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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Biancani, Giuseppe</author> <title>Aristotelis loca mathematica</title> <date>1615</date> <place>Bologna</place> <translator/> <lang>la</lang> <cvs_file>bianc_locam_01_la_1615</cvs_file> <cvs_version/> <locator>009.xml</locator> </info> <text> <front> <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.001.jpg"/><section><p type="head"> <s id="id.000001">ARISTOTELIS<lb/>LOCA MATHEMATICA<lb/>Ex vniuer&longs;is ip&longs;ius Operibus collecta, <lb/>& explicata.</s></p><p type="head"> <s id="id.000002"><emph type="italics"/>Aristotelicæ videlicet expo&longs;itionis complementum <lb/>hactenus de&longs;ideratum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000003">Acce&longs;&longs;ere de Natura Mathematicarum &longs;cientiarum Tractatio; <lb/><expan abbr="atq;">atque</expan> Clarorum Mathematicorum Chronologia.</s></p><p type="head"> <s id="id.000004"><emph type="italics"/>Authore IOSEPHO BLANCANO Bononien&longs;i è Societate Ie&longs;u, <lb/>Mathematicarum in Gymna&longs;io Parmen&longs;i Profe&longs;&longs;ore.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000005">Ad Illu&longs;tri&longs;&longs;imum, ac Nobili&longs;&longs;imum<lb/>PETRVMFRANCISCVM MALASPINAM<lb/>Aedificiorum Marchionem, apud Cæ&longs;. <!-- REMOVE S-->Maie&longs;tatem <lb/>pro Sereni&longs;s. <!-- REMOVE S-->Parmen&longs;ium Duce Legatum.</s> </p><p type="head"> <s id="id.000006">BONONIÆ M. <!-- REMOVE S-->D C. <!-- KEEP S--><!-- REMOVE S-->X V.<!-- KEEP S--></s> </p><p type="head"> <s id="id.000007">Apud Bartholomæum Cochium. <!-- KEEP S--></s> <s id="id.000008">Superiorum permi&longs;&longs;u.</s></p><p type="head"> <s id="id.000009">Sumptibus Hieronymi Tamburini.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.002.jpg"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.003.jpg" pagenum="3"/> </section> <section><p type="head"><s id="id.000010">ILLVSTRISSIMO <lb/>AC NOBILISSIMO<lb/>PETROFRANCISCO <lb/>MALASPINAE<lb/>ÆDIFICIORVM MARCHIONI.</s></p><p type="main"> <s id="id.000011"><emph type="italics"/>En tandem Illustriß. <!-- REMOVE S-->Marchio opus no­<lb/>strum de Locis Mathematicis apud Ari­<lb/>stotelem, vnà cum Tractatione de natura <lb/>&longs;cientiarum Mathematicarum, necnon <lb/>Clarorum <expan abbr="Mathematicorũ">Mathematicorum</expan> Chronologia; <lb/>quod tibi Mecœnati meo munificenti&longs;simo iure meritò <lb/>dicare, ac &longs;ub clarißimi tui nominis patrocinio in lucem <lb/>dare con&longs;titui. </s> <s id="id.000012">primùm quidem, vt mei perpetui erga te <lb/>amoris, & ob&longs;eruantiæ hoc vnum &longs;altem specimen exta­<lb/>ret: tùm vt idoneum, æquumque propo&longs;itæ rei iudicem <lb/>nanci&longs;cerer. </s> <s id="id.000013">cùm enim adiu&longs;tum <expan abbr="arbitrũ">arbitrum</expan> duo potißimùm <lb/>requirantur, rerum &longs;cilicet cognitio, atque prudentia, quem <lb/>te rei, de qua agitur peritiorem, quemuè prudentiorem <lb/>inuenire potuerim? </s> <s id="id.000014">tu enim cùm Phy&longs;iologiæ, ac Mathe­<lb/>maticarum omnium Encyclopædiam mirum in modum<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.004.jpg" pagenum="4"/><emph type="italics"/>excolueris, adintima Mathematicarum penetralia ita <lb/>per&longs;ua&longs;i&longs;ti, vt Archimedis, & Apollonij admirandis, ac <lb/>&longs;ubtilißimis Demon&longs;trationibus detinearis. </s> <s id="id.000015">Quanta por­<lb/>rò in rebus agendis prudentia valeas, toti penè Europæ <lb/>innotuit, cùm pro no&longs;tris Sereniß. <!-- REMOVE S-->Ducibus, non &longs;olùm ad <lb/>omnes ferè Italiæ, atque Germaniæ Principes, verùm etiam <lb/>ad Cæ&longs;aream Maie&longs;tatem rebus fœliciter ge&longs;tis Legatus <lb/>decimùm extiteris; ac demùm à Sereniß. <!-- REMOVE S-->Duce Ranutio <lb/>inter primarios de Rep. <!-- KEEP S--></s> <s id="id.000016">Con&longs;iliorum Authores ad&longs;citus <lb/>fueris. </s> <s id="id.000017">Cæterùm in Clarorum Mathematicorum Chro­<lb/>nologia perlegenda, &longs;æpißimè tibi nobilißimi æquè, ac do­<lb/>ctißimi Viri, tui omnino per&longs;imiles occurrent, quod tibi <lb/>nonni&longs;i gratißimum accidere po&longs;&longs;e arbitror. </s> <s id="id.000018">Complectere <lb/>igitur ea benignitate, atque clementia, qua &longs;oles no&longs;tra stu­<lb/>dia promouere, mea hæc quantulacumque munu&longs;cula. <lb/></s> <s id="id.000019">quæ &longs;i tibi accepta e&longs;&longs;e intellexero, iam tandem ma­<lb/>ximorum munerum loco habenda e&longs;&longs;e cen&longs;e­<lb/>bo. </s> <s id="id.000020">incolumem tibi, ac fœlicem D. Opt. <lb/><!-- REMOVE S-->Max. <!-- REMOVE S-->longæuitatem tueatur. <lb/></s> <s id="id.000021">Vale.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000022"><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/><!-- KEEP S--></s></p></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.005.jpg" pagenum="5"/><section><p type="head"> <s id="id.000023">Liber de &longs;e ip&longs;o.</s></p><p type="head"> <s id="id.000024"><emph type="italics"/>Nec di&longs;cet Lector me &longs;olo interprete totum, <lb/>Nec &longs;ine me totum di&longs;cet Aristotelem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000025">Ego Iordanus Ca&longs;&longs;ini Præpo&longs;itus Prouincialis Prouinciæ Venetæ Societatis <lb/>Ie&longs;u, ex auctoritate Adm. <!-- REMOVE S-->Reuer. <!-- REMOVE S-->P. nc&longs;tro Præpo&longs;iti Generalis P. <!-- REMOVE S-->Claudij <lb/>Aquæuiuæ, facultatem concedo, vt hoc opus P. <!-- REMOVE S-->Io&longs;ephi Blancani eiu&longs;dem <lb/>Societatis, quod in&longs;cribitur, Ari&longs;t. Loca Mathematica ex vniuer&longs;is ip&longs;ius <lb/>operibus collecta, & explicata, à deputatis Patribus recognitum, & ap­<lb/>probatum typis mandari po&longs;&longs;it. </s> <s id="id.000026">Parmæ die 15. Ianuarij 1615.</s></p><p type="main"> <s id="id.000027"><emph type="italics"/>Iordanus Ca&longs;&longs;ini P.<emph.end type="italics"/><!-- REMOVE S-->Don Marcellus Balda&longs;&longs;inus pro Illu&longs;tri&longs;s. </s></p><p type="main"> <s id="id.000028">& Reuerendi&longs;s. <!-- REMOVE S-->Archiepi&longs;c. <!-- REMOVE S-->Bonon.</s> </p><p type="main"> <s id="id.000029">Imprimatur</s></p><p type="main"> <s id="id.000030">Fr. <!-- REMOVE S-->Hieronymus Onuphrius pro Reuerendi&longs;s. <!-- REMOVE S-->P. <!-- REMOVE S-->Inqui&longs;itore Bonon.</s> </p></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.006.jpg" pagenum="6"/><section><p type="head"> <s id="id.000031">LECTORI.</s></p><p type="main"> <s id="id.000032">Qvod pri&longs;cis olim temporibus (humani&longs;&longs;ime Lector) &longs;um­<lb/>mi duo Philo&longs;ophi, Philippus Mendeus, ac Theon Smyr­<lb/>næus in Platonis Dialogis egregiè perfecerunt, vt videli­<lb/>cet quæ pa&longs;&longs;im &longs;ummus hic Philo&longs;ophus de Mathemati­<lb/>cis &longs;cripta reliquit, eadem ip&longs;a ab illis &longs;electa, & in vnum <lb/>qua&longs;i corpus redacta lucubrationibus illu&longs;trarent: idem ego quoque in <lb/>Ari&longs;totelis operibus efficere &longs;um conatus, vt quæ de Mathematicis re­<lb/>bus in vniuer&longs;is eiu&longs;dem monumentis &longs;par&longs;a leguntur, eadem in vnum <lb/>à me collecta, & explicata ijs Philo&longs;ophiæ &longs;tudio&longs;is maxime &longs;eruirent, <lb/>qui pri&longs;ca illa con&longs;uetudine relicta, Mathematicarum omnium ignari <lb/>non &longs;ine graui &longs;tudiorum &longs;uorum detrimento Philo&longs;ophiæ curriculum <lb/>aggrediuntur. </s> <s id="id.000033">Vt autem huius operis nece&longs;&longs;itas, <expan abbr="variæ&qacute;">variæque</expan>; vtilitates pla­<lb/>nius cogno&longs;cantur operæpretium erit initio illius cau&longs;as exponere; quæ <lb/>me poti&longs;&longs;imum ad illud con&longs;cribendum compulerunt, quarum</s></p><p type="main"> <s id="id.000034">Prima &longs;it, quod hæc Ari&longs;t. loca Mathematica, quæ quidem ferè 408. <lb/>numerantur, pe&longs;&longs;imè latinis literis con&longs;ignata &longs;unt v&longs;que adeò, vt Ari­<lb/>&longs;totelem ip&longs;um, vel inuitum (quod po&longs;tea multis in locis planum fiet) in <lb/>ab&longs;urdi&longs;&longs;ima errata &longs;æpi&longs;&longs;imè compellant.</s></p><p type="main"> <s id="id.000035">Secunda, quòd plurima huiu&longs;modi loca à nemine, quod &longs;ciam, adhuc <lb/>declarata in tenebris magno no&longs;trorum malo delite&longs;cunt: cuiu&longs;modi <lb/>&longs;unt ad &longs;exaginta problemata, libellus de lineis in &longs;ecabilibus, libellus <lb/>de mundo, &longs;i tamen Ari&longs;totelis e&longs;t, & Mechanicæ quæ&longs;tiones, quamuis <lb/>enim Picolomineus in eas paraphra&longs;im ediderit, loca tamen earum dif­<lb/>ficiliora non &longs;atis illu&longs;trauit. </s> <s id="id.000036">Vt autem dixi 408. in vniuer&longs;um loca mi­<lb/>nimùm numerantur, quibus illud Platonis in&longs;criptum e&longs;t <foreign lang="greek">agaiome/trhtos <lb/>udei/s eisi/to</foreign>; & in quibus Mathematicæ di&longs;ciplinæ rudes, & imperiti, quem <lb/>&longs;equuntur ducem Ari&longs;t. eum &longs;æpe de&longs;erere non &longs;ine turpi dedecoris no­<lb/>ta coguntur; quo fit vt exempla illa Mathematica lucem rebus aliquan­<lb/>do allatura, tenebras cimmerijs, vt aiunt vmbris cra&longs;&longs;iores ij&longs;dem <lb/>obducant.</s></p><p type="main"> <s id="id.000037">Tertia, quia Græcieorumdem locorum commentatores breuiter, & <lb/>ob&longs;curè admodum ea, quæ ad Mathematicum &longs;pectant, attingunt, hoc <lb/>enim ab ip&longs;is <expan abbr="certũ">certum</expan> ponitur, Lectorem e&longs;&longs;e, vt moris tunc erat, omnium <lb/>Philo &longs;ophorum, Mathematicis imbutum; at verò no&longs;tra ætate magna <lb/>cum Philo&longs;ophiæ iactura, quamplurimi earumdem di&longs;ciplinarum de&longs;ti­<lb/>tuti præ&longs;idijs, ne Græcorum quidem Interpretum explanationes, ne­<lb/>dum Ari&longs;t. ob&longs;curè dicta intelligunt.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.007.jpg" pagenum="7"/><p type="main"> <s id="id.000038">Quarta. </s> <s id="id.000039">Adde, quod etiam &longs;i quis leuiter &longs;it erudito illo Mathemati­<lb/>corum puluere con&longs;per&longs;us, adeò tamen peruer&longs;a e&longs;t eorumdem Græco­<lb/>rum in Latinum tran&longs;latio, <expan abbr="tanta&qacute;">tantaque</expan>; figurarum, quæ nece&longs;&longs;ariæ erant <lb/>confu&longs;io, & deprauatio, vt nec abeo, qui &longs;it Mathematicarum &longs;cientia <lb/>excultus, &longs;ine magno labore percipi po&longs;&longs;int. </s> <s id="id.000040">Quin etiam figuræ illæ, quæ <lb/>omnino nece&longs;&longs;ariæ &longs;unt ob Scriptorum, & Typographorum in&longs;citiam, <lb/>aut inertiam pluribus in locis de&longs;iderantur. </s> <s id="id.000041">Latini verò multo minus, <lb/>quàm Græci Mathematicæ periti, qua ratione eadem loca pertractaue­<lb/>rint, facilius e&longs;t conijcere, quàm vt dici oporteat.</s></p><p type="main"> <s id="id.000042">Quinta. </s> <s id="id.000043">Ex his omnibus in aliud incommodum, vel maximum Phi­<lb/>lo&longs;ophi quidam incidebant; aut enim horum locorum expo&longs;itionem ta­<lb/>citi declinabant: aut eam minime nece&longs;&longs;ariam ad Ari&longs;t. percipiendam <lb/>&longs;ententiam a&longs;&longs;erebant; quo quid ab&longs;urdius, quid &longs;tudio&longs;orum progre&longs;­<lb/>&longs;ibus pernicio&longs;ius excogitari pote&longs;t? </s> <s id="id.000044">Eorum verò nonnulli eorumdem <lb/>locorum expo&longs;itionem audacter nimis aggrediebantur, <expan abbr="atq;">atque</expan> hinc pueri­<lb/>les illæ, ac ridiculæ expo&longs;itiones pa&longs;&longs;im auditæ, cuiu&longs;modi e&longs;t illa, quan­<lb/>do Ari&longs;toteles ait, quod illi frequenti&longs;&longs;imum e&longs;t, omnis triangulus ha­<lb/>bet tres; nihil aliud &longs;ignificari volunt, quàm omnem triangulum habe­<lb/>re tres angulos. </s> <s id="id.000045">quod &longs;i dicat, omnis triangulus habet tres æquales duo­<lb/>bus rectis: hic hærent, hinc anguntur: <expan abbr="cumq;">cumque</expan> ex his angu&longs;tijs, ac tricis <lb/>&longs;e minimè expedire valeant, aurea verba illa, quibus ingentes &longs;apientiæ <lb/>the&longs;auri continentur, alto &longs;ilentij velo contegere Mathematicarum eos <lb/>cogit in&longs;citia: vnde illud, quod Græcæ linguæ imperitis mutata oratio­<lb/>ne acclamandum illis foret, Mathematicum e&longs;t, non legitur. </s> <s id="id.000046">Nec mi­<lb/>nus elegans illa altera expo&longs;itio; Diametrum e&longs;&longs;e incommen&longs;urabilem <lb/>co&longs;tæ; quod &longs;æpe apud Ari&longs;t. legentibus occurrit, nihil aliud &longs;ibi velle, <lb/>quam Diametrum e&longs;&longs;e longiorem co&longs;ta, quam quidem a&longs;ymetriæ huius <lb/>ignorantiam Plato de legibus dial. </s> <s id="id.000047">7. non hominum, &longs;ed &longs;uum, <expan abbr="peco-rumq;">peco­<lb/>rumque</expan> appellare non dubitauit. </s> <s id="id.000048">Quid illa? </s> <s id="id.000049">cum Ari&longs;t. ait duo cubi, cu­<lb/>bus, ip&longs;um loqui putant de duplatione Geometrici cubi, nondum in­<lb/>uenta; non intelligentes, eum ibi de numeris cubis &longs;ermonem habere. <lb/></s> <s id="id.000050">Auerroes ip&longs;e tantus vir 5. Phy&longs;. commen. <!-- REMOVE S-->15. quàm &longs;e Mathematicis, <lb/>reliqui&longs;que Philo&longs;ophis irridendum præbet dum à permutata propor­<lb/>tione putat &longs;erectè in hunc modum pluribus apud ip&longs;um verbis explica­<lb/>tum, argumentari,</s> </p><p type="main"> <s id="id.000051"><emph type="italics"/>Vt &longs;e habet voluntas noua ad effectum nouum, <lb/>It a voluntas antiqua ad effectum antiquum. <lb/></s> <s id="id.000052">Ergo permutatim, vt &longs;e habebit voluntas noua ad effectum <lb/>antiquum, ita voluntas antiqua ad effectum nouum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000053"><expan abbr="Spectatũ">Spectatum</expan> admi&longs;&longs;i ri&longs;um teneatis amici? </s> <s id="id.000054">nego, ait; qui&longs;piam con&longs;equen­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.008.jpg" pagenum="8"/>tiam, non enim hoc e&longs;t argumentari à permutata ratione, deberet enim <lb/>inferre, &longs;ic, ergo ita &longs;e habebit voluntas noua ad antiquam, quemad­<lb/>modum effectus nouus ad antiquum. </s> <s id="id.000055">quæ vitio&longs;a argumentatio quan­<lb/>tumuis læuis &longs;it, & manife&longs;ta, quo&longs;dam tamen magni nominis philo&longs;o­<lb/>phantes adeò tor&longs;it, vt adhuc torqueat.</s></p><p type="main"> <s id="id.000056">Quanta autem mi&longs;eratione digni, qui publicè aliquando apud &longs;uos <lb/>auditores totam Per&longs;pectiuam, qua nihil iucundius e&longs;t, de medio tolle­<lb/>re conati &longs;unt, propterea quod illæ vi&longs;uales lineæ, illi anguli, illæ pyra­<lb/>mides, aut coni, quibus vi&longs;io perficitur nullibi extarent, &longs;ed e&longs;&longs;ent vana <lb/>quædam opticorum figmenta. </s> <s id="id.000057">Quì verò fieri potuit, vt non aduerterint <lb/>i&longs;ti &longs;e Ari&longs;toteli &longs;uo manife&longs;te repugnare, qui &longs;æpius de lineis vi&longs;ualibus <lb/>per&longs;pectiuum pertractare a&longs;&longs;erit, <expan abbr="di&longs;crimen&qacute;">di&longs;crimenque</expan>; inter lineam phy&longs;icam, & <lb/>opticam a&longs;&longs;ignat, <expan abbr="ip&longs;ius&qacute;">ip&longs;iusque</expan>; optices tanquam veræ &longs;cientiæ mentionem <lb/>&longs;æpius facit.</s></p><p type="main"> <s id="id.000058">Alij ex altera parte contra A&longs;tronomos in &longs;urgunt, eccentricos, <expan abbr="atq;">atque</expan> <lb/>epiciclos omnes de cœlo detrahere cupientes. </s> <s id="id.000059">Verum id i&longs;ti nulla ex­<lb/>pre&longs;&longs;a nedum probabili ratione faciunt, falsò exi&longs;timantes A&longs;tronomos <lb/>admirandam illam Cœlorum fabricam a&longs;&longs;erere, non autem &longs;upponere: <lb/>&longs;ed a&longs;tronomi illam &longs;upponunt, <expan abbr="eam&qacute;">eamque</expan>; propterea hypothe&longs;im <expan abbr="appellãt">appellant</expan>, <lb/>non a&longs;&longs;erunt. </s> <s id="id.000060">Quod &longs;i aliqua probabili ratione id facerent, vti nonnulli <lb/>ex recentioribus, quorum Ticho Coripheus e&longs;t, laudandi potius, quam <lb/>vituperandi e&longs;&longs;ent. </s> <s id="id.000061">Impugnant <expan abbr="itaq;">itaque</expan> a&longs;tronomachi i&longs;ti hypothe&longs;im pro <lb/>a&longs;&longs;ertione; <expan abbr="tales&qacute;">talesque</expan>; &longs;æpè hi &longs;unt, vt non &longs;atis intelligant, quid &longs;it Aequa­<lb/>tor, aut Zodiacus, ne dum quid Epiciclus, aut Eccentricus. </s> <s id="id.000062">Nec defuit <lb/>qui viginti duo argumenta excogitarit, <expan abbr="atq;">atque</expan> in medium protulerit, qui­<lb/>bus contra A&longs;tronomos probare conatus e&longs;t, nullo modo Solem, aut <lb/>Lunam moueri po&longs;&longs;e motibus contrarijs, ide&longs;t, ab oriente in <expan abbr="occid&etilde;tem">occidentem</expan> <lb/>motu diurno, & proprio ab occidente in orientem. </s> <s id="id.000063">Sed exi&longs;timandum <lb/>e&longs;t <expan abbr="i&longs;tũ">i&longs;tum</expan> Lunam nouam à Sole quotidie magis, ac magis ver&longs;us orientem <lb/>recedere, nunquam animaduerti&longs;&longs;e; ab ea enim hanc motuum concor­<lb/>diam didici&longs;&longs;et.</s></p><p type="main"> <s id="id.000064">Quid tandem <expan abbr="dic&etilde;dum">dicendum</expan> de quodam magni nominis Philo&longs;opho, om­<lb/>nium tamen <expan abbr="Mathematicarũ">Mathematicarum</expan> experte, qui in publica di&longs;putatione axio­<lb/>ma illud Mathematicum, omne totum e&longs;t maius &longs;ua parte, in &longs;en&longs;u in <lb/>quo à Mathematicis effertur negare non erubuit, eò, quod in infinito, <lb/>vt aiebat non concederetur ab omnibus. </s> <s id="id.000065">&longs;cilicet non intelligebat ma­<lb/>thematicum tantummodo tractare de Quantitate finita, ac terminata, <lb/>in qua axioma prædictum ab omnibus conceditur. </s> <s id="id.000066">Neque vero hic <lb/>nonnullorum infen&longs;us in Mathematicas animus quieuit, verum etiam <lb/>cò progre&longs;&longs;us e&longs;t, vt eas omnes omnino conuellere, atque ex albo &longs;cien­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.009.jpg" pagenum="9"/>tiarum, quamuis non Ari&longs;totele tantum, &longs;ed ip&longs;a etiam veritate repu­<lb/>gnante, expungere conati &longs;int; <expan abbr="idq;">idque</expan> ne&longs;cio an vlla alia de cau&longs;a egerint, <lb/>quàm quod eas non &longs;atis calerent; non &longs;ecus <expan abbr="atq;">atque</expan> Ae&longs;opica illa Vulpes, <lb/>quæ cum cauda mutilata e&longs;&longs;et, caudarum mutilationem reliquis vulpi­<lb/>bus vafrè per&longs;uadere conabatur. </s> <s id="id.000067">Verum enim verò optimè &longs;cio, ea, <lb/>qu&ecedil; hactenus dicta &longs;unt non in omnes no&longs;tri temporis Philo&longs;ophos qua­<lb/>drare, cum non pauci hodie quoque &longs;int, qui more antiquorum Mathe­<lb/>maticis &longs;uffulti, optimè &longs;uis &longs;tudijs con&longs;ulentes, reliquam Philo&longs;ophiam <lb/>non &longs;ine magno compendio aggrediuntur. </s> <s id="id.000068">Quo fit, vt cæteros ageo­<lb/>metretos ita antecellant, vt eorum Magi&longs;tri appellari po&longs;&longs;int, & <expan abbr="debeãt">debeant</expan>; <lb/>tales fuerunt in primis Vicomercatus, Cardanus, Zabarella, Toletus, <lb/>Bonamicus, & alij, quibus paulo antiquiores fuerunt D. Tho. <!-- REMOVE S-->& Scotus, <lb/>Hiomnes Mathematicarum auxilio, quantum inter reliquos philo&longs;o­<lb/>phantes excelluerint, nemo e&longs;t qui non nouerit. </s> <s id="id.000069">Illud hoc loco minimè <lb/>tacendum, Iacobum Zabarellam in &longs;uis logicæ commentarijs te&longs;tari &longs;e <lb/>bis &longs;umma diligentia totum Euclidem perlegi&longs;&longs;e, vt perfectè Ari&longs;t. de <lb/>demon&longs;tratione &longs;ententiam a&longs;&longs;equi po&longs;&longs;et.</s></p><p type="main"> <s id="id.000070">Hi ridiculas illas, ac pueriles expo&longs;itiones &longs;uperius allatas minimè <lb/>effutierunt, neque reliquis &longs;upra recen&longs;itis incommodis obnoxij fue­<lb/>runt, quibus magno etiam cum dedecore alij Mathematicarum ope ca­<lb/>rentes afficiuntur.</s></p><p type="main"> <s id="id.000071">In horum igitur gratiam operam diligenter dedi, vt quantum in me <lb/>e&longs;&longs;et damna à me &longs;upra enarrata aliqua ex parte re&longs;arcirem. </s> <s id="id.000072">Quaprop­<lb/>ter loca hæc mathematica num rectè e&longs;&longs;ent è græco in latinum tran&longs;lata <lb/>diligenter prius expendi. </s> <s id="id.000073">Deinde claritate, quàm potui maxima eadem <lb/>loca interpretatus &longs;um, & in horum, de quibus dixi gratiam, quædam <lb/>fanè tenuia pro&longs;equutus &longs;um, quæ alioquin libenter omi&longs;i&longs;&longs;em. </s> <s id="id.000074">Tum fi­ <lb/>guras omnes, aut correxi, aut re&longs;titui, aut nouas appo&longs;ui. </s> <s id="id.000075">Hocigitur <lb/>no&longs;tro qualicunque labore poterit qui&longs;que omnia illa facile intelligere, <lb/><expan abbr="atq;">atque</expan> enumerata incommoda euitare, vnum tantummodo à Lectore ma­ <lb/>thematicarum experte requiram, vt principia &longs;altem illa, &longs;cilicet defini­ <lb/>tiones, po&longs;tulata, & axiomata, quæ primò Euclideis libro præponuntur, <lb/>diligenter prius perlegat cum illa &longs;ua per&longs;picuitate omnibus &longs;int obuia; <lb/>cætera ego explicanda recipio. </s> <s id="id.000076">Obiter etiam auctaria nonnulla partim <lb/>mathematica, partim naturalia in&longs;erui, quæ ob nouitatem, ac pulchri­ <lb/>tudinem grata Lectori, atque iucunda fore exi&longs;timaui.</s></p><p type="main"> <s id="id.000077">Sciat præterea Lector no&longs;trum in&longs;titutum e&longs;&longs;e loca hæc mathemati­ <lb/>ca, quatenus mathematica &longs;unt declarare, &longs;iue ea &longs;upplere, quæ ex ma­ <lb/>thematicis petenda e&longs;&longs;ent: reliqua autem me tantum attingere, quan­ <lb/>tum harum rerum cum illis connexio po&longs;tulat.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.010.jpg" pagenum="10"/><p type="main"> <s id="id.000078">His omnibus placuit appendices opportune nonnullas addere, qua­ <lb/>rum prima de natura mathematicarum &longs;cientiarum: altera, qua omnes <lb/>demon&longs;trationes primi libri Euclidis breuiter ad Logicam normam ex­ <lb/>penduntur, vt pateat, quonam demon&longs;trationis genere <expan abbr="c&etilde;&longs;eri">cen&longs;eri</expan> <expan abbr="vnaqu&ecedil;q;">vnaqu&ecedil;que</expan> <lb/>debeat, & ex illis de cæteris iudicium fiat. </s> <s id="id.000079">Tandem in gratiam etiam <lb/>Mathematicorum tertiam appendicem appendi, qua omnia loca Ari&longs;t. <lb/><!-- REMOVE S-->Geometrica ad Euclidis ordinem referuntur; vnde & ip&longs;i ad &longs;uas pr&ecedil;le­ <lb/>ctiones exornandas aliquid &longs;ubinde depromere queant.</s> </p><p type="main"> <s id="id.000080">Fruere igitur amice Lector hoc no&longs;tro qualiquali labore, quo ad ple­ <lb/>nam totius Ari&longs;t. intelligentiam, cui adhuc mathematicarum ignoratio <lb/>ob&longs;titit peruenire tandem po&longs;&longs;is: <expan abbr="illud&qacute;">illudque</expan>; experiaris, quod optimus qui­ <lb/>dam Philo&longs;ophus, cum totum hunc librum perlegi&longs;&longs;et, effatus e&longs;t, vide­ <lb/>licet, opus hoc <emph type="italics"/>Aristot elicæ expo&longs;itionis complementum ad hanc v&longs;que <lb/>diem de&longs;ideratum<emph.end type="italics"/> iure ac meritò nuncupari po&longs;&longs;e.</s></p><p type="main"> <s id="id.000081">Illud demum tanquam parergon addam, quod ego his elucubran­ <lb/>dis experientia didici, ad veram &longs;cilicet, ac perfectam to­ <lb/>tius Ari&longs;totelis intelligentiam linguæ in primis <lb/>græcæ, necnon mathematicarum om­ <lb/>nium di&longs;ciplinarum haud medio­ <lb/>crem cognitionem ne­ <lb/>ce&longs;&longs;ariam e&longs;&longs;e. <lb/></s> <s id="id.000082">Vale.<!-- KEEP S--></s></p></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.011.jpg" pagenum="11"/><section><p type="head"> <s id="id.000083">Pr&ecedil;cipua qu&ecedil;dam, aut noua, autre&longs;taurata, <lb/>quæ obiter pertractantur. <lb/><arrow.to.target n="table1"/></s></p><table><table.target id="table1"/><row><cell><emph type="italics"/>1<emph.end type="italics"/></cell><cell><emph type="italics"/>De re&longs;olutione. numero marginali.<emph.end type="italics"/></cell><cell><emph type="italics"/>4<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>2<emph.end type="italics"/></cell><cell><emph type="italics"/>De figuris vacuum replentibus, vbi Aristotelis, & expo-&longs;itorum erratum aperitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>121<emph.end type="italics"/></cell></row><row><cell/><cell><emph type="italics"/>Inibi, Apum mirabilis quædam in cellis &longs;uis hexagonis constituendis indu&longs;tria detegitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>120<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>3<emph.end type="italics"/></cell><cell><emph type="italics"/>De ijs, quæ aquæ in&longs;ident, vnà cum noua demonstratione problema-tis illius Archimedis, quo metallorum mixtionem indi&longs;&longs;oluta Co-rona, explorauit. in additione. ante num.