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Colored diff for /texts/archimedes/xml/Attic/baldi_mecha_01_la_1621.xml between version 1.4 and 1.38

version 1.4, 2002/06/27 17:24:26

version 1.38, 2002/09/20 21:50:12

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<?xml version="1.0"?>

<!DOCTYPE archimedes [

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >

<author>Baldi, Bernardino</author>

<title>In mechanica Aristotelis problemata exercitationes</title>

<place>Mainz</place>

<cvs_file>baldi_mecha_01_la_1621</cvs_file>

<cvs_version/>

</info> <text> <front><section>

<s id="id.000001">BERNARDINI BALDI VRBINATIS <lb/>GVASTALLÆ AB<lb/>BATIS <lb/><emph type="italics"/>IN<emph.end type="italics"/></s>

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<s id="id.000002">MECHANICA ARISTOTE<lb/>LIS PROBLEMATA <lb/>EXERCITATIONES:</s>

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<s id="id.000003"><emph type="italics"/>ADIECTA SVCCINCTA NAR<lb/>ratione de autoris vita & &longs;criptis.<emph.end type="italics"/></s><figure id="id.007.00.002.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.002.1.jpg"></figure>

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<s id="id.000004"><emph type="italics"/><gap/>NTIAE.<emph.end type="italics"/></s>

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<s id="id.000005">Typis & Sumptibus Viduæ Ioannis Albini.</s>

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</section><section>

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<s id="id.000007"><emph type="italics"/>NOBILISSIMO AC GENE<lb/>ROSO DOMINO<emph.end type="italics"/>D. ADAMO PHILIP<lb/>PO BARONI A CRON<lb/>BERG, EQVITI, SACRÆ CÆSA<lb/>REÆ MAIESTATIS, ET SERENISSIMI Principis Archiducis Alberti Camerario intimo & c. <lb/></s>

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<s id="id.000008">Domino meo gratio&longs;i&longs;&longs;imo.</s>

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<s id="id.000009">Opportune &longs;ub hoc ip&longs;um tem<lb/>pus, quo in Belgium ad Scre<lb/>ni&longs;&longs;imos Principes iter ador<lb/>nat.</s>

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<s id="id.000010"> Nobili&longs;&longs;ima & Genero&longs;a <lb/>Dom. V.^{ra}, prodit no&longs;tris for<lb/>mis in publicum editus Com<lb/>mentarius Bernardini Baldi Vrbinatis Gua<lb/>&longs;tallæ Abbatis in Ari&longs;totelis Mechanica. </s>

<!ELEMENT archimedes

(info, text) >

<s id="id.000011">Is <lb/>virin omni &longs;cientiæ genere, at maxime inMa<lb/>thematicis di&longs;ciplinis fuit ver&longs;ati&longs;&longs;imus, quod <lb/>multa ab eo præclare &longs;crip ta te&longs;tantur opera, <lb/>ex quibus paucula edita, reliqua vero &longs;pera<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.004.jpg"/>mus &longs;uo temporein publicam luce<gap/> produ<lb/>cenda. </s>

<!ATTLIST archimedes

<s id="id.000012"> Cum vero nemini&longs;it ob&longs;curum Nobi<lb/>li&longs;&longs;imæ ac Genero&longs;æ Dom. V.^{ræ} id &longs;emper <lb/>extiti&longs;&longs;e familiari&longs;&longs;imum, vttum dome&longs;ticum <lb/>otium, tum maxime peregtinationes, quibus <lb/>totam pæne Europam &longs;umma cum laude <lb/>circum&longs;crip&longs;it, tum variarum linguarum per<lb/>fecto v&longs;u, tum Mathematicarum di&longs;ciplina<lb/>rum notitia & exercitio redderet <expan abbr="iuc&utilde;diores">iucundiores</expan>, <lb/>nulla me tenet dubitatio quin & Baldum Vr<lb/>binatem no&longs;tris typis loquentem in hoc iti<lb/>nere, quod à Deo felici&longs;&longs;imum Nobili&longs;&longs;imæ <lb/>ac Genero&longs;æ Dom. V.^{ræ} precor, in &longs;uum comi<lb/>tatum ac tutelam beneuolo animo &longs;it admi&longs;<lb/>&longs;ura.</s>

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xmlns:xlink CDATA #FIXED "http://www.w3.org/1999/xlink"

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<!ELEMENT author

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<!ATTLIST author

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<s id="id.000013"> Id rogo humillime &longs;imulque precor, vt. <lb/></s>

<!ELEMENT back

<s id="id.000014">hanc meam typographiam plurimis iam re<lb/>tro annis de inclytæ familiæ Cronbergicæ tu<lb/>tela gloriantem, &longs;uo fauore pro&longs;equatur, vi<lb/>duæque afflictæ fortunis beneuole ad&longs;piret.</s>

( section | pb?)+ >

<!ATTLIST back

<s id="id.000015"> <lb/>Sic Deus Nobili&longs;&longs;. & Genero&longs;am Dom. V.^{ram} <lb/>illu&longs;tret omnibus bonis, eamque R.^{mo} & Ill.^{mo} <lb/>Principiac Domino meo Clementi&longs;&longs;imo, D. <lb/>Ioanni Suicardo Archiepi&longs;copo Mogunti<lb/>no Principi Electori ac per <emph type="italics"/>G<emph.end type="italics"/>ermaniam Ar-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.005.jpg"/>chicancellario &c. </s>

id ID #IMPLIED

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(chap, pb?)+ >

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<s id="id.000016">patruo &longs;uo optati&longs;&longs;imo <lb/>&longs;aluo florentique redhibeat &longs;aluum &longs;imili<lb/>ter florentem ac incolumem.</s>

id ID #IMPLIED

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<s id="id.000017"> Moguntiæ è <lb/>typographeio Viduæ Albinianæ, honori No<lb/>bili&longs;&longs;imæ ac <emph type="italics"/>G<emph.end type="italics"/>enero&longs;æ Dom. Ve&longs;træ perpe<lb/>tuum dicato. </s>

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<s id="id.000018">Anno 1621.26.Martij. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.006.jpg"/><figure id="id.007.00.006.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.006.1.jpg"></figure>

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</front>

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<body>

<chap>

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<s id="id.000019">PRAEFATIO.</s>

<s id="id.000020"><emph type="italics"/>Diligenter legenti mihi quæ&longs;tiones il<lb/>las, in quibus ea quæ ad Mecha<lb/>nicam facultatem pertinent, expli<lb/>cantur, multa in mentem venie<lb/>bant; & primum quidem eorum, quæ ibi dispu<lb/>tantur, vtilitatem, &longs;ubtilitatem, copiam admi<lb/>rabar: Tum ex animo dolebam, aureum hunc li<lb/>bellum propè negligi, & ab iis qui pulcherrimis <lb/>hi&longs;ce &longs;tudiis dant operam, assiduè præ manibus <lb/>non haberi: Multas autem Auctori ip&longs;i haben<lb/>das referendasque e&longs;&longs;e gratias, qui tam egregiam, <lb/>vtilem & probè in&longs;tructam &longs;upellectilem Archi<lb/>tectis, Mechanicis, & omnibus ferè Artificibus <lb/>&longs;uppeditauerit. </s>

<!ATTLIST date

<s id="id.000021">Ari&longs;totelis nomini a&longs;cribitur <lb/>Commentarius, licet nonnulli, &longs;itne Philo&longs;ophi <lb/>illius præclarissimi & acutissimi labor, an non, <lb/>adfirmare &longs;ubdubitauerint. </s>

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT editor

<s id="id.000022">Ari&longs;totelis tamen <lb/>e&longs;&longs;e omnes ferè meliores con&longs;entiunt: Idque tum <lb/>ex phra&longs;i, & explicatione, quæ Ari&longs;totelem &longs;a<lb/>piunt, tum iudicio &longs;ubtilitatis & rationum, qui-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.007.jpg"/>bus quæ&longs;tiones ip&longs;æ ingenio&longs;issimè diluuntur. </s>

(#PCDATA) >

<s id="id.000023">Vi<lb/>detur autem mihi, rem accuratius exploranti, &longs;a<lb/>tis veri&longs;imile (nullum enim habeo opinionis hu<lb/>ius a&longs;&longs;ertorem,) &longs;ectionem e&longs;&longs;e hanc, & partem <lb/>quandam eius operis nobilissimi, quod idem au<lb/>ctor De Problematibus edidit, & hanc, ne&longs;cio <lb/>quam ob cau&longs;am; ni&longs;i fortè quod tractatio merè <lb/>Phy&longs;ica non &longs;it, à reliquo corpore di&longs;tractam at<lb/>que reuul&longs;am. </s>

<!ATTLIST editor

<s id="id.000024">Id certè quod ad rem facit, probè <lb/>nouimus, Diogenem Laërtium inter cætera Ari<lb/>&longs;totelici ingenij monumenta Mechanica quoque <lb/>adnumera&longs;&longs;e. </s>

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<!ELEMENT emph

<s id="id.000025">Quibus con&longs;ideratis magnopere <lb/>&longs;ubit mirari, cur ij qui po&longs;t Ari&longs;totelem floruêre <lb/>atque vixere, Mechanici, Archimedes, Athenæus, <lb/>Heron, Pappus, & cæteri, nullam huius libelli fe<lb/>cerint commemorationem: & &longs;anè debuerunt; <lb/>neque enim à vero est dissimile, ip&longs;os per hunc ali<lb/>quatenus profeci&longs;&longs;e. </s>

EMPTY>

<s id="id.000026">Verum enim uero cum inge<lb/>nui illi fuerint homines, & nullatenus obtrecta<lb/>tores, credendum potius est, Comment ariolum i<lb/>&longs;tud, eorum æuo, paucis cognitum, alicubi in Bi<lb/>bliothecis latui&longs;&longs;e: etenim cætera quoque Ari&longs;tote<lb/>lis &longs;cripta, po&longs;t vetu&longs;ta illa tempora, ante Ale<lb/>xandrum Aphrodi&longs;ien&longs;em, à multis fui&longs;&longs;e igno-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.008.jpg"/>rata non dubitamus. </s>

<!ATTLIST emph

<s id="id.000027">Habemus &longs;iquidem, Stra<lb/>bone te&longs;te, lib.

id ID #IMPLIED

n CDATA #IMPLIED>

<!ELEMENT emph.end

13. Ari&longs;totelis, & Theophra&longs;ti bi<lb/>bliothecam, po&longs;t ip&longs;ius Theophra&longs;ti dece&longs;&longs;um, ad <lb/>Neleum quendam Scep&longs;ium, Cori&longs;cifilium, qui <lb/>eius fuerat auditor, perueni&longs;&longs;e; po&longs;t hæc libros, <lb/>blattis olim, & humore corruptos, Apelliconi Te<lb/>io venditos, & ab eo Athenas translatos, tum <lb/>Athenis captis in Syllæ pote&longs;tatem deacni&longs;&longs;e, eo&longs;<lb/>que tandem à Sylla acceptos, Tyrannionem <lb/>Grammaticum, vt potuit meliùs emendatos, <lb/>promulga&longs;&longs;e. </s>

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<s id="id.000028">Exquibus colligimus, mirum non <lb/>e&longs;&longs;e, Archimedi, Heroni, & alijs qui ante Syllam <lb/>vixêre, fui&longs;&longs;e incognitos. </s>

<!ATTLIST emph.end

id ID #IMPLIED

n CDATA #IMPLIED>

<s id="id.000029">quicquid&longs;it, illud cer<lb/>tumest, Ari&longs;totelem eorum omnium quide Me<lb/>chanicis commentaria edidere, e&longs;&longs;e longè vetu<lb/>&longs;tissimum. </s>

<!ELEMENT expan

<s id="id.000030">Pappus enim Heroneiunior, Athe<lb/>næus Archimediæqualis, vterque enim &longs;ub Mar<lb/>cello, cui Athenæus &longs;uum de bellicis Machinis <lb/><expan abbr="libell&umacr;">libellum</expan> dedicauit. </s>

<s id="id.000031">Archimedes verò circa CXL. <lb/></s>

<!ATTLIST expan

<s id="id.000032">Olympiadem floruit, quamobrem po&longs;t Ari&longs;tote<lb/>lem Olympiadas XL. hoc est, annos ferè CLX. <lb/></s>

id ID #IMPLIED

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type CDATA #IMPLIED >

<s id="id.000033">I&longs;thæc autem con&longs;iderantibus, facile e&longs;t cogno&longs;ce<lb/>refacultatis huius nobilitatem, atque dignit atem; <lb/>quippe quod &longs;ummus Philo&longs;ophus non modo eam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.009.jpg"/>probauerit, &longs;cd etiam &longs;uis acutissimis lucubra<lb/>tionibus illu&longs;trauerit. </s>

<!ELEMENT figure

<s id="id.000034">Hanc porro tractationem <lb/>&longs;ubiecto quidem Phy&longs;icam e&longs;&longs;e, demon&longs;tratio<lb/>nibus verò Geometricam, ip&longs;emet nos docuit <lb/>Ari&longs;toteles, cuius etiam naturæ &longs;unt Per&longs;pecti<lb/>ua, Specularia, Mu&longs;ica, & cæteræ eiu&longs;dem <lb/>modi facultates, quas quidem &longs;ubalternas Peri<lb/>patetici appellant. </s>

(#PCDATA) >

<!ATTLIST figure

<s id="id.000035">Vitruuius Architecturæ <lb/>membrum, vt ita dicam, & portionem quan<lb/>dam facit, ait enim Architecturæ partes e&longs;&longs;e tres, <lb/>Ædificationem, Gnomonicam, Machinatio<lb/>nem. </s>

id ID #IMPLIED

place (margin|text) "text"

xlink:type (simple) #FIXED "simple"

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<s id="id.000036">Estautem Architectur â quideminferior, <lb/>paret enim Architecto Mechanicus; attamen &longs;i <lb/>cæteras artes &longs;pectes, Architectonica; hæc enim <lb/>omnesferè &longs;edentariæ, &longs;ellulariæue, quas banau<lb/>&longs;as Græci appellant, ordine &longs;ubijciuntur, & &longs;a<lb/>nè latissimos i&longs;ihæc habet fines; præcipuè autem <lb/>circa eam ver&longs;atur cognitionem, eamque inter <lb/>cæterasferè principem, quam dixere Centrobari<lb/>cam, quæ quidem ad Centri grauitatem, eiu&longs;que <lb/>&longs;peculationem pertinet: quà in &longs;pecie inter vete<lb/>res primum &longs;ibi vindicauit locum Archimedes, <lb/>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.010.jpg"/><emph type="italics"/>tem Commandinus, qui librum de Centro gra<lb/>uitatis &longs;olidorum &longs;crip&longs;it, & po&longs;t eum G. Vbal<lb/>dus è Marchion. Montis, qui non modò ab<lb/>&longs;olutissimum Mechanicorum librum cum maxi<lb/>ma ingenij &longs;ui laude con&longs;crip&longs;it, &longs;ed & Paraphra<lb/>&longs;in in librum Æqueponderantium Archimedis <lb/>egregiè concinnauit Centrobaricam hanc, igno<lb/>tam fui&longs;&longs;e Ari&longs;toteli, &longs;ætis patet. </s>

<!ELEMENT foreign

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<!ELEMENT front

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<!ATTLIST front

<s id="id.000037">nunquam enim <lb/>in Mechanicis demon&longs;trationibus, quod tamen <lb/>est potissimum, grauitatis centrum nominat, e<lb/>iu&longs;ue naturam atque vim &longs;peculatur. </s>

id ID #IMPLIED

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type CDATA #IMPLIED>

<!ELEMENT gap

<s id="id.000038">Diuidi<lb/>tur autem Mechanice tota, te&longs;te Herone apud <lb/>Pappum libro octauo, in Rationalem, hoc est, <lb/>Theoricam & Chirurgicam, id est, manu ope<lb/>ratricem, quam Praxim aptè dicere valemus. <lb/></s>

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<!ATTLIST gap

<s id="id.000039">Rationalis, &longs;peculationi & <expan abbr="dem&omacr;&longs;trationibus">demon&longs;trationibus</expan>, ex <lb/>Geometricis, Arithmeticis & Phy&longs;icis rationi<lb/>bus, dat operam; Chirurgica vero materiam <lb/>tractat, & &longs;e&longs;ein varias artes diffundit, Æra<lb/>riam, Lignariam, Sculptoriam, Pictoriam, Æ<lb/>dificatoriam, Machinariam & Thaumaturgi<lb/>cam, cæterasque eiu&longs;modi. </s>

id ID #IMPLIED

n CDATA #IMPLIED

desc CDATA #IMPLIED>

<s id="id.000040">Machinatoriæ au<lb/>tem &longs;unt partes Manganaria, qua ingentiæ<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.011.jpg"/><emph type="italics"/>transferuntur pondera, tum ip&longs;a Poliorcetica, <lb/>quæ bellicas Machinas ad vrbium expugnatio<lb/>nes, quod velip&longs;o nomine profitetur, ædificat. </s>

<s id="id.000041">At<lb/>qui hac dere plura &longs;cribere &longs;uper&longs;edemus, ne a<lb/>ctum agamus: quis quis enim minutè magis hæc <lb/>cogno&longs;cere de&longs;iderat, is Pappum adeat libro cita<lb/>to, & Guidum Vbaldumin Præfatione quam <lb/>&longs;uo Mechanicorum Operi præpo&longs;uit. </s>

<!ELEMENT info

<s id="id.000042">Vt autem <lb/>ad Ari&longs;totelis, de quo egimus, libellum reuerta<lb/>mur, pauci &longs;unt qui ei ante nos &longs;tilum & operam <lb/>commodauerint: Leonicenus Latinum fecit & <lb/>figuris tum breuissimis, & parui &longs;ane ponderis, <lb/>marginalibus adnotatiunculis, in&longs;truxit. </s>

(author, title, date, place, editor, publisher, translator, lang, chunk,locator) >

<!ATTLIST info

<s id="id.000043">Po&longs;t <lb/>hunc Alexander Picolomineus luculentissima <lb/>Paræphra&longs;i illu&longs;trauit. </s>

id ID #IMPLIED

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<s id="id.000044">Modo, vt audio, Simon <lb/>Sticinus Hollanden&longs;is quædam edidit, quæ ad <lb/>nos minime peruenêre. </s>

<s id="id.000045">Nos demum, omnium, <lb/>tum &longs;cientia, & ingenio, tum ætate, po&longs;tremi huic <lb/>operi manum admouimus; Con&longs;iderantes enim <lb/>Ari&longs;totelem aliis fecerint Mechanici, demon&longs;tra&longs;&longs;e, <lb/>morem huiu&longs;ce facultatis &longs;tudio&longs;is ge&longs;turos nos <lb/>fore arbitrati&longs;umus, &longs;i ea&longs;dem illas quæ&longs;tiones<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.012.jpg"/><emph type="italics"/>Mechanicis, hocest, Archimedeis probationi<lb/>bus confirmaremus; dum per latissimos faculta<lb/>tis huius campos vægantes, alias quoque i&longs;tis af<lb/>fines dubitationes introducentes &longs;olueremus. <lb/></s>

<!ELEMENT lang

<s id="id.000046"><expan abbr="quicquidaut&emacr;&longs;ecerimus">quicquidauten&longs;ecerimus</expan> profecerimu&longs;ue, Lector <lb/>optime, boni con&longs;ule, & quia fax per manus tra<lb/>ditur, tu interim de me accipe, vt alijs tradas.<emph.end type="italics"/></s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.013.jpg"/>

(#PCDATA) >

</chap>

<chap>

<!ATTLIST lang

<s id="id.000047">DE VITA ET SCRI<lb/>PTIS BERNARDINI <lb/>BALDI VRBINATIS</s>

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<!ELEMENT lb EMPTY >

<s id="id.000048"><emph type="italics"/>EX LITERIS FABRITII SCHAR<lb/>loncini ad Illu&longs;trissimum & Reuerendissimum <lb/>Dominum Lælium Ruinum Epi&longs;copum Bal<lb/>neoregien&longs;em ex-Nuntium Apo&longs;tolicum <lb/>ad Poloniæ Regem & c.<emph.end type="italics"/></s>

<!ATTLIST lb

<s id="id.000049">Natus e&longs;t Bern. Baldus Vrbini nobilibus <expan abbr="pa-r&emacr;tibus">pa<lb/>rentibus</expan> po&longs;tridie Non. </s>

ed CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED>

<s id="id.000050">Iunij anno MDLIII. <lb/></s>

<!ELEMENT locator (#PCDATA) >

<s id="id.000051">Genus traxit, quod me &longs;æpè ab eomemini <lb/>audire, à familia Cantagallina, quæ inter <lb/>Peru&longs;inas illu&longs;tris: hocautem cognomen, <lb/>Baldi accepto, vtin varietate temporum fit, <lb/>Abauus reliquit, à teneris vnguiculis <expan abbr="pietat&emacr;">pietatem</expan> erga Deum <lb/>præ&longs;etulit; nam vt mater eius narrabat, &longs;anctorum imagi<lb/>nes & Altariola non cum lætitia &longs;olum, &longs;ed cum venera<lb/>tione anniculus intuebatur. </s>

<!ATTLIST locator

<s id="id.000052">Præceptoribus in adole&longs;cen<lb/>tia v&longs;us fuit laudati&longs;&longs;imis Io. And. Palatio, & Io. Antonio <lb/>Turoneo, qui altero doctior, & Paulo Manutio maxime <lb/>carus ob latinæ & græcæ linguæ peritiam propè &longs;ingula<lb/>rem: ad illorum autem &longs;edulitatem tantum animi ardo<lb/>rem attulit, tantam ingenij ac iudicij vim, vt non tantum <lb/>æqualis &longs;ed omnium vicerit expectationem. </s>

id ID #IMPLIED

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<!ELEMENT p

(s | pb|lb|emph|emph.end|gap )+ >

<!ATTLIST p

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<!ELEMENT pb

<s id="id.000053">Puer adhuc <lb/>Aratiapparitiones Italico carmmered didit. </s>

EMPTY >

<s id="id.000054">Parens hac <lb/>filij laude & gloria motus anno 1573. eum ad maiorem in<lb/>genij cultum cape&longs;&longs;endum Patauium mi&longs;it. </s>

<!ATTLIST pb

<s id="id.000055">Hîcin Ema<lb/>nuelis Margunij familiaritatem &longs;tatim venit, cui porro <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.014.jpg"/>fuit in amotibus. </s>

ed CDATA #IMPLIED

id ID #IMPLIED

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xlink:href CDATA #IMPLIED

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<s id="id.000056">Homeri Iliad. illo Doctore & interpre<lb/>te diligentius quam feci&longs;&longs;et antea, euoluit. </s>

<!ELEMENT place

(#PCDATA) >

<s id="id.000057">priuato autem <lb/>&longs;tudio Anacreonti, Pindaro, Æ&longs;chyli, Euripidi, Sophocli <lb/>operam dedit, &longs;ed præ cæteris Thcocriti Bucolica triuit, <lb/>ad quod &longs;criptionis genus natura magis ferri videbatur: <lb/>centenos græci alicuius poëtæver&longs;us memoriter tenebat, <lb/>&longs;æpeque habebat in ore, in oratoribus græcis ver&longs;andis <lb/>laborem &longs;ealiquem &longs;entire, in poëtis nullum. </s>

<!ATTLIST place

<s id="id.000058">Scrip&longs;it Pa<lb/>tauij libellum de Tormentis Bellicis, & eorum inuentori<lb/>bus, & cum in Tran&longs;alpinorum amicitias incidi&longs;&longs;et, &longs;ibi <lb/>ducebat dedecori ip&longs;os &longs;ua lingua loquentes non intelli<lb/>gere. </s>

id ID #IMPLIED

n CDATA #IMPLIED >

<s id="id.000059">quare incredibili celeritate Gallicam & Germani<lb/>cam didicit. </s>

<s id="id.000060">Pe&longs;tilentia ex co Gymna&longs;io exactus in Pa<lb/>triam redijt, vbi quin quennium integrum Federico <expan abbr="C&omacr;-mandino">Con<lb/>mandino</expan> affixus omnes Mathe&longs;eos partes perdidicit, cui <lb/>viro in delinean dis figuris ad Euclidis, Pappi, & Heronis <lb/>monumenta manum commodauit: ex eiu&longs;dem obitu do<lb/>lorem vix con&longs;olabilem &longs;u&longs;tinuit, &longs;u&longs;ceptoque eius vitam <lb/>&longs;cribendi con&longs;ilio, &longs;ubinde ad omnium Mathematicorum <lb/>vitas con&longs;cribendas animum adplicuit, quod & duode<lb/>cim annorum &longs;patio præ&longs;titit felici&longs;&longs;imè. </s>

<!ELEMENT publisher

<s id="id.000061">cum vero Ma<lb/>thematicarum di&longs;ciplinarum amore torqueretur, ami&longs;&longs;o <lb/>Commandino Præceptore, amicum nactus fuit præ&longs;tan<lb/>ti&longs;&longs;imum & &longs;ymmy&longs;tam Guidum Vbaldum è Marchioni<lb/>bus Montis, in cuius &longs;e con&longs;uetudinem daret: quantum <lb/>profeci&longs;&longs;et, o&longs;tendunt ij commentarij quos anno 1582. in <lb/>Ari&longs;t. Mechanica &longs;crip&longs;it. </s>

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<s id="id.000062">Vt po&longs;tea à grauioribus &longs;tudijs <lb/>ad am&oelig;niora animum abduceret, de re nautica poëma I<lb/>talicè <gap/>onfecit. </s>

<!ATTLIST publisher

<s id="id.000063">quo ab&longs;oluto Paradoxa multa Mathema<lb/>tica explicauit. </s>

id ID #IMPLIED

n CDATA #IMPLIED >

<s id="id.000064">Fama de Baldi virtutibus di&longs;&longs;ipata Ferran<lb/>dus Gonzaga Molfetræ Princeps & Gua&longs;tallæ Dominus <lb/>c&oelig;pit deillo in &longs;uam familiam a&longs;ci&longs;cendo cogitare, vt qui <lb/>ij&longs;dem caperetur artibus, quibus excellere Baldus inci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.015.jpg"/>piebat: Itaque opera Curtij Arditij honorifice fuit in au<lb/>lam euocatus, dum vitam non aulicam viueret totus in <lb/>litteras abditus precibus Ve&longs;pa&longs;iani Gonzagæ Sablonetæ <lb/>Ducis ad explanandos Vitruuij libros adactus fuit. </s>

<!ELEMENT s

<s id="id.000065">quare <lb/><expan abbr="t&umacr;c">tunc</expan> natus de <expan abbr="Verbor&umacr;">Verborum</expan> Vitruuianorum &longs;ignificatione com<lb/>mentarius; in quo minime miran dum &longs;i minuta quæ dam <lb/>pro&longs;equutus fuit, quæ viro magno minus e&longs;&longs;e digna vi<lb/>deantur:illi enim Principi morem ge&longs;&longs;it. </s>

<s id="id.000066">&longs;cio dixi&longs;&longs;e ali<lb/>quando Adrianum Romanum è Polonia reuer&longs;um, vbi <lb/>Vitruuium Palatino cuidam explicauerat, &longs;i commen<lb/>tarium Baldi in Polonia adhibere potui&longs;&longs;em, aurum quod <lb/>mecum attuli emunxi&longs;&longs;em, quia &longs;atis feci&longs;&longs;em muneri la<lb/>borenullo. </s>

<s id="id.000067">Cum Ferrando hero &longs;uo obueni&longs;&longs;et nece&longs;&longs;i<lb/>tas Hi&longs;panias adeundi, illud iter &longs;ine Baldo facere &longs;e po&longs;<lb/>&longs;e non putabat, non tam vt haberet, qui erudito cloquio <lb/>viæ tæ dium leuaret, quam cui po&longs;&longs;et arcana committere, <lb/>atque adeo à quo iuuaretur con&longs;ilio. </s>

<!ATTLIST s

<s id="id.000068">Vix viæ &longs;e dederant <lb/>cum Baldus grauem in morbum delap&longs;us itinere cogitur <lb/>de&longs;i&longs;tere: Mediolanum proinde diuertit, vbi à S. Carolo <lb/>Borromæo & benignè exceptus, & tamdiu detentus do<lb/>nec valetudinem recuperaret. </s>

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT section

( p | pb | figure)+ >

<!ATTLIST section

<s id="id.000069">Gua&longs;tallam po&longs;tea &longs;e re<lb/>cepit, vbi cum ab&longs;ente Domino liberiori otio fruerctur, <lb/>libros &longs;ex de Aula eruditi&longs;&longs;imos methodo analytica con<lb/>&longs;crip&longs;it. </s>

id ID #IMPLIED

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type CDATA #IMPLIED>

<!ELEMENT table

<s id="id.000070">alios non commemoro, quod cum otium erit, o<lb/>mnium &longs;yllabum dabo. </s>

( table.target?, (row, pb?)+ ) >

<!-- order is now:

<s id="id.000071">Anno 1586. ip&longs;o nihil po&longs;tulante <lb/>eligitur Gua&longs;tallæ Abbas, à quo tempore Iuri Can. </s>

i.e. <table> is equiv. to

-->

<!ATTLIST table

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT row

<s id="id.000072">Con<lb/>cilijs, & SS.Patribustotum &longs;e dedit. </s>

( cell+ ) >

<!ATTLIST row

<s id="id.000073">Hebreæ & Chaldææ <lb/>linguarum di&longs;cendarum triennium po&longs;uit. </s>

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT cell

<s id="id.000074">Anno 1593. no<lb/>uæ Gnomonices libros quinque compo&longs;uit. </s>

<!ATTLIST cell

<s id="id.000075">in&longs;equenti <lb/>Chaldæam Onkeli paraphra&longs;in in Pentateuchum vertit <lb/>& commentarios adiunxit; quo exantlato labore in Iob <lb/>ex Heb. fonte paraphra&longs;in texuit, quam & &longs;cholijs illu<lb/>&longs;trauit. </s>

id ID #IMPLIED

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(pb?, front, pb?, body, pb?, back) >

<s id="id.000076">Tabulam Etru&longs;cam Eugubinam interptetatus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.016.jpg"/>fuit:in ca autem diuinatione, vt aiebat, &longs;ubci&longs;iuas vnius <lb/>men&longs;is horas con&longs;ump&longs;it. </s>

<!ATTLIST text

<s id="id.000077">De Firmamento & aquis egre<lb/>gie &longs;crip&longs;it. </s>

type CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED>

<s id="id.000078">Oeconomiam Tropologicam in S.Matthæum <lb/>Card. Baronius, qui non alia Baldi vidit, vehementer pro<lb/>babat. </s>

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<s id="id.000079">Romæ dum viueret, fere ne&longs;ciuit quid gereretur <lb/>in Aulis: Arabicæ enim linguæ cum Io. Bapti&longs;ta Raimon<lb/>do diligenti&longs;&longs;ime &longs;tuduit, & arcana indu&longs;tria Slauonicæ, <lb/>quam perfecte callebat. </s>

(#PCDATA)* >

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<!ENTITY shy "[-]">

<s id="id.000080">Ex Arabico vertit Hortum Geo<lb/>graphicum Anonymi, quem ante &longs;excentos annos flo<lb/>rui&longs;&longs;e arbitrabatur. </s>

<s id="id.000081">Hunc vero extru&longs;i&longs;&longs;et, vtalios Baldi <lb/>libros, Marcus Vel&longs;erus IIvir Aug. &longs;i eo paulo longior <lb/>huius lucis v&longs;ura contigi&longs;&longs;et. </s>

<!ENTITY AElig "[AElig]">

<s id="id.000082">Compo&longs;uit & Dictionarium <lb/>Arabicum. </s>

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<s id="id.000083">atque cum beati&longs;&longs;imam illam vbertatem in<lb/>genij a&longs;&longs;idue diffundi nece&longs;&longs;e e&longs;&longs;et, anno 1603. orbem vni<lb/>uer&longs;um de&longs;eribere aggre&longs;&longs;us fuit; atque ita quidem, vt <lb/>tam quæ ad Hi&longs;toriam, quam quæ ad Geographiam per<lb/>tinerent complecteretur: Neque illu&longs;trare &longs;olum voluit <lb/>quæ nouerunt antiqui, quemadmodum vi&longs;um Ortelio, <lb/>&longs;ed vel oppidula omnia & pagos, de quibus aliquain po<lb/>&longs;tremis &longs;criptoribus mentio. </s>

<s id="id.000084">& profecto totum opus ad <lb/>vmbilicum perduxit: non dige&longs;&longs;it tamen vniuer&longs;um. </s>

<info>

<author>Baldi, Bernadino</author>

<title>In Mechanica Aristotelis problemata exercitationes</title>

<place>Mainz</place>

<publisher>Johannis Albinus</publisher>

</info>

<text>

<front>

</front>

<s id="id.000085">qua<lb/>tuor aut ni fallor quinque tantum Tomi fuerunt ordine <lb/>Alphabetico di&longs;po&longs;iti:&longs;upere&longs;&longs;ent &longs;eptem aut octo di&longs;po<lb/>nendi, quantum ex chartarum & fa&longs;ciculorum mo<gap/>e con<lb/>ijcere licet. </s>

<body>

<chap>

<pb/>

<s id="id.000086">Anno 1617. quarto Idus Octob. </s>

<s>BERNARDINI</s>

<s id="id.000087">po&longs;teaquam <lb/>dies 40. vehementi de&longs;tillatione vexatus fui&longs;&longs;et, &longs;piritum <lb/>Deo reddidit Sacramentis Eccle&longs;iæ omnibus rite muni<lb/>tus. </s>

<s>BALDI VRBINATIS <lb/>GVASTALLÆ AB<lb/>BATIS <lb/><emph type="italics"/>IN<emph.end type="italics"/></s>

<s id="id.000088">Statura procerus fuit, facie oblonga & acribus oculis, <lb/>colore &longs;ubfu&longs;co. </s>

<s>MECHANICA ARISTOTE <lb/>LIS PROBLEMATA <lb/>EXER CITATIONES:</s>

<s id="id.000089">Membrorum ei fuit decens habitudo, & <lb/>compactum corpus. </s>

<s><emph type="italics"/>ADIECTA SVCCINCTA NAR<lb/>ratione de autoris vita & &longs;criptis.<emph.end type="italics"/></s>

<s id="id.000090">Diebus fe&longs;tis omnlbus &longs;acrum facie<lb/>bat, ieiunabat bis in hebdomada, eleemo&longs;yni&longs;que paupe<lb/>res &longs;ubleuabat. </s>

<s><emph type="italics"/><gap/>NTIAE.<emph.end type="italics"/></s>

<s id="id.000091">In&longs;tudijs &longs;ica&longs;&longs;iduus fuit, vt &longs;æpe & legeret <lb/>& comederet. </s>

<s>Typis & Sumptibus Viduæ Ioannis Albini.</s>

<s id="id.000092">S.Augu&longs;tini libros de Ciuitate Dei ter in-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.017.jpg"/>ter prandium euoluit. </s>

<s id="id.000093">Statim à noctis meridie dum ei vi<lb/>res firmiores e&longs;&longs;ent ad lucubrandum &longs;urgebat. </s>

<pb/>

<s><emph type="italics"/>NOBILISSIMO AC GENE<lb/>ROSO DOMINO<emph.end type="italics"/></s>

<s id="id.000094">à prandio <lb/>Euclidem Arabice editum, vel libellum aliquem germa<lb/>nicum aut gallicum in manus &longs;umebat. </s>

<s>D. ADAMO PHILIP<lb/>PO BARONI A CRON<lb/>BERG, EQVITI, SACRÆ CÆSA<lb/>REÆ MAIESTATIS, ET SERENISSIMI</s>

<s id="id.000095">Suauitate morum <lb/>& mode&longs;tia, etiam &longs;i ceteræ dotes abfui&longs;&longs;ent, quemlibet <lb/>ad amorem &longs;ui allicere potui&longs;&longs;et. </s>

<s>Principis Archiducis Alberti Camerario intimo & c. <lb/>Domino meo gratio&longs;i&longs;&longs;imo.</s>

<s id="id.000096">Sermo modicus ei fuit, <lb/>itemque cultus. </s>

<s>Opportune &longs;ub hoc ip&longs;um tem<lb/>pus, quo in Belgium ad Scre<lb/>ni&longs;&longs;imos Principes iter ador<lb/>nat. Nobili&longs;&longs;ima & Genero&longs;a <lb/>Dom. V.^{ra}, prodit no&longs;tris for<lb/>mis in publicum editus Com<lb/>mentarius Bernardini Baldi Vrbinatis Gua<lb/>&longs;tallæ Abbatis in Ari&longs;totelis Mechanica. Is <lb/>virin omni &longs;cientiæ genere, at maxime inMa<lb/>thematicis di&longs;ciplinis fuit ver&longs;ati&longs;&longs;imus, quod <lb/>multa ab eo præclare &longs;crip ta te&longs;tantur opera, <lb/>ex quibus paucula edita, reliqua vero &longs;pera

<s id="id.000097">Nullos vnquam honores petijt, qui à <lb/>Clem. </s>

<pb/>mus &longs;uo temporein publicam luce<gap/> produ<lb/>cenda. Cum vero nemini&longs;it ob&longs;curum Nobi<lb/>li&longs;&longs;imæ ac Genero&longs;æ Dom. V.^{ræ} id &longs;emper <lb/>extiti&longs;&longs;e familiari&longs;&longs;imum, vttum dome&longs;ticum <lb/>otium, tum maxime peregtinationes, quibus <lb/>totam pæne Europam &longs;umma cum laude <lb/>circum&longs;crip&longs;it, tum variarum linguarum per<lb/>fecto v&longs;u, tum Mathematicarum di&longs;ciplina<lb/>rum notitia & exercitio redderet <expan abbr="iuc&utilde;diores">iucundiores</expan>, <lb/>nulla me tenet dubitatio quin & Baldum Vr<lb/>binatem no&longs;tris typis loquentem in hoc iti<lb/>nere, quod à Deo felici&longs;&longs;imum Nobili&longs;&longs;imæ <lb/>acGenero&longs;æ Dom. V.^{ræ} precor, in &longs;uum comi<lb/>tatum ac tutelam beneuolo animo &longs;it admi&longs;<lb/>&longs;ura. Id rogo humillime &longs;imulque precor, vt. <lb/>hanc meam typographiam plurimis iam re<lb/>tro annis de inclytæ familiæ Cronbergicæ tu<lb/>tela gloriantem, &longs;uo fauore pro&longs;equatur, vi<lb/>duæque afflictæ fortunis beneuole ad&longs;piret. <lb/>Sic Deus Nobili&longs;&longs;. & Genero&longs;am Dom. V.^{ram} <lb/>illu&longs;tret omnibus bonis, eamque R.^{mo} & Ill.^{mo} <lb/>Principiac Domino meo Clementi&longs;&longs;imo, D. <lb/>Ioanni Suicardo Archiepi&longs;copo Mogunti<lb/>no Principi Electori ac per <emph type="italics"/>G<emph.end type="italics"/>ermaniam Ar-

<pb/>chicancellario &c. patruo &longs;uo optati&longs;&longs;imo <lb/>&longs;aluo florentique redhibeat &longs;aluum &longs;imili<lb/>ter florentem ac incolumem. Moguntiæ è <lb/>typographeio Viduæ Albinianæ, honori No<lb/>bili&longs;&longs;imæ ac <emph type="italics"/>G<emph.end type="italics"/>enero&longs;æ Dom. Ve&longs;træ perpe<lb/>tuum dicato. Anno 1621.26.Martij. </s>

<pb/>

<s>PRAEFATIO.</s>

<s id="id.000098">8. ampli&longs;&longs;imi promi&longs;&longs;i fuerant; nullum emolumen<lb/>tum quæ&longs;iuit &longs;uo ceniu contentus. </s>

<s><emph type="italics"/>Diligenter legenti mihi quæ&longs;tiones il<lb/>las, in quibus ea quæ ad Mecha<lb/>nicam facultatem pertinent, expli<lb/>cantur, multa in mentem venie<lb/>bant; & primum quidem eorum, quæibi di<foreign lang="greek">c</foreign>pu<lb/>tantur, vtilitatem, &longs;ubtilitatem, copiam admi<lb/>rabar: Tum ex animo dolebam, aureum hunc li<lb/>bellum propènegligi, & ab iis qui pulcherrimis <lb/>hi&longs;ce &longs;tudiis dant operam, assiduè præ manibus <lb/>non haberi: Multas autem Auctoriip&longs;ihaben<lb/>das referendasque e&longs;&longs;e gratias, qui tam egregiam, <lb/>vtilem & probèin&longs;tructam &longs;upellectilem Archi<lb/>tectis, Mechanicis, & omnibus ferè Artificibus <lb/>&longs;uppeditauerit. Ari&longs;toetlis nomini a&longs;cribitur <lb/>Commentarius, licet nonnulli, &longs;itne Philo&longs;ophi <lb/>illius præclarissimi & acutissimilabor, an non, <lb/>adfirmare &longs;ubdubitauerint. Ari&longs;totelis tamen <lb/>e&longs;&longs;e omnes ferè meliores con&longs;entiunt: <gap/>dquetum <lb/>ex phra&longs;i, & explicatione, quæ Ari&longs;totelem &longs;a<lb/>piunt, tum iudicio &longs;ubtilitatis & rationum, qui-

<s id="id.000099">facile parcendum e&longs;&longs;e <lb/>dicebat, ijs maxime qui in re leui impegi&longs;&longs;ent, quoniam &longs;i <lb/>quos cen&longs;emus optimos, nudos con&longs;piceremus, nullum <lb/>eorum non iudicaremus multis dignum verberibus. </s>

<pb/>bus quæ&longs;tiones ip&longs;æ ingenio&longs;issimè diluuntur. Vi<lb/>detur autem mihi, rem accuratius exploranti, &longs;a<lb/>tis veri&longs;imile (nullum enim habeo opinionis hu<lb/>ius a&longs;&longs;ertorem,) &longs;ectionem e&longs;&longs;e hanc, & partem <lb/>quandam eius operis nobilissimi, quod idem au<lb/>ctor De Problematibus edidit, & hanc, ne&longs;cio <lb/>quam ob cau&longs;am; ni&longs;i fortè quod tractatio merè <lb/>Phy&longs;icanon&longs;it, àreliquo corpore di&longs;tractam at<lb/>que reuul&longs;am. Jd certè quod ad rem facit, probè <lb/>nouimus, Diogenem Laërtium inter cætera Ari<lb/>&longs;totelici ingenij monumenta Mechanica quoque <lb/>adnumera&longs;&longs;e. Quibus con&longs;ideratis magnopere <lb/>&longs;ubit mirari, cur ij qui po&longs;t Ari&longs;totelem floruêre <lb/>atque vixere, Mechanici, Archimedes, Athenæus, <lb/>Heron, Pappus, & cæteri, nullam huius libelli fe<lb/>cerint commemorationem: & &longs;anè debuerunt; <lb/>neque enim à vero est dissimile, ip&longs;os per hunc ali<lb/>quatenus profeci&longs;&longs;e. Verum enimuero cum inge<lb/>nuiilli fuerint homines, & nullatenus obtrecta<lb/>tores, credendum potius est, Comment ariolum i<lb/>&longs;tud, eorum æuo, paucis cognitum, alicubi in Bi<lb/>bliothecis latui&longs;&longs;e: etenim cætera quoque Ari&longs;tote<lb/>lis &longs;cripta, po&longs;t vetu&longs;tailla tempora, ante Ale<lb/>xandrum Aphrodi&longs;ien&longs;em, àmultis fui&longs;&longs;e igno-

<pb/>rata non dubitamus. Habemus &longs;iquidem, Stra<lb/>bone te&longs;te, lib. 13. Ari&longs;totelis, & Theophra&longs;ti bi<lb/>bliothecam, po&longs;t ip&longs;ius Theophra&longs;ti dece&longs;&longs;um, ad <lb/>Neleum quendam Scep&longs;ium, Cori&longs;cifilium, qui <lb/>eius fuerat auditor, perueni&longs;&longs;e; po&longs;t hæc libros, <lb/>blattis olim, & humore corruptos, Apelliconi Te<lb/>io venditos, & ab eo Athenas translatos, tum <lb/>Athenis captis in Syllæ pote&longs;tatem deacni&longs;&longs;e, eo&longs;<lb/>que tandem à Sylla acceptos, Tyrannionem <lb/>Grammaticum, vt potuit meliùs emendatos, <lb/>promulga&longs;&longs;e. Exquibus colligimus, mirum non <lb/>e&longs;&longs;e, Archimedi, Heroni, & alijs qui ante Syllam <lb/>vixêre, fui&longs;&longs;e incognitos. quicquid&longs;it, illud cer<lb/>tumest, Ari&longs;totelem eorum omnium quide Me<lb/>chanicis commentaria edidere, e&longs;&longs;e longè vetu<lb/>&longs;tissimum. Pappus enim Heroneiunior, Athe<lb/>næus Archimediæqualis, vterque enim &longs;ub Mar<lb/>cello, cui Athenæus &longs;uum de bellicis Machinis <lb/><expan abbr="libell&umacr;">libellum</expan> dedicauit. Archimedes verò circa CXL. <lb/>Olympiadem floruit, quamobrem po&longs;t Ari&longs;tote<lb/>lem Olympiadas XL. hoc est, annos ferè CLX. <lb/>I&longs;thæc autem con&longs;iderantibus, facile e&longs;t cogno&longs;ce<lb/>refacultatis huius nobilitatem, atque dignit atem; <lb/>quippe quod &longs;ummus Philo&longs;ophus non modo eam

<pb/>probauerit, &longs;cd etiam &longs;uis acutissimis lucubra<lb/>tionibus illu&longs;trauerit. Hanc porro tractationem <lb/>&longs;ubiecto quidem Phy&longs;icam e&longs;&longs;e, demon&longs;tratio<lb/>nibus verò Geometricam, ip&longs;emet nos docuit <lb/>Ari&longs;toteles, cuius etiam naturæ &longs;unt Per&longs;pecti<lb/>ua, Specularia, Mu&longs;ica, & cæteræ eiu&longs;dem <lb/>modifacultates, quas quidem &longs;ubalternas Peri<lb/>patetici appellant. Vitruuius Architecturæ <lb/>membrum, vt ita dicam, & portionem quan<lb/>dam facit, ait enim Architecturæ partes e&longs;&longs;e tres, <lb/>Ædificationem, Gnomonicam, Machinatio<lb/>nem. Estautem Architectur â quideminferior, <lb/>paret enim Architecto Mechanicus; attamen &longs;i <lb/>cæteras artes &longs;pectes, Architectonica; hæc enim <lb/>omnesferè &longs;edentariæ, &longs;ellulariæue, quas banau<lb/>&longs;as Græci appellant, ordine &longs;ubijciuntur, & &longs;a<lb/>nè latissimos i&longs;ihæc habetfines; præcipuè autem <lb/>circa eam ver&longs;atur cognitionem, eamque inter <lb/>cæterasferè principem, quam dixere Centrobari<lb/>cam, quæ quidem ad Centri grauitatem, eiu&longs;que <lb/>&longs;peculationem pertinet: quà in &longs;pecie inter vete<lb/>res primum &longs;ibi vindicauit locum Archimedes, <lb/>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"/>

<pb/><emph type="italics"/>tem Commandinus, qui librum de Centro gra<lb/>uitatis &longs;olidorum &longs;crip&longs;it, & po&longs;t eum G. Vbal<lb/>dus è Marchion. Montis, qui non modò ab<lb/>&longs;olutissimum Mechanicorum librum cum maxi<lb/>maingenij &longs;ui laude con&longs;crip&longs;it, &longs;ed & Paraphra<lb/>&longs;in in librum Æqueponderantium Archimedis <lb/>egregiè concinnauit Centrobaricam hanc, igno<lb/>tam fui&longs;&longs;e Ari&longs;toteli, &longs;ætis patet. nunquam enim <lb/>in Mechanicis demon&longs;trationibus, quod tamen <lb/>est potissimum, grauitatis centrum nominat, e<lb/>iu&longs;ue naturam atque vim &longs;peculatur. Diuidi<lb/>tur autem Mechanice tota, te&longs;te Herone apud <lb/>Pappum libro octauo, in Rationalem, hoc est, <lb/>Theoricam & Chirurgicam, id est, manu ope<lb/>ratricem, quam Praxim aptè dicere valemus. <lb/>Rationalis, &longs;peculationi & <expan abbr="dem&omacr;&longs;trationibus">demon&longs;trationibus</expan>, ex <lb/>Geometricis, Arithmeticis & Phy&longs;icis rationi<lb/>bus, dat operam; Chirurgica vero materiam <lb/>tractat, & &longs;e&longs;ein varias artes diffundit, Æræ<lb/>riam, Lignariam, Sculptoriam, Pictoriam, Æ<lb/>dificatoriam, Machinariam & Thaumaturgi<lb/>cam, cæterasque eiu&longs;modi. Machinatoriæ au<lb/>tem &longs;unt partes Manganaria, qua ingentiæ<emph.end type="italics"/>

<pb/><emph type="italics"/>transferuntur pondera, tum ip&longs;a Poliorcetica, <lb/>quæ bellicas Machinas ad vrbium expugnatio<lb/>nes, quod velip&longs;o nomine profitetur, ædificat. At<lb/>qui hac dere plura &longs;cribere &longs;uper&longs;edemus, ne a<lb/>ctum agamus: quis quis enim minutè magis hæc <lb/>cogno&longs;cere de&longs;iderat, is Pappum adeat libro cita<lb/>to, & Guidum Vbaldumin Præfatione quam <lb/>&longs;uo Mechanicorum Operi præpo&longs;uit. Vt autem <lb/>ad Ari&longs;totelis, de quo egimus, libellum reuerta<lb/>mur, pauci &longs;unt qui ei ante nos &longs;tilum & operam <lb/>commodauerint: Leonicenus Latinum fecit & <lb/>figuris tum breuissimis, & parui &longs;ane ponderis, <lb/>marginalibus adnotatiunculis, in&longs;truxit. Po&longs;t <lb/>hunc Alexander Picolomineus luculentissima <lb/>Paræphra&longs;i illu&longs;trauit. Modo, vt audio, Simon <lb/>Sticinus Hollanden&longs;is quædam edidit, quæ ad <lb/>nos minime peruenêre. Nos demum, omnium, <lb/>tum &longs;cientia, & ingenio, tum ætate, po&longs;tremi huic <lb/>operi manum admouimus; Con&longs;iderantes enim <lb/>Ari&longs;totelem æly<gap/>s principijs v&longs;um, ac probatissi<lb/>mipo&longs;t eum fecerint Mechanici, demon&longs;tra&longs;&longs;e, <lb/>morem huiu&longs;ce facultatis &longs;tudio&longs;is ge&longs;turos nos <lb/>fore arbitrati&longs;umus, &longs;i ea&longs;dem illas quæ&longs;tiones<emph.end type="italics"/>

<pb/><emph type="italics"/>Mechanicis, hocest, Archimedeis probationi<lb/>bus confirmaremus; dum per latissimos faculta<lb/>tis huius campos vægantes, alias quoque i&longs;tis af<lb/>fines dubitationes introducentes &longs;olueremus. <lb/><expan abbr="quicquidaut&emacr;&longs;ecerimus">quicquidauten&longs;ecerimus</expan> profecerimu&longs;ue, Lector <lb/>optime, boni con&longs;ule, & quia fax per manus tra<lb/>ditur, tu interim de me accipe, vt alijs tradas.<emph.end type="italics"/></s>

<pb/>

<s>DE VITA ET SCRI<lb/>PTIS BERNARDINI <lb/>BALDI VRBINATIS</s>

<s id="id.000100">Bi<lb/>bliothecam habuit non locupletem, &longs;ed &longs;electis <expan abbr="in&longs;tructã">in&longs;tructam</expan> <lb/>codicibus. </s>

<s><emph type="italics"/>EX LITERIS FABRITII SCHAR<lb/>loncini ad Illu&longs;trissimum & Reuerendissimum <lb/>Dominum Lælium Ruinum Epi&longs;copum Bal<lb/>neoregien&longs;em ex-Nuntium Apo&longs;tolicum <lb/>ad Poloniæ Regem & c.<emph.end type="italics"/></s>

<s id="id.000101">Verum ire per &longs;ingula longum e&longs;&longs;et. </s>

<s>Natus e&longs;t Bern. Baldus Vrbini nobilibus <expan abbr="pa-r&emacr;tibus">pa<lb/>rentibus</expan> po&longs;tridie Non. Iunij anno MDLIII. <lb/>Genus traxit, quod me &longs;æpè ab eomemini <lb/>audire, à familia Cantagallina, quæ inter <lb/>Peru&longs;inas illu&longs;tris: hocautem cognomen, <lb/>Baldiaccepto, vtin varietate temporum fit, <lb/>Abauus reliquit, à teneris vnguiculis <expan abbr="pietat&emacr;">pietatem</expan> erga Deum <lb/>præ&longs;etulit; nam vt mater eius narrabat, &longs;anctorum imagi<lb/>nes & Altariola non cum lætitia &longs;olum, &longs;ed cum venera<lb/>tione anniculus intuebatur. Præceptoribus in adole&longs;cen<lb/>tia v&longs;us fuit laudati&longs;&longs;imis Io. And. Palatio, & Io. Antonio <lb/>Turoneo, qui altero doctior, & Paulo Manutio maxime <lb/>carus ob latinæ & græcæ linguæ peritiam propè &longs;ingula<lb/>rem: adillorum autem &longs;edulitatem tantum animi ardo<lb/>rem attulit, tantam ingenij aciudicij vim, vt non tantum <lb/>æqualis &longs;ed omnium vicerit expectationem. Puer adhuc <lb/>Aratiapparitiones Italico carmmered didit. Parens hac <lb/>filij laude & gloria motus anno 1573. eum ad maiorem in<lb/>genij cultum cape&longs;&longs;endum Patauium mi&longs;it. Hîcin Ema<lb/>nuelis Margunij familiaritatem &longs;tatim venit, cui porro

<s id="id.000102">Satis mihi <lb/>de incomparabili Baldi doctrina, & &longs;umma innocentia, ô <lb/>rarum connubium, pauca dixi&longs;&longs;e, quæ for&longs;itan ad imitan<lb/>dum nimis multa. </s></chap>

<pb/>fuit in amotibus. Homeri Iliad. illo Doctore & interpre<lb/>te diligentius quam feci&longs;&longs;et antea, euoluit. priuato autem <lb/>&longs;tudio Anacreonti, Pindaro, Æ&longs;chyli, Euripidi, Sophocli <lb/>operam dedit, &longs;ed præ cæteris Thcocriti Bucolica triuit, <lb/>ad quod &longs;criptionis genus natura magis ferri videbatur: <lb/>centenos græci alicuius poëtæver&longs;us memoriter tenebat, <lb/>&longs;æpeque habebat in ore, in oratoribus græcis ver&longs;andis <lb/>laborem &longs;ealiquem &longs;entire, in poëtis nullum. Scrip&longs;it Pa<lb/>tauij libellum de Tormentis Bellicis, & eoruminuentori<lb/>bus, & cum in Tran&longs;alpinorum amicitias incidi&longs;&longs;et, &longs;ibi <lb/>ducebat dedecori ip&longs;os &longs;ua lingua loquentes non intelli<lb/>gere. quare incredibili celeritate Gallicam & Germani<lb/>cam didicit. Pe&longs;tilentia ex co Gymna&longs;io exactus in Pa<lb/>triam redijt, vbi quin quennium integrum Federico <expan abbr="C&omacr;-mandino">Con<lb/>mandino</expan> affixus omnes Mathe&longs;eos partes perdidicit, cui <lb/>viro in delinean dis figuris ad Euclidis, Pappi, & Heronis <lb/>monumenta manum commodauit: ex eiu&longs;dem obitu do<lb/>lorem vix con&longs;olabilem &longs;u&longs;tinuit, &longs;u&longs;ceptoque eius vitam <lb/>&longs;cribendi con&longs;ilio, &longs;ubinde ad omnium Mathematicorum <lb/>vitas con&longs;cribendas animum adplicuit, quod & duode<lb/>cim annorum &longs;patio præ&longs;titit felici&longs;&longs;imè. cum vero Ma<lb/>thematicarum di&longs;ciplinarum amore torqueretur, ami&longs;&longs;o <lb/>Commandino Præceptore, amicum nactus fuit præ&longs;tan<lb/>ti&longs;&longs;imum & &longs;ymmy&longs;tam Guidum Vbaldum è Marchioni<lb/>bus Montis, in cuius &longs;e con&longs;uetudinem daret: quantum <lb/>profeci&longs;&longs;et, o&longs;tendunt ij commentarij quos anno 1582. in <lb/>Ari&longs;t. Mechanica &longs;crip&longs;it. Vt po&longs;tea à grauioribus &longs;tudijs <lb/>ad am&oelig;niora animum abduceret, de re nautica poëma I<lb/>talicè <gap/>onfecit. quo ab&longs;oluto Paradoxa multa Mathema<lb/>tica explicauit. Fama de Baldi virtutibus di&longs;&longs;ipata Ferran<lb/>dus Gonzaga Molfetræ Princeps & Gua&longs;tallæ Dominus <lb/>c&oelig;pit deillo in &longs;uam familiam a&longs;ci&longs;cendo cogitare, vt qui <lb/>ij&longs;dem caperetur artibus, quibus excellere Baldus inci-

