobrem factum eſt vt <lb></lb>in his inueniendis tum vetuſtiſſimo tum etiam in feriori æ<lb></lb>uo, clariſſimi Viri magnopere laborauerint. </s> <s id="s.002056">Plato etenim, <lb></lb>Eudoxus (cuius modum repudiauit Eutocius) Heron A<lb></lb>lexandrinus, Philon Byzantius, Apollonius, clariſſimi <lb></lb>Geometræ, Diocles, Pappus, Sporus, Menæchmus, Ar<lb></lb>chytas Tarentinus, Platoni æqualis: Eratoſthenes, & Ni<lb></lb>comedes ad has inueniendas varias rationes <expan abbr="excogitarūt">excogitarunt</expan>, <lb></lb>quorum omnium modos, & inſtrumenta, <expan abbr="demonſtratio-neſq;">demonſtratio<lb></lb>neſque</expan> diligentiſſime collegit, & in illos <expan abbr="Cōmentarios">Commentarios</expan> con<lb></lb>iecit idemmet Eutocius, quos elegantiſſimos in Archime<lb></lb>dis libros de Sphæra & Cylindro ſcripſit. </s> <s id="s.002057">Nos autem ijs o<lb></lb>mnibus accurate perſpectis, & diligentiſſime ponderatis, <lb></lb>inuenimus eos fere omnes tentando negotium abſolue<lb></lb>re, quod ſane laborioſum valde eſt & operantibus permo<lb></lb>leſtum. </s> <s id="s.002058">Itaque cum modum praximue inueniſſemus, ex <lb></lb>qua is qui operatur tutiſſime & facillime ad quæ ſitas ipſas <lb></lb>medias manu ducitur, hunc pulcherrimæ huius facultatis <lb></lb>ſtudio ſis inuidere nefarium iudicauimus. </s> <s id="s.002059">Quod ſi <expan abbr="quiſpiã">quiſpiam</expan> <lb></lb>dixerit, Balliſtarum, Catapultarum, Scorpionum, & cæ<lb></lb>terarum eiuſcemodi Machinarum vſum, olim apud nos <lb></lb>deſijſſe, & ideo Problema hoc videri ſuperuacaneum, Re<lb></lb>ſpondemus, nulla alia ratione æneorum tormentorum pi<lb></lb>las augeri imminuiue ſeruata ponderis ratione poſſe, in<lb></lb>numeraque eſſe, quæ vt rite perficiantur, hæc penitus in<lb></lb>digent ſpeculatione. </s> <s id="s.002060">Nos rem Mechanicis vtilem, Me. <pb xlink:href="007/01/211.jpg"></pb>chanicis noſtris Exercitationibus annectere, haud im<lb></lb>portunum iudicauimus. </s> <s id="s.002061">Sed tempus eſt, vt his breuiter <lb></lb>præfatis, ad rem ipſam <expan abbr="explicandã">explicandam</expan> commode accedamus. </s> </p> <p type="head"> <s id="s.002062"><emph type="italics"></emph>Datis duabus proportionalibus prima, & quarta duas inter eas <lb></lb>medias in continua proportione inuenire.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002063">Esto prima datarum AB, quarta BC, inter quas <expan abbr="ſecundã">ſecundam</expan> <lb></lb>& tertiam oportet inuenire. </s> <s id="s.002064">Ducatur recta DE, cui à <lb></lb>puncto F, vtcunque ſumpto, perpendicularis demittatur <lb></lb>FG, Tum ab F verſus D duplicetur quarta BC, ſitque FH, <lb></lb>deinde ab H ipſi FG parallela demittatur HI, & ab HF <lb></lb>abſcindatur HK, ipſius BC quartæ medietati æqualis. <lb></lb></s> <s id="s.002065">Poſthæc puncto K ſpatio autem medietati, primæ data<lb></lb>rum æquali, in linea HI notetur punctum L, & ipſi HL <lb></lb>fiat æqualis FM, & KM iungatur. </s> <s