<emph.end type="italics"/></cell><cell><emph type="italics"/>124<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>4<emph.end type="italics"/></cell><cell><emph type="italics"/>De Cometa, recentiorum &longs;ententia. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>136<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>5<emph.end type="italics"/></cell><cell><emph type="italics"/>De altitudine montis Cauca&longs;i.<emph.end type="italics"/></cell><cell><emph type="italics"/>148<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>6<emph.end type="italics"/></cell><cell><emph type="italics"/>De Terræ rotunditate, ac mundi duratione.<emph.end type="italics"/></cell><cell><emph type="italics"/>151<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>7<emph.end type="italics"/></cell><cell><emph type="italics"/>De Iride. in additione.<emph.end type="italics"/></cell><cell><emph type="italics"/>181<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>8<emph.end type="italics"/></cell><cell><emph type="italics"/>Scytala quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>250<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>9<emph.end type="italics"/></cell><cell><emph type="italics"/>Securis antiqua quæ, & qua ratione fieret.<emph.end type="italics"/></cell><cell><emph type="italics"/>258<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>10<emph.end type="italics"/></cell><cell><emph type="italics"/>Statera antiqua quæ,<emph.end type="italics"/></cell><cell><emph type="italics"/>259<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>11<emph.end type="italics"/></cell><cell><emph type="italics"/>De Ae&longs;tu Maris.<emph.end type="italics"/></cell><cell><emph type="italics"/>272<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>12<emph.end type="italics"/></cell><cell><emph type="italics"/>Araneorum indu&longs;tria nuper patefacta: vbi Democritus contra Ari&longs;t. defenditur.<emph.end type="italics"/></cell><cell><emph type="italics"/>293<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>13<emph.end type="italics"/></cell><cell><emph type="italics"/>De Lucis figuratione, & rerum &longs;imulacris in ob&longs;curo loco.<emph.end type="italics"/></cell><cell><emph type="italics"/>345<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>14<emph.end type="italics"/></cell><cell><emph type="italics"/>De Pupilla oculi.<emph.end type="italics"/></cell><cell><emph type="italics"/>408<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>15<emph.end type="italics"/></cell><cell><emph type="italics"/>De Mathematicarum natura. propè finem operis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>16<emph.end type="italics"/></cell><cell><emph type="italics"/>Clarorum Mathematicorum Chronologia, in fine operis.<emph.end type="italics"/></cell><cell/></row></table></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.012.jpg" pagenum="12"/><section><p type="head"> <s id="id.000084"><emph type="italics"/>PRIMVS INDEX LOCORVM ARIST.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000085"><emph type="italics"/>Quæ in hoc opere explicantur, iuxta ordine librorum <lb/>ip&longs;ius ex vulgata editione Lugdunen&longs;i.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000086">In Prædicamentis.</s></p><p type="main"> <s id="id.000087"><emph type="italics"/>Gapite s. </s> <s id="id.000088">de Relatione, vbi de Quadratura circuli.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000089"><emph type="italics"/>Cap. de Priori, vbi de Principijs Mathematicarum,<emph.end type="italics"/></s></p><p type="main"> <s id="id.000090"><emph type="italics"/>Cap. de Motu, vbi de Gnomone.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000091">In Primo Priorum Re&longs;olutoriorum.</s></p><p type="main"> <s id="id.000092"><emph type="italics"/>Ad titulum libri de Re&longs;olutione.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000093"><emph type="italics"/>Cap. 23. &longs;ect 1. libri 1. de Incommen &longs;ur abilibus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000094"><emph type="italics"/>Cap. 24. &longs;ecti 1. lib. 1. de De&longs;criptionibus.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000095"><emph type="italics"/>Cap. 2. &longs;ect 2. lib. 1. de De&longs;criptionibus.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000096"><emph type="italics"/>Cap. 3. &longs;ecti 2. lib. 1. de Incommen&longs;urabili.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000097"><emph type="italics"/>Cap. 1. &longs;ecti 3. lib. 1. de eo, quod est, omnis triangulus habet tres angulos æquales <lb/>æquales duobus rectis: Aequalitas Geometrica, quæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000098"><emph type="italics"/>Cap. eodem, de exemplis, quibus vtuntar Geometræ.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000099">In &longs;ecundo Priorum Re&longs;ol.<!-- REMOVE S--><emph type="italics"/>Cap. 21. de lineis Paralellis, &longs;eu Coalternis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000100"><emph type="italics"/>Cap. eodem. </s> <s id="id.000101">de Paralellis, & de triangulo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000102"><emph type="italics"/>Cap. 26. Quod omnis triangulus habet tres, & c.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000103"><emph type="italics"/>Cap. 31. de Abductione.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000104"><emph type="italics"/>Cap. codem, de circuli Quadratura, &longs;ecundum Hippocratem Chium.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000105">In primo Po&longs;teriorum.</s></p><p type="main"> <s id="id.000106"><emph type="italics"/>Textu primo, De Præcognitis Mathematicarum.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000107"><emph type="italics"/>T. 2. Omnis triangulus habet tres, & c.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000108"><emph type="italics"/>T. 5. De Diametro incommen&longs;urabili. </s> <s id="id.000109">Item De Mathematicarum Principijs.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000110"><emph type="italics"/>T. eodem, De Indiui&longs;ibilitate vnitatis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000111"><emph type="italics"/>T. 9. De Puncto, & linea. </s> <s id="id.000112">Item de recto, & circulari. </s> <s id="id.000113">Item de numero pari, impari; <lb/>primo, & compo&longs;ito; æquilatero, & altera parte longiore.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000114"><emph type="italics"/>T. 11. Lineæ punctum inest per &longs;e, & c.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000115"><emph type="italics"/>T. 13. De Parallelis. <!-- KEEP S--></s> <s id="id.000116">De I&longs;o&longs;cele. </s> <s id="id.000117">De Alterna Proportione, <lb/>Item quod omnis triangulus habet tres, & c.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000118"><emph type="italics"/>T. 14. De ij&longs;dem cum præcedentibus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000119"><emph type="italics"/>T. 20. Magnitudines euadunt numeri. </s> <s id="id.000120">Item, quod non duo cubi cubus. </s> <s id="id.000121">Item de <lb/>Mathematicis &longs;ubalternatis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000122"><emph type="italics"/>T. 23. Quadratura circuli &longs;ecundum Bry&longs;onem. </s> <s id="id.000123">Item per&longs;ectam illam e&longs;&longs;e Demon­ <lb/>&longs;trationem, qua Geometræ o&longs;tendunt, quod Omnis triangulus habet tres, & c. <lb/><!-- KEEP S--></s> <s id="id.000124">Per&longs;pectinam, & Mechanicam &longs;ubalternari Geometriæ, Mu&longs;icam, Arithmeticæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000125"><emph type="italics"/>T. 24. De numero pari, impari, quodrangulo, cubo. </s> <s id="id.000126">In Geometria quid irrationale, <lb/>refrangi, concurrere. </s> <s id="id.000127">Quid Astronomia con&longs;ideret.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000128"><emph type="italics"/>T. 25. Geometram non mentiri in &longs;uis exemplis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000129"><emph type="italics"/>T. 28. De Parallelis.<emph.end type="italics"/><!-- KEEP S--></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.013.jpg" pagenum="13"/><p type="main"> <s id="id.000130"><emph type="italics"/>T. 29. Cur in Mathematicis non &longs;it Paralogi&longs;mus. <!-- KEEP S--></s> <s id="id.000131">Item quid multiplicata propor­ <lb/>tio. </s> <s id="id.000132">Quid Cæneus dixerit. </s> <s id="id.000133">Cur Affectiones <expan abbr="Mathematicorũ">Mathematicorũ</expan> maximè <expan abbr="conuertãtur">conuertantur</expan>.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000134"><emph type="italics"/>T. 30. De Lunæ &longs;phæricitate. </s> <s id="id.000135">Quid &longs;tereometria. </s> <s id="id.000136">& De &longs;ubalternatione, &c. </s> <s id="id.000137">& Ma­ <lb/>thematicorum e&longs;t &longs;cire Propter quid: &longs;en&longs;itiuorum verò &longs;cire Quod.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000138"><emph type="italics"/>T. 37. I&longs;o&longs;celes, & Scalenum habere tres æquales, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000139"><emph type="italics"/>T. 38. Quid Mina, quid Die&longs;is.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000140"><emph type="italics"/>T. 39. Habere tres angulos æquales, &c. </s> <s id="id.000141">Item, quod omnis figura habet &longs;uos angu­ <lb/>los externos æquales quatuor tantum rectis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000142"><emph type="italics"/>T. 43. Triangulum tres æquales, &c. </s> <s id="id.000143">De Eclyp&longs;i.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000144"><emph type="italics"/>De combu&longs;tione per refractionem ex &longs;phæra vitrea. </s> <s id="id.000145">De principijs &longs;cientiarum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000146"><emph type="italics"/>T. 44. Diameter incommen&longs;urabilis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000147">In 2. Po&longs;teriorum.</s></p><p type="main"> <s id="id.000148"><emph type="italics"/>T. 1. Aequalitas, & inæqualitas. </s> <s id="id.000149">Terram e&longs;&longs;e in medio mundi ab A&longs;tronomis per­ <lb/>fectè demon&longs;tratur. </s> <s id="id.000150">Item Quid con&longs;onantia.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000151"><emph type="italics"/>T. 2. Omnis triangulus habet tres, &c. </s> <s id="id.000152">Item de Definitionibus Mathematicarum.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000153"><emph type="italics"/>T. 7. Geometra, quædam accipit, quædam demon&longs;trat.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000154"><emph type="italics"/>T. 11. Angulum in &longs;emicirculo rectum e&longs;&longs;e probari à Geometra per cau&longs;am materia­ <lb/>lem. </s> <s id="id.000155">Zabarella correctus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000156"><emph type="italics"/>T. 24. Echo, Imago è &longs;peculo, Iris.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000157"><emph type="italics"/>T. 25. Permutatim proportionale quid; exemplum in triangulis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000158">In primo lib. <!-- REMOVE S-->Topicorum.</s> </p><p type="main"> <s id="id.000159"><emph type="italics"/>Cap. 13. Diameter est incommen&longs;urabilis. </s> <s id="id.000160">Vox acuta velox, cur. </s> <s id="id.000161">&c. </s> <s id="id.000162">Colores in <lb/>Mu&longs;ica, qui. </s> <s id="id.000163">tria genera veteris Mu&longs;icæ.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000164">In 4. libro.</s></p><p type="main"> <s id="id.000165"><emph type="italics"/>Cap. 1. loco 1. lineæ in&longs;ecabiles.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000166">In 6. libro.</s></p><p type="main"> <s id="id.000167"><emph type="italics"/>Cap. 2. loco 32. Definitio lineæ.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000168">In 8. libro.</s></p><p type="main"> <s id="id.000169"><emph type="italics"/>Cap. 2. loco 41. V&longs;us Definitionum in Mathematicis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000170"><emph type="italics"/>Cap. 4. loco 86. Elementa geometrica: Numeri capitales.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000171">In Elenchorum lib. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.000172"><emph type="italics"/>Cap. 10. Quid P&longs;eudographia. </s> <s id="id.000173">Quadraturarur&longs;us Hippocratis, & Bry&longs;enis. </s> <s id="id.000174">Mathe­ <lb/>maticæ non contentio&longs;æ. </s> <s id="id.000175">Quadratio Antiphontis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000176">Ex 1. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000177"><emph type="italics"/>T. 11. De Quadraturis circuli Hippocratis, & Antiphontis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000178">Ex 2. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000179"><emph type="italics"/>T. 20. Quatenus Per&longs;pectmus con&longs;ideret lineam.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000180"><emph type="italics"/>T. 28. Quid con&longs;onantia Diapa&longs;on.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000181"><emph type="italics"/>T. 68. Mathematicas Demon&longs;trationes babere cau&longs;am, quæ reducitur ad defini­ <lb/>tionem.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000182"><emph type="italics"/>T. 8. De nece&longs;&longs;ario, quod e&longs;t in Mathematicis. <!-- KEEP S--></s> <s id="id.000183">& omnis triangulus habet tres an­ <lb/>gulos, &c.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000184">Ex 3. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000185"><emph type="italics"/>T. 76. Quinam numeri dicantur Gnomones.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000186"><emph type="italics"/>T. 31. Quonam infinito vtantur Mathematici.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000187"><emph type="italics"/>T. 71. De infinito Mathematica.<emph.end type="italics"/><!-- KEEP S--></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.014.jpg" pagenum="14"/><p type="head"> <s id="id.000188">Ex 4. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000189"><emph type="italics"/>T. 120. De commen&longs;urab. </s> <s id="id.000190">& incommen&longs;.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000191">Ex 5. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000192"><emph type="italics"/>T. 6. De chordis, graui, acuta; media, & vltima.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000193">Ex 8. Phy&longs;ic.<!-- KEEP S--></s></p><p type="main"> <s id="id.000194"><emph type="italics"/>T. 15. Omnis triangulus habet tres æquales, &c.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000195">Ex 1. de Cœlo.<!-- KEEP S--></s></p><p type="main"> <s id="id.000196"><emph type="italics"/>T. 33. De minimo indiui&longs;ibili.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000197"><emph type="italics"/>T. 36. Ratione vtitur lineari; vt probet mundum e&longs;&longs;e finitum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000198"><emph type="italics"/>T. 48. Commen&longs;urab. <!-- REMOVE S-->& incommen&longs;urab. </s> <s id="id.000199">quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000200"><emph type="italics"/>T. 119. Omnis triangulus habet tres, &c. </s> <s id="id.000201">Item de commen&longs;urabili.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000202">Ex 2. de Cœlo.<!-- KEEP S--></s></p><p type="main"> <s id="id.000203"><emph type="italics"/>T. 24. Plato ex planis &longs;olida componebat, quì.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000204"><emph type="italics"/>T. 25. Ordo figurarum planarum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000205"><emph type="italics"/>T. 31. Aquæ &longs;uperficiem e&longs;&longs;e &longs;phæricam.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000206"><emph type="italics"/>T. 46. Maiorem circulum velocius moueri. </s> <s id="id.000207">Recentiorum ob&longs;eruationes.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000208"><emph type="italics"/>T. 57. De ordine Cœlorum ex &longs;ententia A&longs;tronomorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000209"><emph type="italics"/>T. 59. De rotunditate Lunæ, bis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000210"><emph type="italics"/>T. 107. Centrum duplex grauit: & molis. </s> <s id="id.000211">Qua ratione grauia ad mundi centrum <lb/>aptarentur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000212"><emph type="italics"/>T. 109. Terram e&longs;&longs;e rotundam. </s> <s id="id.000213">alio item modo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000214"><emph type="italics"/>T. 110. Terram e&longs;&longs;e paruam re&longs;pectu Cœli.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000215"><emph type="italics"/>T. 111. Mare occidentale coniungi indico.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000216"><emph type="italics"/>T. 112. De quantitate Terræ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000217">Ex 3. de Cœlo.<!-- KEEP S--></s></p><p type="main"> <s id="id.000218"><emph type="italics"/>T. 40. Vt componatur &longs;phæra.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000219"><emph type="italics"/>T. 66. Omne corpus diui&longs;ibile.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000220"><emph type="italics"/>T. 66. Quænam planarum figurarum totum &longs;patium repleant. </s> <s id="id.000221">Hinc de admirabili <lb/>Apum mgenio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000222"><emph type="italics"/>T. eodem. </s> <s id="id.000223">Num plures Pyramides locum replere valeant, vbi Ari&longs;totiles, & omnes <lb/>expo&longs;itores erra&longs;&longs;e o&longs;tenduntur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000224"><emph type="italics"/>T. 71. Terram e&longs;&longs;e cubum, cur dictum &longs;it.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000225">Ex 4. de Cœlo.<!-- KEEP S--></s></p><p type="main"> <s id="id.000226"><emph type="italics"/>T. 33. Extare centrum Mundi, quò grauia tendunt, à quo leuia abeant.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000227"><emph type="italics"/>T. 44. & <expan abbr="&longs;eq.">&longs;eque</expan> Cur quædam grauiora quàm aqua, &longs;upernatent.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000228">Ex 2. de Generatione, & Corruptione.<!-- KEEP S--></s></p><p type="main"> <s id="id.000229"><emph type="italics"/>Tex. 56. Cur Planetæ duobus motibus moueri dicantur.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000230">Ex 1. Meteororum.<!-- KEEP S--></s></p><p type="main"> <s id="id.000231"><emph type="italics"/>Summa prima cap. 3. De magnitudine Terræ ad a&longs;tra, & &longs;olem collata.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000232"><emph type="italics"/>Cap. eodem. </s> <s id="id.000233">De magnitudine A&longs;trorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000234"><emph type="italics"/>Cap. 4. De ordine Luminarium Solis, & Lunæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000235"><emph type="italics"/>Summa 2. cap. 3. de Mercurij stella. </s> <s id="id.000236">Item de Cometa: e&longs;&longs;e in Cœlo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000237"><emph type="italics"/>Cap. 5. De Magnitudine Solis, & de vmbra Terræ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000238"><emph type="italics"/>Cap. 5. De Glaxia.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000239"><emph type="italics"/>Cap. 6. Sententia Ari&longs;totelis de Glaxia, partim defenditur: vera, deinde <lb/>aperitur.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.015.jpg" pagenum="15"/><p type="main"> <s id="id.000240"><emph type="italics"/>Summa 4. cap 1. De Monte Parna&longs;&longs;o, dubia. </s> <s id="id.000241">Mare extraneum, quod. </s> <s id="id.000242">Errata quæ­ <lb/>dam veterum Geographorum, & Ari&longs;t. corriguntur. </s> <s id="id.000243">Altitudo montis Cauca&longs;i.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000244"><emph type="italics"/>Cap. 2. De permutatione Aquarum, & continentis. </s> <s id="id.000245">Noua ob&longs;eruatio de rotundi­ <lb/>tate Terræ, <expan abbr="atq;">atque</expan> Mundi duratione.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000246">Ex 2. Meteororum.<!-- KEEP S--></s></p><p type="main"> <s id="id.000247"><emph type="italics"/>Summa 1. cap. 1. De Mari rubro.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000248"><emph type="italics"/>Summa 2. cap. 2. De ortu stellarum fixarum: Item de occa&longs;u earumdem.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000249"><emph type="italics"/>Cap. eodem, De Canicula. </s> <s id="id.000250">De Zonis temperatis. </s> <s id="id.000251">Corona Ariadnæ. </s> <s id="id.000252">Zonam torridam <lb/>falsò putabant inho&longs;pitalem. </s> <s id="id.000253">cur habitabilis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000254"><emph type="italics"/>Cap. 3. De Ventorum &longs;itu.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000255">Ex 3. Meteor.<!-- KEEP S--></s></p><p type="main"> <s id="id.000256"><emph type="italics"/>Summa 2. cap. 2. De Halone, &longs;eu Area, &longs;eu Corona, Mathematica demon&longs;tratio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000257"><emph type="italics"/>Cap. 4. De Iridis figura Mathematica demon&longs;tratio, &longs;ed deficiens. </s> <s id="id.000258">Noua de eadem <lb/>tractatio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000259"><emph type="italics"/>Cap. 5. De Parelio. <!-- KEEP S--></s> <s id="id.000260">Rationes Ari&longs;totelis refelluntur.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000261">Ex 1. De Anima.<!-- KEEP S--></s></p><p type="main"> <s id="id.000262"><emph type="italics"/>Tex. 11. Quid rectum, quid obliquum. </s> <s id="id.000263">& omnis triangulus babet tres, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000264"><emph type="italics"/>T. 13. Sphæra planum tangit in puncto.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000265">Ex 2. De Anima.<!-- KEEP S--></s></p><p type="main"> <s id="id.000266"><emph type="italics"/>T. 12. Definitioncm formalem, & cau&longs;alem explicat exemplo Quadrationis Geo­ <lb/>metricæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000267"><emph type="italics"/>T. 86. Acutum, & Graue, vt differant.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000268"><emph type="italics"/>T. 159. De Solis magnitudine ad terram.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000269">Ex 3. De Anima.<!-- KEEP S--></s></p><p type="main"> <s id="id.000270"><emph type="italics"/>T. 21. Incommen&longs;urabile.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000271"><emph type="italics"/>T. 25. Indiui&longs;ibilia e&longs;&longs;e priuationes.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000272"><emph type="italics"/>T. 32. Permutata proportio.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000273">Ex lib. <!-- REMOVE S-->De Sen&longs;u.</s> </p><p type="main"> <s id="id.000274"><emph type="italics"/>Capite 6. Die&longs;is.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000275"><emph type="italics"/>Cap. 8. Nete. </s> <s id="id.000276">Diapa&longs;on. <!-- KEEP S--></s> <s id="id.000277">Diapen&longs;e.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000278">Ex lib. <!-- REMOVE S-->De Memoria, & Rem.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000279"><emph type="italics"/>Cap. 1. Omnis triangulus habet tres, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000280"><emph type="italics"/>Cap. 3. Mathemata facile remini&longs;cibilia.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000281">Ex lib. <!-- REMOVE S-->De Somnijs.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000282"><emph type="italics"/>Cap. 2. Terra, cur nauigantibus moueri videatur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000283"><emph type="italics"/>Cap. 3. Cur Oculus digito dimotus res geminatas videat.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000284">Ex 1. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000285"><emph type="italics"/>Cap. 1. Initium Mathematicarum ab Aegyptiorum Saterdotibus. </s> <s id="id.000286">Item, Automata, <lb/>quæ &longs;olstitia. </s> <s id="id.000287">Diameter incommen&longs;.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000288"><emph type="italics"/>Summa 2. cap. 3. Pythagorei Mathematicas cæteris præferebant.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000289"><emph type="italics"/>T. 47. Geometria habet &longs;uas præcognitiones.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000290">Ex 2. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000291"><emph type="italics"/>T. 14. Leges apud Mu&longs;icos quid.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000292">Ex 3. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000293"><emph type="italics"/>Tex. <!-- KEEP S--></s> <s id="id.000294">Mathematicas puras carere cau&longs;is efficiente, & finali. </s> <s id="id.000295">Ari&longs;tippus, vt Mathe­ <lb/>maticas &longs;ugillaret. </s> <s id="id.000296">Tetragoni&longs;mus est inuentio mediæ.