<pb/>piebat: Itaque opera Curtij Arditij honorifice fuit in au<lb/>lam euocatus, dum vitam non aulicam viueret totus in <lb/>litteras abditus precibus Ve&longs;pa&longs;iani Gonzagæ Sablonetæ <lb/>Ducis ad explanandos Vitruuij libros adactus fuit. quare <lb/><expan abbr="t&umacr;c">tunc</expan> natus de <expan abbr="Verbor&umacr;">Verborum</expan> Vitruuianorum &longs;ignificatione com<lb/>mentarius; in quo minime miran dum &longs;i minuta quæ dam <lb/>pro&longs;equutus fuit, quæ viro magno minus e&longs;&longs;e digna vi<lb/>deantur:illi enim Principi morem ge&longs;&longs;it. &longs;cio dixi&longs;&longs;e ali<lb/>quando Adrianum Romanum è Polonia reuer&longs;um, vbi <lb/>Vitruuium Palatino cuidam explicauerat, &longs;i commen<lb/>tarium Baldi in Polonia adhibere potui&longs;&longs;em, aurum quod <lb/>mecum attuli emunxi&longs;&longs;em, quia &longs;atis feci&longs;&longs;em muneri la<lb/>borenullo. Cum Ferrando hero &longs;uo obueni&longs;&longs;et nece&longs;&longs;i<lb/>tas Hi&longs;panias adeundi, illud iter &longs;ine Baldo facere &longs;e po&longs;<lb/>&longs;e non putabat, non tam vt haberet, qui erudito cloquio <lb/>viæ tæ dium leuaret, quam cui po&longs;&longs;et arcana committere, <lb/>atque adeo à quo iuuaretur con&longs;ilio. Vix viæ &longs;e dederant <lb/>cum Baldus grauem in morbum delap&longs;us itinere cogitur <lb/>de&longs;i&longs;tere: Mediolanum proinde diuertit, vbi à S. Carolo <lb/>Borromæo & benignè exceptus, & tamdiu detentus do<lb/>necvaletudinem recuperaret. Gua&longs;tallam po&longs;tea &longs;e re<lb/>cepit, vbi cum ab&longs;ente Domino liberiori otio fruerctur, <lb/>libros &longs;ex de Aula eruditi&longs;&longs;imos methodo analytica con<lb/>&longs;crip&longs;it. alios non commemoro, quod cum otium erit, o<lb/>mnium &longs;yllabum dabo. Anno 1586. ip&longs;o nihil po&longs;tulante <lb/>eligitur Gua&longs;tallæ Abbas, à quo tempore Iuri Can. Con<lb/>cilijs, & SS.Patribustotum &longs;e dedit. Hebreæ & Chaldææ <lb/>linguarum di&longs;cendarum triennium po&longs;uit. Anno 1593. no<lb/>uæ Gnomonices libros quinque compo&longs;uit. in&longs;equenti <lb/>Chaldæam Onkeli paraphra&longs;in in Pentateuchum vertit <lb/>& commentarios adiunxit; quo exantlato labore in Iob <lb/>ex Heb. fonte paraphra&longs;in texuit, quam & &longs;cholijs illu<lb/>&longs;trauit. Tabulam Etru&longs;cam Eugubinam interptetatus

<pb/>fuit:in ca autem diuinatione, vt aiebat, &longs;ubci&longs;iuas vnius <lb/>men&longs;is horas con&longs;ump&longs;it. De Firmamento & aquis egre<lb/>gic &longs;crip&longs;it. Oeconomiam Tropologicam in S.Matthæum <lb/>Card. Baronius, qui non alia Baldi vidit, vehementer pro<lb/>babat. Romæ dum viueret, fere ne&longs;ciuit quid gereretur <lb/>in Aulis: Arabicæ enim linguæ cum Io. Bapti&longs;ta Raimon<lb/>do diligenti&longs;&longs;ime &longs;tuduit, & arcana indu&longs;tria Slauonicæ, <lb/>quam perfecte callebat. Ex Arabico vertit Hortum Geo<lb/>graphicum Anonymi, quem ante &longs;excentos annos flo<lb/>rui&longs;&longs;e arbitrabatur. Hunc vero extru&longs;i&longs;&longs;et, vtalios Baldi <lb/>libros, Marcus Vel&longs;erus IIvir Aug. &longs;i eo paulo longior <lb/>huius lucis v&longs;ura contigi&longs;&longs;et. Compo&longs;uit & Dictionarium <lb/>Arabicum. atque cum beati&longs;&longs;imam illam vbertatem in<lb/>genij a&longs;&longs;idue diffundi nece&longs;&longs;e e&longs;&longs;et, anno 1603. orbem vni<lb/>uer&longs;um de&longs;eribere aggre&longs;&longs;us fuit; atque ita quidem, vt <lb/>tam quæ ad Hi&longs;toriam, quam quæ ad Geographiam per<lb/>tinerent complecteretur: Neque illu&longs;trare &longs;olum voluit <lb/>quæ nouerunt antiqui, quemadmodum vi&longs;um Ortelio, <lb/>&longs;ed vel oppidula omnia & pagos, de quibus aliquain po<lb/>&longs;tremis &longs;criptoribus mentio. & profecto totum opus ad <lb/>vmbilicum perduxit: non dige&longs;&longs;it tamen vniuer&longs;um. qua<lb/>tuor aut ni fallor quinque tantum Tomi fuerunt ordine <lb/>Alphabetico di&longs;po&longs;iti:&longs;upere&longs;&longs;ent &longs;eptem aut octo di&longs;po<lb/>nendi, quantum ex chartarum & fa&longs;ciculorum mo<gap/>e con<lb/>ijcere licet. Anno 1617. quarto Idus Octob. po&longs;teaquam <lb/>dies 40. vehementi de&longs;tillatione vexatus fui&longs;&longs;et, &longs;piritum <lb/>Deo reddidit Sacramentis Eccle&longs;iæ omnibus rite muni<lb/>tus. Statura procerus fuit, facie oblonga & acribus oculis, <lb/>colore &longs;ubfu&longs;co. Membrorum ei fuit decens habitudo, & <lb/>compactum corpus. Diebus fe&longs;tis omnlbus &longs;acrum facie<lb/>bat, ieiunabat bis in hebdomada, eleemo&longs;yni&longs;que paupe<lb/>res &longs;ubleuabat. In&longs;tudijs &longs;ica&longs;&longs;iduus fuit, vt &longs;æpe & legeret <lb/>& comederet. S.Augu&longs;tinilibros de Ciuitate Dei ter in-

<pb/>ter prandium euoluit. Statim à noctis meridie dum ei vi<lb/>res firmiores e&longs;&longs;ent ad lucubrandum &longs;urgebat. à prandio <lb/>Euclidem Arabice editum, vel libellum aliquem germa<lb/>nicum aut gallicum in manus &longs;umebat. Suauitate morum <lb/>& mode&longs;tia, etiam &longs;i ceteræ dotes abfui&longs;&longs;ent, quemlibet <lb/>ad amorem &longs;ui allicere potui&longs;&longs;et. Sermo modicus ei fuit, <lb/>itemque cultus. Nullos vnquam honores petijt, qui à <lb/>Clem. 8. ampli&longs;&longs;imi promi&longs;&longs;i fuerant; nullum emolumen<lb/>tum quæ&longs;iuit &longs;uo ceniu contentus. facile parcendum e&longs;&longs;e <lb/>dicebat, ijs maxime qui in re leui impegi&longs;&longs;ent, quoniam &longs;i <lb/>quos cen&longs;emus optimos, nudos con&longs;piceremus, nullum <lb/>eorum non iudicaremus multis dignum verberibus. Bi<lb/>bliothecam habuit non locupletem, &longs;ed &longs;electis <expan abbr="in&longs;tructã">in&longs;tructam</expan> <lb/>codicibus. Verum ire per &longs;ingula longum e&longs;&longs;et. Satis mihi <lb/>de incomparabili Baldi doctrina, & &longs;umma innocentia, ô <lb/>rarum connubium, pauca dixi&longs;&longs;e, quæ for&longs;itan ad imitan<lb/>dum nimis multa. </s>

<s>SYLLABVS LIBRORVM</s>

<chap>

<s>omnium B.Abb.Baldi.</s>

<s id="id.000103">SYLLABVS LIBRORVM</s>

<s>Arati apparitiones è gr.in Ital. vertit. </s>

<s id="id.000104">omnium B.Abb.Baldi.</s>

<s>De Tormentis Bellicis & eorum Inuentoribus lib. </s>

<s id="id.000105">Arati apparitiones è gr.in Ital. vertit. </s>

<s>Heronis automata vertit. </s>

<s>Vitas omnium Mathematicorum &longs;crip&longs;it, & trib. in Tom. <lb/>2.1.P^{s}.à Thalete ad Chri&longs;tum.2.àChri&longs;toad &longs;ua tem<lb/>pora. </s>

<s id="id.000106">De Tormentis Bellicis & eorum Inuentoribus lib. Heronis automata vertit. </s>

<s>Earumdem vitarum Epitomen Chronologicum confecit. </s>

<s id="id.000107">Vitas omnium Mathematicorum &longs;crip&longs;it, & trib. </s>

<s>In Ari&longs;tot. Mechan. Commentar. </s>

<s id="id.000108">in Tom. <lb/>2.1.P^{s}.à Thalete ad Chri&longs;tum.2.à Chri&longs;to ad &longs;ua tem<lb/>pora. </s>

<s>De Renautica Poëmation. </s>

<s>Paradoxorum Mathematicorum liber. </s>

<s id="id.000109">Earumdem vitarum Epitomen Chronologicum confecit. </s>

<s>De&longs;criptio Palatij Ducum Vrbinarum quod e&longs;t Vrbini. </s>

<s id="id.000110">In Ari&longs;tot. Mechan. Commentar. </s>

<s>Poema cui titulus, Lamus. </s>

<s id="id.000111">De Renautica Poëmation. </s>

<pb/>

<s>Carmina pia, quæ in&longs;cribuntur, Anni Corona. </s>

<s id="id.000112">Paradoxorum Mathematicorum liber. </s>

<s>De Verborum Vitruuianorum &longs;ignificatione. </s>

<s id="id.000113">De&longs;criptio Palatij Ducum Vrbinarum quod e&longs;t Vrbini. </s>

<s>Carmina varia & eclogæ mixtæ. </s>

<s id="id.000114">Poema cui titulus, Lamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.018.jpg"/>

<s>Apologi centum, quos &longs;crip&longs;it æmulatus Leonem Bapt. <lb/>Albertum. </s>

<s id="id.000115">Carmina pia, quæ in&longs;cribuntur, Anni Corona. </s>

<s>De Humanitate Dialogus qui in&longs;cribitur Go&longs;elinus. </s>

<s id="id.000116">De Verborum Vitruuianorum &longs;ignificatione. </s>

<s>Compatatio Vitæ Mona&longs;ticæ cum &longs;eculari. </s>

<s id="id.000117">Carmina varia & eclogæ mixtæ. </s>

<s>De Aula libri &longs;ex. </s>

<s id="id.000118">Apologi centum, quos &longs;crip&longs;it æmulatus Leonem Bapt. <lb/>Albertum. </s>

<s>De felicitate Principis Dialogus. </s>

<s>De Dignitate Dial. </s>

<s id="id.000119">De Humanitate Dialogus qui in&longs;cribitur Go&longs;elinus. </s>

<s>Carmina Romana. </s>

<s id="id.000120">Compatatio Vitæ Mona&longs;ticæ cum &longs;eculari. </s>

<s>Mo&longs;æi fabulam vertit. </s>

<s id="id.000121">De Aula libri &longs;ex. </s>

<s>De Italici carminis natura Dial. qui in&longs;cribitur Ta&longs;&longs;us. </s>

<s id="id.000122">De felicitate Principis Dialogus. </s>

<s>De vniuer&longs;ali Diluuio poemation. </s>

<s id="id.000123">De Dignitate Dial. </s>

<s>Nouæ Gnomonices lib. quin que. </s>

<s id="id.000124">Carmina Romana. </s>

<s>Hieremiæ Threnos vertit, & ex Heb. fonte annotat. ad<lb/>iecit. </s>

<s id="id.000125">Mo&longs;æi fabulam vertit. </s>

<s>Poemation in&longs;eriptum, Deiphobe, quod &longs;erip&longs;it æmula<lb/>tus Lycophonem in Ca&longs;&longs;andra. </s>

<s id="id.000126">De Italici carminis natura Dial. qui in&longs;cribitur Ta&longs;&longs;us. </s>

<s>Scala c&oelig;le&longs;tis.1.Sermones pij & carmina. </s>

<s>Onkeli paraphra&longs;in Chaldæam in Pentateuchum ver<lb/>tit & vberes commentarios adiecit. </s>

<s id="id.000127">De vniuer&longs;ali Diluuio poemation. </s>

<s>In Iob Paraphra&longs;is latina ex fonte Heb. additis Scholijs. </s>

<s id="id.000128">Nouæ Gnomonices lib.

<s>De &longs;camillis imparibus Vitruuij. </s>

quin que. </s>

<s>De firmamento & aquis. </s>

<s id="id.000129">Hieremiæ Threnos vertit, & ex Heb. fonte annotat. </s>

<s>Quincti Calabri Paralipomena vertit. </s>

<s>Tabulæ Etru&longs;cæ Eugubinæ Interpretatio. </s>

<s>Oeconomía Tropologicain S.Matthæum. </s>

<s id="id.000130">ad<lb/>iecit. </s>

<s>Vrbini encomium. </s>

<s id="id.000131">Poemation in&longs;eriptum, Deiphobe, quod &longs;erip&longs;it æmula<lb/>tus Lycophonem in Ca&longs;&longs;andra. </s>

<s>Horti geographici ex Arab. ver&longs;io. </s>

<s id="id.000132">Scala c&oelig;le&longs;tis.1.Sermones pij & carmina. </s>

<s>Aduer&longs;us Aulam Carmina. </s>

<s id="id.000133">Onkeli paraphra&longs;in Chaldæam in Pentateuchum ver<lb/>tit & vberes commentarios adiecit. </s>

<s>Luciani de mi&longs;erijs.Aulicorum ver&longs;io. </s>

<s id="id.000134">In Iob Paraphra&longs;is latina ex fonte Heb. additis Scholijs. </s>

<s>Oratio ad Romæ con&longs;eruatores pro antiquitatum eius <lb/>Vrbis cu&longs;todia. </s>

<pb/>

<s>Vniuer&longs;i orbis geographica & Hi&longs;torica de&longs;criptio con<lb/>texta ex &longs;eptingentis & eo amplius &longs;criptoribus. </s>

<s id="id.000135">De &longs;camillis imparibus Vitruuij. </s>

<s>Fede ici Vrbini Ducis Vita. </s>

<s id="id.000136">De firmamento & aquis. </s>

<s>Guidi Vbaldi Vibini Ducis Vita. </s>

<s id="id.000137">Quincti Calabri Paralipomena vertit. </s>

<s>Epigrammaton & Odarum libritres. </s>

<s id="id.000138">Tabulæ Etru&longs;cæ Eugubinæ Interpretatio. </s>

<s>Aliorum Carminum liber. </s>

<s id="id.000139">Oeconomía Tropologicain S.Matthæum. </s>

<s>Sententiarum moralium liber. </s>

<s id="id.000140">Vrbini encomium. </s>

<s>Dictionarium Arabicum. </s>

<s id="id.000141">Horti geographici ex Arab. ver&longs;io. </s>

<s>Pro Procopio contra Flauium Blondum. </s>

<s>Horographium vniucr&longs;ale. </s>

<s id="id.000142">Aduer&longs;us Aulam Carmina. </s>

<s>Epigrammata alia. </s>

<s id="id.000143">Luciani de mi&longs;erijs.Aulicorum ver&longs;io. </s>

<s>Heronis lib. de Balli&longs;tis conuer&longs;io. </s>

<s id="id.000144">Oratio ad Romæ con&longs;eruatores pro antiquitatum eius <lb/>Vrbis cu&longs;todia. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.019.jpg"/>

<s>Exercitationes in Ari&longs;totelis Mechan. </s>

<s id="id.000145">Vniuer&longs;i orbis geographica & Hi&longs;torica de&longs;criptio con<lb/>texta ex &longs;eptingentis & eo amplius &longs;criptoribus. </s>

<s>Templi Ezechielis noua de&longs;criptio. </s>

<s id="id.000146">Fede ici Vrbini Ducis Vita. </s>

<s>Antiquitatum Gua&longs;tallen&longs;ium liber. </s>

<s id="id.000147">Guidi Vbaldi Vibini Ducis Vita. </s>

<s>Hi&longs;toriæ &longs;cribendæ leges. </s>

<s id="id.000148">Epigrammaton & Odarum libri tres. </s>

<s>Etalia quædam. </s>

<s id="id.000149">Aliorum Carminum liber. </s>

<s>IN MECHANICA ARISTOTE<lb/>LIS PROBLEMATA</s>

<s id="id.000150">Sententiarum moralium liber. </s>

<s>EXER CITATIONES.</s>

<s id="id.000151">Dictionarium Arabicum. </s>

<s><emph type="italics"/>Mechanices de&longs;criptio, natura, finis.<emph.end type="italics"/></s>

<s id="id.000152">Pro Procopio contra Flauium Blondum. </s>

<s>MECHANICE, facultas quædam e&longs;t, quæ <lb/>naturalimateriâ, Geometricisque; demon<lb/>&longs;trationibus v&longs;a, ex centrobaricâ, & <expan abbr="eor&umacr;">eorum</expan> <lb/>quæ ad vectem & libram rediguntur, &longs;pe<lb/>culatione; humanæ con&longs;ulens nece&longs;&longs;itati, <lb/>commoditatiqueue, &longs;uapte vi, Naturam i<lb/>p&longs;am vel &longs;ecundans, vel &longs;uperans, varia, caqueue mirabilia <lb/>operatur. Hac diffinitione de&longs;criptionéue brcuiter ca fe<lb/>rè omnia complexi &longs;umus, quæ fu&longs;i&longs;&longs;imè ab Ari&longs;totele, <lb/>Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. </s>

<s id="id.000153">Horographium vniucr&longs;ale. </s>

<s><emph type="italics"/>Mechanices Obiectum.<emph.end type="italics"/></s>

<s id="id.000154">Epigrammata alia. </s>

<s>Con&longs;ideratautem Mechanicus Graue & Leue. </s>

<s id="id.000155">Heronis lib.

<s>Graue duplex, Naturâ, Violentiâ. </s>

de Balli&longs;tis conuer&longs;io. </s>

<s>Graue Naturâ dicitur, quod in&longs;ita propen&longs;ione in <lb/>centrum mundifertnr. Graue autem Violentiâ, quod im<lb/>pre&longs;&longs;o extrin&longs;ecus pondere ab impellente pellitur. </s>

<s id="id.000156">Exercitationes in Ari&longs;totelis Mechan. </s>

<s>Leue contrà, quòd Naturâ à centro fertur. </s>

<s id="id.000157">Templi Ezechielis noua de&longs;criptio. </s>

<s>Gæterùm quicquid graue e&longs;t, &longs;ecundum punctum <lb/>e&longs;t, quod Grauitatis centrum dicitur, & hoc duplex, vt <lb/>duplex e&longs;t grauitas, Naturæ, Violentiæ. </s>

<s id="id.000158">Antiquitatum Gua&longs;tallen&longs;ium liber. </s>

<s>Grauitatis centrum in triplici magnitudine con&longs;i<lb/>deraripote&longs;t, lineari, planà, &longs;olidâ. </s>

<s id="id.000159">Hi&longs;toriæ &longs;cribendæ leges. </s>

<s>De centro grauitatis linearum nemo &longs;crip&longs;it, &longs;impli<lb/>ci&longs;&longs;imi enim illud e&longs;t contemplationis. </s>

<s id="id.000160">Etalia quædam. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.020.jpg"/><figure id="id.007.00.020.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.020.1.jpg"></figure>

<s>De centro grauitatis linearum egregiè tractauit Ar<lb/>chimedes in libro Æ queponderantium, & de quadratu<lb/>ra Parabole, tum in co quem de his quæ vehuntur in<lb/>&longs;crip&longs;it. </s>

</chap>

<chap>

<s>De centro grauitatis &longs;olidorum ipíemet olim &longs;cri<lb/>p&longs;erat Archimedes, &longs;ed ea quæ protulit, temporis iniuriâ <lb/>deperdita, fuâ diligentiâ re&longs;tituit Iedericus Commandi<lb/>nus. </s>

<s>E&longs;&longs;e autem & Leuitatis centrum in rerum natura, <lb/>palam e&longs;t. Punctum enim illud e&longs;t, &longs;ecundum quod lcuia <lb/>rectà à centro &longs;ur&longs;um feruntur. Huius autem non memi<lb/>nêre Mechanici, propterea quod aut nihil, aut parum ad <lb/>eorum rem faciat. </s>

<s id="id.000161">IN MECHANICA ARISTOTE<lb/>LIS PROBLEMATAEXERCITATIONES.</s>

<s>Porro Grauitatis centrum ita definit Heron, & qui <lb/>ab Herone Pappus 1.8. Collectionum Mathematicarum. </s>

<s>Centrum grauitatis <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> corporis e&longs;t pun<lb/>ctum quod dam intra po&longs;itum, à quo &longs;i graue, mente ap<lb/>pen&longs;um concipiatur, dum fertur, quie&longs;cit, & &longs;eruat eam <lb/>quam in prin cipio habuit po&longs;itionem; neque in ip&longs;a latio<lb/>ne circumuertitur. Commandinus verò in lib. de centro <lb/>grauitatis &longs;olidorum hoc pacto: Centrum grauitatis v<lb/>niu&longs;cuiu&longs;que &longs;olidæ figuræ, e&longs;t punctum illud intra po&longs;i<lb/>tum, circa quod vndique partes æqualium momentorum <lb/>ad&longs;i&longs;tunt. Sienim per tale centrum ducatur planum, fi<lb/>guram quomodolibet &longs;ecans, in partes æ què ponderantes <lb/>eam diuidit. Nos verò quàm breui&longs;&longs;imè dicimus: <expan abbr="Centr&umacr;">Centrum</expan> <lb/>grauitatis, <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> magnitudinis punctum e&longs;&longs;e intra <lb/>extraue magnitudinem po&longs;itum, per quod &longs;i plano linea <lb/>punctoue diuidatur, in partes &longs;ecatur æqueponderantes. </s>

<s>Diximus, Magnitudinis vtlineæ, plani &longs;olidique; cen<lb/>trum complecteremur. Eritigitur, vt in præ&longs;enti figura, li<lb/>neæ quidem centrum A, plani B, &longs;olidi verò C. quod &longs;i ob<lb/>ijciat qui&longs;piam, lineam & &longs;uperficiem nullam habere gra<lb/>uitatem; is &longs;ciat, <expan abbr="neq;">neque</expan> corpora Mathematica grauitatem <lb/>habere, Mechanicum verò funes, ha&longs;tas, vectes pro lineis <lb/>&longs;umere; tabulas verò, & eiu&longs;modi plana ad &longs;uperficierum <lb/>naturam referre. </s>

<s>Diximus in&longs;uper, intra extraue. Aliquando enim <lb/>grauitatis centrum extra molem corporis cuius corporis <lb/>centrum e&longs;t, cadit, vtin &longs;equenti figura. </s>

<s>E&longs;to corpus aliquod <lb/>&longs;uperficiesue ABCDE, <lb/>ducatur linea CF, <expan abbr="diuid&emacr;s">diuidens</expan> <lb/>figuras in partes hinc inde <lb/>æqueponderantes ABC, <lb/>EDC. Ducatur & GH. <lb/>diuídens item in partes æ<lb/>queponderantes GCH, & GAB, EDH. &longs;ecentautem <lb/>&longs;eip&longs;as in I. eritigitur centrum I extra figuræ terminos & <lb/>molemip&longs;am. Attamen licet hoc verum &longs;it, intra e&longs;&longs;e dici <lb/>pote&longs;t, quippe quod imaginario quodam, & vtita dicam, <lb/>virtuali ambitu ACDA contineatur. </s>

<s>Dicebamus, duplex e&longs;&longs;e grauitatis centrum, Natu

<s id="id.000162"><emph type="italics"/>Mechanices de&longs;criptio, natura, finis.<emph.end type="italics"/></s>

<pb pagenum="4"/>râ, Violentià: a&longs;firmamus modò, hæc re quidem vnum e&longs;<lb/>&longs;e, & ratione &longs;olum, non autem reip&longs;a ac&longs;i duo e&longs;&longs;ent con<lb/>&longs;iderari. </s>

<s>E&longs;to enim grauitatis na<lb/>turalis centrum B, corporis A, <lb/>&longs;ecundum quod dimi&longs;&longs;um, &longs;ua<lb/>pte naturâ cadet in C, &longs;i verò <lb/>corpus violenter impellatur in <lb/>D, aiiud acquiret centrum gra<lb/>uitatis ex violentia &longs;ecundum <lb/>quam fertur, motum, in D, <expan abbr="id&emacr;">idem</expan> <lb/>autem &longs;untre, nempe vnum B, <lb/>duo autem &longs;i violentia & natura &longs;eor&longs;um con&longs;ideren<lb/>tur. </s>

<s id="id.000163">MECHANICE, facultas quædam e&longs;t, quæ <lb/>naturalimateriâ, Geometricisque; demon<lb/>&longs;trationibus v&longs;a, ex centrobaricâ, & <expan abbr="eor&umacr;">eorum</expan> <lb/>quæ ad vectem & libram rediguntur, &longs;pe<lb/>culatione; humanæ con&longs;ulens nece&longs;&longs;itati, <lb/>commoditatiqueue, &longs;uapte vi, Naturam i<lb/>p&longs;am vel &longs;ecundans, vel &longs;uperans, varia, caqueue mirabilia <lb/>operatur. </s>

<s>Hæc centra, duo motus &longs;equuntur, rectus vterque, <lb/>Naturalis videlicet, & Violentus. Tertius ex his mixtus, & <lb/>is quidem non rectus, &longs;ed curuus. </s>

<s id="id.000164">Hac diffinitione de&longs;criptionéue brcuiter ca fe<lb/>rè omnia complexi &longs;umus, quæ fu&longs;i&longs;&longs;imè ab Ari&longs;totele, <lb/>Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. </s></subchap1>

<s>Proijciatur enim violen<lb/>ter corpus graue A &longs;uperante <lb/>igitur violentia, rectà feretur <lb/>in B; ea autem elangue&longs;cente <lb/>paullatim per curuam & mi<lb/>xtam <expan abbr="lineã">lineam</expan> &longs;ecetur in C, qua<lb/>tenus enim ad anteriora fer<lb/>tur, violentia e&longs;t; quatenus ve<lb/>rò ad inferiores partes, naturæ. Vbi verò peruenit in C, <lb/>violentiâ ce&longs;&longs;ante, naturâ verò manente, rectà deor&longs;um <lb/>fertur DCD. </s>

<s>Cæteiùm hæc centra, hiqueue motus, naturalis nem<lb/>pe, & violentus diuer&longs;imode &longs;e habent adinuicem. Sie<lb/>nim graue corpus externâ vi adhibita, centrum mundi <lb/>ver&longs;us impellatur, adiuuabunt &longs;e inuicem Natura, Vio<lb/>lentia, Si autem contra, altera alteri re&longs;i&longs;ter, in motibus

<pb pagenum="5"/>autem ad latus, eo magis pugnabunt, quo magis ab infe<lb/>rioribus ad &longs;uperiora fiet motus. </s>

<s><emph type="italics"/>Mechanices præcipua in&longs;trumenta.<emph.end type="italics"/></s>

<s id="id.000165"><emph type="italics"/>Mechanices Obiectum.<emph.end type="italics"/></s>

<s>Hic ira con&longs;titutis dicimus, in&longs;trumenta, quibusad <lb/>varias operationes Mechanici vtuntur, e&longs;&longs;e inter &longs;e qui<lb/>dem diuer&longs;a, multiplicia, & &longs;i varietatem &longs;pectes, penè in<lb/>numerabilia; quod quamuis verum&longs;it, ea omnia Ari&longs;tote<lb/>les ad vectem re ducit, & libram: quod etiam G. Vbaldus <lb/>in libris Mechanicoiumfecit. Cæterum qui po&longs;t Ari&longs;to<lb/>telem floruere Mechanici, omnia ad quinque, quas ap<lb/>pellant, Potentias, redegêre. Sunt autem ex Herone, Pap<lb/>po, Guido Vbaldo, qui eos &longs;ecutus e&longs;t, Vectis, Trochlea, <lb/>Axis in Peritrochio, Cuneus, Cochlèa. Videtur autemi<lb/>p&longs;e G. Vbaldus &longs;extam addere, nempe Libram, de qua & <lb/>primus ip&longs;e Mechanicorum tractatum in &longs;tituit. Verum <lb/>enimuero idem ferè &longs;unt Vectis & Libra, ni&longs;i forte quod <lb/>Libra tunc dicitur, cum brachia &longs;unt æ qualia. Vectis vero <lb/>quomodocun que ea &longs;e habeant; quinque harum <expan abbr="Poten-tiar&umacr;">Poten<lb/>tiarum</expan> imagines ita ob oculos ponimus. Vectis A. Trochlea <lb/>B, Axisin Peritrochio C. Cuneus D. Cochlea vero E. </s>

<s id="id.000166">Con&longs;ideratautem Mechanicus Graue & Leue. </s>

<s>Porro, Cuneum ad libram reducere conatur Ari<lb/>&longs;toteles, quod facit & G. Vbaldus, qui cò refert & Co<lb/>chleam, quippe quod nihil aliud &longs;it Cochlea, quàm Cu<lb/>neus Cylindro inuolutus. Nos autem duas tantùm Po<lb/>tentias ad vectem reduci po&longs;&longs;e arbitramur, Trochleam <lb/>nempe, & Axem in Peritrochio. Nequaquam autem Cu<lb/>neum & Cochleam. quod latiùs quidem o&longs;tendemus, <lb/>cùm de Cuneo eritnobis &longs;ermo peculiaris. </s>

<s id="id.000167">Graue duplex, Naturâ, Violentiâ. </s>

<s><emph type="italics"/>De Vecte & Libra &longs;ecundum Ari<lb/>&longs;totelem.<emph.end type="italics"/></s>

<s id="id.000168">Graue Naturâ dicitur, quod in&longs;ita propen&longs;ione in <lb/>centrum mundifertnr. </s>

<s>Ari&longs;toteles in ip&longs;o Mechanicorum ingre&longs;&longs;uita &longs;cri<lb/>bit, Mirum videri ab exigua virtute magnum pondus mo-

<s id="id.000169">Graue autem Violentiâ, quod im<lb/>pre&longs;&longs;o extrin&longs;ecus pondere ab impellente pellitur. </s>

<pb pagenum="7"/>ueri, addito nimirum ponderi pondere, &longs;iquidem & vectis <lb/>e&longs;t pondus. Duplex ergo illi admiratio, &longs;cilicet quòd exi<lb/>gua potentia moucat ingens pondus, idqueue etiam addito <lb/>vectis ip&longs;ius pondere, fiat. Hoc &longs;ecundum adieci&longs;&longs;e vide<lb/>tur, amplificationis alicuius gratiâ. Erenim quatenus <lb/>ad rem pertinet, &longs;i mouendis ponderibus vectis ip&longs;ius <lb/>pondus compares, nullius ferè e&longs;&longs;e momenti proculdu<lb/>bio affirmaueris. Sed & illud quoque notandum, aliquan<lb/>do vectis pondus mouenti auxilium ferre, quod fit vbi <lb/>fulcimento inter potentiam mouentem, & pondus ip&longs;um <lb/>collocato, vectis pars quæ à fulcimento ad potentiam e&longs;t, <lb/>premitur. Tunc enim, vt dicebamus, vectis pondere &longs;uo <lb/>potentiam adiuuat. Contra verò accidit, cum pondus i<lb/>p&longs;um inter fulcimentum e&longs;t & potentiam vel potentia i<lb/>p&longs;a inter fulcimentum & pondus. tunc cnim vectis vnâ <lb/>cum pondere attollitur. quæ licet vera &longs;int, non tamen in<lb/>de &longs;e quitur, vectis pondus, quicquam quod curandum &longs;it, <lb/>in operatione efficere, aut impedire. </s>

<s>Porrò vectem ita finire po&longs;&longs;umus, longitudinem e&longs;<lb/>&longs;e quandam inflexibilem, quæ fulcimento dato, datâ po<lb/>tentiâ datum pondus mouetur. </s>

<s id="id.000170">Leue contrà, quòd Naturâ à centro fertur. </s>

<s>Ip&longs;a quoque Libra, vt diximus, vectis e&longs;t: eius autem <lb/>naturæ, vt&longs;emper fulcimentum medium obtineat locum <lb/>inter pondus & pondus. Statera autem merus e&longs;t vectis, &longs;i <lb/>&longs;par&longs;um pro fulcimento; appendiculum verò currens pro <lb/>potentia mouente deputaueris. </s>

<s id="id.000171">Gæterùm quicquid graue e&longs;t, &longs;ecundum punctum <lb/>e&longs;t, quod Grauitatis centrum dicitur, & hoc duplex, vt <lb/>duplex e&longs;t grauitas, Naturæ, Violentiæ. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.021.jpg"/>

<s><emph type="italics"/>De Circulo eiusque natura Ari&longs;totelis doctri<lb/>na examinata.<emph.end type="italics"/></s>

<s id="id.000172">Grauitatis centrum in triplici magnitudine con&longs;i<lb/>deraripote&longs;t, lineari, planà, &longs;olidâ. </s>

<s>Ari&longs;toteles, quicquid mirum in Mechanicis opera<lb/>tur, id totum admirabili circuli naturæ e&longs;&longs;e tribuen dum <lb/>arbitratur. Aitautem, ab&longs;urdum nullatenus e&longs;&longs;e, &longs;i ex re <lb/>mirabili mirandum quippiam oriatur. In circulo autem

<s id="id.000173">De centro grauitatis linearum nemo &longs;crip&longs;it, &longs;impli<lb/>ci&longs;&longs;imi enim illud e&longs;t contemplationis. </s>

<pb pagenum="8"/>quatuorinueniri qualitates admiratione dignas. <expan abbr="Primã">Primam</expan>, <lb/>quod ex contrarijs con&longs;tituatur, mouente videlicet & <lb/>moto. Secundam, quòd contraria in eius circumferentia <lb/>inueniantur, quippe quæ cum vnica linea &longs;it, concaua &longs;i<lb/>mul e&longs;t & conuexa. Tettiam, quod contrarijs feratur mo<lb/>tionibus, antror&longs;um nimirum, retror&longs;um, &longs;ur&longs;um, atque <lb/>deor&longs;um. Quartam, quod vnicâ exi&longs;tente &longs;emidiametro, <lb/>nullum in ca punctum &longs;umi po&longs;&longs;it, æqualis alteri, in latio<lb/>ne, velocitatis. Sit enim circulus AB, cuius centrum C, <lb/>&longs;emidiameter AC, &longs;umatur autem in ea punctum D, i<lb/>temqueue punctum E. Erit itaque in ip&longs;a circulatione D <lb/>tardius E, ip&longs;um verò E tardius A, & ita citius id feretur <lb/>&longs;emper, quod remotius à mouente termino accipitur. </s>

<s>Hæc ex illo, quibus ne vltro a&longs;<lb/>&longs;en&longs;um præbeamus non vnica de cau<lb/>&longs;a cohibemur. Dicimus igitur, videri <lb/>nobis, circulum non ex contrarijs <expan abbr="c&omacr;-&longs;titui">con<lb/>&longs;titui</expan>, puta ex manente & moto, &longs;ed ex <lb/>moto &longs;impliciter. Nulla e&longs;t enim &longs;e<lb/>midiametri pars, quæ non moueatur. <lb/>Punctum autem, quod &longs;tat, &longs;emidia<lb/>metri pars nulla e&longs;t. Et &longs;anè cur moto <lb/><expan abbr="&longs;emidiam&emacr;tro">&longs;emidiamentro</expan> fiat circulus, non ideo accidit, quod <expan abbr="alter&umacr;">alterum</expan> <lb/>extremum &longs;tet, alterum verò moueatur:led ideo quòd &longs;e<lb/>midiameter perpetuò candem &longs;eruct longitudinem. Elli<lb/>p&longs;is &longs;anè centrum habet, &longs;ed ab eo ad circumferentiam <lb/>quatuor tantùm &longs;emidiametri quomodolibet &longs;umpti du<lb/>cuntur æ quales. Si quis igitur &longs;emidiametrum daret pro<lb/>portione cre&longs;centem & decre&longs;centem, &longs;tante altero ex<lb/>tremoru<gap/> Ellip&longs;is de&longs;criberetur. Præterea & &longs;piralis li<lb/>nea, quæ mixta e&longs;t, altero &longs;emidiametri extremo manen<lb/>te, altero vero moto producitur. Legem itaque circulo

<s id="id.000174">De centro grauitatis linearum egregiè tractauit Ar<lb/>chimedes in libro Æ queponderantium, & de quadratu<lb/>ra Parabole, tum in co quem de his quæ vehuntur in<lb/>&longs;crip&longs;it. </s>

<pb pagenum="9"/>prælcribit, non quidem quòd hæc extremitas &longs;ter, illa ve<lb/>rò moueatur, &longs;ed quod &longs;ua circulatione &longs;em per &longs;emidia<lb/>meter eandem &longs;eruet longitudinem, quod vel ex ip&longs;a cir<lb/>culi definitionc colligitur. </s>

<s>Ad &longs;ecundum miraculum, &longs;cilicet, quòd in circulo <lb/>circum ferentia, quæ vacua linea e&longs;t, concaua &longs;imul&longs;it, & <lb/>conuexa. Diceret qui&longs;piam id, &longs;i modò mirabile e&longs;t non <lb/>circulari tantum, &longs;ed cui ibet curuæ lineæ primo compe<lb/>tere, etenim & Elhp&longs;is & Hyperbole, & Parabolc, & &longs;pi<lb/>ra, tum Cy&longs;&longs;ois, Conchois, & infinitæ aliæ irregulares <lb/>concauæ &longs;imul &longs;unt & conuexæ. Sed & hæcin &longs;uperficie<lb/>bus quoque de&longs;iderantur. </s>

<s id="id.000175">De centro grauitatis &longs;olidorum ipíemet olim &longs;cri<lb/>p&longs;erat Archimedes, &longs;ed ea quæ protulit, temporis iniuriâ <lb/>deperdita, fuâ diligentiâ re&longs;tituit Iedericus Commandi<lb/>nus. </s>

<s>Ad tertium, quod contrarijs feratur lationibus, an<lb/>tror&longs;um, retror&longs;um, &longs;ur&longs;um & deor&longs;um. Dicimus, facilè <lb/>&longs;olui, Nullus enim, re bene per&longs;pectâ, affirmauerit circu<lb/>lum contrarijs lationibus moueri. </s>

<s id="id.000176">E&longs;&longs;e autem & Leuitatis centrum in rerum natura, <lb/>palam e&longs;t. </s>

<s>E&longs;to enim circulus ABCD, <lb/>circa centrum E; ponamus ro<lb/>tari, & A ver&longs;us B, exempli gra<lb/>tiâ, antror&longs;um, mouebitur <expan abbr="aut&emacr;">autem</expan> <lb/>& B ver&longs;us C, & C ver&longs;us D, tum <lb/>D ver&longs;us A. Non puto <expan abbr="quenquã">quenquam</expan> <lb/>dicturum, circulum hunc an<lb/>tror&longs;um codem tempore, & re<lb/>tror&longs;um ferri nec &longs;ur&longs;um aut de<lb/>or&longs;um, &longs;i enim qui&longs;piam per eius circuli circumferentiani <lb/>ambularet, is certè centrum ip&longs;um &longs;emper ad dexteram <lb/>haberet, vel ad &longs;ini&longs;tram, &longs;i ad dexteram, antror&longs;um ibit, &longs;i <lb/>ad &longs;ini&longs;tram, tetror&longs;um. Sed nec &longs;ur&longs;um vel deor&longs;um, e&longs;t <lb/>manife&longs;tum. Nihil autem prohibet eundem motum va<lb/>rio re&longs;pectu contrarium dici po&longs;&longs;e, id tamen profectò fie<lb/>rinequaquam pote&longs;t, nempe A moueriver&longs;us B, hoc e&longs;t,

<s id="id.000177">Punctum enim illud e&longs;t, &longs;ecundum quod lcuia <lb/>rectà à centro &longs;ur&longs;um feruntur. </s>

<pb pagenum="10"/>antro r&longs;um, & eandem codem tempore ver&longs;us B, id e&longs;t, re<lb/>tror&longs;um; repugnat enim naturæ. </s>

<s>De quarto circuli miraculo, ibi erit nobis &longs;ermo, vbi <lb/>ca perpenderimus primò, quæ Philo&longs;ophus de Circuli <lb/>productione di&longs;&longs;erens in medium profert. Sunt autem e<lb/>iu&longs;modi: </s>

<s id="id.000178">Huius autem non memi<lb/>nêre Mechanici, propterea quod aut nihil, aut parum ad <lb/>eorum rem faciat. </s>

<s>Circulum quidem duplici notione produci, Natu<lb/>rali videlicet altera, & altera quæ e&longs;t præter naturam, & <lb/>ideo circularem lineam in ter mixtas computari. </s>

<s id="id.000179">Porro Grauitatis centrum ita definit Heron, & qui <lb/>ab Herone Pappus 1.8. Collectionum Mathematicarum. </s>

<s>Motus mixtus ait, vel proportione &longs;eruata fit, aut <lb/>non; Si proportione &longs;eruatâ, rectam lineam; ea verò non <lb/>&longs;eruata, circularem lineam produci. </s>

<s id="id.000180">Centrum grauitatis <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> corporis e&longs;t pun<lb/>ctum quod dam intra po&longs;itum, à quo &longs;i graue, mente ap<lb/>pen&longs;um concipiatur, dum fertur, quie&longs;cit, & &longs;eruat eam <lb/>quam in principio habuit po&longs;itionem; neque in ip&longs;a latio<lb/>ne circumuertitur. </s>

<s>E&longs;to enim rectangu<lb/>lum ABCD, cuius late<lb/>ra in datâ &longs;int proportio<lb/>ne, AD cum AB. Mo<lb/>ueatur A, duplici motu, <lb/>Altero quidem tendens <lb/>in B, altero vero ad mo<lb/>tum lineæ AB, feratur <lb/>ver&longs;us D, &longs;eruata inte<lb/>rim laterum proportione. Itaque ponatur ex motu ab A <lb/>ver&longs;us B, perueni&longs;&longs;e in E, ex motu autem quo proportio<lb/>naliter fertur cum linea AB, facta ip&longs;a AB, in FH, perue<lb/>ni&longs;&longs;e in G, & EG connectatur. Eritigitur Parallelogram<lb/>mum AEGF, Parallelogrammo ABCD proportiona<lb/>le &longs;imile, & circa eandem diametrum AGC. Semperigi<lb/>tur punctum A &longs;i duabus lationibus feratur, laterum pro<lb/>portione &longs;eruata, lineam producet rectam, diametrum <lb/>nempe AGC. Et hoc &longs;anè nullam habet dubitationem, <lb/>ex ijs quæ docet Euclides 1. 6. prop. 24. </s>

<s id="id.000181">Commandinus verò in lib.

<s>His ita demon&longs;tratis hac vti videtur Philo&longs;ophus

de centro <lb/>grauitatis &longs;olidorum hoc pacto: Centrum grauitatis v<lb/>niu&longs;cuiu&longs;que &longs;olidæ figuræ, e&longs;t punctum illud intra po&longs;i<lb/>tum, circa quod vndique partes æqualium momentorum <lb/>ad&longs;i&longs;tunt. </s>

<pb pagenum="11"/>argumentatione: Si mixtus motus proportione &longs;emotâ, <lb/>rectam producir, &longs;i nun quam &longs;emota, efficiet circulum; &longs;i <lb/>enim modo &longs;eruaretur, modo non, partim recta partim <lb/>non recta produceretur. Ingenio&longs;a quidem argumenta<lb/>tio, ni vitium contineret. non enim mixtus motus, qui <lb/>nun quam &longs;eruatâ proportione fit, &longs;emper ci, culum pro<lb/>ducit, &longs;ed & Elli &longs;im pote&longs;t, & quamlibet aliam lineam, <lb/>cuius nulla pars &longs;it recta. Hanc difficultatem vidit Pico<lb/>lomineus in &longs;ua Paraphra&longs;i, & eam &longs;oluere conatus e&longs;t, <lb/>&longs;ed quàm bene, aliorum e&longs;to <gap/>dicium. Cæterùm fal&longs;um <lb/>e&longs;t, a&longs;&longs;erere circulum ex mixto motu nun quam &longs;eruatâ <lb/>proportione produci. &longs;eruat enim a&longs;&longs;iduè mixtus motus <lb/>quo producitur (&longs;i cum mixto motu producere velimus) <lb/>aliquam proportionem, &longs;ed non eandem. </s>

<s>E&longs;to enim recta AB, cui ad rectos <lb/>angulos AC. Moueatur autem A, ver<lb/>&longs;us C per lineam AC, & eodem tempo<lb/>re linea AC, ver&longs;us B, ita tamen, vt &longs;em<lb/>per ip&longs;i AB, &longs;it perpendicularis. feratur <lb/>autem eâ lege, vt quam proportionem <lb/>habet motus lineæ AC ver&longs;us B, ad mo<lb/>tum puncti A ve, &longs;us C, eandem habeat <lb/>ip&longs;e motus ab A ver&longs;us C, ad re&longs;iduum <lb/>lineæ AB, demptâ nempe ea parte quam <lb/>peragrauit linea AC mota ver&longs;us B. Sit <lb/>autem, cum AC &longs;uo motu peruenerit <lb/>in D, punctum A, &longs;imiliter &longs;uo motu per eam latum perue<lb/>nitle in E erit eigo ex mixto motu, non quidem in D, nec <lb/>in E, &longs;ed in F, eritque punctum F in circum ferentia circu<lb/>li, cuius e&longs;t diameter ip&longs;a linea AB, quod quidem demon<lb/>&longs;tratur ex conuer&longs;a propo&longs;. 13. lib. 6. Elem. E&longs;t enim AE <lb/>hoce&longs;t DF media proportionalis inter EF, hoc e&longs;t, AD, <lb/>& DB. Iterum &longs;i &longs;iat motus AC in GH, ad motum H per

<s id="id.000182">Sienim per tale centrum ducatur planum, fi<lb/>guram quomodolibet &longs;ecans, in partes æ què ponderantes <lb/>eam diuidit. </s>

<pb pagenum="12"/>lineam AC, v&longs;que in C, vt &longs;e habet proportio AG ad <lb/>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb/>eiu&longs;dem circuli circum ferentia AFHB. ex quibus ha<lb/>bemus, circulum ex mixto motu fieri po&longs;&longs;e proportioni<lb/>bus quidem mediarum &longs;eruatis, &longs;ed nun quam ij&longs;dem. </s>

<s>Vera hæc pro culdubio &longs;unt; nihilominus, veluti ad <lb/>rectam producendam mixtus motus non e&longs;t nece&longs;&longs;arius, <lb/>licet mixto motu produci po&longs;&longs;it, ita ne que ad circularem, <lb/>& ideo verum non e&longs;&longs;e quod a&longs;&longs;erebat Philo&longs;ophus, cir<lb/>culum ex mixto motu proportione nun quam &longs;eruatâ ne<lb/>ce&longs;&longs;ariò produci. </s>

<s id="id.000183">Nos verò quàm breui&longs;&longs;imè dicimus: <expan abbr="Centr&umacr;">Centrum</expan> <lb/>grauitatis, <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> magnitudinis punctum e&longs;&longs;e intra <lb/>extraue magnitudinem po&longs;itum, per quod &longs;i plano linea <lb/>punctoue diuidatur, in partes &longs;ecatur æqueponderantes. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.022.jpg"/><figure id="id.007.00.022.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.022.1.jpg"></figure>

<s>Conatur po&longs;t hæc Ari&longs;toteles rationem afferre, cur <lb/>circuli partes, quò propiores centro fuerint, eo &longs;int tar<lb/>diores. Ait autem; &longs;i duobus ab eadem potentia latis hoc <lb/>quidem plus repellatur, illud verò minus, æquum e&longs;t tar<lb/>diùs id moueri quod plus repellitur, eo quod minus. De<lb/>trahi autem plus lineam, cuius extremum prepius e&longs;t cen<lb/>tro illa quæ &longs;uum habet terminum à centro remotiorem. </s>

<s id="id.000184">Diximus, Magnitudinis vtlineæ, plani &longs;olidique; cen<lb/>trum complecteremur. </s>

<s id="id.000185">Eritigitur, vt in præ&longs;enti figura, li<lb/>neæ quidem centrum A, plani B, &longs;olidi verò C. quod &longs;i ob<lb/>ijciat qui&longs;piam, lineam & &longs;uperficiem nullam habere gra<lb/>uitatem; is &longs;ciat, <expan abbr="neq;">neque</expan> corpora Mathematica grauitatem <lb/>habere, Mechanicum verò funes, ha&longs;tas, vectes pro lineis <lb/>&longs;umere; tabulas verò, & eiu&longs;modi plana ad &longs;uperficierum <lb/>naturam referre. </s>

<s id="id.000186">Diximus in&longs;uper, intra extraue. </s>

<s id="id.000187">Aliquando enim <lb/>grauitatis centrum extra molem corporis cuius corporis <lb/>centrum e&longs;t, cadit, vtin &longs;equenti figura. </s><figure id="id.007.00.022.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.022.2.jpg"></figure>

<s id="id.000188">E&longs;to corpus aliquod <lb/>&longs;uperficiesue ABCDE, <lb/>ducatur linea CF, <expan abbr="diuid&emacr;s">diuidens</expan> <lb/>figuras in partes hinc inde <lb/>æqueponderantes ABC, <lb/>EDC. </s>

<s id="id.000189">Ducatur & GH. <lb/>diuídens item in partes æ<lb/>queponderantes GCH, & GAB, EDH. &longs;ecentautem <lb/>&longs;eip&longs;as in I. eritigitur centrum I extra figuræ terminos & <lb/>molem ip&longs;am. </s>

<s id="id.000190">Attamen licet hoc verum &longs;it, intra e&longs;&longs;e dici <lb/>pote&longs;t, quippe quod imaginario quodam, & vtita dicam, <lb/>virtuali ambitu ACDA contineatur. </s>

<s id="id.000191">Dicebamus, duplex e&longs;&longs;e grauitatis centrum, Natu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.023.jpg"/>râ, Violentià: affirmamus modò, hæc re quidem vnum e&longs;<lb/>&longs;e, & ratione &longs;olum, non autem reip&longs;a ac &longs;i duo e&longs;&longs;ent con<lb/>&longs;iderari. </s><figure id="id.007.00.023.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.023.1.jpg"></figure>

<s id="id.000192">E&longs;to enim grauitatis na<lb/>turalis centrum B, corporis A, <lb/>&longs;ecundum quod dimi&longs;&longs;um, &longs;ua<lb/>pte naturâ cadet in C, &longs;i verò <lb/>corpus violenter impellatur in <lb/>D, aliud acquiret centrum gra<lb/>uitatis ex violentia &longs;ecundum <lb/>quam fertur, motum, in D, <expan abbr="id&emacr;">idem</expan> <lb/>autem &longs;untre, nempe vnum B, <lb/>duo autem &longs;i violentia & natura &longs;eor&longs;um con&longs;ideren<lb/>tur. </s>

<s id="id.000193">Hæc centra, duo motus &longs;equuntur, rectus vterque, <lb/>Naturalis videlicet, & Violentus. </s>

<s id="id.000194">Tertius ex his mixtus, & <lb/>is quidem non rectus, &longs;ed curuus. </s><figure id="id.007.00.023.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.023.2.jpg"></figure>

<s id="id.000195">Proijciatur enim violen<lb/>ter corpus graue A &longs;uperante <lb/>igitur violentia, rectà feretur <lb/>in B; ea autem elangue&longs;cente <lb/>paullatim per curuam & mi<lb/>xtam <expan abbr="lineã">lineam</expan> &longs;ecetur in C, qua<lb/>tenus enim ad anteriora fer<lb/>tur, violentia e&longs;t; quatenus ve<lb/>rò ad inferiores partes, naturæ. </s>

<s id="id.000196">Vbi verò peruenit in C, <lb/>violentiâ ce&longs;&longs;ante, naturâ verò manente, rectà deor&longs;um <lb/>fertur DCD. </s>

<s id="id.000197">Cæteiùm hæc centra, hiqueue motus, naturalis nem<lb/>pe, & violentus diuer&longs;imode &longs;e habent adinuicem. </s>

<s id="id.000198">Sie<lb/>nim graue corpus externâ vi adhibita, centrum mundi <lb/>ver&longs;us impellatur, adiuuabunt &longs;e inuicem Natura, Vio<lb/>lentia, Si autem contra, altera alteri re&longs;i&longs;ter, in motibus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.024.jpg"/>autem ad latus, eo magis pugnabunt, quo magis ab infe<lb/>rioribus ad &longs;uperiora fiet motus. </s>

</subchap1><subchap1>

<s id="id.000199"><emph type="italics"/>Mechanices præcipua in&longs;trumenta.<emph.end type="italics"/></s>

<s id="id.000200">Hic ira con&longs;titutis dicimus, in&longs;trumenta, quibusad <lb/>varias operationes Mechanici vtuntur, e&longs;&longs;e inter &longs;e qui<lb/>dem diuer&longs;a, multiplicia, & &longs;i varietatem &longs;pectes, penè in<lb/>numerabilia; quod quamuis verum&longs;it, ea omnia Ari&longs;tote<lb/>les ad vectem re ducit, & libram: quod etiam G. Vbaldus <lb/>in libris Mechanicoiumfecit. </s>

<s id="id.000201">Cæterum qui po&longs;t Ari&longs;to<lb/>telem floruere Mechanici, omnia ad quinque, quas ap<lb/>pellant, Potentias, redegêre. </s>

<s id="id.000202">Sunt autem ex Herone, Pap<lb/>po, Guido Vbaldo, qui eos &longs;ecutus e&longs;t, Vectis, Trochlea, <lb/>Axis in Peritrochio, Cuneus, Cochlèa. </s>

<s id="id.000203">Videtur autem i<lb/>p&longs;e G. Vbaldus &longs;extam addere, nempe Libram, de qua & <lb/>primus ip&longs;e Mechanicorum tractatum in &longs;tituit. </s>

<s id="id.000204">Verum <lb/>enimuero idem ferè &longs;unt Vectis & Libra, ni&longs;i forte quod <lb/>Libra tunc dicitur, cum brachia &longs;unt æ qualia. </s>

<s id="id.000205">Vectis vero <lb/>quomodocun que ea &longs;e habeant; quinque harum <expan abbr="Poten-tiar&umacr;">Poten<lb/>tiarum</expan> imagines ita ob oculos ponimus. </s>

<s id="id.000206">Vectis A. </s>

<s id="id.000207">Trochlea <lb/>B, Axisin Peritrochio C. </s>

<s id="id.000208">Cuneus D. </s>

<s id="id.000209">Cochlea vero E. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.025.jpg"/><figure id="id.007.00.025.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.025.1.jpg"></figure>

<s id="id.000210">Porro, Cuneum ad libram reducere conatur Ari<lb/>&longs;toteles, quod facit & G. Vbaldus, qui cò refert & Co<lb/>chleam, quippe quod nihil aliud &longs;it Cochlea, quàm Cu<lb/>neus Cylindro inuolutus. </s>

<s id="id.000211">Nos autem duas tantùm Po<lb/>tentias ad vectem reduci po&longs;&longs;e arbitramur, Trochleam <lb/>nempe, & Axem in Peritrochio. </s>

<s id="id.000212">Nequaquam autem Cu<lb/>neum & Cochleam. </s>

<s id="id.000213">quod latiùs quidem o&longs;tendemus, <lb/>cùm de Cuneo erit nobis &longs;ermo peculiaris. </s>

<s id="id.000214"><emph type="italics"/>De Vecte & Libra &longs;ecundum Ari<lb/>&longs;totelem.<emph.end type="italics"/></s>

<s id="id.000215">Ari&longs;toteles in ip&longs;o Mechanicorum ingre&longs;&longs;uita &longs;cri<lb/>bit, Mirum videri ab exigua virtute magnum pondus mo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.026.jpg"/>ueri, addito nimirum ponderi pondere, &longs;iquidem & vectis <lb/>e&longs;t pondus. </s>

<s id="id.000216">Duplex ergo illi admiratio, &longs;cilicet quòd exi<lb/>gua potentia moucat ingens pondus, idqueue etiam addito <lb/>vectis ip&longs;ius pondere, fiat. </s>

<s id="id.000217">Hoc &longs;ecundum adieci&longs;&longs;e vide<lb/>tur, amplificationis alicuius gratiâ. </s>

<s id="id.000218">Erenim quatenus <lb/>ad rem pertinet, &longs;i mouendis ponderibus vectis ip&longs;ius <lb/>pondus compares, nullius ferè e&longs;&longs;e momenti proculdu<lb/>bio affirmaueris. </s>

<s id="id.000219">Sed & illud quoque notandum, aliquan<lb/>do vectis pondus mouenti auxilium ferre, quod fit vbi <lb/>fulcimento inter potentiam mouentem, & pondus ip&longs;um <lb/>collocato, vectis pars quæ à fulcimento ad potentiam e&longs;t, <lb/>premitur. </s>