id="s.002066">His ita conſtitutis pare<lb></lb>tur ſeorſum ſcheda regulaue quæpiam NO, in cuius late<lb></lb>re accipiatur OP, æqualis medietati primæ datarum ſeu <lb></lb>ipſi KL. </s> <s id="s.002067">Tum regulæ latus aptetur puncto L, extremum <lb></lb>vero O, feratur aſſidue per rectam EK, verſus K, nunquam <lb></lb><figure id="id.007.01.211.1.jpg" xlink:href="007/01/211/1.jpg"></figure><pb xlink:href="007/01/212.jpg"></pb>interim regulæ latere ON amoto à puncto L, idque do<lb></lb>nec punctum P, obuians incidat in lineam KM, puta vbi <lb></lb>Q extremum vero O inueniatur in R, notato igitur in li<lb></lb>nea EK puncto R habebitur, quod quærebatur. </s> <s id="s.002068">Erunt i<lb></lb>gitur AB prima, RK ſecunda, QL tertia, BC quarta. </s> </p> <p type="main"> <s id="s.002069">Hæc praxis ijſdem principijs demonſtratur, quibus <lb></lb>ſuam ex Conchoide oſtendit Nicomedes. </s> <s id="s.002070">Conficit ille <lb></lb>inſtrumentum, ex quo deſcribit <expan abbr="Conchoidē">Conchoidem</expan>, ex qua poſt<lb></lb>ea duas medias venatur. </s> <s id="s.002071">Nos autem nec inſtrumentum <lb></lb>conſtruimus nec Conchoidem deſcribimus, & duabus fe<lb></lb>re lineis rem abſoluimus, vt nemo fere non dixerit, hoci<lb></lb>ſtud quod docemus, à Nicomedea praxi eſſe prorſus a<lb></lb>lienum. </s> </p> <p type="main"> <s id="s.002072">Sed nos, vt eius, quam oſtendimus, operationis de<lb></lb>monſtratio habeatur; ipſius Nicomedis ex Pappi libro 3. <lb></lb>propoſ. </s> <s id="s.002073">5. deſumptam in medio afferemus, quippe quod <lb></lb>iſthæc ea quam in ſuis in Archimedem commentarijs re<lb></lb>fert Eutocius, ſit lucidior. </s> </p> <p type="main"> <s id="s.002074">Datis duabus rectis lineis CD, DA; duæ mediæ in <lb></lb>continua proportione hoc modo aſſumuntur. </s> </p> <p type="main"> <s id="s.002075">Compleatur ABCD parallelogrammum, & <expan abbr="vtraq;">vtraque</expan> <lb></lb>ipſarum AB, BC, bifariam ſecetur in punctis L, E, iuncta<lb></lb>que LD producatur; & occurrat productæ CB, in G, ipſi <lb></lb>vero BC ad rectos angulos ducatur EF, & CF iungatur, <lb></lb>quæ ſit æqualis AL. </s> <s id="s.002076">Iungatur præterea FG & ipſi paralle<lb></lb>la ſit CH, eritque angulus KCH, æqualis angulo CGF. <lb></lb></s> <s id="s.002077">Tum à dato puncto F ducatur FHK, quae faciat KH æqua<lb></lb>lem ipſi AL vel CF. </s> <s id="s.002078">Hoc enim per lineam Conchoidem <lb></lb>fieri poſſe oſtendit Nicomedes, & iuncta KD producatur, <lb></lb>occurratque ipſi BA, productæ in puncto M. </s> <s id="s.002079">Dico vt DC <lb></lb>ad CK ita CK ad MA & MA ad AD. </s> <s id="s.002080">Quoniam enim BC <lb></lb>bifariam ſecta eſt in E, & ipſi adijcitur CK. </s> <s id="s.002081">Rectangulum <lb></lb>BKC per 6. ſecundi: vna cum quadrato ex CE, æquale eſt <pb xlink:href="007/01/213.jpg"></pb><figure id="id.007.01.213.1.jpg" xlink:href="007/01/213/1.jpg"></figure><lb></lb>quadrato ex EK. commune apponatur ex EF quadratum, <lb></lb>ergo rectangulum