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.016.jpg" pagenum="16"/><p type="main"> <s id="id.000297"><emph type="italics"/>Tex. 8. Geodæ&longs;ia quid.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000298">Ex 4. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000299"><emph type="italics"/>T. 4. Quæ &longs;int primæ, & quæ &longs;ecundæ inter Mathematicas.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000300"><emph type="italics"/>T. 28. Diameter, commen&longs;urabilis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000301">Ex 5. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000302"><emph type="italics"/>T. 2. Exemplum cau&longs;æ formalis ex Diapa&longs;on. <!-- KEEP S--></s> <s id="id.000303">Quæ &longs;int proportiones Mu&longs;icales.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000304"><emph type="italics"/>T. 3. Quæ &longs;it Materia in Mathematicis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000305"><emph type="italics"/>T. 4. Quidnam &longs;int elementa apud Geometras.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000306"><emph type="italics"/>T. 12. Die&longs;is.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000307"><emph type="italics"/>T. 17. Diameter incommen&longs;urab. </s> <s id="id.000308">Quid potentia vnius lineæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000309"><emph type="italics"/>T. 34. Diameter incommen&longs;urabilis.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000310">Ex 6. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000311"><emph type="italics"/>T. 1. Principia, elementa, & cau&longs;æ in Mathem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000312"><emph type="italics"/>T. 8. Diameter. </s> <s id="id.000313">commen&longs;urab.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000314"><emph type="italics"/>T. 20. De&longs;criptiones. </s> <s id="id.000315">Omnis triangulus habet tres, &c. </s> <s id="id.000316">Cur Angulus in &longs;emicir­ <lb/>culo rectus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000317"><emph type="italics"/>T. 22. Omnis triangulus habet tres, &c.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000318">Ex 10. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000319"><emph type="italics"/>T. 4. Motum diurnum men&longs;uram reliquorum. </s> <s id="id.000320">Die&longs;is.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000321"><emph type="italics"/>T. 11. Similes figuræ quæ. </s> <s id="id.000322">Diuer&longs;um in Math. quid.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000323">Ex 11. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000324"><emph type="italics"/>Cap. 2. Ortus punctorum, linearum, &longs;uperficierum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000325">Ex 12. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000326"><emph type="italics"/>T. 44. Peculiari&longs;&longs;imam Philo&longs;ophiam, Mathematicorum videlicet, A&longs;tronomiam <lb/>pluralitatem Cœlorum docere.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000327"><emph type="italics"/>T. 45. Numerus orbium cœle&longs;t ium &longs;ecundum Eudoxum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000328"><emph type="italics"/>T. 46. Itidem ex Eudoxo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000329"><emph type="italics"/>T. 47. Orbium cœle&longs;tium numerus, & fabrica ex Calippo.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000330">Ex 13. Methaphy&longs;.<!-- KEEP S--></s></p><p type="main"> <s id="id.000331"><emph type="italics"/>Cap. 3. Qua ratione Mathematici tractant de Bono.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000332">In Mechanicas Quæ&longs;tiones.</s></p><p type="main"> <s id="id.000333"><emph type="italics"/>Cap. 1. Quæ &longs;it Mechanica facultas.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000334"><emph type="italics"/>Cap. 2. De Admirandis circuli.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000335"><emph type="italics"/>Quæ&longs;tio 1. De Libra. <!-- KEEP S--></s> <s id="id.000336">cur maior, exactior. </s> <s id="id.000337">inibi Ari&longs;t. lap&longs;us.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000338"><emph type="italics"/>Quæ&longs;t. 2. Duplex Libra. <!-- KEEP S--></s> <s id="id.000339">Piccolomineus reiectus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000340"><emph type="italics"/>Quæ&longs;t. 3. De Vecte.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000341"><emph type="italics"/>Quæ&longs;t. 4. De Remo; Petri Nonÿ in Arist. correctio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000342"><emph type="italics"/>Quæ&longs;t. 5. De Temone Nauis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000343"><emph type="italics"/>Quæ&longs;t. 6. De Antenna.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000344"><emph type="italics"/>Quæ&longs;t. 8 De Rota.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000345"><emph type="italics"/>Quæ&longs;t. 9. De Trochlea, & Scytali. </s> <s id="id.000346">figura antiquæ &longs;cytalis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000347"><emph type="italics"/>Quæ&longs;t. 10. De Libra vacua.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000348"><emph type="italics"/>Qùæ&longs;t. <!-- REMOVE S-->11. De Curru, & &longs;cytala.<emph.end type="italics"/></s> </p><p type="main"> <s id="id.000349"><emph type="italics"/>Quæ&longs;t. 13. De lugo. </s> <s id="id.000350">De Succula.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000351"><emph type="italics"/>Quæ&longs;t. 15. De Vmbelicis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000352"><emph type="italics"/>Quæ&longs;t. 16. De ligni oblongi, ac breuis flexura.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.017.jpg" pagenum="17"/><p type="main"> <s id="id.000353"><emph type="italics"/>Quæ&longs;t. 17. De Cuneo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000354"><emph type="italics"/>Quæ&longs;t. 18. De Trochlea; error Piccolominei.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000355"><emph type="italics"/>Quæ&longs;t. 19. De Securi. </s> <s id="id.000356">Securis veteris figura, & con&longs;tructio; vnà cum affectione <lb/>eius mirabili.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000357"><emph type="italics"/>Quæ&longs;t. 20. De Statera. <!-- KEEP S--></s> <s id="id.000358">Veteris stateræ figura restaurata.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000359"><emph type="italics"/>Quæ&longs;t. 21. De Dentiforcipe.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000360"><emph type="italics"/>Quæ&longs;t. 22. De Nucifrago.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000361"><emph type="italics"/>Quæ&longs;t. 23. De Motibus in Rhombo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000362"><emph type="italics"/>Quæ&longs;t. 24. De duobus circulis concentricis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000363"><emph type="italics"/>Quæ&longs;t. 25. De funibus lectulorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000364"><emph type="italics"/>Quæ&longs;t. 26. De ligno humeris gestato.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000365"><emph type="italics"/>Quæ&longs;t. 27. De ponderibus humero ge&longs;tatis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000366"><emph type="italics"/>Quæ&longs;t. 28. De Tollenone.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000367"><emph type="italics"/>Quæ&longs;t. 29. De onere à duobus phalanga ge&longs;tato.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000368"><emph type="italics"/>Quæ&longs;t. 30. De &longs;urgente à &longs;e&longs;&longs;ione.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000369">In libello De Mundo ad Alex.<!-- REMOVE S--><emph type="italics"/>Cap. 2. Ordo Planetarum.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000370"><emph type="italics"/>Cap. 3. De Cometis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000371"><emph type="italics"/>Cap. 5. De fluxu maris. </s> <s id="id.000372">noua de maris æ&longs;tu &longs;ententia.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000373">In libro De Admirandis audit.</s></p><p type="main"> <s id="id.000374"><emph type="italics"/>Num. <!-- REMOVE S-->8. De nouo orbe.<emph.end type="italics"/></s> </p><p type="main"> <s id="id.000375"><emph type="italics"/>Nu. </s> <s id="id.000376">100. De I&longs;tro, error Ari&longs;t. & veterum Geographorum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000377">In libello De lineis in&longs;ecabilibus.</s></p><p type="main"> <s id="id.000378"><emph type="italics"/>Primus locus. </s> <s id="id.000379">De commen&longs;urabili, & incommen&longs;urabili.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000380"><emph type="italics"/>2. locus. </s> <s id="id.000381">De figuris incommen&longs;.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000382"><emph type="italics"/>3. locus. </s> <s id="id.000383">Quæ linea rationalis, quæ irrationalis. </s> <s id="id.000384">Binomio, Apotome.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000385"><emph type="italics"/>4. locus. </s> <s id="id.000386">De communi men&longs;ura.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000387"><emph type="italics"/>5. locus. </s> <s id="id.000388">Lineæ rectæ motus in &longs;emicirculum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000389"><emph type="italics"/>6. locus. </s> <s id="id.000390">Circulorum æqualium ab inuicem motus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000391"><emph type="italics"/>7. locus. </s> <s id="id.000392">Multum Mathematicis demon&longs;trationibus tribuitur ab Ari&longs;totele.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000393"><emph type="italics"/>8. locus. </s> <s id="id.000394">Si extarent indiuidua, omnes lineæ e&longs;&longs;ent commen&longs;.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000395"><emph type="italics"/>9. locus. </s> <s id="id.000396">Idem probat aliteŕ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000397"><emph type="italics"/>10. locus. </s> <s id="id.000398">Idem ex triangulo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000399"><emph type="italics"/>11. locus. </s> <s id="id.000400">Idem ex quadrato.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000401"><emph type="italics"/>12. Ex diui&longs;ione lineæ idem confirmatur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000402"><emph type="italics"/>13. Idem eodem ferè modo cum præcedenti.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000403"><emph type="italics"/>14. A quadrato cuiu&longs;uis lineæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000404"><emph type="italics"/>15. Idem probat ex &longs;uperficie, & ex corpore.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000405"><emph type="italics"/>16. Idem ex contactu circuli cum linea recta.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000406">Ex lib. 9. Hi&longs;toriæ Animalium.</s></p><p type="main"> <s id="id.000407"><emph type="italics"/>Cap. 39. error Ari&longs;t. & noua ob&longs;eruatio de admiranda quadam Aranearum indu&longs;tria.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000408">De Ince&longs;&longs;u Animal.<!-- KEEP S--></s></p><p type="main"> <s id="id.000409"><emph type="italics"/>Cap. 7. qua ratione in gre&longs;&longs;u fiat bypotenu&longs;a. </s> <s id="id.000410">& ea quid &longs;it.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000411">De Motu Animal.<!-- KEEP S--></s></p><p type="main"> <s id="id.000412"><emph type="italics"/>Cap. 1. in flexuris animalium e&longs;&longs;e centrum, & circulum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000413"><emph type="italics"/>Cap. 3. Automata.<emph.end type="italics"/><!-- KEEP S--></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.018.jpg" pagenum="18"/><p type="head"> <s id="id.000414">De Generatione Animal.<!-- KEEP S--></s></p><p type="main"> <s id="id.000415"><emph type="italics"/>Lib. 2. cap. 1. Automata.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000416"><emph type="italics"/>Lib. 2. cap. 4. Omnis triangulus habet tres, &c. </s> <s id="id.000417">Ibidem Diametrum e&longs;&longs;e incommen­ <lb/>&longs;urabilem co&longs;tæ, habet cau&longs;am, & demon&longs;trationem.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000418">In Ethicis ad Nicom.</s></p><p type="main"> <s id="id.000419"><emph type="italics"/>Lib. 1. cap. 7. Faber, & Geometra diuersè con&longs;iderant angulum rectum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000420"><emph type="italics"/>Lib. 2. cap. 6. De Arithmetica proportione.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000421"><emph type="italics"/>cap. 9. Centrum circuli reperire.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000422"><emph type="italics"/>Lib. 3. cap. 3. Diameter, & latus incommen&longs;urabilis: Item quid re&longs;olutio Geome­ <lb/>trica: Quid de&longs;ignatio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000423"><emph type="italics"/>Lib. 5. cap. 3. Vnitarius numerus. </s> <s id="id.000424">Quid Proportionalitas. <!-- KEEP S--></s> <s id="id.000425">Eam in 4. terminis con­ <lb/>&longs;i&longs;tere. </s> <s id="id.000426">Item quid Permutata proportio. </s> <s id="id.000427">Item quid Geometrica proportio. </s> <s id="id.000428">Propor­ <lb/>tio continuata, & di&longs;iuncta quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000429"><emph type="italics"/>cap. 6. Proportio Geometrica, & Arithmetica.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000430"><emph type="italics"/>Lib. 6. cap. 5. Omnis triangulus, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000431"><emph type="italics"/>cap. 8. Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000432"><emph type="italics"/>Lib. 7. cap. 8. De principijs Mathem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s id="id.000433">Ex 1. Magnorum Moralium.</s></p><p type="main"> <s id="id.000434"><emph type="italics"/>Cap. 1. Numerus pariter par.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000435"><emph type="italics"/>Cap. 2. Omnis triangulus habet, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000436"><emph type="italics"/>Cap. 10 Omnis triangulus habet, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000437"><emph type="italics"/>Cap. 16. Quadratum quatuor rectis æquales habere.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000438"><emph type="italics"/>Cap. 30. Proportionale in quatuor terminis con&longs;i&longs;tit.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000439">Ex 1. lib. <!-- REMOVE S-->Moralium Eudemiorum.</s> </p><p type="main"> <s id="id.000440"><emph type="italics"/>Cap. 5. Duplum inter multiplices rationes primum tenet locum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000441">Ex 1. lib. <!-- REMOVE S-->Mor. <!-- REMOVE S-->Eudemiorum.</s> </p><p type="main"> <s id="id.000442"><emph type="italics"/>Cap. Omnis triangulus habet tres, &c.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000443"><emph type="italics"/>Cap. 10. Diametrum commen&longs;. </s> <s id="id.000444">e&longs;&longs;e. </s> <s id="id.000445">Circuli quadratio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000446"><emph type="italics"/>Cap. 12. Triangulus habet tres, &c.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000447">Ex 7. lib. <!-- REMOVE S-->Mor. <!-- REMOVE S-->Eudemiorum.</s> </p><p type="main"> <s id="id.000448"><emph type="italics"/>Cap. 12. Diametralis oppo&longs;itio.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000449">Ex 3. lib. <!-- REMOVE S-->Politicorum.</s> </p><p type="main"> <s id="id.000450"><emph type="italics"/>Cap. 2. Modi Dorius, & Phrygius apud Mu&longs;icos, quì.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000451">Ex 4. lib. <!-- REMOVE S-->Polit.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000452"><emph type="italics"/>Cap. 3. Modus Doricus, & Phrygius.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000453">Ex 5. lib. <!-- REMOVE S-->Polit.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000454"><emph type="italics"/>Cap. 1. Aequitas Arithmetica, & quæ &longs;ecundum dignitatem.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000455">Ex 8. Polit.<!-- KEEP S--></s></p><p type="main"> <s id="id.000456"><emph type="italics"/>Cap. 5. Mu&longs;ica nuda, & cum melodia. </s> <s id="id.000457">Rithmus quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000458"><emph type="italics"/>Harmonia lydia. </s> <s id="id.000459">Rithmus quid &longs;it dicetur in Problematibus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000460"><emph type="italics"/>Cap. 7. Harmoniæ, & Rithmi, vt in præcedenti.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000461">Ex Problematibus.</s></p><p type="main"> <s id="id.000462"><emph type="italics"/>Sectione 1. num. </s> <s id="id.000463">3. De ortu &longs;yderum innerrantium: Succulæ, Hypades, Atlantides, <lb/>Virgiliæ, Pleiades. </s> <s id="id.000464">num. </s> <s id="id.000465">17. De occa&longs;u affixarum &longs;tellarum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000466"><emph type="italics"/>Sectione 15. num. </s> <s id="id.000467">1. Diametri ethymon.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000468"><emph type="italics"/>num. </s> <s id="id.000469">2. Iterum Diametri ethymologia.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.019.jpg" pagenum="19"/><p type="main"> <s id="id.000470"><emph type="italics"/>num. </s> <s id="id.000471">3. Denarius numerus cur perfectus. </s> <s id="id.000472">eius dignitates. </s> <s id="id.000473">Petri Apponen&longs;is deceptio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000474"><emph type="italics"/>4. De inæquali &longs;olis vmbrarum incremento.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000475"><emph type="italics"/>5. Cur Solis illuminationes &longs;emper rotundæ, quamuis per angulo&longs;a foramina ingre­ <lb/>diantur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000476"><emph type="italics"/>6. Cur Luna &longs;emiplena videtur linea recta terminari? </s> <s id="id.000477">vbi de illuminatione Lunæ, <lb/>quæ experientia docetur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000478"><emph type="italics"/>7. Cur Sol, & Luna videantur plana?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000479"><emph type="italics"/>8. De vmbris Solis orientis, occidentis, meridiantis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000480"><emph type="italics"/>9. Cur Lunæ, quàm Solis minores vmbræ?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000481"><emph type="italics"/>10. Cur in defectu Solis etiam illuminationes ip&longs;ius defectiuæ &longs;unt? </s> <s id="id.000482">modus commodè <lb/>videndi eclyp&longs;im Solis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000483"><emph type="italics"/>Sect. <!-- REMOVE S-->16. nu. </s> <s id="id.000484">1. Cur ba&longs;es bullarum in aquis &longs;unt albæ?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000485"><emph type="italics"/>3. Opplumbati tali.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000486"><emph type="italics"/>4. De re&longs;ultu cadentium in terram.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000487"><emph type="italics"/>5. Cur conus, & cylindrus diuersè moueantur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000488"><emph type="italics"/>6. De voluminum &longs;ectione.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000489"><emph type="italics"/>12. Idem cum præcedenti 3.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000490"><emph type="italics"/>13. Idem cum 4 &longs;uperiori. </s> <s id="id.000491">reflexio radiorŭ pulchrè comparatur corporŭ re&longs;ultationi.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000492">Ex &longs;ectione 19. De Mu&longs;ica.<!-- KEEP S--></s></p><p type="main"> <s id="id.000493"><emph type="italics"/>num. </s> <s id="id.000494">2. Lineæ duplæ quadratum quadruplum. </s> <s id="id.000495">Hoc loco &longs;equentium probl. </s> <s id="id.000496">cau&longs;a, <lb/>præmittitur totius Mu&longs;icæ ortus breuis tractatio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000497"><emph type="italics"/>3. Vox tam in hypate, quam in nete cantando rumpitur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000498"><emph type="italics"/>4. Cur facilius hypate, quam nete canitur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000499"><emph type="italics"/>5. Cur &longs;uauius notam cantilenam audimus?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000500"><emph type="italics"/>7. Cur veteres hypatem omittebant.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000501"><emph type="italics"/>8. Cur grauis &longs;onum potest acutæ?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000502"><emph type="italics"/>9. Cur cantus ad tibiam vnam, aut lyram &longs;uauior?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000503"><emph type="italics"/>10. Teretizare, quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000504"><emph type="italics"/>11. Vox de&longs;inens acutior fit.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000505"><emph type="italics"/>12. Grauior è fidibus contilenam &longs;u&longs;cipit.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000506"><emph type="italics"/>13. In Diapa&longs;on graue e&longs;t acuti Antiphonum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000507"><emph type="italics"/>14. Cur Diapa&longs;on vnica vox videtur. </s> <s id="id.000508">Punicum quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000509"><emph type="italics"/>15. Leges Mu&longs;icæ, quæ. </s> <s id="id.000510">Genera, Diatonicum, Chromaticum, Encharmonium. <lb/></s> <s id="id.000511">Tetrachorda quæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000512"><emph type="italics"/>16. Antiphonum &longs;uauius est &longs;ymphono, cur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000513"><emph type="italics"/>17. Cur &longs;ola Diapa&longs;on canitur. </s> <s id="id.000514">Magadis quid. </s> <s id="id.000515">Magadare.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000516"><emph type="italics"/>18. De Antiphonis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000517"><emph type="italics"/>19. Cur Diapente, & Diabe&longs;&longs;acon non canunt in Antiphonis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000518"><emph type="italics"/>20. Me&longs;e &longs;ola di&longs;&longs;onante, totum de&longs;&longs;onat p&longs;alterium.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000519"><emph type="italics"/>21. Vocum grauium errores manifestiores, cur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000520"><emph type="italics"/>23. Cur nete duplo acutior, quam hypate?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000521"><emph type="italics"/>24. Nete interpellata, hypate re&longs;onare videtur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000522"><emph type="italics"/>25. Cur Me&longs;e &longs;ic appellata e&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000523"><emph type="italics"/>27. Cur &longs;ola audibilia mores obtinent. </s> <s id="id.000524">Rithmus quid.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000525"><emph type="italics"/>28. Cur cantilenæ quædam leges de cebantur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000526"><emph type="italics"/>30. De Harmonijs, &longs;eu Modis, &longs;eu Tonis pri&longs;corum.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.020.jpg" pagenum="20"/><p type="main"> <s id="id.000527"><emph type="italics"/>31. Vetustiores fui&longs;&longs;e magis Melopæos.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000528"><emph type="italics"/>32. De ip&longs;ius Diapa&longs;on ethymo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000529"><emph type="italics"/>33. Cur aptè de acuto in graue, non è contra canitur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000530"><emph type="italics"/>34. Cur bi&longs;diapente, aut bi&longs;diate&longs;&longs;aron con&longs;onantia non e&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000531"><emph type="italics"/>35. Cur diapa&longs;on omnium pulcherrima e&longs;t con&longs;onantia?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000532"><emph type="italics"/>36. Me&longs;e &longs;ola di&longs;&longs;onante, tota perit harmonia.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000533"><emph type="italics"/>37. Cur difficilius acutum canere, quam graue?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000534"><emph type="italics"/>38. Cur Rythmo, & harmonij omnes gaudent?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000535"><emph type="italics"/>39. Cur &longs;uauius e&longs;t &longs;ymphonum vni&longs;ono?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000536"><emph type="italics"/>40. Cur &longs;olam Diapa&longs;on magadari &longs;olent?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000537"><emph type="italics"/>41. Idem cum 5.