<s id="id.000220">Tunc enim, vt dicebamus, vectis pondere &longs;uo <lb/>potentiam adiuuat. </s>

<s id="id.000221">Contra verò accidit, cum pondus i<lb/>p&longs;um inter fulcimentum e&longs;t & potentiam vel potentia i<lb/>p&longs;a inter fulcimentum & pondus. </s>

<s id="id.000222">tunc cnim vectis vnâ <lb/>cum pondere attollitur. </s>

<s id="id.000223">quæ licet vera &longs;int, non tamen in<lb/>de &longs;e quitur, vectis pondus, quicquam quod curandum &longs;it, <lb/>in operatione efficere, aut impedire. </s>

<s id="id.000224">Porrò vectem ita finire po&longs;&longs;umus, longitudinem e&longs;<lb/>&longs;e quandam inflexibilem, quæ fulcimento dato, datâ po<lb/>tentiâ datum pondus mouetur. </s>

<s id="id.000225">Ip&longs;a quoque Libra, vt diximus, vectis e&longs;t: eius autem <lb/>naturæ, vt&longs;emper fulcimentum medium obtineat locum <lb/>inter pondus & pondus. </s>

<s id="id.000226">Statera autem merus e&longs;t vectis, &longs;i <lb/>&longs;par&longs;um pro fulcimento; appendiculum verò currens pro <lb/>potentia mouente deputaueris. </s>

<s id="id.000227"><emph type="italics"/>De Circulo eiusque natura Ari&longs;totelis doctri<lb/>na examinata.<emph.end type="italics"/></s>

<s id="id.000228">Ari&longs;toteles, quicquid mirum in Mechanicis opera<lb/>tur, id totum admirabili circuli naturæ e&longs;&longs;e tribuen dum <lb/>arbitratur. </s>

<s id="id.000229">Ait autem, ab&longs;urdum nullatenus e&longs;&longs;e, &longs;i ex re <lb/>mirabili mirandum quippiam oriatur. </s>

<s id="id.000230">In circulo autem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.027.jpg"/>quatuorinueniri qualitates admiratione dignas. </s>

<s id="id.000231"><expan abbr="Primã">Primam</expan>, <lb/>quod ex contrarijs con&longs;tituatur, mouente videlicet & <lb/>moto. </s>

<s id="id.000232">Secundam, quòd contraria in eius circumferentia <lb/>inueniantur, quippe quæ cum vnica linea &longs;it, concaua &longs;i<lb/>mul e&longs;t & conuexa. </s>

<s id="id.000233">Tettiam, quod contrarijs feratur mo<lb/>tionibus, antror&longs;um nimirum, retror&longs;um, &longs;ur&longs;um, atque <lb/>deor&longs;um. </s>

<s id="id.000234">Quartam, quod vnicâ exi&longs;tente &longs;emidiametro, <lb/>nullum in ca punctum &longs;umi po&longs;&longs;it, æqualis alteri, in latio<lb/>ne, velocitatis. </s>

<s id="id.000235">Sit enim circulus AB, cuius centrum C, <lb/>&longs;emidiameter AC, &longs;umatur autem in ea punctum D, i<lb/>temqueue punctum E. </s>

<s id="id.000236">Erit itaque in ip&longs;a circulatione D <lb/>tardius E, ip&longs;um verò E tardius A, & ita citius id feretur <lb/>&longs;emper, quod remotius à mouente termino accipitur. </s><figure id="id.007.00.027.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.027.1.jpg"></figure>

<s id="id.000237">Hæc ex illo, quibus ne vltro a&longs;<lb/>&longs;en&longs;um præbeamus non vnica de cau<lb/>&longs;a cohibemur. </s>

<s id="id.000238">Dicimus igitur, videri <lb/>nobis, circulum non ex contrarijs <expan abbr="c&omacr;-&longs;titui">con<lb/>&longs;titui</expan>, puta ex manente & moto, &longs;ed ex <lb/>moto &longs;impliciter. </s>

<s id="id.000239">Nulla e&longs;t enim &longs;e<lb/>midiametri pars, quæ non moueatur. <lb/></s>

<s id="id.000240">Punctum autem, quod &longs;tat, &longs;emidia<lb/>metri pars nulla e&longs;t. </s>

<s id="id.000241">Et &longs;anè cur moto <lb/><expan abbr="&longs;emidiam&emacr;tro">&longs;emidiamentro</expan> fiat circulus, non ideo accidit, quod <expan abbr="alter&umacr;">alterum</expan> <lb/>extremum &longs;tet, alterum verò moueatur:led ideo quòd &longs;e<lb/>midiameter perpetuò eandem &longs;eruct longitudinem. </s>

<s id="id.000242">Elli<lb/>p&longs;is &longs;anè centrum habet, &longs;ed ab eo ad circumferentiam <lb/>quatuor tantùm &longs;emidiametri quomodolibet &longs;umpti du<lb/>cuntur æ quales. </s>

<s id="id.000243">Si quis igitur &longs;emidiametrum daret pro<lb/>portione cre&longs;centem & decre&longs;centem, &longs;tante altero ex<lb/>tremoru<gap/> Ellip&longs;is de&longs;criberetur. </s>

<s id="id.000244">Præterea & &longs;piralis li<lb/>nea, quæ mixta e&longs;t, altero &longs;emidiametri extremo manen<lb/>te, altero vero moto producitur. </s>

<s id="id.000245">Legem itaque circulo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.028.jpg"/>prælcribit, non quidem quòd hæc extremitas &longs;ter, illa ve<lb/>rò moueatur, &longs;ed quod &longs;ua circulatione &longs;em per &longs;emidia<lb/>meter eandem &longs;eruet longitudinem, quod vel ex ip&longs;a cir<lb/>culi definitionc colligitur. </s>

<s id="id.000246">Ad &longs;ecundum miraculum, &longs;cilicet, quòd in circulo <lb/>circum ferentia, quæ vacua linea e&longs;t, concaua &longs;imul&longs;it, & <lb/>conuexa. </s>

<s id="id.000247">Diceret qui&longs;piam id, &longs;i modò mirabile e&longs;t non <lb/>circulari tantum, &longs;ed cui ibet curuæ lineæ primo compe<lb/>tere, etenim & Elhp&longs;is & Hyperbole, & Parabolc, & &longs;pi<lb/>ra, tum Cy&longs;&longs;ois, Conchois, & infinitæ aliæ irregulares <lb/>concauæ &longs;imul &longs;unt & conuexæ. </s>

<s id="id.000248">Sed & hæcin &longs;uperficie<lb/>bus quoque de&longs;iderantur. </s>

<s id="id.000249">Ad tertium, quod contrarijs feratur lationibus, an<lb/>tror&longs;um, retror&longs;um, &longs;ur&longs;um & deor&longs;um. </s>

<s id="id.000250">Dicimus, facilè <lb/>&longs;olui, Nullus enim, re bene per&longs;pectâ, affirmauerit circu<lb/>lum contrarijs lationibus moueri. </s><figure id="id.007.00.028.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.028.1.jpg"></figure>

<s id="id.000251">E&longs;to enim circulus ABCD, <lb/>circa centrum E; ponamus ro<lb/>tari, & A ver&longs;us B, exempli gra<lb/>tiâ, antror&longs;um, mouebitur <expan abbr="aut&emacr;">autem</expan> <lb/>& B ver&longs;us C, & C ver&longs;us D, tum <lb/>D ver&longs;us A. </s>

<s id="id.000252">Non puto <expan abbr="quenquã">quenquam</expan> <lb/>dicturum, circulum hunc an<lb/>tror&longs;um codem tempore, & re<lb/>tror&longs;um ferri nec &longs;ur&longs;um aut de<lb/>or&longs;um, &longs;i enim qui&longs;piam per eius circuli circumferentiani <lb/>ambularet, is certè centrum ip&longs;um &longs;emper ad dexteram <lb/>haberet, vel ad &longs;ini&longs;tram, &longs;i ad dexteram, antror&longs;um ibit, &longs;i <lb/>ad &longs;ini&longs;tram, tetror&longs;um. </s>

<s id="id.000253">Sed nec &longs;ur&longs;um vel deor&longs;um, e&longs;t <lb/>manife&longs;tum. </s>

<s id="id.000254">Nihil autem prohibet eundem motum va<lb/>rio re&longs;pectu contrarium dici po&longs;&longs;e, id tamen profectò fie<lb/>rinequaquam pote&longs;t, nempe A moueriver&longs;us B, hoc e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.029.jpg"/>antro r&longs;um, & eandem codem tempore ver&longs;us B, id e&longs;t, re<lb/>tror&longs;um; repugnat enim naturæ. </s>

<s id="id.000255">De quarto circuli miraculo, ibi erit nobis &longs;ermo, vbi <lb/>ca perpenderimus primò, quæ Philo&longs;ophus de Circuli <lb/>productione di&longs;&longs;erens in medium profert. </s>

<s id="id.000256">Sunt autem e<lb/>iu&longs;modi: </s>

<s id="id.000257">Circulum quidem duplici notione produci, Natu<lb/>rali videlicet altera, & altera quæ e&longs;t præter naturam, & <lb/>ideo circularem lineam in ter mixtas computari. </s>

<s id="id.000258">Motus mixtus ait, vel proportione &longs;eruata fit, aut <lb/>non; Si proportione &longs;eruatâ, rectam lineam; ea verò non <lb/>&longs;eruata, circularem lineam produci. </s><figure id="id.007.00.029.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.029.1.jpg"></figure>

<s id="id.000259">E&longs;to enim rectangu<lb/>lum ABCD, cuius late<lb/>ra in datâ &longs;int proportio<lb/>ne, AD cum AB. </s>

<s id="id.000260">Mo<lb/>ueatur A, duplici motu, <lb/>Altero quidem tendens <lb/>in B, altero vero ad mo<lb/>tum lineæ AB, feratur <lb/>ver&longs;us D, &longs;eruata inte<lb/>rim laterum proportione. </s>

<s id="id.000261">Itaque ponatur ex motu ab A <lb/>ver&longs;us B, perueni&longs;&longs;e in E, ex motu autem quo proportio<lb/>naliter fertur cum linea AB, facta ip&longs;a AB, in FH, perue<lb/>ni&longs;&longs;e in G, & EG connectatur. </s>

<s id="id.000262">Eritigitur Parallelogram<lb/>mum AEGF, Parallelogrammo ABCD proportiona<lb/>le &longs;imile, & circa eandem diametrum AGC. </s>

<s id="id.000263">Semperigi<lb/>tur punctum A &longs;i duabus lationibus feratur, laterum pro<lb/>portione &longs;eruata, lineam producet rectam, diametrum <lb/>nempe AGC. </s>

<s id="id.000264">Et hoc &longs;anè nullam habet dubitationem, <lb/>ex ijs quæ docet Euclides 1. 6. prop.

24. </s>

<s id="id.000265">His ita demon&longs;tratis hac vti videtur Philo&longs;ophus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.030.jpg"/>argumentatione: Si mixtus motus proportione &longs;emotâ, <lb/>rectam producir, &longs;i nun quam &longs;emota, efficiet circulum; &longs;i <lb/>enim modo &longs;eruaretur, modo non, partim recta partim <lb/>non recta produceretur. </s>

<s id="id.000266">Ingenio&longs;a quidem argumenta<lb/>tio, ni vitium contineret. </s>

<s id="id.000267">non enim mixtus motus, qui <lb/>nun quam &longs;eruatâ proportione fit, &longs;emper ci, culum pro<lb/>ducit, &longs;ed & Elli &longs;im pote&longs;t, & quamlibet aliam lineam, <lb/>cuius nulla pars &longs;it recta. </s>

<s id="id.000268">Hanc difficultatem vidit Pico<lb/>lomineus in &longs;ua Paraphra&longs;i, & eam &longs;oluere conatus e&longs;t, <lb/>&longs;ed quàm bene, aliorum e&longs;to <gap/>dicium. </s>

<s id="id.000269">Cæterùm fal&longs;um <lb/>e&longs;t, a&longs;&longs;erere circulum ex mixto motu nun quam &longs;eruatâ <lb/>proportione produci. </s>

<s id="id.000270">&longs;eruat enim a&longs;&longs;iduè mixtus motus <lb/>quo producitur (&longs;i cum mixto motu producere velimus) <lb/>aliquam proportionem, &longs;ed non eandem. </s><figure id="id.007.00.030.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.030.1.jpg"></figure>

<s id="id.000271">E&longs;to enim recta AB, cui ad rectos <lb/>angulos AC. </s>

<s id="id.000272">Moueatur autem A, ver<lb/>&longs;us C per lineam AC, & eodem tempo<lb/>re linea AC, ver&longs;us B, ita tamen, vt &longs;em<lb/>per ip&longs;i AB, &longs;it perpendicularis. </s>

<s id="id.000273">feratur <lb/>autem eâ lege, vt quam proportionem <lb/>habet motus lineæ AC ver&longs;us B, ad mo<lb/>tum puncti A ve, &longs;us C, eandem habeat <lb/>ip&longs;e motus ab A ver&longs;us C, ad re&longs;iduum <lb/>lineæ AB, demptâ nempe ea parte quam <lb/>peragrauit linea AC mota ver&longs;us B. </s>

<s id="id.000274">Sit <lb/>autem, cum AC &longs;uo motu peruenerit <lb/>in D, punctum A, &longs;imiliter &longs;uo motu per eam latum perue<lb/>nitle in E erit ergo ex mixto motu, non quidem in D, nec <lb/>in E, &longs;ed in F, eritque punctum F in circum ferentia circu<lb/>li, cuius e&longs;t diameter ip&longs;a linea AB, quod quidem demon<lb/>&longs;tratur ex conuer&longs;a propo&longs;. </s>

<s id="id.000275">13. lib.

6. Elem. </s>

<s id="id.000276">E&longs;t enim AE <lb/>hoce&longs;t DF media proportionalis inter EF, hoc e&longs;t, AD, <lb/>& DB. </s>

<s id="id.000277">Iterum &longs;i &longs;iat motus AC in GH, ad motum H per <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.031.jpg"/>lineam AC, v&longs;que in C, vt &longs;e habet proportio AG ad <lb/>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb/>eiu&longs;dem circuli circum ferentia AFHB. ex quibus ha<lb/>bemus, circulum ex mixto motu fieri po&longs;&longs;e proportioni<lb/>bus quidem mediarum &longs;eruatis, &longs;ed nun quam ij&longs;dem. </s>

<s id="id.000278">Vera hæc pro culdubio &longs;unt; nihilominus, veluti ad <lb/>rectam producendam mixtus motus non e&longs;t nece&longs;&longs;arius, <lb/>licet mixto motu produci po&longs;&longs;it, ita ne que ad circularem, <lb/>& ideo verum non e&longs;&longs;e quod a&longs;&longs;erebat Philo&longs;ophus, cir<lb/>culum ex mixto motu proportione nun quam &longs;eruatâ ne<lb/>ce&longs;&longs;ariò produci. </s>

<s id="id.000279">Conatur po&longs;t hæc Ari&longs;toteles rationem afferre, cur <lb/>circuli partes, quò propiores centro fuerint, eo &longs;int tar<lb/>diores. </s>

<s id="id.000280">Ait autem; &longs;i duobus ab eadem potentia latis hoc <lb/>quidem plus repellatur, illud verò minus, æquum e&longs;t tar<lb/>diùs id moueri quod plus repellitur, eo quod minus. </s>

<s id="id.000281">De<lb/>trahi autem plus lineam, cuius extremum prepius e&longs;t cen<lb/>tro illa quæ &longs;uum habet terminum à centro remotiorem. </s><figure id="id.007.00.031.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.031.1.jpg"></figure>

<s id="id.000282">E&longs;to, inquit, circulus <lb/>BCDE & alter in eo minor <lb/>MNOP circa idem centrum <lb/>A. Ducanturque; Diametrima<lb/>ioris quidem CD, EB, mino<lb/>ris verò MO, NP. </s>

<s id="id.000283">Itaque vbi <lb/>AB circulata eò peruenerit <lb/>vnde e&longs;t gre&longs;&longs;a, ip&longs;a quoque <lb/>AM eo vnde moueri c&oelig;pe<lb/>rat, perueniet. </s>

<s id="id.000284">Tardiùs autem <lb/>fertur AM, quam AD, pro<lb/>pterea quòd AM à centro <lb/>magis retrahatur quàm ip&longs;a AB. </s>

<s id="id.000285">Ducatur igitur ALF & <lb/>à puncto L, ip&longs;i AB perpendicularis L q, cadens in mino-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.032.jpg"/>ri circulo, & rur&longs;us ab codem L ip&longs;i AB, parailela duca<lb/>tur LS, Ab S verò eidem perpendicularis ST, & ab F i<lb/>tem FX. </s>

<s id="id.000286">Sunt ergo q L, ST, quidem æquales, nempeillæ, <lb/>per qua<gap/>, &longs;ecundum naturam, mouentur puncta BM. </s>

<s id="id.000287">Mo<lb/>tu verò retractionis ad centrum, hoc e&longs;t, præter naturam, <lb/>plus motum e&longs;t M quàm B. </s>

<s id="id.000288">Maior enim e&longs;t M q, ip&longs;a BT, <lb/>quod, ceu notum, &longs;uppo&longs;uit Ari&longs;toteles. </s>

<s id="id.000289">nos autem inf. </s>

<s id="id.000290">à <lb/>demon&longs;trabimus. </s>

<s id="id.000291">Si igitur fiat vt motus præter naturam <lb/>ad motum præter naturam, ita motus <expan abbr="&longs;ec&umacr;dum">&longs;ecundum</expan> naturam, <lb/>ad motum &longs;ecundum naturam, punctum B; cum M fuerit <lb/>in L, non eritin S, &longs;ed in F. tunc enim, vt e&longs;t FX motus &longs;e<lb/>cundùm naturam ad XB, præter naturam, ita e&longs;t q L &longs;e<lb/>cundum naturam ad q M præter naturam; &longs;ed BF maior <lb/>e&longs;t ML, ergo proportione &longs;eruatâ, velociùs mouetur B <lb/>quàm M circa idem centrum A. </s>

<s id="id.000292">Hæc autem &longs;umma e&longs;t <lb/>eorum quæ præfert Ari&longs;toteles. </s>

<s id="id.000293">Cæterùm nos parallelo<lb/>grammum, quod in figura eius habetur prætermi&longs;imus, <lb/>quippe quod nihil ad eam quæ affertur, demon&longs;tratio<lb/>nem faciat. </s>

<s id="id.000294">Modò quod pollicebamur, nempe minorem e&longs;&longs;e <lb/>BT, quàm q M, ita demon&longs;tramus. </s>

<s id="id.000295"><expan abbr="quoniã">quoniam</expan> ST. ex prop.

13. <lb/>1. 6. media proportionalis e&longs;t inter BT & TE, erit qua<lb/>dratum TS æquale <expan abbr="parallelogrãmo">parallelogrammo</expan> &longs;eu rectangulo BT, <lb/>TE, item, quoniam q L media proportionalis e&longs;t inter <lb/>M q, & q O. erit quadratum q L æquale rectangulo M q, <lb/>q O, æqualia ergo &longs;unt rectangula BTE, M q O, itaque <lb/>reciprocalatera habent proportionalia. </s>

<s id="id.000296">quare, vt TE, ad <lb/>q O, ita M q ad TB, &longs;ed TE maior e&longs;t ip&longs;a q O, quippe <lb/>quòd pars &longs;it q O ip&longs;ius TE, maior ergo & M q ip&longs;a TB, <lb/>quod o&longs;tendendum fuerat. </s>

<s id="id.000297">Cæterùm &longs;ubtilia & ingenio&longs;a i&longs;thæc e&longs;&longs;e non nega<lb/>mus, & longè faciliori & explicatiori modo veritas hæc <lb/>demon&longs;trari pote&longs;t, reiectis nem peillis, &longs;ecundùm, & prae<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.033.jpg"/>ter naturam motibus, qui <expan abbr="quid&emacr;">quidem</expan> in &longs;implici circulo nece&longs;<lb/>&longs;ario non cadunt: caderent autem forta&longs;&longs;e, &longs;i de circulo <lb/>res e&longs;&longs;et à <expan abbr="p&omacr;deribus">ponderibus</expan> circumlatis ex &longs;tabili centro de&longs;eri<lb/>pto, qua de re agit G. Vbaldus in Mechanicis ttactatu de <lb/>libra. </s>

<s id="id.000298">tunc enim dici pote&longs;t, pondus quod aliâs rectà ad <lb/>mundi centrum tenderet, à circuli centro in circulatio<lb/>ne retrahi, &longs;ed hæc ad circuli naturam, quatenus circulus <lb/>e&longs;t, ne quaquam &longs;pectant. </s><figure id="id.007.00.033.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.033.1.jpg"></figure>

<s id="id.000299">E&longs;to igitur circum ferentia <lb/>AFBH, cuius centrum C, dia<lb/>meter ACB, &longs;emidiameter AC. <lb/>&longs;umatur in AC punctum quod<lb/>libet, D, & centro C, &longs;patio CD, <lb/>circumferentia de&longs;cribatur <lb/>DGEI. </s>

<s id="id.000300">Dico punctum A velo<lb/>cius moueri puncto D eâdem <lb/>circulatione rotato. </s>

<s id="id.000301">etenim vt <lb/>diameter ad diametrum, & &longs;emidiameter ad &longs;emidiame<lb/>trum, ita circumferentia ad circumferentiam: igitur vt <lb/>AC ad CD, ita circumferentia AFHB ad circumferen<lb/>tiam DGEI. </s>

<s id="id.000302">At mota linea CA circa centrum C mo<lb/>uetur &longs;imul & CD, eodem igitur tempore rotationem <lb/>complent puncta AD, maius ergo &longs;patium eodem tem<lb/>pore metitur A, ip&longs;a D, quare velocior. </s>

<s id="id.000303">Ita igitur &longs;e ha<lb/>bet velocitas ad velocitatem, vt circumferentia ad cir<lb/>cumferentiam, & diameter ad diametrum, quare id quod <lb/>mouetur in puncto à centro remotiori, velocius illo mo<lb/>uetur quod ab eo di&longs;tat minus, quod fuerat <lb/>demon&longs;trandum. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.034.jpg"/>

</subchap1>

</chap>

<chap>

<s id="id.000304">QVÆSTIONES <lb/>MECHANICÆ.</s>

<s id="id.000305">QVÆSTIO I.</s>

<s id="id.000306"><emph type="italics"/>Cur maiores libræ exactiores &longs;int mi<lb/>noribus?<emph.end type="italics"/></s>

<s id="id.000307">Prioríbus, ceu fundamentis quibu&longs;dam iactis, oppor<lb/>tunè ad quæ&longs;tiones proponendas, eas queue diluendas &longs;e <lb/>confert Ari&longs;toteles. </s>

<s id="id.000308">Porro in propo&longs;ita quæ&longs;tione vide<lb/>tur prima fi onte cau&longs;&longs;am quæri de re quæ non e&longs;t: etenim <lb/>quis affirmauerit vnquam, lances quibus Apothecarij & <lb/>Macellarij vtuntur, magnas eas quidem, illis exactiores <lb/>e&longs;&longs;e quibus Gemmatij, atque Argentarij &longs;iliquis, & &longs;eru<lb/>pulis minuti&longs;&longs;ima appendunt, quæ tamen perexiguæ &longs;unt, <lb/>& &longs;i illis comparentur minimæ? </s>

<s id="id.000309">Veruntamen, ita pror&longs;us <lb/>res habet, vt a&longs;&longs;erit Ari&longs;toteles. </s>

<s id="id.000310">Non enim propterea <lb/>quòd illæ magnæ &longs;int, hæ verò exiguæ, hæ &longs;unt illis exa<lb/>ctiores; &longs;ed quoniam magnæ, rudes &longs;unt, minores verò ex<lb/>qui&longs;ita diligentia elaboratæ, & à materiæ pertina cia libe<lb/>riores. </s>

<s id="id.000311">Cæteris ergo paribus, exactiores e&longs;&longs;e maiores, ex <lb/>Philo&longs;ophimente, ita docebimus. </s><figure id="id.007.00.034.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.034.1.jpg"></figure>

<s id="id.000312">E&longs;to libra maior AB, <lb/>cuius fulcimentum C. <lb/></s>

<s id="id.000313">Minor verò libra DE, <lb/>circa idem <expan abbr="fulcim&emacr;tum">fulcimentum</expan> <lb/>C, vnà cum maiori, ima<lb/>ginatione, conuer&longs;a. </s>

<s id="id.000314">Ap<lb/>ponatur queduis pon<lb/>dus maiori libræ in A, <lb/>de clinetque; exempli gratiâ in F, erit queue minor libra in G, <lb/>in eadem enim linea &longs;unt CGF. <expan abbr="Vnaq;">Vnaque</expan> igitur ex eodem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.035.jpg"/>centro C portionem circuli de&longs;cribet GD, AF, eritqueue <lb/>ACF &longs;ector circuli, cuius diameter AB, &longs;ed DCG &longs;e<lb/>ctor circuli, cuius diameter DE. </s>

<s id="id.000315">Itaque vt diameter ad <lb/>diametrum, ita portio ad portionem: maior autem dia<lb/>meter AB diametro DE: maior ergo portio AF, portio<lb/>ne DG. quod autem maius e&longs;t, minus obtutum fallit, ex<lb/>qui&longs;itius itaque tractum ex maiori AB quàm ex ip&longs;a mi<lb/>nori DE cogno &longs;cemus, quod fuerat o&longs;tendendum. </s>

<s id="id.000316">Cæterùm hac eadem de cau&longs;&longs;a, A&longs;tronomica in<lb/>&longs;trumenta, puta A&longs;trolabia, Armillæ, & alia eiu&longs;modi, <lb/>quo ampliora eò ex qui&longs;itiora, & certiora probantur. </s><figure id="id.007.00.035.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.035.1.jpg"></figure>

<s id="id.000317">E&longs;to enim A<lb/>&longs;trolabium magnum, <lb/>cuius diameter AB, <lb/>paruum autem CD, <lb/>circa idem centrum <lb/>E. </s>

<s id="id.000318">Ducatur à centro <lb/>recta EF tangens ma<lb/>iorem circulum in F, <lb/><expan abbr="minor&emacr;">minorem</expan> verò <expan abbr="&longs;ecãs">&longs;ecans</expan> in <lb/>G, vt igitur GD ad to<lb/>tum circulum GCD, <lb/>ita FB. ad totum cir<lb/>culum FAB, vt ergò <lb/>GD ad FB, ita gradus <lb/>&longs;ignati in GD, ad eos qui &longs;ignantur in BF, maiores ergo <lb/>&longs;unt qui in FB, & minutarum partium capaciores. </s>

<s id="id.000319">Hinc <lb/>itaque apparet, <expan abbr="in&longs;trum&emacr;ta">in&longs;trumenta</expan> quælibet quò maiora fuerint, <lb/>eò e&longs;&longs;e & exqui&longs;itiora, quod propo&longs;uerat Ari&longs;toteles, in <lb/>hac quæ&longs;tione de Libra. </s>

<s id="id.000320">Quod autem addit de fraudibus Purpurariorum, <lb/>inquiens; quamobrem machinánturij qui purpuram ven<lb/>dunt, vt <expan abbr="p&emacr;dendo">pendendo</expan> defraudent, dum ad medium, &longs;partum, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.036.jpg"/>non ponentes; tum plumbum in alterutram libræ partem <lb/>infundentes; aut ligni quod ad radicem vergebat, in eam <lb/>quam deferri volunt partem con&longs;tituentes, aut &longs;i nodum <lb/>habucrit, ligni enim grauior ea e&longs;t pars, in qua e&longs;t radix, <lb/>nodus verò radix quæ dam e&longs;t. </s>

<s id="id.000321">Hinc quæri po&longs;&longs;et: </s>

<s id="id.000322"><emph type="italics"/>Vtrum libræ quæ ponderibus vacuæ æquilibrant, <lb/>omni pror&longs;us careant fraude?<emph.end type="italics"/></s>

<s id="id.000323">Videri cuipiam po&longs;&longs;et, libras, quæ ponderibus va<lb/>cuæ, æquilibrant, omm pror&longs;us fraude carere, verunta<lb/>men ita non e&longs;t, quod diligentiùs (res enim magni mo<lb/>menti e&longs;t) di&longs;quiremus. </s><figure id="id.007.00.036.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.036.1.jpg"></figure>

<s id="id.000324">E&longs;to enim libra AB, ita diui&longs;a <lb/>in C, vt AC &longs;it partium IS, CB ve<lb/>rò carundem &longs;it 10. apponatur parti <lb/>A lanx ponderans 10, parti vero B <lb/>lanx ponderans 15. ex permutata i<lb/>gitur proportione libra &longs;u&longs;pen&longs;a in <lb/>C, aequè ponderabit; &longs;i autem appo<lb/>natur lanci B &longs;acoma vnciarum 6, & in lance A con&longs;titua<lb/>tur purpura, quæ ita &longs;e habeat ad vncias 6, vt 10 ad 15, ite<lb/>rum æqueponderabit, &longs;ed vt 10 ad 15, ita 4 ad 6. Purpura<lb/>rius ergo fraudulentus, ponens in lance A vncias purpuræ <lb/>4, facto æquilibrio petet pretium vnciarum 6, & ita em<lb/>ptorem decipiet, quod &longs;anè innuerat, non autem demon<lb/>&longs;trauerat Ari&longs;toteles. </s>

<s id="id.000325">Hæc autem faciliora fient ex ijs, <lb/>quæ in &longs;equentibus quæ &longs;tionibus, vbi de vecte agetur, ex<lb/>plicabuntur. </s>

<s id="id.000326">Detegitur autem fraus, &longs;i alternatim &longs;acoma in pon<lb/>derando, modo huic, modò illi lanci apponatur. </s>

<s id="id.000327">Si enim <lb/>in lance A con&longs;tituatur &longs;acoma, in B verò purpura non fit <lb/>æquilibrium. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.037.jpg"/>

</subchap1>

<s id="id.000328">QVÆSTIO II.</s>

<s id="id.000329"><emph type="italics"/>Cur, &longs;i &longs;ur &longs;um libræ fulcimentum &longs;it, appo&longs;ito ad alteram partem <lb/>pondere, de &longs;cendat libra, & eo amoto, iterum a&longs;cendat, & ad æqui<lb/>librium reuertatur. </s>

<s id="id.000330">Si verò deor &longs;um fulcimentum fuerit, de<lb/>pre&longs;&longs;a ad æquilibrium non reuertatur?<emph.end type="italics"/></s>

<s id="id.000331">Bimembrem proponit Philo&longs;ophus quæ&longs;tionem, quam <lb/>trimembrem debuit, triplici &longs;i quidem loco fulcimen<lb/>tum aptari pote&longs;t, &longs;uperiori, medio, inferiori. </s>

<s id="id.000332">Nos de o<lb/>mnibus ver ba faciemus. </s>

<s id="id.000333">Prima Quæ&longs;tionis pars.</s>

<s id="id.000334"><emph type="italics"/>De Libra &longs;ur &longs;um fulcimentum habente.<emph.end type="italics"/></s>

<s id="id.000335">Ari&longs;toteles primam quæ&longs;tionis partem ita &longs;oluit: An <lb/>quia &longs;ur&longs;um parte quidem exi&longs;tente, plus libræ extra per<lb/>pend culum &longs;it? </s>

<s id="id.000336">Spartum enim perpendiculum e&longs;t: quare <lb/>nece&longs;&longs;<gap/> e&longs;t deor&longs;um ferriid quod plus e&longs;t, donec a&longs;cendat <lb/>qua bifariam libram diuidit ad ip&longs;um perpendiculum, <lb/>cum onus in cum bat ad libræ partem &longs;ur&longs;us raptam. </s><figure id="id.007.00.037.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.037.1.jpg"></figure>

<s id="id.000337">Sit libra recta (hoc e&longs;t, in æquilibrio con&longs;tituta) BC, <lb/>&longs;partum autem AD, <lb/>fulcimentum autem <lb/>D, de&longs;uper: &longs;parto au<lb/>tem deor&longs;um proie<lb/>cto ad M perpendicu<lb/>laris erit vbi ADM. <lb/></s>

<s id="id.000338">Si igitur in ip&longs;o B po<lb/>natur onus, erit B qui<lb/>dem vbi E, C autem <lb/>vbi H, quamobrem <lb/>ea quæ bifariam <expan abbr="librã">libram</expan> <lb/>&longs;ecat, primo quidem erit DM, ip&longs;ius perpendiculi; in <expan abbr="c&umacr;-bente">cun<lb/>bente</expan> <expan abbr="aut&emacr;">autem</expan> onere, erit DG. quare libræ ip&longs;ius EH, quod <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.038.jpg"/>extra perpendiculum, e&longs;t AM, vbi e&longs;t q P maius e&longs;t dimi<lb/>dio. </s>

<s id="id.000339">Si igitur amoueatur onus ab E, nece&longs;&longs;e e&longs;t deor&longs;um <lb/>ferri H, minus e&longs;t enim E: &longs;iquidem igitur habuerit &longs;par<lb/>tum &longs;ur&longs;um, propter hoc a&longs;cendit libra. </s>

<s id="id.000340">Pe&longs;&longs;imè omnes &longs;chema hoc lineârunt, ita vt difficil<lb/>limum &longs;it auctoris inde &longs;en&longs;um a&longs;&longs;equi. </s>

<s id="id.000341">Nos autem cla<lb/>rius rem ob oculos ponimus. </s>

<s id="id.000342">Id ergo &longs;ibi vult Ari&longs;toteles, <lb/>propterea quòd pars iugi HDG maior e&longs;t parte ED q, <lb/>eam eleuatam nece&longs;&longs;e e&longs;t de&longs;cendere, & iterum à perpen<lb/>diculari ADM bifariam diui&longs;am ad æquilibrium reuer<lb/>ti, Po&longs;&longs;umus nos idem &longs;impliciori figura demon&longs;trare. </s><figure id="id.007.00.038.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.038.1.jpg"></figure>

<s id="id.000343">E&longs;to libra AB, bi<lb/>fariam, diui&longs;a in G, <lb/><expan abbr="fulciment&umacr;">fulcimentum</expan> verò &longs;ur<lb/>&longs;um vbi D, prod<gap/>ca<lb/>tur perpendicularis <lb/>DC in E. </s>

<s id="id.000344">Stante igi<lb/>tur libra AB, in æqui<lb/>librio æqualis e&longs;t pars <lb/>CH, ip&longs;i parti CB <lb/>apponatur pondus in <lb/>B. </s>

<s id="id.000345">Declinabit igitur <lb/>libra mota circa centrum D, fiat autem in FG, &longs;ecetqueue <lb/>perpendicularem in I. </s>

<s id="id.000346">Punctum vero C eodem motu cir<lb/>ca idem centrum D erit in H. amoueatur pondus appo&longs;i<lb/>tum: Dico libram à &longs;itu FG declinaturam & iterum re<lb/>uer&longs;uram in &longs;itum pri&longs;tinum ACB. quoniam enim parti <lb/>GH, quæ æqualis e&longs;t parti HF, additur pars IH, quæ à <lb/>perpendiculari e&longs;t v&longs;que ad H, ip&longs;i verò HF eadem pars <lb/>detrahitur, erit IF minor GI. </s>

<s id="id.000347">Superabiturita que IF à <lb/>GI, de&longs;cendetque FI, a&longs;cendet verò IF, donec iterum li<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.039.jpg"/>bra ín partes æquales, vt antea, diuidatur in C, &longs;iat que æ<lb/>quilibrium. </s>

<s id="id.000348">Hæc Philo&longs;ophi demon &longs;tratio e&longs;t vera illa quidem, <lb/>&longs;ed non ex Mechanicis principijs, hoc e&longs;t, ex centri graui<lb/>tatis &longs;pe culatione; nos igitur clariùs rem exponemus, his <lb/>quæ &longs;equuntur con&longs;ideratis. </s>

<s id="id.000349">Si pondus circa &longs;tabile centrum conuertatur, dimi&longs;<lb/>&longs;um non &longs;tabit, ni&longs;i &longs;ecundum grauitatis centrum fuerit <lb/>in perpendiculari, quæ per centrum, circa quod conuer<lb/>titur, ad mundi centrum cadit. </s>

<s id="id.000350">Stabit autem in ea per<lb/>pendiculari in duobus punctis, altero à centro mundi <lb/>remoti&longs;&longs;imo; altero verò cidem quantum licuerit pro<lb/>ximo. </s><figure id="id.007.00.039.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.039.1.jpg"></figure>

<s id="id.000351">E&longs;to corpus A, cuius graui<lb/>tatis centrum B, nixum lineae in<lb/>flexibili BC, cum qua liberè <lb/>conuertatur circa centrum C. <lb/></s>

<s id="id.000352">Ducatur autem per mundi cen<lb/>trum perpendicularis BCD. <lb/></s>

<s id="id.000353">Sit igitur primò pondus A <expan abbr="&longs;ec&umacr;-dum">&longs;ecun<lb/>dum</expan> gracilis B centrum, in per<lb/>pendiculari ip&longs;a &longs;upra centrum <lb/>C, puta in B. </s>

<s id="id.000354">Moueatur & <expan abbr="de&longs;c&emacr;-dat">de&longs;cen<lb/>dat</expan> in E. </s>

<s id="id.000355">Po&longs;t hæc verò in F, hoc <lb/>e&longs;t iterum in ip&longs;a perpendiculari <lb/>infra centrum C. </s>

<s id="id.000356">De&longs;cribet er<lb/>go circulum ex centro C, nem<lb/>pe BEF &longs;ecantem perpendicu<lb/>larem in duobus punctis oppo<lb/>&longs;itis BF, dico, pondus libe è di-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.040.jpg"/>mi&longs;&longs;um in duobus tantum punctis &longs;uapte naturâ perman<lb/>&longs;urum, BF, in B, primò, quoniam cum corpus ip&longs;um A à <lb/>perpendiculari, quæ &longs;upei ficiei loco intelligitur ABCD <lb/>per centrum grauitatis diuidatur, in partes diuiditur æ<lb/>queponderantes, quare in neutram partem inclinabit. <lb/></s>

<s id="id.000357">Stabit igitur erectum, lineæ ip&longs;i fultum, inflexibili BC, <lb/>quæ nititur puncto C. </s>

<s id="id.000358">In E verò non &longs;tabit, quippe quod <lb/>eo &longs;itu centrum ip&longs;um grauitatis &longs;it extra perpendicula<lb/>rem, & ideo extra fulcimentum &longs;tabile C. </s>

<s id="id.000359">In F verò ite<lb/>rum &longs;tabit, pendens à centro C, propterea quòd & ibi ab <lb/>eadem perpendiculari diuidatur per grauitatis centrum <lb/>in partes æqueponderantes. </s>

<s id="id.000360">E&longs;t igitur re&longs;pectu B, ip&longs;um <lb/>punctum C, ful cimentum deor&longs;um, re&longs;pectu verò F, ful<lb/>cimentum &longs;ur&longs;um. </s>

<s id="id.000361">At quia linea DFCB, à centro mundi, <lb/>quod e&longs;t extra circulum, BEF, circulum ip&longs;um per cen<lb/>trum C &longs;ecat, erit pars eius DF quidem breui&longs;&longs;ima, ip&longs;a <lb/>verò DB longi&longs;&longs;ima, ex propo&longs;. </s>

<s id="id.000362">8. lib.

3. Elem. </s>

<s id="id.000363">Pondus igi<lb/>tur A conuer&longs;um &longs;eu liberè motum circa centrum C, in <lb/>duobus tantum locis perpendicularis lineæ &longs;tabit remo<lb/>ti&longs;&longs;imo altero, vt e&longs;t B, altero verò cidem quamproximo, <lb/>vt e&longs;t F. </s>

<s id="id.000364">Hoc idem egregiè demon&longs;trauit G. Vbald. in &longs;uis <lb/>Mechanicis, Tractatu de Libra prop.1.<gap/></s>

<s id="id.000365">Ad hæc autem dubitare quis po&longs;&longs;et, cur experientiâ <lb/>docente, pondera quæ infra fulcimentum habent, vt lan<lb/>cea &longs;ari&longs;&longs;aue ad planum horizontis perpendiculariter e<lb/>recta, licet eo ca&longs;u grauitatis centrum in ip&longs;a perpendicu<lb/>lari con&longs;tituatur, non &longs;tet quidem, &longs;ed altrin&longs;ecus ca<lb/>dat? </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.041.jpg"/><figure id="id.007.00.041.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.041.1.jpg"></figure>

<s id="id.000366">Sit enim horizontis <lb/>planum AB, cui in puncto <lb/>C perpendiculariter ere<lb/>cta &longs;tatuatur &longs;ari&longs;&longs;a DC, <lb/>cuius grauitatis centrum <lb/>E, in ip&longs;a perpendiculari. <lb/></s>

<s id="id.000367">Stabit ergo, ex præmi&longs;&longs;is, <lb/>& certè &longs;tare debuit, &longs;ta<lb/>retqueue, ni vitium ob&longs;taret <lb/>materiæ; non &longs;tat autem, <lb/>quia difficillimum e&longs;t gra<lb/>uitatis centrum, &longs;uapte naturâ indiui&longs;ibile, ita ad amu&longs;&longs;im <lb/>&longs;i&longs;tere, vt in neutram partem à perpendiculari declinet. <lb/></s>

<s id="id.000368">Hæc igitur ex ijs &longs;peculationibus e&longs;t, quæ ad praxim, ma<lb/>teriæ vitio impediente, aut vix aut nun quam rediguntur. </s>

<s id="id.000369">Hinc autem ea quæ&longs;tio &longs;oluitur, Cur ij qui &longs;ari&longs;&longs;am <lb/>erectam digito &longs;ummo &longs;u&longs;tinere conantur, non &longs;tent qui<lb/>dem, &longs;ed digiti motu, &longs;ari&longs;&longs;æ motum &longs;equantur. </s>

<s id="id.000370">Id certè agit, qui nutantis &longs;ari&longs;&longs;æ, digito, motum &longs;e<lb/>quitur; vt in ip&longs;o motu digitum a&longs;&longs;iduè centro grauitatis <lb/>&longs;ari&longs;&longs;æ &longs;upponat, vnde &longs;it vt nun quam extra fulcimentum <lb/>permanens, nun quam cadat. </s>

<s id="id.000371">Similis huic alia quo que dubitatio &longs;oluitur: Nempe, <lb/>Cur turbines, quibus pueri ludunt, dum quidem rotan<lb/>tur, &longs;tent erecti, rotationevero ce&longs;&longs;ante, cadant. </s><figure id="id.007.00.041.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.041.2.jpg"></figure>

<s id="id.000372">E&longs;to enim Turbo AB, cu<lb/>ius grauitatis centrum C, planum <lb/>horizontis DE, linea Horizonti <lb/>perpendicularis ABC, tran&longs;iens <lb/>per centrum grauitatis C, &longs;it au<lb/>tem fulcimentum in B. <expan abbr="Itaq;">Itaque</expan> cum <lb/>centrum grauitatis C &longs;it in ip&longs;a <lb/>perpendiculari, &longs;tabit ex demon-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.042.jpg"/>&longs;tratis, at ex vitio materiæ non &longs;tabit. </s>

<s id="id.000373">Modò, vt a&longs;&longs;olet, ra<lb/>pido motu rotetur. </s>

<s id="id.000374">Dico, Turbinem, motu &longs;eu rotatione <lb/>durante &longs;tare. </s>

<s id="id.000375">ea autem paullatim elangue&longs;cente ín ca<lb/>&longs;um vergere; ce&longs;&longs;ante verò penitus cadere. </s>

<s id="id.000376">fit enim ex in<lb/>æqualitate materiæ, vel operis ruditate, vel aliâ quauis <lb/>ex cau&longs;&longs;a, grauitatis centrum non e&longs;&longs;e in C, &longs;ed exempli <lb/>gratiâ vbi F, notentur autem hinc inde Turbinis latera <lb/>notis GH. </s>

<s id="id.000377">Vtique cum F extra perpendicularem fuerit, <lb/>cadet Turbo ad partem G; at id ne &longs;iat, efficitur velocita<lb/>te motus, quo centrum F transfertur in contrariam par<lb/>tem, vbi I. non autem cadit ver&longs;us H, quoniam eadem ve<lb/>locitate iterum transfertur in F, quamobrem cum huius<lb/>cemodi centri a&longs;&longs;idua circa perpendicularem fiat trans<lb/>latio, ad nullam partem Turbo cadere pote&longs;t; elangue<lb/>&longs;cente verò motu rotans, paullatim in cipit inclinari, do<lb/>nec eo penitus ce&longs;&longs;ante, ad eam partem cadit, ad quam à <lb/>per pendiculari grauitatis centrum vergit. </s>

<s id="id.000378">De&longs;cribit au<lb/>tem in rotatione grauitatis centrum, quod in medio non <lb/>e&longs;t paruum circulum, per cuius centrum ip&longs;a perpendi<lb/>cula<gap/>is pertingit. </s>

<s id="id.000379">Modò redeuntes ad libram, cuius ful cimentum e&longs;t <lb/>&longs;ur&longs;um, alio principio, nempe Mechanico, cur depre&longs;&longs;a <lb/>ad æqualitatem reuertatur, demon&longs;trabimus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.043.jpg"/><figure id="id.007.00.043.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.043.1.jpg"></figure>

<s id="id.000380">Sit igitur, vt &longs;u<lb/>periùs, libra AB, cu<lb/>ius centrum grauita<lb/>tis C, fulcimentum, <lb/>verò &longs;ur&longs;um, in D li<lb/>bræ quidem in C per<lb/>pendiculariter con<lb/>iunctum. </s>

<s id="id.000381">Perpendi<lb/>cularis verò quæ per <lb/>fulcimentum, & gra<lb/>uitatis <expan abbr="c&emacr;trum">centrum</expan> tran&longs;<lb/>iens ad mundi cen<lb/>trum tendit DLE. &longs;tante igitur librâ in &longs;ua æqualitate, e<lb/>rit centrum grauitatis C in ip&longs;a perpendiculari infra qui<lb/>dem fulcimentum D. </s>

<s id="id.000382">Loco verò, mundi centro quàm <lb/>proximo. </s>

<s id="id.000383">Pondus po&longs;t hæc apponatur in B, Declinabit au<lb/>tem pars CB, in HF, eleuatâ interim parte AC, in GH. <lb/></s>

<s id="id.000384">Mota igitur libra tota, circa fulcimentum D mouebitur <lb/>circa idem centrum, & grauitatis centrum C, de&longs;cribens <lb/>portionem circuli CH, fi etque; C in H, & quoniam H, hoc <lb/>e&longs;t C, extra per pendicularem fit, amoto pondere, ex lan<lb/>ce B, cuius pre&longs;&longs;ione libra declinauerat, centrum grauita<lb/>tis per eandem circulì portionem HC, ad perpendicula<lb/>rem de&longs;cendet, donec iterum in ea quie&longs;cat, quo ca&longs;u li<lb/>bra AB ad æquilibrium reuertetur: quod fuerat demon<lb/>&longs;trandum. </s>

<s id="id.000385">His ita declaratis, o&longs;tendemus, (quod nullus ante <lb/>nos animaduertit) harum librarum, quæ fulcimentum <lb/>habent &longs;ur&longs;um, eam e&longs;&longs;e naturam, vt non à quouis ponde<lb/>re appo&longs;ito moueantur, vel penitus declinent. </s>

<s id="id.000386">Ij&longs;dem enim &longs;tantibus, addatur quoduis pondus lan<lb/>ci B; Itaque &longs;i tale fuerit quod &longs;uperet re&longs;i&longs;tentiam, quam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.044.jpg"/>illi facit centrum grauitatis contra naturam elatum in H <lb/>mouebitur quædam libra. </s>

<s id="id.000387">Sin autem tam parui momenti <lb/>&longs;it, vt eam re&longs;i&longs;tentiam non vincat, &longs;tante circa locum in<lb/>fimum centro C, non mouebitur aut &longs;altem parum, ip&longs;a <lb/>libra. </s>

<s id="id.000388">Hinc colligimus &longs;ieri po&longs;&longs;e, libras illas, quæ non <gap/><lb/>quouis, quantumuis paruo pondere declinant, cas fulci-<gap/><lb/>mentum habere &longs;ur&longs;um. </s>

<s id="id.000389">His ad dimus, cæteris paribus, re&longs;i&longs;tentiam eò e&longs;&longs;e <lb/>maiorem, quo minus grauitatis centrum di&longs;tat à fulci<lb/>mento &longs;ur&longs;um, circa quod ip&longs;a libra aduertitur. </s><figure id="id.007.00.044.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.044.1.jpg"></figure>

<s id="id.000390">E&longs;to libra AB, cuius gra<lb/>uitatis centrum C, & primò <lb/>quidem eius ful cimentum <lb/>&longs;ur&longs;um &longs;it vbi D, itaque &longs;i ap<lb/>po&longs;ito pondere de clinauerit <lb/>libra ad partes B, punctum <lb/>C, dum a&longs;cendet de&longs;cribet <lb/>portionem circuli CE. fulciatur iterum &longs;ur&longs;um puncto F, <lb/>& iterum declinet ad partes B, & iterum punctum C, dum <lb/>a&longs;cendet, circuli portionem de&longs;cribet CG. </s>

<s id="id.000391">E&longs;t autem <lb/>minor angulus contactus ACE, angulo ACG, magis er<lb/>go &longs;ur&longs;um, hoc e&longs;t, ad naturam &longs;ui feretur C, per CG, ex <lb/>centro F, quàm per CE, ex centro D, quod fuerat de<lb/>mon&longs;trandum. </s>

<s id="id.000392">Hæc autem re&longs;i&longs;tentia ex eodem fulcimento & eo<lb/>dem pondere eo faciliùs &longs;uperabitur, quo longius bra<lb/>chium libræ fuerit. </s>

<s id="id.000393">E&longs;to enim iterum libra AB, cuius fulcimentum D, <lb/>centrum grauitatis C, &longs;it & alia libra, cuius brachia bre<lb/>uiora EF, idem habens centrum C, & eidem puncto &longs;u<lb/>&longs;pen&longs;a D. </s>

<s id="id.000394">Dico igitur, eodem pondere appo&longs;ito, faciliùs <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.045.jpg"/><figure id="id.007.00.045.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.045.1.jpg"></figure><lb/>declinaturam libram ad <lb/>partes B, quàm &longs;i idem ap<lb/>poneretur in F. </s>

<s id="id.000395">Demit<lb/>tatur enim, à puncto B <lb/>horizonti perpendicula<lb/>ris BG, & ab F item per<lb/>pendicularis FH, Tum <lb/>iuncta DB, centro D, eo<lb/>dem vero &longs;patio DB, circuli portio de&longs;cribatur BI, item <lb/>iuncta DF eodem centro D, &longs;patio DF, portio circuli de<lb/>&longs;cribatu: FK. e&longs;t autem maior DB ip&longs;a DF ex propo&longs;. <lb/></s>

<s id="id.000396">21. lib.