BKC vna cum quadrato CF æquale <lb></lb>eſt quadratis ex KE, EF, hoc eſt, quadrato ex FK. </s> <s id="s.002082">Et quo<lb></lb>niam vt MA ad AB, ita eſt MD ad DK, vt autem MD ad <lb></lb>DK per 2. ſexti, ita BC ad C<emph type="italics"></emph>K<emph.end type="italics"></emph.end> erit vt MA ad AB, ita BC <lb></lb>ad C<emph type="italics"></emph>K<emph.end type="italics"></emph.end>. </s> <s id="s.002083">Atque eſt ipſius AB dimidia AL, & ipſius BC, du<lb></lb>pla CG, eſt igitur vt MA ad AL, ita GC ad C<emph type="italics"></emph>K<emph.end type="italics"></emph.end>. </s> <s id="s.002084">Sed vt GC <lb></lb>ad C<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, ita FH ad H<emph type="italics"></emph>K<emph.end type="italics"></emph.end> propter lineas parallelas GF, CH. <lb></lb>quare & componendo vt ML, ad LA, ita F<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, ſed <lb></lb>AL ponitur æqualis H<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, quoniam & ipſi CF, ergo & ML <lb></lb>per 9. lib. 5. æqualis erit F<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, & quadratum ex ML, æquale <lb></lb>quadrato ex F<emph type="italics"></emph>K<emph.end type="italics"></emph.end>. </s> <s id="s.002085">eſt autem quadrato ex ML, æquale re<lb></lb>ctangulum BMA vna cum quadrato ex AL & quadrato <lb></lb>ex Fk æquale oſtenſum eſt rectangulum BkC vna cum. <pb xlink:href="007/01/214.jpg"></pb>quadrato ex CF, quorum quidem quadratum ex AL æ<lb></lb>quale eſt quadrato ex CF, ponitur enim AL, ipſi CF æ<lb></lb>qualis, ergo reliquum BMA rectangulum æquale eſt reli<lb></lb>quo BkC. </s> <s id="s.002086">Vt igitur MB ad Bk, ita Ck ad MA. </s> <s id="s.002087">Sed vt MD <lb></lb>ad Bk, ita DC ad Ck. </s> <s id="s.002088">quare vt DC ad Ck, ita eſt Ck ad <lb></lb>MA. vt autem MD ad Bk, ita MA, ad AD. </s> <s id="s.002089">Ergo vt DC, <lb></lb>prima, ad Ck ſecundam, ita Ck ſecunda ad MA tertiam, <lb></lb>& MA tertia ad AD quartam, quod fuerat demonſtran<lb></lb>dum. </s> <s id="s.002090">Hæc Pappus. </s> <s id="s.002091">Quod autem in noſtra Praxi diximus, <lb></lb>QL eſſe tertiam, ea ratio eſt, quod LR vt in prima figura <lb></lb>eſt, ſit æqualis ipſi LM ſecundæ figuræ, in demonſtratio<lb></lb>ne Pappi, ex quibus demptis QR & LA, quæ ſunt æqua<lb></lb>les, reliqua QL primæ figuræ æqualis eſt AM ſecundæ fi<lb></lb>guræ, hoc eſt, ipſi tertiæ proportionali: Eſt igitur, vt in pri<lb></lb>ma figura dicebamus, AB prima, kR ſecunda, QL tertia, <lb></lb>BC quarta. </s> </p> <p type="main"> <s id="s.002092">Vides igitur tu qui legis, nos ex Nicomedis demon<lb></lb>ſtratione (quatenus ad praxin pertinet) ſuperflua reſecaſ<lb></lb>ſe, & abſque Conchoidis inſtrumento lineaue rem ipſam <lb></lb>confeciſſe, idque non tentantes, vt alij, ſed progre<lb></lb>dientes, & quaſi manuductos quæſi<lb></lb>tum inueſtigaſſe. </s> </p> <p type="head"> <s id="s.002093">FINIS.<lb></lb> </s> </p> </chap> </body> <back></back> </text> </archimedes>