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000538"><emph type="italics"/>42. Idem cum 34.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000539"><emph type="italics"/>43. Idem cum 24. Iugum in lyra veteri quodnam fuerit, vna cum figura vete­ <lb/>ris lyræ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000540"><emph type="italics"/>44. Cur &longs;uauius ad tibiam, quam ad lyram cantatur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000541"><emph type="italics"/>45. Idem cum 25. &longs;uperiori.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000542"><emph type="italics"/>46. Idem cum 22.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000543"><emph type="italics"/>47. Idem cum 26.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000544"><emph type="italics"/>48. Idem cum 7. quid Grauiden&longs;um.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000545"><emph type="italics"/>49. Idem cum 30. In choris tragœdiarum, nec &longs;ubdorius, nec &longs;ubphrygius modus <lb/>erat in v&longs;u.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000546"><emph type="italics"/>50. Cur grauior Melodia e&longs;t etiam mollior?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000547"><emph type="italics"/>51. Dolia duo æqualia, quorum alterum plenum &longs;it, alterum dimidium, Diapa&longs;on <lb/>re&longs;onant.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000548">Ex &longs;ectione 23.</s></p><p type="main"> <s id="id.000549"><emph type="italics"/>De immer&longs;ione Nauigij.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000550">Ex &longs;ectione 30.</s></p><p type="main"> <s id="id.000551"><emph type="italics"/>6. Omnis triangulus habet tres æquales, &c.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000552">Ex &longs;ectione 31.</s></p><p type="main"> <s id="id.000553"><emph type="italics"/>7. Cur o culos, <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla vi, ab inuicem di&longs;&longs;ociari nequimus?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000554"><emph type="italics"/>Cur duobus oculis res vna tantum videatur. </s> <s id="id.000555">Cur aliquando rei vi&longs;æ gemina­ <lb/>tio accidat.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000556"><emph type="italics"/>11. Cur di&longs;tractis oculis res vna duæ apparent?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000557"><emph type="italics"/>17. Oculo in latera contorto, cur non fit geminatio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000558"><emph type="italics"/>21. Cur &longs;olam rectitudinem vnico oculo in&longs;piciamus.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000559">Auctarium De Oculi Pupilla.<!-- KEEP S--></s></p><p type="main"> <s id="id.000560"><emph type="italics"/>Oculi fabrica præmittitur, colores oculi vbi &longs;int: vnde qui noctu vident.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000561"><emph type="italics"/>Primo. </s> <s id="id.000562">De pupillæ voce.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000563"><emph type="italics"/>2. Cur in oculo appareat.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000564"><emph type="italics"/>3. Cur non in tota cornea.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000565"><emph type="italics"/>4. Pupillæ definitio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000566"><emph type="italics"/>5. Cur mgra in omnibus hominibus.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000567"><emph type="italics"/>6. Cur in Sole euane&longs; cat.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000568"><emph type="italics"/>7. Quantitas ip&longs;ius num videatur?<emph.end type="italics"/></s></p><p type="main"> <s id="id.000569"><emph type="italics"/>8. Cur modo maior, modo minor videatur, & cui&longs;dam lepida deceptio.<emph.end type="italics"/></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.021.jpg" pagenum="21"/><p type="main"> <s id="id.000570">Additamentum de natura Mathematicarum di&longs;ciplinarum.</s></p><p type="main"> <s id="id.000571"><emph type="italics"/>Primo. </s> <s id="id.000572">De &longs;ubiecto Mathem. &longs;eu de materia intelligibili: vbi o&longs;tenduntur definitio­ <lb/>nes Mathematicæ e&longs;&longs;e perfecti&longs;&longs;imæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000573">2. <emph type="italics"/>Demon&longs;trationes Mathematicas e&longs;&longs;e perfecti&longs;&longs;imas.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000574">3. <emph type="italics"/>Obiectiones: <expan abbr="atq;">atque</expan> etiam calumniæ diluuntur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000575">4. <emph type="italics"/>De præstantia cognitionis, quam Geometria, & Arithmetica gignunt.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000576">5. <emph type="italics"/>De 4. Mathematicis medijs: A&longs;tronomia, Per&longs;pectiua, Mu&longs;ica, Mechanica.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000577">6. <emph type="italics"/>Appendix de re&longs;olutione omnium demonstrationum primi Euclidis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000578">7. <emph type="italics"/>Clarorum Mathematicorum Chronicon.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000579"><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s></p></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.022.jpg" pagenum="22"/><section><p type="head"> <s id="id.000580"><emph type="italics"/>ALTER INDEX<emph.end type="italics"/></s></p><p type="main"> <s id="id.000581"><emph type="italics"/>Quo loca Aristotelis Geometrica, in hoc Opere explicata, <lb/>ad Euclidem, &longs;ecundum propo&longs;itionum ordinem refe­ <lb/>runtur; vt Mathematicarum Profe&longs;&longs;ores habeant, <lb/>vnde &longs;uas prælectiones aliquando valeant locupletare.<emph.end type="italics"/></s></p><p type="head"> <s id="id.000582"><emph type="italics"/>In Primo Elem. Euclidis.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000583">Ad verbum ip&longs;um <emph type="italics"/>(Elementum Euclidis)<emph.end type="italics"/> vide infra tex. <!-- REMOVE S-->4. quinti <lb/>Methaph.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000584">Ad principia primi elementorum, vide infra tex. <!-- REMOVE S-->5. pri. <!-- REMOVE S-->Po&longs;ter.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000585">Ad definitionem 10. pri. <!-- REMOVE S-->pro angulo recto, vide 30. quæ&longs;t. </s> <s id="id.000586">Mecha­ <lb/>nic. <!-- REMOVE S-->& cap. 7. lib. 1. Eth.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000587">Ad axioma 10. quamuis Ari&longs;toteles nihil hac de re dicat; &longs;cias tamen velim <lb/>hoc vno axiomate qu&ecedil;&longs;tionem <expan abbr="quãdam">quandam</expan> inter Philo&longs;ophos valdè difficilem, <lb/>facile di&longs;&longs;olui. </s> <s id="id.000588">ea e&longs;t, vtrum marmor, aut adamas, aliudue quidpiam infle­ <lb/>xibile &longs;ucce&longs;&longs;iuè findi, & aperiri po&longs;&longs;it. </s> <s id="id.000589">qui enim aiunt, &longs;ic refelluntur, quia <lb/>nimirum &longs;equeretur, duas rectas lineas habere &longs;egmentum commune: in­ <lb/>telligantur enim duæ lineæ, vna in vna &longs;uperficie, altera vero in altera, quæ <lb/>antequam incipiat apertio, congruant; Quoniam igitur tota illa apertio <lb/>non fit in in&longs;tanti, &longs;ed &longs;ucce&longs;&longs;iuè, facta iam aliqua apertionis parte con&longs;i­ <lb/>derentur prædictæ lineæ, erit igitur earum pars aliqua ab inuicem &longs;epara­ <lb/>ta, altera verò adhuc alteri congruens, ergo &longs;equetur, duas lineas habere <lb/>&longs;egmentum commune, quod e&longs;t impo&longs;&longs;ibile, quia contra 10. axioma.</s></p><p type="main"> <s id="id.000590">Ad Calcem axiomatum primi accommodetur tex. <!-- REMOVE S-->1. primi Po&longs;ter.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000591">Ad primam primi, po&longs;t ip&longs;ius explicationem, commodè declarari pote&longs;t, cur <lb/>Ari&longs;t. Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna­ <lb/>tiones, vide cap. de Priori, & cap. 24. &longs;ecti primi, libri primi Priorum, & <lb/>tex. <!-- REMOVE S-->4. quinti Methaph. <!-- REMOVE S-->& tex. <!-- REMOVE S-->20. &longs;exti Methaph. <!-- REMOVE S-->& cap. 3. lib. 3. Ethic. <lb/><!-- KEEP S--></s> <s id="id.000592">Item ad primam primi, vide tex. <!-- REMOVE S-->7. &longs;ecundi Po&longs;ter. loco 2.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000593">Ad 5. primi, vide cap. 24. &longs;ecti 1 lib. 1. Priorum.</s></p><p type="main"> <s id="id.000594">Ad 21. primi, vide tex. <!-- REMOVE S-->20. primi Po&longs;ter. <!-- REMOVE S-->loco 2.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000595">Ad 22. primi, vide locum 10. de lineis in&longs;ecabilibus.</s></p><p type="main"> <s id="id.000596">Ad 28. primi, vide cap. 21. & cap. 22. &longs;ecundi <expan abbr="Priorũ">Priorum</expan>, & tex. <!-- REMOVE S-->13. primi Po&longs;ter.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000597">Ad 32. primi, vide cap. 1. &longs;ecti 3. lib. 1. Prior. <!-- REMOVE S-->& cap. 26. &longs;ecundi <expan abbr="Priorũ">Priorum</expan>, & tex. <!-- REMOVE S-->2. <lb/>primi Po&longs;ter. loco 4. & tex. <!-- REMOVE S-->23. primi Po&longs;ter. <!-- REMOVE S-->vbi ait hanc e&longs;&longs;e poti&longs;&longs;imam <lb/><expan abbr="demõ&longs;trationem">demon&longs;trationem</expan>. </s> <s id="id.000598">& tex. <!-- REMOVE S-->37. primi Po&longs;ter. & tex. <!-- REMOVE S-->39. primi Po&longs;ter. <!-- KEEP S--></s> <s id="id.000599">Ibidem <lb/>loco 4. & tex. <!-- REMOVE S-->43. primi Po&longs;ter. <!-- REMOVE S-->& tex. <!-- REMOVE S-->2. &longs;ecundi Po&longs;ter. <!-- REMOVE S-->bis. </s> <s id="id.000600">& tex. <!-- REMOVE S-->89. &longs;e­ <lb/>cundi Phy&longs;. & tex. <!-- REMOVE S-->15. octaui Phy&longs;. <!-- REMOVE S-->& tex. <!-- REMOVE S-->119. primi de Cœlo. <!-- KEEP S--></s> <s id="id.000601">& tex. <!-- REMOVE S-->25. <lb/>&longs;ecundi de Cœlo. <!-- KEEP S--></s> <s id="id.000602">tex 11. primi de Anima. <!-- REMOVE S-->& cap. 1. de mem. </s> <s id="id.000603">& remini&longs;c. <lb/></s> <s id="id.000604">& tex. <!-- REMOVE S-->35. quinti Methaphy&longs;. & tex. <!-- REMOVE S-->20. &longs;exti Methaphy&longs;. <!-- REMOVE S-->& tex. <!-- REMOVE S-->22. &longs;exti <lb/>Methaphy&longs;. <!-- REMOVE S-->& cap. 4. lib. 2. de Generat. <!-- REMOVE S-->animal. <!-- KEEP S--></s> <s id="id.000605">& cap. 5. lib. 6. Ethic. <!-- REMOVE S-->& <lb/>cap. 2. Magnorum Moral. & cap. 10. Mag. Moral. & cap. 16. Mag. Moral. <lb/><!-- REMOVE S-->& cap. <!-- REMOVE S-->7. &longs;ecundi Eudem. & cap. <!-- REMOVE S-->12. &longs;ecundi Eudem. <!-- REMOVE S-->& problema 6. &longs;ectio­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.023.jpg" pagenum="23"/>nis 30. tot Ari&longs;totelis loca illu&longs;trat vnica hæc Euclidis Demon&longs;tratio.</s> </p><p type="main"> <s id="id.000606">Ad &longs;cholion præcedentis 32. primi, vide tex. <!-- REMOVE S-->39. primi Po&longs;ter. loco 3. Item <lb/>tex. <!-- REMOVE S-->25. &longs;ecundi Po&longs;ter. <!-- REMOVE S-->loco vlt.</s> </p><p type="main"> <s id="id.000607">Ad 45. primi, vide locum 9. de lineis in&longs;ecabilibus.</s></p><p type="main"> <s id="id.000608">Ad 46. primi, vide locum 11. de lineis in&longs;ecabilibus.</s></p><p type="main"> <s id="id.000609">Ad 47. primi, vide locum 11. de lineis in&longs;ecab. </s> <s id="id.000610">Item locum 14. de ij&longs;dem.</s></p><p type="head"> <s id="id.000611"><emph type="italics"/>In &longs;ecundo Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000612">Ad 2. definitionem 2. Gnomonis, vide cap. de Motu in Po&longs;tprædicam. </s> <s id="id.000613">Qua­ <lb/>dratum augetur Gnomone circumpo&longs;ito.</s></p><p type="main"> <s id="id.000614">Ad 14. propo&longs;. </s> <s id="id.000615">2. opportunum e&longs;t Auditores de Quadratura circuli erudire, <lb/>vide igitur cap. de relatione in prædicam. </s> <s id="id.000616">& cap. 31. &longs;ecundi Priorum, & <lb/>tex. <!-- REMOVE S-->23. primi Po&longs;ter. & finem 1. cap. primi Elenchorum. </s> <s id="id.000617">lege primam Ar­ <lb/>chimedis de dimen&longs;ione circuli.</s></p><p type="head"> <s id="id.000618"><emph type="italics"/>In tertio Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.000619">Ad primam 3. vide cap. 9. lib. 2. Ethycorum.</s></p><p type="main"> <s id="id.000620">Ad 2. tertij, vide tex. <!-- REMOVE S-->13. lib. 1. de Anima. <!-- REMOVE S-->& locum 16. de lineis in&longs;ecab.</s> </p><p type="main"> <s id="id.000621">Ad 31. tertij, vide tex. <!-- REMOVE S-->11. &longs;ecundi Po&longs;ter. & tex. <!-- REMOVE S-->20. &longs;exti Methaph. <!-- REMOVE S-->loco 2.<!-- KEEP S--></s> </p><p type="head"> <s id="id.000622"><emph type="italics"/>In quarto.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000623">Ad commentarium P. <!-- REMOVE S-->Clauij extremum lib. 4. elementorum. </s> <s id="id.000624">lege tex. <!-- REMOVE S-->66. <lb/>tertij de Cœlo.<!-- KEEP S--></s> </p><p type="head"> <s id="id.000625"><emph type="italics"/>In quinto.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000626">Ad 4. definitionem 5. vide cap. 3. lib. 2. Ethyc.<!-- KEEP S--></s></p><p type="main"> <s id="id.000627">Ad 9. definitionem 5. vide cap. 3. lib. 5. Ethyc. <!-- REMOVE S-->loco 4. & cap. 31. primi Ma­ <lb/>gnorum Moralium.</s> </p><p type="main"> <s id="id.000628">Ad 10. definitionem 5. vide tex. <!-- REMOVE S-->29. primi Po&longs;ter. loco 2.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000629">Ad 12. definitionem 5. vide tex. <!-- REMOVE S-->13. primi Po&longs;ter. <!-- REMOVE S-->loco 3. & tex. <!-- REMOVE S-->25. &longs;ecundi <lb/>Po&longs;ter. <!-- REMOVE S-->& tex. <!-- REMOVE S-->32. tertij de Anima. <!-- REMOVE S-->& cap. 3. lib. 5. Ethyc. <!-- REMOVE S-->loco 4.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000630">Ad 16. propo&longs;. </s> <s id="id.000631">5. vide tex. <!-- REMOVE S-->25. &longs;ecundi Po&longs;ter. loco 2. ex hac Euclidis demon­ <lb/>&longs;tratione patet, vitio&longs;am e&longs;&longs;e illam Auerrois argumentationem, 8. Phy&longs;. <lb/><!-- REMOVE S-->comm. <!-- REMOVE S-->15. &longs;cilicet.</s> </p><p type="main"> <s id="id.000632">Vt &longs;e habet voluntas antiqua ad antiquum effectum, <lb/>Ita &longs;e habet etiam voluntas noua ad effectum nouum: <lb/>Ergo permutando, ita &longs;e habebit voluntas antiqua ad effectum nouum. <lb/></s> <s id="id.000633">Quemadmodum voluntas noua ad effectum antiquum.</s></p><p type="main"> <s id="id.000634">Non enim in permutando confert antecedentem ad antecedentem, & con­ <lb/>&longs;equentem ad con&longs;equentem, vt par erat, &longs;ed confert antecedentem ad <lb/>con&longs;equentem, quod non licet.</s></p><p type="head"> <s id="id.000635"><emph type="italics"/>In &longs;exto.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000636">Ad 2. propo&longs;it. </s> <s id="id.000637">6. vide cap. 2. lib. 8. Topicorum loco 41.</s></p><p type="main"> <s id="id.000638">Ad 13. &longs;exti, vide tex. <!-- REMOVE S-->12. &longs;ecundi de Anima, & tex. <!-- REMOVE S-->3. tertij Methaphy&longs;.<!-- REMOVE S--><emph type="italics"/>In &longs;eptimo.<emph.end type="italics"/></s> </p><p type="main"> <s id="id.000639">Ad primam definitionem 7. vide tex. <!-- REMOVE S-->5. primi Po&longs;ter.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000640">Ad 8. definitionem 7. vide cap. 1. lib. 1. Magnorum Moral.<!-- KEEP S--></s></p><p type="head"> <s id="id.000641"><emph type="italics"/>In octauo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000642">Ad 4. propo&longs;. </s> <s id="id.000643">9. vide tex. <!-- REMOVE S-->20. primi Po&longs;ter. loco 2.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000644">Ad 8. propo&longs;. </s> <s id="id.000645">9. vide problem. </s> <s id="id.000646">3. &longs;ectionis 15. loco 4.<!-- KEEP S--></s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.024.jpg" pagenum="24"/><p type="head"> <s id="id.000647"><emph type="italics"/>In decimo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000648">Ad primam definitionem 10. vide cap. 23. &longs;ecti 1. primi Priorum. </s> <s id="id.000649">& tex. <!-- REMOVE S-->48. <lb/>primi de Cœlo.<!-- KEEP S--></s> </p><p type="main"> <s id="id.000650">Ad 118. decimi, vide cap. 23. &longs;ecti 1. libri 1. Priorum. </s> <s id="id.000651">& &longs;ecto 2. cap. 23. li­ <lb/>bri 1. Priorum. </s> <s id="id.000652">& cap. 22. lib. 2. Priorum. </s> <s id="id.000653">& tex. <!-- REMOVE S-->5. primi Po&longs;ter. & tex. <!-- REMOVE S-->44. <lb/>primi Po&longs;ter. <!-- REMOVE S-->& cap. 15. primi Po&longs;ter. <!-- REMOVE S-->& tex. <!-- REMOVE S-->119. primi de Cœlo. <!-- KEEP S--></s> <s id="id.000654">& tex. <lb/><!-- REMOVE S-->120. quarti Phy&longs;. & tex. <!-- REMOVE S-->21. tertij de Anima. <!-- REMOVE S-->& cap. 1. primi Methaphy&longs;. <lb/><!-- REMOVE S-->& tex. <!-- REMOVE S-->28. quarti Met. <!-- REMOVE S-->& tex. <!-- REMOVE S-->34. quinti Met. <!-- REMOVE S-->& tex. <!-- REMOVE S-->8. &longs;exti Met. <!-- REMOVE S-->& cap. 4. <lb/>lib. 2. de Generat. <!-- REMOVE S-->animal. <!-- KEEP S--></s> <s id="id.000655">& lib. 3. cap. 3. Ethyc. <!-- REMOVE S-->& cap. 10. &longs;ecundi Eu­ <lb/>dem. <!-- KEEP S--></s> <s id="id.000656">tot Ari&longs;t. loca ab hac vna Euclidis Demon&longs;tratione illu&longs;trantur.</s></p><p type="head"> <s id="id.000657"><emph type="italics"/>In decimotertio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000658">Ad primam propo&longs;. </s> <s id="id.000659">13. &longs;ecundum editionem Commandini, aut Zamberti. <lb/><!-- KEEP S--></s> <s id="id.000660">vide initio Priorum, in verbum (Re&longs;olutio)</s></p><p type="main"> <s id="id.000661">Atqne hæc &longs;unt, quæ ex Elementorum opere Ari&longs;toteles pa&longs;&longs;im v&longs;urpauit, <lb/>quæque nos infra explicabimus.</s></p><p type="head"> <s id="id.000662"><emph type="bold"/>Finis Secundi Indicis.<emph.end type="bold"/></s></p><p type="main"> <s id="id.000663">Quæ verò ad alias Mathematicas, Mu&longs;icam &longs;cilicet, Per&longs;pe­ <lb/>ctiuam, Mechanicam, & A&longs;tronomiam pertinens, facilè <lb/>poterunt ex primo Indice ad vnamquamque earum &longs;eor­ <lb/>&longs;um cum libuerit, &longs;ecerni.</s></p></section><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.025.jpg" pagenum="25"/><section><p type="head"> <s id="id.000664"><emph type="italics"/>TERTIVS INDEX ALPHABETICVS,<emph.end type="italics"/></s></p><p type="head"> <s id="id.000665"><emph type="italics"/>cuius numeri re&longs;pondent numeris marginalibus Operis.<emph.end type="italics"/> <lb/><arrow.to.target n="table2"/></s></p><table><table.target id="table2"/><row><cell><emph type="italics"/>A<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Abductio quid. eius inuentor. numero 16. marginali.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Acuti den&longs;um quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>Aequalitas mathematica, quæ.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell><emph type="italics"/>Ae&longs;tus maris natura.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Aequitas arithmetica, & æquitas &longs;ecundum dignitatem.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Agathir&longs;i populi.<emph.end type="italics"/></cell><cell>382</cell></row><row><cell><emph type="italics"/>Angulus quid. vt nominari debeat 10. angulum in &longs;emicirculo e&longs;&longs;e rectum o&longs;tendi per cau&longs;am materialem.<emph.end type="italics"/></cell><cell>71</cell></row><row><cell><emph type="italics"/>Angulus rectus variè con&longs;ideratur à Geometra, & à Fabro.<emph.end type="italics"/></cell><cell>301</cell></row><row><cell><emph type="italics"/>Antiphontis quadratura circuli.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell><emph type="italics"/>Antennæ nauis problema.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell><emph type="italics"/>Antipbonæ voces.<emph.end type="italics"/> 358. 363. 370. 371.</cell><cell>373</cell></row><row><cell><emph type="italics"/>Apum mirabilis indu&longs;tria.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Apotome linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Aquæ &longs;uperficiem e&longs;&longs;e &longs;phæricam ratione mathematica.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell><emph type="italics"/>Arithmetica proportio.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Aranei industria patefacta, qua ad res inaca&longs;&longs; as tran&longs;eat.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>filum emittit ex &longs;ece&longs;&longs;u contra Ari&longs;totelem pro Democrito.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Astronomiæ principia duo, Apparentia, & Ob&longs;eruatio.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell><emph type="italics"/>Automata, quæ.<emph.end type="italics"/> 199. 298.</cell><cell><emph type="italics"/>a. b.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>B<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ba&longs;is ballarum in aqua, cur &longs;it alba, & cur non faciat vmbram.<emph.end type="italics"/></cell><cell>351</cell></row><row><cell><emph type="italics"/>Binomium linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Bra&longs;ilien&longs;es, qua ratione numer are &longs;oliti.<emph.end type="italics"/></cell><cell>340</cell></row><row><cell><emph type="italics"/>Bry&longs;onis quadratura circuli.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell><emph type="italics"/>C<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Calippi opinio de numero Cœlorum.<emph.end type="italics"/></cell><cell>236</cell></row><row><cell><emph type="italics"/>Cantilenam notam &longs;uauius, quam ignotam audimus.<emph.end type="italics"/></cell><cell>362</cell></row><row><cell><emph type="italics"/>Centrum circuli reperire. 303. Centrum mundi mathematicè o&longs;tenditur. 123. Cen-trum grauitatis, & molis.<emph.end type="italics"/> 38.</cell><cell>112</cell></row><row><cell><emph type="italics"/>Chordarum veterum nomina.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Circuli quadratura quid. an po&longs;&longs;ibilis. num. 1. Circulorum concentricorum maiores velocius moueri. 108. Circuli admiranda. 239. De circulorum concentricorum volutatione.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell><emph type="italics"/>Coalternæ lineæ, quæ.<emph.end type="italics"/> 12. 14.</cell><cell>44</cell></row><row><cell><emph type="italics"/>Cœlorum ordinem petendum ex A&longs;tronomis 109. item numerum<emph.end type="italics"/></cell><cell>233</cell></row><row><cell><emph type="italics"/>Colores in mu&longs;ica 78. Colores oculorum vnde.<emph.end type="italics"/></cell><cell>408</cell></row><row><cell><emph type="italics"/>Cometarum tractatiuncula ex recentioribus, qui eas in Cœlo collocant. 136. e&longs;&longs;e &longs;u-pra aerem longi&longs;&longs;imo &longs;altem interuallo o&longs;tenditur mathematicè. 129. in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Con&longs;onantia, quid. 64. quibus numeris con&longs;onantiæ con&longs;tent.<emph.end type="italics"/></cell><cell>210</cell></row><row><cell><emph type="italics"/>Conus, & cylindrus, cur variè mouentur.<emph.end type="italics"/></cell><cell>355</cell></row><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.026.jpg" pagenum="26"/><row><cell><emph type="italics"/>Cubus numerus. duo cubi cubus, quid &longs;ignificet.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Curru problema.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Cunei problema.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell><emph type="italics"/>Cylindri, & coni motus comparatio problematica.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>D<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Definitiones mathematicæ e&longs;&longs;e e&longs;&longs;entiales, & perfe&longs;&longs;imas. cap. 1. de nat. Math. definitionum v&longs;us in Mathematicis.<emph.end type="italics"/></cell><cell>81</cell></row><row><cell><emph type="italics"/>De&longs;criptio, & de&longs;cribere, quid.<emph.end type="italics"/> 2. 6. 7.</cell><cell>205</cell></row><row><cell><emph type="italics"/>De&longs;ignatio pro demon&longs;tratione mathematica.<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Demon&longs;trationis perfectæ exemplum. 36. demon&longs;trationum mathematicarum præ-&longs;tantia. cap. 4. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Dentiforcipis problema.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell><emph type="italics"/>Denarij numeri perfectio. 339. cur <expan abbr="v&longs;q;">v&longs;que</expan> ad denariŭ omnes <expan abbr="g&etilde;tes">gentes</expan> <expan abbr="numer&etilde;t">numerent</expan>.<emph.end type="italics"/></cell><cell>339. 8. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Diameter incommen&longs;urabilis costæ. 5. diametri etymon.<emph.end type="italics"/></cell><cell>337</cell></row><row><cell><emph type="italics"/>Diapa&longs;on quid. 90. 