1. Elem.

quare maioris circuli portio e&longs;t BI quàm <lb/>FK. </s>

<s id="id.000397">Obliquior autem, hoc e&longs;t, à perpendiculariremotior <lb/>e&longs;t motus per FK quàm per BI. maior &longs;i quidem e&longs;t angu<lb/>lus KFH angulo IBG. quod nos ita probamus. </s>

<s id="id.000398">Ducatur <lb/>perpendicularis ip&longs;i DF linea LF contingens circulum <lb/>FK in F, item ip&longs;i DB, perpendicularis MB, contingens <lb/>circulum BI in B, & quia angulus contingentiæ maioris <lb/>circuli minor e&longs;t angulo contingentiæ minoris, erit KFL <lb/>maior IBM, Recti autem &longs;unt DFL, DBM, minor ergo <lb/>DFK re&longs;idua ip&longs;o DBI re&longs;iduo. </s>

<s id="id.000399">Maior autem DFC ex <lb/>iam citata propo&longs;. </s>

<s id="id.000400"><expan abbr="quã">quam</expan> DBC, erit igitur re&longs;iduum CFK, <lb/>multo minus re&longs;iduo FBI, &longs;ed recti &longs;unt CFH, FBG, ex <lb/>quibus &longs;i detrahantur CFK, FBI, erit re&longs;iduum KFH, <lb/>maius re&longs;iduo IBG, plus ergo retra hitur à perpendicula<lb/>ri po<gap/>dus de&longs;cendens per FK quàm per BI, minus igitur <lb/>præ<gap/>alebit re&longs;i&longs;tentiæ in C pondus appen&longs;um in F, quàm <lb/>&longs;i appendatur in B. quod fuerat demon&longs;trandum. </s>

<s id="id.000401">Po&longs;&longs;<gap/>mus & idem quoque aliter o&longs;tendere. </s>

<s id="id.000402">Sint enim &longs;eor&longs;um duæ libræ, maior AB, mïnor EF, <lb/>quàm commune grauitatis centrum C, fulcimentum ve<lb/>rò &longs;ur&longs;um D. </s>

<s id="id.000403">Producatur perpendicularis DC, in G & fiat <lb/>CG æqualis CB, CH verò æqualis CF. </s>

<s id="id.000404">Sunt igitur duo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.046.jpg"/><figure id="id.007.00.046.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.046.1.jpg"></figure><lb/>vectes DG, DH, quo<lb/>rum quidem commu<lb/>ne fulcimentum D, <lb/>pondus verò C, poten<lb/>tiæ vbi HG. </s>

<s id="id.000405">Sunt au<lb/>tem hi vectes cius na<lb/>turæ, in quibus <expan abbr="p&omacr;dus">pondus</expan> <lb/>e&longs;t inter fulcimentum <lb/>& potentiam, itaque <lb/>vt &longs;e habet DC, ad <lb/>DG, ita potentia in G <lb/>ad pondus in C, item vt DC ad DH ita potentia in H ad <lb/>idem pondus C, &longs;ed minor e&longs;t propo&longs;itio DC, ad DG <lb/>quàm DC ad DH. minor ergo potentia requiritur in G, <lb/>hoc e&longs;t, in B, quàm in H, hoc e&longs;t in F. </s>

<s id="id.000406">Data igitur ponderis <lb/>æqualitate faciliùs &longs;uperabitur re&longs;i&longs;tentia C in B, quàm <lb/>in F: quod o&longs;tendendum fuerat. </s>

<s id="id.000407">Ad huius libræ naturam illæ quoque rediguntur, <lb/>quarum iugum non rectum quidem, &longs;ed curuum, vel ex <lb/>rectis &longs;ur&longs;um in angulum ad fulcimentum detinentibus, <lb/>nec refert vtrum curuitas &longs;it circuli portio quælibet, aut <lb/>ellip&longs;is &longs;ecundum alterum diametrorum; quod ita de<lb/>mon&longs;tramus. </s><figure id="id.007.00.046.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.046.2.jpg"></figure>

<s id="id.000408">E&longs;to libra, cuius iugum <lb/>curuum <expan abbr="angulat&umacr;ue">angulatumue</expan> ABC, <lb/>cuius fulcimentum B, æqua<lb/>lia autem brachia AB, BC, <lb/>& pondera item <expan abbr="vtrinq;">vtrinque</expan> ap<lb/>pen&longs;a æqualia. </s>

<s id="id.000409">Demittatur <lb/>ex puncto B ad mundi cen<lb/>trum perpendicularis BD. <lb/></s>

<s id="id.000410">Stante igitur libra ABC in <lb/>æquilibrio, erit eius graui<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.047.jpg"/>tatis centrum in ip&longs;a perpendiculari BD, puta in E. </s>

<s id="id.000411">Ap<lb/>ponatur pondus in C, declinabit autem libra, &longs;it autem <lb/>iuxta po&longs;itionem FBG. </s>

<s id="id.000412">Centrum igitur grauitatis E per <lb/>portionem EH, erit in H. </s>

<s id="id.000413">A&longs;cendit ergo centrum graui<lb/>tatis in H, hoc e&longs;t, &longs;ur&longs;um, id e&longs;t, contra cius naturam; a<lb/>moto igitur pondere ex C, grauitatis centrum extra per<lb/>pendicularem con&longs;titutum rur&longs;us de&longs;cendet, & iterum <lb/>libra ABC ad æquilibrium reuertetur. </s>

<s id="id.000414">Hoc idem egre<lb/>giè o&longs;tendit G. Vbald. in tractatu de libra, propo&longs;. </s>

<s id="id.000416">Hinc ratio pendet earum imaguncularum, quas ex <lb/>contu&longs;a papyro ligneaue leui materia compingunt, per<lb/>queue manus earum ambas, ferreum filum trajicientes, v<lb/>trinque plumbea appendunt pondera æqualia, ea <expan abbr="quid&emacr;">quidem</expan> <lb/>lege, vt centrum grauitatis infra pedes imaguncula &longs;ta<lb/>tuatur. </s>

<s id="id.000417">Tunc enim exten &longs;o filo imponentes ceu funam<lb/>bulos per illud, vltrò citroque; decurrere faciunt, imagun<lb/>cula interim erecta & in neutram partem cadente, quod <lb/>vt figurâ clarius fiat; </s><figure id="id.007.00.047.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.047.1.jpg"></figure>

<s id="id.000418">E&longs;to imaguncu<lb/>la AB, per cuius ma<lb/>nus traij ciatur filum <lb/>ferreum curuum <expan abbr="c&umacr;">cum</expan> <lb/>æ qualibus ponderi<lb/>bus hinc inde <expan abbr="app&emacr;-&longs;is">appen<lb/>&longs;is</expan> CD. </s>

<s id="id.000419">Nitatur au<lb/>tem pedibus filo HI <lb/>in <emph type="italics"/>B<emph.end type="italics"/>, &longs;itque; totìus ma<lb/>chinæ grauitatis <expan abbr="c&emacr;-trum">cen<lb/>trum</expan> E, &longs;itque <expan abbr="per-p&emacr;dicularis">per<lb/>pendicularis</expan> per gra<lb/>uitatis <expan abbr="centr&umacr;">centrum</expan> tran&longs;i<lb/>ens A<emph type="italics"/>B<emph.end type="italics"/> E. </s>

<s id="id.000420">Itaque in<lb/>clinata imaguncula, & conuer&longs;a circa punctum <emph type="italics"/>B<emph.end type="italics"/>, &longs;i de-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.048.jpg"/>clinet ad partes I, centrum grauitatis eleuabitur in F. </s>

<s id="id.000421">Si <lb/>verò ad partes H eleuabitur in G. quare cum FG loca <lb/>&longs;intremotiora à mundi centro, quàm &longs;it E, non &longs;tabit gra<lb/>uitatis centrum in punctis FG, &longs;ed ad infimum locum re<lb/>uertecur, hoc e&longs;t, in ip&longs;a perpendiculari in E, & imagun<lb/>cula ad perpendiculum ip&longs;i H<emph type="italics"/>B<emph.end type="italics"/>E filo, hoc e&longs;t, ip&longs;i hori<lb/>zonti reuertetur. </s>

<s id="id.000422">Hinc etiam Arictum, T e&longs;tudinun<gap/> queue demolito<lb/>riatum Machinarum vis pendet, nempe ex ratione libra<lb/>rum, quæ fulcimentum habent &longs;ur&longs;um. </s><figure id="id.007.00.048.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.048.1.jpg"></figure>

<s id="id.000423">E&longs;to enim Aries A<emph type="italics"/>B<emph.end type="italics"/> <lb/>funi appen&longs;us CD, cu<lb/>ius grauitatis centrum, <lb/>D, perpendicularis verò <lb/>quæ ad mundi centrum <lb/>ip&longs;a CDE. </s>

<s id="id.000424">Stante igitur <lb/>in æquilibrio machina, <lb/>centrum grauitatis erit <lb/>in ip&longs;a perpendiculari. <lb/></s>

<s id="id.000425">Applicetur alicubi po<lb/>tentia retropellens, eleuabitur igitur centrum grauitatis <lb/>per circuli portionem DF, cuius &longs;emidiameter e&longs;t CD, <lb/>&longs;i etqueue iuxta po&longs;itionem CF. </s>

<s id="id.000426">Aries verò in GFH. </s>

<s id="id.000427">Di<lb/>mi&longs;&longs;a itaque Machina centrum F vtpote graue, non &longs;tabit, <lb/>&longs;ed &longs;uapte naturâ reuertetur in D. </s>

<s id="id.000428">Quadruplici autem <lb/>de cau&longs;&longs;a motus Arietis violenti&longs;&longs;imus e&longs;t ex vi naturalis <lb/>ponderis, quo deor&longs;um fertur, tum velocitate naturalis <lb/>motus in de&longs;cendendo auctæ, tum ex vi pote<gap/> tiæ impel<lb/>lentis, & naturalem motum adiuuantis, tum ex velocita<lb/>te ex motu violento deor&longs;um & antror&longs;um impellente <lb/>acqui&longs;itâ. </s>

<s id="id.000429">Id etiam addimus, eo validiores fore ictus, quò <lb/>grauior fuerit Machina, & maius &longs;patium, quo retrotra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.049.jpg"/>hitur, grauitate ip&longs;a & &longs;patio tum virium vnione opera<gap/><lb/>tionem mirum in modum adiuuantibus. </s>

<s id="id.000430">Hæc nos de Libra &longs;ur&longs;um fulcimentum habente, dí<lb/>cta voluimus, nunc de ea, cuius fulcimentum deor&longs;um, <lb/>e&longs;t, verba faciemus. </s>

<s id="id.000431">Altera quæ&longs;tionis pars:</s>

<s id="id.000432"><emph type="italics"/>De Libra cuius fulcimentum deor&longs;um e&longs;t.<emph.end type="italics"/></s>

<s id="id.000433">Si deor&longs;um fuerit, inquit Ari&longs;toteles, id quod &longs;ub<lb/>&longs;tat, contrarium facit illi quæ &longs;ur&longs;um habet, nempe ad æ<lb/>quilibrium non reuertitur. </s>

<s id="id.000434">Plus enim, ait, dimidio fit li<lb/>bræ, quæ deor&longs;um e&longs;t pars, quàm quod perpendiculum <lb/>&longs;ecet, quapropter non a&longs;cendit. </s>

<s id="id.000435">eleuata enim pars leuior <lb/>e&longs;t. </s>

<s id="id.000436">Hæc ille, qui &longs;chemate quo que rem aperit, at eo a<lb/>pud interpretes, & Picolomineum Paraphra&longs;tem, ita <expan abbr="m&emacr;-dosè">men<lb/>dosè</expan> lineato, vt inde ob&longs;curitas lucis loco, legentibus of<lb/>fundatur. </s>

<s id="id.000437">Nos, quod & &longs;uprà quo que fecimus, no&longs;tra fi<lb/>gurâ, &longs;ole ip&longs;o clariorem, ex Ari&longs;to telis ip&longs;ius mente rem <lb/>totam efficiemus. </s><figure id="id.007.00.049.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.049.1.jpg"></figure>

<s id="id.000438">Sit libra recta, (hoc <lb/>e&longs;t, in æquilibrio con<lb/>&longs;tituta) vbi NG. </s>

<s id="id.000439">Per<lb/>pendiculum autem (id <lb/>e&longs;t, perpendicularis <lb/>quæ ad mundi <expan abbr="centr&umacr;">centrum</expan>) <lb/>KLM. </s>

<s id="id.000440">Bifariam igitur <lb/>&longs;ecatur NG. impo&longs;ito <lb/>po&longs;thæc onere in ip&longs;o <lb/>N, erit quidem N, vbi <lb/>O. ip&longs;um autem G vbi <lb/>R. KL autem vbi LP. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.050.jpg"/>quare maius e&longs;t KO, quam LR, ip&longs;a parte PKL. </s>

<s id="id.000441">Amoto <lb/>igitur onere nece&longs;&longs;e e&longs;t manere. </s>

<s id="id.000442">Incumbit enim onus ex<lb/>ce&longs;&longs;us medietatis eius, vbi e&longs;t F. </s>

<s id="id.000443">Sen&longs;us e&longs;t igitur, idcirco <lb/>partem iugi KLO inclinatam, ad æquilibrium non re<lb/>uerti, propterea quòd maior &longs;it ip&longs;a KLO pars quæ tra<lb/>hit, ip&longs;a<gap/> RKL, quæ trahitur & eleuatur. </s><figure id="id.007.00.050.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.050.1.jpg"></figure>

<s id="id.000444">Pote&longs;t hoc idem longè <lb/>&longs;impliciori themate demon<lb/>&longs;trari. </s>

<s id="id.000445">E&longs;to enim libra AB, <lb/>cuius centrum C, fulcimen<lb/>tum vero deor&longs;um D, Per<lb/>pendicularis per centrum & <lb/>fulcimentum tran&longs;iens EF. <lb/></s>

<s id="id.000446">Apponatur pondus in B, de<lb/>clinabitque; puta ad GH, cen<lb/>trum verò C, ex &longs;tabili fulci<lb/>mento D, circuli portionem de&longs;cribet CI, libra autem <lb/>&longs;ecabit EF perpendicularem in K. Æquales autem &longs;unt <lb/>IG, IH, at ex parte HI de&longs;umpta e&longs;t KI, addita queue ip&longs;i <lb/>IG, maior e&longs;t ergo tota KG, torâ KH. </s>

<s id="id.000447">Non igitur KH <lb/>habet KG, &longs;ed libra, ni&longs;i impedita fuerit, cum centro C <lb/>de&longs;cendente per <gap/>in M, ad ip&longs;am perpendicularem dela<lb/>ta, ad in feriorem partem, mutatis vicibus quie&longs;cet, facto <lb/>nempe fulcimento &longs;ur&longs;um, fietque; horizonti æque di&longs;tans <lb/>iuxta po&longs;itionem LMN. </s>

<s id="id.000448">Demon&longs;tratio <expan abbr="quid&emacr;">quidem</expan> e&longs;t hæc, &longs;ed non ex proprijs prin<lb/>cipijs Mechanicis, <expan abbr="n&emacr;pe">nempe</expan> ex ratione <expan abbr="c&emacr;t">cent</expan><gap/>i grauitatis petitâ. <lb/></s>

<s id="id.000449">Ii&longs;dem enim &longs;tantibus, <expan abbr="c&umacr;">cum</expan> centrum grauitatis C fiat extra <lb/>perpendicularem, de&longs;cendens ad I, nun quam reuert<gap/> tur <lb/>in C, a&longs;cen deret enim; &longs;ed &longs;i liberè circa centrum D con<lb/>uerteretur, de&longs;cendens vt dictum e&longs;t per circulum CIM <lb/>pondus B, fieret in L, A vero in N adepta po&longs;itione <lb/>LMN. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.051.jpg"/>

<s id="id.000450">Cur autem huius libræ, quæ aliàs inutilis e&longs;t, memi<lb/>nerit Philo&longs;ophus, ea videtur cau&longs;&longs;a, quòd inde vectis vir<lb/>tutem eliciat, vt &longs;uo loco videbimus. </s>

<s id="id.000451">Id autem valde mi<lb/>rum, hominem acuti&longs;&longs;imum nihil pror&longs;us de ea libra egi&longs;<lb/>&longs;e, quæ fulcimentum nec &longs;ur&longs;um habet, nec deor&longs;um, &longs;ed <lb/>in ip&longs;o exqui&longs;itè medio, ita vt centrum grauitatis in ip&longs;o<lb/>met fulcimento con&longs;i&longs;tat. </s>

<s id="id.000452">Nos igitur de hac quod operæ <lb/>pretium fuerit, & ad rem, qua de agimus, vtile, in medium <lb/>proferemus. </s>

<s id="id.000453"><emph type="italics"/>De libra cuius fulcimentum est in medio.<emph.end type="italics"/></s>

<s id="id.000454">Dicimus itaque, libram, cuius fulcimentum nec &longs;ur<lb/>&longs;um e&longs;t, nec deor&longs;um, &longs;ed pror&longs;us in medio, nempe in ip&longs;o <lb/>grauitatis centro, vbi brachia & pondera vtrinque appo<lb/>&longs;ita fuerint æqualia, &longs;i ab æquilibrio mouentur, quomo<lb/>docunque po&longs;ita, &longs;tare nec ab eo, quem adepta e&longs;t, &longs;itu di<lb/>moueri. </s>

<s id="id.000455">Quæ&longs;tionem hanc perperam tractârunt recentio<lb/>res quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij <lb/>nonnulli, qui Iordani Nemoracij a&longs;&longs;ertiones &longs;unt &longs;ecuti, <lb/>quorum demon&longs;trationes vel paralogi&longs;mos potiùs egre<lb/>giè confutauit in libr. </s>

<s id="id.000456">Mechanicor. Tractatu de libra pro<lb/>po&longs;. </s>

<s id="id.000457">4. Guid. Vbald. ad cuius probati&longs;&longs;ima &longs;cripta Lecto<lb/>rem ablegamus. </s>

<s id="id.000458">fu&longs;i&longs;&longs;imè enim ibi hac de re & ab&longs;oluti&longs;&longs;i<lb/>mè agit. </s>

<s id="id.000459">Nos autem quidem paucis ea, quæ ad hanc co<lb/>gnitionem pertinent, explicabimus. </s><figure id="id.007.00.051.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.051.1.jpg"></figure>

<s id="id.000460">E&longs;to enim libra A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>cuius brachia æqualia, <lb/>& centrum grauitatis <lb/>in C, brachijs verò <lb/>AC, C<emph type="italics"/>B<emph.end type="italics"/> æqualibus, æ<lb/>qualia pondera hinc <lb/>inde <expan abbr="apponãtur">apponantur</expan>. </s>

<s id="id.000461">Tum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.052.jpg"/>fulcimento in medio, hoc e&longs;t, vbi grauitatis centrum C <lb/>applicato per centrum ip&longs;um C ducatur perpendicularis, <lb/>quæ ad mundi centrum, DCE, &longs;itque primum libra æ<lb/>quedi&longs;tans horizonti, con&longs;tituta. </s>

<s id="id.000462">Tum ex altera parte <lb/>pre&longs;&longs;a moueatur & fiat iuxta po&longs;itionem FCG. </s>

<s id="id.000463">Dico eam <lb/>dimi&longs;&longs;am permanere, etenim cum grauitatis centrum &longs;it <lb/>in ip&longs;a perpendiculari, in neutram partem verget, &longs;ed nec <lb/>vergere pote&longs;t, quippe quod non circa fulcimentum ceu <lb/>centrum motus, moueatur grauitatis centrum, &longs;ed in ip&longs;o <lb/>&longs;it ful cimento; &longs;itum ergo non mutat. </s>

<s id="id.000464">Præterea cum per<lb/>pendicularis DCE per grauitatis centrum ducatur, cor<lb/>pus ip&longs;um ex ponderibus & libra con&longs;tans ab ea in partes <lb/>çque ponderantes &longs;ecatur, & ideo ex centri grauitatis dif<lb/>finitione, quam protulit Pappus, corpus ip&longs;um centro <lb/>grauitatis appen&longs;um, dum fertur quie&longs;cit, & &longs;eruat eam, <lb/>quam à principio habuit po&longs;ition<gap/>. </s>

<s id="id.000465">Et &longs;anè &longs;i partes quo<lb/>modo libet librâ per grauitatis centrum diuisâ, &longs;untæ<lb/>queponderantes nec trahent inuicem, nec trahentur, &longs;ta<lb/>bit ergo libra, & quam adepta fuerat po&longs;itionem, eam &longs;er<lb/>uabit. </s>

<s id="id.000466">Id tamen non negamus, difficile e&longs;&longs;e libras eiu&longs;ce<lb/>modi ex materia fabricare, quippe quod non omnia quæ <lb/>vera &longs;unt, & euidenti&longs;&longs;imis demon&longs;trationibus patent, <lb/>commodè ad praxim, ex artis & materiæ imperfectione, <lb/>reducuntur. </s>

<s id="id.000467">Cæterùm harum librarum ea e&longs;t virtus, vt vel mini<lb/>mo pondere altrin&longs;ecus appo&longs;ito, declinet; quod illis quæ <lb/>centrum &longs;u &longs;um habent, non euenire, demon&longs;trauimus. </s>

<s id="id.000468">Circa hæc po&longs;&longs;et cuipiam oriri Dubium, num chor<lb/>dulæ, quibus lances appenduntur, variationem aliquam <lb/>circa ea quæ demon&longs;trata &longs;unt, inducere valeant. </s>

<s id="id.000469">Dicimus nullam inde fieri: E&longs;to enim libra AB, cu<lb/>ius centrum & fulcimentum C, ab cuius extremitate A <lb/>dependeat, funiculus AD, ab alia verò <emph type="italics"/>B<emph.end type="italics"/>, funiculus <emph type="italics"/>B<emph.end type="italics"/>E, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.053.jpg"/><figure id="id.007.00.053.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.053.1.jpg"></figure><lb/>quibus appen&longs;æ &longs;int æ<lb/>qualis ponderis lances <lb/>DE. </s>

<s id="id.000470">Moueatur libra, <lb/>fiatque in ICH, funi<lb/>culi verò in lancibus in <lb/>IK, HL. &longs;ecet autem fu<lb/>niculus IK libram A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>in M, LH verò produ<lb/>catur & eandem &longs;ecer <lb/>in N. quoniam igitur <lb/>IC, æqualis e&longs;t CH, pa<lb/>rallelæ autem KI, LN æquales <expan abbr="er&umacr;t">erunt</expan> alterni anguli MIC, <lb/>NHC, &longs;ed & anguli ad verticem ICH, BCH æquales <lb/>&longs;unt, quare triangulum IMC, æquale triangulo HNC, <lb/>& latera lateribus, quæ æqualibus angulis &longs;ubtenduntur. <lb/></s>

<s id="id.000471">Æqualis e&longs;t igitur linea MC lineæ NC. </s>

<s id="id.000472">Itaque &longs;i ponde<lb/>ra lancesue, KL mente concipiantur appen&longs;æ in punctis <lb/>MN, ex brachiorum & ponderum æqualitate æquepon<lb/>derabunt. </s>

<s id="id.000473">quod fuerat demon&longs;trandum. </s>

</subchap1>

<s id="id.000474">QVÆSTIO III.</s>

<s id="id.000475"><emph type="italics"/>Cur exiguæ vires (quod etiam à principio dixerat) vecte magna <lb/>mouent pondera, vectes in&longs;uper onus accipientes, cum facilius <lb/>&longs;it, minorem mouere grauitatem, minor est au<lb/>tem &longs;ine vecte?<emph.end type="italics"/></s>

<s id="id.000476">Ari&longs;toteles ita quæ&longs;tionem proponit, vt eam Rheto<lb/>rico quodam fuco admirabiliorem f<gap/>ciat. </s>

<s id="id.000477">Soluit au<lb/>tem hoc pacto, <expan abbr="inqui&emacr;s">inquiens</expan>, fieri po&longs;&longs;e eam e&longs;&longs;e cau&longs;&longs;am, quod <lb/>vectis &longs;it libra, eius nempe generis quod fulcimentum ha<lb/>bet deor&longs;um, atque id circo in ip&longs;a pre&longs;&longs;ione in partes in<lb/>æquales vectem diuidi. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.054.jpg"/><figure id="id.007.00.054.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.054.1.jpg"></figure>

<s id="id.000478">Figura quam ex<lb/>hibet, vix ferè quid &longs;i<lb/>bi velit explicat. </s>

<s id="id.000479">Nos <lb/>ad eius <expan abbr="m&emacr;tem">mentem</expan> aliam <lb/>proponemus <expan abbr="eamq;">eamque</expan> <lb/>longè clariorem. </s>

<s id="id.000480">E&longs;to vectis A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>cuius fulcimentum, <lb/>deor&longs;um in C, pon<lb/>dus D, potentia ex vecte, pondus &longs;u&longs;tinens E. </s>

<s id="id.000481">Perpendi<lb/>cularis per fulcimentum FCG. </s>

<s id="id.000482">Itaque quoniam poten<lb/>tia in E non &longs;uperat pondus D, nec ab eo &longs;uperatur, &longs;tat <lb/>vectis cum potentia Horizonti æquidi&longs;tans, hoc e&longs;t, in æ<lb/>quilibrio, vectis autem in puncto C diuiditur in partes æ<lb/>queponderantes. </s>

<s id="id.000483">Modo præualeat potentia ponderi, & <lb/>vectem deprimat, fiat autem in LCH, erit igitur <emph type="italics"/>B<emph.end type="italics"/>, in L, <lb/>A in H, D in K, & CF, quæ vectem in partes æque ponde<lb/>rantes diuidebat, in CI. </s>

<s id="id.000484">Iam igitur non æqueponderant <lb/>partes, &longs;i quidem pars vectis FCI, aufertur parti HCI, & <lb/>adiungitur parti ICL, quæ ideo &longs;it pondero&longs;ior, vnde & <lb/>potentia ad ponderis eleuationem adiuuatur. </s>

<s id="id.000485">Eadem i<lb/>gitur vtitur hic demon&longs;tratione, quam in explicando ef<lb/>fectu libræ, cuius fulcimentum deor&longs;um e&longs;t, adhibuerat. <lb/></s>

<s id="id.000486">Nec alia de cau&longs;&longs;a, vt &longs;uprà notauimus, videtur eius libræ <lb/>in &longs;uperiori quæ&longs;tione, con&longs;iderationem introduxi&longs;&longs;e. </s>

<s id="id.000487">Et <lb/>&longs;anè verum e&longs;t quod concludit, Veruntamen minimi e&longs;t <lb/>momenti ad tantam vim parua illa adiectio, quæ parti ve<lb/>ctis depre&longs;&longs;æ in ip&longs;a depre&longs;&longs;ione adiungitur. </s>

<s id="id.000488">Aliunde igi<lb/>tur tantæ rei cau&longs;&longs;a e&longs;t petenda, quod & nos deinceps fa<lb/>ciemus. </s>

<s id="id.000489">Videtur autem ip&longs;e quoque Ari&longs;toteles non &longs;ibi <lb/>pror&longs;us in a&longs;&longs;ignata ratione &longs;atis feci&longs;&longs;e, & ideo &longs;ubiungit: <lb/>quoniam ab æquali pondere celerius mouetur maior ca<lb/>rum quæ à centro &longs;unt duo verò pondera; quod mouet & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.055.jpg"/>quod mouetur, quod igitur motum pondus ad mouens <lb/>longitudo patitur ad longitudinem, &longs;emper autem <expan abbr="quã-tum">quan<lb/>tum</expan> ab hypomochlio (id e&longs;t, fulcimento) di&longs;tabit magis, <lb/>tanto facilius mouebit. </s>

<s id="id.000490">Cau&longs;&longs;a autem <gap/>, quæ retro com<lb/>memorata e&longs;t, quoniam quæ plus à centro di&longs;tat <expan abbr="maior&emacr;">maiorem</expan> <lb/>de&longs;cribit circulum. </s>

<s id="id.000491">quare ab eadem potentia plus &longs;upera<lb/>biturid quod mouetur, quæ plus à fulcimento di&longs;&longs;at. </s>

<s id="id.000492">H&ucedil;c <lb/>ille, qui a&longs;&longs;erit duo pondera in vecte con&longs;iderari, Pondus <lb/>nempe motum, & mouentem Potentiam (hanc enim <expan abbr="p&omacr;-deris">pon<lb/>deris</expan> habere vim <expan abbr="atq;">atque</expan> rationem certum e&longs;t) Vires autem <lb/>potentiam acquirere ex brachij longitudine, & ex inde <lb/>con&longs;equenti velocitate, quo enim brachia longiora, eo <lb/>in extremitate velociora, atque idcirco ita &longs;e habere mo<lb/>tum pondus ad potentiam mouentem, vt brachij longi<lb/>tudo ad brachij longitudinem: brachia autem vocamus, <lb/>partes illas vectis, quæ à fulcimento ad vtranque vectis <lb/>extremitatem pertingunt, & ideo quantum à fulcimento <lb/>potentia di&longs;tabit magis, eo faciliùs pondus mouebit. </s>

<s id="id.000493">Vera vtique & explorati&longs;&longs;ima hæc a&longs;&longs;ertio e&longs;t. </s>

<s id="id.000494">Ve<lb/>runtamen, cau&longs;&longs;am huiu&longs;ce mirabilis effectus, e&longs;&longs;e velo<lb/>citatem, quæ brachij longitudinem con&longs;equitur, non af<lb/>firmamus. </s>

<s id="id.000495">quæ enim velocitas in re &longs;tante? </s>

<s id="id.000496">Stant autem <lb/>vectis, & libra dum manent in æquilibrio, & nihilo &longs;ecius <lb/>parua potentia ingens &longs;u&longs;tinet pondus. </s>

<s id="id.000497">Dicet ad hæc qui&longs;piam, velocitatem in longiori bra<lb/>chio &longs;i non actu, &longs;altem potentiâ e&longs;&longs;e maiorem. </s>

<s id="id.000498">At quæ&longs;o <lb/>quid in re quæ e&longs;t actu, momenti habet potentia? </s>

<s id="id.000499">actu e<lb/>nim &longs;u&longs;tinet, &longs;u&longs;tinens. </s>

<s id="id.000500">Con&longs;equìtur, (id vtique fatemur) <lb/>nece&longs;&longs;ariò velocitas maior motu brachij maioris; non ta<lb/>men cau&longs;&longs;a e&longs;t cur vis loco vbi velocitas maior &longs;it, appo&longs;i<lb/>ta magis moueat. </s>

<s id="id.000501">Sanè ex velocitate, dum mouentur, <expan abbr="p&omacr;-dus">pon<lb/>dus</expan> acquirere corpora, tum proiecta, tum cadentia cer<lb/>tum e&longs;t, quod etiam in quæ&longs;tione 19. cum Philo&longs;opho <expan abbr="c&omacr;-">con-</expan><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.056.jpg"/>&longs;i derabimus. </s>

<s id="id.000502">Sed hoc ex velocitate & motu &longs;it, quæ &longs;unt <lb/>actu. </s>

<s id="id.000503">At brachia in ip&longs;o æquilibrio &longs;u&longs;tinent actu quidem, <lb/>&longs;ed non mouentur. </s>

<s id="id.000504">Cæterum videtur A riftoteles id &longs;ub<lb/>odora&longs;&longs;e, quod po&longs;tea Archimedes, Mechanicorum prin<lb/>ceps, in propo&longs;. </s>

<s id="id.000505">6. primi Æqueponderantium explicitè <lb/>protulit & probauit: nempe in æquilibrio ita e&longs;&longs;e pondus <lb/>ad pondus, vt brachium ad brachium, ratione permutata. </s><figure id="id.007.00.056.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.056.1.jpg"></figure>

<s id="id.000506">E&longs;to enim vectis <lb/>AB, quomodolibet <lb/>fulcimento diui&longs;us in <lb/>C. <expan abbr="app&emacr;datur">appendatur</expan> autem <lb/>in A, pondus D, in B <lb/>verò pondus E, ita &longs;e <lb/>habens ad pondus D, vtip&longs;a AC ad CB. </s>

<s id="id.000507">Stabit igitur ve<lb/>ctis, & neutram in partem verget, erit enim centrum gra<lb/>uitatis in C, diui&longs;o nempe ibi vecte in partes æque ponde<lb/>rantes. </s>

<s id="id.000508">Hoc po&longs;t Archimedem, & in&longs;ignes illos veteres <lb/>Mechanicos præclari&longs;&longs;imè demon&longs;trauit G. Vbaldus in <lb/>Mechanicis, Tractatu de Libra propo&longs;. </s>

<s id="id.000509">6. nec non de Ve<lb/>cte propo&longs;. </s>

<s id="id.000511">Cæterùm vt aliquid interim, quod no&longs;trum &longs;it, affe<lb/>ramus, liceat nobis egregios illos viros interrogare, quæ<lb/>nam mirabilis eius effectionis &longs;it cau&longs;&longs;a? </s>

<s id="id.000512">Dicent permu<lb/>tatam proportionem. </s>

<s id="id.000513">Teneo, at nondum acquie&longs;co: pe<lb/>tam enim, Cur ea rationis permutatio mirabilem illum <lb/>effectum pariat. </s>

<s id="id.000514">Hoc quod illi non do cent, puto nos, i<lb/>gnorantiæ &longs;omno &longs;epultos, &longs;omnia&longs;&longs;e. </s><figure id="id.007.00.056.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.056.2.jpg"></figure>

<s id="id.000515">Æqualitatem &longs;tatus <lb/>e&longs;&longs;e cau&longs;&longs;am, nemo, vt <lb/>puto, inficiabitur. </s>

<s id="id.000516">res e&longs;t <lb/>enim per &longs;e clara. </s>

<s id="id.000517">E&longs;to &longs;i<lb/>quidem linea quæpiam AB, applicetur extremitati A po<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.057.jpg"/>tentia quæ dam quæ lineam ad &longs;e trahat ad partes nempe <lb/>A, Tum in B quædam alia potentia ip&longs;i quæ in A potentiae, <lb/>æqualis, quæ lin eam trahat &longs;imili modo ad partes B. </s>

<s id="id.000518">Datâ <lb/>igitur harum potentiarum æqualitate, linea AB, nec ad <lb/>partes A, nec ad partes B transferetur, &longs;ed pror&longs;us immo<lb/>bilis &longs;tabit. </s>

<s id="id.000519">His ita con&longs;titutis, Dico vecte quomodolibet diui&longs;o, <lb/>ponderibu&longs;que vtrinque appo&longs;itis, permutatâ propor<lb/>tione &longs;ibi inuicem re&longs;pondentibus, rem e&longs;&longs;e redactam ad <lb/>æqualitatem, & inde &longs;tatum fieri, hoc e&longs;t, æquilibrium. </s><figure id="id.007.00.057.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.057.1.jpg"></figure>

<s id="id.000520">E&longs;to enim vectis AB, quo modo libet diui&longs;us in C, & <lb/>ip&longs;i quidem C fulcimentum &longs;upponatur. </s>

<s id="id.000521">Appendantur <lb/>quo que vtrinque pondera ex ratione brachiorum AC, <lb/>CB, &longs;ibi inuicem permutatim re&longs;pondentia, &longs;intque; DE. <lb/></s>

<s id="id.000522">Dico vectem ex æqualitate, in neutram partem <expan abbr="inclina-tur&umacr;">inclina<lb/>turum</expan>, &longs;ed perman&longs;urum in æquilibrio. </s>

<s id="id.000523">quoniam enim <expan abbr="P&omacr;-dus">Pon<lb/>dus</expan> D idem pote&longs;t quod brachium CB, addatur in dire<lb/>ctum ip&longs;i AC, recta AF æqualis ip&longs;i CB, item quoniam <lb/>Pondus E id pote&longs;t quod brachium AC, rectæ CB ad<lb/>datur in directum BG, ip&longs;i AC æqualis. </s>

<s id="id.000524">Igitur cum par<lb/>tes CA, AF totius FC, æquales &longs;int partibus CB, BG, <lb/>totius CG, erit totum FC, toti CG æquale. </s>

<s id="id.000525">Diui&longs;us ita-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.058.jpg"/>que erit vectis FG in partes æquales FC, CG in puncto <lb/>fulcimenti C. </s>

<s id="id.000526">Et quoniam æquale in æquale non agit, <lb/>&longs;tabit vectis & in neutram partem inclinabit. </s>

<s id="id.000527">Rur&longs;um <lb/>quoniam ad partem FC, duæ &longs;unt brachiorum potentiæ <lb/>FA, HC, appendantur puncto F, duo pondera H, I, ip&longs;is <lb/>DE æqualia, item puncto G, alia duo pondera ij&longs;dem DE <lb/>æqualia KL, iterum æqueponderabit, quippe quod æ<lb/>qualibus brachijs FCCG æqualia appen&longs;a &longs;int pondera <lb/>HI KL. </s>

<s id="id.000528">Cur igitur &longs;eruata permutatim brachiorum & <lb/>ponderum proportione fiat æquilibrium, ex his quæ de<lb/>mon&longs;trauimus, clarè patet. </s>

<s id="id.000529">Sed forte dicet qui&longs;piam, &longs;i brachia, pondera &longs;unt, <lb/>vel ponderibus æquipollentia, &longs;u&longs;tinenti duplicabitur <lb/>pondus. </s><figure id="id.007.00.058.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.058.1.jpg"></figure>

<s id="id.000530">E&longs;to enim vectis AB, <lb/>ita diui&longs;us in C, vt pars <lb/>maior CB minori AC &longs;it <lb/>in proportione quintu<lb/>pla. </s>

<s id="id.000531">Appendatur autem <lb/>in A pondus D, <expan abbr="quintupl&umacr;">quintuplum</expan> <lb/>ponderi E appen&longs;o in B. </s>

<s id="id.000532">Si <lb/>igitur brachio AC, quod <lb/>e&longs;t vnum, ad datur pondus <lb/>D, quod e&longs;t quinque, fi ent &longs;ex, item &longs;i brachio CB, quod <lb/>e&longs;t quinque, addatur pondus E, quod e&longs;t vnum, fient &longs;ex. <lb/></s>

<s id="id.000533">Fulcimentum igitur &longs;u&longs;tinebit duodecim, quod e&longs;t ab<lb/>&longs;urdum ex ijs quæ clarè demon&longs;trauit G. Vbald. in Me<lb/>chan. tractatu de Libra propo&longs;. </s>

<s id="id.000534">5. His re&longs;pondemus, bra<lb/>chia quidem operari non pondere, &longs;ed potentiâ, quæ vis <lb/>quædam e&longs;t, non autem pondus. </s>

<s id="id.000535">Et&longs;i & illud verum &longs;it, da<lb/>to vecte pondero&longs;o, fulcimentum rum ponderum appen<lb/>&longs;orum, tum vectis ip&longs;ius pondus &longs;u&longs;tinere. </s>

<s id="id.000536">Iacta huiu&longs;cemodi, quam diximus, æqualitate, &longs;e-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.059.jpg"/>quitur nece&longs;&longs;ariò, centrum grauitatis ip&longs;ius vectis cum <lb/>appen&longs;is ponderibus, ac &longs;i vnum idemqueue e&longs;&longs;et corpus <lb/>cadere in perpen diculari quæ per centrum ip&longs;um & ful<lb/>cimentum tran&longs;iens ad mundi centrum pertingit. </s>

</subchap1><subchap1>

<s id="id.000537">QVÆSTIO IV.</s>

<s id="id.000538"><emph type="italics"/>Quærit hic Ari&longs;toteles, cur ij qui in nauis medio &longs;unt remiges ma<lb/>ximè nauem moueant?<emph.end type="italics"/></s>

<s id="id.000539">Ait, ideo forta&longs;&longs;e fieri, quòd remus vectis &longs;it, fulcimen<lb/>tum verò &longs;calmus, &longs;tat enim. </s>

<s id="id.000540">Pondus autem marei<lb/>p&longs;um, quod à remo propellitur, mouens verò ip&longs;um remi<lb/>gem, &longs;emper autem plus mouere ponderis quimouet, <lb/>quo magis di&longs;tatà fulcimento. </s>

<s id="id.000541">Ita enim maiorem fieri <lb/>quæ ex centro; Scalmum verò centrum e&longs;&longs;e. </s>

<s id="id.000542">Cæterùmin <lb/>medio nauis plurimum remi intus e&longs;&longs;e. </s>

<s id="id.000543">Ibi enim nauem <lb/>e&longs;&longs;e lati&longs;&longs;imam. </s>

<s id="id.000544">Moueri autem nauim, quoniam <expan abbr="appell&emacr;-te">appellen<lb/>te</expan> mariremo, <expan abbr="extrem&umacr;">extremum</expan> illius quod intus e&longs;t anterius pro<lb/>mouctur, cuius motum nauis &longs;equitur, cui &longs;calmus alliga<lb/>tur. </s>

<s id="id.000545">Vbiautem plurimum maris diuidit remus, eo maximè <lb/>nece&longs;&longs;e e&longs;&longs;e propelli. </s>

<s id="id.000546">Plurimum autem diuidi vbi plurima <lb/>pars remi à &longs;calmo e&longs;t. </s>

<s id="id.000547">Rem facilem, eo quod verbis potu<lb/>erit, &longs;chemate non declarauit, nos autem apponemus. </s><figure id="id.007.00.059.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.059.1.jpg"></figure>

<s id="id.000548">E&longs;to enim nauis AB, mare CD, <lb/>remorum alter, quiad proram EF, cu<lb/>ius &longs;calmus G, alter verò in medio na<lb/>uis, HI, circa &longs;calmum K. </s>

<s id="id.000549">Ait igitur, <lb/>remos e&longs;&longs;e vectes, &longs;calmos verò fulci<lb/>menta, pondus quod remo, ceu vecte, <lb/>mouetur mare ip&longs;um. </s>

<s id="id.000550">Itaque quoniam <lb/>nauis lata e&longs;t in medio vbi Scalmus K <lb/>maior pars KH intra nauim e&longs;t, minor <lb/>verò KI, extra. </s>

<s id="id.000551">Contra autem remiad <lb/>proram, nempe EF pars minor EG <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.060.jpg"/>intra nauim, pars verò maior GF extra nauim e&longs;t. </s>

<s id="id.000552">Pondus <lb/>autem cò faciliùs mouctur, quo maior e&longs;t vectis pars, quæ <lb/>à fulcimento e&longs;t ad mouentem potentiam. </s>

<s id="id.000553">Acutè &longs;anè Philo&longs;ophus. </s>

<s id="id.000554">Ego autem &longs;i per mode&longs;tiam <lb/>liceret, dicerem, non quidem e&longs;&longs;e fulcimentum <expan abbr="&longs;calm&umacr;">&longs;calmum</expan>, <lb/>&longs;ed mare ip&longs;um, pondus vero nauim, ad locum &longs;calmi, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> inter mouentem potentiam, & fulcimentum po&longs;itum, <lb/>etenim & eo pacto po&longs;&longs;umus vti vecte, quod ob&longs;eruat & <lb/>demon&longs;trat G. Vbaldus tractatu de vecte propo&longs;. </s>

<s id="id.000555">2. Erunt <lb/>igitur in de&longs;cripta figura puncta FI, quæ in mari&longs;unt, ful<lb/>cimenta, quibus remorum extrema in ip&longs;aim pul&longs;ione ni<lb/>tuntur, pondera verò &longs;eu pondus pluribus vectibus & po<lb/>tentijs impul&longs;um nauis ip&longs;a, quæ &longs;calmis e&longs;t annexa. </s>

<s id="id.000556">Re&longs;i<lb/>&longs;tente igitur mari, cedente autem impul&longs;ionibus &longs;calmo, <lb/>nauis eo transfertur, quo &longs;calmi ab ip&longs;a potentia mouen<lb/>te in anteriorem partem pelluntur. </s>

<s id="id.000557">quoniam autem vt <lb/>FG ad FE ita potentia mouens in E ad pondus motum <lb/>in G. item vt IK ad IH ita potentia mouens in H ad pon<lb/>dus motum in K, maior autem e&longs;t proportio FG ad FE <lb/>quàm proportio IK ad IH. </s>

<s id="id.000558">Maiori indiget potentia vt <lb/>pellatur pondus in G quàm pondus in K. </s>

<s id="id.000559">Hæc certè vti diximus ita &longs;e habent. </s>

<s id="id.000560">Philo&longs;ophi au<lb/>tem ratio tunc procederet, &longs;i &longs;tante naui immobili, vt fit <lb/>vbi à Remoræ occulta vi aut ab alio impedimento reti<lb/>netur, remiges in ip&longs;o remigandi actu mare pul&longs;arent, <lb/>Tunc enim verè &longs;calmus fieret fulcimentum, mare autem <lb/>pondus, remex verò ip&longs;e mouens. </s>

<s id="id.000561">Addimus, fal&longs;um videri quod a&longs;&longs;erit Ari&longs;toteles, <lb/>nempeillos qui in media naui &longs;unt, remiges, maximè na<lb/>uim mouere; facilius, melius dixi&longs;&longs;et. </s>

<s id="id.000562">Si enim maximè, <lb/>quod ait, denorat, maximo &longs;patio, & velocius pror&longs;us fal<lb/>&longs;um, etenim tardius mouent & minori &longs;patio, quod nos i<lb/>ta demon&longs;tramus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.061.jpg"/><figure id="id.007.00.061.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.061.1.jpg"></figure>

<s id="id.000563">E&longs;to enim Remus AB <lb/>qui marí fulcitur in B, Scal<lb/>mus remi qui ad <expan abbr="prorã">proram</expan> pup<lb/>pimue C, qui in media naui <lb/>D, maior autem remi pars <lb/>e&longs;t à &longs;calmo Dad A quami<lb/>p&longs;ius C 2d A, Pellantur remi & &longs;tante ceu centro BA, in <lb/>E. eodem igitur tempore C eritin F, & D in G, &longs;ed maiu<gap/><lb/>e&longs;t &longs;patium CF &longs;patio DG, Ergo vnica impul&longs;ione, plus <lb/>mouit &longs;calmum, hoc e&longs;t, nauim, potentia ad puppim pro<lb/>ramue remigans, quàm ea quæ operatur in media naui vt <lb/>&longs;entire vid<gap/>batur (&longs;i modo is e&longs;t eius &longs;en&longs;us) Ari&longs;toteles. <lb/></s>

<s id="id.000564">Nece&longs;&longs;arium igitur e&longs;t, quodait, maximè intelligendum, <lb/>faciliùs, Veritatem hanc cogno&longs;centes Triremium præ<lb/>fecti robu&longs;tiores quidem remiges ad proram & puppim, <lb/>inualidiores vcrò circa mediam triremem collocant. </s>

</subchap1><subchap1>

<s id="id.000565">QVÆSTIO V.</s>

<s id="id.000566"><emph type="italics"/>Dubitatur, Cur paruum exi&longs;tens gubernaculum, & in extremo <lb/>nauigio tantas habeat vires, vt ab exiguo temone, & ab hominis <lb/>vnius viribus alioqui modicè vtentis magnæ nauigiorum <lb/>moueantur moles?<emph.end type="italics"/></s>

<s id="id.000567">AN, inquit, quoniam gubernaculum vectis e&longs;t, onus <lb/>autem mare, Gubernator vero mouens e&longs;t? </s>

<s id="id.000568">Non au<lb/>tem &longs;ecundùm latitudinem veluti remus, mare accipit <lb/>gubernaculum; non enim in ante nauigium mouet, &longs;edi<lb/>p&longs;um commotum mare accipiens inclinat obliquè. </s>

<s id="id.000569">quo<lb/>niam enim pondus e&longs;t mare contrario innixum modo na<lb/>uem inclinat. </s>

<s id="id.000570">fulcimentum enimin contrarium ver&longs;atur, <lb/>mare vetò interius, & illud exterius. </s>

<s id="id.000571">illud autem &longs;equitur <lb/>nauis quæ illi e&longs;t alligata & remus quidem &longs;ecundum la<lb/>titudinem onus propellens & ab eodem repul&longs;us in re-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.062.jpg"/>ctum propellit, Gubernaculum verò, vt obliquum iacet <lb/>hinc inde in obliquum motionem facit. </s>

<s id="id.000572">in extremo <expan abbr="aut&emacr;">autem</expan>, <lb/>non in medio iacet, quoniam mouenti fa cillimum e&longs;t mo<lb/>tum moucre: prima enim pars celerrimè fertur, & quo<lb/>niam, quemadmodum in ijs quæ feruntur in fine deficit <lb/>latio, &longs;ic ip&longs;ius continui in finem, imbecillima e&longs;t latio. <lb/></s>

<s id="id.000573">Imbecillima autem ad expellendum e&longs;t facilis. </s>

<s id="id.000574">Propter <lb/>hæc igitur in puppi gubernaculum ponitur, nec minus, <lb/>quoniam paruaibi motione facta, multo maior fit in vlti<lb/>mo, quia æqualis angulus &longs;emper maiorem ad&longs;pectat, <expan abbr="tã-to">tan<lb/>to</expan> queue magis, quanto maiores fuerint illæ, quæ continent. <lb/></s>

<s id="id.000575">Exijs ctiam manife&longs;tum e&longs;t, quam ob cau&longs;&longs;am magis in <lb/>contrarium procedit nauigium, quam remi ip&longs;ius palmu<lb/>la, eadem enim magnitudo ij&longs;dem mota viribus in aëre <lb/>plus quàm in aqua progreditur. </s>

<s id="id.000576">Hæc Philo&longs;ophus, qui <lb/>haudquaquam ex more &longs;uo, quod duobus ferè poterat, <lb/>&longs;excentis verbis expo&longs;uit. </s>

<s id="id.000577">Licebat enimid tantum dicere, <lb/>Gubernaculum (ita vocatid totum quod gubernaculo & <lb/>temone con&longs;tat) e&longs;&longs;e ceuremum, quo nauis non antror<lb/>&longs;um, &longs;ed obliquè & ad latus mouetur. </s>

<s id="id.000578">quamobrem omnia <lb/>ferè quæ de Temone dicenda fuerant, de remo loquens <lb/>proponit. </s>

<s id="id.000579">Aitautem. </s><figure id="id.007.00.062.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.062.1.jpg"></figure>

<s id="id.000580">Sit remus AB, <lb/>&longs;calmus vero C, remi <lb/>in nauigio <expan abbr="principi&umacr;">principium</expan> <lb/>A, palmula autem, <lb/>quæ in mari B. </s>

<s id="id.000581">Si igi<lb/>tur A, vbi D transla<lb/>tum e&longs;t, non erit B v<lb/>bi E. æqualis enim, <lb/>BE ip&longs;i AD, æquale <lb/>igitur translatum erit, &longs;ed erat minus. </s>

<s id="id.000582">eritigitur vbi F, mi<lb/>nor enim BF, ip&longs;a AD, quareip&longs;o GF ip&longs;a DG. </s>

<s id="id.000583">Hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.063.jpg"/>demon&longs;tratio licet vera videatur, rei ta men, de qua e&longs;t <lb/>&longs;ermo, minimè aptatur. </s>

<s id="id.000584">Si enim aptaretur in ip&longs;ius remi <lb/>motu, cum palmula e&longs;&longs;et in F, &longs;calmus ficret in G, excur<lb/>reretergo vel &longs;calmus per remum, vel remus per <expan abbr="&longs;calm&umacr;">&longs;calmum</expan>, <lb/>facta nempe ciu&longs;modi translatione de C in G, & &longs;ic intra <lb/>nauim modo e&longs;&longs;et pars remi DC, modò verò GD, quod <lb/>tamen non &longs;ieri ipsâ experientia docemur. </s>

<s id="id.000585">Illud quoque <lb/>fal&longs;um e&longs;t, nauim ip&longs;am tantum moueri in aëre, quantum <lb/>e&longs;t &longs;patium AD, hoc e&longs;t, remi extremum quod e&longs;t in naui, <lb/>&longs;iquidem &longs;calmi motu, non autem manubrij remi, nauis <lb/>agatur. </s>

<s id="id.000586">Aliter igitur res &longs;e habet, & forte hoc pacto. </s><figure id="id.007.00.063.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.063.1.jpg"></figure>

<s id="id.000587">Sit remus AB, cuíus <lb/>manubrium A, palmula <lb/>B, &longs;calmus C. </s>

<s id="id.000588">Pellatur an<lb/>tror&longs;us A, fiatque; in D, tunc <lb/>&longs;i æqualiter mouerentur <lb/>manubrium & palmula, i<lb/>p&longs;a palmula ficret in G, at <lb/>minus mouetur: fiet ergo <lb/>in E. ip&longs;e verò &longs;calmus C <lb/>translatus erit in F, motaque; erit nauis à C in F, non autem <lb/>ab A in D. </s>

<s id="id.000589">P o&longs;uitautem Ari&longs;toteles &longs;calmum ad medium <lb/>remi, &longs;ed non ad medium collocari &longs;olet, maior enim pars <lb/>in mare propendet puta HB, quo ca&longs;u translationis &longs;pa<lb/>tium fit maius, nempe ab H in I. fit autem motus &longs;calmi ex <lb/>centris qui &longs;unt in &longs;patio ip&longs;o BE, quatenus autem ad te<lb/>monem pertinet, quem remum ait, obliquè puppim ip&longs;am <lb/>propellentem, ita &longs;e res habet. </s>

<s id="id.000590">E&longs;to nauis carina AB, prora A, puppis B, Temonis <lb/>ala BC, gubernaculum BD, cardo verò fulcimentumue <lb/>B; factaitaque impul&longs;ione obliquâ gubernaculi à D in E, <lb/>minor fiet motus in mari à C in F, eritqueue temo vbi EGF, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.064.jpg"/><figure id="id.007.00.064.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.064.1.jpg"></figure><lb/>cardo verò vbi G, translata igitur e<lb/>rit eo motu, puppis ip&longs;a à B in G. facta <lb/>itaque paruâ motione puppis ex B in <lb/>G, prora ip&longs;a quæ longè di&longs;tat à pup<lb/>pi B maiori &longs;patio &longs;uperato translata <lb/>erit in H facta proræ in contrariam <lb/>partem ab ea quæ facta e&longs;t guberna<lb/>culi motione. </s>

<s id="id.000591">Porrò quod & in præ<lb/>cedente quæ&longs;tione adnotauimus, <expan abbr="l&omacr;-gè">lon<lb/>gè</expan> meliùs procedet demon&longs;tratio &longs;i <lb/><expan abbr="fulciment&umacr;">fulcimentum</expan> mare intelligatur, quàm <lb/>&longs;calmus, neque enim mare ceu pon<lb/>dus, &longs;ed &longs;calmus ip&longs;e Temonisuecardo, ponderum in&longs;tar <lb/>transferuntur. </s>

<s id="id.000592">Cæterùm in hac &longs;peculatione liceat nobis aliquan<lb/>tulum à Philo&longs;opho di&longs;&longs;entire. </s>

<s id="id.000593">Certè &longs;i breuitas Temo<lb/>nis, è puppi eminentis, re&longs;pectu longitudinis totius nauis <lb/>con&longs;ideretur, & parua motio, quæ temone guberna culo<lb/>ue moto fit, nullius ferè momenti erit ad eam quæ in pro. <lb/></s>

<s id="id.000594">ra fit translationem. </s>

<s id="id.000595">aliter ergo &longs;e rem habere non dubi<lb/>tamus, & quæ&longs;tionis &longs;olutionem aliunde petendam. </s>

<s id="id.000596">Na<lb/>uinon currentenullum ferè, aut qui vix curandus &longs;it ex <lb/>gubernaculi conuer&longs;ione nauis ad dextram &longs;ini&longs;tramue <lb/>motum fieri. </s>

<s id="id.000597">at eâ currente maximum, experientiâ doce<lb/>mur. </s>

<s id="id.000598">Obliqui igitur motus qui validèin puppi &longs;it, cau&longs;&longs;a <lb/>e&longs;t non quidem ex conuer&longs;ione temonis percu&longs;&longs;io maris, <lb/>&longs;ed mare ip&longs;um, cuius fluctus naui currente obliquam te<lb/>monis alam ad eam partem quæ mari obuertitur, impel<lb/>lentes temonem cum puppiad contrariam partem vali<lb/>di&longs;&longs;imè transferunt. </s>

<s id="id.000599">E&longs;to nauis carina AB, prora B, puppis A, Temo AC, <lb/>gubernaculum AD; Itaque currentenaui, Temone in<lb/>terim & guberna culo in eadem carinæ linea exi&longs;tentibus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.065.jpg"/><figure id="id.007.00.065.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.065.1.jpg"></figure><lb/>Temo quidem mare &longs;ecat, nulla fa<lb/>ctâ in puppi, nauis ad &longs;ini&longs;tram dex<lb/>tramue translatione. </s>

<s id="id.000600">Si verò mouea<lb/>tur gubernaculum à D in E, co moto <lb/>mouebitur aliquantulum & puppis <lb/>ad partes E, quod voluit Ari&longs;toteles. <lb/></s>

<s id="id.000601">Sedminimi, vt diximus, ea res ad tan<lb/>tum effectum e&longs;t momenti. </s>

<s id="id.000602">Temone <lb/>autem in obliquum <expan abbr="c&omacr;&longs;tituto">con&longs;tituto</expan> vt AF, <lb/>naui interim, ventorum aut remorum <lb/>vi pul&longs;a proram ver&longs;us currente te<lb/>monis latus à fluctibus obliquam par<lb/>tem alamue in ip&longs;o cur&longs;u ferientibus, <lb/>in contrariam partem transfertur, ad <lb/>eam nempe, ad quam ip&longs;um gubernaculum vergit. </s>

<s id="id.000603">facta i<lb/>gitur nauis ceu circa centrum centraue quæ in carina in<lb/>ter puppim proramue con&longs;i derantur A, fertur in G, prora <lb/>verò in H. ex quibus manife&longs;tè apparet, duo ad nauis ex <lb/>temone in puppi conuer&longs;ione motionem e&longs;&longs;e ne ce&longs;&longs;aria; <lb/>Temonis nempe obliquationem, & nauis cur&longs;um, <expan abbr="quor&umacr;">quorum</expan> <lb/>&longs;i alterum &longs;ine altero adhibeatur, nullam fieri quæ alicu<lb/>ius momenti &longs;it, nauis conuer&longs;ionem. </s>

<s id="id.000604">Illud quoque nota<lb/>mus, carinam in nauis conuer&longs;ione vectis in&longs;tar &longs;e habere, <lb/>cuius pars mota ad puppim, & mouens potentia e&longs;t; fulci<lb/>mentum verò circa proram, potentia autem mouens ma<lb/>reip&longs;um, temonem in nauis cur&longs;u oblique feriens. </s>

<s id="id.000605">Vnde <lb/>colligimus naues, quo longiores &longs;unt in mouente ad Te<lb/>monem adhibita maiori facilitate ad dextram &longs;ini&longs;tram<lb/>ue propelli: quod &longs;anè ip&longs;emet con&longs;iderauit Ari&longs;toteles, <lb/>quì idcirco inquir, in extremo, non autem in medio temo<lb/>nem poni eo quod mouenti facilimum &longs;it ab extremo <lb/>motum mouere. </s>

<s id="id.000606">Ex hac no&longs;trâ &longs;peculatione ratio habetur eius ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.066.jpg"/>chinationis, quâ in magnis fluminibus, ceu Pado, Abdua <lb/>& &longs;imilibus, Portitores, equos, currus, viatore&longs;que; ip&longs;os, è <lb/>ripa in ripam transferunt. </s>

<s id="id.000607">Pulcherrima enim res e&longs;t, & <lb/>nobis per&longs;pecti&longs;&longs;ima, qui Gua&longs;tallâ re&longs;identiæ olim no<lb/>&longs;træ oppido ad Padum, Mantuam pergentes &longs;æpi&longs;&longs;imè ad <lb/>Ca&longs;trum B<gap/>rgi Iu&longs;is ea qua diximus machinatione lati&longs;<lb/>&longs;imum eiu&longs;dem Padi aluum tran&longs;ie cimus. </s>

<s id="id.000608">Habet autem <lb/>&longs;e hoc pacto. </s><figure id="id.007.00.066.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.066.1.jpg"></figure>

<s id="id.000609">E&longs;to fluminis citerior <lb/>ripa AB, vlterior CD. </s>

<s id="id.000610">Pon<lb/>tones duo tabulis &longs;trati, & v<lb/>nà firmiter juncti EF, Temo <lb/>inter eorum puppes extans <lb/>GH, locus in ripa &longs;tabilis A, <lb/>funis, quo pontones, & ma<lb/>china tota continetur AI. <lb/>fluuij decur&longs;us ver&longs;us BD, <lb/>&longs;tantibus itaque pontonibus <lb/>ad ripam citeriorem AB, Te<lb/>mone in <expan abbr="neutrã">neutram</expan> partem pul<lb/>&longs;o, cum aqua decurrens eum <lb/>re&longs;i&longs;tentem non inueniat, <lb/>&longs;cinditur quidem ab eo, &longs;ed <lb/>non propellit, eo autem con<lb/>uer&longs;o & in GK con&longs;tituto, a<lb/>la eius GK ab aqua defluente propul&longs;a machinam &longs;ecum <lb/>trahit ver&longs;us ripam CD, factâ motione circa centrum &longs;eu <lb/>&longs;tabilem locum A, otio&longs;is interim portitoribus, donec per <lb/>circuli portionem ML deuenerit ad vlteriorem ripam in <lb/>L. </s>

<s id="id.000611">Vnde iterum temone in contrariam partem conuer&longs;o, <lb/>aquâ &longs;imiliter temonem propellente, per eandem circuli <lb/>portionem ad ripam citeriorem reuertitur, à qua paullo <lb/>antè di&longs;ce&longs;&longs;erat. </s>

<s id="id.000612">Ex quibus apparet, motus cau&longs;&longs;am non <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.067.jpg"/>e&longs;&longs;e &longs;olam cam, quæ ab ala temonis fit, aquæ <expan abbr="percu&longs;&longs;ion&emacr;">percu&longs;&longs;ionem</expan>, <lb/>vt &longs;en&longs;erat Ari&longs;toteles, &longs;ed currentis a quæ temonis alam <lb/>ferientis impul&longs;ion<gap/>m: nihil autem referre, vtrum &longs;tante <lb/>naui a qua currat, vel câ currente a qua &longs;tet, vt in mari fit, <lb/>idem enim vtroque modo temo patitur. </s>

<s id="id.000613">Vt autem machi<lb/>næ huius & totius negotij &longs;pecies facilius animo concipia<lb/>tur, &longs;chema hoc &longs;tudio &longs;orum oculis &longs;ubijciemus. </s><figure id="id.007.00.067.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.067.1.jpg"></figure>

<s id="id.000614">Lembi nauiculæueideo appo&longs;itæ &longs;unt, vt oblongum <lb/>funem &longs;u&longs;tineant; id etenim nî fieret, aquæ immer&longs;us a<lb/>quam &longs;cindens machinæ motum impediret, ideo etiam <lb/>apponuntur, ne funis madens celeriter maceretur & pu<lb/>tre&longs;cat. </s>

<s id="id.000615">Huic &longs;peculationi affinis e&longs;t ea, velorum eorum, <lb/>quæ obliquè ventum, excipientia frumentarijs molis <lb/>dant motum, item verticillorum ex papyro, quibus con<lb/>tra ventum currentes per lu&longs;um pueri vtuntur. </s>

<s id="id.000616">vnicum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.068.jpg"/>enim horum emnium principium, & eadem, ratio. </s>

<s id="id.000617">Diximus enim, Temonem currente naui, lateraliter <lb/>conuer&longs;um obuios fluctus ex cipientem puppim ip&longs;am ob<lb/>liquè in alteram partem transferre. </s>

<s id="id.000618">Porrò ea vela, de qui<lb/>bus loquimur, ventorum flatibus obliquè oppo&longs;ita can<lb/>dem ob cau&longs;&longs;am circulariter agitantur, quodvt figurâ eui<lb/>dentius fiat, </s><figure id="id.007.00.068.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.068.1.jpg"></figure>

<s id="id.000619">E&longs;to velum AB, brachio <lb/>CE obliquè affixum, ita vt <lb/>angulus ACE maior &longs;it an<lb/>gulo BCE, ventus obliquè <lb/>velum feriens FG. <expan abbr="Itaq;">Itaque</expan> quo<lb/>niam ventus in velum obli. <lb/></s>

<s id="id.000620">quum incidit, elabitur velum, <lb/>& circa centrum E vnà cum <lb/>brachio circumuertitur, in <lb/>cuius locum &longs;uccedit velum <lb/>HI, ex qua a&longs;&longs;idua velorum <lb/>&longs;ucce&longs;&longs;ione, brachiorum & a<lb/>xis cui adhærent, rotatio fit <lb/>perpetua. </s>

<s id="id.000621">Sed enim de Te<lb/>mone agentes non e&longs;t interim cur de caudis auium pi&longs;ci<lb/>umque taceamus, in&longs;tar enim remonum &longs;unt à Naturai<lb/>p&longs;a opportunis animalium partibus, po&longs;tremis videlicet, <lb/>appo&longs;iti, quanquam nec&longs;olum Temonis v&longs;um præ&longs;tent, <lb/>vt videbimus. </s>

<s id="id.000622">E&longs;to pi&longs;cis AB, cuius caput A, cauda verò CB. </s>

<s id="id.000623">Hac <lb/>igitur neutram in partem reflexâ, pi&longs;cis pinnarum motu <lb/>rectâ in anteriorem partem progreditur. </s>

<s id="id.000624">Si autem nece&longs;<lb/>&longs;e ei fuerit ad dextram &longs;ini&longs;tramqueue conuerti non pote<lb/>rit, ni&longs;i cauda ip&longs;a iuuetur. </s>

<s id="id.000625">Omnis enim motus progre&longs;&longs;i<lb/>uus quiete indiget, nec <expan abbr="ab&longs;q;">ab&longs;que</expan> &longs;tabili fulcimento progredi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.069.jpg"/><figure id="id.007.00.069.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.069.1.jpg"></figure><lb/>pote&longs;t, quod in libris de ani<lb/>malium ince&longs;&longs;u docet ip&longs;e<lb/>met Philo&longs;ophus. </s>

<s id="id.000626">Sit igitur, <lb/>pi&longs;cem conuerti velle, & fie<lb/>ri capite in D, deflectet illi<lb/>co caudam in E, caque; aquam <lb/>ceu &longs;tabile quippiam <expan abbr="feri&emacr;s">feriens</expan> <lb/>eiqueue quod<gap/>mmodo fultus, <lb/>reliquum corpus CA refle<lb/>ctet in D, &longs;i autem conuerti <lb/>velit in F, caudam defle ctet in G, & eadem ratione <gap/> cte<lb/>tur in F. </s>

<s id="id.000627">Sed & Temonis quoque v&longs;um præ&longs;tat natatili<lb/>bus & volatilibus cauda. </s>

<s id="id.000628">Sit enim rectus pi&longs;cis, hoc e&longs;t, re<lb/>ctâ pergens IKL, caudam obliquet in KM itaque ex a<lb/>quæ in ip&longs;o motu colli&longs;ione, eius po&longs;teriora pellentur vbi <lb/>INO. </s>

<s id="id.000629">Hæc itaque nos de Temone, quatenus ad hanc <lb/>quæ&longs;tionem pertinet, con&longs;idera&longs;&longs;e &longs;it &longs;atis. </s>

</subchap1><subchap1>

<s id="id.000630">QVÆSTIO VI.</s>

<s id="id.000631"><emph type="italics"/>Dubitatur, Cur quanto Antenna &longs;ublimior fuerit, ÿ&longs;dem velis, & <lb/>vento eodem celeriùs ferantur nauigia?<emph.end type="italics"/></s>

<s id="id.000632">Soluit Philo&longs;ophus, inquiens: An quia malus quidem <lb/>&longs;it vectis, fulcimentum verò mali &longs;edes, in qua colloca<lb/>tur, pondus autem quod moueri debet, ip&longs;um nauigium: <lb/>mouens verò is, qui vela tendit &longs;piritus? </s>

<s id="id.000633">Si igitur quanto <lb/>remotior fuerit fulcimentum facilius eadem potentia, & <lb/>citiùs idem mouet pondus, altius certè &longs;ublatâ antennâ, <lb/>velum à mali &longs;ede, quae fulcimentum e&longs;t remotius faciens, <lb/>id efficiet. </s>

<s id="id.000635">quæ &longs;ic figurâ explicamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.070.jpg"/><figure id="id.007.00.070.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.070.1.jpg"></figure>

<s id="id.000636">E&longs;to nauis AB, malus CD, <lb/>mali &longs;edes D, locus antennæ <lb/>&longs;ublimior C, depre&longs;&longs;ior E: ita<lb/>que quoniam CD vectis e&longs;t, <lb/>quo mouens remotior fuerit à <lb/>fulcimento D, co citiùs & vio<lb/>lentiùs pellet, velocius ergo <lb/>nauis mouebitur antenna in <lb/>C, quàm in E, con&longs;tituta. </s>

<s id="id.000637">Plau&longs;ibilia &longs;unt hæc, at certè per veritatem ip&longs;am, <lb/>non vera. </s>

<s id="id.000638">Rogo, Si fulcimentum dum vectis mouetur, <expan abbr="c&emacr;-trum">cen<lb/>trum</expan> e&longs;t, centrum vtique motus erit D. &longs;pirante igitur va<lb/>lidè vento inclinabitur malus, fietque; vbi FGD, quæ qui<lb/>dem in clinatio vio lentius fiet, vento pellentein F q uàm <lb/>in G, vtpote puncto à fulcimento remotiore. </s>

<s id="id.000639">Impul&longs;o ma<lb/>lo, duo nece&longs;&longs;ariò <expan abbr="c&omacr;&longs;equentur">con&longs;equentur</expan>, vel enim ad ip&longs;am &longs;edem <lb/>D. frangetur vel puppis ip&longs;a circa D punctum conuer&longs;a, <lb/>vt mali &longs;e quatur motum eleuabitur. </s>

<s id="id.000640">Prora verò &longs;ubmer<lb/>getur facta naui in HDI. </s>

<s id="id.000641">Quod &longs;i qui&longs;piam funem ad ma<lb/>li &longs;ummitatem annexam ad ip&longs;am puppim alligauerit in <lb/>B, impe dietur &longs;anè mali in clinatio ad partes F, & ideo nul<lb/>la vis pror&longs;us fiet in D ex vectis ratione. </s>

<s id="id.000642">Attamen nihilo <lb/>&longs;ecius, quo &longs;ublimior fuerit antenna, eo faciliùs à &longs;pirante <lb/>vento puppis eleuabitur. </s>

<s id="id.000643">quatenus igitur malus vectis <lb/>e&longs;t, hoc tantum quod dicimus operatur. </s>

<s id="id.000644">Quod &longs;i contrà <lb/>obiectum fuerit, experientiam docere, quo &longs;ublimior an<lb/>tenna fuerit, eo citiùs nauigium, &longs;piritu flante moueri. <lb/></s>

<s id="id.000645">Re&longs;pon&longs;io facilis, nempe, mirum non e&longs;&longs;e, &longs;i mali pars &longs;ub<lb/>limior validius à vento feriatur. </s>

<s id="id.000646">Videmus enim, & turres <lb/>quo &longs;ublimiores fuerint, eo magis à ventorum impetuo&longs;is <lb/>flatibus infe&longs;tari, quod &longs;anè ad vectis longitudinem refer<lb/>re, e&longs;&longs;et ridiculum. </s>

<s id="id.000647">Cætcrùm quod ad puppis faciliorem <lb/>eleuationem ex mali ip&longs;ius altitudine pertinet, ad vectis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.071.jpg"/>contemplationem reducimus. </s>

<s id="id.000648">e&longs;t enim quæ dam vectium <lb/>&longs;pecies ab alijs non con&longs;iderata, cuius brachia in angu<lb/>lum de&longs;inunt, vtip&longs;e angulus in operatione &longs;it fulcimen<lb/>tum. </s><figure id="id.007.00.071.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.071.1.jpg"></figure>

<s id="id.000649">E&longs;to enim vectis, de quo agimus, <lb/>ABC, cuius brachia AB, BC. iuncta <lb/>ad angulum B, &longs;itqueue B in operatione <lb/>fulc mentum. </s>

<s id="id.000650">Nec quicquam refert <lb/>quatenus ad v&longs;um pertinet, vtrum an<lb/>gulus ip&longs;e rectus &longs;it, acutus vel obtu<lb/>&longs;us. </s>

<s id="id.000651">&longs;it autem modò rectus. </s>

<s id="id.000652">Ponaturi<lb/>gitur pondus aliquod in C, tum po<lb/>tentia quædam applicetur in A, quae i<lb/>p&longs;am vectis extremitatem A propel<lb/>lat in D. erit igitur AB in DB & an<lb/>gulo &longs;eruato BC in BE. </s>

<s id="id.000653">Pondus igi<lb/>tur cum parte vectis BC eleuabitur in E. </s>

<s id="id.000654">In hoc autem <lb/>vectis genere attenditur proportio quam habet AB ad <lb/>BC. </s>

<s id="id.000655">Si enim potentia quæ applicatur in A ita &longs;e habet ad <lb/>pondus in C vt CB, ip&longs;i BA, fiet æ quilibrium. </s>

<s id="id.000656">Si maior <lb/>autem fuerit proportio potentiæ in A, ad pondus in C, ea <lb/>quam habet AB ad BC, &longs;uperatâ ponderis re&longs;i&longs;tentiâ fiet <lb/>motus. </s>

<s id="id.000657">Res autem haud aliter &longs;e habet, ac &longs;i producta in <lb/>F, fieret BF æqualis BC. </s>

<s id="id.000658">Tunc enim vectis ad rectitudi<lb/>nem, &longs;eruatâ proportione, redigeretur, & ita potentia in <lb/>A, fulcimento B operaretur in F, vt operabatur in C. </s>

<s id="id.000659">Ad huius vectis naturam referuntur fabrorum mal<lb/>lei, quibus clauos reuellunt, forcipes item quæ tenaci <lb/>mor&longs;u clauorum capita vmbellasue apprendentes, vio<lb/>lenterè tabulis extrahunt. </s>

<s id="id.000660">In malleo itaque &longs;ubtili, vt in <lb/>figura videre e&longs;t, AB vectis e&longs;t pars quæ à fulcimento ad <lb/>potentiam; ac verò quæ à fulcimento ad pondus, ponderi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.072.jpg"/><figure id="id.007.00.072.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.072.1.jpg"></figure><lb/>&longs;iquidem æquiparatur re&longs;i<lb/>&longs;tentia quae fit in C. </s>

<s id="id.000661">I dem ob<lb/>&longs;eruamus in forcipe, in quo <lb/>duo quidem brachia AD, <lb/>CB, quatenus ad appren&longs;io<lb/>nem pertinet, fulcimentum, <lb/>habent in ip&longs;o <expan abbr="c&emacr;tro">centro</expan> &longs;eu ver<lb/>rebra, & ideo quo longiores <lb/>fuerint, eo tenaciùs appre<lb/>hendunt & retinent. </s>

<s id="id.000662">quate<lb/>nus autem ad extractionem, <lb/>facit, pro vnico forceps totus habetur vecte, cuius <expan abbr="quid&emacr;">quidem</expan> <lb/>pars à potentia ad fulcimentum AB. quæ verò à <expan abbr="fulcim&emacr;-to">fulcimen<lb/>to</expan> ad hoc e&longs;t clauum ip&longs;um qui reuellitur AC. </s>

<s id="id.000663">Violenti&longs;<lb/>&longs;imè autem extrahunt forcipes, propterea quod maxima <lb/>&longs;it proportio longitudinis brachij BA, ad eam quæ e&longs;t ab <lb/>A ad C. </s>

<s id="id.000664">His igitur hoc pacto examinatis, ad nauim & malum <lb/>reuertentes, dicimus, tunc facillimam fieri puppis eleua<lb/>tionem, proræ verò demer&longs;ionem, cum maxima fuerit <lb/>proportio, quam habet altitudo mali, ad eam nauis <expan abbr="part&emacr;">partem</expan> <lb/>quæ à malo ad ip&longs;am puppis extremitatem, pertingit. <lb/></s>

<s id="id.000665">Quamobrem prudentes nauium fabri, vt huic difficultati <lb/>occurrant, malum non in medio quidem nauis, &longs;ed in ter<lb/>tia ferè parte longitudinis quæ à prora e&longs;t, puppim ver&longs;us <lb/>con&longs;tituunt. </s><figure id="id.007.00.072.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.072.2.jpg"></figure>

<s id="id.000666">E&longs;to enim nauis AB; cuius <lb/>malus CD: prora A: puppis B; <expan abbr="v&emacr;-to">ven<lb/>to</expan> igitur velum impellente, <expan abbr="mal&umacr;">malum</expan> <lb/>ad partem contrariam vergit, pu<lb/>ta in FD. </s>

<s id="id.000667">At <expan abbr="quoniã">quoniam</expan> ca<gap/>che&longs;ium <lb/>funi ad puppim vnitur in B, nauim, <lb/>hoc e&longs;t, ip&longs;am puppim trahatne<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.073.jpg"/>ce&longs;&longs;e e&longs;t. </s>

<s id="id.000668">non pote&longs;t autem; quoniam &longs;uburræ grauitas & <lb/>onera, quæ naui impo&longs;ita inter D. & <emph type="italics"/>B.<emph.end type="italics"/> grauitatis centrum <lb/>circa punctum E con&longs;tituunt, quod quidem vi ventorum <lb/>inclinante malo ab E, in G eleuaretur, quo igitur minor <lb/>fuerit proportio CD ad DE & maius pondus ip&longs;um cu<lb/>ius grauitatis centrum in E minus præualebit potentia <lb/>pellens in C ad eleu<gap/>tionem partis nauigij, quæ à mali &longs;e<lb/>de ad puppim intercedit, An igitur malus &longs;it vectis, pesve<lb/>rò fulcimentum, pondus autem quodvecte mouetur, <expan abbr="ips&umacr;">ipsum</expan> <lb/>nauigium, vt placuit Ari&longs;toteli, & qua item ratione malus <lb/>in nauim vt vectis operetur, exijs quae dicta &longs;unt, facilè pa<lb/>tet. </s>

</subchap1><subchap1>

<s id="id.000669">QVÆSTIO VII.</s>

<s id="id.000670"><emph type="italics"/>Quaritur, Cur quando ex puppi nauigare voluerint, non flante ex <lb/>puppi vento, veli quidem partem, quæ ad gubernatorem vergit, <lb/>con&longs;tringunt; illam verò quæ proram ver&longs;us e&longs;t, pedem <lb/>facientes, relaxant?<emph.end type="italics"/></s>

<s id="id.000671">Mirabilis huius effectionis cau&longs;&longs;am explicat Ari&longs;tote<lb/>les. </s>

<s id="id.000672">inquit enim, An quia retrahere quidem multo <lb/>exi&longs;tente vento gubernaculum non pote&longs;t, pauco autem <lb/>pote&longs;t, quem con&longs;tringunt? </s>

<s id="id.000673">propellit igitur quidem ip&longs;e <lb/>ventus, in puppim verò illum con&longs;tituit gubernaculum, <lb/>retrahens, & mare compellens: &longs;imul & nautæ ip&longs;i cum <lb/>vento contendunt; in contrariam enim &longs;e reclinant par<lb/>tem. </s>

<s id="id.000675">Cuius &longs;en&longs;um breuitate &longs;ubob&longs;curum, mirâ facilita<lb/>te explicat Picolomineus. </s>

<s id="id.000676">Nos autem vt rem lucidiorem <lb/>faciamus, &longs;chema, quod necip&longs;e fecit, nec Philo&longs;ophus, <lb/>proponemus. </s>

<s id="id.000677">E&longs;to nauis A <emph type="italics"/>B<emph.end type="italics"/>, cuius prora A, puppis verò D, guber<lb/>naculum C<emph type="italics"/>B<emph.end type="italics"/>, temonis ala <emph type="italics"/>B<emph.end type="italics"/>D, veli &longs;inus EF, velum vero <lb/>ita con&longs;titutum, vt directè ex puppi flantem ventum exci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.074.jpg"/><figure id="id.007.00.074.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.074.1.jpg"></figure><lb/>piat. </s>

<s id="id.000678">Hoc vbi euenerit, naui<lb/>gium, rectâ è puppi mouetur <lb/>in proram; Si autem ventus la<lb/>teraliter &longs;pirat, puta à parte <lb/>G ver&longs;us H & nihilo &longs;ecius na<lb/>uigium, ac &longs;i ventus ex pup<lb/>pi e&longs;&longs;et antror&longs;um propelle<lb/>re volunt, velum quidem obli<lb/>quant partem cius infimam, <lb/>pedem nempe, quæ e&longs;t in F <lb/>contrahentes, Cornu verò <lb/>antennæ vbi E, proram ver&longs;us <lb/>laxantes ventumque; ip&longs;um obliquè ex cipientesid <expan abbr="effici&umacr;t">efficiunt</expan>, <lb/>vt ventus minus violenter feriat, & minori &longs;ui parte <expan abbr="vel&umacr;">velum</expan> <lb/>impleat, & quoniam ventus velum pellit in partem con<lb/>trariam, nempe in H, ip&longs;ivt vento re&longs;i&longs;tant conuer&longs;o gu<lb/>bernaculo ex C in L, & temone <emph type="italics"/>B<emph.end type="italics"/>D, in <emph type="italics"/>B<emph.end type="italics"/>M compellunt <lb/>proram ad partem à qua ventu<gap/> ip&longs;e &longs;pirat. </s>

<s id="id.000679">Sit igitur inter <lb/>ventum & temonem pugna, illo proram in dextram, hoc <lb/>verò eandem in &longs;ini&longs;tram pellente, <expan abbr="itaq;">itaque</expan> cum neuter præ<lb/>ualeat, nece&longs;&longs;ario nauis mediam viam, quæ inter <expan abbr="vtramq;">vtramque</expan> <lb/>e&longs;t, &longs;uo cur&longs;u tenet. </s>

<s id="id.000680">Nautæ autem ideo in partem nauis <lb/>AE<emph type="italics"/>B<emph.end type="italics"/>, quæ ver&longs;us ventum e&longs;t, &longs;e conferunt, vt vento æqui<lb/>librium faciant, ne &longs;cilicetnaui in <expan abbr="c&omacr;trariam">contrariam</expan> partem pel<lb/>lente &longs;piritu, eam demergat. </s>

<s id="id.000681">Cæterùm quod nec Ari&longs;to<lb/>teles nec Picolomineus animaduerterunt, velum obli<lb/>què con&longs;titutum à vento in anteriora impellitur eandem <lb/>ob cau&longs;&longs;am, quam retulimus, vbi de temone & velis, qui<lb/>bus farin ariæ molæ <expan abbr="c&omacr;uertuntur">conuertuntur</expan>, verba faceremus. </s>

<s id="id.000682">Quod <lb/>autem addit Picolomineus rem ad vectem reduci po&longs;&longs;e, <lb/>non e&longs;t cur &longs;ub &longs;ilentio prætereamus. </s>

<s id="id.000683">Ventus, in quit, pon<lb/>deris gubernaculum mouentis vicem obtinet; centrum <lb/>verò (fulcimentum intelligit) in medio nauis e&longs;t, quod ta-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.075.jpg"/>men ad proram vergit, vt faciliùs ip&longs;i vento re&longs;i&longs;tere po&longs;<lb/>&longs;it. </s>

<s id="id.000684">Tunc enim in rectum mouebitur nauis, cum &longs;ibi inui<lb/>cem æ quatæ vires, qua&longs;i libramentum con&longs;tituerint. </s>

<s id="id.000685">Hæc <lb/>ille, cuius &longs;en&longs;um figurâ propo&longs;itâ facilè aperiemus. </s><figure id="id.007.00.075.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.075.1.jpg"></figure>

<s id="id.000686">E&longs;to carina AB, cuius prora <lb/>A, puppis, B temo BC, ventus verò <lb/>obliquè feriens H. </s>

<s id="id.000687">Conuer&longs;us ita<lb/>que temo vt in BC vndarum vi cur<lb/>rente naui repul&longs;us &longs;it in EF ten<lb/>dens ver&longs;us I, quo ca&longs;u prora con<lb/>uertitur in D, nempe contra <expan abbr="vent&umacr;">ventum</expan> <lb/>qui &longs;pirat ex H. fit autem conuer<lb/>&longs;io circa punctum G, quod fulcimenti locum obtinet. </s>

<s id="id.000688"><expan abbr="V&emacr;-tus">Ven<lb/>tus</expan> verò ad contrariam <expan abbr="part&emacr;">partem</expan> proram impellit, repugnans <lb/>Temonis violentiæ contra ip&longs;am proram dirigentis. </s>

<s id="id.000689">E&longs;t i<lb/>gitur AB, &longs;eu DE carina, in&longs;tar vectis, cuius fulcimentum <lb/>G, vis mouens mare quo temo EF repellitur, pondus ve<lb/>ro, ventus premens in D; quo igitur remotior erittemo à <lb/>fulcimento G, D autem vbi pondus ei vicinius, eo magis <lb/>temo venti vim&longs;uperabit. </s>

<s id="id.000690">Hæc Picolominei ratio, quam <lb/>explicauimus, &longs;anè ingenio&longs;a e&longs;t, verum enimuero, quo<lb/>niam fulcimentum &longs;ui naturâ &longs;tare debet, hic verò <expan abbr="nullã">nullam</expan> <lb/>habeat &longs;tabilitatem, difficultatem patitur. </s>

</subchap1><subchap1>

<s id="id.000691">QVÆSTIO VIII.</s>

<s id="id.000692"><emph type="italics"/>Quæritur, Cur ex figuris omnibus rotundæ faciliùs <lb/>moueantur?<emph.end type="italics"/></s>

<s id="id.000693">Trifariam, in quit Ari&longs;toteles, circulum rotari contin<lb/>git; Aut &longs;ecundum ab&longs;idem <expan abbr="c&emacr;tro">centro</expan> &longs;imul moto, quem<lb/>admodum plau&longs;tri vertitur rota; aut circa manens cen<lb/>trum, velutitrochleæ puteorum, &longs;tante centro: Aut in pa<lb/>uimento manente centro, &longs;icuti figuli rota conuertitur. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.076.jpg"/>Cau&longs;&longs;am verò explicans, ait, celerrima eiu&longs;modi corpora <lb/>e&longs;&longs;e, eo quod paruâ &longs;ui parte planum contingunt, vti cir<lb/>culus &longs;e cundum punctum, item quoniam non offen&longs;ant: <lb/>Non offen&longs;andi vero e&longs;&longs;e cau&longs;&longs;am, quod &longs;emotum à terra <lb/>habeant angulum. </s>

<s id="id.000694">Item propterea quod corpus, cui fiunt <lb/>obuiam, &longs;ecundum pu&longs;illum tangunt. </s>

<s id="id.000695">Rectilineo autem <lb/>aliter euenire, quippe quod rectitudine &longs;uâ, multum pla<lb/>ni contingat. </s>

<s id="id.000696">Ad hæc, quo nutat pondus eo mouentem <lb/>mouere. </s>

<s id="id.000697">Hæc ferè Philo&longs;ophus, cuius rationes ad eum &longs;olum<lb/>modo circularem motum faciunt, qui fit &longs;e cundum ab&longs;i<lb/>dem, vt in carrorum rotis v&longs;u venit, nec aptantur rotis fi<lb/>gulorum trochlei&longs;queue, cuiu&longs;modi &longs;unt illæ, quæ &longs;upra <lb/>puteos appenduntur. </s>

<s id="id.000698">Nos igitur, ad Ari&longs;totelis mentem, <lb/>primam rotationis &longs;peciem, quæ e&longs;t &longs;ecundum ab&longs;idem, <lb/>examinabimus. </s><figure id="id.007.00.076.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.076.1.jpg"></figure>

<s id="id.000699">E&longs;to rota &longs;phæ<lb/>raue AB, cuius cen<lb/>trum C; Horizontis <lb/>planum DE; conta<lb/>ctus circuli in plano <lb/>B. <expan abbr="perp&emacr;">perpem</expan> dicularis ho<lb/>rizonti à puncto <expan abbr="c&omacr;-tactus">con<lb/>tactus</expan> B ip&longs;a <emph type="italics"/>B<emph.end type="italics"/>CA, <lb/>tran&longs;iens per <expan abbr="centr&umacr;">centrum</expan> <lb/>C, partes rotæ circa <lb/>perpendicularem AF<emph type="italics"/>B<emph.end type="italics"/>, AG<emph type="italics"/>B<emph.end type="italics"/>, angulus contactus G<emph type="italics"/>B<emph.end type="italics"/>E. <lb/></s>

<s id="id.000700">Primo itaque id con&longs;tat, circulum in puncto planum, &longs;eu <lb/>lineam contingere. </s>

<s id="id.000701">At quoniam, vt Mechanici, de circulis <lb/>roti&longs;queue &longs;eu &longs;phæris agimus materialibus, rectè Philo&longs;o<lb/>phus non in puncto planum præ cisè tangere dixit, &longs;ed &longs;e<lb/>cundum partem &longs;ui minimam. </s>

<s id="id.000702">Angulum porro, quem à <lb/>terra &longs;emotum dicit, ip&longs;e angulus e&longs;t contingentiae. </s>

<s id="id.000703">cleua<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.077.jpg"/>tur enim ex <emph type="italics"/>B<emph.end type="italics"/> in G. </s>

<s id="id.000704">Si autem corpus quodpiam in plano <lb/>fuerit, puta HI in puncto illud tanget ci culus ei occur<lb/>rens, exempli gratiâ in K. </s>

<s id="id.000705">Hæc igitur accidunt circulari <lb/>figuræ. </s>

<s id="id.000706">In lateratis autem &longs;ecus fit, quippe quænec in <expan abbr="p&umacr;-cto">pun<lb/>cto</expan> &longs;eu &longs;ecundum paruam &longs;ui partem, planum tangunt, <lb/>nec &longs;emotum vt circulus à plano habent angulum, nec <lb/>impingentes offen diculum in puncto tangunt. </s>

<s id="id.000707">Cæterùm <lb/>poti&longs;&longs;imam facilitatis motus in rotatione quæ fit &longs;ecun<lb/>dum ab&longs;idem, e&longs;&longs;e cau&longs;&longs;am dixit, nempe quò nutat pon<lb/>dus cò à mouente impelli ac moueri. </s>

<s id="id.000708">Primò igitu circu<lb/>laris &longs;phæricaue figura in æquilibrio &longs;tat; æquales enim <lb/>&longs;unt partes quæ circa perpendicularem: ceu &longs;unt AF<emph type="italics"/>B<emph.end type="italics"/>, <lb/>AG<emph type="italics"/>B.<emph.end type="italics"/> &longs;i enim impul&longs;us fiat ex parte F, pars oppo&longs;ita nuta<lb/>bit, & propendet in pa<gap/>tem G, & &longs;uo nutu motuque; &longs;ecum <lb/>trahet partem AF<emph type="italics"/>B<emph.end type="italics"/>, fietqueue progre&longs;&longs;us. </s>

<s id="id.000709">Si enim ducatur <lb/>FCG diameter, ip&longs;i horizonti æ que di&longs;tans, erit veluti li<lb/>bra, cuius pondera vtrinque AF<emph type="italics"/>B<emph.end type="italics"/>, AG<emph type="italics"/>B<emph.end type="italics"/>, brachia verò <lb/>æqualia CF, CG. </s>

<s id="id.000710">Potentia autem quâ trahitur pellitur<lb/>ue ad in&longs;tar ponderis &longs;e habet, quo addito partium alteri, <lb/>facto queue rece&longs;&longs;u ab æquilibrio, &longs;equetur motus. </s>

<s id="id.000711">Putauêre <lb/>quidam, vt refert Philo&longs;ophus, <expan abbr="circular&emacr;">circularem</expan> lineam, ita per<lb/>peti motu ver&longs;atumiri, vt manentia, propte<gap/> contrarium <lb/>nixum, manent, neque enim circulus in plano contrarium <lb/>nixum habet, cum &longs;it, veluti dicebamus, in æquilibrio & <lb/>facilis in vtramuis partem moueri. </s>

<s id="id.000712">Veruntamen perpe<lb/>tuum e&longs;&longs;e non po&longs;&longs;e horum corporum motum, ea e&longs;t cau&longs;<lb/>&longs;a, quod violentum accidat naturæ, & ideo non durabile. <lb/></s>

<s id="id.000713">Ad hæc, addit Philo&longs;ophus, Maiores circulos ad minores <lb/>nutum habere <expan abbr="qu&emacr;dam">quendam</expan>; & nutum maioris ad minoris nu<lb/>tum, &longs;e habere vt angulos ad angulos, & <expan abbr="diametr&umacr;">diametrum</expan> ad dia<lb/>metrum. </s>

<s id="id.000714">Angulos autem hî c&longs;ectores ip&longs;os vocat; oportet <lb/>enim circulos tum maiores tum minores circa idein cen<lb/>trum e&longs;&longs;e con&longs;titutos. </s>

<s id="id.000715">Hæc autem non ab&longs;imili ab eo <lb/>quod &longs;uprà po&longs;uimus &longs;chemate explicantur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.078.jpg"/><figure id="id.007.00.078.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.078.1.jpg"></figure>

<s id="id.000716">E&longs;to enim circulus <lb/>AB circa centrum, C, <lb/>Horizontis planum DE, <lb/>tangens circulum in B, <lb/>linea verò perpendicu<lb/>laris per centrum BCA. <lb/></s>

<s id="id.000717">Sit autem circa idem <expan abbr="c&emacr;-trum">cen<lb/>trum</expan> C, minor circulus <lb/>FG, ducatur queue CH &longs;e<lb/>cu<gap/> minorem circulum in I, tangens verò maiorem in H, <lb/>con&longs;tituen&longs;queue cum AC linea angulum ACH, duos an<lb/>gulos, ex Ari&longs;totelis mente comprehendentem, hoc e&longs;t, <lb/>duos &longs;ectores ACH, FCI. quoniam igitur &longs;ector &longs;eu an<lb/>gulus ACH, &longs;uo &longs;patio &longs;uperat angulum &longs;eu &longs;ectorem <lb/>FGI, facilè ex nutu quem maior &longs;upra minorem habet, <lb/>maior ip&longs;e mìnorem mouet. </s>

<s id="id.000718">Videtur autem tacitè Philo<lb/>&longs;ophus hæc ad vectis naturam referre, cuius altera extre<lb/>mitatum in centro &longs;it, altera verò in ab &longs;ide, & ita &longs;e habe<lb/>renutum maioris &longs;upra minorem, vt vectis ad vectem, hoc <lb/>e&longs;t, &longs;emid<gap/>ameter ad &longs;emidiametrum, &longs;eu &longs;ector ad &longs;ecto<lb/>rem, quos quidem &longs;ectores, vt vidimus, angulos appellat. <lb/></s>

<s id="id.000719">Hæc autem quæ de nutu refert, licet &longs;ubtilia &longs;int, vera e&longs;<lb/>&longs;e non videntur. </s>

<s id="id.000720">Si enim in figura producatur ad oppo&longs;i<lb/>tam partem &longs;emidiameter HC in K &longs;ecans minorem cir<lb/>culum in L, duos alios &longs;ectores angulosue habebimus, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> KCB, LCG, ip&longs;is ACHFCI æ quales. </s>

<s id="id.000721"><expan abbr="Itaq;">Itaque</expan> quan<lb/>tum adiuuat motum anguli ACH maioris nutus, in de<lb/>&longs;cendendo ad partes B, tantundem retardat anguli item <lb/>maioris KCB, contra nutus (vtita appellem) in <expan abbr="a&longs;cend&emacr;-do">a&longs;cenden<lb/>do</expan> ad partes A. & &longs;anè quatenus ad reinaturam pertinet <lb/>& ad ip&longs;um æquilibrium, non differunt maiores circuli à <lb/>minoribus, nec &longs;unt maiores minoribus mobiliores, imo <lb/>ex ali quaratione minores videntur fore ad motum faci<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.079.jpg"/>liores, tum quia data materiæ æqualitate &longs;unt leuiores, <lb/>tum etiam quod maior e&longs;t angulus contactus ad planuin <lb/>circum ferentiae minoris quàm maioris circuli, vt in &longs;ubie<lb/><figure id="id.007.00.079.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.079.1.jpg"></figure><lb/>cta figura angulus ABC maior <lb/>e&longs;t angulo DBC, in materiali i<lb/>gitur circulo rotaue maiore &longs;ui <lb/>parte tanget planum DB circu<lb/>lus, ip&longs;o AB. quicquid tamen fit, <lb/>mobiliores &longs;unt maiores circuli <lb/>non quidem ex natura circuli, <lb/>quæ tam in maioribus quàm in <lb/>ip&longs;is minoribus e&longs;t par, &longs;ed alijs de cau&longs;&longs;is, quas &longs;uo loco <lb/>examin abimus. </s>

<s id="id.000722">Cæterùm vt aliquid de motu qui &longs;e cundum ab&longs;idem <lb/>fit, ex no&longs;tro penu promamus, Dicimus, Circulos, rota&longs;ue, <lb/>quæ hoc pacto mouentur, vel per horizontis planum mo<lb/>ueri, vel per accliue, aut decliue. </s>

<s id="id.000723">Siautem perhorizontis <lb/>planum, ideo facilem e&longs;&longs;e motum, quòd nunquam, cæte<lb/>ris paribus, centrum grauitatis ip&longs;ius corporis à centro <lb/>mundi, in ip&longs;a rotatione, fiat remotius. </s><figure id="id.007.00.079.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.079.2.jpg"></figure>

<s id="id.000724">E&longs;to enim planum, <lb/>horizontis AB, cui circu<lb/>lus in&longs;i&longs;tat AD, circa cen<lb/>trum C, diui&longs;us per <expan abbr="centr&umacr;">centrum</expan> <lb/>ip&longs;um à perpendiculari <lb/>ACD; Ducatur autem per <lb/>centrum C recta linea ho<lb/>rizonti æquidi&longs;tans, ECFG: dum diuidatur circulus vt<lb/><gap/>unque in partes AH, HF, FI, ID, & CI, CH iungan<lb/>tur. </s>

<s id="id.000725">Po&longs;thæc intelligatur circulum &longs;ecundum ab&longs;idem <lb/>moueri ad partes G, erit igitur aliquando punctum H, <lb/>rangens horizontis planum, tangat autem in K, tum F in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.080.jpg"/>L, I in N. D verò in O. </s>

<s id="id.000726">Ducanturqueue KP, LQ, NR, OS <lb/>ip&longs;i AC parallelæ horizonti autem perpen diculares. <lb/></s>

<s id="id.000727">Centrum ergo circuli, quod idem & grauitatis e&longs;t <expan abbr="centr&umacr;">centrum</expan>, <lb/>feretur per rectam CPQRS, &longs;unt enim KP, LQ, NR, <lb/>OS ip&longs;i AC &longs;emidiam etro æquales, <expan abbr="n&umacr;quam">nunquam</expan> igitur cen<lb/>trum ip&longs;um C in circuli rotatione ab horizontis plano e<lb/>leuabitur, nec à mundi centro fietremotius. </s>

<s id="id.000728">Hoc autem longè aliter cæteris figuris contingit, <lb/>quarum motus ideo in æ qualis, quòd non &longs;em per in rota<lb/>tione centium grauitatis eandem &longs;eruet à mundi centro <lb/>di&longs;tantiam. </s><figure id="id.007.00.080.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.080.1.jpg"></figure>

<s id="id.000729">E&longs;to enim Ellip&longs;is <lb/>ABCD, cuius <expan abbr="c&emacr;trum">centrum</expan> <lb/>E, diameter longior <lb/>BED, breuior AEC, <lb/>Horizontis planum, <lb/>FCG. locus contactus <lb/>C perpendicularis à <lb/>contactu per centrum i<lb/>p&longs;a CEA diuidens El<lb/>lip&longs;im in partes æquales, & æqueponderantes ABC, <lb/>ADC. </s>

<s id="id.000730">Sumantur in quadrante CD, <expan abbr="p&umacr;cta">puncta</expan> HI, tum EH, <lb/>HI iungantur, eritautem EH longior ip&longs;a EC, tum EI, <lb/>ip&longs;a EH & ED, p&longs;a EI. </s>

<s id="id.000731">Rotetur ellip&longs;is &longs;ecun dum ab&longs;i<lb/>dem, fiet igitur punctum H in K, & à puncto K horizonti <lb/>perpendicularis erigatur KL, quæ fiat æ qualis EH. </s>

<s id="id.000732">P o&longs;t <lb/>hæc punctum I eritin M, & ab M perpen dicularis, æqua<lb/>lis EI. rui&longs;us D fiat in O, & ip&longs;i ED, æqualis perpendicu<lb/>laris OP. </s>

<s id="id.000733">Mota igitur ellip&longs;ià C in K, haud ita difficilis e<lb/>rit motus, quippe quod haud multum EH &longs;uperet EC, at <lb/>difficilior erit translatio in M, difficillima verò in O. Val<gap/><lb/>de enim à &longs;itu E, ibi attollitur grauitatis centrum, a&longs;cen<lb/>dens nempe vbi P. </s>

<s id="id.000734">Videmus igitur ex his eandem poten<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.081.jpg"/>tiam in mouendo ellip&longs;im, haud pariter &longs;e habere, vt in <lb/>mouendo circulum. </s>

<s id="id.000735">ibi enim centrum grauitatis fertur <lb/>per æquidi&longs;tantem horizonti, hic verò modò attollitur, <lb/>modò deprimitur, quod &longs;anè mole&longs;tiam & difficultatem <lb/>facit. </s>

<s id="id.000736">Sed idem alijs figuris contingere, & maximè latera<lb/>tis, ita docebimus. </s><figure id="id.007.00.081.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.081.1.jpg"></figure>

<s id="id.000737">E&longs;to enim triangulum <lb/>æquilaterum ABC, cuius <lb/>grauitatis centrum E hori<lb/>zontis planum BD. </s>

<s id="id.000738">Demit<lb/>tatur à vertice A perpendi<lb/>cularis horizonti AF tran&longs;<lb/>ibit autem per centrum E, <lb/>& bifariam diuidet ba&longs;im <lb/>BC in F. </s>

<s id="id.000739">Sunt autem trianguli ABF, ACF, æquales & <lb/>æqueponderantes. </s>

<s id="id.000740">angulus verò AFC rectus. </s>

<s id="id.000741">lungatur <lb/>EC, erit igitur maior EC, ip&longs;a EF. </s>

<s id="id.000742">Rotetur iraque trian<lb/>gulum circa punctum C, fiatque; EC horizonti perpendi<lb/>cularis, &longs;itqueue GH, & per E horizonti parallela ducatur <lb/>EK, moto igitur triangulo, centrum grauitatis E transla<lb/>tum erit in H, &longs;ed KC æqualis e&longs;t EF, minor autem ip&longs;a <lb/>CH, eleuatur ergo centrum grauitatis ab Ein H, nempe <lb/>&longs;upra K, totum &longs;patium KH. ex qua eleuatione fit in mo<lb/>tu difficultas. </s>

<s id="id.000743">Idem pror&longs;us eadem demon&longs;tratione o&longs;ten<lb/>deretur fieri in quadrato & alijs lateratis figuris. </s>

<s id="id.000744">Curigi<lb/>tur in plano horizontis facillimè circularia, difficile <expan abbr="aut&emacr;">autem</expan> <lb/>laterata & quæ inæquales habent &longs;emidiametros, mo<lb/>ueantur, ex dictis clarè patet. </s>

<s id="id.000745">Ad hanc quæ&longs;tionem illud quoque facit, cur per de<lb/>cliue planum grauiora corpora, & rotunda maximè; ma<lb/>gno impetu dimi&longs;&longs;a, delabantur. </s>

<s id="id.000746">E&longs;to enim rota &longs;phæraue aut Cylindrus CD, cuius <lb/>centrum E, tangens decliue planum AB in D, quæritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.082.jpg"/>cur dimi&longs;&longs;a hæc magno impetu deferantur ad partes B, <lb/>Ducatur per grauitatis centrum E ad horizontem, BK <lb/>perpendicularis FEL &longs;ecans decliue planum in G, cir<lb/>cum ferentiam verò in H. opponitur autem EG angulo <lb/>recto EDG, maior ergo EG ip&longs;a ED, hoc e&longs;t, EH, inter <lb/><figure id="id.007.00.082.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.082.1.jpg"></figure><lb/>circumferentiam igitur & pla<lb/>num decliue, &longs;patium interce<lb/>dit HG. </s>

<s id="id.000747">Ducatur item DI ip&longs;i <lb/>FG æquidi&longs;tans. </s>

<s id="id.000748">non tran&longs;ibit <lb/>igitur per centrum E. minor e<lb/>rit igitur diametro CD, quare <lb/>circulum in partes inæquales <lb/>&longs;ecabit, & non per grauitatis <lb/>centrum, quod idem cum ma<lb/>gnitudinis &longs;eu figuræ centro &longs;upponitur. </s>

<s id="id.000749">Dimi&longs;&longs;a igitur <lb/>rota, contingit quidem planum decliue in puncto D. </s>

<s id="id.000750">At <lb/>centrum grauitatis premit &longs;e cun dam per lineam perpen<lb/>dicularem FG, non &longs;u&longs;tentatur autem in H, quippe quod <lb/>inter planum & circum <expan abbr="ferentiã">ferentiam</expan> intercedat &longs;patium HG, <lb/>nec H locum habeat cui innitatur, corpus autem ita per <lb/>lineam DI e&longs;t diui&longs;um, vt longè maior &longs;it pars IFCHD <lb/>ip&longs;a DI, & centrum in ea parte eadat quæ non fulcitur. </s>

<s id="id.000752">taque &longs;uopte nutu, cum extra ful cimentum &longs;it D & per<lb/>pendicularem DI ad inferiores partes rapidè rotans de<lb/>labitur. </s>

<s id="id.000753">Ducatur autem perpen dicularis GL, parallela <lb/>MN, & quoniam BN breuior e&longs;t BL, erit MN ip&longs;a GL <lb/>breuior. </s>

<s id="id.000754">E&longs;t igitur punctum M mundi centro propius <lb/>quàm D & G, quare eò non impedita rota ip&longs;a &longs;uo nutu <lb/>feretur, nec&longs;tabit donec in fimum <expan abbr="loc&umacr;">locum</expan> vbi quie&longs;catnan. <lb/></s>

<s id="id.000755">ci&longs;catur. </s>

<s id="id.000756">Po&longs;&longs;umus etiam Rota &longs;phæraue in plano decliui <lb/>collocata, datam potentiam inuenire, quæ extremitati <lb/>diametri ad eam partem quavergit applicata ip&longs;am rotam <lb/>&longs;phæramue impediatne delabatur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.083.jpg"/><figure id="id.007.00.083.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.083.1.jpg"></figure>

<s id="id.000757">E&longs;to planum in clinatum <lb/>AB, cui Rota &longs;phæraue in&longs;i<lb/>&longs;tat tangatque; illud in C. </s>

<s id="id.000758">Rota <lb/>verò ip&longs;a &longs;phæraue DC, cu<lb/>ius centrum E, diameter ve<lb/>rò DEC ip&longs;i BA ad <expan abbr="punct&umacr;">punctum</expan> <lb/>contactus C, perpendicula<lb/>ris. </s>

<s id="id.000759">Ducatur per C ip&longs;i hori<lb/>zonti perpendiculatis FCG <lb/>circulum <expan abbr="&longs;ecãs">&longs;ecans</expan> in G tum per <lb/>E ip&longs;i CG perpendicularis, ip&longs;i verò BF horizonti æ qui<lb/>di&longs;tans HEI ceu vectis, cuius fulcimentum I re&longs;pondens <lb/>ip&longs;i C, pondus verò in E, vbi grauitatis e&longs;t centrum. </s>

<s id="id.000760">Ap<lb/>plicata igitur potentia in H erit pondus inter fulcimen<lb/>tum & potentiam, quare vt IE ad IH ita potentla &longs;u&longs;ti<lb/>nens in H ad pondus in E, quod demon&longs;trandum fuerat. </s>

<s id="id.000761">Quippiam &longs;imile o&longs;ten dit Pappus 1. 8. prop.

9. alijs <lb/>tamen &longs;uppo&longs;itis & con&longs;ideratis. </s>

<s id="id.000762">Dico præterea, ij&longs;dem <lb/>&longs;tantibus angulum ECI æqualem e&longs;&longs;e angulo inclinatio<lb/>nis CBF. </s>

<s id="id.000763">Producatur HI concurrens cum ip&longs;a AB in K, <lb/>concurret autem propterea, quod CIK rectus &longs;it, ICA <lb/>minorrecto, & quoniam HK parallela e&longs;t horizonti BF <lb/>alterni anguli IKC, CBF, æquales erunt. </s>

<s id="id.000764">Similes autem <lb/>&longs;unt ECI, ECK, trianguli, e&longs;tqueue ECI angulus æqualis <lb/>angulo EKC, hoc e&longs;t, ip&longs;i CBF. vnde &longs;equitur, quo mi<lb/>nor fuerit inclinationis angulus, eo facilius rotam &longs;phæ<lb/>ramue in piano inclinato &longs;u&longs;tineri. </s>

<s id="id.000765">quo enim minor fuerit <lb/>angulus ECI, eo minus latus EI & minor proportio EI <lb/>ad IH, & ideo minor potentia &longs;u&longs;tinens requiratur in H. <lb/></s>

<s id="id.000766">Cæterùm accliue & decliue planum nihil differunt ni&longs;i <lb/>re&longs;pectu. </s>

<s id="id.000767">His ita con&longs;ideratis, admonet nos locus, vt pulcher<lb/>rimam dubitationem diluamus. </s>

<s id="id.000768">Quæritur, Cur maiores <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.084.jpg"/>rotae impingentes, facilius offendicula &longs;uperent quàm mi<lb/>nores. </s>

<s id="id.000769">Neque enim &longs;atisfacere videtur quod ait Ari&longs;tote<lb/>les, ex contactu in puncto eo anguli à plano eleuatione id <lb/>fieri, alijs ergo principijs dubitatio &longs;oluitur. </s><figure id="id.007.00.084.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.084.1.jpg"></figure>

<s id="id.000770">E&longs;to rota quidem maior <lb/>AB, circa centrum C minor <lb/>vero DB circa centrum, E, <lb/><expan abbr="tãgentes">tangentes</expan> horizontis planum <lb/>in B. </s>

<s id="id.000771">Diameter maioris AB, <lb/>minoris DB, offen diculum, <lb/>horizonti perpendiculare <lb/>FG. </s>

<s id="id.000772">Ducatur per F horizonti <lb/>parallela FK &longs;ecans minoris <lb/>rotæ peripheriam in H, dia<lb/>metrum verò AB in K, & à <lb/>puncto H ad <expan abbr="plan&umacr;">planum</expan> horizon<lb/>tis perpendicularis demittatur HI: erit autem HI æqua<lb/>lis ip&longs;i offendiculo FG, & iungantur BH, BF. <expan abbr="Itaq;">Itaque</expan> quo<lb/>niam BH ab extremo B cadit in triangulum KFB, erit <lb/>KHB angulus maior angulo KFB. </s>

<s id="id.000773">Parallelæ autem &longs;unt <lb/><emph type="italics"/>K<emph.end type="italics"/>F, BG, pares ergo anguli <emph type="italics"/>K<emph.end type="italics"/>HB, HBG, pares item <emph type="italics"/>K<emph.end type="italics"/>FB, <lb/>FBG, Maior ergo HBI, ip&longs;o FBC. </s>

<s id="id.000774">At minoris rotæ gra<lb/>uitatis centrum mouetur &longs;ecundum lineam BH, maius <lb/>verò &longs;ecun dum literam BF, difficilius ergo mouebitur, & <lb/>&longs;uperabit offen diculum minorrota, quàm maior: quod <lb/>fuerat demon&longs;trandum. </s>

<s id="id.000775">Po&longs;&longs;umus idem o&longs;tendere magis mechanicè, hoc <lb/>e&longs;t, tem ad vectem reducendo. </s>

<s id="id.000776">E&longs;to horizontis planum <lb/>AB, rota maior CD planum tangens in D. rotæ verò ma<lb/>ioris centrum E. </s>

<s id="id.000777">Rota verò minor FD, tangens itidem <lb/>planum in D. rotæ autem centrum G, offendiculi verò re<lb/>ctitudo DH. </s>

<s id="id.000778">Ducatur per H ip&longs;i AB horizonti æquidi<lb/>&longs;tans HI &longs;ecans minorem circulum in K, maiorem verò <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.085.jpg"/><figure id="id.007.00.085.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.085.1.jpg"></figure><lb/>in I. </s>

<s id="id.000779">Ducantur etiam dia<lb/>metri maioris quidem <lb/>LEM, minoris NGO, <lb/>Tum à puncto K perpen<lb/>dicularis ducatur ad <lb/>GO, ip&longs;a KP, item à pun<lb/>cto I ad EM perpendi<lb/>cularis <expan abbr="Iq.">Ique</expan> Dico EQ ad <lb/>QL, minorem habere <lb/>proportionem quam GP, <lb/>ad PN. </s>

<s id="id.000780">Connectatur <lb/>GK, & ei per E parallela <lb/>ducatur ER, &longs;ecans maiorem circulum in R, & ab Rip&longs;i <lb/>EM perpen dicularis ducatur RS. quoniam igitur ER <lb/>parallela e&longs;t ip&longs;i GK, erit GER angulus HGK angulo <lb/>æqualis. </s>

<s id="id.000781">Recti autem &longs;unt HGP, GES reliqui ergo KGP, <lb/>RES ad inuicem &longs;unt æquales. </s>

<s id="id.000782">Sed & ESR, GPK recti <lb/>&longs;unt, quare ERSGKP anguli æquales &longs;unt, & trianguli <lb/>GPKESR, per pr. diff. </s>

<s id="id.000783">1.6. &longs;imiles. </s>

<s id="id.000784">Vtergo GK hoc e&longs;t <lb/>GN ad GP, ita ER hoc e&longs;t EL ad ES. </s>

<s id="id.000785">Componendo igi<lb/>tur vt NP ad PG, ita LS ad SE. quamobrem &longs;i fulcimen<lb/>tum e&longs;&longs;etin S, pondus in E, <expan abbr="pot&emacr;tia">potentia</expan> in L, idem &longs;ieret ac fiat <lb/>fulcimento in P, pondere in G, potentia verò in N con&longs;ti<lb/>tuta. </s>

<s id="id.000786">& id quidem &longs;i eiu&longs;dem ponderis vtraque rota &longs;up<lb/>ponatur. </s>

<s id="id.000787">Rur&longs;us quoniam vt DK ad totum circulum DF, <lb/>ita DR ad totum DC. </s>

<s id="id.000788">Minor e&longs;t autem proportio DI ad <lb/>totum circulum DC, ergo minor e&longs;t DI ip&longs;a DR. </s>

<s id="id.000789">Maior <lb/>ergo MI ip&longs;a MR, maior ergo QI ip&longs;a SR, propius ergo <lb/>centro E e&longs;t Q ip&longs;o puncto S, minor e&longs;t igitur proportio <lb/>EG ad LQ quàm ES ad SL. </s>

<s id="id.000790">Minor ergo potentia requi<lb/>ritur in L ad &longs;u&longs;tinendum pondus E ex fulcimento Q hoc <lb/>e&longs;t I, quàm requiratur in N ad &longs;u&longs;tinendum pondus G ex <lb/>fulcimento P, hoc e&longs;t K. </s>

<s id="id.000791">Minor ergo potentia requiritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.086.jpg"/>ad transferendam maiorem retam CD vltra offendicu<lb/>lum IV, hoc e&longs;t, DH, quàm requiratur ad trans ferendam <lb/>minorem vltra offendiculum KT, hoc e&longs;t HD, quod fue<lb/>rat o&longs;ten dendum. </s>

<s id="id.000792">Ad hæc, quæri pote&longs;t, quo pacto plau&longs;trorum rotæ <lb/>in ip&longs;a plau&longs;tri conuer&longs;ione &longs;e habeant, nempe quæ &longs;it li<lb/>neailla curua, quam in conuer&longs;ione de&longs;cribunt. </s><figure id="id.007.00.086.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.086.1.jpg"></figure>

<s id="id.000793">E&longs;to rotarum in <lb/>plano orbita, <expan abbr="d&umacr;">dum</expan> plau<lb/>&longs;trum rectâ procedit <lb/>AB, CD, Sunt autem i<lb/>p&longs;æ lineæ, quod o&longs;ten<lb/>demus po&longs;tea, æquedi<lb/>&longs;tantes. </s>

<s id="id.000794">Sit itaque pun<lb/>ctum. </s>

<s id="id.000795">B illud in quod <lb/>rota quæ per AB &longs;er<lb/>tur, eò delata planum <lb/>tangit. </s>

<s id="id.000796">D verò alterius rotæ at que plani contactus. </s>

<s id="id.000797">Igitur <lb/>dum plau&longs;tri fit conuer&longs;io, punctum D conuer&longs;ionis fit <lb/>centrum. </s>

<s id="id.000798">Stat enim interim rota & circa lineam conuer<lb/>titur, quæ å puncto contactus D per rotæ centrum ducta <lb/>horizontis plano e&longs;t perpendicularis. </s>

<s id="id.000799">ea autem &longs;tante, ro<lb/>ta quæ in B circa centrum D <expan abbr="&longs;emicircul&umacr;">&longs;emicirculum</expan> pertran&longs;it DEF, <lb/>vbi autem rota B, peruenerit in F, plau&longs;tro iam in oppo&longs;i<lb/>tam partem conuer&longs;o, rota quæ e&longs;t in D per lineam DC, <lb/>quæ verò in F per rectam FG mouetur, plau&longs;triqueue fit re<lb/>gre&longs;&longs;us. </s>

<s id="id.000800">Et quoniam vel D in ip&longs;a conuer&longs;ione &longs;tat omnino <lb/>nec quicquam progreditur, vt in prima figura, vel non &longs;tat <lb/>vt in &longs;ecunda, quo ca&longs;u portionem parui circuli de&longs;cribit, <lb/>ip&longs;i maiori circulo & exteriori concentricam. </s>

<s id="id.000801">Vnde col<lb/>ligimus, Plau&longs;trorum conuer&longs;iones flexione<gap/>queue &longs;emper <lb/>circa centrum, & con centricorum circulorum portiones <lb/>fieri, <emph type="italics"/>H<emph.end type="italics"/>inc etiam di&longs;cimus, cur veteres, vt ex antiquis co<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.087.jpg"/>gno&longs;cimus ve&longs;tigijs, circos in quibus cur&longs;us quadrigarum <lb/>fiebant ea forma quæ apparet, efformauerint. </s>

<s id="id.000802">Hoc etiam <lb/>theorema probamus. </s>

<s id="id.000803">Cylindros, quorum ba&longs;es axi &longs;unt perpendiculares, <lb/>dum in æquato plano conuoluuntur, rectâ incedere & <lb/>per parallelas, quarum di&longs;tantia axis &longs;eu latoris longitudi<lb/>ne præfinitur. </s><figure id="id.007.00.087.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.087.1.jpg"></figure>

<s id="id.000804">E&longs;to enim Cylin<lb/>drus ABCD, cuius a<lb/>xis GH, <expan abbr="horiz&omacr;tis">horizontis</expan> pla<lb/>no in&longs;i&longs;tens &longs;ecundum <lb/>latus AB, cui latus op<lb/>po&longs;itum & aequale CD. <lb/></s>

<s id="id.000805">Moueatur Cylindrus <lb/>rotans, donec latus <lb/>CD, in plano &longs;it vbi EF. </s>

<s id="id.000806">De&longs;cribat autem circuli CB <expan abbr="lineã">lineam</expan> <lb/>BF. </s>

<s id="id.000807">Circulo verò AD lineam AE. </s>

<s id="id.000808">Dico eas rectas e&longs;&longs;e, & <lb/>parallelas. </s>

<s id="id.000809">Si enim &longs;uperficies ba&longs;ium DA, CB, extendan<lb/>tur ita vt horizontis planum &longs;ecent, illud &longs;eca bunt iuxta <lb/>lineas AE BF, recta ergo e&longs;t vtraque. </s>

<s id="id.000810">Sed & parallelas e&longs;&longs;e <lb/>ad inuicem ita o&longs;tendimus. </s>

<s id="id.000811">quoniam &longs;emicirculus AD, <lb/>æqualis e&longs;t &longs;emicirculo BC, erit linea AE, æqualis lineæ <lb/>BF, &longs;ed & AB, æqualis e&longs;t ip&longs;i DC, quare & ip&longs;i EF. </s>

<s id="id.000812">Oppo<lb/>&longs;ita igitur quadrilateri figura ABFE latera æqualia &longs;unt, <lb/>quare EF æquedi&longs;tat ip&longs;i AB, tum AE ip&longs;i BF, quod fue<lb/>rat demon&longs;trandum. </s>