350. omnium con&longs;onantiarum pulcberrima.<emph.end type="italics"/></cell><cell>388</cell></row><row><cell><emph type="italics"/>Diapa&longs;on diapente.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diapente con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diate&longs;&longs;aron con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Di&longs;d apa&longs;on con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Die&longs;is, quid.<emph.end type="italics"/> 53.</cell><cell>226</cell></row><row><cell><emph type="italics"/>Dolia duo, quomodo aliquando Diapa&longs;on re&longs;onent.<emph.end type="italics"/></cell><cell>402</cell></row><row><cell><emph type="italics"/>Duplum inter multiplicia primum e&longs;t.<emph.end type="italics"/></cell><cell>322</cell></row><row><cell><emph type="italics"/>E<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Elementa mundi non componi ex figuris geometricis.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Elementa geometrica, quæ.<emph.end type="italics"/> 82.</cell><cell>213</cell></row><row><cell><emph type="italics"/>Eudoxi opinio de numero Cœlorum.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell><emph type="italics"/>Exempla mathematicorum, qualia. 11. non e&longs;&longs;e fal&longs;a.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell><emph type="italics"/>Exemplorum veritas, & conformitas, quatenus requirantur.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell><emph type="italics"/>F<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Figuram omnem planam habere &longs;uos angulos externos <expan abbr="quotcŭq;">quotcŭque</expan> æquales quatuor rectis angulis, quæ e&longs;t mira proprietas.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell><emph type="italics"/>Figuræ &longs;imiles, quæ.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell><emph type="italics"/>Figurarum planarum ordo 88. quæ nam totum locum repleant.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell><emph type="italics"/>Figurarum &longs;olidarum, quænam totum locum repleant: vbi Ari&longs;t. & omnium expo&longs;i-torum rratum aperitur.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell><emph type="italics"/>Figuratio lucis.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Figurationes pro demonftrationibus Mathem.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell><emph type="italics"/>Filum Araneorum, ex qua parte corporis exeat, & ex qua materia.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>Fluxus, ac refluxus maris.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Funium lectorum problema.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell><emph type="italics"/>G<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Galaxia quid. 131. Ari&longs;toteles defen&longs;us.<emph.end type="italics"/> 132.</cell><cell>140</cell></row><row><cell><emph type="italics"/>Galibei recens ob&longs;eruatio.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell><emph type="italics"/>Generatria Mu&longs;icæ veteris. 78. Fusè explicantur.<emph.end type="italics"/></cell><cell>371</cell></row><row><cell><emph type="italics"/>Geodæ&longs;ia.<emph.end type="italics"/></cell><cell>207</cell></row><row><cell><emph type="italics"/>Geographiæ veteris plura errata,<emph.end type="italics"/> 145. 146. 147. 148.</cell><cell>149</cell></row><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.027.jpg" pagenum="27"/><row><cell><emph type="italics"/>Gnomon, quid. 3. &<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Gnomones numeri.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Graue qua ratione ad centrum mundi de&longs;cenderet, eiqué aptaretur.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell><emph type="italics"/>Grauiden&longs;um, quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>H<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Halonis demonfiratio.<emph.end type="italics"/></cell><cell>161</cell></row><row><cell><emph type="italics"/>Hippocratis chij quadratura circuli. 17. <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> quadratura lunulæ optima.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell><emph type="italics"/>Hyades, Atlantides, & Succulæ.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Hypate, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Hypotenu&longs;a in ince&longs;&longs;u animalium.<emph.end type="italics"/></cell><cell>294</cell></row><row><cell><emph type="italics"/>I<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Illuminationes Solis deficientis per foramina tran&longs;euntes, eur &longs;int defectiuæ.<emph.end type="italics"/></cell><cell>350</cell></row><row><cell><emph type="italics"/>modus videndi eclyp&longs;im facilis, ac iucundus.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Ince&longs;&longs;us animalium lineis explicatur.<emph.end type="italics"/></cell><cell>294. <emph type="italics"/>& <expan abbr="&longs;eqq.">&longs;eqque</expan><emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Incommen&longs;urabilia, quæ, & eorum inuentores.<emph.end type="italics"/></cell><cell>5</cell></row><row><cell><emph type="italics"/>Indiui&longs;ibilia mathematica e&longs;&longs;e priuationes. 189. oriri ex diui&longs;ione. 231. eorum duo genera.<emph.end type="italics"/></cell><cell>276</cell></row><row><cell><emph type="italics"/>Infinito, qua ratione vtantur Mathematici.<emph.end type="italics"/> 94.</cell><cell>96</cell></row><row><cell><emph type="italics"/>Iridis demon&longs;tratio &longs;ecundum Ari&longs;t. 163. & &longs;equentibus. Item noua de Iride tra-ctatio. ibidem in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Iugum in lyra quid, & eius figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>L<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Leges mu&longs;icales.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell><emph type="italics"/>Libra maior, cur exactior. initio Mechanicarum quæ&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Linea, quæ terminus est illuminationis in Luna, cur modo recta, modo curua vi-deatur.<emph.end type="italics"/></cell><cell>346</cell></row><row><cell><emph type="italics"/>Lineæ rationales, & irrationales, &c.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Lumen oculorum noctu videntium, in qua oculi parte manet. in addit. de Pupilla.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Lumen Solis, cur &longs;it circulare, quamuis per foramina angulo&longs;a ingrediatur.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Luna plana, cur appareat, cum &longs;it &longs;phærica. 347. cur in eadem altitudine cum Sole &longs;upra horizontem, maiorem vmbram efficiat.<emph.end type="italics"/></cell><cell>349</cell></row><row><cell><emph type="italics"/>Lunam e&longs;&longs;e &longs;phæricam. 48. illuminari &longs;phæricè quid: ibidem & de illuminatione Lu-næ. iterum e&longs;&longs;e &longs;phæricam ab eclyp&longs;ibus.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell><emph type="italics"/>Luminarium Solis, & Lunæ ordo.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell><emph type="italics"/>Lychanos, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Lyræ veteris figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>M<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Magalis, &longs;eu magas, & magadi&longs;&longs;are.<emph.end type="italics"/> 373.</cell><cell>393</cell></row><row><cell><emph type="italics"/>Materia intelligibilis. fusè verò. explicatur in tract. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mathematicæ mediæ, &longs;eu &longs;ubalternatæ habent propter quid &longs;uarum demon&longs;iratio-nem.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell><emph type="italics"/>Mathematici negant reperiri quantitatem indiui&longs;ibilem, &longs;eu minimam.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell><emph type="italics"/>Mathematicæ non &longs;unt contentio&longs;æ. 83. ostendunt per cau&longs;am formalcm.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell><emph type="italics"/>Mathematicas inuenerunt Aegyptij Sacerdotes.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell><emph type="italics"/>Mathematicæ o&longs;tendunt per cau&longs;am materialem, & formalem. 205. earum vtilitas. ibidem. De earum natura. in proprio tractatu.<emph.end type="italics"/></cell><cell/></row> <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.028.jpg" pagenum="28"/><row><cell><emph type="italics"/>Mathematicæ maximè tractant de Bono, & Pulcro, qua ratione.<emph.end type="italics"/></cell><cell>237</cell></row><row><cell><emph type="italics"/>Mechanica facultas, quæ.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell><emph type="italics"/>Melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Melopeia quid.<emph.end type="italics"/></cell><cell>384</cell></row><row><cell><emph type="italics"/>Medium Demonstrationum Mathem. in earum tractatu.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Me&longs;e quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Mina in men&longs;uris quid.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell><emph type="italics"/>Monochordium.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Motus nauigij, & remi comparatio. 247. Pulchra P. Nonij in id annotatio con-tra Ari&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Modi mu&longs;ici.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Modorum antiquorum ordo, numerus, &c.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Motus primi mobilis, &longs;eu diurnus e&longs;t men&longs;ura cœlestium motuum.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell><emph type="italics"/>Mu&longs;ici recentiores reprehen&longs;i. 331. & in fine Chronologiæ.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mu&longs;icæ totius elementa.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Mu&longs;ica nuda, & cum melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>N<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Nete quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Nucifragi in&longs;trumenti problema.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell><emph type="italics"/>Numerus, par, impar, primus, & compo&longs;itus, quadratus, &longs;eu æquilaterus, altera parte longior. 24. Cubus num.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Numeri capitales, qui.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell><emph type="italics"/>Numerum parem e&longs;&longs;e cau&longs;am infiniti: imparem verò finiti.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Numerorum parium alij &longs;unt primi, alij non.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell><emph type="italics"/>Numerus vnitarius.<emph.end type="italics"/></cell><cell>307</cell></row><row><cell><emph type="italics"/>O<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Oculi cur moueantur con&longs;imiliter.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Oculi anathome.<emph.end type="italics"/></cell><cell>408. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Omophonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Oppo&longs;itio diametralis e&longs;t omnium maxima.<emph.end type="italics"/></cell><cell>327</cell></row><row><cell><emph type="italics"/>Ortus, & occa&longs;us &longs;yderum, quid, & quotuplex: vbide Orione, & Canicula.<emph.end type="italics"/></cell><cell>153</cell></row><row><cell><emph type="italics"/>P<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Paranete quæ voces, aut chordæ.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Parame&longs;e<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parhypate<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parelia, cur appareant nondum &longs;atis explicari.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell><emph type="italics"/>Parna&longs;&longs;us mons, vbinam &longs;it. Item paropame&longs;&longs;us.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell><emph type="italics"/>Partes quantitatis &longs;unt materia illius.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell><emph type="italics"/>Per&longs;pectiuus, quatenus con&longs;ideret lineam.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell><emph type="italics"/>Pa&longs;&longs;iones Mathematicorum cum &longs;ubiecto conuertuntur.<emph.end type="italics"/></cell><cell>47</cell></row><row><cell><emph type="italics"/>Pila chri&longs;tallina, vel vitrea, qua ratione comburat.<emph.end type="italics"/></cell><cell>60</cell></row><row><cell><emph type="italics"/>Planetæ, qua ratione moueantur duplici motu.<emph.end type="italics"/></cell><cell>130</cell></row><row><cell><emph type="italics"/>Plato &longs;olida ex planis componebat. 105. cur Elementis figuras Geometricas attri-bueret.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Planetarum ordo.<emph.end type="italics"/></cell><cell>271</cell></row><row><cell><emph type="italics"/>Principia Mathematicorum.<emph.end type="italics"/> 2.</cell><cell>118</cell></row><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.029.jpg" pagenum="29"/><row><cell><emph type="italics"/>Principia &longs;cientiarum duplicia. Ex quibus, & circa quod.<emph.end type="italics"/></cell><cell>61</cell></row><row><cell><emph type="italics"/>Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></cell><cell>315</cell></row><row><cell><emph type="italics"/>Proportio alterna. 28. multiplicata, &longs;eu multiplex &longs;ecundum Cæneum.<emph.end type="italics"/></cell><cell>46</cell></row><row><cell><emph type="italics"/>P&longs;eudographia quid.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell><emph type="italics"/>Proportionalitas quid.<emph.end type="italics"/></cell><cell>308</cell></row><row><cell><emph type="italics"/>Proportio continuata, & di&longs;iuncta quid. 310. alterna, &longs;eu permutata quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>inibi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Proportio Geometrica. 311. Arithmetica.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Proportio &longs;ecundum dignitatem, e&longs;t Geometrica.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Problemata mu&longs;icalia varia à 360. <expan abbr="v&longs;q;">v&longs;que</expan> ad finem &longs;ectionis 19. problematum.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Punicum, mu&longs;icum in&longs;trumentum.<emph.end type="italics"/></cell><cell>370</cell></row><row><cell><emph type="italics"/>Pupillæ oculi etymon., & natura. 408. cur in oculo no&longs;tro imago pupillæ appareat. problem.<emph.end type="italics"/> 2.</cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Cur nigra in omnibus hominibus. probl.<emph.end type="italics"/> 5.</cell><cell/></row><row><cell><emph type="italics"/>Cur in Sole euane&longs;cat. probl.<emph.end type="italics"/> 6.</cell><cell/></row><row><cell><emph type="italics"/>Cur modo maior, modo minor appareat.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Pythagorici primi Mathematicis <expan abbr="operã">operam</expan> dedere, easqué ceteris &longs;cientijs præponebăt.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell><emph type="italics"/>Q<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Qvadratura circuli. vide circulus. de quadratura figurarum rectangularum, & quadraturæ duplex definitio cau&longs;alis, & formalis.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell><emph type="italics"/>Quantitas an conctet ex indiui&longs;ibili. toto libello de lineis in&longs;ecabilibus argumentis mathematicis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>R<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Remi problema.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell><emph type="italics"/>Re&longs;olutio logica, & mathematica, vt conueniant. 4. &<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Re&longs;ultus cadentium in terram, quibus angulis fiat.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell><emph type="italics"/>Rythmus fusè explicatur.<emph.end type="italics"/></cell><cell>381</cell></row><row><cell><emph type="italics"/>Rubrum mare duplex.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell><emph type="italics"/>S<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Scythala quid, & eius figura 250. &<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Securis problema, vbi de antiquæ &longs;euris figura, & angulo pulchra demon&longs;tran-tur.<emph.end type="italics"/></cell><cell>258</cell></row><row><cell><emph type="italics"/>Semitonium, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Solem e&longs;&longs;e terra multo maiorem: probatur.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell><emph type="italics"/>Sphæram planum tangit in puncto. demon&longs;tratur.<emph.end type="italics"/></cell><cell>184</cell></row><row><cell><emph type="italics"/>Statera antiqua, quæ: eius figura, & problema.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell><emph type="italics"/>Stereomatria, vt differat à Geometria.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell><emph type="italics"/>Succula.<emph.end type="italics"/></cell><cell>253</cell></row><row><cell><emph type="italics"/>Symphonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Symphonia.<emph.end type="italics"/></cell><cell>391</cell></row><row><cell><emph type="italics"/>T<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Temonis nauis problema.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell><emph type="italics"/>Terram e&longs;&longs;e rotundam ex eclyp&longs;i. 114. Item aliter. 115. e&longs;&longs;e re&longs;pectu Cœli paruam valde. 115. e&longs;&longs;e cubum cur Plato voluerit.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Terræ quantitas.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell><emph type="italics"/>Terram paulatim reduci ad pefféctam rotunditatem.<emph.end type="italics"/></cell><cell>151</cell></row><row><cell><emph type="italics"/>Tetragoni&longs;mus. vide Quadratura.<emph.end type="italics"/></cell><cell/></row><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.030.jpg" pagenum="30"/><row><cell><emph type="italics"/>Teretizare, quid.<emph.end type="italics"/></cell><cell>366</cell></row><row><cell><emph type="italics"/>Tetrachordon, quid.<emph.end type="italics"/></cell><cell>386</cell></row><row><cell><emph type="italics"/>Tollenonis problema.<emph.end type="italics"/></cell><cell>267</cell></row><row><cell><emph type="italics"/>Tonus mu&longs;icus, qui; vnde oriatur.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Trochleæ problemata.<emph.end type="italics"/> 249. 250.</cell><cell>251</cell></row><row><cell><emph type="italics"/>Tunicæ oculi. 408. in tractatu de Pupilla.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>V<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ventorum nomina, & &longs;itus.<emph.end type="italics"/></cell><cell>160. <emph type="italics"/>a.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Vectis quotuplex, & c.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell><emph type="italics"/>Veteres canere &longs;olitos non &longs;olum in choris, &longs;ed etiam in &longs;cenis.<emph.end type="italics"/> 371. 384.</cell><cell>400</cell></row><row><cell><emph type="italics"/>Virgiliæ, Pleiades.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Vi&longs;æ res gemmantur di&longs;tractis oculis.<emph.end type="italics"/></cell><cell>406</cell></row><row><cell><emph type="italics"/>Vi&longs;æ rei geminatio non fit altero oculo in latera torto, cur.<emph.end type="italics"/></cell><cell>407</cell></row><row><cell><emph type="italics"/>Vi&longs;us res vi&longs;as, cur non duplicet, etiam &longs;i duos oculos habeamus.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Vmbelici litoralis problema.<emph.end type="italics"/></cell><cell>254</cell></row><row><cell><emph type="italics"/>Vmbram terræ parum &longs;upra Lunam tran&longs;cendere.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell><emph type="italics"/>Vmbrarum incrementa, & decrementa, cur inæqualia.<emph.end type="italics"/> 344.</cell><cell>348</cell></row><row><cell><emph type="italics"/>Vi&longs;us res geminat, &longs;i alter oculorum digito pellatur, cur.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell><emph type="italics"/>Vocum mu&longs;icalium antiquæ appellationes.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Vox acuta velocior, grauis verò tarda, cur.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell><emph type="italics"/>Voluminum &longs;ectio modo rectam lineam, modo curuam refert, cur.<emph.end type="italics"/></cell><cell>356</cell></row><row><cell><emph type="italics"/>Vnitas, cur indiui&longs;ibilis.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell><emph type="italics"/>Z<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Zonas terræ, vt Arist. de&longs;ignet: & quæ &longs;ecundum ip&longs;um &longs;int habitabiles.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell><emph type="italics"/>Zonam torridam quatuor reddunt habitabilem.<emph.end type="italics"/></cell><cell>159</cell></row></table><p type="head"><s id="id.000666">Finis Tertij Indicis.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.031.jpg" pagenum="31"/> </section><section><p type="main"> <s id="id.000667">Vi&longs;um e&longs;t etiam opportunum Lectori fore, ea &longs;imul in vnum <lb/>loca colligere, in quibus Ari&longs;toteles mihi vi&longs;us e&longs;t in Ma­ <lb/>thematicis &longs;copum non attigi&longs;&longs;e, vt alij pr&ecedil;&longs;ertim Peripa­ <lb/>tetici facilius ea inuenire, <expan abbr="atq;">atque</expan> de ij&longs;dem iudicium ferre <lb/>po&longs;&longs;int. <lb/><arrow.to.target n="table3"/></s></p><table><table.target id="table3"/><row><cell><emph type="italics"/>121<emph.end type="italics"/></cell><cell><emph type="italics"/>Nvmero marginali: vbi ait plura Octaedra, &longs;eu Pyramides re-plere locum: in quo omnes pariter expo&longs;itores lap&longs;i &longs;unt.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>124<emph.end type="italics"/></cell><cell><emph type="italics"/>Latitudinem figura, ait, cau&longs;am e&longs;&longs;e &longs;upernatationis. & aquam re&longs;i&longs;tere &longs;impliciter diui&longs;ioni.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>136<emph.end type="italics"/></cell><cell><emph type="italics"/>Cometas in &longs;uprema aeris regione collocat; cuius contrarium ibi line ari demonctratione ostenditur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>147<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Tanaim, & Indum oriri ex monte Paropami&longs;&longs;o. & c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>148<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait, tertia parte noctis Cauca&longs;i verticem illuminari à Sole.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>149<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium ex Pyreneo monte defluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>150<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait fluuium quendam non minorem Rhodano in Liguria ab&longs;orberi, & iterum egredi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>152<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Rubrum mare parum Atlantico Oceano commi&longs;ceri.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>159<emph.end type="italics"/></cell><cell><emph type="italics"/>Zonam torridam inhabit abilem exi&longs;timat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>164<emph.end type="italics"/></cell><cell><emph type="italics"/>Putat Iridis angulos non po&longs;&longs;e vnum &longs;upra alterum collocari, &longs;ed tantummodo in orbem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>182<emph.end type="italics"/></cell><cell><emph type="italics"/>Rationes, quas in Parelij dubitationibus affert, videntur inanes.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>236<emph.end type="italics"/></cell><cell><emph type="italics"/>In &longs;ubducendo cœle&longs;tium orbium numero, memoria labitur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>243<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait lineam O L, &longs;uperare lineam L R, quantitate P L, vt in figura.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>245<emph.end type="italics"/></cell><cell><emph type="italics"/>Remum ad vectem primi generis reducit.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>246<emph.end type="italics"/></cell><cell><emph type="italics"/>Temonem nauis reducit ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>247<emph.end type="italics"/></cell><cell><emph type="italics"/>In motu Remi collato cum motu Nauigij, à Nonio erroris manifestè arguitur. eodem modo in motu Temonis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>250<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait maioribus trochleis, aut rotulis facilius onera &longs;ubleuari.