<s id="id.000813">Probabimus etiam &longs;i cylindri ba&longs;es axi perpendicu<lb/>lares non fuerint, & ideo ellip&longs;es in ip&longs;a rotatione perpla<lb/>num, parallelas quidem de&longs;cribere, &longs;ed non rectas. </s>

<s id="id.000814">E&longs;to enim Cylindrus ABCD, cuius ba&longs;es ellip&longs;es <expan abbr="inuic&emacr;">inuicem</expan> <lb/><expan abbr="æquedi&longs;tãtes">æquedi&longs;tantes</expan>, quarum axes longiores AB, CD, Commu<lb/>nis autem &longs;ectio cylindri & plani ad axem & horizontem <lb/>planum perpendicularis EHF. </s>

<s id="id.000815">Diuidatur autem &longs;emicir-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.088.jpg"/>culus EHF in partes æquales quatuor FI, IH, HG, GE. <lb/><figure id="id.007.00.088.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.088.1.jpg"></figure><lb/>Tum per diui&longs;ionum puncta lateri parallelae, rectæ ducan<lb/>tur KGL, M<emph type="italics"/>H<emph.end type="italics"/>N, OIP, quæ quidem <expan abbr="c&umacr;">cum</expan> ba&longs;es AMB, DNC <lb/>parallelæ &longs;int, eruntinuicem æ quales, cumqueue circum<lb/>ferentia E<emph type="italics"/>H<emph.end type="italics"/>F æquales, eosqueue rectos angulos <expan abbr="c&omacr;&longs;tituent">con&longs;tituent</expan>. <lb/></s>

<s id="id.000816">Ducatur po&longs;t hæc &longs;eor&longs;um recta QR, & eidem perpendi<lb/>cularis ST eam &longs;ecans in V. applicetur autem rectæ ST <lb/>æqualis Cylindri lateri BC, ip&longs;a <foreign lang="greek">hz. </foreign></s>

<s id="id.000817">ita tamen vt punctum <lb/>E congruat puncto V, &longs;itqueue V<foreign lang="greek">h</foreign> æqualis EB, V<foreign lang="greek">z</foreign>verò æ<lb/>qualis EC. </s>

<s id="id.000818">Tum fiant VX, XY, YZ, Z<foreign lang="greek">a</foreign> æ quales ip&longs;is EG, <lb/>G<emph type="italics"/>H<emph.end type="italics"/>, <emph type="italics"/>H<emph.end type="italics"/>I, IF, & per puncta X, Y, Z, <foreign lang="greek">a</foreign> & paralleli ip&longs;i ST du<lb/>cantur <foreign lang="greek">o a p, n *z c, l g m, k x q</foreign>, tum & his ex altera parte re<lb/>&longs;pondentes parallelæ per puncta <foreign lang="greek">b, g, d, e. </foreign></s>

<s id="id.000819">Sit autem <foreign lang="greek">o a</foreign> æ<lb/>qualis AF, <foreign lang="greek">a</foreign> <11> æqualis FD, item <foreign lang="greek">e</foreign> <10>, æqualis EC, <foreign lang="greek">e s</foreign> æqualis <lb/>EB, &longs;ed & <foreign lang="greek">n *z</foreign> aequalis OI, <foreign lang="greek">*z c</foreign> ip&longs;i P, <foreign lang="greek">l</foreign>yi<gap/> &longs;i MH, y <foreign lang="greek">m</foreign> verò ip&longs;i <lb/>HN, <expan abbr="t&umacr;">tum</expan> <foreign lang="greek">k x</foreign> ip&longs;i KG. & <foreign lang="greek">x q</foreign>, ip&longs;i GL & ip&longs;is æquales & aequa<lb/>liter po&longs;itæ ad partes R, aliæ paralle læ <expan abbr="apt&emacr;tur">aptentur</expan> per <foreign lang="greek">b, g, d, c</foreign>, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.089.jpg"/>quibusita di&longs;po&longs;itis per puncta <foreign lang="greek">o, n, l, k, h</foreign>, item per <foreign lang="greek">p, c, m, q, z</foreign>. <lb/></s>

<s id="id.000820">ducantur lineæ <foreign lang="greek">oh, pz</foreign>, curuæ quidern & codem pacto a<lb/>liæ curuæ illis re&longs;pondentes <foreign lang="greek">h <10>, zs</foreign>, Erunt igitur <foreign lang="greek">o, h, <10>, <lb/>p, z, s</foreign>, parallelæ quidem eo quod lincae quæ inter ip&longs;as du<lb/>cuntur, parallelæ &longs;int & æquales, non tamen rectæ illæ, <lb/>&longs;ed curuæ. </s>

<s id="id.000821">Moto igitur Cylindro circulus EHF rectam <lb/>de&longs;cribet<foreign lang="greek">ae</foreign>, ellip&longs;is verò AMB, curuam <foreign lang="greek">ohr</foreign>, ellip&longs;is au<lb/>rem DNC, ip&longs;am curuam <foreign lang="greek">pzs. </foreign></s>

<s id="id.000822">In hoc <expan abbr="aut&emacr;">autem</expan> Cylindri mo<lb/>tuillud mirabile, velociores nempe, in ip&longs;a rotatione e&longs;&longs;e <lb/>ellip&longs;es ip&longs;o circulo EHF. </s>

<s id="id.000823">Ducatur enim recta<foreign lang="greek">o<10></foreign> quæ oc<lb/>currat ip&longs;i VS in S, & <foreign lang="greek">oh</foreign> iungatur, fietqueue triangulum <lb/><foreign lang="greek">oh</foreign>S. c&longs;t autem, angulus <foreign lang="greek">o</foreign> S <foreign lang="greek">h</foreign> rectus, maior erg. <foreign lang="greek">oh</foreign> i<lb/>p&longs;a <foreign lang="greek">o</foreign> S, &longs;ed recta <foreign lang="greek">o</foreign> S æqualis e&longs;t ip&longs;i<foreign lang="greek">an</foreign>, hoc e&longs;t, &longs;emicircu<lb/>lo FHE. multo maior e&longs;t autem curua, <foreign lang="greek">o, n, l, k, h</foreign>, ip&longs;a recta <lb/><foreign lang="greek">oh</foreign>, &longs;ed eodem tempore quo &longs;emicirculus EHF conficit <lb/>in rotatione <expan abbr="&longs;pati&umacr;">&longs;patium</expan> <foreign lang="greek">a</foreign> V, eodem dimidia ellip&longs;is BMA me<lb/>titur curuam <foreign lang="greek">onlkh. </foreign></s>

<s id="id.000824">velocior igitur e&longs;t ellip&longs;is ip&longs;o cir<lb/>culo. </s>

<s id="id.000825">Hæc quo que &longs;peculatio ad motum qui &longs;ecundum <lb/>ab&longs;idem fit, manife&longs;tè pertinet. </s>

<s id="id.000826">Coni, quorum ba&longs;es cir<lb/>culi &longs;unt, &longs;i in plano &longs;ecundum latus rotentur, ba&longs;i circu<lb/>lum de&longs;cribunt, cuius centrum immobile coni ip&longs;ius e&longs;t <lb/>vertex, &longs;emidiameter verò ip&longs;um latus. </s><figure id="id.007.00.089.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.089.1.jpg"></figure>

<s id="id.000827">E&longs;to conus ABC cu<lb/>ius vertex C ba&longs;is AB, axis <lb/>DC, ba&longs;is verò centrum, <lb/>D, latus quo planum tan<lb/>git BC, &longs;ecatur itaque Co<lb/>nus per latus BC & axem <lb/>DE à plano horizonti per<lb/>pendiculari, cuius & coni <lb/>communis &longs;ectio e&longs;t ABC <lb/>triangulum, & quoniam coni grauitatis centrum e&longs;t in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.090.jpg"/>axe ip&longs;o, conus in partes æque <expan abbr="p&omacr;derantes">ponderantes</expan> &longs;ecatur AEBC, <lb/>AFBC, &longs;tat ergo conus &longs;ibimet æquilibris. </s>

<s id="id.000828">Si autem à po<lb/>tentia quadam moueatur, puta ab A ver&longs;us F, trahitur &longs;e<lb/>micirculus BEA, à &longs;emicirculo AFB, & ita fit rotatio. </s>

<s id="id.000829">Ita<lb/>que &longs;i imaginemur, in finitos v&longs;que ad verticem parallelos <lb/>ba&longs;i cir culos, eorum &longs;emicirculi in ip&longs;o motu & trahent & <lb/>trahentur; at cum ad verticem circuli de&longs;inant, nec ibi &longs;e<lb/>micirculi &longs;int qui trahant & trahantur, motus rotationis <lb/>pror&longs;us ce&longs;lat & vertex ip&longs;e immobilis fit rotationis cen<lb/>trum. </s>

<s id="id.000830">Quoniam igitur lateris BC, punctum C &longs;tat, B verò <lb/>circa ip&longs;um mouetur, in ip&longs;o motu circulus de&longs;cribitur <lb/>BHIK, cuius &longs;emidiameter BC, & eodem pacto alij cir<lb/>culi in cono, qui ba&longs;i HEBF &longs;unt æquedi&longs;tantes, circulos <lb/>in plano circa idem centrum de&longs;cribent, vt facile videre <lb/>e&longs;t in obiecto &longs;chemate. </s>

<s id="id.000831">Huic &longs;imilem demon &longs;trationem <lb/>affert Heron in libello Automatum, quem nos Tyrones <lb/>adhuc vernacule è Græco translatum, Ven<gap/>e tijs prælo <lb/>&longs;ubiecimus. </s>

<s id="id.000832">Porrò &longs;i conus rotundus pro ba&longs;i ellip&longs;im habeat, <lb/>&longs;ectionem videlicet per planum axi non perpendiculare, <lb/>in ip&longs;a rotatione, &longs;tante vertice, ellip&longs;is ba&longs;is, ellip&longs;im de<lb/>&longs;cribit in plano, cuius maior diameter à puncto quod co<lb/>nivertex e&longs;t, ita diuiditur, vt diametri pars maior æqualis <lb/>&longs;it lateri maximo; minor verò æqualis lateri minimo. </s>

<s id="id.000833">Sed <lb/>hæc ad aliam pertinent &longs;peculationem. </s>

<s id="id.000834">His itaque de moturotundorum, qui circa ab&longs;idem <lb/>fit, con&longs;ideratis, reliquum e&longs;&longs;et de motu trochlearum, qui <lb/>circa centrum &longs;it, opportunè agere, &longs;ed cùm in &longs;equenti <lb/>quæ&longs;tione de hoc &longs;ermonem faciat Philo&longs;ophus, ad ea <lb/>quæ ibi di&longs;puta buntur, lectorem ablegamus. </s>

<s id="id.000835">Modò de tertia motus &longs;pecie nobis erit &longs;ermo; in <lb/>qua quidem &longs;pecienonnulla perpendemus, quæ omi&longs;it A<lb/>ri&longs;toteles. </s>

<s id="id.000836">Agitur autem hîc de rotundorum corporum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.091.jpg"/>motu, qui fit çirca axem horizonti perpendicularem, axis <lb/>altera extremitate in codem horizontis plano manente, <lb/>vti videre e&longs;t in ip&longs;is figulorum rotis. </s>

<s id="id.000837">Hanc motus &longs;peciem in extrema quæ&longs;tionis parte <lb/>cum duabus alijs &longs;peciebus comparans ait, cam quæ in <lb/>obliquo fit motionem (ita enim hanc, de qua agimus, ap<lb/>pellat) ip&longs;am impellere mouentem, hoc e&longs;t, nullum ex&longs;e <lb/>ad motum propen&longs;ionem habere, nutumue, & omnia illi <lb/>e&longs;&longs;e à motore, &longs;ecundum verò eam motionem, quæ &longs;upra <lb/>diametrum e&longs;t, &longs;e ip&longs;um mouere circulum. </s>

<s id="id.000838">Dixerat enim, <lb/>ea referens quæ &longs;uperiùs circa principium de circulo ver<lb/>ba faciens, examinauerat, circulum ex duabus fieri latio<lb/>nibus, altera præter, altera verò &longs;ecundum naturam, & <lb/>ideo hanc &longs;emper nutum habere, & ceu continuo motam <lb/>ab eo moueri quimouet. </s>

<s id="id.000839">Videtur autem clarè profiteri, <lb/>ideo difficiliorem e&longs;&longs;e huius terriæ &longs;peciei motum, eo <lb/>quòd nutu <gap/>areat proprio & t<gap/>m ab alieno, vt ita di<lb/>cam, motore, moueatur. </s>

<s id="id.000840">Veruntamen motum hunc facilitate alijs illis duo<lb/>bus nequaquam cedere, facilè ex &longs;equentibus o&longs;tende<lb/>mus. </s>

<s id="id.000841">Primo, quia pondus totum rotati corporis, ex graui<lb/>tatis centro quodin ip&longs;o axe e&longs;t à plano cuinititur, &longs;u&longs;ti<lb/>netur: minima quidem &longs;ui parte axe ip&longs;o tangente <expan abbr="plan&umacr;">planum</expan> <lb/>vndefit, nullam ferè dum rotatur corpus, circa centrum <lb/>vbi nititur, frictionem partium fieri. </s>

<s id="id.000842">Præterea grauitatis <lb/>centrum &longs;emper &longs;tat, nec minimum quidem in ip&longs;a rota<lb/>tione attollitur, quod &longs;anè cum naturæ &longs;it repugnans, dif<lb/>ficultatem facit. </s>

<s id="id.000843">Ad hæc circa axem ita libratur rota, vt <lb/>quantumuis exigua potentia alteri parti applicetur, alte<lb/>ra illico &longs;uperata moueatur. </s>

<s id="id.000844">Licet enim propliè ea <expan abbr="tant&umacr;">tantum</expan> <lb/>corpora æquilibrare dicantur, quæ ob ponderis hinc in de <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.092.jpg"/>æqualitatem horizonti fiunt æquidi&longs;tantes, nihilominus <lb/>& hic aliquam e&longs;&longs;e æquilibrij &longs;imilitudinem patebit. </s><figure id="id.007.00.092.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.092.1.jpg"></figure>

<s id="id.000845">E&longs;to enim rota ABCD, <lb/>cuius axis horizonti perpendi<lb/>cularis FEG tran&longs;iens per cen<lb/>trum E, tangens autem planum <lb/>in puncto G. </s>

<s id="id.000846">Ducatur diame<lb/>ter BED, Itaque &longs;i per diame<lb/>trum BED, & axem FEG cor<lb/>pus diuidatur, eo quòd <expan abbr="centr&umacr;">centrum</expan> <lb/>grauitatis in axe inueniatur, <lb/>corpus ip&longs;um in duas partes <expan abbr="t&umacr;">tum</expan> <lb/>mole tum <expan abbr="p&omacr;dere">pondere</expan> æquales &longs;ecabitur, nempe BAD, BCD. <lb/></s>

<s id="id.000847">Nullaigitur adhibita vi extranea &longs;tabit corpus in <expan abbr="quodã">quodam</expan>, <lb/>vt diximus, æquilibrio. </s>

<s id="id.000848">At alteri partium potentiâ quauis <lb/>licet exigua appo&longs;itâ, puta in C, præualebit pars BCD, & <lb/>partem BAD vel impellet vel rapiet, alterâ interim eius <lb/>motui ob&longs;equente. </s>

<s id="id.000849">Potentia igitur quæ in C, nullam rem <lb/>quæ impediat inueniens, veloci&longs;&longs;imè rotam mouet, quod <lb/>eo faciliùs velocius queue fit, quo magis rota e&longs;t in motu, e<lb/>ius verò diameter maior & potentia mouens à centro re<lb/>motior, & &longs;anè motus <expan abbr="facilitat&emacr;">facilitatem</expan> inde cogno&longs;cimus, quòd <lb/>ip&longs;o impul&longs;ore ab impul&longs;u ce&longs;&longs;ante, diuti&longs;<gap/>è rota im<lb/>pre&longs;&longs;um motum &longs;eruet, nec ni&longs;i po&longs;t longam rotationem <lb/>omnino quie&longs;cat. </s>

<s id="id.000850">Cæterùm quia &longs;icco, vtaiunt, pede Ari&longs;toteles quæ <lb/>ad hunc motum <expan abbr="pertin&emacr;t">pertinent</expan> pertran&longs;ijt, nos quædam quæ ad <lb/>hanc rem faciunt, diligentiùs expendemus. </s>

<s id="id.000851">Quærimus igitur primò; Cur ea quæ hoc pacto <expan abbr="ro-tãtur">ro<lb/>tantur</expan>, in ip&longs;a rotatione locum non mutent, ni&longs;i extrin&longs;eca <lb/>aliqua id fiat ex cau&longs;&longs;a. </s>

<s id="id.000852">E&longs;to enim rota aut aliud quippiam rotundum ccu <lb/>Turbines &longs;unt, quibus pueri ludunt, quod circa axem ho<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.093.jpg"/><figure id="id.007.00.093.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.093.1.jpg"></figure><lb/>rizonti perpendicularem mo<lb/>ueatur, ABCD, cuius centrum <lb/>E, Diameter AEC. </s>

<s id="id.000853">Modò circa <lb/>centrum E in finiti imagin entur <lb/>circuli, alij alijs minores v&longs;que <lb/>ad <expan abbr="centr&umacr;">centrum</expan> ip&longs;um, vti &longs;unt FGH; <lb/>ibi enim circuli e&longs;&longs;e de&longs;inunt, <lb/>vbi nullum amplius e&longs;t &longs;patium. <lb/></s>

<s id="id.000854">Applicetur itaque potentia in <lb/>B, quæ rotam v. geat ver&longs;us A. <lb/>codem igitur tempore & in&longs;imul A ver&longs;us D, D ver&longs;us C, <lb/>& Cver&longs;us B mouebitur. </s>

<s id="id.000855">quantum enim &longs;emicirculorum <lb/>à parte CBA tran&longs;it vltra diametrum AEC, tantundem <lb/>&longs;emicir culorum, qui &longs;unt ad partem ADC, tran&longs;ibit ad <lb/>partes CBA. </s>

<s id="id.000856">At vbi de&longs;ierit motus, ibi de&longs;init rotatio; vbi <lb/>autem de&longs;init &longs;patium, de&longs;init motus, &longs;ed vbi de&longs;inunt cir<lb/>culi, de&longs;init &longs;patium, quare in centro cum non &longs;int circuli, <lb/>nec &longs;patium ibi de&longs;init motus. </s>

<s id="id.000857">nulla enim ade&longs;t ratio, cur <lb/>ip&longs;um corpus alio à loco in quo e&longs;t, ex rotatione transfe<lb/>ratur. </s>

<s id="id.000858">Statergo rotans, quod fuerat demon&longs;trandum. </s>

<s id="id.000859">E&longs;t <lb/>autem hæc demon&longs;tratio ei &longs;imilis, quam &longs;uprà retuli<lb/>mus de coni in plano circa verticem rotatione, quam ab <lb/>Herone in Automatis excogitatam diximus. </s>

<s id="id.000860">Addimus in hoc rotationis genere corpus in ip&longs;o<lb/>motu fieri leuius, idqueue eo magis, quo rotatio velocior. <lb/></s>

<s id="id.000861">Cau&longs;&longs;a e&longs;t, quod lateralis motus eum motum aliqualiter <lb/>impedit, qui ex naturali grauitate fit ad centrum, idcirco <lb/>experientiâ docemur, leui&longs;&longs;imos e&longs;&longs;e turbines, quibus pu<lb/>eri ludunt, &longs;i manus teneantur palmâ, dum citi&longs;&longs;ima rota<lb/>tione mouentur. </s>

<s id="id.000862">Ad hæc alia proponitur, & &longs;oluitur quæ&longs;tio, Cur ro<lb/>tunda corpora huic motionis generi &longs;int aptiora. </s>

<s id="id.000863">Explorati&longs;&longs;imum e&longs;t, corporum, quæ ita mouentur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.094.jpg"/>partes eo e&longs;&longs;e velociores, quo magis à centro, circa quod <lb/>mouentur, fuerint remotiores. </s>

<s id="id.000864">maius enim eodem tem<lb/>pore &longs;patium pertran&longs;eunt. </s>

<s id="id.000865">quo igitur figura ijs partibus, <lb/>quæ longius à centro ab&longs;unt, abundauerit magis, eo faci<lb/>lius, & velocius in circulum rotata mouebitur. </s>

<s id="id.000866">Modò o<lb/>&longs;tendemus, circularem cæteras omnes ea qua diximus <lb/>partium à centro remoti&longs;&longs;imarum copiâ abundare. </s><figure id="id.007.00.094.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.094.1.jpg"></figure>

<s id="id.000867">E&longs;to triangulum puta æqui<lb/>laterum, ABC circa centrum D. <lb/></s>

<s id="id.000868">Ducantur Catheti per centrum ab <lb/>oppo&longs;itis angulis ad oppo&longs;ita late<lb/>ra ADG, BDF, CDE, erunt autem <lb/>lateribus perpendiculares. </s>

<s id="id.000869"><expan abbr="quoniã">quoniam</expan> <lb/>igitur latera AD, DB, DC, rectis <lb/>angulis &longs;ubtenduntur, maiora <expan abbr="er&umacr;t">erunt</expan> <lb/>lateribus DE, DF, DG. tres igitur <lb/>lineæ in hoc triangulo &longs;unt longi&longs;&longs;imæ DA; DB, DC. tres <lb/>verò breui&longs;&longs;imæ DE, DG, DF, quamobrem rotato &longs;uper <lb/>centrum D triangulo, tres tantum partes eius ABC velo<lb/>ci&longs;&longs;imæ erunt, tres verò tardi&longs;&longs;imæ E, G, F. </s>

<s id="id.000870">Minus igitur a<lb/>pta e&longs;t motui huic triangularis figura, quam quadrata, in <lb/>qua partes à centro remoti&longs;&longs;imè, & ideo veloci&longs;&longs;imè &longs;unt <lb/>quatuor. </s>

<s id="id.000871"><expan abbr="Itaq;">Itaque</expan> quo <gap/>agis laterata figura angulis abunda<lb/>bit, eo magis erit ad hunc, & cæteros omnes circulares <lb/>motus aptior. </s>

<s id="id.000872">At circulus infinitas, vt ita dicam, partes à <lb/>centro remoti&longs;&longs;imas habet, itaque nulla figura e&longs;t circu<lb/>lari, in ip&longs;a rotatione, commodior atque velocior. </s>

<s id="id.000873">Alia <lb/>quoque de cau&longs;&longs;a id fit, quod dum circularis figura mo<lb/>uetur, nullis eminentibus angulis aërem verberet <expan abbr="circ&umacr;-&longs;tãtem">circun<lb/>&longs;tantem</expan>, ex qua verberatione motus impeditus &longs;it tardior. <lb/></s>

<s id="id.000874">Quæri etiam pote&longs;t, Num axe in clinato, rotæ motus ali<lb/>qualiter impediatur? </s>

<s id="id.000875">Nos negatiuam partem amplecti<lb/>mur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.095.jpg"/><figure id="id.007.00.095.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.095.1.jpg"></figure>

<s id="id.000876">E&longs;to enim tota ABCD, cuius cen<lb/>trum E axis inclin itus, circa quem <lb/>conuertitur EGF. </s>

<s id="id.000877">Duobus aute pun<lb/>ctis fulcitur GF. </s>

<s id="id.000878">Sit autem tum gra<lb/>uius tum figuræ centrum E, Perpen<lb/>dicularis vero per inferius fulcimen<lb/>tum tran&longs;iens HFI. </s>

<s id="id.000879">Conuer&longs;a igitur <lb/>rota, grauitatis centrum &longs;tabit nec à <lb/>&longs;uo &longs;itu &longs;ur&longs;um deor&longs;umue mouebi<lb/>tur. </s>

<s id="id.000880">E&longs;t autem axis FEG, ceu vectis in <lb/>quo pondus in E, potentiæ &longs;u&longs;tinentes GF; non enim hic <lb/>vt in axe perpendiculari pondus totum ab inferiori fulci<lb/>mento &longs;u&longs;tinetur. </s>

<s id="id.000881">quo igitur minor erit proportio FE ad <lb/>FG, eo minori in digebit potentiâ is qui pondus &longs;u&longs;tinet in <lb/>G. </s>

<s id="id.000882">Et hæc &longs;anè ita &longs;e habent, grauitatis çentro in axe ip&longs;o <lb/>con&longs;tituto, &longs;i enim extra fuerit motus impeditur & moto<lb/>re ce&longs;&longs;ante citò quie&longs;cit. </s>

<s id="id.000883">E&longs;to enim grauitatis centrum in <lb/>K. </s>

<s id="id.000884">Dum igitur circa axem fit motus, centrum circulatum <lb/>aliquando erit in L; Secetautem rotæ diameter AC per<lb/>pendicularem Hl in M. </s>

<s id="id.000885">Porrò à punctis LK ad ip&longs;am <expan abbr="per-p&emacr;dicularem">per<lb/>pendicularem</expan> ducantur ad rectos angulos lineæ LN, KO. <lb/></s>

<s id="id.000886">Maior e&longs;t autem MK ip&longs;a ML, maior ergo MO, ip&longs;a MN. <lb/>magis igitur à mundi centro di&longs;tat punctum N puncto O. <lb/></s>

<s id="id.000887">Centrum ergo grauitatis K &longs;i liberè dimittatur, requie&longs;cet <lb/>in K & contranaturam transferetur in L. </s>

<s id="id.000888">Ce&longs;&longs;ante igitur <lb/>violentiâ & præualentenaturâ citò rota &longs;uâ &longs;ponte quie<lb/>&longs;cet, quod fuerat o&longs;tendendum. </s>

</subchap1><subchap1>

<s id="id.000889">QVÆSTIO IX.</s>

<s id="id.000890"><emph type="italics"/>Quæritur, Cur ea quæ per maiores cir culos tolluntur, & trahuntur <lb/>faciliùs, & celeriùs moueri contingat, veluti maioribus tro<lb/>chleis, & &longs;cytalis &longs;imiliter?<emph.end type="italics"/></s>

<s id="id.000891">Re&longs;pondet ad hæc Philo&longs;ophus, forteid cuenire, quo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.096.jpg"/>niam quanto maior fucrit illa quæ à centro e&longs;t, in æquali <lb/>tempore maius mouetur &longs;patium. </s>

<s id="id.000892">quamobrem æquali <lb/>exi&longs;tente onere idem faciet. </s>

<s id="id.000893">Ita enim dixerat de <expan abbr="librar&umacr;">librarum</expan> <lb/>natura, & differentijs agens, maiores minoribus exactio<lb/>res e&longs;&longs;e. </s>

<s id="id.000894">Circulos verò jibras, in quibus centrum &longs;partum, <lb/>&longs;emidiametri hinc in de æqualia brachia. </s>

<s id="id.000895">Quod vltimo loco affirmauit, trochleas e&longs;&longs;e in&longs;tar <lb/>librarum, verum e&longs;t. </s>

<s id="id.000896">Quo d autem dixit, faciliùs & cele<lb/>rius mouere maiores libras ijs quæ minores &longs;unt, &longs;i &longs;inipli<lb/>citer intelligatur, fal&longs;um, quippe quod facilitas motus, in <lb/>tractorijs machinis velocitati &longs;it contraria, quod demon<lb/>&longs;trauit Guid. Vbald. in tractatu de Trochlea in 2. Corol<lb/>lario propo&longs;itione vltima. </s>

<s id="id.000897">Ad id autem quod dixit, quo <expan abbr="maior&emacr;s">maiorens</expan> fuerint tro<lb/>chleæ, eo faciliùs mouere, non e&longs;t, vt dicebamus, &longs;impli<lb/>citer verum, quod facilè o&longs;tendemus. </s><figure id="id.007.00.096.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.096.1.jpg"></figure>

<s id="id.000898">Efto enim trochlea AB circa centrum C, appen&longs;a in <lb/>puncto D, perpendicularis quæ ad mundi centrum DCE, <lb/>pondera æqualia vtrinque appen&longs;a FG. </s>

<s id="id.000899">E&longs;to item alia <lb/>Trochlea, eaque; maior HI, circa centrum K appen&longs;a in L, <lb/>perpen dicularis, quæ ad mundi centrum LKM, æqualia <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.097.jpg"/>pondera vtrinque appen&longs;a N, O. </s>

<s id="id.000900">Dico maiorem Hl ip&longs;a <lb/>minori DE facilius pondera non mouere, eo quòd &longs;it ma<lb/>ior, illa verò difficiliùs, propterea quòd &longs;it minor. </s>

<s id="id.000901">Etcnim, <lb/>quoniam vtraque trochlea per centrum grauitatis à per<lb/>pendiculari diui ditur, erunt partes DAE, DBE, æque <expan abbr="p&omacr;-derantes">pon<lb/>derantes</expan>. </s>

<s id="id.000902">Eadem ratione ip&longs;æ quoque LHM, L<gap/>M æquè <lb/>ponderabunt. </s>

<s id="id.000903">Itaque &longs;i quantumuis pu&longs;illa pondera ad<lb/>das, <expan abbr="vtriq;">vtrique</expan> earum ad alteram partem tolletur <expan abbr="æquilibri&umacr;">æquilibrium</expan>, <lb/>nec minus requiritur pondus vt recedat ab æquilibrio <lb/>Trochlea minor, quàm maior. </s>

<s id="id.000904">Vnico autem verbo con<lb/>cludi pote&longs;t di&longs;putatio, <expan abbr="tã">tam</expan> in minori quàm in maiori, bra<lb/>chia &longs;iqui dem bifariam diuiduntur, ergo in <expan abbr="vtri&longs;q;">vtri&longs;que</expan> eadem <lb/>brachiorum proportio, & eadem ponderum ratio. </s>

<s id="id.000905">Explorati&longs;&longs;ima &longs;unt hæc. </s>

<s id="id.000906">Veruntamen cùm res ip&longs;a <lb/>doceat, verum e&longs;&longs;e quod &longs;cribit Ari&longs;toteles, huius effe<lb/>ctus cau&longs;&longs;a aliunde à nobis, nempe à mechanicis princi<lb/>pijs, e&longs;t mutuanda. </s>

<s id="id.000907">Dico igitur, Axium, circa quos tro<lb/>chleæ rotæue conuertuntur ad rotas ip&longs;as, varias habere <lb/>proportiones. </s>

<s id="id.000908">O&longs;tendemus autem <expan abbr="rotã">rotam</expan> illam, trochleam<lb/>ue faciliùs moueri, & mouere pondera, quo rotæ diame<lb/>ter ad axis diametrum maiorem habuerit proportionem, <lb/>& ideo fieri po&longs;&longs;e rotam maiorem ad &longs;uum axem <expan abbr="minor&emacr;">minorem</expan> <lb/>habere proportionem quam rotam minorem ad &longs;uum. </s><figure id="id.007.00.097.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.097.1.jpg"></figure>

<s id="id.000909">E&longs;to enim <lb/>trochlea AB cir<lb/>ca centrum C, <lb/>cuius diameter <lb/>DCE &longs;it in ip&longs;a <lb/>quæ ad mundi <lb/>centrum <expan abbr="perp&emacr;-diculari">perpen<lb/>diculari</expan>: &longs;it au<lb/>tem appen&longs;a in D. </s>

<s id="id.000910">Alia &longs;imiliter ei æqualis &longs;it trochlea F <lb/>G circa centrum H, cuius diameter IHK, conueniens <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.098.jpg"/>cum perpen diculari quæ ad mundi centrum. </s>

<s id="id.000911">appendatur <lb/>autem in I. </s>

<s id="id.000912">Habeant autem & axes, circa quos conuertan<lb/>tur. </s>

<s id="id.000913">Hi&longs;i æquales fuerint, proportione non mutatâ idem <lb/>operabuntur. </s>

<s id="id.000914">Modò ponanturinæquales, &longs;itqueue axis ro<lb/>tç AB, cra&longs;&longs;ior axe rotæ FG, &longs;itqueue cra&longs;&longs;ioris quidem &longs;emi<lb/>diameter CL, &longs;ubtilioris autem HM. </s>

<s id="id.000915">Dico per tro chleam <lb/>FG facilius attolli pondera æqualia quàm per AB, licet <lb/>altera tro chlearum alteri &longs;it æqualis. </s>

<s id="id.000916">Quoniam enim me<lb/>chanica corpora &longs;ine materia & pondere non &longs;unt, onera <lb/><expan abbr="app&emacr;&longs;a">appen&longs;a</expan> & trochlearum ip&longs;arum grauitas ex &longs;uperiori par<lb/>te prement axes, vbi puncta L, M, quæres, &longs;ecutâ in uicem <lb/>corporum &longs;olidorum fricatione, motum ip&longs;um trochlea<lb/>rum difficiliorem & a&longs;periorem facit. </s>

<s id="id.000917">Succedit igitur im<lb/>pedimentum loco ponderis. </s>

<s id="id.000918">Duos igitur habemus vectes <lb/>DC, IH, quorum fulcimenta contra ip&longs;a C, H. </s>

<s id="id.000919">Pondera <lb/>verò inter fulcimenta & potentiasin L, M. </s>

<s id="id.000920">Intelligantur <lb/>autem potentiæ applicatæ punctis DI. </s>

<s id="id.000921">Igitur ex natura e<lb/>iu&longs;modi vectis, in quo pondus inter fulcimentum e&longs;t & <lb/>potentiam erit vt CL, ad CD, ita potentia in D ad <expan abbr="p&omacr;dus">pondus</expan>, <lb/>hoc e&longs;t, re&longs;i&longs;tentiam fricationis, quæ fit in L. </s>

<s id="id.000922">Sed maior <lb/>e&longs;t proportio CL ad CD quàm HM ad HI. </s>

<s id="id.000923">Maior igitur <lb/>ad &longs;uperandum idem &longs;eu æquale impedimentum poten<lb/>tia requiritur in D, qu<gap/>m in I. </s>

<s id="id.000924">Itaque cum vis tota in rota<lb/>rum & axium, diametrorum proportione con&longs;i&longs;tat, fieri <lb/>pote&longs;t, quod dicebamus, minorem trochleam dari, quæ <lb/>maiorem habeat proportionem ad &longs;uum axem, quàm, <lb/>maior ad &longs;uum, quo ca&longs;u minor rota facilius imp edimen<lb/>tum, quod diximus, ip&longs;a maiori rota &longs;eu trochlea &longs;upera<lb/>bit. </s>

<s id="id.000925">Veruntamen quoniam ex materia fiunt tum axes tum <lb/>rotæ, nec rei natura patitur axes &longs;ubtiles, & imbecilles <lb/>magna <expan abbr="p&omacr;dera">pondera</expan> &longs;u&longs;tinere po&longs;&longs;e, idcirco cra&longs;&longs;iores fiunt, quç <lb/>cra&longs;&longs;itudo cum proportione magis à magnarum rotarum <lb/>diametris &longs;uperetur; fit hinc maiores rotas datâ axium pa-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.099.jpg"/>ritate facilius impedimentum &longs;uperare quàm minores, & <lb/>hoc videtur &longs;en&longs;i&longs;&longs;e Philo&longs;ophus in ip&longs;a quæ&longs;tionis huius <lb/>propo&longs;itione, Hinc aurigæ vulgo axungiâ (quæ inde no<lb/>men trahit) axium a&longs;peritates mitigant, vt minor in rotan. <lb/></s>

<s id="id.000926">do, ex fricatione fiat re&longs;i&longs;tentia. </s>

<s id="id.000927">Concludimus igitur, fa<lb/>cillimè trochleam illam pondus trahere, quæ cum maxi<lb/>ma &longs;it, axem habet minimum, cumqueue axungiâ aliaue vn<lb/>ctuo &longs;a materia perfu&longs;um. </s>

<s id="id.000928">De manubrijs, quæ rotarum a<lb/>xibus aptantur, nemo ferè verba fecit; nos igitur de his a<lb/>liquid; &longs;iquidem res ad &longs;peculationem, qua de agimus, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> Mechanicam pertinet. </s>

<s id="id.000929">Manubria vectes &longs;unt, & ad vectium naturam redu<lb/>cuntur, corum &longs;cilicet, in quibus fulcimentum e&longs;tinter <lb/>pondus & potentiam. </s>

<s id="id.000930">In his autem attenditur proportio, <lb/>quam habet manubrij longitudo ad ip&longs;um axis &longs;emidia<lb/>metrum, eo enim faciliùs mouent, quo eorum longitudo <lb/>ad axium &longs;emidia metros proportionem, habuerit ma<lb/>iorem. </s>

<s id="id.000931">Duabus autem partibus con&longs;tant, alterâ, quæ ab <lb/>axe ad angulum; quæ verè vectis e&longs;t; alterâ, cui manusi<lb/>p&longs;a admouetur, ex qua res tota manubrium dicitur. </s>

<s id="id.000932">Fiunt <lb/>autem manubria hæc vt plurimum amouibilia, &longs;unt <expan abbr="tam&emacr;">tamen</expan> <lb/>ceu rotarum ip&longs;arum partes, & rotis ip&longs;is commodè affi<lb/>gerentur, ni&longs;i in rotatione à tran&longs;uer&longs;arijs, quibus rotæ &longs;u<lb/>&longs;tinentur, impedimentum fieret. </s><figure id="id.007.00.099.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.099.1.jpg"></figure>

<s id="id.000933">E&longs;to enim rota AB, cu<lb/>ius axis E, terebretur autem <lb/>in F, ibiqueue paxillus affigatur <lb/>FK. </s>

<s id="id.000934">Sit & alia rota CD, cu<lb/>ius axis G, manubrium axi <lb/>appo&longs;itum GHI. </s>

<s id="id.000935">Sint autem <lb/>rotæ æquales & axes æqua<lb/>les. </s>

<s id="id.000936">Sint etiam æqualia ip&longs;a <lb/>&longs;patia EF, GH, hoc e&longs;t, ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.100.jpg"/>nubrij GHI longitudo. </s>

<s id="id.000937">Dico, câdem facilitate moueri AB <lb/>rotam à potentia in FK, quâ mouetur CB, à potentia po<lb/>&longs;ita in HI, datis ip&longs;i nempe potentijs æqualibus. </s>

<s id="id.000938">Produca<lb/>tur enim IH, v&longs;que ad rotæ CD latus in L, & LG ducatur, <lb/>& FE in rota AB iungatur. </s>

<s id="id.000939">Erunt igitur FE LG inter &longs;e æ<lb/>quales. </s>

<s id="id.000940">Sunt autem eorum circulorum &longs;emidiametri, qui <lb/>à punctis FL, in ip&longs;a rotatione de&longs;cribuntur. </s>

<s id="id.000941">Ita igitur &longs;e <lb/>habebit potentia applicata in L ad diametrum &longs;emidia<lb/>metrumue axis rotæ CD, vt &longs;e habet potentia applicata <lb/>in F, ad diametrum &longs;emidiame trumue axis E rotæ AB, &longs;ed <lb/>&longs;patia &longs;unt æqualia & potentiæ æquales, quare nihil re<lb/>fert, vtrum manubrium lateria ffigatur, vel axi à latere ro<lb/>tæ &longs;eparatum applicetur. </s><figure id="id.007.00.100.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.100.1.jpg"></figure>

<s id="id.000942">Duplex autem e&longs;t ma<lb/>nubriorum forma; altera e<lb/>nim rectis partibus con&longs;tat, <lb/>altera verò curua e&longs;t tota, <lb/>&longs;ed rectis vtimur vt mani<lb/>bus apprendamus, curuis <lb/>verò vt locum illis appona<lb/>mus, & pedis pre&longs;&longs;ione ceu <lb/>in molis lapideis, quibus <lb/>gladij acuuntur, fieri a&longs;&longs;olet, conuertantnr. </s>

<s id="id.000943">Cur autem <lb/>manubria hæc curua fiant, ea videtur ratio, ne videlicet <lb/>manubrij capite &longs;upra centrum in linea quæ per centrum <lb/>tran&longs;it, <expan abbr="c&omacr;&longs;tituto">con&longs;tituto</expan>, factâ interim pre&longs;&longs;ione motus à centro, <lb/>ad quod directè fieret pre&longs;&longs;io, impediretur. </s>

<s id="id.000944">Curuitas <expan abbr="aut&emacr;">autem</expan> <lb/>facilitatem quan dam habet, ex qua factâ modicâ flexione <lb/>axis caput, dum premitur ab ip&longs;a perpendiculari linea le<lb/>niter abducitur. </s>

<s id="id.000945">quæ cum ce&longs;&longs;ent in manubrijs quæ manu <lb/>aguntur, ideo alia forma, nempe ex rectis partibus pa&longs;&longs;im <lb/>fiunt. </s>

<s id="id.000946">E&longs;to igiturillud quod ex rectis partibus AB, curuum <lb/>verò CD, linea verò, &longs;ecundum quam pede fit pre&longs;&longs;io <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.101.jpg"/>CDE. </s>

<s id="id.000947">Hæc itaque de manubrijs &longs;eu vectibus nos con&longs;i<lb/>dera&longs;&longs;e &longs;it &longs;atis. </s>

<s id="id.000948">Quæriinterim po&longs;&longs;et, Cur duabus datis rotis æqua<lb/>lis magnitudinis in æqualis ponderis, circa æquales axes <lb/>con&longs;titutis leuior faciliùs moueatur & citiùs quie&longs;cat; <lb/>grauior verò di&longs;ficilius moucatur & tardiùs ce&longs;&longs;et à mo<lb/>tu, ea videtur ratio, quod grauior re&longs;i&longs;tens magis cum &longs;u<lb/>peratur impre&longs;&longs;am vim &longs;u&longs;cipit, & diutiùs retinet, quod <lb/>ce&longs;&longs;at in leuiore. </s>

</subchap1><subchap1>

<s id="id.000949">QVÆSTIO X.</s>

<s id="id.000950"><emph type="italics"/>Dubitat Ari&longs;toteles, Cur faciliùs, quando &longs;ine pondere est, mouca<lb/>tur libra, quàm cum pondus habet. </s>

<s id="id.000951">Simili modo rota, & eiu&longs;modi<lb/>quidpiam, quod grauius quidem est, item quod maius & <lb/>grauius minori, & leutori?<emph.end type="italics"/></s>

<s id="id.000952">Breuiter autem &longs;oluit, ait enim, An quia non &longs;olum in <lb/>contrarium quod graue e&longs;t, &longs;ed in obliquam etiam dif<lb/>ficulter mouetur? </s>

<s id="id.000953">In contrarium enim ei ad quod vergit <lb/>onus mouere difficile e&longs;t, quo autem vergit, e&longs;t facile. </s>

<s id="id.000954">In <lb/>obliquum autem haudqua quam vergit. </s>

<s id="id.000955">Nos quod ip&longs;e <lb/>non fecit figurâ ip&longs;a appo&longs;itâ rem clariorem faciemus. </s><figure id="id.007.00.101.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.101.1.jpg"></figure>

<s id="id.000956">Efto libra AB, cuius ful<lb/>cimentum C, pondera vtrin<lb/>que appen&longs;a AB, quorum v<lb/>trumque ponderet 10. Item <lb/>libra DE, cuius fulcimentum <lb/>F pondere vero appen&longs;a D, E, <lb/>ip&longs;is A, B, dimidio leuiora, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> S. </s>

<s id="id.000957">Addatur ponderi B pon<lb/>dus G, & ponderi E pondus <lb/>H, quorum &longs;imiliter <expan abbr="vtrumq;">vtrumque</expan> <lb/>ponderet S, nutabunt igitur <lb/>libræ ponderibus appo&longs;itis, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.102.jpg"/>BG &longs;ecetur in K, EH verò in N, grauius e&longs;t autem GB, e&longs;t <lb/>enim IS, ip&longs;o EH, quod e&longs;t 10. Difficiliùs autem de&longs;cen<lb/>det BG, quàm EH. hoc autem ex doctrina Ari&longs;totelis, <lb/>quia non &longs;olum in contrarium quod graue e&longs;t, &longs;ed in obli<lb/>quum etiam difficulter mouetur, in contrarium enim ei <lb/>ad quod vergit onus mouere difficile e&longs;t, quò autem ver<lb/>git facilè in obliquum autem puta per lineas BK, EN non <lb/>vergit onus. </s>

<s id="id.000958">Difficiliùs ergo in obliquum mouebitur pon<lb/>dus BG ip&longs;o pondere EH. vtrumque autem in de&longs;cen&longs;u <lb/>retrahitur nempe à perpendicularibus BI, EM & retra<lb/>ctionis quidem anguli &longs;unt æquales & æquales ip&longs;æ retra<lb/>ctioncs. </s>

<s id="id.000959">Sedgrauius e&longs;t pondus GB. quod autem grauius <lb/>e&longs;t, violentius <expan abbr="de&longs;c&emacr;dit">de&longs;cendit</expan> eo quod e&longs;t leuius. </s>

<s id="id.000960">maiori igitur ni<lb/>&longs;u atque impetu cum cætera paria &longs;int, de&longs;cendet pondus <lb/>BG, ip&longs;o EH, quod è diametro Ari&longs;totelis a&longs;&longs;ertioni e&longs;t <lb/>contrarium. </s>

<s id="id.000961">ex alijs igitur principijs veritas ip&longs;a e&longs;t eruen<lb/>da. </s>

<s id="id.000962">Dicimus autem id ex proportionum fieri inæqualita<lb/>te; quia enim is ad 10. proportionem habet &longs;e&longs;quialteram, <lb/>10. verò ad 5. duplam, maiorem proportionem habet EH <lb/>ad oppo&longs;itum pondus D, quàm BG ad pondus A, facilius <lb/>ergo trahet libra DE lcuior pondus D, quàm ip&longs;a AB, gra<lb/>uior pondus A, quod vtique fuerat o&longs;tendendum. </s>

<s id="id.000963">Alia <lb/>quoque cau&longs;&longs;a & hæc accidentalis ad hunc effectum pa<lb/>riendum concurrit, axium nempe ad fulcimenta, in qui<lb/>bus rotantur, fricatio. </s>

<s id="id.000964">quo enim maius e&longs;t pondus cæteris <lb/>paribus, quod nos in præ cedente quæ&longs;tione demon&longs;tra<lb/>uimus, eò maiìor fit ip&longs;a collifio. </s>

<s id="id.000965">Porrò huius <expan abbr="quoq;">quoque</expan> &longs;peculationis e&longs;t, Cur æqualia & <lb/>&longs;imilia corpora in æqualibus &longs;imilibu&longs;queue bafibus con&longs;ti<lb/>tuta eodem &longs;imiliqueue plano fulta, ponderibus tamen in<lb/>æqualia, non eâdem facilitate euertantur, &longs;ed horum gra<lb/>uiora difficilius. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.103.jpg"/><figure id="id.007.00.103.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.103.1.jpg"></figure>

<s id="id.000966">Sit enim Pri&longs;ma &longs;eu <lb/>Cylindrus ABCD, cuius <lb/>grauitatis centrum E in <lb/>plano Cl, ba&longs;i fultus CD. <lb/></s>

<s id="id.000967">Sit & alter Cylindrus <lb/>FGHI, cuius grauitatis <lb/>centrum K fultus ba&longs;i HI <lb/>æqualis quidem & &longs;imilis <lb/>ip&longs;i AD. </s>

<s id="id.000968">Sit autem grauior FGHI, ip&longs;o ABCD. Dico, pari <lb/>potentiâ vtrumque impellente, facilius euer&longs;um iri Cy<lb/>lindrum AD, ip&longs;o Fl. </s>

<s id="id.000969">Ducantur EC, KH, & æquales po<lb/>tentiæ applicentur punctis BG, pellentes Cylindros ad <lb/>partes AF. </s>

<s id="id.000970">Euer&longs;io autem non fiet donec facta corporis <lb/>conuer&longs;ione circa puncta CH, grauitatis centra E, K <expan abbr="trãs-ferunturin">trans<lb/>ferunturin</expan> L, M, in ip&longs;is &longs;cilicet <expan abbr="perp&emacr;dicularibus">perpendicularibus</expan> ACFH. <lb/></s>

<s id="id.000971">Demit tantur EN, KO, perpendiculares ip&longs;is CD, HF. </s>

<s id="id.000972">Et <lb/>quoniam CNE, HOK anguli recti &longs;unt, erunt EC KH i<lb/>p&longs;is EN, KO, maiores, quare & LC, MH ip&longs;is EN KO, ma<lb/>iores atto lluntur ergo in ip&longs;a cuer&longs;ione, grauitatum cen<lb/>tra E in L, K in M. </s>

<s id="id.000973">At quod grauius e&longs;t, difficilius contra <lb/>&longs;ui naturam mouetur, ideo difficilius euertetur corpus <lb/>FI, ip&longs;o AD, quod fuerat demon&longs;trandum. </s>

</subchap1><subchap1>

<s id="id.000974">QVÆSTIO XI.</s>

<s id="id.000975"><emph type="italics"/>Dubitat Philo&longs;ophus, Cur &longs;uper &longs;cytalas facilius portentur onera <lb/>quàm &longs;uper currus, cum tamen ij magnas habeant rotas, <lb/>illæ verò pu&longs;illas?<emph.end type="italics"/></s>

<s id="id.000976">Optimè re&longs;pondet dubitationi. </s>

<s id="id.000977">An, in quiens, quoniam <lb/>in &longs;cytalis nulla e&longs;t offen&longs;atio; in curribus verò axis <lb/>e&longs;t, ad quem offen&longs;ant. </s>

<s id="id.000978">De&longs;uper enim illum premunt, & <lb/>à lateribus. </s>

<s id="id.000979">quod autem e&longs;t in &longs;cytalis ad i&longs;thæc duo mo<lb/>uetur & inferiori &longs;ub&longs;trato &longs;patio, & onere &longs;uperimpo&longs;i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.104.jpg"/>to, in vtri&longs;que enim ijs <gap/>euoluitur locis circulus, & motus <lb/>impellitur. </s>

<s id="id.000980">Tam appo&longs;itè paucis verbis veritatem expli<lb/>cauit, vt ferè quicquid in&longs;uper ad datur, &longs;uperuacaneum <lb/>videri po&longs;&longs;it. </s>

<s id="id.000981">quicquid tamen &longs;it, ad maiorem claritatem <lb/>aliquantulum in hac ip&longs;a quæ&longs;tione immorabimur. </s>

<s id="id.000982">Rotatas &longs;cytalas proponit hîc Ari&longs;toteles. </s>

<s id="id.000983">Coniun<lb/>ctasautem e&longs;&longs;e rotas ip&longs;is &longs;cytalis e&longs;t intelligen dum, nem<lb/>pe, vt &longs;imul rotæ cum &longs;cytalis conuertantur. </s>

<s id="id.000984">Secus enim <lb/>axium & Rotarum fieret offen&longs;atio, cuius offen&longs;ationis <lb/>vim & effectum cum nouerit Ari&longs;toteles, vel hoc ip&longs;o lo<lb/>co te&longs;te, mirum e&longs;t, nihil de ea egi&longs;&longs;e quæ&longs;tione 9. vbi nos <lb/>hac de re fu&longs;i&longs;&longs;imè tractauimus. </s>

<s id="id.000985">Cæterùm quod de rotatis &longs;cytalis &longs;cribit Philo&longs;o<lb/>phus, notandum, à Pappo quidem lib.