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>256<emph.end type="italics"/></cell><cell><emph type="italics"/>Reducit cuneum ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>270<emph.end type="italics"/></cell><cell><emph type="italics"/>Cur res in vorticibus ad medium ferantur, veram cau&longs;am a&longs;signa-re non videtur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>275<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium altero ramo in Mediterraneum, altero in Pontum effluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>293<emph.end type="italics"/></cell><cell><emph type="italics"/>Negat Araneum filum ab intrin&longs;eco emittere, Democritum iniuria refellens. quamuis hoc vltimum ad Phy&longs;icum pertineat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>403<emph.end type="italics"/></cell><cell><emph type="italics"/>Problem. 2. &longs;ect. 2 3. non benè videtur a&longs;signare cau&longs;am variæ im-mer&longs;ionis nauigij.<emph.end type="italics"/></cell></row></table><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.032.jpg"/><!--blank page --></section></front><body><chap> <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.033.jpg" pagenum="33"/> <p type="head"><s id="id.000668">LOCA <lb/>MATHEMATICA <lb/>EX LIBRO <lb/>PRÆDICAMENTORVM <lb/>Per ordinem declarata.</s></p><p type="main"> <s id="id.000669"><arrow.to.target n="marg1"/></s></p><p type="margin"> <s id="id.000670"><margin.target id="marg1"/>1</s></p> <p type="main"> <s id="id.000671">Ex c. <!-- REMOVE S-->3. De his, quæ ad aliquid. </s> <s id="id.000672">Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita <lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g. <lb/><!-- REMOVE S-->angulus B A C, vbi ait <emph type="italics"/>(Scientia verò &longs;i non &longs;it, <lb/>nihil probibet e&longs;&longs;e &longs;cibile, vt circuli quadratura, &longs;i e&longs;t &longs;cibilis, <lb/>&longs;cientia quidem eius nondum e&longs;t)<emph.end type="italics"/> Cum velit Ari&longs;t. o&longs;tendere, <lb/>nó omnia correlatiua &longs;imul e&longs;&longs;e natura, id de &longs;cibili, & &longs;cien­ <lb/>tia variè probat, præ&longs;ertim verò, quia multa &longs;int &longs;cibilia, <lb/>quæ tamen nondum &longs;ciantur, vt patet, inquit, in Quadratu­ <lb/>ra circuli, & &longs;cientia ip&longs;ius, quia quamuis ip&longs;a circuli quadratura &longs;it &longs;cibi­ <lb/>lis, nondum tamen &longs;imul cum ip&longs;a, &longs;cientia illius extat. </s> <s id="id.000673">Quæ vt perfectè <lb/>intelligantur, &longs;ciendum e&longs;t, quadraturam circuli, quæ à Græcis tetrago­ <lb/>ni&longs;mus dicitur, nihil aliud e&longs;&longs;e, quàm propo&longs;ito cuilibet circulo exhibere <lb/>quadratum æquale. </s> <s id="id.000674">Quæ æqualitas debet intelligi de areis, &longs;eu &longs;patijs, ita <lb/>vt area circuli, &longs;eu &longs;patium illud, &longs;iue &longs;uperficies illa circularis, &longs;it æqualis <lb/>areæ, &longs;eu &longs;uperficiei quadratæ. </s> <s id="id.000675">Qua in re plurimi decipiuntur exi&longs;timantes <lb/>per quadraturam cir culi inquiri æqualitatem linearum, ita vt circumferen­ <lb/>tia circuli debeat e&longs;&longs;e æqualis ambitui, &longs;eu quatuor lateribus quadrati: <lb/>quod omnino fal&longs;um e&longs;t.</s></p><p type="main"> <s id="id.000676">Quadratio porrò circuli dupliciter proponi pote&longs;t, vel tanquam Theo­ <lb/>rema, vel tanquam Problema <emph type="italics"/>(theorema autem e&longs;t propo&longs;itio, in qua nihil fa­ <lb/>ciendum proponitur; problema verò aliquid fieri expo&longs;cit)<emph.end type="italics"/> neutrum autem tem­ <lb/>pore Ari&longs;t. erat adinuentum nam theorema inuentum e&longs;t po&longs;t ip&longs;um ducen­ <lb/>tis circiter annis ab Archimede: problema verò nondum à quoquam per­ <lb/>fectè potuit reperiri. </s> <s id="id.000677">qua di&longs;tinctione &longs;aluari po&longs;&longs;unt nonnulli, vt Boetius <lb/>hocloco, qui aiunt, &longs;e vidi&longs;&longs;e Demon&longs;trationem quadraturæ huius, &longs;i nimi­ <lb/>rum intelligant theorema. </s> <s id="id.000678">& alij etiam verum a&longs;&longs;erunt, dum negant hacte­ <lb/>nus repertam e&longs;&longs;e, &longs;i nimirum de problemate loquantur, theorema Archi­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.034.jpg" pagenum="34"/>medis e&longs;t propo&longs;itio prima acuti&longs;&longs;imi libelli de Dimen&longs;ione circuli; e&longs;t au­ <lb/>tem huiu&longs;modi. </s> <s id="id.000679">Quilibet circulus æqualis e&longs;t triangulo rectangulo, cuius <lb/>quidem &longs;emidiameter vni laterum, quæ circa rectum angulum &longs;unt, ambi­ <lb/>tus verò ba&longs;i eius e&longs;t æqualis.</s></p><figure id="id.009.01.034.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.034.1.jpg" place="text"/><p type="main"> <s id="id.000680">Sit, v.g. <!-- REMOVE S-->datus circulus, cuius &longs;emidiameter A B; & fit trian gulum rectangu­ <lb/>lum A B C, cuius angulus B, &longs;it rectus, & latus B A, <expan abbr="con&longs;titu&etilde;s">con&longs;tituens</expan> angulum re­ <lb/>ctum B, cum ba&longs;i B C, &longs;it æquale &longs;emidiametro A B; ba&longs;is verò B C, &longs;it æqua­ <lb/>lis peripheriæ eiu&longs;dem circuli dati. </s> <s id="id.000681">demon&longs;trat iam ibi Archimedes acuta <lb/>æquè, ac euidenti demon&longs;tratione triangulum i&longs;tud æquale e&longs;&longs;e circulo illi. <lb/></s> <s id="id.000682">quod perinde e&longs;t, ac &longs;i o&longs;tendi&longs;&longs;et cuinam quadrato &longs;it æqualis, cum per vl­ <lb/>timam 2. Eucl. <!-- REMOVE S-->po&longs;&longs;imus triangulo huic quadratum æquale con&longs;truere, quod <lb/>con&longs;equenter dato circulo æquale erit. </s> <s id="id.000683">Quod &longs;i in modum Problematis ita <lb/>proponatur: Dato circulo æquale quadratum con&longs;truere, nondum inuenta <lb/>e&longs;t ratio, quæ demon&longs;tratione confirmetur, qua id geometricè penitus, hoc <lb/>e&longs;t ad æqualitatem mathematicam, &longs;eu exacti&longs;&longs;imam effici po&longs;&longs;it, <expan abbr="tota&qacute;">totaque</expan>; dif­ <lb/>ficultas po&longs;ita e&longs;&longs;e videtur in inue&longs;tigando, quonam modo exhibeamus li­ <lb/>neam rectam B C, æqualem peripheriæ circuli dati. </s> <s id="id.000684">quam nullus hactenus <lb/>geometricè illi æqualem potuit exhibere, <expan abbr="atq;">atque</expan> exhibita <expan abbr="euid&etilde;ti">euidenti</expan> demon&longs;tra­ <lb/>tione comprobare; Quamuis Archimedes acumine &longs;anè mirabili in lib. de <lb/>lineis &longs;piralibus, eam <expan abbr="quoq;">quoque</expan> theorematicè, non tamen problematicè inue­ <lb/>&longs;tigauit. </s> <s id="id.000685">nam propo&longs;itione 18. illius <expan abbr="admirãdi">admirandi</expan> operis inuenit lineam rectam <lb/>æqualem circumferentiæ primi circuli &longs;piralis lineæ; propo&longs; verò 19. repe­ <lb/>rit aliam rectam æqualem circumferentiæ &longs;ecundi circuli. </s> <s id="id.000686">tu ip&longs;um con&longs;ule, <lb/>&longs;i admirandarum rerum contemplatione delectaris. </s> <s id="id.000687">Multa hac de re Pap­ <lb/>pus Alexandrinus lib. 4. Math. coll. </s> <s id="id.000688">& Ioannes Buteo vnico volumine om­ <lb/>nes quadraturas tain pri&longs;corum, quam recentiorum <expan abbr="cõprehen&longs;us">comprehen&longs;us</expan> e&longs;t. </s> <s id="id.000689">Qua­ <lb/>re qui plura cupit, eos adeat; nos tamen infra &longs;uis locis explicabimus tres <lb/>illas celebres antiquorum Antiphontis, Bri&longs;&longs;onis, & Hippocratis quadra­ <lb/>turas, quamuis fal&longs;as, <expan abbr="quarũ">quarum</expan> &longs;æpe meminit Ari&longs;t. & alij. </s> <s id="id.000690">&longs;olet autem à non­ <lb/>nullis di&longs;putari, vtrum quadratura i&longs;ta problematica &longs;it po&longs;&longs;ibilis, nec ne, <lb/>cum videant eam à nemine, quamuis diu magno labore perqui&longs;itam, hacte­ <lb/>nus adinuentam e&longs;&longs;e. </s> <s id="id.000691">ego quidem e&longs;&longs;e po&longs;&longs;ibilem exi&longs;timo, quis enim dubi­ <lb/>tare pote&longs;t, po&longs;&longs;e exi&longs;tere quadratum æquale circulo propo&longs;ito? </s> <s id="id.000692">Quod &longs;i po­ <lb/>te&longs;t fieri, quare non etiam demon&longs;trari? </s> <s id="id.000693">pr&ecedil;fertim cum videamus ab Archi­ <lb/>mede iam inuentam e&longs;&longs;e, quatenus Theorema e&longs;t. </s> <s id="id.000694">& præterea con&longs;tet, Hip­ <lb/>pocratem quadra&longs;&longs;e lunulam, vt &longs;uo loco dicemus, & Archimedem in libel­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.035.jpg" pagenum="35"/>lo de quadratura Paraboles, quadra&longs;&longs;e ip&longs;am Parabolem, quæ tamen duæ fi­ <lb/>guræ, lunula &longs;cilicet, & parabola &longs;unt curuilineæ.</s></p><p type="main"> <s id="id.000695"><arrow.to.target n="marg2"/></s></p><p type="margin"> <s id="id.000696"><margin.target id="marg2"/>2</s></p><p type="main"> <s id="id.000697">Ex cap. de Priori <emph type="italics"/>(in &longs;cientijs demon&longs;tratiuis e&longs;t prius, & po&longs;terius ordine, <lb/>elementa enim priora &longs;unt ijs, quæ de&longs;cribuntur, nam principia prior a &longs;unt theore­ <lb/>matibus ordine)<emph.end type="italics"/> verba illa, nam principia, &c. </s> <s id="id.000698">quæ non &longs;unt in antiqua tran­ <lb/>&longs;latione de&longs;ump&longs;imus ex ca&longs;tigati&longs;&longs;imo græco codice editionis Francfor­ <lb/>dien&longs;is, propterea quod totum hunc locum declarant; &longs;unt autem i&longs;ta, <lb/><foreign lang="greek">ai/gar arxai/ pro/terai tw_n qewrhma/twn th ta/xei.</foreign> per &longs;cientias autem demon&longs;tra­ <lb/>tiuas intelligendas e&longs;&longs;e hoc loco ip&longs;as Mathematicas ex eo patet, quod illis <lb/>a&longs;&longs;ignet Ari&longs;t. De&longs;criptiones; nam hoc verbo, De&longs;criptiones, &longs;eu figuratio­ <lb/>nes, &longs;olet ip&longs;e Mathematicas Demon&longs;trationes innuere, quod in ip&longs;is figu­ <lb/>rationes, & De&longs;criptiones adhibeantur, vt alijs locis patebit: idcirco ver­ <lb/>ba illa à nobis addita ex græco, optim è <expan abbr="præced&etilde;tia">præcedentia</expan> exponunt, cum per ele­ <lb/>menta intelligantur principia, qualia &longs;unt initio Euclidis, & per de&longs;criptio­ <lb/>nes exponant theoremata. </s> <s id="id.000699">quod autem principia illa ordine priora &longs;int de­ <lb/>mon&longs;trationibus, &longs;iue ip&longs;as præcedant, ex ip&longs;a primi Euclidis in&longs;pectione <lb/>patere pote&longs;t.</s></p><p type="main"> <s id="id.000700"><arrow.to.target n="marg3"/></s></p><p type="margin"> <s id="id.000701"><margin.target id="marg3"/>3</s></p><p type="main"> <s id="id.000702">Ex cap. de motu <emph type="italics"/>(Quadratum augetur Gnomone circumpo&longs;ito)<emph.end type="italics"/> Gnomon vox <lb/>græca inter alia &longs;ignificat in&longs;trumentum illud, quod Latini tum amu&longs;&longs;im, <lb/><figure id="id.009.01.035.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.035.1.jpg" place="text"/> <lb/>tum normam appellant, Itali verò, Squadra, ad <lb/>cuius &longs;imilitudinem Geometræ denominarunt fi­ <lb/>guram quandam, &longs;eu portionem cuiu&longs;uis paralle­ <lb/>logrammi, vt videre e&longs;t in definitione &longs;ecunda <lb/>2. elem. </s> <s id="id.000703">& in præ&longs;enti figura, in qua quadratum <lb/>A B C D, circumpo&longs;ito gnomone E F G, augetur, <lb/>& fit maius quadratum H B I L.<!-- KEEP S--></s></p><p type="main"> <s id="id.000704">Idem etiam verum e&longs;t in quadrato arithmeti­ <lb/>co, &longs;iue in numero quadrato: is enim pariter ad­ <lb/>dito Gnomone augetur. </s> <s id="id.000705">i. </s> <s id="id.000706">addito numero impari. <lb/></s> <s id="id.000707">quemadmodum infra 3. Phy&longs;. tex. <!-- REMOVE S-->26. fusè explicabimus.</s> </p><p type="head"> <s id="id.000708"><emph type="italics"/>Ex Primo Priorum re&longs;olutoriorum.<emph.end type="italics"/></s></p> </chap><chap> <p type="main"> <s id="id.000709"><arrow.to.target n="marg4"/></s></p><p type="margin"> <s id="id.000710"><margin.target id="marg4"/>4</s></p><p type="main"> <s id="id.000711">Aliquorum opinio e&longs;t, Ari&longs;totelem ho&longs;ce libros appella&longs;&longs;e re&longs;olu­ <lb/>torios, quod per illos doceat &longs;yllogi&longs;mum, ac demon&longs;trationem <lb/>iam factam in &longs;ua immediata principia re&longs;oluere, quam opinio­ <lb/>nem meum non e&longs;t, nunc refellere. </s> <s id="id.000712">per&longs;ua&longs;um tamen mihi e&longs;t, rem <lb/>multo aliter &longs;e habere, veram rationem huius tituli petendam e&longs;&longs;e ex peni­ <lb/>tiori Mathematicorum eruditione. </s> <s id="id.000713">Sciendum <expan abbr="itaq;">itaque</expan> id, quod tradit Pappus <lb/>Alex. initio &longs;eptimi Mathem. collect. </s> <s id="id.000714">antiqui&longs;&longs;imos videlicet Geometras, <lb/>Euclidem, Apollonium Pergæum, & Ari&longs;t&ecedil;um &longs;crip&longs;i&longs;&longs;e libros de re&longs;olutio­ <lb/>ne, in quibus ars tradebatur, qua propo&longs;ito quouis theoremate, aut proble­ <lb/>mate po&longs;&longs;ent facile ex eo, tanquam vero accepto inue&longs;tigare aliquam veri­ <lb/>tatem, per quam deinde componerent illius, quod quærebatur, Demon&longs;tra­ <lb/>tionem; inue&longs;tigationem illam appellabant re&longs;olutionem: compo&longs;itionem <lb/>verò nominabant di&longs;cur&longs;um <expan abbr="illũ">illum</expan>, quo ex vero illo per re&longs;olutionem inuento, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.036.jpg" pagenum="36"/>o&longs;tendebant conclu&longs;ionem. </s> <s id="id.000715">Porrò Diogenes Laert. <!-- REMOVE S-->huius re&longs;olutionis in­ <lb/>uentorem facit Platonem: à quo eam Leodamas Tha&longs;ius didicit, cuius be­ <lb/>neficio, pluries deinde Geometricas demon&longs;trationes adinuenit. </s> <s id="id.000716">definitio <lb/><expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> e&longs;t a pud Euclidem ad primam propo&longs;. </s> <s id="id.000717">13. Elem. iuxta tran&longs;latio­ <lb/>nem Zamberti, & Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri­ <lb/>mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan­ <lb/>quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. </s> <s id="id.000718">&longs;unt præterea fre­ <lb/>quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap­ <lb/>pi. </s> <s id="id.000719">extat adhuc liber Datorum Euclidis, qui geometricis re&longs;olutionibus in­ <lb/>&longs;eruiebat. </s> <s id="id.000720">vtinam extarent etiam alij de re&longs;olutione, quorum auxilio non <lb/>tantopere recentiores Mathematici in inueniendis Demo&longs;trationibus la­ <lb/>borarent; hanc re&longs;olutionem, &longs;ic Pappus fu&longs;ius, quam Euclides explicat; <lb/>re&longs;olutio e&longs;t via à quæ&longs;ito tanquam conce&longs;&longs;o per ea, quæ exip&longs;o con&longs;equun­ <lb/>tur ad aliquod certum, & conce&longs;&longs;um: in re&longs;olutione enim id, quod quæritur <lb/>tanquam factum, & verum &longs;upponentes, quid ex hoc &longs;equatur, con&longs;idera­ <lb/>mus, quou&longs;que incidamus in aliquod iam cognitum, vel quod &longs;it è numero <lb/>principiorum. </s> <s id="id.000721">Quod quidem erat fignum euidens, quæ&longs;itum quoque verum <lb/>e&longs;&longs;e. </s> <s id="id.000722">eadem omnino habet Proclus in comm. <!-- REMOVE S-->ad &longs;extam primi elem. </s> <s id="id.000723">Quod <lb/>porrò Ari&longs;t. ip&longs;e hanc re&longs;olutionem Mathematicam cognouerit e&longs;&longs;e medij <lb/>inqui&longs;itionem manife&longs;tum e&longs;t ex cap. 3. lib. 3. Ethyc. <!-- REMOVE S-->vbi &longs;ic ait <emph type="italics"/>(Qui enim <lb/>con&longs;ultat, quærere videtur, & re&longs;oluere prædicto modo, quemadmodum de&longs;igna­ <lb/>tiones)<emph.end type="italics"/> vbi per de&longs;ignationes intelligit Geometricas demon&longs;trationes, vt <lb/>&longs;upra innuimus, & infra probabimus; cum ergo con&longs;ultatio nihil aliud &longs;it, <lb/>quam medij idonei ad finem in rebus agendis inqui&longs;itio, eamque dicat e&longs;&longs;e <lb/>&longs;imilem re&longs;olutioni Geometricæ, manife&longs;tum e&longs;t, ip&longs;am <expan abbr="quoq;">quoque</expan> re&longs;olutionem <lb/>e&longs;&longs;e medij in rebus &longs;peculatiuis idonei perue&longs;tigationem. </s> <s id="id.000724">Exi&longs;timo igitur <lb/>cum docti&longs;&longs;imis Zabarella, Burana, Toleto, & alijs, Ari&longs;totilem non &longs;olum <lb/>hanc &longs;uam logicam ad mathematicarum &longs;cientiarum typum compegi&longs;&longs;e, <lb/>verum potius imitatum e&longs;&longs;e opus illud Euclidis de re&longs;olutione, atque ex eo <lb/>non &longs;olum plurima exempla Geometrica, verum etiam titulum de&longs;ump&longs;i&longs;&longs;e, <lb/>præ&longs;ertim cum argumentum e&longs;&longs;et ferè idem vtrobique, &longs;ed Ari&longs;t. intentio <lb/>fuerit accommodare re&longs;olutionem omnibus <expan abbr="&longs;ci&etilde;tijs">&longs;cientijs</expan>; Euclidis verò, & alio­ <lb/>rum Geometriæ &longs;oli. </s> <s id="id.000725">hinc patere pote&longs;t, cur hi libri re&longs;olutorij in&longs;cribantur, <lb/>quod &longs;cilicet tradunt methodum, qua valeamus quæ&longs;itum quoduis re&longs;olue­ <lb/>re, ide&longs;t, ex quæ&longs;ito tanquam vero inue&longs;tare aliquam veritatem, per quam <lb/>deinde propo&longs;itæ quæ&longs;tionis rationem methodo compo&longs;itiua reddamus. </s> <s id="id.000726">Et <lb/>verò cum reliquas appellationes Problematis, Theorematis, Propo&longs;itionis, <lb/>definitionum, po&longs;tulatorum, axiomatum, & alia huiu&longs;modi ex Geometri­ <lb/>cis ad omnes &longs;cientias tran&longs;tulerit, quid ni etiam re&longs;olutionem? </s> <s id="id.000727">maximè <lb/>verò, quia &longs;i horum lib. intentio e&longs;&longs;et docere iam factum &longs;yllogi&longs;mum in &longs;ua <lb/>principia re&longs;oluere, parum e&longs;&longs;et vtilis; imò nec vtilis, &longs;ed &longs;uperfluum quid. <lb/></s> <s id="id.000728">at verò vbinam docuit hanc re&longs;olutionem? </s> <s id="id.000729">profecto nullibi. </s> <s id="id.000730">quid opus e&longs;t <lb/>iam factum &longs;yllogi&longs;mum re&longs;oluere? </s> <s id="id.000731">at verò propo&longs;itam quæ&longs;tionem re&longs;ol­ <lb/>uere veterum mathematicorum more, hoc opus, hic labor e&longs;t.</s></p><p type="main"> <s id="id.000732">Hanc porrò re&longs;olutionem attendendam e&longs;&longs;e primò penes formam, quam <lb/>docet primis duobus analyticis; &longs;ecundò penes materiam, quam tradit duo­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.037.jpg" pagenum="37"/>bus vltimis, non prætereundum. </s> <s id="id.000733">reliquas duas logicæ partes, Topicam &longs;ci­ <lb/>licet, & Elenchos, quæ &longs;yllogi&longs;mos probabilem, & apparentem docent, no­ <lb/>luit appellare re&longs;olutorios, quamuis inuentionem mediorum doceant, quia <lb/>iam mos i&longs;te inoleuerat apud Philo&longs;ophos, & Mathematicos, vt illa &longs;ola <lb/>pars, quæ ex materia nece&longs;&longs;aria doceret &longs;yllogi&longs;mum demon&longs;tratiuum con­ <lb/>&longs;truere, diceretur re&longs;olutio: cum Mathematici, qui primi de re&longs;olutione <lb/>&longs;crip&longs;erunt, talem materiam &longs;olum con&longs;iderent.</s></p><p type="main"> <s id="id.000734"><arrow.to.target n="marg5"/></s></p><p type="margin"> <s id="id.000735"><margin.target id="marg5"/>5</s></p><p type="main"> <s id="id.000736">Ex cap. 23. &longs;ecti primi lib. 1. <emph type="italics"/>(Vt quod diameter incommen&longs;urabilis eo, quod <lb/>imparia æqualia paribus fiant, &longs;i fuerit po&longs;ita commen&longs;ur abilis. </s> <s id="id.000737">æqualia igitur fieri <lb/>imparia paribus ratiocinantur, diametrum vtrò incommen&longs;urabilem e&longs;&longs;e ex &longs;uppo­ <lb/>&longs;itione <expan abbr="mon&longs;trãt">mon&longs;trant</expan>, quoniam fal&longs;um accidit propter contradictionem)<emph.end type="italics"/> Euclides pri­ <lb/>mis duabus definitionibus 10. elem. </s> <s id="id.000738">definit, quæ nam &longs;int magnitudines <lb/>commen&longs;. </s> <s id="id.000739">& quæ incommen&longs;. </s> <s id="id.000740">&longs;ic; commen&longs;. </s> <s id="id.000741">magnitudines dicuntur, quas <lb/><figure id="id.009.01.037.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.037.1.jpg" place="text"/> <lb/>eadem men&longs;ura metitur, vt &longs;i fuerint duæ magnitu­ <lb/>dines, A, & B, quas eadem men&longs;ura C, ide&longs;t quan­ <lb/>titas C, metiatur, ide&longs;t <expan abbr="quãtitas">quantitas</expan> C, applicata quan­ <lb/>titati A, & per ip&longs;am aliquoties replicata ip&longs;am ad­ <lb/>æquatè ab&longs;umat, vt &longs;i linea C, quinquies &longs;uper li­ <lb/>neam A, replicata eam præcisè, & perfectè omninò <lb/>adæquaret: & eadem linea C, applicata lineæ B, & &longs;uper illam ter, v.g. <!-- REMOVE S-->re­ <lb/>petita ip&longs;am con&longs;umeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, & proinde duas lineas <lb/>A, & B, e&longs;&longs;e comm. <!-- REMOVE S-->definit po&longs;tea <expan abbr="incomm&etilde;&longs;">incommen&longs;</expan> hoc modo, incomm, autem, qua­ <lb/>rum nullam contingit communem men&longs;uram reperiri; vt &longs;i duarum linea­ <lb/><figure id="id.009.01.037.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.037.2.jpg" place="text"/> <lb/>rum, A, B, nunquam po&longs;&longs;et reperiri aliqua men&longs;u­ <lb/>ra, quæ <expan abbr="vtranq;">vtranque</expan> adæquatè metiretur, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i linea <lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta <lb/>non adæquaret omnino <expan abbr="lineã">lineam</expan> B, &longs;ed deficeret, vel ex­ <lb/>cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia <lb/>men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue <lb/>minor ip&longs;a C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, e&longs;&longs;ent duæ illæ lineæ <lb/>incommen&longs;. </s> <s id="id.000742">Extare porrò tales lineas, & &longs;uperficies, & corpora, <expan abbr="ea&qacute;">eaque</expan>; quam­ <lb/>plurima, ac penè infinita ex 10. Elem. manife&longs;tum e&longs;t. </s> <s id="id.000743">inuentum autem hu­ <lb/>ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um <lb/>e&longs;t omni maius admiratione, cum nulla experientia, <expan abbr="nullus&qacute;">nullusque</expan>; effectus in ip­ <lb/>&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. </s> <s id="id.000744">Quapropter <lb/>non immeritò diuinus ille Plato lib. 7. de legib. </s> <s id="id.000745">huius a&longs;ymmetriæ ignora­ <lb/>tionem, adeo dete&longs;tatus e&longs;t, vt eam non hominum, &longs;ed &longs;uum, pecorumque <lb/>ignorantiam cen&longs;uerit. </s> <s id="id.000746">inter lineas incommen&longs;. </s> <s id="id.000747">&longs;unt diameter, & latus eiu&longs;­ <lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti <lb/><figure id="id.009.01.037.3.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.037.3.jpg" place="text"/> <lb/>e&longs;t lineola E, in præ&longs;enti quadrato, etiam&longs;i illam in <lb/>infinitum &longs;ubdiuidas, quæ <expan abbr="vtranq;">vtranque</expan> lineam, diame­ <lb/>trum &longs;cilicet A C, & latus quoduis ex quatuor, v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->latus B C, præcisè omnino metiatur. </s> <s id="id.000748">theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. eodem me­ <lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex <lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.038.jpg" pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm. <lb/><!-- REMOVE S-->deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t. fal&longs;um ratiocinatur, quod &longs;ci­ <lb/>licet idem numerus e&longs;&longs;et par, & impar, quod Ari&longs;t. &longs;ignificat, quando ait, <lb/>imparia æqualia paribus fiunt. </s> <s id="id.000749">ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi­ <lb/>ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. <!-- REMOVE S-->& proinde altera pars con­ <lb/>tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. <!-- REMOVE S-->vera a&longs;truitur. </s> <s id="id.000750">ex quibus &longs;atis videtur ex­ <lb/>plicari hic locus. </s> <s id="id.000751">videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis <lb/>ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. <!-- REMOVE S-->co&longs;tæ, nihil <lb/>aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione <lb/>nihil ineptius. </s> <s id="id.000752">Aduerte tandem figuram vulgatæ editionis e&longs;&longs;e ineptam, <lb/>cum habeat duo quadrata alterum &longs;uper diametro alterius, quorum maius <lb/>&longs;uperuacaneum e&longs;t.</s></p><p type="main"> <s id="id.000753"><arrow.to.target n="marg6"/></s></p><p type="margin"> <s id="id.000754"><margin.target id="marg6"/>6</s></p><p type="main"> <s id="id.000755">Et cap. 24. &longs;ecti primi libri primi <emph type="italics"/>(Sed magis efficitur manife&longs;tum in de&longs;cri­ <lb/>ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B, <lb/>A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. </s> <s id="id.000756">&longs;timans <lb/>æquales, qui &longs;emicirculorum, & rur&longs;us G, ip&longs;i D, non omnem a&longs;&longs;umens eum, qui &longs;e­ <lb/>cti. </s> <s id="id.000757">amplius ab æqualibus existentibus totis angulis, & ablatorum æquales e&longs;&longs;e re­ <lb/>tiquos E, F, quod ex principio petet, ni&longs;i acceperit ab æqualibus demptis æqualia <lb/>derelinqui.)