8. & à no&longs;tris Me<lb/>chanicis pa&longs;&longs;im ab&longs;que rotis Cylindrica &longs;implici videli<lb/>cet, & tereti formâ ad v&longs;um adhiberi. </s>

<s id="id.000986">E&longs;to igitur Ari<lb/><figure id="id.007.00.104.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.104.1.jpg"></figure><lb/>&longs;totelis quidem &longs;cytala <lb/>AB, Pappi verò &longs;eu vul<lb/>garis, & communis CD. <lb/></s>

<s id="id.000987">His non modò lapicidæ <lb/>pa&longs;&longs;im, &longs;ed & nautæ na<lb/>uiumqueue fabri &longs;ubdu<lb/>cendis & mari inducen<lb/>dis nauibus vtuntur, quod varare dicunt vernaculè, Hi<lb/>&longs;panico, vt arbitror, vocabulo. </s>

<s id="id.000988">ca enim natio teres lignum <lb/>baculumue appellat Varam. </s>

<s id="id.000989">Quæriautem po&longs;&longs;et, vtra harum formatum &longs;it vti<lb/>lior atque commodior? </s>

<s id="id.000990">Nos rotatas laudamus magis in <lb/>plano duroqueue &longs;olo, minus enim tangunt & minus offen<lb/>&longs;ant; in molliori autem & minus duro proponimus non <lb/>rotatas, &longs;iquidem rotæ &longs;ui naturâ pondere pre&longs;&longs;æ &longs;olum, <lb/>facillimè &longs;cindunt & ab&longs;orbentur. </s>

<s id="id.000991">Quatenus autem ad v&longs;um pertinet. </s>

<s id="id.000992">E&longs;to horizontis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.105.jpg"/><figure id="id.007.00.105.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.105.1.jpg"></figure><lb/>planum AB, &longs;cytalae duae<lb/>CD, EF, Pondus verò <lb/>eis impo&longs;itum G, tan<lb/>gens ip&longs;as in <expan abbr="p&umacr;ctis">punctis</expan> CE, <lb/>&longs;cytalæ autem planum <lb/>in punctis D, F, Pellatur <lb/>à potentia quapiam <expan abbr="p&omacr;-dus">pon<lb/>dus</expan> Gad anteriora, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> ad partes E. rotabuntur igitur &longs;cytalæ & pars quædam <lb/>&longs;cytalæ D, in qua &longs;it contactus a&longs;cendet in I, C verò de<lb/>&longs;cendetin H, nulla re motum impediente, quippe quòd <lb/>nulla ponderis &longs;cytalarum, & plani ad inuicem fiat offen<lb/>&longs;atio. </s>

<s id="id.000993">Præterea cum &longs;cytalarum centra ab horizontis pla<lb/>no æqualiter di&longs;tent, pondus quidem horizonti æquidi<lb/>&longs;tanter mouetur, & ideo cius centrum grauitatis nequa<lb/>quam, in motu qui &longs;it, eleuatur. </s>

<s id="id.000994">Cæterùm materiæ imperfectione remota nihil re<lb/>fert ad facilitatem, vtrum maioris minorisue diametri <lb/>&longs;int&longs;cytalæ, vt ea po&longs;ita eo quod maiores circuli faciliùs <lb/>offendicula &longs;uperent, quod demon&longs;tratum e&longs;t in quæ&longs;tio<lb/>ne 8. eo vtiliores &longs;unt &longs;cytalæ, quo cra&longs;&longs;iores. </s>

<s id="id.000995">Quatenus <lb/>autem ad plau&longs;tri naturam &longs;pectat, cuius ad &longs;cytalas Phi<lb/>lo&longs;ophus fecit comparationem, vt o&longs;ten da mus difficilius <lb/>ex eo moueri pondera. </s><figure id="id.007.00.105.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.105.2.jpg"></figure>

<s id="id.000996">E&longs;to plau&longs;tri rota <lb/>KL, cuius centrum M, a<lb/>xis verò NO circa quem <lb/>rotaip&longs;a conuertiturKL. <lb/></s>

<s id="id.000997">Funis quo rota ex axis <lb/>centro M trahitur MP, <lb/>pondus vero QR. </s>

<s id="id.000998">Quo<lb/>niam igitur pondus axem <lb/>premit in N, axis autem rotæ modiolum in O, & codem, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.106.jpg"/>tempore potentia quæ trahitin P, axem admouet modio<lb/>lo in parte V. duplex itaque fit ex fricatione &longs;eu offen&longs;a<lb/>tione impedimentum, in fra nempe, vbi O, & ad latus vbi <lb/>V. quæ quidem offen&longs;iones currus motum reddunt diffi<lb/>ciliorem, quæ quidem difficultas eo maior erit, quo ma<lb/>ior fuerit pondus axem premens, & minor proportio &longs;e<lb/>midiametri rotæ KM, ad axis &longs;emidiametrum MO. </s>

<s id="id.000999">Cur <lb/>igitur &longs;cytalis facilius pondera transferantur quam plau<lb/>&longs;tris, apertè ex dictis ad Ari&longs;to telis mentem demon&longs;tra<lb/>uimus. </s>

<s id="id.001000">Cætetùm quod ip&longs;e reticuit, n<gap/>s dicemus, nempe <lb/>validi&longs;&longs;imè enormia pondera per &longs;cytalas moueri, &longs;i &longs;cy<lb/>talis ip&longs;is vectes adiungantur. </s>

<s id="id.001001">Et &longs;anè motus erit tar di&longs;&longs;i<lb/>mus, veruntamen tar ditas ip&longs;a facilitate, quæ in de fit, v<lb/>berrimè compen&longs;atur. </s><figure id="id.007.00.106.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.106.1.jpg"></figure>

<s id="id.001002">E&longs;to igitur horizontis planum AB, &longs;cytalæ CD, fo<lb/>ramina in &longs;cytalis EFGH, vectes foraminibus in&longs;erti IE, <lb/>KF, LG, MH. </s>

<s id="id.001003">Pondus vero &longs;cytalis impo&longs;itum N. </s>

<s id="id.001004">Appli<lb/>catis igìtur quatuor potentijs extremitatibus vectium I, <lb/><emph type="italics"/>K<emph.end type="italics"/>, L, M, ij&longs;que in anteriora propul&longs;is, fiet &longs;cytalarum rota-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.107.jpg"/>tio, & ponderis N translatio ad anteriores partes B. </s>

<s id="id.001005">E&longs;to <lb/>item &longs;eor&longs;um &longs;cytala PR, cuius centrum Q, vectis eidem <lb/>per centrum in&longs;ertus O, P, Q, R. facto igitur vectis motu <lb/>O P Q R fiet ex O; centro <expan abbr="aut&emacr;">autem</expan> Q circuli quadrans O T. <lb/>exi&longs;tente igitur O in T erit P in S. facta quartæ partisip&longs;ius <lb/>&longs;cytalæ rotatione. </s>

<s id="id.001006">Et quoniam ex eodem centro &longs;unt qua<lb/>drantes PSOT. erit vt OQ ad QP. ita quadrans OT, ad <lb/>quadrantem PS. </s>

<s id="id.001007">Maxima autem e&longs;t proportio OQ, ad <lb/>QP. </s>

<s id="id.001008">Maxima igitur proportio OT ad PS. </s>

<s id="id.001009">Ex magno igitur <lb/>motu O ad T, paruus &longs;it &longs;cytalæ motus à P in S. tardius i<lb/>gitur progreditur &longs;cytala, quæ longioribus vectibus rota<lb/>tur, vis tamen maxima, quippe quod vt &longs;o habet QP, hoc <lb/>e&longs;t, QR ad QO, ita potentia in O ad pondus quod premit <lb/>in P vel in V. </s>

<s id="id.001010">Facillimè itaque pondera vectibus & &longs;cyta<lb/>lis per horizontis planum transferri, exi&longs;tis patet. </s>

</subchap1><subchap1>

<s id="id.001011">QVAESTIO XII.</s>

<s id="id.001012"><emph type="italics"/>Quæritur, Cur Mi&longs;&longs;ilia longius funda mittantur quam manu, <lb/>præ&longs;ertim cum proijcienti fundæ pondus addatur lapidis &longs;eu mi&longs;&longs;i<lb/>lis ponderi: & minus mi&longs;&longs;ili, manu proiecto, com<lb/>prehendatur?<emph.end type="italics"/></s>

<s id="id.001013">Soluit Philo&longs;ophus, inquiens, fortè ita fieri, quòd fun<lb/>ditor mi&longs;&longs;ile proijciatiam ex funda commotum, &longs;iqui<lb/>dem fundam circulo &longs;ubinderotans, iaculatur, ex manu <lb/>autem à quiete e&longs;t initium. </s>

<s id="id.001014">Oinnia autem cum in motu <lb/>&longs;unt, quàm cum quie&longs;cunt, facilius mouentur. </s>

<s id="id.001015">Addit præ<lb/>terea, An & ob eam cau&longs;&longs;am e&longs;t, &longs;ed nec minus etiam, quia <lb/>in fundç v&longs;a manus quidem fit centrum, funda verò quod <lb/>à centro exit? </s>

<s id="id.001016">quantò igitur productius fuerit quod à cen<lb/>tro e&longs;t, tanto citiùs mouetur; iactus autem, qui manu fit, <lb/>fundæ re&longs;pectu breuior e&longs;t. </s>

<s id="id.001017">Hæc Philo&longs;ophus. </s>

<s id="id.001018">Et &longs;anè perquàm appo&longs;itè, <expan abbr="itaq;">itaque</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.108.jpg"/>illi pror&longs;us a&longs;&longs;entirer, ni&longs;i pro comperto haberem, in lactu <lb/>qui fundâ fit, non e&longs;&longs;e manum ip&longs;am motus centium, &longs;ed <lb/>potius partem illam brachij, quæ humero iungitur, & id<lb/>co motum eo fieri velociotem, quo longior e&longs;t linea quæ <lb/>ab humero ad &longs;ummitatem fundæ e&longs;t, ea quæ ab humero <lb/>ad manum ip&longs;am. </s>

<s id="id.001019">Illud quo que mirabilc e&longs;t, quod non <lb/>ob&longs;etuat Ari&longs;toteles, nempe à funditoribus in ip&longs;o eiacu<lb/>landi actu, tardam fieri circa caput fundæ rotationem. <lb/></s>

<s id="id.001020">Quamobrem con&longs;iderandum e&longs;t, quo pacto fiat à tardi<lb/>tate velocitas. </s>

<s id="id.001021">Re&longs;pondemus, velocitatem acquiri non ex <lb/>&longs;implici, quæ circa funditoris caput &longs;it, rotatione, &longs;ed ex <lb/>co impetu qui fit in ip&longs;a lapidis emi&longs;&longs;ione, qui quidem im<lb/>petus &longs;i ante vel po&longs;t illud tempus fiat, quod à funditore <lb/>captatur, ca&longs;&longs;a pror&longs;us & inualida fit ip&longs;a iaculatio. </s><figure id="id.007.00.108.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.108.1.jpg"></figure>

<s id="id.001022">E&longs;to funda AB, manus <lb/>B, brachium BC. </s>

<s id="id.001023">Vt igitur&longs;e <lb/>habet CH, ad CB, ita veloci<lb/>tas AD ad velocitatem, BE; <lb/>Vidimus nos pueros, arundi<lb/>ni ad caput &longs;ci&longs;&longs;æ, paruos la<lb/>pides in&longs;erentes, arundinem<lb/>queue manu rotantes longi&longs;&longs;i<lb/>mè lapides ip&longs;os proijcere; A<lb/>rundo FG, lapis F, manus G, <lb/>brachium GH. </s>

</subchap1><subchap1>

<s id="id.001024">QVÆSTIO XIII.</s>

<s id="id.001025"><emph type="italics"/>Quæritur, Cur circa idemiugum, maiores collopes (vectes &longs;unt, <lb/>quos alij &longs;cytalas appellant, vt Pappus & Heron) faciliùs quàm mi<lb/>nores mouentur: & item &longs;uculæ, quæ graciliores &longs;unt eadem <lb/>vi quam cra&longs;&longs;iores?<emph.end type="italics"/></s>

<s id="id.001026">Ideo hoc fieri po&longs;&longs;e docet Philo&longs;ophus, quòd <expan abbr="tamiug&umacr;">tamiugum</expan> <lb/>quam &longs;ucula <expan abbr="c&emacr;trum">centrum</expan> &longs;it, prominenres autem collopum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.109.jpg"/>longitudines eæ lineæ quæ &longs;unt à centro. </s>

<s id="id.001027">Celeriùs autem <lb/>moueri & plus ab eadem vi quæ maiorum &longs;unt <expan abbr="circulor&umacr;">circulorum</expan> <lb/>quàm quæ minorum. </s>

<s id="id.001028">quippe quod ab ea dem vi plus <expan abbr="trã&longs;-feratur">tran&longs;<lb/>feratur</expan> illud extremum quod longius à centro di&longs;tat. </s>

<s id="id.001029">In <lb/>gracilioribus verò &longs;uculis datâ collopum paritate plus e&longs;<lb/>&longs;e id quod à ligno di&longs;tat. </s>

<s id="id.001030">E&longs;to iugum &longs;ucu<lb/>laue maior, AB circa <lb/>centrum C, minor verò <lb/>circa idem <expan abbr="centr&umacr;">centrum</expan> DE. <lb/></s>

<s id="id.001031">Collops <expan abbr="aut&emacr;">autem</expan> AF, pon<lb/>dus quod per iugum at<lb/>tollitur G. </s>

<s id="id.001032">A it igitur A<lb/>ri&longs;toteles, &longs;uculas, iu<lb/>gaue AB, DE ceu cen<lb/>tra e&longs;&longs;e, à quibus extat colops AB, ex maiori quidem, totâ <lb/>&longs;ui parte BF, ex minori autem EF. quo igitur, ait, longior <lb/>fuerit collops extans, eo maior, & deo velocior ad <expan abbr="part&emacr;">partem</expan> <lb/>F per maiorem circulum FH, fiet collopis motus & pon<lb/>deris eleuatio, at maior e&longs;t collops EF ip&longs;o BF, facil. </s>

<s id="id.001033">us er<lb/>go mouebitur pondus per &longs;uculam DE, ex collope EF, ab <lb/>cadem vi, quam per &longs;uculam AB, & collopem BF. </s>

<s id="id.001034">Hæc &longs;en&longs;i&longs;&longs;e videtur Ari&longs;to teles, qui cra&longs;&longs;a, vt aiunt, <lb/>Minerua rem pulchram & &longs;ubtilem e&longs;t pro&longs;equutus. </s>

<s id="id.001035">Di<lb/>cimus igitur primò, in&longs;trumentum illud quod Latini &longs;u<lb/>culam, id e&longs;t, &longs;ero&longs;ulam, à &longs;tridore arbitror qui in conuer<lb/>&longs;ione fit, appellauere, Græci verò <foreign lang="greek">o)/non</foreign>, id e&longs;t, A &longs;inum, quip<lb/>pe quod ceu A &longs;inus pondera &longs;u&longs;tineat portetque. </s>

<s id="id.001036">Hanc <lb/>eandem Machinam veteres Mechanici vocauere Axem <lb/>in Peritrochio, cuius nos imaginem, è Pàppo in 8. Col<lb/>lect. Mathematicarum de&longs;umptam in ip&longs;o huius no&longs;trio<lb/>peris initio, inter quinque Potentias propo&longs;uimus. </s>

<s id="id.001037">Huius <lb/>vim inter antiquos diligenti&longs;&longs;ime examinauêre Heron, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.110.jpg"/>ip&longs;emet Pappus, inter iuniores verò Guilibaldus co Tra<lb/>ctatu quem hac de Potentia Mechanicis &longs;uis in&longs;eruit. <lb/></s>

<s id="id.001038">Summa e&longs;t, hanc Machinam ad vectem reduci. </s>

<s id="id.001039">Nec ve<lb/>rum e&longs;t quod &longs;cribit Ari&longs;to teles, iugum &longs;uculamue cen<lb/>tra e&longs;&longs;e, hæc enim centrum habent, quod in figura &longs;upe<lb/>rius po&longs;ita notatur &longs;igno C. igitur vt &longs;e habet FC, ad CA, <lb/>ita pondus G ad potentiam in F; e&longs;t autem maior propor<lb/>tio FC ad CD, quàm FC, ad CA. faciliùs ergo mouebit <lb/>potentia quæ in F, pondus in D, quàm eadem potentia F, <lb/>pondus in A, hoc e&longs;t, G. </s>

<s id="id.001040">Huius naturæ &longs;unt quo que Erga<lb/>tæ, quas machinas no&longs;tr<gap/>, Græco luxato vocabulo Arga<lb/>nos appellant. </s>

<s id="id.001041">Suculæ enim reuera &longs;unt, po&longs;itione <expan abbr="tant&umacr;">tantum</expan> <lb/>ab eis differentes, non enim plano horizontis ergatæ æ<lb/>quidi&longs;tant, ceu &longs;uculæ & Axis in Peritrochio, &longs;ed eidem <lb/>fiunt perpendiculares. </s>

<s id="id.001042">Cæterùm facilitatem à velocitate <lb/>non oriri &longs;uperius demon&longs;trauimus. </s></subchap1><subchap1>

<s id="id.001043">QVAESTIO XIV.</s>

<s id="id.001044"><emph type="italics"/>Proponitur dubitatio, Cur eiu&longs;dem magnitudinis lignum facilius <lb/>genu frangatur &longs;i qui&longs;piam æque diductis manibus extrema com<lb/>prehendens fregerit, quàm &longs;i iuxta genu. </s>

<s id="id.001045">Et &longs;i terræ applicans pede <lb/>&longs;uperpo&longs;ito manu hinc inde diducta confregerit <lb/>quàm propè.<emph.end type="italics"/></s>

<s id="id.001046">Soluitur à Philo&longs;opho paucis verbis, An quia ibi genu <lb/>centrum e&longs;t, hic verò ip&longs;e pe? </s>

<s id="id.001047">quanto autem remotius <lb/>à centro fuerit, facilius mouetur quodcunque: Moueri <lb/>autem quod frangitur nece&longs;&longs;e e&longs;t. </s>

<s id="id.001048">E&longs;to lignum quod frangi debet AB, genu vel pedis <lb/>locus C, manuum latè diductarum &longs;itus DE, minus didu<lb/>ctarum FG; ltaque quoniam DE magis à centro C di&longs;tant <lb/>quàm FG, velocius mouebuntur puncta DE ip&longs;is FG, er<lb/>go inde facilius fiet fractio quam ex FG. </s>

<s id="id.001049">Hæc ille ex &longs;uis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.111.jpg"/><figure id="id.007.00.111.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.111.1.jpg"></figure><lb/>principijs. </s>

<s id="id.001050">Nos dili<lb/>gentius, &longs;i fieri poterit, <lb/>effectus huius cau&longs;&longs;am <lb/>per&longs;crutemur. </s>

<s id="id.001051">E&longs;to igi<lb/>tur in &longs;ecunda figura <lb/>lignum oblongum AB, <lb/>cuius medium C, linea <lb/>ducatur CD perpen<lb/>dicularis ip&longs;i AB. </s>

<s id="id.001052">Ad<lb/>moueatur genu <expan abbr="p&umacr;cto">puncto</expan> <lb/>C, manus verò diuari<lb/>centur in AB, facta i<lb/>gitur vtrinque impre&longs;<lb/>&longs;ione, lignum non <expan abbr="frã-getur">fran<lb/>getur</expan>, ni&longs;i partium in CD coniunctarum &longs;eparatio fiat, <lb/>&longs;itqueue altera in E, altera verò in F, fractum ergo erit <expan abbr="lign&umacr;">lignum</expan>, <lb/>& centro C immobili permanente, partes facto angulo <lb/>GCH erunt in GC, HC: Modò lignum &longs;uæ integritati re<lb/>&longs;tituetur, & denuò admoto genu puncto C, manus didu<lb/>cantur in I, K, quæ lo ca viciniora &longs;intip&longs;i C, quam AB, Di<lb/>co hinc difficilius fractionem fieri quam ex AB. </s>

<s id="id.001053">Con&longs;ide<lb/>ramus enim in integro ligno AB, duos vectes ACD, BCD, <lb/>quorum anguli concurrunt in commune fulcimentum C, <lb/>Sunt autem vectes angulati, & eius naturæ, quam exami<lb/>nauimus in quæ&longs;tiones. </s>

<s id="id.001054">E&longs;t igitur re&longs;i&longs;tentia, qua ligni <lb/>partes vniuntur in D, loco ponderis: &longs;uperanda hæc e&longs;t, vt <lb/>ligni fiat fractio. </s>

<s id="id.001055">Dico id facilius ce&longs;&longs;urum, &longs;i fiat ex pun<lb/>ctis A, B, remotioribus quam ex IK, ip&longs;i puncto C propio<lb/>ribus: etenim vt AC, ad CD, itare&longs;i&longs;tentia quæ fit in Dad <lb/>potentiam in A, item vt &longs;e habet IC ad CD, ita re&longs;i&longs;tentia <lb/>in Dad potentiam in I, &longs;ed minor e&longs;t proportio IC ad CD, <lb/>quam AC ad CD. ergo facilius potentia quæ e&longs;t in A, re<lb/>&longs;i&longs;tentiam &longs;uperabit, quæ e&longs;t in D, quam ea quæ e&longs;t in I, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.112.jpg"/>quod fuerat demon&longs;trandum. </s>

<s id="id.001056">Idem autem <expan abbr="intelligend&umacr;">intelligendum</expan> <lb/>e&longs;t de parte CB; eadem enim e&longs;t ratio. </s>

<s id="id.001057">Curigitur longio<lb/>ra & graciliora ligna facilè frangantur, ex i&longs;tis clare patet: <lb/>nempe quia maxima e&longs;t proportio longitudinis ad cra&longs;&longs;i<lb/>tudinem, cuius quidem cra&longs;&longs;itudinis &longs;patium loco partis <lb/>illius in vecte &longs;uccedit, quæ pertingit à fulcimento ad <expan abbr="p&omacr;-dus">pon<lb/>dus</expan>, hoc e&longs;t, ad ip&longs;am re&longs;i&longs;tentiam. </s>

<s id="id.001058">Sed nos hac eadem de <lb/>re nonnulla in declaranda quæ&longs;tione 16. perpendemus. </s>

</subchap1>

<s id="id.001059">QVAESTIO XV.</s>

<s id="id.001060"><emph type="italics"/>Proponitur inueftigandum, Cur litterales crocæ (glareas dicunt <lb/>Latini, velcalculos, quos vmbilicos appellat Cicero lib.

2. de Orat.) <lb/>rotundâ&longs;int figurâ, cum aliquando ex magnis &longs;int la<lb/>pidibus te&longs;tisue?<emph.end type="italics"/></s>

<s id="id.001061">A It Philo&longs;ophus, ideo forta&longs;&longs;e fieri, quòd ca quæ à me<lb/>dio magis recedunt, in motionibus, celerius feran<lb/>tur; me dium e&longs;&longs;e centrum, interuallum vero quæ à cen<lb/>tro, &longs;emper autem maiorem ab æ quali motione maiorem <lb/>de&longs;cribere circulum; quod autem maius in æquali tem<lb/>pore &longs;patium tran&longs;it, celerius ferri; quæ autem celerius ex <lb/>æquali feruntur &longs;patio vehementius impetere, quæ <expan abbr="aut&emacr;">autem</expan> <lb/>impetunt, impeti magis, & ideo quæ magis à centro di<lb/>&longs;tant, nece&longs;&longs;e e&longs;&longs;e confringi, quod cum glareæ &longs;eu croc æ <lb/>patiantur, nece&longs;&longs;ariò rotundas fieri. </s>

<s id="id.001062">Hactenus ille, & &longs;anè <lb/>p<gap/>obabiliter. </s>

<s id="id.001063">Verum enimuerò aliter &longs;eres habere vide<lb/>tur: &longs;iquidem enim à rotatione ex maiori à centro di&longs;tan<lb/>tia id fieret, maiores quidem glareæ crocæue e&longs;&longs;ent ro<lb/>tundiores, at nos non maximas modò, &longs;ed & minimas, <lb/>ea&longs;que magis angulis carere, & ad rotunditatem accede<lb/>re videmus. </s>

<s id="id.001064">Præterea non moueri eas circa centrum pa<lb/>lam e&longs;t, imò vt varia &longs;unt figura, ita varijs quo que motio<lb/>nibus, ex agitatione moueri. </s>

<s id="id.001065">Id &longs;anè explorati&longs;&longs;imum e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.113.jpg"/>angulos omnes, & emin entias quaslibet in corporibus e&longs;<lb/>&longs;e infirmiores, offen&longs;ionibus enim expo&longs;itæ &longs;unt, necre&longs;i<lb/>&longs;ten di habent facultatem. </s>

<s id="id.001066">Itaque in attritione quæ fit in <lb/>eorum agitatione perpetua, eminentiæ contunduntur, & <lb/>&longs;uperficies ip&longs;a paullatim leuigatur. </s>

<s id="id.001067">E&longs;to angulatus lapis ABCD. <lb/></s>

<s id="id.001068">Dum igitur perpeti motione <expan abbr="atq;">atque</expan> <lb/>a&longs;&longs;iduâ ver&longs;atione agitatur, fer<lb/>turqueue, eminentiæ anguliqueue, vt<lb/>pote debiles & imbecilli, conte<lb/>runtur, & inde figura fit quædam <lb/>irregularis, ad primam quidem la<lb/>pidis <expan abbr="formã">formam</expan> accedens, leuis tamen <lb/>& quouis angulo carens, qualis e&longs;t E remotis ABCD, an<lb/>gularibus eminentijs. </s>

<s id="id.001069">Hanc eandem ob cau&longs;&longs;am, &longs;culptores antequam mar<lb/>moribusvltimum læuorem inducant, dentato malleo pri<lb/>mum quidem vtuntur, tum demum eminentiores parti<lb/>culas radula facilè amouentes &longs;uperficiem ip&longs;am læuem <lb/>& adæquatam reddunt. </s>

<s id="id.001070">Hinc etiam no&longs;trates Architecti, in arcium propu<lb/>gnaculis efformandis a cutos angulos <expan abbr="deuitãt">deuitant</expan>, vtpote de<lb/>biliores, & magis offen&longs;ionibus obnoxios. </s>

<s id="id.001071">quod nec Vi<lb/>truuium latuit, qui ideo lib.

1. cap.

5. ita &longs;cribit: <emph type="italics"/>Turres itaque<lb/>rotundæ aut polygoniæ &longs;unt faciendæ, quadrat as enim machinæ <lb/>celerius di&longs;&longs;ipant; & angulos, Arietes tundendo frangunt, inro<lb/>tundationibus autem, vti cuneos ad centrum adigendo lædere non <lb/>po&longs;&longs;unt.<emph.end type="italics"/> Hæc ille. </s>

<s id="id.001072">Cur autem no&longs;tri rotundas figuras alias <lb/>vtiles reijciant, ab ijs petendum qui in ea facultate ver<lb/>&longs;antur. </s>

<s id="id.001073">Porrò quod ad hanc eandem &longs;peculationem facit, <lb/>videmus, antiquas &longs;tatuas, vt &longs;æpius auribus, na&longs;o, digitis, <lb/>manibu&longs;ue atque pedibus carere, quippe quod imbecillæ <lb/>&longs;int partes, & facilè quouis occur&longs;u mutilentur. </s>

<s id="id.001074">Quæo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.114.jpg"/>m<gap/>a cùm vera &longs;int, nemo, vt arbitror, dixerit, ab&longs;olutè, <lb/>quod voluit Ari&longs;toteles, id ex rotatione velociori & par<lb/>tium à centro remotione, fieri. </s>

</subchap1><subchap1>

<s id="id.001075">QVAESTIO XVI.</s>

<s id="id.001076"><emph type="italics"/>Dubitatur, quare, quò longiora &longs;unt ligna, <gap/><expan abbr="tãto">tanto</expan> imbecilliora fiant, <lb/>& &longs;itolluntur, inflectuntur magis: tamet&longs;i quod breue est ceu bi<lb/>cubit umfuerit, tenue, quod verò cubitorum cen<lb/>tum cra&longs;&longs;um?<emph.end type="italics"/></s>

<s id="id.001077">Ex &longs;uis principijs &longs;oluit Ari&longs;toteles. </s>

<s id="id.001078">Inquit enim: An <lb/>quia & vectis & ont s & hypomochlium, id e&longs;t, fulci<lb/>mentum in leuando, fit ip&longs;a ligni proceritas? </s>

<s id="id.001079">Prior <expan abbr="namq;">namque</expan> <lb/>illius pars ceu hypomochlium fit, quod verò in extremo <lb/>e&longs;t, pondus: quamobrem quanto exten&longs;ius fuerit id quod <lb/>à fulcimento e&longs;t, in flecti nece&longs;&longs;e e&longs;t magis; quo enim plus <lb/>à fulcimento di&longs;tat, eo magis incuruarinece&longs;&longs;e e&longs;t. </s>

<s id="id.001080">Ne<lb/>ce&longs;&longs;ariò igitur extrema vectis eleuantur. </s>

<s id="id.001081">Si igitur flexilis <lb/>fuerit vectis, ip&longs;um inflectimagis cum extollitur nece&longs;&longs;e <lb/>e&longs;t, quod longis accidit lignis, in breuibus autem quod vl<lb/>timum e&longs;t, quie&longs;centi hypomochlio depropè fit. </s>

<s id="id.001082">Hæc <lb/>&longs;ubiectâ figurâ ob oculos ponimus. </s>

<s id="id.001083">E&longs;to longum ac fle<lb/>xile lignum AB, manu ele<lb/>uetur in A, fle ctetur <expan abbr="itaq;">itaque</expan> <lb/>in B, & declinabit in C. et<lb/>enim manus quæ &longs;u&longs;tin et <lb/>in A, fulcimenti loco &longs;uccedit: longitudo vero AB ponde<lb/>ris vices refert, at que vectis, quare quo longius abfuerit à <lb/>fulcimento, id e&longs;t, manu extremum B, eo magis flectetur; <lb/>&longs;i autem lignum breuius fuerit, nempe terminatum in D, <lb/>nequaquam fle ctetur, eò quòd eius extremum D minus à <lb/>fulcimento quod e&longs;t in A &longs;it remotum. </s>

<s id="id.001084">Hæc igitur e&longs;t <expan abbr="m&emacr;s">mens</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.115.jpg"/>Ari&longs;totelis, cuius quidem &longs;ententiam non damnamus; <lb/>quippiam tamen addimus. </s>

<s id="id.001085">Dicimus autem materiam, <lb/>quatenus ad hanc contemplationem &longs;pectat, in duplici <lb/>e&longs;&longs;e differentia. </s>

<s id="id.001086">aut enim rarefactionis & con&longs;tipationis <lb/>e&longs;t incapax, vt in chalybe videmus, nitro, metallo, mar<lb/>more, aut capax quidem, & hæc duplex: Vel enim natura <lb/>nata e&longs;t ad rectitudinem quandam, vt ar borum flagella <lb/>virgæque, aut non item, ceu &longs;tannum, plumbum, & cæte<lb/>ra eiu&longs;modi. </s>

<s id="id.001087">E&longs;to primò vitreum <lb/>corpus gracile, procerum, <lb/>teres AB, manu capiaturin <lb/>A, <expan abbr="itaq.">itaque</expan> pondere ip&longs;ius cor<lb/>poris præualente ad partes <lb/>B, quia in C puncto, quod <lb/>circa medium e&longs;t, ex parte <lb/>&longs;uperiori non fit rarefactio, <lb/>nec in in feriori con&longs;tipatio, <lb/>nec interim datur penetra<lb/>tio corporum, fit fractio à <lb/>&longs;uperiori parte, & pars CB à <lb/>reliqua parte AC, auul&longs;a & <lb/>&longs;eparata cadit in D, &longs;uccedit autem ip&longs;a &longs;eparatio rarefa<lb/>ctioni. </s>

<s id="id.001088">Porrò quod materias ha&longs;ce non flexibiles diximus, <lb/>&longs;ed frangibiles, non ideo negamus vel &longs;en&longs;u docente, ali<lb/>quam inijs fieri flexionem. </s>

<s id="id.001089">Si autem lignea fuerit mate<lb/>ria, caque; flexibilis, vt EF, &longs;i manu eleuetur in E, præualen<lb/>te pondere in F flectetur vbi G. ibi enim à parte &longs;uperiori <lb/>fitrarefactio, ab in feriori verò con&longs;tipatio, & pars GF de<lb/>clinabitin H, quæ declinatio eò v&longs;que procedet, quo ra<lb/>refactio & con&longs;tipatio competens naturæ illius materiæ, <lb/>quæ flectitur ad &longs;ummam inten&longs;ionem deuenerint; tunc <lb/>&longs;ivis maior ingruerit, frangetur omnino: &longs;i &longs;ecus factaibi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.116.jpg"/>re&longs;i&longs;tentia, vbi rarefactio fit & con&longs;tipatio pe&longs;t inclina<lb/>tionem &longs;ur&longs;um ferctur pars in clinata & nutans, tum in <lb/>contrariam partem tendens reflectetur, vt videre e&longs;t in <lb/>virga IN. </s>

<s id="id.001090">Declinans enim in KL, repellente ea quæ infra <lb/>K fit materiæ conden&longs;atione, impetu ex de&longs;cen&longs;u a cqui<lb/>&longs;ito facta reflexione a&longs;cendit in KM, donec paullatim cir<lb/>ca pri&longs;tinam rectitudinem reuertatur, & hic quidem mo<lb/>tus vibratio dicitur, agitatioue. </s>

<s id="id.001091">Si autem virga plumbea <lb/>fuerit, naturâ non factâ ad rectitudinem, puta OP, pro<lb/>prio vincente pondere, ad partes declinabit QS, fietque; in <lb/>QR rarefacta, nempe &longs;uperiori parte ea con&longs;tipata infe<lb/>riori in Q, nec reflectetur, quippe quòd eius natura con<lb/>den&longs;ationem & rarefactionem commodè patiatur, nec <lb/>facta &longs;it ad rectitudinem. </s>

<s id="id.001092">Porrò tripliciter fieri pote&longs;t horum oblongorum <lb/>corporum eleuatio, nempe vel extremorum alteio, aut &longs;i <lb/>ambobus, &longs;i vtrinque &longs;u&longs;pen datur, vel alicubi inter extre<lb/>ma. </s>

<s id="id.001093">De priori modo iam egimus. </s>

<s id="id.001094">Modò &longs;u&longs;pendatur in <lb/>medio vt AB, in C. eo igitur ca&longs;u cum fulcimentum &longs;it in <lb/>C, <expan abbr="vtrinq;">vtrinque</expan> fit flexio in D, & E, & id quidem &longs;i materia fle<lb/>xionem patitur: &longs;in minus, fractio fit in C. </s>

<s id="id.001095">Si autem ab ex<lb/><figure id="id.007.00.116.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.116.1.jpg"></figure><lb/>tremis fiat &longs;u&longs;pen&longs;io, vt in <lb/>AB, tunc ceu duo vectes <lb/>fient, quorum fulcimenta in <lb/>extremis AB. </s>

<s id="id.001096">Pondera au<lb/>tem communia in medio vbi <lb/>Cremoti&longs;&longs;ima enim ea pars e&longs;t ab extremis AB. </s>

<s id="id.001097">Cedente <lb/><figure id="id.007.00.116.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.116.2.jpg"></figure><lb/>igitur materia &longs;uomet pon<lb/>deri, &longs;iquidein in flexibilis fu<lb/>erit, frangetur, & fiet <expan abbr="parti&umacr;">partium</expan> <lb/>&longs;eparatio in C, duoque in de <lb/>corpora AD, BE. </s>

<s id="id.001098">Si autem fle<lb/>xionis capax, vt AB in po&longs;tre<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.117.jpg"/>ma figura, facta ex contrario, nempe in in feriori parte cir<lb/>ca C rarefactione, in &longs;uperiori verò conden&longs;atione, pon<lb/>dere præualente curuabitur, fietque; lignum quidue aliud <lb/>huiu&longs;modi, vt ADB, nec amplius pondere &longs;uapte naturâ <lb/>inferiùs vergente ad rectitudinem reuertetur. </s>

<s id="id.001099">Cæterùm cur oblonga & graciliora corpora facilius <lb/>illis, quæ contrario &longs;e habent modo, frangantur, ex me<lb/>chanicis principijs in quæ&longs;tione 14. apertè demon&longs;traui<lb/>mus. </s>

<s id="id.001100">Modò vt ex hac contemplatione, quæ aliàs inutilis <lb/>videtur, aliquam vtilitatem capiamus, & ex his quæ con<lb/>templabimur, Architecti prudentiotes fiant, i&longs;thæcip&longs;a, <lb/>de quibus agimus, ad rem ædificatoriam commodè apta<lb/>bimus. </s>

<s id="id.001101">Transferamus igitur cogitationem ad eam <expan abbr="trabi&umacr;">trabium</expan> <lb/>comp gem, quæ ad tecta &longs;u&longs;tinenda ex tran&longs;uer&longs;ario ar<lb/>rectarioque; &longs;it, & duobus cauterijs, quam no&longs;trià Latinis <lb/>detorto vocabulo Bi&longs;cauterium dicunt. </s>

<s id="id.001102">Per&longs;crutabimur <lb/>enim, vnde illi tanta ad &longs;u&longs;tin endum vis, & quæ compa<lb/>gem hanc con&longs;equantur pa&longs;&longs;iones. </s>

<s id="id.001103">quamuis enim fabri <lb/>meræ praxi, quod vtile e&longs;t efficiant, nos meliorum inge<lb/>niorum gratiâ, rei ip&longs;ius cau&longs;&longs;as diligenter examinatas in <lb/>medium proferemus; nec de hac re tantùm agemus, &longs;ed <lb/>de Cameris quoque, fornicibus eorumqueue vitijs & virtu<lb/>tibus quatenus ad Mechanicum pertinet, &longs;ermonem ha<lb/>bebimus. </s>

<s id="id.001104">Quærimus primo, cur perpendiculariter erectae<lb/>trabes &longs;uperimpo&longs;ita pondera validi&longs;&longs;ime &longs;u&longs;tineant? </s>

<s id="id.001105">Et <lb/>&longs;ane hoc omnes norunt, &longs;ed non per cau&longs;&longs;as. </s>

<s id="id.001106">E&longs;to horizontis planum, illudqueue &longs;olidi&longs;&longs;imum, & <lb/>impenetrabile AB, trabs eidem ad perpendiculum erecta <lb/>CD fulta ba&longs;i vbi C grauitatis centrum F. pondus &longs;uper<lb/>impo&longs;itum FG, cuius grauitatis centrum H: Sint autem <lb/>H & E in eadem perpendiculari, quæ ad mun di centrum <lb/>HEC. </s>

<s id="id.001107">Itaque eo quod tum penderis tum trabis centra <lb/>grauitent in perpendiculari, illa verò fulciatur in C, to-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.118.jpg"/><figure id="id.007.00.118.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.118.1.jpg"></figure><lb/>tius ponderis moles recumbet <lb/>in C: non de&longs;cendet autem in I, <lb/>propterea quod &longs;upponatur i<lb/>p&longs;um planum AB, impenetrabi<lb/>le. </s>

<s id="id.001108">Igitur vt pondus H de&longs;cen<lb/>dat in C, alterum duorum e&longs;t <lb/>nece&longs;&longs;arium, nempe vel trabem <lb/>&longs;ubiectam comminui, aut eius <lb/>partes &longs;e&longs;e penetrare, & plura <lb/>corpora e&longs;&longs;e in eodem loco, pu<lb/>ta KC, quorum hoc &longs;ecun dum <lb/>naturæ penitus repugnat, illud <lb/>vero primum, penè impo&longs;&longs;ibile. </s>

<s id="id.001109">Diuidatur enim trabs in <lb/>partes æquales tres, lineis KL, ip&longs;a igitur KC infima &longs;u&longs;ti<lb/>net mediam KL, hæc verò &longs;upremam LD, hæc autem <expan abbr="p&omacr;-dus">pon<lb/>dus</expan>, ip&longs;um &longs;uperpo&longs;itum in H. </s>

<s id="id.001110">Seigitur &longs;u&longs;tinent partes. <lb/></s>

<s id="id.001111">Sed illud totum partibus con&longs;tat. </s>

<s id="id.001112">ergo pondus totum à <lb/>trabe tota, hoc e&longs;t, à &longs;e toto &longs;u&longs;tinetur. </s>

<s id="id.001113">Præterea in præcedenti quæ&longs;tione mon&longs;trauimus <lb/>tunc facilem e&longs;&longs;e gracilis & oblongi ligni fractionem, <expan abbr="c&umacr;">cum</expan> <lb/>maxima e&longs;t longitudinis ad cra&longs;&longs;itudinem proportio. </s>

<s id="id.001114">Hîc <lb/>verò contrà accidit, etenim MD pars vectis quæ à fulci<lb/>mento e&longs;t ad potentiam minimam habet proportionem <lb/>ad rectam DC, quæ à fulcimento ad locum fractionis ex<lb/>tenditur, vbi C, quod vt euidentius pateat, </s>

<s id="id.001115">E&longs;to &longs;eor&longs;umtrabs AB, <lb/>cuius medium C. </s>

<s id="id.001116">Sit autem <lb/>pondus D impo&longs;itum pun<lb/>cto C. facilè igitur frange<lb/>tur lignum AB, propterea <lb/>quòd maxima &longs;it proportio <lb/>AC ad CE; re&longs;i&longs;tentia verò <lb/>fiat in E, addatur vniatu que; <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.119.jpg"/>ligno AB lignum FH. </s>

<s id="id.001117">Cra&longs;&longs;ius igitur e&longs;t totum AL, ip&longs;o <lb/>A<gap/>, & ideo minor proportio AC ad CG quàm AC, ad <lb/>CE. </s>

<s id="id.001118">Addatur adhuc & IM. </s>

<s id="id.001119">Longè itaque difficilius fran<lb/>getur in K propterea quòd longè minor &longs;it proportio AC <lb/>ad CK quàm ciuidem ad CE & CG. </s>

<s id="id.001120">His igitur con&longs;ide<lb/>ratis, & demon&longs;tratis concludimus, impo&longs;libile e&longs;&longs;e ere<lb/>ctam trabem ponderi cedere, & frangi. </s>

<s id="id.001121">Dicet autem qui&longs;piam, haec&longs;i vera &longs;unt, quo gracilius <lb/>fuerit fulcrum, eo validiùs &longs;u&longs;tinebit, & frangetur minus, <lb/>quod oppido fal&longs;um e&longs;t. </s>

<s id="id.001122">Re&longs;pondemus, id non ex propor<lb/>tionum naturâ, &longs;ed ex materiæ ip&longs;ius infirmitate fieri. </s>

<s id="id.001123">Ita <lb/>quoque invecte non materiam, quatenus ad vim pertinet, <lb/>&longs;ed proportiones partium con&longs;ideramus. </s>

<s id="id.001124">Vtiumqueigi<lb/>turrequiritur ad fulcti validitatem proportio longitudi<lb/>nis ad cra&longs;&longs;itudinem debita, & materiæ ip&longs;ius robur & <lb/>fortitudo. </s>

<s id="id.001125">Præterea, quoniam pondus, cuifulcrum re&longs;i<lb/>&longs;tit, vel ex natura premit, vel ex violentia, illud quidem <lb/>per lincam perpen dicularem, quæ ad mundi <expan abbr="c&emacr;trum">centrum</expan>, hoc <lb/>autem lateraliter & diuer&longs;imodè, varia fit fulcrorum di&longs;<lb/>po&longs;itio. </s>

<s id="id.001126">Cuius rei &longs;umma hæce&longs;t, vt &longs;emper contra impe<lb/>tum &longs;upponantur. </s>

<s id="id.001127">E&longs;to enim horizontis planum <lb/>AB, <expan abbr="eid&emacr;">eidem</expan> perpendiculares CADB, <lb/>ítaque &longs;i naturaliter pondus pre<lb/>matex C, fulcrum &longs;upponetur AE. <lb/></s>

<s id="id.001128">Siautem ex F ip&longs;um GE, &longs;i verò ex <lb/>H, &longs;upponaturiuxta BE. </s>

<s id="id.001129">Si verò &longs;e<lb/>cundum I ponderi opponatur KE. <lb/></s>

<s id="id.001130">Hæc nos de arrectarijs fulcrisue; <lb/>nunc de tran&longs;uer&longs;arijs, & inclinatis agemus, & primum <lb/>de tran&longs;uer&longs;arijs, quatenus ad tectorum trabeationes &longs;pe<lb/>ctat. </s>

<s id="id.001131">E&longs;to tran&longs;uer&longs;aria trabs AB, muris <expan abbr="vtrinq;">vtrinque</expan> fulta CD, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.120.jpg"/><figure id="id.007.00.120.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.120.1.jpg"></figure><lb/>cuius grauitatis centrum <lb/>E, in <expan abbr="perp&emacr;diculari">perpendiculari</expan> FEG, <lb/>quæ quidem ad mundi <lb/>centrum vergit. </s>

<s id="id.001132"><expan abbr="Itaq;">Itaque</expan> eo<lb/>dem tendente grauitatis <lb/>contro, &longs;i pondus quod <lb/>premit in E, non præua<lb/>leat vnioni <expan abbr="parti&umacr;">partium</expan> ip&longs;ius <lb/>materiæ quæ e&longs;t in E, re&longs;i&longs;tet trabs &longs;uomet ponderi, nec <lb/>frangetur. </s>

<s id="id.001133">Si autem vel in firmitate materiæ, aut vitio, vel <lb/>maxima exiftente proportione AF ad FE, fractio fiet in E, <lb/>& &longs;ecutâ partium &longs;epaiatione duæ fient vtrin que trabes <lb/>AH, Bl, quorum grauitatis centra KL. </s>

<s id="id.001134">Erunt igitur duo <lb/>vectes AE, BE, quorum fulcimenta MN, quamobrem &longs;i <lb/>proportio EM ad MH ita præualeat, vt pondus quod e &longs;t <lb/>in E, &longs;uperet pondus muri O &longs;uperimpo&longs;iti, & item muri <lb/>P, corruent quidem trabes, & murorum fiet hinc inde di&longs;<lb/>&longs;ipatio. </s>

<s id="id.001135">Si autem non præualuerit ea, quam diximus, pro<lb/>portio, &longs;u&longs;pen&longs;æ remanebunt vtrin que trabes vt AHBI. </s>

<s id="id.001136">Huic difficultati egregiè occurrunt Architecti, ali<lb/>quando autem hoc modo: </s>

<s id="id.001137">E&longs;to tran&longs;uer&longs;aria <lb/>trabs &longs;uâ gracilitate, alia<lb/>ue de cau&longs;&longs;a imbecilla <lb/>AB, muri quibus <expan abbr="vtrinq;">vtrinque</expan> <lb/>&longs;u&longs;tinetur CD, Trabis i<lb/>p&longs;ius grauitatis centrum <lb/>G. </s>

<s id="id.001138">Itaque adpactis trabi <lb/>lignis EF, caprcolos ad<lb/>dunt muro vtrinque ful<lb/>tos CE, DF, corum capita adpactis lignis admouentes EF, <lb/>&longs;ed & tunc validi&longs;&longs;ima fit colligatio, &longs;i inter E & F capreo<lb/>lorum capita integrum lignum trabi &longs;upponatur EF. </s>

<s id="id.001139">Ra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.121.jpg"/>tio autem validitatis patet; premente enim grauitatis <expan abbr="c&emacr;-tro">cen<lb/>tro</expan> in G, fulcra hinc inde &longs;uccurrunt CE, DF, quæ cum &longs;e<lb/>ip&longs;is fieri non valeant breuiora, ne corpori detur penetra<lb/>tio, re&longs;i&longs;tunt & robu&longs;ti&longs;&longs;imè ip&longs;i ponderi &longs;uperimpo&longs;ito <lb/>contranituntur. </s>

<s id="id.001140">Videntur autem in hoc opere duo con<lb/>&longs;iderari vectes, GH, GB, quorum fulcimenta EF, potentia <lb/>premens vtrinque G. </s>

<s id="id.001141">Pondera autem parietum partes ca<lb/>pitibus trabis impo&longs;itæ in A & B. </s>

<s id="id.001142">Quoniam igitur parua <lb/>e&longs;t proportio GE ad EH, parua potentia premens in G, <lb/>maximè autem pondus in A, fieri non pote&longs;t trabem fran<lb/>gi aut muros vtrin que di&longs;&longs;ipare in AB. </s>

<s id="id.001143">Po&longs;&longs;unt etiam to<lb/>tius trabis tres partes con&longs;iderari AE, EF, FB, quarum ful<lb/>cimenta quatuor A, E, F, B, Diui&longs;o igitur pondere & mul<lb/>tiplicatis fulcimentis impo&longs;&longs;ibile e&longs;t trabem conuelli & <lb/>vitium facere. </s>

<s id="id.001144">Sed & tectorum contignationes imbecillaque; tran&longs;<lb/>uer&longs;aria Mechanici corroborare &longs;olent, additis nempe <lb/>arrectaria trabe atque cauterijs. </s>

<s id="id.001145">E&longs;to enim tran&longs;<lb/>uer&longs;aria trabs AB <lb/>parietibus vtrinque <lb/>fulta I, K, <expan abbr="arrectari&umacr;">arrectarium</expan> <lb/>CD. </s>

<s id="id.001146">Cauterij vtrin<lb/>que AD, BD, ita <lb/>tran&longs;uer&longs;ariæ trabi <lb/>in AB, & arrectario <lb/>in D in&longs;erti, vt ne<lb/>quaquam inde ela<lb/>bi valeant. </s>

<s id="id.001147">Tum ferrea fa&longs;cia EF mediam tran&longs;uer&longs;ariam <lb/>trabem AB, à parte inferiori ip&longs;i arrectario connectens, <lb/>Debet autem arrectarij pes vbi C, aliquantulum à tran&longs;<lb/>uer&longs;aria trabe di&longs;tare, ne deor&longs;um ex pondere vergente <lb/>paululum arrectario ip&longs;am tran&longs;uer&longs;ariam premat. </s>

<s id="id.001148">His i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.122.jpg"/>gitur ita con&longs;titutis pondus quidem tran&longs;uer&longs;ariæ trabis, <lb/>quod &longs;uapte naturâ premit in medio vbi C, ferrea fa&longs;cia, <lb/>arrectariæ trabi affixa di&longs;tinetur, Arrectariam cauterij &longs;u<lb/>&longs;tinent, hos verò tran&longs;uer&longs;ariæ capita AB, quibus indun<lb/>tur. </s>

<s id="id.001149">Tota igitur eiu&longs;cemodi operis vis in eo con&longs;i&longs;tit, vt <lb/>probè cauterij tran&longs;uersariæ & arrectariæ trabi in&longs;eran<lb/>tur. </s>

<s id="id.001150">fixis enim cauteriorum pedibus in AB, non <expan abbr="de&longs;cend&emacr;t">de&longs;cendent</expan> <lb/>à partibus &longs;eu capitibus D, ijs verò &longs;tantibus &longs;tabit & arre<lb/>ctarium, quo inde &longs;u&longs;pen&longs;o tran&longs;uer&longs;aria trabs ei ex ferrea <lb/>fa&longs;cia alligata nequaquam pendebit. </s>

<s id="id.001151">Stabit ergo compa<lb/>ges tota & &longs;uapte vi robu&longs;ti&longs;&longs;imè connexa totius tecti <lb/>pondus &longs;u&longs;tinebit. </s>

<s id="id.001152">Quoniam autem v&longs;u venire &longs;olet, cauterios nimia <lb/>longitudine debiles, aliquando tum proprio tum extra<lb/>neo cedentes ponderi deor&longs;um vergentes pandare, Ar<lb/>chitecti capreolis hinc inde &longs;uppo&longs;itis, ceu fulcris, huic <lb/>medentur infirmitati. </s>

<s id="id.001153">Sint enim cauterij <lb/>debiles hinc inde AB, <lb/>AC, media trabs arre<lb/>ctaria, quam <expan abbr="Monach&umacr;">Monachum</expan> <lb/>dicimus AD. </s>

<s id="id.001154">Cauterio<lb/>rum mediæ partes E, F, <lb/>in punctis igitur EF, vtpote maximè ab extremis di&longs;tanti<lb/>bus debiles cauterij val de laborant. </s>

<s id="id.001155">Itaque &longs;uppo&longs;itis v<lb/>trin que arrectariolis EH, Fl, eorum capitibus E, F, duos <lb/>cauteriolos &longs;ibi ip&longs;is ad pedem arrectarij in D, re&longs;i&longs;tentes <lb/>apponunt. </s>

<s id="id.001156">quibus ita con&longs;titutis nec E, nec F ad partes H, <lb/>I, de&longs;cendere valent. </s>

<s id="id.001157">Capiatur enim inter EH, quoduis <lb/>punctum G, & BG, DG, connectantur, erunt autem BG, <lb/>DG ip&longs;is BE ED breuiores ex 21. primi elem. </s>

<s id="id.001158">Tuncigitur <lb/>punctum E fiet in G cum BE, ED fient in BG, DG, quod <lb/>non cedentibus B, D, & &longs;ibi ip&longs;is breuioribus factis parti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.123.jpg"/>bus BE, ED, pror&longs;us e&longs;t impo&longs;&longs;ibile. </s>

<s id="id.001159">&longs;tabuntigitur in co<lb/>rum rectitudine cauterij AB, AC, nec pandabunt, quod <lb/>fieri querebatur. </s>

<s id="id.001160">Hîc autem damnandi veniunt ij, quitran&longs;uer&longs;ariæ <lb/>quidem trabis capitibus cauteriorum pedes non <expan abbr="in&longs;er&umacr;t">in&longs;erunt</expan>, <lb/>&longs;ed ea vice tran&longs;uer&longs;ariolo quodam medios cauterios v<lb/>trin que connectunt ad in&longs;tar elementi A, quam compa<lb/>gem, capram, appellant. </s>

<s id="id.001161">Sint enim cauterij hinc inde AB, <lb/>AC, quorum medias partes connectit tran&longs;uer&longs;ariolum, <lb/>DE. </s>

<s id="id.001162">Dico igitur colligationem i&longs;tam magnopere impro<lb/>bandam. </s>

<s id="id.001163">Sunt enim AB, AC vectes, quorum commune <lb/>fulcimentum A, potentiæ hinc inde diuaricantes B, C, <lb/>pondera inter fulcimentum & potentias DE. quoniami<lb/>gitur vt DH ad AB, ita potentia in B, ad pondus in D, par<lb/>ua quidem potentia, pondus in D di&longs;trahet & &longs;uperabit: <lb/>facillimaque; in de fiet tran&longs;uer&longs;ariolì à capreolis ip&longs;is vtrin<lb/>que reuul&longs;io: Et quoniam centrum quidem e&longs;t A, fact, in <lb/>D, E, parua diuaricatione, maxima fit in BC, vtpote parti<lb/>bus ab ip&longs;o centro A quam remotis. </s>

<s id="id.001164">Calcitrant igitur li<lb/>beri prope cauteriorum pedes, & muros ip&longs;os &longs;ummos, <lb/>non &longs;ine magno operis totius vitio, &longs;ua calcitratione pro<lb/>pellunt. </s>

<s id="id.001165">Hæc nos de trabeationibus, modò ad fornicum ca<lb/>merarumque; naturam &longs;tilum transferemus; id enim &longs;uadet <lb/>vtilitas, imo & nece&longs;&longs;itas ip&longs;a. </s>

<s id="id.001166">Pauci enim ante nos hæc <lb/>tractarunt, & &longs;anè his probè non cognitis aut neglectis, <lb/>Architecti fabriqueue ingentes per&longs;æpe incurrunt, & inex<lb/>plicabiles difficultates. </s>

<s id="id.001167">Dicimus igitur primò, coctiles la<lb/>teres, & non cuneatos lapides ad rectam lineam di&longs;po&longs;i. <lb/></s>

<s id="id.001168">tos, non &longs;tare. </s>

<s id="id.001169">Sint enim muri vtrinque AC, BD. </s>

<s id="id.001170">Ducatur hori<lb/>zonti æquidi&longs;tans CD, iuxta quam lateres lapide&longs;ue non <lb/>cuneati, &longs;eriatim collocentur EF. </s>

<s id="id.001171">Dicimus amoto arma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.124.jpg"/><figure id="id.007.00.124.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.124.1.jpg"></figure><lb/>mento, hoc e&longs;t, pro<lb/>hibente ip&longs;o lateres <lb/>ruere. </s>

<s id="id.001172">Producantur <lb/>enim AC in G, BD <lb/>verò in H, cum ip&longs;is <lb/>CG, DH, æquales <lb/>fiant CI, DK, & recta <lb/>IK iungatur, crit igi<lb/>tur GD &longs;patium ip&longs;i <lb/>CK &longs;patio &longs;imile qui<lb/>dem & æquale, quod <lb/>cùm ita &longs;it, nihil prohibet quin tota laterum GD moles in <lb/>&longs;patium CK transferatur, & corruat. </s>

<s id="id.001173">Si autem cunei ip&longs;i latere&longs;ue, cuneatim di&longs;po&longs;iti, ita <lb/>&longs;int vt ad vnum centrum tendant, licet ad rectam lineam <lb/>collocentur, non delabentur, &longs;ed &longs;tabunt; quod ita o&longs;ten<lb/>demus. </s>

<s id="id.001174">Sint cunci latere&longs;ue <lb/>cuneatim di&longs;po&longs;iti ABCD, <lb/>tendentes ad centrum, &longs;eu <lb/>commune punctum E, Du<lb/>cantur CAE, DBE, &longs;intqueue <lb/>muri vtrinque ponderire&longs;i<lb/>&longs;tentes CL, DM, Demitta<lb/>tur perpendicularis, quæ ad <lb/>mundi centrum FGE &longs;ecans AB, in G. </s>

<s id="id.001175">Tum fiat GK aequa<lb/>lis GF & per K ip &longs;i AGB parallela ducatur, HKI claudens <lb/>&longs;patium AHIB. </s>

<s id="id.001176">Quoniam igitur vt EC, ad EA, ita CD ad <lb/>AB per 4. propo&longs;. </s>

<s id="id.001177">lib.

6. maior erit CD ip&longs;a AB, & eâdem <lb/>de cau&longs;&longs;a maior AB, ip&longs;a HI, & idcirco maius ABDC &longs;pa<lb/>tium, &longs;patio AHIB. </s>

<s id="id.001178">Non igitur pote&longs;t linea CD, fieri in <lb/>AB, neque AB, in HI, neque &longs;patium totum CABD, tran&longs;<lb/>ferri in &longs;patium AHIB non data (quod naturæ ip&longs;i repu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.125.jpg"/>gnat) corporum penetratione. </s>

<s id="id.001179">Stabunt ergo cunei, quod <lb/>fuerat demon&longs;trandum. </s>

<s id="id.001180">Verumenimuero, debilis hæc &longs;tructura e&longs;t, & eo de<lb/>bilior, quo vani latitudo fucrit maior, cuneorum verò al<lb/>titudo minor. </s>

<s id="id.001181">Idem enim patitur quod epi&longs;tylia in &longs;pecie <lb/>Aræos&longs;tyla, quæ, vt&longs;cribit Vitruuius lib.

3. c. 2. propter in<lb/>teruallorum magnitudinem franguntur. </s>

<s id="id.001182">Id quoque ha<lb/>bet vitij, quod cunei ita di&longs;po&longs;iti &longs;uo pondere in cumbas <lb/>vtrinque violenti&longs;&longs;imè pellant. </s>

<s id="id.001183">Vtilis tamen e&longs;&longs;c pote&longs;t <lb/>ad portarum & fene&longs;trarum, quæ in medijs muris &longs;unt, & <lb/>mediocri vano aperiuntur, &longs;uperliminaria. </s>

<s id="id.001184">Si verò ad minorem circuli portionem curuetur Ca<lb/>mera, vtilior quidem erit &longs;tructura ea ip&longs;a, de qua locuti <lb/>&longs;umus; non tamen omninò &longs;ine vitio. </s>

<s id="id.001185">E&longs;to fornix ex minori <lb/>circuli portione AB, cuius in<lb/>cumbæ AF, BH muris fultæ <lb/>AC, BD. </s>

<s id="id.001186">Con&longs;tet autem vel <lb/>ex lapidibus cuneatis, vel ex <lb/>coctilibus lateribus ad E <expan abbr="c&emacr;-trum">cen<lb/>trum</expan> tendentibus. </s>

<s id="id.001187">Sitque; for<lb/>nicis linea exterior FGH, in<lb/>terior AIB. </s>

<s id="id.001188">Ducantur EA, <lb/>ED, & producantur in M, N. <lb/></s>

<s id="id.001189">Quoniam igitur vt EM ad EA, ita MGN ad AIB, maior e<lb/>rit MGN lineaip&longs;a AIB, quamobrem fieri non pote&longs;t vt <lb/>aptetur lineæ AIB, & in eius locum de&longs;cendat. </s>

<s id="id.001190">Stabit igi<lb/>tur, in cumbis vtrinque non cedentibus. </s>

<s id="id.001191">Validè autem <lb/>&longs;peciem hanc, loca quibus incumbit, propellere, ita o<lb/>&longs;tendemus. </s>

<s id="id.001192">Producatur in eadem figura CA in K, & DB in L. <lb/></s>

<s id="id.001193">Partes igitur quæ muris ad perpendiculum fulciuntur, <lb/>&longs;unt AKF, BLH, minimæ illæ quidem, maxima verò pars <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.126.jpg"/>e&longs;t extra fulcimenta, nempetota AKLB quæ idcircó &longs;uo<lb/>pte pondere deor&longs;um vergens & in incumbas <expan abbr="vtrinq;">vtrinque</expan> pel<lb/>lens aperitur, & facillimè vitium facit. </s>

<s id="id.001194">Eiu&longs;dem ferè na<lb/>turæ ea &longs;pecies e&longs;t, quæ vel ex media, vel ex minori ellip&longs;is <lb/>&longs;ecundum maiorem diametrum fit &longs;egmento. </s>

<s id="id.001195">Vtilior ta<lb/>men hæc e&longs;t, præcipuè circa incumbas, propterea quod <lb/>partes habeat erectiores, & circulari illa de qua egimus, <lb/>magis fultas. </s>

<s id="id.001196">circa medium autem pote&longs;t videri debilior, <lb/>quippe quod ellip&longs;isibi circulo curuetur minus. </s>

<s id="id.001197">Ea verò forma, qua mirum in modum delectati &longs;unt <lb/>Barbari, qui declinante imperio Italiam inua&longs;erunt, & <lb/>bonam emendati&longs;&longs;imamqueue antiquorum ædificandi ra<lb/>tionem deturparunt, ex duobus con&longs;tat circuli portioni<lb/>bus, quamobrem Albertus lib.

3. ho&longs;ce arcus, compo&longs;itos, <lb/>appellat. </s>

<s id="id.001198">Circinantur autem hoc pacto, diui&longs;a nempe <lb/>&longs;ubten&longs;a, in partes tres, ea&longs;que æquales, ponitur circini <lb/>pes in altero diui&longs;ionum puncto & pars circuli de&longs;cribi<lb/>tur, mox in altero puncto circini pede collocato alia cir<lb/>culi portio lineatur, quibus arcus ip&longs;e integratur. </s>

<s id="id.001199">Appel<lb/>lant autem tertium acutum, eo quod ex &longs;ubten&longs;a in tres <lb/>partes diui&longs;a, arcus non fiatrotundus, &longs;ed in acutum an<lb/>gulum ex duabus circuli portionibus de&longs;inens. </s>

<s id="id.001200">Sint igitur muri <lb/>AC, BD, in quibus v<lb/>trinque incumbæ KA, <lb/>BI. </s>

<s id="id.001201">Ducatur itaque &longs;ub<lb/>ten&longs;a horizonti æquidi<lb/>&longs;tans AP, quæ in tres æ<lb/>quales partes diuidatur <lb/>punctis E, F, tum centris <lb/>EF, circulorum portio<lb/>nes de&longs;cribantur hinc <lb/>AG, HK, inde verò BG, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.127.jpg"/>IH, ex quibus arcus totus integratur. </s>

<s id="id.001202">Vtilis hæc quidem <lb/>&longs;pecies e&longs;t, licet inuenu&longs;ta, propterea quod haud violen<lb/>ter incumbas vtrinque repellat, & in &longs;ummo magnis &longs;u&longs;ti<lb/>nendis oneribus &longs;it apta. </s>

<s id="id.001203">Producantur CH in N, DB verò <lb/>in O, &longs;itqueue centrum grauitatis AG in L, partis vero BG <lb/>in M. </s>

<s id="id.001204">Quoniam igitur centra hæc ob elatam portionum <lb/>con&longs;titutionem quam proxima lineis AN, BO, fulcimen<lb/>torum fiunt, maximè <expan abbr="&longs;u&longs;tin&emacr;tur">&longs;u&longs;tinentur</expan>, & deor&longs;um potius quam <lb/>lateraliter in cumbas ip&longs;as premunt. </s>

<s id="id.001205">Si quid tamen <expan abbr="hab&emacr;t">habent</expan> <lb/>vitij, illud e&longs;t quod grauitatis centra momentum haben<lb/>tia ad interiorem partem ver&longs;us PQ vim faciant, & nifi <lb/>partes magno &longs;uperimpo&longs;ito pondere comprimantur, <lb/>partes quæ &longs;unt circa HG, &longs;ur&longs;um pellentes aliquali &longs;ibi <lb/>rectitudine comparata corruunt, facta nempe circa L, M, <lb/>coniunctarum partium &longs;eparatione. </s>

<s id="id.001206">His hoc pacto explicatis de &longs;emicirculari fornice a<lb/>gemus, quæ cæteris omnibus vtilior e&longs;t, & longè pulcher<lb/>rima, quamobrem Antiquis Architectis omnibus inpri<lb/>mis admodum familiaris: </s>

<s id="id.001207">E&longs;to vanum <lb/>ABCD, muris v<lb/>trinque clau&longs;um. <lb/></s>

<s id="id.001208">Ducatur per <expan abbr="s&umacr;-mitates">sun<lb/>mitates</expan> <expan abbr="muror&umacr;">murorum</expan> <lb/>horizonti æqui<lb/>di&longs;tans recta AD, <lb/>hac bifariam &longs;e<lb/>cta in E, eodem <lb/>centro E, &longs;patio <lb/>verò EA &longs;emicir<lb/>culus de&longs;cribatur <lb/>AFD, concaua <lb/>nempeip&longs;ius for-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.128.jpg"/>nicis pars; tum eodem centro, &longs;patio verò EG, circinetur <lb/>GHI eiu&longs;dem fornicis pars conuexa. </s>

<s id="id.001209">Po&longs;t hæc productis <lb/>lineis BH, CD, in OP, &longs;ecetur fornix tota in tres æquales <lb/>partes AGKM, MNLK, NDIL, & KME, LNE iungantur, <lb/>&longs;int autem partium ip&longs;arum grauitatis centra QRS. </s>

<s id="id.001210">E&longs;t <lb/>autem R in ip&longs;a perpendiculari HE. </s>

<s id="id.001211">Quoniam igitut <lb/>partium AGKM, DILN, quæ <expan abbr="vtrinq;">vtrinque</expan> &longs;unt grauitatis cen<lb/>tra QS, in ip&longs;is &longs;unt fulcimentorum lineis OH PD, &longs;uâ <lb/>&longs;ponte fulcimentis eas &longs;u&longs;tinentibus partes ip&longs;æ &longs;tabunt. <lb/></s>

<s id="id.001212">Pars autem media KMNL deor&longs;um vergente perip&longs;am <lb/>HE lineam grauitatis centro, &longs;i parumper vel incumbæ <lb/>vel partes vtrinque AG<emph type="italics"/>K<emph.end type="italics"/>M, DILN cedant, vtpote quæ à <lb/>fulcimentis e&longs;t remoti&longs;&longs;ima, magno impetu &longs;uopte pon<lb/>dere deor&longs;um feretur. </s>

<s id="id.001213">quæ igitur in his &longs;emicircularibus <lb/>fornicibus partes &longs;tabiliores &longs;int, quæ verò ca&longs;ibus obno<lb/>xiæ, ex his quæ diximus, clarè patet. </s>

<s id="id.001214">Cæterùm cur incumbis manentibus fornix &longs;tet, ea <lb/>cau&longs;&longs;a e&longs;t, quod partes exteriores G<emph type="italics"/>K<emph.end type="italics"/>, <emph type="italics"/>K<emph.end type="italics"/>L, LI, maiores &longs;int <lb/>in ferioribus & oppo&longs;itis AM, MN, NG; quod &longs;uprà de<lb/>mon&longs;trauimus. </s>

<s id="id.001215">Si quid autem vitij in hac &longs;pecie e&longs;t, illud quidem <lb/>e&longs;t, quod &longs;ummapars <emph type="italics"/>K<emph.end type="italics"/>MNL deor&longs;um vergens magnâ vi <lb/>partes, quæ vtrinque &longs;unt, repellat, ex qua re &longs;olidarum <lb/>partium fit &longs;olutio, & inde ruina. </s>

<s id="id.001216">Huic difficultati vt occurrerent peritiores Archite<lb/>cti, plura excogitârunt remedia. </s>

<s id="id.001217">Primum enim parietes <lb/>hinc inde ita &longs;olidos, cra&longs;&longs;os & firmos faciunt, vt &longs;uapte vi <lb/>re&longs;i&longs;tentes dimoueri loco nequeant, vel para&longs;tatas <expan abbr="add&umacr;t">addunt</expan> <lb/>vtin figura TX, VY. </s>

<s id="id.001218">Præterea & ferrea claui ex incumba <lb/>in incumbam ducta & vtrinque firmata contrarias partes <lb/>validi&longs;&longs;imè connectunt, quæ calcitrantes (ita enim lo<lb/>quuntur no&longs;trates <emph type="italics"/>A<emph.end type="italics"/>rchitecti,) fornicis pedes cohibent, & <lb/>&longs;olidum ne &longs;oluatur impediunt. </s>

<s id="id.001219">qua in &longs;pecie dubitan <expan abbr="d&umacr;">dum</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.129.jpg"/>e&longs;&longs;et, an optimo loco &longs;it a &longs;it clauis, quæ per centrum? </s>

<s id="id.001220">Et <lb/>&longs;anè videtur, quippe quod circa incumbas impetus fiat <lb/>maior. </s>

<s id="id.001221">Ego autem vtiliusibi poni arbitror, vbi <expan abbr="punctaq.">punctaque</expan> <lb/>5. hoc e&longs;t, in medio tertiarum illarum partium, quæ vtrin<lb/>que incumbis in&longs;i&longs;tunt, propterea quod primus impul&longs;us <lb/>ex media parte quæ impendet, ibi fiat. </s>

<s id="id.001222">Rarò tamen boni <lb/>Architecti eo loco aptare &longs;olent, eo quòd eiu&longs;modi cla<lb/>ues vel pulcherrimis ædificijs minuant gratiam. </s>

<s id="id.001223">Vnde fit <lb/>vt nunquam &longs;atis laudetur Lucianus ille Benuerardus <lb/>Lauranen&longs;is Dalmata, qui nullibi apparentes eas po&longs;uit <lb/>in admirabili illa Vrbini Aula, quam Federico Feltrio, fe<lb/>lici&longs;&longs;imo æquè & inuicti&longs;&longs;imo Duci, ædificauit. </s>

<s id="id.001224">Tertio denique modo huic infirmitati me dentur, <lb/>vt videre e&longs;t in &longs;equenti figura, in qua vanum ADBC, mu<lb/>ri vtrinque AF, BH, fornix verò FGH. </s>

<s id="id.001225">Itaque dum muros <lb/><figure id="id.007.00.129.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.129.1.jpg"></figure><lb/>ex&longs;truunt, arre<lb/>ctarias trabes, ro<lb/>bore aliaue mate<lb/>ria firmi&longs;&longs;ima, illis <lb/>in&longs;erunt, quales <lb/>&longs;unt IF<emph type="italics"/>K<emph.end type="italics"/> LHM, <lb/>ea proceritate vt <lb/>futuri fornicis &longs;u<lb/>perent &longs;ummita<lb/>tem. </s>

<s id="id.001226">Con&longs;umma<lb/>to enim fornice, <lb/>nondum tamen, <lb/>exarmato, tran&longs;<lb/>uer&longs;ariam <expan abbr="trab&emacr;">trabem</expan> à <lb/>&longs;ummo fornicis <lb/>dor&longs;o parumper <lb/>eminentem in punctis I, L, arrectarijs trabibus validi&longs;&longs;i<lb/>mis clauibus connectunt, tum punctis NP, Oq, capreolos <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.130.jpg"/>tran&longs;uer&longs;ario, & arrectarijs ferreis, clauis affigunt. </s>

<s id="id.001227">Qui<lb/>bus ita concinnatis, facta fornicis validâ pre&longs;&longs;ione in G, <lb/>in cumbi&longs;queue F, H, ad exteriora repul&longs;is, AB &longs;patium non <lb/>fit maius. </s>

<s id="id.001228">Repul&longs;is enim in cumbis & muros propelli ne<lb/>ce&longs;&longs;e e&longs;t, & cum muris ip&longs;as in&longs;ertas trabes, I<emph type="italics"/>K<emph.end type="italics"/>, LM. </s>

<s id="id.001229">At va<lb/>ricari non po&longs;&longs;unt, nî &longs;ecum trahant puncta PQ, quod fie<lb/>ri non pote&longs;t, propterea quod in punctis N, O, validè di&longs;<lb/>tineantur. </s>

<s id="id.001230">Itaque &longs;patio AB non dilatato nulla fit ip&longs;ius <lb/>fornicis di&longs;&longs;olutio, quod vtique à principio ceu propo&longs;i<lb/>tus finis quærebatur. </s>

<s id="id.001231">Sed dicet qui&longs;piam, Nonne pende<lb/>bit tran&longs;uer&longs;aria trabs in ip&longs;a di&longs;tractione arrectariorum, <lb/>pre&longs;&longs;a in punctis N, O? aut parum dicimus, aut nihil. </s>

<s id="id.001232">Cum <lb/>enim PQ proxima &longs;int punctis FH, quæ cum arrectarijs à <lb/>muro di&longs;tinentur, magna in ijs fit vtrobique re&longs;i&longs;tentia. </s>

<s id="id.001233">Rebus igitur ita &longs;e habentibus cum ob&longs;erua&longs;&longs;ent Ar<lb/>chitecti, ob enormitatem ponderis fornices in tertia illa <lb/><figure id="id.007.00.130.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.130.1.jpg"></figure><lb/>parte quæ &longs;umma e&longs;t <lb/>laborare, <expan abbr="quãtum">quantum</expan> ter<lb/>tijs vtrinque partibus <lb/>&longs;oliditatis addunt, tan<lb/>tundem ex illa parte <lb/>&longs;uprema demere <expan abbr="&longs;ol&emacr;t">&longs;olent</expan>, <lb/>vt videre e&longs;t in &longs;ubie<lb/>cta figura, in qua par<lb/>tes A, B, &longs;olidæ & cra&longs;<lb/>&longs;iores, quibus hærent <lb/>partes, quæ CE, DG <lb/>cra&longs;&longs;æ quidem & illæ, <lb/>tum vero &longs;umma EFG, <lb/>alijs &longs;ubtilior. </s>

<s id="id.001234">Minus <lb/>igitur grauante ponde<lb/>re in F, minor fit ad in cumbas pre&longs;&longs;io, aut &longs;i qua fit, à <expan abbr="parti&umacr;">partium</expan> <lb/>ACE, BDG &longs;oliditate haud inualidè &longs;u&longs;tinetur. </s>

<s id="id.001235">Cæterùm admonet nos locus, vt aliquid de forni<lb/>cum di&longs;&longs;olutionibus in medium afferamus: cau&longs;&longs;is enim <lb/>morborum cognitis, facilius periti medici adhibere &longs;o<lb/>lent remedia. </s>

<s id="id.001236">E&longs;to enim &longs;emicircula<lb/>ris fornix ABC, cuius cen<lb/>trum E, perpendicularis ve<lb/>rò quæ per centrum DBE, &longs;e<lb/>micirculi ABC, diameter <lb/>AEC, incumbæ <expan abbr="vtrinq;">vtrinque</expan> A, C. <lb/></s>

<s id="id.001237">Itaque &longs;i nulla fiat incumba<lb/>rum repul&longs;io, &longs;tabit fornix; &longs;i verò fiat, ruinam facict. </s>

<s id="id.001238">Pellanturitaque ad exteriores partes, vt in &longs;ecunda <lb/><figure id="id.007.00.131.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.131.2.jpg"></figure><lb/>figura, H in F, & C in G, <lb/>ex qua pul&longs;ione cum ma<lb/>ius fiat &longs;patium quod in<lb/>tegro fornice impleba<lb/>tur, iam di&longs;tractis <expan abbr="vtrinq;">vtrinque</expan> <lb/>fornicis partibus <expan abbr="n&omacr;">non</expan> im<lb/>pletur, Diuiditur igitur <lb/>locus maior factus in tres partes, quarum hincinde duas <lb/>replent fornicis partes, tertiam verò quæ media e&longs;t, re<lb/>plet in&longs;ertus, ne vacuum detur, aër, vt in figura videre e&longs;t, <lb/>in qua &longs;olutæ vtrinque fornicis partes HIKF, PMNG, aër <lb/>autem medius &longs;patium replens IKMN. </s>

<s id="id.001239">Diuidantur &longs;in<lb/>guli quadrantes FK, GN, in partes tres, quarum duæ &longs;int <lb/>hincinde FQ, GR, & à centris, quæ &longs;eparatis quadranti<lb/>bus facta &longs;untin ST, rectæ ducantur SQV. TRX. </s>

<s id="id.001240">Quo<lb/>niam igitur tertiæ partes vtrinque VIKQ MNRX pro<lb/>pria grauitate depre&longs;&longs;æ, nullum quo &longs;u&longs;tineantur fulci<lb/>mentum habent, corruent quidem. </s>

<s id="id.001241">Ducantur autem re<lb/>ctæ QI, RM, con&longs;tituentes cum ip&longs;is QV, RX pares an<lb/>gulos VQI MRX. </s>

<s id="id.001242">Itaque centris QR partes QIRM ad <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.132.jpg"/>inferiores partes deuoluentur, fientqueue QI, RM, vbi QZ, <lb/>RZ. </s>

<s id="id.001243">Siautem QI, RM perpendicularibus quæ à punctis <lb/>QR ad perpendicularem DE ducuntur, fuerint maiores <lb/>conuenient alicubi in ip&longs;a perpendiculari, & altera alte<lb/>ram &longs;u&longs;tinebit; &longs;i autem æquales tangent &longs;e & nihilomi<lb/>nus fiet ruina, &longs;i minores nec &longs;e inuicem tangent, & nullà <lb/>re prohibente deor&longs;um corruent. </s>

<s id="id.001244">tangant tautem &longs;e in <expan abbr="p&umacr;-cto">pun<lb/>cto</expan> Z. quo pacto igitur fornices incumbis cedentibus in <lb/>medio aperti, <expan abbr="di&longs;&longs;oluãtur">di&longs;&longs;oluantur</expan> & ruinam faciant, ex i&longs;tis patet. </s>

<s id="id.001245">Ex demon&longs;tratis qua&longs;i ex con&longs;ectario habemus for<lb/>nices quo fuerint cra&longs;&longs;iores dato pari incumbarum &longs;ece&longs;<lb/>&longs;u, ruinæ minus e&longs;&longs;e obnoxios quàm tenuiores, hoc e&longs;t, <lb/>maiori aperitione indigere ad ruinam cra&longs;&longs;iores quam te<lb/>nuiores, quod licet ex iam dictis re&longs;ultet, nos tamen cla<lb/>rius ex &longs;ubiecto &longs;chemate demon&longs;trabimus. </s>

<s id="id.001246">E&longs;to enim cra&longs;&longs;ioris <lb/>fornicis pars <expan abbr="quid&emacr;">quidem</expan> ABCD, <lb/>tenuioris EFCD circa <expan abbr="id&emacr;">idem</expan> <lb/>centrum R. </s>

<s id="id.001247">Ducatur au<lb/>tem RM, &longs;ecans CD in G. <lb/>EF in H AB, in M. </s>

<s id="id.001248">Centro <lb/>igitur G fiet euer&longs;io portio<lb/>num fornicum, MD, HD, <lb/>Ducantur GA, GE & producta AD in N ip&longs;i AN perpen<lb/>dicularis ducatur GN. quoniam igitur GE cadit in trian<lb/>gulo AGN erit ex 21. propo&longs;. </s>

<s id="id.001249">lib.

1. elem. </s>

<s id="id.001250">GA, maior GE. <lb/></s>

<s id="id.001251">Corruente igitur maioris fornicis portione MD, recta <lb/>GA centro G punctum A de&longs;cribet portionem AI, mino<lb/>ris interim ex GE, de&longs;cribente EL, at cadenti angulo A <lb/>occurrit in perpendiculari IK in puncto I angulus oppo<lb/>&longs;itæ portionis, O, ip&longs;i autem E cadenti per EL non occur<lb/>ret punctum P, cadens per Pq eo quod neutrum eorum <lb/>pertingat ad perpendicularem IK. Tenuioris ergo forni<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.133.jpg"/>cis partes è &longs;uis locis auul&longs;æ ex eadem aperitione ruinam <lb/>facient, quod non contingit partibus cra&longs;&longs;ioris. </s>

<s id="id.001252">quod &longs;a<lb/>nè fuerat de clarandum. </s>

<s id="id.001253">Quæritur adhuc, quare grauiores fornices in &longs;um<lb/>mis ædificijs non &longs;ine vitio fiant? </s>

<s id="id.001254">E&longs;to ædificium ABGH, cuius <expan abbr="vtrinq;">vtrinque</expan> muri ABCD, <lb/>EFGH, maiorum &longs;ummitates AD, EH, mediæ murorum <lb/>partes KL, fornicum &longs;ummus quidem DIE, medius verò <lb/><figure id="id.007.00.133.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.133.1.jpg"></figure><lb/>KML. Dico, magis cedere pul<lb/>&longs;os muros &longs;ummos circa DE, <lb/>quam in medio circa KL. </s>

<s id="id.001255">Sunt <lb/>enim muri BA, GH ceu vectes <lb/>quidam, <expan abbr="quor&umacr;">quorum</expan> extremis par<lb/>tibus à fulcimentis BG remo<lb/>ti&longs;&longs;imis potentia admouetur, <lb/>hoc e&longs;t, ip&longs;ius fornicis DIE ad <lb/>DE in cumbans repul&longs;io; lon<lb/>gior e&longs;t autem pars à <expan abbr="fulcim&emacr;-to">fulcimen<lb/>to</expan> ad potentiam AB, ip&longs;a BK. <lb/></s>

<s id="id.001256">Data igitur paritate potentia<lb/>rum plus operabitur ea quæ in <lb/>D, illa quæ K. facilius crgo re<lb/>pellentur muri in DE quàm in <lb/>KL. </s>

<s id="id.001257">Alia quo que ratio intercedit, &longs;iquidem pondus muri <lb/>&longs;uperioris ADK, premens inferiorem murum KBC, cum <lb/>&longs;ua grauitate firmiorem, & pul&longs;ionibus minus obnoxium <lb/>reddit. </s>

<s id="id.001258">Difficilius enim propellitur id quod graue e&longs;t <expan abbr="quã">quam</expan> <lb/>quod leue, vt nos quæ&longs;tione 10. demon&longs;trauimus. </s>

</subchap1><subchap1>

<s id="id.001259">QVÆSTIO XVII.</s>

<s id="id.001260"><emph type="italics"/>Quærit Ari&longs;toteles, Cur paruo exi&longs;tente cuneo magna &longs;cindantur <lb/>pondera & corporum moles, validaque, fiat impre&longs;&longs;io?<emph.end type="italics"/></s>

<s>E&longs;to, inquit, circulus <lb/>BCDE & alter in eo minor <lb/>MNOP circa idem centrum <lb/>A. Ducanturque; Diametrima<lb/>ioris quidem CD, EB, mino<lb/>ris verò MO, NP. Itaque vbi <lb/>AB circulata eò peruenerit <lb/>vnde e&longs;t gre&longs;&longs;a, ip&longs;a quoque <lb/>AM eo vnde moueri c&oelig;pe<lb/>rat, perueniet. Tardiùs antem <lb/>fertur AM, quam AD, pro<lb/>pterea quòd AM à centro <lb/>magis retrahatur quàm ip&longs;a AB. Ducatur igitur ALF & <lb/>à puncto L, ip&longs;i AB perpendicularis L q, cadens in mino-

<s id="id.001261">In parua re magnum negotium. </s>

<pb pagenum="13"/>ri circulo, & rur&longs;us ab codem L ip&longs;i AB, parailela duca<lb/>tur LS, Ab S verò eidem perpendicularis ST, & ab F i<lb/>tem FX. Sunt ergo q L, ST, quidem æquales, nempeillæ, <lb/>per qua<gap/>, &longs;ecundum naturam, mouentur puncta BM. Mo<lb/>tu verò retractionis ad centrum, hoc e&longs;t, præter naturam, <lb/>plus motum e&longs;t M quàm B. Maior enim e&longs;t M q, ip&longs;a BT, <lb/>quod, ceu notum, &longs;uppo&longs;uit Ari&longs;toteles. nos autem inf. à <lb/>demon&longs;trabimus. Si igitur fiat vt motus præter naturam <lb/>ad motum præter naturam, ita motus <expan abbr="&longs;ec&umacr;dum">&longs;ecundum</expan> naturam, <lb/>ad motum &longs;ecundum naturam, punctum B; cum M fuerit <lb/>in L, non eritin S, &longs;ed in F. tunc enim, vt e&longs;t FX motus &longs;e<lb/>cundùm naturam ad XB, præter naturam, ita e&longs;t q L &longs;e<lb/>cundum naturam ad q M præter naturam; &longs;ed BF maior <lb/>e&longs;t ML, ergo proportione &longs;eruatâ, velociùs mouetur B <lb/>quàm M circa idem centrum A. Hæc autem &longs;umma e&longs;t <lb/>eorum quæ præfert Ari&longs;toteles. Cæterùm nos parallelo<lb/>grammum, quod in figura eius habetur prætermi&longs;imus, <lb/>quippe quod nihil ad eam quæ affertur, demon&longs;tratio<lb/>nem faciat. </s>

<s id="id.001262">Etenim quæ&longs;tio hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.134.jpg"/>clari&longs;&longs;imorum virorum ingenia magnopere fatigauit. </s>

<s id="id.001263">Ex <lb/>quibus Ari&longs;toteles inter veteres, Guid. Vbald. inter re<lb/>centiores ad vectis naturam (ne quid in Mechanicis ad <lb/>vectem non reduci putaretur) cuneum ip&longs;um trahere co<lb/><figure id="id.007.00.134.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.134.1.jpg"></figure><lb/>nati &longs;unt. </s>

<s id="id.001264">Nos autem pro <lb/>veritate certantes, &longs;i in <lb/>horum &longs;ententiam vltrò <lb/>non tran&longs;ierimus, multa <lb/>venia digni à non iniquo <lb/>iudice exi&longs;timabimur. </s>

<s id="id.001265">A<lb/>ri&longs;totelis mentem clarè <lb/>& fusè explicat G. V<lb/>bald. in Mechan. vbi de <lb/>Cuneo peculiariter a<lb/>git. </s>

<s>Modò quod pollicebamur, nempe minorem e&longs;&longs;e <lb/>BT, quàm q M, ita demon&longs;tramus. <expan abbr="quoniã">quoniam</expan> ST. ex prop. 13. <lb/>1. 6. media proportionalis e&longs;t inter BT & TE, erit qua<lb/>dratum TS æquale <expan abbr="parallelogrãmo">parallelogrammo</expan> &longs;eu rectangulo BT, <lb/>TE, item, quoniam q L media proportionalis e&longs;t inter <lb/>M q, & q O. erit quadratum q L æquale rectangulo M q, <lb/>q O, æqualia ergo &longs;unt rectangula BTE, M q O, itaque <lb/>reciprocalatera habent proportionalia. quare, vt TE, ad <lb/>q O, ita M q ad TB, &longs;ed TE maior e&longs;t ip&longs;a q O, quippe <lb/>quòd pars &longs;it q O ip&longs;ius TE, maior ergo & M q ip&longs;a TB, <lb/>quod o&longs;tendendum fuerat. </s>

<s id="id.001266">E&longs;to igitur &longs;cindendum quippiam ABCD, Cuneus <lb/>EFG, cuius pars HFI &longs;ci&longs;&longs;uræ in&longs;erta HI, facta igitur vali<lb/>da percu&longs;&longs;ione in EG, fiet vt cum EG fuerit in NO, H &longs;it v<lb/>bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit <lb/>&longs;ci&longs;&longs;io NSO, toti nempe cuneo EFG, æqualis. </s>

<s id="id.001267">Vultigitur <lb/>Ari&longs;toteles, duos in cunco vectes con&longs;iderari EF, GF, quo<lb/>rum alterius, nempe EF, fulcimentum &longs;it in H, pondus ve<lb/>ro in F; alterius autem, hoc e&longs;t, GF fulcimentum quidem <lb/>&longs;it in I, pondus verò itidem &longs;it in F. </s>

<s id="id.001268">His nequaquam con<lb/>&longs;entiens G. Vbald. aliam viam ingreditur. </s>

<s id="id.001269">Ait enim EHF <lb/>vectes quidem e&longs;&longs;e, quorum commune fulcimentum F, <lb/>potentias verò mouentes in EG. </s>

<s id="id.001270">Pondera vtrinque inter <lb/>fulcimenta & potentias, vbi HI, idemque; e&longs;&longs;e ac &longs;i EF, GF, <lb/><gap/>eor&longs;um à cuneo con&longs;iderati in puncto F, adinuicem fulti <lb/>atque di&longs;tracti pondera pellerent H in NP, I verò in O, <lb/><expan abbr="q.">que</expan> Verumenimuerò quoniam cunei angulus non muta<lb/>tur, nec vertex ip&longs;e centri vllum pror&longs;us præbet v&longs;um, nec <lb/>eius latera vtrinque di&longs;tracta ad contrarias partes didu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.135.jpg"/>cuntur, vectes in cuneo hoc pacto con&longs;iderare videtur à <lb/>veritate alienum. </s>

<s id="id.001271">Ari&longs;totelis autem &longs;olutionem fal&longs;am e&longs;<lb/>&longs;e, clarè patet. </s>

<s id="id.001272">quo pacto enim F pellet ex fulcimento Hi<lb/>p&longs;am ligni partem OS, & idem F ex fulcimento I pellet <lb/>oppo&longs;itam partem NS, &longs;i inuicem contendentes extremæ <lb/>vectium partes in F, altera alteri ne quicquam operentur, <lb/>e&longs;t impedimento? </s>

<s id="id.001273">Et &longs;anè opinionis fal&longs;itas inde patet, <lb/>quòd videamus materiæ partes &longs;ci&longs;&longs;as, in ip&longs;o &longs;ei&longs;&longs;ionis a<lb/>ctu facta di&longs;tractione à cunei vertice nequaquam tangi. <lb/></s>

<s id="id.001274">At eiu&longs;modi operationes per contactum fieri nulli e&longs;t i<lb/>gnotum. </s>

<s id="id.001275">Solutio igitur i&longs;ta mco iudicio, tanto Philo&longs;o<lb/>pho pror&longs;us videtur indigna. </s>

<s id="id.001276">Porrò G. Vbald. ijs quæ de diuaricatis vectibus in <lb/>medium adduxerat non acquie&longs;cens alias quærit cau&longs;&longs;as, <lb/>cur cuneus minoris anguli validiùs &longs;cindat. </s>

<s id="id.001277">Idque; ex quo<lb/>dam lemmate demon&longs;trare conatur, figura autem eius ita <lb/>ferè &longs;e habet. </s>

<s id="id.001278">E&longs;to cuneus ABC, <lb/>item alius DEF. <expan abbr="Dem&omacr;-&longs;trauit">Demon<lb/>&longs;trauit</expan> igitur ex a&longs;&longs;um<lb/>pto, quo acutior fuerit <lb/>angulus BIM, eo facilius <lb/>pondera moueri, & ideo <lb/>facilius ceu vecte AB <lb/>moueri pondus I quàm <lb/>vecte DE pondus <expan abbr="q.">que</expan> In<lb/>geniosè quidem. </s>

<s id="id.001279">At ma<lb/>gnam hæc apud me ha<gap/><lb/>bent difficultatem. </s>

<s id="id.001280">Si e<lb/>nim ita &longs;e habet AB, ad BI, vt DE, ad EQ (ip&longs;æ enim DE, <lb/>EQ &longs;upponuntur æquales) ergo eadem æquali&longs;ue poten<lb/>tia æqualiter mouebit pondera I & <expan abbr="q.">que</expan> quod ip&longs;i eiu&longs;dem <lb/>demon&longs;trationi pror&longs;us concludit contrarium. </s>

<s id="id.001281">Nec meo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.136.jpg"/>quidem iudicio id &longs;equi videtur, propterea quod ex Pap<lb/>po ea quæ in planis inclinatis mouentur, redigantur ad li<lb/>bram. </s>

<s id="id.001282">Ratio enim valde e&longs;t diuer&longs;a, &longs;iquidem pondera <lb/>quæ in planis inclinatis mouentur, certa habent fulci<lb/>menta & determinatas tum brachiorum tum ponderum <lb/>proportiones, quæ omnia in cuneo, nec quidem mente <lb/>concipi po&longs;&longs;e, clarè patet. </s>

<s id="id.001283">His igitur difficultatibus con&longs;ideratis, Nos cunei <lb/>vim, ad alia e&longs;&longs;e principia referendam pro comperto ha<lb/>bemus. </s>

<s id="id.001284">Ordimur igitur hoc pacto. </s>

<s id="id.001285">Cuneo quidem res di<lb/>uidi certum e&longs;t. </s>

<s id="id.001286">Cæ terùm quæ natura diuidere apta &longs;unt, <lb/>tria &longs;unt, punctum, linea, &longs;uperficies, Puncto enim linea, <lb/>lineâ &longs;uperficies, &longs;uperficie autem corpus ip&longs;um diuidi<lb/>tur. </s>

<s id="id.001287">quæ omnia à Mathematico ab&longs;que materia con&longs;ide<lb/>rantur. </s>

<s id="id.001288">De diui&longs;ione autem quæ fit ex puncto, nihil agit <lb/>Mechanicus, qui corporibus quidem vtitur, ad cuius na<lb/>turam non trahitur punctum, cuius partes &longs;unt nullæ. </s>

<s id="id.001289">At <lb/>non lineis & &longs;uperficiebus modò corpora diuiduntur, &longs;ed <lb/>etiam corporibus, quod verum e&longs;t, at ea corpora ad linea<lb/>rum & &longs;uperficierum naturam quodammodo aptari faci<lb/>lè docebimus. </s>

<s id="id.001290">Dicimus igitur, duplicem e&longs;&longs;e Cuneorum <lb/>&longs;peciem, linearem vnam, &longs;uperficialem alteram. </s>

<s id="id.001291">linearem <lb/>appello, quæ ad lineæ naturam magnopere accedit. </s>

<s id="id.001292">Tales <lb/>&longs;unt orbiculares illæ cu&longs;pides, quibus ad perforandum v<lb/>timur, & ideo vernaculè Pantirolos vocamus. </s>

<s id="id.001293">Acus item <lb/>&longs;utorij, & cætera quæ nen &longs;ecus ac linea in punctum de&longs;i<lb/>nunt, & imagina<gap/>iam quandam lineam ceu axem in eo <lb/>puncto de&longs;inen<gap/>em continent. </s>

<s id="id.001294">Ad lineam quo que refe<lb/>runtur lateratæ cu&longs;pides oblongæ, & &longs;ubtiles ceu&longs;ubulæ, <lb/>claui, en&longs;es, pugiones, & his &longs;imilia, quæ cum adacta vali<lb/>dam faciant partium &longs;eparationem ad cunei naturam <expan abbr="n&omacr;">non</expan> <lb/>referre magnæ videretur dementiæ. </s>

<s id="id.001295">Ettunc quantoma<lb/>gis corpora hæc ad linearem naturam accedunt, eo ma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.137.jpg"/>gis penetrant. </s>

<s id="id.001296">Sed & hocidem in rebus non ab arte, &longs;ed <lb/>ab ip&longs;anatura productis facile e&longs;t cogno&longs;cere. </s>

<s id="id.001297">Quis enim <lb/>non experitur, quàm validè culex, infirmi&longs;&longs;imum animal, <lb/>& ea paruitate qua e&longs;t, hominum & cæterorum <expan abbr="animali&umacr;">animalium</expan>, <lb/>cutes aculeata probo&longs;cide penetret? </s>

<s id="id.001298">Id vtique non alia de <lb/>cau&longs;&longs;a fit, quod ad imaginariæ lineæ &longs;ubtilitatem quam, <lb/>proximè accedat. </s>

<s id="id.001299">Ve&longs;pæ quoque, Apes, Scorpiones a<lb/>culeis i&longs;tis ceu linearibus cuneis vtuntur. </s>

<s id="id.001300">Nec refert, vt <lb/>diximus, vt um laterati &longs;int, ceu &longs;ubulæ, & claui, vel ro<lb/>tundi & vtrum plura paucioraue latera habeant, dummo<lb/>do in punctum & aculeatam aciem de&longs;inant. </s>

<s id="id.001301">Altera por<lb/>ro cuneorum &longs;pecies &longs;uperficiei naturam &longs;apit, acie &longs;iqui<lb/>dem in lineam de&longs;init, quæ &longs;uperficiei e&longs;t terminus, <expan abbr="quã">quam</expan>. <lb/></s>

<s id="id.001302">obrem huc ea omnia referuntur, quæ acie ipsâ &longs;cindunt, <lb/>ceu &longs;unt cunei propriè dicti, de quibus hoc loco e&longs;t &longs;er<lb/>mo, cultra, en&longs;es, a&longs;ciæ, &longs;ecures, &longs;calpra lata, & cætera e<lb/>in&longs;modi, quibus corpora acie &longs;cinduntur. </s>

<s id="id.001303">Quidam his ad<lb/>dunt &longs;erras, quibus haud pror&longs;us a&longs;&longs;entimur. </s>

<s id="id.001304">Etenim alia <lb/>ratione diuidunt, &longs;icut & limæ &longs;olent, deterendo enim, <expan abbr="n&omacr;">non</expan> <lb/>&longs;cindendo ferri, ligni, & marmorum duritiem diuidunt & <lb/>domant. </s>

<s id="id.001305">His igitur <expan abbr="c&omacr;&longs;ideratis">con&longs;ideratis</expan>, &longs;i daretur ex materia qua<lb/>piam in frangibili cuneus, qui maximè ad &longs;uperfi ciei natu<lb/>ram accederet, vel paruo labore tenaci&longs;&longs;ima ligna validi&longs;<lb/>&longs;imè &longs;cinderet, & ideo optimè res gladijs illis diuiditur, <lb/>qui magis ad &longs;uperficiei naturam accedunt. </s>

<s id="id.001306">Ex quibus o<lb/>mnibus, nî fallimur, clarè patet, curacutiores angulo cu<lb/>nei obtu&longs;ioribus facilius &longs;cindant, quæ quidem ratio lon<lb/>gè ab ea di&longs;tat, ex qua cæteri ferè omnes Cuneum ad ve<lb/>ctis naturam referre hactenus contenderunt. </s>

<s id="id.001307">Cæterùm vtramque eorum quos diximus, <expan abbr="cuneor&umacr;">cuneorum</expan> <lb/>&longs;peciem &longs;olerti&longs;&longs;ima cognouit Natura, & ideo quoniam <lb/>res vel contu&longs;ione vel perforatione, vel &longs;ecatione con&longs;i<lb/>ciuntur, triplicem dentium qualitatem dentatis animali-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.138.jpg"/><figure id="id.007.00.138.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.138.1.jpg"></figure><lb/>bus dedit, Molares, <lb/>qui & Maxillares ap<lb/>pellantur, quibus <lb/>cibus contunditur, <lb/>Canini, quibus fit <lb/>perforatio, Anterio<lb/>res, quibus cibus <lb/>&longs;cinditur, quos ideo <lb/><foreign lang="greek">temnikou\s</foreign>, id e&longs;t, &longs;ecan<lb/>tes appellant Graeci. </s>

<s id="id.001308">Molares KK, <lb/>CaniniL, L, Temni<lb/>ci &longs;eu &longs;ecantes M. </s>

<s id="id.001309">Cuneus orbicularis lineari&longs;queue AB, in <lb/>quo axis linea e&longs;t, ad cuius naturam accedit AB cuneus <lb/>&longs;uperficialis CD, accedens ad &longs;uperficiei naturam, quam <lb/>vitro imaginamur EFGD, in aciem cunei de&longs;inentem, <lb/>GD, Lateratus lineari&longs;que cuneus, clauus HI. </s>

<s id="id.001310">Cunei autem omnes dupliciter &longs;unt efficaces, vel e<lb/>nim malleo, vt in ijs fit, quibus lìgna &longs;cin duntur & &longs;calpris <lb/>fieri &longs;olet, adiguntur, vel impul&longs;u & pre&longs;&longs;ione, vt in gla<lb/>dijs fit, pugionibus, cælatorum &longs;calpris, &longs;ubulis, & cæteris <lb/>eiu&longs;modi. </s>

<s id="id.001311">Quidam etiam &longs;unt, qui licet mallei ictu non <lb/>adigantur, malleum coniunctum habent, ceu &longs;unt &longs;ecu<lb/>res, ligones, A&longs;ciæ, & his &longs;imilia, quæ ex percu&longs;&longs;ione &longs;e<lb/>metip&longs;a &longs;cindendis rebus in&longs;erunt & validè penetrant. <lb/></s>

<s id="id.001312">De vi autem & efficacia ictus &longs;eu percu&longs;&longs;ionis hic &longs;uper<lb/>&longs;ed emus aliquid, ea de re, in &longs;equenti quæ&longs;tione verba fa<lb/>cturi. </s>

<s id="id.001313">Multa hîc addere potui&longs;&longs;emus ad Cochleam &longs;pe<lb/>ctantia, quippe quòd Cochlea cuneus &longs;it Cylindro inuo<lb/>lutus, qui quidem ad mallei, &longs;ed vectis virtute &longs;ibi adiun<lb/>ctâ, validi&longs;&longs;imè operatur, & &longs;excentis in&longs;eruit v&longs;ibus. </s>

<s id="id.001314">Ve<lb/>runtamen cùm de hac &longs;pecie egregiè di&longs;&longs;erat G. Vbaldus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.139.jpg"/>con&longs;ultò hanc di&longs;putationem omittimus; idque hac quo<lb/>que de cau&longs;&longs;a, quod nihil de cochlea, ac &longs;i eam non noui&longs;<lb/>&longs;et, locutus &longs;it Ari&longs;toteles. </s>

<s id="id.001315">Po&longs;&longs;umus autem in actu &longs;ci&longs;&longs;ionis, quæ cuneo fit, a<lb/>liâ tamen ratione vectem con&longs;iderare, nempe non in cu<lb/>neo quidem, &longs;ed in ip&longs;a re quæ &longs;cinditur. </s>

<s id="id.001316">E&longs;to enim quip<lb/>piam &longs;ci&longs;&longs;ile ABCD, <lb/>cui alteri extremita<lb/>tum, puta BD, cuneus <lb/>adigatur EFG, <expan abbr="fiatq;">fiatque</expan> <lb/>&longs;ci&longs;&longs;io per longitudi<lb/>nem &longs;ecundum <expan abbr="lineã">lineam</expan> <lb/>EH. facta igitur ex <lb/>cunei ingre&longs;&longs;u <expan abbr="parti&umacr;">partium</expan> &longs;eparatione B, expelletur in I, D ve<lb/>rò in K. fient igitur materiæ &longs;ci&longs;&longs;æ partes AIBH, CKDH, <lb/>ceu duo vectes, quorum hinc inde in corpore ip&longs;o fulci<lb/>menta L, M potentiæ vtrinque dilatantes BD, pondus ve<lb/>rò materiæ re&longs;i&longs;tentia, in &longs;eparationis loco vbi N. </s>

<s id="id.001317">Duca<lb/>tur NL, quanto itaque BN maiorem habebit proportio<lb/>nem ad LN, eo faciliùs re&longs;i&longs;tentia quæ in N, &longs;uperabitur. <lb/></s>

<s id="id.001318">Mutatur <expan abbr="aut&emacr;">autem</expan> a&longs;&longs;iduè in ip&longs;a &longs;ci&longs;&longs;ione fulcimentum, & <expan abbr="c&umacr;">cum</expan> <lb/>fulcimento ip&longs;a proportio. </s>

<s id="id.001319">Pertingente enim &longs;ci&longs;&longs;ione in <lb/>O, <expan abbr="fulcim&emacr;tum">fulcimentum</expan> fit in P. quo ca&longs;u &longs;ci&longs;&longs;ura e&longs;t facilior, quip<lb/>pe quod maiorem habeat proportionem BO ad OP, <expan abbr="quã">quam</expan> <lb/>BN ad NL. </s>

<s id="id.001320">Hoc autem experiuntur materiarij, qui primis <lb/>ictibus, &longs;ecuriculâ nondum probè adactâ, & nondum fa<lb/>ctâ notabili &longs;ci&longs;&longs;ione difficultatem &longs;entiunt, mox <expan abbr="factaiã">factaiam</expan> <lb/>&longs;eparatione faoillima paullatim fit materiæ totius &longs;epara<lb/>tio. </s>

<s id="id.001321">Hocidem & nos ab&longs;que cunei v&longs;u experimur, cum ba<lb/>culum aut quippiam tale manibus diductis &longs;cin dimus. </s>

<s id="id.001322">à <lb/>principio enim difficultatem &longs;entimus, deinde ex ea <expan abbr="quã">quam</expan> <lb/>diximus proportionc &longs;ci&longs;&longs;io ip&longs;a fit apprime facilis. </s>

<s id="id.001323">Vti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.140.jpg"/>mur etiam vecte cuneato ad &longs;cindendum & aperiendum: <lb/>adacto enim &longs;ci&longs;&longs;uræ cuneo, idqueue manu malleoue, tum <lb/>ab altera extremitate pre&longs;&longs;o, valida fit ex vectis vi <expan abbr="c&omacr;tinui">continui</expan> <lb/><figure id="id.007.00.140.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.140.1.jpg"></figure><lb/>corporis &longs;eparatio. </s>

<s id="id.001324">Ma<lb/>teria &longs;ci&longs;&longs;ilis AB <expan abbr="&longs;calpr&umacr;">&longs;calprum</expan> <lb/>ceu vectis cuneatus CD, <lb/>cuius fulcimentum, E, <lb/>pondus verò vbi C, po<lb/>tentia vbi D, quo ca&longs;u <lb/>quo maior e&longs;t proportio <lb/>DE ad EC, eo e&longs;t ip&longs;a &longs;ci&longs;&longs;io leuior & facilior. </s>

</subchap1><subchap1>

<s id="id.001325">QV AESTIO XVIII.</s>

<s id="id.001326"><emph type="italics"/>Quærit hic Ari&longs;toteles, Cur per Trochleas ab exiguapotentia in<lb/>gentia moueantur pondera?<emph.end type="italics"/></s>

<s id="id.001327">De Trochlea Pappus, & veteres: inter recentiores e<lb/>gregiè admodum, vt omnia examinauit in Mechani<lb/>cis G. Vbaldus. </s>

<s id="id.001328">Nos tamen interim po&longs;t clari&longs;&longs;imos illos <lb/>viros aliquid quod nouitatem & &longs;ubtilitatem &longs;apiat, de <lb/>no&longs;tro penu promemus. </s>

<s id="id.001329">Et &longs;anè inuentis quidem addere <lb/>res e&longs;t fa cilis, at quod inuentis addas inuenire haud adeo <lb/>facile. </s>

<s id="id.001330">Sed nos primum Philo&longs;ophi ip&longs;ius dicta ad <expan abbr="trutinã">trutinam</expan> <lb/>reuocemus. </s>

<s id="id.001331">Ita autem quæ&longs;tionem proponit; Cur &longs;i qui&longs;<lb/>piam Trochleas componens duas, in &longs;ignis duobus, ad &longs;e <lb/>inuicem iunctis contrario ad Trochleas modo circulo fu<lb/>nem circumduxerit, cuius alterum quidem caput tigno<lb/>rum appendatur alteri, alterum verò Trochleis &longs;it <expan abbr="innix&umacr;">innixum</expan> <lb/>& à funis initio trahere c&oelig;perit, magna trahit pondera, li<lb/>cet imbecillium fuerit virium? </s>

<s id="id.001332">Ob&longs;euri&longs;&longs;ima expo&longs;itio, & nî res e&longs;&longs;et vulgò per &longs;e <lb/>nota, dequeue ea Vitruuius & Mechanici non egi&longs;&longs;ent, diffi<lb/>cile vtique e&longs;&longs;et ex eius verbis &longs;en&longs;um a&longs;&longs;equi. </s>

<s id="id.001333">Tigna &longs;anè voca&longs;&longs;e videtur ea ligna, quæ à Vitruuio <lb/>Rechami dicuntur, in quibus nempe ip&longs;i in&longs;eruntur orbi<lb/>culi. </s>

<s id="id.001334">Et&longs;i de tignis eiu&longs;mo di aliud quippiam &longs;entire videa<lb/>tur Picolomineus. </s>

<s id="id.001335">Græea lectio pro tignis habet <foreign lang="greek">cu/la</foreign>, id <lb/>e&longs;t, ligna; item vbi Leoniceni ver&longs;io legit, ad &longs;e inuicem <lb/>iunctis, textus habet <foreign lang="greek">snm<gap/>ai/nousin e(autoi_s e)ranti/ws</foreign>, hoc e&longs;t, in<lb/>uicem ex oppo&longs;ito concurrunt. </s>

<s id="id.001336">Certè locum totum ita <lb/>redderem: Cur &longs;i quis duas Trochleas fecerit, in duobus <lb/>lignis &longs;ibi ex oppo&longs;ito concurrentibus, ei&longs;queue Trochleis <lb/>circumpo&longs;uerit funem, cuius alterum caput alteri ligno<lb/>rum &longs;it annexum, alterum verò Trochleis cohæreat, vel <lb/>apponatur. </s>

<s id="id.001337">Si quis alterum funis principium trahat, ma<lb/>gna trahat pondera, et&longs;i trahens potentia &longs;it exigua? </s>

<s id="id.001338">Nos <lb/>verbis figuram, & figurâ verba ip&longs;a elucidabimus. </s>

<s id="id.001339">Sint duo ligna ex oppo&longs;ito concurrentia, <lb/>in quibus Trochleæ, hoc e&longs;t, orbiculi AB, fu<lb/>nis ductarius DABC, cuius alterum caput re<lb/>ligatum e&longs;t ligno trochleæ A, vbi e&longs;t C. </s>

<s id="id.001340">Tro<lb/>chlea A loco &longs;tabili commendata, vbi E. </s>

<s id="id.001341">Pon<lb/>dus alteri ligno Trochleæ appen&longs;um F. </s>

<s id="id.001342">Tra<lb/>cto itaque fune DABC, eleuatur & trahitur <lb/>pondus F. </s>

<s id="id.001343">Ex quibus clarè patet, <expan abbr="Philo&longs;oph&umacr;">Philo&longs;ophum</expan> <lb/>propo&longs;ui&longs;&longs;e Trochleam duobus tantum orbi<lb/>culis munitam, quod vtique &longs;atis erat ad ex<lb/>plicationem. </s>

<s id="id.001344">Inquit autem, faciliùs vecte <expan abbr="quã">quam</expan> <lb/>manu pondus moueri. </s>

<s id="id.001345">Trochleam vero (id <lb/>e&longs;t, orbiculum; ita enim e&longs;t intelligendum) e&longs;<lb/>&longs;e vectem, aut vectis virtute operari. </s>

<s id="id.001346">Ita autem <lb/>videtur argumentari. </s>

<s id="id.001347">Si vnicâ Trochleâ plus trahitur <lb/>quàm manu, multo faci ius & velocius id fiet duobus, <lb/>quibus plus, vt ip&longs;e ait, quàm in duplici velocitate pon<lb/>dus leuabitur. </s>

<s id="id.001348">Summa dictorum e&longs;t, ex multiplicatione <lb/>orbiculorum pondus ip&longs;um imminui, & minori difficul-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.142.jpg"/>tate leuari, quod &longs;anè verum e&longs;t. </s>

<s id="id.001349">Nos tamen nonnulla <expan abbr="c&omacr;-&longs;iderabimus">con<lb/>&longs;iderabimus</expan>. </s>

<s id="id.001350">quod ait, vecte facilius moueri pondera <lb/>quam manu, &longs;emper non e&longs;t verum. </s>

<s id="id.001351">Si enim vectis pars <lb/>quæ à fulcimento ad manum breuior fuerit illâ, quæ à <lb/>fulcimento ad pondus difficilius vecte pondus mouebi<lb/>tur quam manu. </s>

<s id="id.001352">Idem quoque accidet, &longs;i eo modo vecte <lb/>vtamur, quem ob&longs;eruat Guidus Vbald. Tract. </s>

<s id="id.001353">de Vecte <lb/>prop.

3. Po&longs;ita nempe inter fulcimentum & pondus &longs;u&longs;ti<lb/>nente potentiâ. </s>

<s id="id.001354">Præterea quod a&longs;&longs;eruit Ari&longs;toteles, Tro<lb/>chleas ad vectem reduci, verum quidem e&longs;t, &longs;ed aptius di<lb/>xi&longs;&longs;et ad libram, etenim vectis vtcunque à ful cimento di<lb/>uiditur. </s>

<s id="id.001355">Libra verò quod & orbiculis ex centro accidit, <lb/>&longs;emper bifariam. </s>

<s id="id.001356">Ad hæc videtur ille ad orbiculorum <lb/>multiplicitatem Trochlearum vim referre. </s>

<s id="id.001357">Si enim, ait, <lb/>vnicâ Trochleâ pondus facile trahitur, id multo validius <lb/>pluribus fiet. </s>

<s id="id.001358">Veruntamen non ab&longs;olutè ex orbiculorum <lb/>multiplicationeid fieriita o&longs;tendemus. </s>

<s id="id.001359">Sint duæ op<lb/>po&longs;itæ lineae rectae, <lb/>vtpote trabes AB, <lb/>CD, <expan abbr="inuic&emacr;">inuicem</expan> æqui<lb/>di&longs;tantes & ip&longs;æ <lb/>&longs;tabiles: &longs;uperiori <lb/>tres appendantur <lb/>orbiculi ex <expan abbr="p&umacr;ctis">punctis</expan> <lb/>E, F, G, <expan abbr="n&emacr;pe">nempe</expan> ML, <lb/>PQ, TV. inferiori <lb/><expan abbr="aut&emacr;">autem</expan> duobus pun<lb/>ctis IH, nempe <lb/>NO, RS. </s>

<s id="id.001360">Erunti<lb/>gitur invniuer&longs;um <lb/>quinque, indatur pereos funis ductarius KLMNOP <lb/>QRSTVX, ex cuius extremitate pendeat pondus X, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.143.jpg"/>Trahatur funis in K. </s>

<s id="id.001361">Dico ex multiplicatione <expan abbr="orbiculor&umacr;">orbiculorum</expan>, <lb/>trahentipondus nequaquam minui. </s>

<s id="id.001362">Sint autem orbicu<lb/>lorum diametri, LM, NO, PQ, RS, TV, applicetur poten<lb/>tîa in S. </s>

<s id="id.001363">Erit igitur ad hoc vt &longs;u&longs;tineat æqualis ponderi X, <lb/>orbiculi enim TV &longs;emidiametri &longs;unt æquales. </s>

<s id="id.001364">Transfe<lb/>ratur <expan abbr="pot&emacr;tia">potentia</expan> in q, & ita deinceps donec perueniatur in K, <lb/>vbi funis ip&longs;ius e&longs;t principium, Idem e&longs;t igitur &longs;eruata &longs;em<lb/>per &longs;emidiametrorum æqualitate ac &longs;i potentia quæ e&longs;t in <lb/>K, applicata intelligatur in T vel in V. vbicunque enim <lb/>collocetur, ponderi erit æqualis. </s>

<s id="id.001365">Nihil igitur rebus ita <lb/>di&longs;po&longs;itis, orbiculorum multiplicatio ad facilitatem ope<lb/>ratur. </s>

<s id="id.001366">Alia itaque ratio quærenda e&longs;t, quam non &longs;atis ex<lb/>plica&longs;&longs;e videtur Ari&longs;toteles. </s>

<s id="id.001367">Probabimus autem, nullam <lb/>ex &longs;uperioribus orbiculis fieri ponderum imminutionem, <lb/>&longs;ed totam vim in inferioribus con&longs;i&longs;tere. </s>

<s id="id.001368">At nos interim <lb/>quippiam quod ad rem faciat, proponamus. </s>

<s id="id.001369">E&longs;to punctum A, cuirectæ ap<lb/>pendantur lineæ BAC, diui&longs;æ qui<lb/>dem in A, &longs;it autem lineæ BA caput <lb/>B, ip&longs;ius verò CA caput C. </s>

<s id="id.001370">Modò <lb/>intelligantur vnitæ in A, &longs;itqueue vni<lb/>ca linea à puncto A ceu funiculus <lb/>dependens BAC; Appendatur capi<lb/>ti B pondus B. </s>

<s id="id.001371">Capiti vero C, <expan abbr="p&omacr;dus">pondus</expan> <lb/>C, inter &longs;e æqualia. </s>

<s id="id.001372">Potentia igitur <lb/>in A, duo &longs;u&longs;tinebit pondera BC. <lb/></s>

<s id="id.001373">Pondera verò ex æqualitate æque<lb/>ponderabunt. </s>

<s id="id.001374">Quod &longs;i B potentia <lb/>dicatur &longs;u&longs;tinens pondus C, aut C <lb/>potentia &longs;u&longs;tinens pondus D, vel <lb/>duæ potentiæ inter &longs;e æquales, nihil <lb/>refert. </s>

<s id="id.001375">Vtcunque enim id &longs;it, fiet æquilibrium. </s>

<s id="id.001376">Habemus <lb/>igitur ex i&longs;tis ad &longs;u&longs;tinendum pondus ex &longs;uperiori parte <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.144.jpg"/>appen&longs;um potentiam requiri ip&longs;i ponderi æqualem. </s>

<s id="id.001377">Ani<lb/>mo po&longs;thæc concipiatur alia recta linea DEF, cuius inte<lb/>gra longitudo &longs;i exten deretur, e&longs;&longs;et DE, EF. </s>

<s id="id.001378">Appendatur <lb/>in E pondus E æ quale alteri ponderum B vel, C, &longs;int autem <lb/>duæ potentiæ pondus E &longs;u&longs;tinentes D, F. </s>

<s id="id.001379">Vtraque igitur <lb/>dimidium &longs;u&longs;tinebit ponderis E, &longs;ed potentia quæ &longs;u&longs;ti<lb/>nebat pondus B, in C erat ip&longs;i B æqualis, vbi appen&longs;io pon<lb/>deris erat in &longs;uperiori parte in A, hîc autem, vbi appen&longs;io <lb/>e&longs;t in parte in feriori, vtraque potentia dimidium &longs;u&longs;tinet <lb/>appen&longs;i ponderis. </s>

<s id="id.001380">Videmus igitur illam appen&longs;ionem <lb/>quidem pondus nullatenus imminuere, hanc verò pon<lb/>dus ip&longs;um, bifariam diui&longs;um, &longs;u&longs;tinentibus potentijs im<lb/>partiri. </s>

<s id="id.001381">Hæ cin lineis, Mathematicâ v&longs;i ab&longs;tractione, con<lb/>&longs;iderauimus, nunc verò eadem mechanicè perpenda<lb/>mus. </s>

<s id="id.001382">Sit igitur <lb/>punctum A, vt <lb/>in &longs;e quenti figu<lb/>ra clauus paxil<lb/>lu&longs;ue, cui appen<lb/>&longs;us funiculus <lb/>BAC, & funicu<lb/>li capitibus pon<lb/>dera BC, &longs;it quo<lb/>que anulus D, <lb/>per quem traìe<lb/>ctus funiculus <lb/>EDF. </s>

<s id="id.001383">Anulo au<lb/>tem <expan abbr="c&omacr;iunctum">coniunctum</expan> <lb/>pondus G. </s>

<s id="id.001384">His igiturita con&longs;titutis, eadem demon&longs;tra<lb/>buntur q