<emph.end type="italics"/> Primum &longs;cias characteres vulgatæ editionis, vna cum figura ip­ <lb/>&longs;is re&longs;pondente, e&longs;&longs;e mendo&longs;os; propterea ex textu græco vtrunque corri­ <lb/>gendum putaui in hunc, quem vidi&longs;ti modum. </s> <s id="id.000758">Secundo, per de&longs;criptiones <lb/>Ari&longs;t. intelligere <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> Geometricas &longs;upra diximus, quod ex hoc <lb/>loco euidenter confirmatur, vbi manife&longs;tè loco de&longs;criptionis &longs;upponit li­ <lb/>nearem demon&longs;trationem. </s> <s id="id.000759">In hoc <expan abbr="itaq;">itaque</expan> exemplo vult Ari&longs;t. illud demon­ <lb/>&longs;trare, quod Euclides in 5. primi o&longs;tendit, alio tamen modo, &longs;cilicet I&longs;o&longs;ce­ <lb/>lium triangulorum, qui ad ba&longs;im &longs;unt anguli, inter &longs;e &longs;unt æquales. </s> <s id="id.000760">e&longs;t au­ <lb/>tem figura in omnibus textibus deprauata, quam &longs;ic puto <expan abbr="rè&longs;tītuendam">rè&longs;tintuendam</expan> e&longs;&longs;e <lb/>ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. </s> <s id="id.000761">&longs;it I&longs;o&longs;ce­ <lb/><figure id="id.009.01.038.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.038.1.jpg" place="text"/> <lb/>les C A B, cuius ba&longs;is C B, Dico angulos &longs;upra ba&longs;im, <lb/>in quibus literæ E F, e&longs;&longs;e inuicem æquales. </s> <s id="id.000762">facto centro <lb/>in A, de&longs;cribatur circulus A B C, tran&longs;iens per puncta <lb/>C B, iam &longs;ic. </s> <s id="id.000763">omnes anguli &longs;emicirculi &longs;unt æquales in­ <lb/>ter &longs;e, ergo anguli A C G, A B D, &longs;unt æquales. </s> <s id="id.000764">Præte­ <lb/>rea cùm anguli ciu&longs;dem &longs;ectionis &longs;int æquales ad inui­ <lb/>cem, erunt anguli &longs;ectionis C B D G, nimirum anguli, <lb/>in quibus &longs;unt G, & D, inter &longs;e æquales: <expan abbr="cum&qacute;">cumque</expan>; hi duo <lb/>anguli &longs;ectionis &longs;int partes <expan abbr="angulorũ">angulorum</expan> &longs;emicirculi A C G, <lb/>A B D, &longs;i illi ab his auferantur, auferuntur æquales anguli ab æqualibus an­ <lb/>gulis, ergo anguli, qui remanent, &longs;cilicet E, & F, erunt æquales, quod erat <lb/>demon&longs;trandum. </s> <s id="id.000765">hinc Ari&longs;t. infert manife&longs;tum e&longs;&longs;e oportere in omni &longs;yllo­ <lb/>gi&longs;mo, reperiri vniuer&longs;ales, & affirmatiuas propo&longs;itiones, vt Factum e&longs;t in <lb/>præcedenti aliter e&longs;&longs;et petitio principij. </s> <s id="id.000766">Quænam vero &longs;it æqualitas, quam <lb/>Geometræ con&longs;iderant, infra cap. 1. &longs;ecti 3. explicabicur.</s></p><p type="main"> <s id="id.000767"><arrow.to.target n="marg7"/></s></p><p type="margin"> <s id="id.000768"><margin.target id="marg7"/>7</s></p><p type="main"> <s id="id.000769">Ex cap. 2. &longs;ecti 2. lib. 1. <emph type="italics"/>(Secundum veritatem quidem ex ijs, quæ &longs;ecundum <lb/>veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus <lb/>&longs;ecundum opinionem)<emph.end type="italics"/> verba illa; ex ijs, quæ <expan abbr="&longs;ecundũ">&longs;ecundum</expan> veritatem de&longs;cribuntur <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.039.jpg" pagenum="39"/>ine&longs;&longs;e; &longs;ic græcè, <foreign lang="greek">e/a tw_n xata\ aleiq/ei/an diagegramme/non,</foreign> vbi manife&longs;tè vtitur <lb/>verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. &longs;ignificari <lb/>Geometricas demon&longs;trationes, nam eas opponit dialecticis &longs;yllogi&longs;mis, &longs;e­ <lb/>quentibus verbis, cum dixit (ad diale cticos autem &longs;yllogi&longs;mos ex propo&longs;i­ <lb/>tionibus &longs;ecundum opinionem) hac adhibita con&longs;ideratione, quam inter­ <lb/>pres non videtur adhibui&longs;&longs;e, &longs;en&longs;us huius loci non erit ob&longs;curus.</s></p><p type="main"> <s id="id.000770"><arrow.to.target n="marg8"/></s></p><p type="margin"> <s id="id.000771"><margin.target id="marg8"/>8</s></p><p type="main"> <s id="id.000772">Ex eodem loco paulo po&longs;t <emph type="italics"/>(Quare principia quidem, quæ &longs;ecundum <expan abbr="vnum-quodq;">vnum­ <lb/>quodque</expan> &longs;unt experimenti est tradere: dico autem, vt a&longs;trologicam experientiam <lb/>a&longs;trologicæ &longs;cientiæ: acceptis enim apparentibus <expan abbr="&longs;uffici&etilde;ter">&longs;ufficienter</expan>, ita inuentæ &longs;unt a&longs;tro­ <lb/>logicæ demonstrationes)<emph.end type="italics"/> Cum rationem tradat inueniendorum mediorum ad <lb/>quodlibet problema demon&longs;trandum; nunc docet, non omnia in &longs;eientijs <lb/>po&longs;&longs;e probari, aut demou&longs;trari: principia enim &longs;cientiarum <expan abbr="nõ">non</expan> demon&longs;tran­ <lb/>tur, &longs;ed &longs;ola experientia manife&longs;ta &longs;unt; vt patet in A&longs;tronomia, quæ ab ex­ <lb/>perientia &longs;ua &longs;olet &longs;tabilire principia: principijs autem <expan abbr="experim&etilde;to">experimento</expan> con&longs;ti­ <lb/>tutis ex ip&longs;is reliqua problemata <expan abbr="demon&longs;trãtur">demon&longs;trantur</expan>. </s> <s id="id.000773">duo autem &longs;unt apud a&longs;tro­ <lb/>nomos genera experimenti, primum dicitur Phænomena, ide&longs;t, <expan abbr="appar&etilde;tiæ">apparentiæ</expan>; <lb/>& &longs;unt ea, quæ vulgo omnibus patent, vt Solem oriri, & occidere; a&longs;tra fer­ <lb/>ri circulariter, diem augeri modo, modo minui: & his &longs;imilia. </s> <s id="id.000774">alterum ge­ <lb/>nus dicitur ob&longs;eruationes, quæ tantummodo a&longs;tronomiæ peritis per ob&longs;er­ <lb/>uationem innote&longs;cunt, vt Solem inæqualiter ferri proprio motu per Zodia­ <lb/>cum; aliquando maiorem, aliquando minorem videri; plures dies immo­ <lb/>rari citra æquatiorem in parte Zodiaci boreali, quam in altera vltra æqua­ <lb/>torem au&longs;trali. </s> <s id="id.000775">dies naturales e&longs;&longs;e inuicem inæquales, &c. </s> <s id="id.000776">ex quibus deinde <lb/>ponunt eccentricos, & augem, ad &longs;aluandas tum apparentias, tum ob&longs;erua­ <lb/>tiones; & hac ratione a&longs;trologica &longs;cientia paulatim reperta e&longs;t, ac in dies <lb/>reperitur.</s></p><p type="main"> <s id="id.000777"><arrow.to.target n="marg9"/></s></p><p type="margin"> <s id="id.000778"><margin.target id="marg9"/>9</s></p><p type="main"> <s id="id.000779">Ex cap. 3. &longs;ecti 2. lib. 1. <emph type="italics"/>(Vt an ne diameter incomm.)<emph.end type="italics"/> loquitur de a&longs;ymme­ <lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in <lb/>cap. 23. &longs;ecti 1. huius libri; quæ &longs;i repetantur, optimè hunc <expan abbr="locũ">locum</expan> declarant.</s></p><p type="main"> <s id="id.000780"><arrow.to.target n="marg10"/></s></p><p type="margin"> <s id="id.000781"><margin.target id="marg10"/>10</s></p><p type="main"> <s id="id.000782">Ex cap. 1. &longs;ecti 3. lib. 1. <emph type="italics"/>(Sit A, duo recti, in quo B, triangulus, in quo C, <lb/>æquicrus, ip&longs;i <expan abbr="itaq;">itaque</expan> C, ine&longs;t A. per B; ip&longs;i vero B, non amplius per aliud, per &longs;e <lb/>namque triangulus habet duos rectos)<emph.end type="italics"/> nullum aliud exemplum tam frequenter <lb/>v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu­ <lb/>lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an­ <lb/>gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem. quod, vt probè intelliga­ <lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, & <lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio­ <lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt <lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="illũ">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="id.009.01.039.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.039.1.jpg" place="text"/> <lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s> <s id="id.000783">&longs;olum igitur duo anguli erunt æqua­ <lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon­ <lb/>giores lineis alterum angulum con&longs;tituentibus, quia <lb/>quantitas anguli non attenditur penes longitudinem <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.040.jpg" pagenum="40"/><expan abbr="linearũ">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i <lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum­ <lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur, <lb/>erit &longs;emper eadem quantitas anguli A. <!-- KEEP S--></s> <s id="id.000784">Aduertendum præterea rationem <lb/>anguli non po&longs;&longs;e &longs;aluari in &longs;olo puncto A, in quo lineæ concurrunt, &longs;ed ne­ <lb/>ce&longs;&longs;ariam e&longs;&longs;e aliquam quantitatem, quamuis exiguam, linearum A B, A C. <lb/><!-- KEEP S--></s> <s id="id.000785">Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite­ <lb/>ras, &longs;emper literam illam e&longs;&longs;e medio loco proferendam, quæ ad acumen ip­ <lb/>&longs;um po&longs;ita e&longs;t, vt in &longs;uperiori, litera A, debet &longs;emper media proferri, dicen­ <lb/>do angulum B A C, &longs;iue C A B, <expan abbr="nũquam">nunquam</expan> tamen licet dicere angulum A C B, <lb/>vel C B A. <!-- KEEP S--></s> <s id="id.000786">Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita <lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g. <lb/><!-- REMOVE S-->angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu­ <lb/><figure id="id.009.01.040.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.040.1.jpg" place="text"/> <lb/>li partiales B A D, D A C, erunt æquales totali angulo <lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua­ <lb/>les. </s> <s id="id.000787">pariter tres anguli po&longs;&longs;unt æquari & vni, & duobus <lb/>alijs angulis, quando nimirum a cumina, &longs;iue mucrones il­ <lb/>li &longs;imul ad vnum punctum con&longs;tituti <expan abbr="adæquar&etilde;tur">adæquarentur</expan> mucro­ <lb/>niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres <lb/>&longs;unt pares, v.g. <!-- REMOVE S-->&longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure id="id.009.01.040.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.040.2.jpg" place="text"/> <lb/>quos linea perpendicularis D E, facit cum li­ <lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G, <lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua­ <lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones <lb/>trianguli fimul &longs;umpti, & vniti ad punctum <lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum <lb/><figure id="id.009.01.040.3.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.040.3.jpg" place="text"/> <lb/>rectorum coeunt, congruent omnino duobus <lb/>prædictis angulis rectis, &longs;iue duobus illis mu­ <lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti­ <lb/>tuent lineam rectam F E G, &longs;icuti faciunt <lb/>etiam duo illi anguli recti; &longs;iue etiam dica­ <lb/>mus, occupabunt idem &longs;patium omninò, & <lb/>præcisè, quod occupant duo recti: v.g. <!-- REMOVE S-->&longs;i mucro B, ibi poneretur, faceret <lb/>angulum F E H, & &longs;i ibi iuxta ip&longs;um apponeretur mucro A, faceret angulum <lb/>H E I. quem &longs;i deinceps &longs;ub&longs;equetur reliquus angulus C, con&longs;titueret <expan abbr="reli-quũ">reli­ <lb/>quum</expan> angulum I E G. iam, vt vides, illi tres anguli ad E, tran&longs;lati, &longs;unt æqua­ <lb/>les duobus rectis ad E, pariter con&longs;titutis, cum illi tres fiant partes duorum <lb/>rectorú, vel quia occupant idem &longs;patium, vel eandem lineam rectam F E G, <lb/>con&longs;tituant. </s> <s id="id.000788">habet igitur omne triangulum &longs;iue &ecedil;quilaterum, &longs;iue &longs;calenum, <lb/>&longs;iue I&longs;o&longs;celes mirabilem hanc proprietatem, vt tres anguli, cuiu&longs;uis trian­ <lb/>guli &longs;int æquales duobus rectis angulis. </s> <s id="id.000789">Quam demon&longs;trationem primi om­ <lb/>nium Pythagorici perfecerunt, vt refert Proclus ad 32. primi Elem. Eucli­ <lb/>des deinde ibidem aliter, quam Pythagorici idem demon&longs;trauit. </s> <s id="id.000790">Quod &longs;i <lb/>quis huius rei <expan abbr="experi&etilde;tiam">experientiam</expan> aliquam velit; etiam&longs;i non exactam (cum æqua­ <lb/>litas mathematica non cadat &longs;ub &longs;en&longs;um, &longs;ed &longs;ola intelligentia percipiatur, <lb/>quippe quæ in materia intelligibili, non autem &longs;en&longs;ibili ver&longs;atur, & cuius <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.041.jpg" pagenum="41"/>æqualitas nullum di&longs;crimen, quantumuis minimum admittat, quod &longs;en&longs;ui <lb/>vitare ob &longs;ui imperfectionem non licet: vnde inter eæ, quæ mathematicè <lb/>&longs;unt æqualia, nullus intellectus aliquam valeat reperire differentiam) &longs;umat <lb/>inquam triangulum quodpiam materiale, vt ex charta, quantum fieri po­ <lb/>ce&longs;t perfectum, deinde ducat lineam vnam perpendicularem &longs;uper aliam, <lb/>quæ &longs;cilicet faciat, cum illa duos angulos rectos. </s> <s id="id.000791">po&longs;tea ab&longs;cindat tres an­ <lb/>gulos trianguli materialis, <expan abbr="eos&qacute;">eosque</expan>; ita &longs;imul componat, vt mucrones illorum <lb/>&longs;int vniti, & contigui ad punctum lineæ perpendicularis cum altera, vti e&longs;t <lb/>in &longs;uperiori figura punctnm E; & illicò apparebit tres illos angulos mate­ <lb/>riales obtegere adæquatè totum illud &longs;patium duorum rectorum, quos per­ <lb/>pendicularis con&longs;tituit. </s> <s id="id.000792">Hoc autem experiri poteris in diuer&longs;is admodum <lb/>triangulis Scalenis, Rectangulis, I&longs;o&longs;celibus, Aequilateris, &c. </s> <s id="id.000793">non &longs;ine de­ <lb/>lectatione, atque hic e&longs;t &longs;en&longs;us illorum verborum, omnis triangulus habet <lb/>tres &ecedil;quales duobus rectis. </s> <s id="id.000794">Ab&longs;tineo à demon&longs;trationibus geometricis, quo­ <lb/>niam ij, qui Mathematicis &longs;unt imbuti, no&longs;tra hac opera parum indigent. <lb/></s> <s id="id.000795">&longs;i quis tamen volet, con&longs;ulat 32. primi Elem. <!-- KEEP S--></s> <s id="id.000796">Ex hac igitur declaratione <lb/>licet cogno&longs;cere nonnullos ageometretos locum hunc, & &longs;imiles &longs;ub&longs;equen­ <lb/>tes non &longs;atis intelligere, dicentes, nihil aliud verba illa Ari&longs;t. velle &longs;ignifi­ <lb/>care, quàm omnem triangulum habere tres angulos, quod inquiunt, noti&longs;­ <lb/>&longs;imum e&longs;t. </s> <s id="id.000797">Sed &longs;i incidant in &longs;equentia; æquales duobus rectis, tunc, cum <lb/>hæc non intelligant, ab&longs;tinent etiam à priorum declaratione, quibus præ­ <lb/>mi&longs;&longs;is facile e&longs;t Ari&longs;t. textum percipere. </s> <s id="id.000798">&longs;it A, duo recti, ide&longs;t, duo anguli <lb/>recti &longs;int pa&longs;&longs;io demon&longs;tranda, in quo B, triangulus, in quo C, æquicrus. </s> <s id="id.000799">ip&longs;i <lb/>itaque C, ide&longs;t triangulo æquicru&longs;i, ine&longs;t A, &longs;cilicet duo recti, hoc e&longs;t, ine&longs;t <lb/>æquicru&longs;i hæc, pa&longs;&longs;io habere tres angulos æquales duobus rectis per B, ide&longs;t <lb/>per <expan abbr="triangulũ">triangulum</expan> vniuer&longs;ale, quia hæc proprietas e&longs;t trianguli propria, & <expan abbr="cõpe-tit">compe­ <lb/>tit</expan> æquicru&longs;i, non vt æquicrus e&longs;t, &longs;ed, vt triangulum e&longs;t; quare B, non crit <lb/>medium ip&longs;ius A, quia prædicta pa&longs;&longs;io. </s> <s id="id.000800">A, non competit triangulo B, per <lb/>aliud, &longs;ed per &longs;e, de eo enim primo, & per &longs;e demon&longs;tratur in 32. primi Elem. <lb/>optimè Aegydius, & Niphus in hunc locum.</s></p><p type="main"> <s id="id.000801"><arrow.to.target n="marg11"/></s></p><p type="margin"> <s id="id.000802"><margin.target id="marg11"/>11</s></p><p type="main"> <s id="id.000803">Ex eodem cap. <emph type="italics"/>(Non oportet autem exi&longs;timare penes id, quod exponimus, ali­ <lb/>quid accidere ab&longs;urdum, nihil enim vtimur eo, quod e&longs;t hoc aliquid e&longs;&longs;e. </s> <s id="id.000804">&longs;ed &longs;icut <lb/>Geometra pedalem, & rectam hanc, & &longs;ine latitudine dicit, quæ non &longs;unt. </s> <s id="id.000805">verum <lb/>non &longs;ic vtitur, tanquam ex his ratiocinans)<emph.end type="italics"/> Quoniam Ari&longs;t. in exemplis affert <lb/>pro rebus characteres, A, B, C, po&longs;&longs;et qui&longs;piam &longs;u&longs;picari aliquod propterea <lb/>ab&longs;urdum accidere: cui &longs;u&longs;picioni Ari&longs;t. re&longs;pondet, dicens, nihil inde ab&longs;ur­ <lb/>di accidere po&longs;&longs;e, quoniam ip&longs;e vtitur hi&longs;ce literis, <expan abbr="nõ">non</expan> quatenus literæ &longs;unt, <lb/>&longs;ed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum <lb/>etiam Geometræ faciunt, qui lineam, quæ pedalis non e&longs;t, pedalem, & quæ <lb/>non e&longs;t recta, rectam; & quæ lata e&longs;t, non latam, &longs;upponunt, & tamen nihil <lb/>inde ab&longs;urdi contingit. </s> <s id="id.000806">Ex quibus intelligimus per lineas illas &longs;en&longs;ibiles, & <lb/>phy&longs;icas, quas Geometræ in &longs;uis figuris ducunt, intelligendas e&longs;&longs;e lineas ve­ <lb/>rè Mathematicas omni latitudine carentes; vtitur enim inquit Ari&longs;t. <!-- REMOVE S-->Geo­ <lb/>metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis <lb/>lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. </s> <s id="id.000807">idem dicen­ <lb/>dum e&longs;t de &longs;uperficiebus, necnon de corporibus, quæ ijdem Goometræ de­ <lb/>&longs;cribunt, vt per ea, de verè mathematicis di&longs;currant.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.042.jpg" pagenum="42"/> <chap><p type="head"> <s id="id.000808"><emph type="italics"/>Ex Libro &longs;ecundo Priorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000809"><arrow.to.target n="marg12"/></s></p><p type="margin"> <s id="id.000810"><margin.target id="marg12"/>12</s></p><p type="main"> <s id="id.000811">Ex cap. 21. <emph type="italics"/>(Quod faciunt, qui coalternas putant &longs;cribere, latent enim ip&longs;i <lb/>&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exi&longs;tentibus <lb/>coalternis)<emph.end type="italics"/> Vult Ari&longs;t. exemplo mathematico explicare, quid &longs;it pe­ <lb/>titio principij. </s> <s id="id.000812">vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s> <s id="id.000813">quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="id.009.01.042.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.042.1.jpg" place="text"/> <lb/>probat Euclides in 28. primi Elem. quod &longs;i <lb/>linea recta quædam, vti E F, cadens &longs;uper <lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu­ <lb/>los alternos &ecedil;quales, angulos <expan abbr="nimirũ">nimirum</expan> A G H, <lb/>G H D, ij enim dicuntur alterni; &longs;iue alios <lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt <lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li­ <lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s> <s id="id.000814">Iam &longs;i quis vellet probare, &longs;e duas <lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al­ <lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa­ <lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="probandũ">probandum</expan> erat, <lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe­ <lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon­ <lb/>&longs;trare propo&longs;ueramus. </s> <s id="id.000815">aduerte, quod characteres, qui &longs;unt in &longs;equentibus <lb/>verbis huius loci, non appellant characteres figuræ appo&longs;itæ; in quo quidam <lb/>decepti, nullo pacto poterant locum hunc intelligere.</s></p><p type="main"> <s id="id.000816"><arrow.to.target n="marg13"/></s></p><p type="margin"> <s id="id.000817"><margin.target id="marg13"/>13</s></p><p type="main"> <s id="id.000818">Ex cap. 22. lib. 2. Priorum <emph type="italics"/>(Vt &longs;i volens mon&longs;trare, quod diameter e&longs;t incom­ <lb/>men&longs;. argueret Zenonis rationem, quod non e&longs;t moueri)<emph.end type="italics"/> &longs;uperius &longs;ecto 3. lib. 1. <lb/>fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea­ <lb/>dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> com­ <lb/>munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando <lb/>infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum, <lb/>e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea <lb/>non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban­ <lb/>tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t. non cau&longs;am pro cau&longs;a.</s></p><p type="main"> <s id="id.000819"><arrow.to.target n="marg14"/></s></p><p type="margin"> <s id="id.000820"><margin.target id="marg14"/>14</s></p><p type="main"> <s id="id.000821">Ex eodem cap. <emph type="italics"/>(Quoniam idem <expan abbr="vtiq;">vtique</expan> fal&longs;um per plures petitiones accidere <lb/>nihil forta&longs;&longs;e inconueniens, veluti coalternas coincidere; & &longs;i maior e&longs;t extrin&longs;ecus <lb/>angulus intrin&longs;eco; & &longs;i triangulus habet plures rectos duobus)<emph.end type="italics"/> per plures po&longs;i­ <lb/>tiones &longs;ubaudi fal&longs;as. </s> <s id="id.000822">per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa­ <lb/>rallelas, vt in &longs;uperiori cap. monuimus. </s> <s id="id.000823">Cæterum Euclides propo&longs;. </s> <s id="id.000824">28. pri­ <lb/>mi Elem. o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex­ <lb/>trin&longs;ecum E G B, v. <!-- REMOVE S-->g. <!-- REMOVE S-->æqualem interno, & oppo&longs;ito, & ad ea&longs;dem partes, <lb/>angulo videlicet G H D. &longs;i ergo inquit Ari&longs;t &longs;upponamus i&longs;tud fal&longs;um, an­ <lb/>gulum &longs;cilicet E G B, externum e&longs;&longs;e maiorem angulo interno G H D, &longs;equi­ <lb/>tur etiam fal&longs;um, videlicet lineas <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> A B, C D, concurrere. </s> <s id="id.000825">& pro­ <lb/>batur con&longs;equentia hoc modo, quia &longs;i angulus E G B, maior e&longs;t angulo <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.043.jpg" pagenum="43"/>G H D, appo&longs;ito <expan abbr="vtiq;">vtique</expan> communi angulo B G H, erant primum, duo anguli <lb/>E G B, B G H, maiores, quam &longs;int duo B G H, G H D, quia &longs;i inæqualibus <lb/>æqualia addantur, tota erunt inæqualia, vt prius per 4, axioma: hoc loco <lb/>communis angulus additur &longs;emel maiori angulo, & &longs;emel minori; & ideo <lb/>totum illud, in quo e&longs;t maior angulus, adhuc maius e&longs;t altero toto, in quo <lb/>minor angulus continetur. </s> <s id="id.000826">at illi duo E G B, B G H, per 13. primi, &longs;unt <lb/>æquales duobus rectis angulis, ergo duo <expan abbr="quoq;">quoque</expan> recti erunt maiores duobus <lb/>internis B G H, D H G, &longs;iue hi duo interni erunt minores duobus rectis. <lb/></s> <s id="id.000827">At quando hi duo interni &longs;unt minores duobus rectis, tunc lineæ A B, C D, <lb/>&longs;unt concurrentes, &longs;i protrahantur ad partes prædictorum <expan abbr="angulorũ">angulorum</expan>. </s> <s id="id.000828">quod <lb/>P. <!-- REMOVE S-->Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi <lb/>demon&longs;trauit. </s> <s id="id.000829"><expan abbr="Atq;">Atque</expan> hoc pacto ex prima fal&longs;a &longs;uppo&longs;itione, nimirum angu­ <lb/>lum illum externum e&longs;&longs;e maiorem interno, & oppo&longs;ito; &longs;equitur fal&longs;um, ni­ <lb/>mirum lineas parallelas concurrere.</s></p><p type="main"> <s id="id.000830">Præterea &longs;i &longs;upponamus aliam fal&longs;itatem, &longs;cilicet triangulum habere tres <lb/>angulos maiores duobus rectis, &longs;equetur eadem iterum fal&longs;itas, &longs;cilicet pa­ <lb/><figure id="id.009.01.043.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.043.1.jpg" place="text"/> <lb/>rallelas coincidere, & probatur &longs;ic; &longs;int enim <lb/><expan abbr="triãguli">trianguli</expan> A B C, tres anguli maiores, quam duo <lb/>recti anguli, & per punctum C, ducta &longs;it recta <lb/>C D, parallela lateri B A. quia ergo angulus <lb/>A, æqualis e&longs;t angulo &longs;ibi alterno A C D, per <lb/>29. primi, & quia totalis angulus B C D, æqua­ <lb/>lis e&longs;t duobus angulis B C A, A C D, quos tanquam &longs;uas partes adæquatas <lb/>continet, quorum alter, &longs;cilicet A C D, e&longs;t æqualis angulo A. erit idem to­ <lb/>talis angulus B C D, æqualis duobus angulis A, & A C B, trianguli propo&longs;i­ <lb/>ti. </s> <s id="id.000831">ergo totus i&longs;te angulus B C D, &longs;imul cum reliquo <expan abbr="triãguli">trianguli</expan> angulo B. con­ <lb/>flabit compo&longs;itionem ex tribus angulis trianguli dati: & con&longs;equenter ta­ <lb/>lis compo&longs;itio trium angulorum erit maior, quam &longs;int duo anguli recti. </s> <s id="id.000832">ex <lb/>quo &longs;equitur duas rectas B A, C D, &longs;uper quas cadit linea B C, faciens duos <lb/>angulos internos, & ad ea&longs;dem partes, &longs;cilicet A B D, maiores duobus re­ <lb/>ctis non e&longs;&longs;e parallelas, &longs;ed concurrentes (vt patet ex nuper citata demon­ <lb/>&longs;tratione P. Clauij) quod fal&longs;um e&longs;t. </s> <s id="id.000833">& &longs;equitur ex &longs;ecunda fal&longs;a &longs;uppo&longs;itio­ <lb/>ne. </s> <s id="id.000834">ex quibus textus Ari&longs;t. videtur &longs;atis clarus.</s></p><p type="main"> <s id="id.000835"><arrow.to.target n="marg15"/></s></p><p type="margin"> <s id="id.000836"><margin.target id="marg15"/>15</s></p><p type="main"> <s id="id.000837">Ex cap. 26. <emph type="italics"/>(Vt &longs;i A, duo recti, in quo autem P., triangulus, in quo vero C, <lb/>&longs;en&longs;ibuis triangulus, &longs;u&longs;picari <expan abbr="namq;">namque</expan> po&longs;&longs;et aliquis non e&longs;&longs;e C, &longs;ciens, quod omnis <lb/>triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. </s> <s id="id.000838">no&longs;ce enim <lb/>omnem triangulum, quod duobus rectis, non &longs;implex e&longs;t: &longs;ed hoc quidem eo, quod <lb/>vniuer&longs;alem habet &longs;cientiam: illud autem eo, quod &longs;ingularem. </s> <s id="id.000839">&longs;ic igitur, vt vni­ <lb/>uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit <lb/>contrarias)<emph.end type="italics"/> vide, quæ diximus lib. 1. &longs;ecto 3. cap. 1. ex quibus quidquid Ma­ <lb/>thematicum e&longs;t hic, clarum redditur. </s> <s id="id.000840">reliqua verò, quæ ad Logicum &longs;pe­ <lb/>ctant, huius loci commentatores pro&longs;equuntur.</s></p><p type="main"> <s id="id.000841">In cap. 31. de Abductione.<!-- KEEP S--></s></p><p type="main"> <s id="id.000842"><arrow.to.target n="marg16"/></s></p><p type="margin"> <s id="id.000843"><margin.target id="marg16"/>16</s></p><p type="main"> <s id="id.000844">Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc <lb/>cap. agitur e&longs;&longs;e vocem mathematicam, <expan abbr="cam&qacute;">camque</expan>; Ari&longs;t. quemadmodum multa <lb/>alia à Mathematicis mutuatum ad omnes alias &longs;cientias tran&longs;tuli&longs;&longs;e. </s> <s id="id.000845">e&longs;&longs;e <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.044.jpg" pagenum="44"/>autem terminum mathematicum colligitur manife&longs;tè ex Proelo, qui lib. 3. <lb/>in comm. <!-- REMOVE S-->Elem. Euclidis ad primam propo&longs;itionem primi Elementi, pag. <lb/></s> <s id="id.000846">121. &longs;ic ait, Abductio verò e&longs;t tran&longs;itus à propo&longs;ito problemate, vel theo­ <lb/>remate ad aliud, quo cognito, aut comparato Propo&longs;itum quoque per&longs;pi­ <lb/>cuum e&longs;t. </s> <s id="id.000847">Exempli cau&longs;a, cum cubi duplicatio propo&longs;ita e&longs;&longs;et ad inue&longs;ti­ <lb/>gandam quæ&longs;tionem in aliud tran&longs;tulere, quod illud propo&longs;itum con&longs;equi­ <lb/>tur, ad duarum nempe mediarum linearum inuentionem tran&longs;lata e&longs;t quæ­ <lb/>&longs;tio, & &longs;ic quærebant deinceps, quonam modo datis duabus rectis lineis, <lb/>duæ mediæ proportionales reperirentur. </s> <s id="id.000848">Primum autem dicunt Hippocra­ <lb/>tem Chium pr&ecedil;dictorum titulorum, Abductionem feci&longs;&longs;e, qui & lunulæ qua­ <lb/>dratum fecit æquale, & alia multa in Geometria inuenit. </s> <s id="id.000849">hæc Proclus. <!-- KEEP S--></s> <s id="id.000850">vbi <lb/>non di&longs;&longs;imulandum nos re&longs;titui&longs;&longs;e verbum, Abductionem, cuius loco inter­ <lb/>pres Procli vtitur inductionis voce, &longs;equuti & rationem, & græcum textum, <lb/>qui no&longs;tram hanc expo&longs;itionem euidenter po&longs;tulat, <foreign lang="greek">apagwgh\</foreign> enim valet & <lb/>inductionem, & abductionem, &longs;ed abductio omnino rei propo&longs;itæ quadrat.</s></p><p type="main"> <s id="id.000851">Notandum præterea Hippoetatem Chium fui&longs;&longs;e auctorem huius Abdu­ <lb/>ctionis, <expan abbr="eum&qacute;">eumque</expan>; feci&longs;&longs;e Abductionem à propo&longs;ito Problemate quadrandi cir­ <lb/>culi, vnde manife&longs;tè apparet, Ari&longs;totelem ex Mathematicis hunc terminum <lb/>mutuò accepi&longs;&longs;e, quandoquidem ex ij&longs;dem accepit etiam exemplum Abdu­ <lb/>ctionis Mathematicæ, imò etiam exemplum ip&longs;ius authoris Abductionis <lb/>Mathematicæ. </s> <s id="id.000852">&longs;yllogi&longs;mus autem Hippocratis, quo o&longs;tendebat circuli qua­ <lb/>draturam reducebatur ad has propo&longs;itiones, omnis rectilinea figura qua­ <lb/>dratur, &longs;ed circulus reducitur ad figuram rectilineam, ergo circulus qua­ <lb/>dratur. </s> <s id="id.000853">in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re­ <lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrinũ">Alexandrinum</expan>, & apud P. <!-- REMOVE S-->Cla­ <lb/>uium in fine &longs;exti Elem. & alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro­ <lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;itũ">propo&longs;itum</expan> pro­ <lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;imus in cap. 3. Præ­ <lb/>dicam. <!-- REMOVE S-->de hac re, quia plurimum hunc conferunt. </s> <s id="id.000854">&longs;ed iam ad textus expli­ <lb/>cationem veniamus.</s></p><p type="main"> <s id="id.000855"><arrow.to.target n="marg17"/></s></p><p type="margin"> <s id="id.000856"><margin.target id="marg17"/>17</s></p><p type="main"> <s id="id.000857">Ex eodem cap. <emph type="italics"/>(Veluti &longs;i K, e&longs;&longs;et quadrari, in quo autem E, rectilineum, in <lb/>quo verò F, circulus, &longs;i ip&longs;ius E F, vnum &longs;olum e&longs;&longs;et medium, hoc, quod e&longs;t, cum <lb/>lunulis æqualem fieri circulum rectilineo, e&longs;&longs;e po&longs;&longs;et propè ip&longs;um cogno&longs;cere, cum <lb/>vero B C, neque credibilius &longs;it, quam A C, <expan abbr="neq;">neque</expan> pauca media, non dico Abductio­ <lb/>nem: <expan abbr="neq;">neque</expan> quando B C, &longs;it immediatum, tale enim &longs;cientia est)<emph.end type="italics"/> Aduerte figuram <lb/>vulgatæ editionis e&longs;&longs;e mendo&longs;am, & propterea re&longs;tituendam e&longs;&longs;e, qualis pri­ <lb/>ma &longs;equens ex Simplicio ad tex. <!-- REMOVE S-->11. primi Phy&longs;ic. <!-- REMOVE S-->hoc modo Hippocrates <lb/>Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua­ <lb/>drandus; con&longs;tituatur <expan abbr="itaq;">itaque</expan> &longs;uper diametro cius B C, quadratum B C D F, <lb/>cuius diameter B D, &longs;ecatur bifariam in G, à circumferentia circuli dati, <lb/>quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre­ <lb/>mo puncto G, &longs;ecat bifariam, & <expan abbr="diametrũ">diametrum</expan> B D, & circumferentiam B G C. <lb/>facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F, <expan abbr="conne-ctatur&qacute;">conne­ <lb/>ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, &longs;ubtenditur <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.045.jpg" pagenum="45"/><figure id="id.009.01.045.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.045.1.jpg" place="text"/> <lb/>angulo recto C, ergo quadratum eius ex eorol­ <lb/>lario 47. primi, duplum erit quadrati B C, quare <lb/>etiam circulus B C D F, duplus erit circuli A B­ <lb/>G C, per 2. duodecimi, & &longs;emicirculus B C D, <lb/>duplus erit &longs;emicirculi B A C: & quadrans B E­ <lb/>C G, æqualis erit &longs;emicirculo B A C: ablato igi­ <lb/>tur communi &longs;egmento B E C H, remanet lunu­ <lb/>la B A C E, æqualis triangulo B C G, quod trian­ <lb/>gulum &longs;i per vltimam &longs;ecundi quadretur, erit lu­ <lb/>nula B A C, con&longs;equenter quadrata. </s> <s id="id.000858"><expan abbr="hucu&longs;q;">hucu&longs;que</expan> be­ <lb/>nè procedit Hippocrates. <!-- KEEP S--></s> <s id="id.000859">&longs;ed vt reliquum circu­ <lb/>li quadret, &longs;ic pergit, ponatur recta L M, dupla <lb/>ip&longs;ius B C, &longs;upra quam &longs;emicirculus de&longs;cribatur <lb/><figure id="id.009.01.045.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.045.2.jpg" place="text"/> <lb/>L O M, cui in&longs;cribatur hexagoni <lb/>æquilateri dimidium L Q S M, & &longs;u­ <lb/>per tribus hexagoni lateribus, &longs;int <lb/>tres &longs;emicirculi, vt in figura. </s> <s id="id.000860">& <expan abbr="quo-niã">quo­ <lb/>niam</expan> diameter L M, dupla e&longs;t <expan abbr="vniu&longs;-cuiu&longs;q;">vniu&longs;­ <lb/>cuiu&longs;que</expan> <expan abbr="diametrorũ">diametrorum</expan> B C, L Q, Q S, <lb/>S M, erit &longs;emicirculus L O M, &ecedil;qua­ <lb/>lis quatuor &longs;emicirculis prædictis <lb/>per 2. duodecimi, & per 4. &longs;ecundi <lb/>ablatis igitur tribus <expan abbr="&longs;egm&etilde;tis">&longs;egmentis</expan> com­ <lb/>munibus L N Q, Q O S, S P M, relinquetur trapezium L Q S M, æquale &longs;e­ <lb/>micirculo B A C, & tribus lunulis L H Q N, Q R S O, S X M P, ab&longs;cindan­ <lb/>tur <expan abbr="itaq;">itaque</expan> detrapezio tria triangula æqualia tribus lunulis, eo modo, quo &longs;u­ <lb/>pra in prima figura factum e&longs;t, & quod relinquetur æquale erit &longs;emicirculo <lb/>B A C. quod deinde quadretur per vlt. </s> <s id="id.000861">&longs;ecundi, &longs;ed aduerte, quod quando <lb/>ait, ab&longs;cindantur de trapezio tria triangula æqualia lunulis, eo modo, quo <lb/>&longs;upra, committit deceptionem, quia eodem modo, quo &longs;upra minimè id fa­ <lb/>cere po&longs;&longs;umus, quia in &longs;uperiori figura triangula erant con&longs;tituta &longs;uper la­ <lb/>tus B C, quadrati B C D F, intra circulum de&longs;cripti, qui circulus facit cum <lb/>B C, maius <expan abbr="&longs;egmentũ">&longs;egmentum</expan>, quam faciat &longs;emicirculus L O M, cum lateribus L Q, <lb/>Q S, S M. & propterea &longs;emicirculus i&longs;te non habet eandem proportionem <lb/>ad vnamquamque lunularum &longs;uarum, quam habet &longs;emicirculus &longs;uperior <lb/>B C D, ad lunulam B A C E. <expan abbr="atq;">atque</expan> hæc e&longs;t fallacia, quam authorem &longs;uum mi­ <lb/>nimè latui&longs;&longs;e putandum, cuius Ari&longs;t. &longs;æpius mentionem in &longs;equentibus fa­ <lb/>ciet : quì enim fieri pote&longs;t, vt tam acutus inuentor, adeo manife&longs;tum erro­ <lb/>rem non vidi&longs;&longs;et, verum propter adinuenti excellentiam, authori &longs;uo pla­ <lb/>cuit paralogy&longs;mus. </s> <s id="id.000862">mirabilis tamen &longs;emper habita e&longs;t illa &longs;uperior lunulæ <lb/>quadratio. </s> <s id="id.000863">Ex quibus &longs;atis clara e&longs;&longs;e po&longs;&longs;unt ea, quæ ad <expan abbr="Mathematicũ">Mathematicum</expan> per­ <lb/>tinent, ad locum hunc de Abductione declarandum. </s> <s id="id.000864">facta e&longs;t igitur abdu­ <lb/>ctio ab Hippocrate in quadratione trium po&longs;teriorum lunularum, in qua­ <lb/>rum quadratione diu immoratus, nunquam ni&longs;i cum paralogy&longs;mo quadra­ <lb/>re valuit. </s> <s id="id.000865">Hæc pluribus, vt &longs;equentibus etiam textibus, in quibus huius te­ <lb/>tragoni&longs;mi fit mentio &longs;atisfacere po&longs;&longs;imus. </s> <s id="id.000866">Hippocrates i&longs;te Chius e&longs;t alter <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.046.jpg" pagenum="46"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre <lb/>Aphrod. <!-- REMOVE S-->in Primum Meteororum de Cometis.<!-- KEEP S--></s> </p></chap><chap><p type="head"> <s id="id.000867"><emph type="italics"/>Ex Primo Posteriorum re&longs;olutoriorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.000868"><arrow.to.target n="marg18"/></s></p><p type="margin"> <s id="id.000869"><margin.target id="marg18"/>18</s></p><p type="main"> <s id="id.000870">Textu primo <emph type="italics"/>(Omnis doctrina, & omnis di&longs;ciplina di&longs;cur&longs;iua ex præexi­ <lb/>&longs;tenti fit cognitione. </s> <s id="id.000871">manife&longs;tum autem hoc &longs;peculantibus in omnibus, <lb/>Mathematicæ <expan abbr="namq;">namque</expan> &longs;cientiarum per hunc modum accedunt)<emph.end type="italics"/> quo mo­ <lb/>do Mathematicæ fiant ex præcedenti cognitione, &longs;cilicet Princi­ <lb/>piorum per&longs;picuè quilibet videbit, qui &longs;altem primum <expan abbr="Elem&etilde;torum">Elementorum</expan> Eucli­ <lb/>dis, vel è ianuis in&longs;pexerit; pr&ecedil;cedunt enim primo principiorum tria gene­ <lb/>ra, quorum primum continet definitiones &longs;ubiecti Geometriæ, vt definitio­ <lb/>nes lineæ, &longs;uperficiei, trianguli, &c: Secundum continet Po&longs;tulata. </s> <s id="id.000872">Tertium <lb/>Axiomata, &longs;eu communes omnium conceptiones, & &longs;ententias, ex quibus <lb/>tanquam ex vberrimis, & chri&longs;taltinis fontibus Demon&longs;trationes Geome­ <lb/>tricæ deriuantur. </s> <s id="id.000873">Idem vìdere licet in operibus aliorum Geometrarum, <lb/>Archimedis, Apollonij, Pappi, & cæterorum. </s> <s id="id.000874">Aliæ &longs;iniliter mathematicæ, <lb/>vt Arithmetica, Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia, non ni&longs;t ex <lb/>præmi&longs;&longs;is, ac manife&longs;ti&longs;simis principijs &longs;uas demon&longs;trationes deducunt. <lb/></s> <s id="id.000875">Nulla porrò alia &longs;cientia tam di&longs;tinctè &longs;ua præmittit principia, <expan abbr="tam&qacute;">tamque</expan>; per­ <lb/>&longs;picua, &longs;icuti Mathematicæ, vt non immeritò Philo&longs;ophus eas, tamquam <lb/>veræ &longs;cientiæ <expan abbr="typũ">typum</expan>, <expan abbr="eum&qacute;">eumque</expan>; omnibus numeris ab&longs;olutum &longs;ibi ob oculos pro­ <lb/>po&longs;uerit, ex quo veræ &longs;cientiæ de&longs;criptionem hi&longs;ce libris complecteretur.</s></p><p type="main"> <s id="id.000876"><arrow.to.target n="marg19"/></s></p><p type="margin"> <s id="id.000877"><margin.target id="marg19"/>19</s></p><p type="main"> <s id="id.000878">Tex. 2. <emph type="italics"/>(Quod enim omne triangulum habet duobus rectis æquales, præ&longs;ciuit: <lb/>quod autem hoc, quod e&longs;t in &longs;emicirculo triangulum e&longs;t, &longs;imul inducens cognouit)<emph.end type="italics"/> <lb/>vide primo, quæ &longs;upra libro 1. Prior. <!-- REMOVE S-->&longs;ecto 3. cap. 1. explicaui de angulis <lb/>trianguli. </s> <s id="id.000879">deinde &longs;cias, quod quando Ari&longs;t. ait, hoc, quod e&longs;t in &longs;emicir cu­ <lb/>lo triangulum, &c. </s> <s id="id.000880">alludit ad demon&longs;trationem quandam, quam ip&longs;e infe­ <lb/>rius in exemplum adducet, & quæ e&longs;t in 3. Elem. Euclidis 31. in qua talis fi­ <lb/>gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e­ <lb/><figure id="id.009.01.046.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.046.1.jpg" place="text"/> <lb/>micirculo. </s> <s id="id.000881">tunc autem dicitur triangulum in <lb/>&longs;emicirculo, quando ba&longs;is ip&longs;ius e&longs;t diameter <lb/>&longs;emicirculi, & reliqua duo latera ita concur­ <lb/>runt &longs;imul in angulum B, vt ip&longs;um paricer in <lb/>circumferentia con&longs;tituant, quibus pr&ecedil;mi&longs;sis <lb/>&longs;ic textum explicaueris: quod enim omne <lb/>triangulum habet tres angulos æquales duo­ <lb/>bus rectis angulis præ&longs;ciuit vniuer&longs;aliter per <lb/>32. primi; quod autem hoc particulare triangulum A B C, quod e&longs;t in &longs;e­ <lb/>micirculo habeat eandem proprietatem, &longs;imul, ac qui&longs;piam animaduertit <lb/>illud e&longs;&longs;e triangulum cogno&longs;cit, <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla demon&longs;tratione, &longs;ed &longs;olum virtu­ <lb/>te illius maioris propo&longs;itionis; omne triangulum habet tres, &c.</s></p><p type="main"> <s id="id.000882"><arrow.to.target n="marg20"/></s></p><p type="margin"> <s id="id.000883"><margin.target id="marg20"/>20</s></p><p type="main"> <s id="id.000884">Tex. 5. <emph type="italics"/>(Vera quidem igitur oporter e&longs;&longs;e, quoniam non e&longs;t non ens &longs;cire, vt quod <lb/>diameter &longs;it commen&longs;urabi is)<emph.end type="italics"/> con&longs;ule ea, quæ &longs;crip&longs;imus ad cap. 23. primi <lb/>Priorum, &longs;ecto 1. &longs;ine quibus locus hic &longs;atis intelligi nequit; ijs autem per­ <lb/>ceptis &longs;ic <expan abbr="locũ">locum</expan> hunc explicare po&longs;&longs;umus, cum diameter quadrati &longs;it incom­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.047.jpg" pagenum="47"/>men&longs;urabilis lateri &longs;ui quadrati, fal&longs;um erit dicere diametrum e&longs;&longs;e com­ <lb/>men&longs;urabilem prædicto lateri, quod autem fal&longs;um e&longs;t, illud non e&longs;t; igitur <lb/>impo&longs;sibile e&longs;t &longs;cire diametrum e&longs;&longs;e commen&longs;urabile.</s></p><p type="main"> <s id="id.000885"><arrow.to.target n="marg21"/></s></p><p type="margin"> <s id="id.000886"><margin.target id="marg21"/>21</s></p><p type="main"> <s id="id.000887">Hoc eodem cap. plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien­ <lb/>tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo <lb/>&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con­ <lb/>templatione primi libri Elem. Euclidis percipi pote&longs;t. </s> <s id="id.000888">vt propterea benè ij <lb/>&longs;entiant, inter quos præcipui &longs;unt Toletus, & Zabarella, qui a&longs;&longs;erunt, Ari&longs;t. <lb/>Mathematicas &longs;cientias tamquam typum perfecti&longs;simarum &longs;cientiarum <lb/>&longs;ibi ob oculos propo&longs;ui&longs;&longs;e; ex quo typo veræ &longs;cientiæ de&longs;criptionem his li­ <lb/>bris complectaretur.</s></p><p type="main"> <s id="id.000889"><arrow.to.target n="marg22"/></s></p><p type="margin"> <s id="id.000890"><margin.target id="marg22"/>22</s></p><p type="main"> <s id="id.000891">Eodem tex. <!-- REMOVE S-->5. <emph type="italics"/>(Ponit enim Arithmeticus vnitatem indiui&longs;ibilem e&longs;&longs;e &longs;ecun­ <lb/>dum quantum)<emph.end type="italics"/> hoc quamquam non ponatur ab Arithmeticis expre&longs;sè, præ­ <lb/>&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme­ <lb/>ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan­ <lb/>titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. </s> <s id="id.000892">Verum dubitabit forrè qui&longs;­ <lb/>piam hoc modo, &longs;i vnitas minimum, <expan abbr="atq;">atque</expan> indiui&longs;ibile e&longs;t in quanto di&longs;creto, <lb/>qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien­ <lb/>tem, in quadrantem, & alijs &longs;imiliter modis, vnde numeri illi, qui fractio­ <lb/>nes appellantur, exurgunt? </s> <s id="id.000893">Re&longs;pondemus, <expan abbr="quotie&longs;eunq;">quotie&longs;eunque</expan> vnitas diuiditur ab <lb/>Arithmeticis, tunc ip&longs;i eam accipiunt tanquam totum quoddam <expan abbr="cõtinuum">continuum</expan> <lb/>in plures partes diui&longs;ibile: &longs;iue tanquam aggregatum quoddam vnitatum, <lb/>quæ vnitates &longs;unt partes illius, vt quando dicunt, vnum horæ quadrantem, <lb/>vel duos horæ quadrantes, vel tres horæ quadrantes, accipiunt horam tan­ <lb/>quam aggregatum quatuor quadrantum, & propterea numeri illi 1/4. 2/4. 3/4. <lb/>& &longs;imiles fractiones, nihil aliud &longs;unt, quam numeri partium vnius horæ: ex <lb/>quo patet huiu&longs;modi fractiones omnes reduci ad numeros integros, qui <lb/>enim dicit tres quadrantes 3/4. dicit tres partes alicuius totius, quod intel­ <lb/>ligitur diui&longs;um e&longs;&longs;e in 4. æquales partes, ex quibus illæ tres tantummodo <lb/>numerat.</s></p><p type="main"> <s id="id.000894"><arrow.to.target n="marg23"/></s></p><p type="margin"> <s id="id.000895"><margin.target id="marg23"/>23</s></p><p type="main"> <s id="id.000896">Tex. 9. <emph type="italics"/>(Per &longs;e autem, <expan abbr="quæcunq;">quæcunque</expan> & in&longs;unt in eo, quod quid e&longs;t, vt triangulo li­ <lb/>nea, & lineæ punctum; &longs;ub&longs;tantia <expan abbr="namq;">namque</expan> ip&longs;orum ex his e&longs;t, & in oratione dicen­ <lb/>te, quid e&longs;t, in&longs;unt)<emph.end type="italics"/> aggreditur explicare quænam &longs;int ea, quæ per &longs;e dicun­ <lb/>tur: <expan abbr="quot&qacute;">quotque</expan>; modis dicatur aliquid per &longs;e. </s> <s id="id.000897">quorum primus e&longs;t, ea &longs;cilicet, <lb/>per &longs;e de aliquo &longs;ubiecto dici, <expan abbr="quæcunq;">quæcunque</expan> in definitione illius ponuntur, cu­ <lb/>iu&longs;modi &longs;unt linea, & punctum, quæ per &longs;e prædicantur, illa de triangulo, <lb/>i&longs;tud de linea; in de&longs;initione enim trianguli ponitur linea recta, quia linea <lb/>recta dum terminat illam &longs;uperficiem, quæ dicitur triangulus illi trianguli <lb/>naturam impertitur, & ideo triangulus definitur &longs;ic, triangulus e&longs;t figura <lb/>tribus lineis rectis terminata. </s> <s id="id.000898">&longs;imiliter in definitione lineæ, non in&longs;initæ, <lb/>&longs;ed finitæ, & terminatæ ponitur punctum, quia duo puncta, quæ &longs;unt extre­ <lb/>ma illius, faciunt, vt ea &longs;it line a finita, & definitur &longs;ic, linea finita e&longs;t lon­ <lb/>gitudo, caius extrema &longs;unt puncta. </s> <s id="id.000899">quamuis autem hæc definitio apud Eu­ <lb/>clidem expre&longs;&longs;a non habeatur, tamen ex definitionibus ip&longs;ius præ&longs;ertim &longs;e­ <lb/>cunda, tertia, & quarta elici pote&longs;t.</s></p><p type="main"> <s id="id.000900"><arrow.to.target n="marg24"/></s></p><p type="margin"> <s id="id.000901"><margin.target id="marg24"/>24</s></p><p type="main"> <s id="id.000902">Eodem tex. <!-- REMOVE S-->9. <emph type="italics"/>(Et <expan abbr="quibu&longs;cunq;">quibu&longs;cunque</expan> iuexi&longs;tentium ip&longs;is, ip&longs;æ &longs;unt in oratione, quid<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.048.jpg" pagenum="48"/><emph type="italics"/>est declarante, quemadmodum rectum ine&longs;t lineæ, & circulare: & impar, & par <lb/>numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. </s> <s id="id.000903">& <lb/><expan abbr="oĩbus">oimbus</expan> bis in&longs;unt in oratione, quid e&longs;t <expan abbr="declarãte">declarante</expan>, ibi quidem linea, hic vero numerus)<emph.end type="italics"/> <lb/>quia locus hic benè exponitur à Toleto, & melius etiam à Conymbr. </s> <s id="id.000904">addam <lb/>tantummodo quædam, quæ ad perfectam eius intelligentiam de&longs;iderantur. <lb/></s> <s id="id.000905">Sciendum igitur primò, nu&longs;quam ab Euclide definiri rectum, circulare, <lb/>impar, par, primum, compo&longs;itum, æquilaterum, nec altera parte longius: <lb/><expan abbr="verũ">verum</expan> ab ip&longs;o in definitionibus primi definiri lineam rectam, non tamen cir­ <lb/>cularem expre&longs;sè. </s> <s id="id.000906">in definitionibus deinde &longs;eptimi definiri <expan abbr="numerũ">numerum</expan> parem, <lb/>& imparem, item numerum primum, & compofitum, & æquilaterum, & al­ <lb/>tera parte longiorem. </s> <s id="id.000907">ex quibus definitionibus po&longs;&longs;unt erui definitiones re­ <lb/>cti, circularis, imparis, & cæterorum, quorum hic Ari&longs;toteles meminit. <lb/></s> <s id="id.000908">Cæterum Euclides definitione 11. &longs;eptimi, &longs;ic definit numerum primum: <lb/>primus numerus e&longs;t, quem vnitas &longs;ola metitur. </s> <s id="id.000909">numerus autem, vel vnitas <lb/>metiri dicitur alium numerum, quando &longs;æpius repetita ip&longs;um omnino ad­ <lb/>æquat, vt ternarius metitur nouenarium, quia ter repetitus ip&longs;um ad vn­ <lb/>guem explet. </s> <s id="id.000910">illi igitur numeri dicuntur ab Arithmeticis primi, qui à nullo <lb/>alio, præterquam ab vnitate men&longs;urantur, quales &longs;unt, 2. 3. 5. 7. &c. </s> <s id="id.000911">Defi­ <lb/>nitione verò 13. definit numerum compo&longs;itum &longs;ic; compo&longs;itus numerus e&longs;t, <lb/>quem numerus qui&longs;piam metitur, vt &longs;enarius erit compo&longs;itus, quia ip&longs;um <lb/>binarius metitur, nam ter repetitus, ip&longs;i perfectè adæquatur.</s></p><p type="main"> <s id="id.000912">Per æquilaterum, intelligit quadratum, quadratus autem numerus defi­ <lb/>nitione 18. &longs;eptimi &longs;ic explicatur: Quadratus numerus e&longs;t, qui &longs;ub duobus <lb/>æqualibus numeris continetur, ide&longs;t, qui fit ex ductu vnius numeri in &longs;e ip­ <lb/><figure id="id.009.01.048.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.048.1.jpg" place="text"/> <lb/>&longs;um, vt &longs;i ducantur 3. in 3. fient 9. qui continetur &longs;ub duobus <lb/>ternarijs; omnes autem ternarij &longs;unt æquales. </s> <s id="id.000913">is autem nu­ <lb/>merus dicetur quadratus, quia, vt apparet in figura, nouem <lb/>ip&longs;ius vnitates po&longs;&longs;unt in plano ita ad inuicem collocari, vt <lb/>referant quadratum; & &longs;icuti quadratum geometricum ha­ <lb/>bet latera æqualia, ita etiam quadratum arithmeticum: &longs;i­ <lb/>ue numerus quadratus, habet &longs;ua latera æqualia, quot enim vnitates &longs;unt <lb/>in vno, tot etiam &longs;unt in reliquis, vt in præ&longs;enti &longs;unt tres vnitates in &longs;ingulis <lb/>lateribus. </s> <s id="id.000914">pr&ecedil;terea quemadmodum quadratum geometricum re&longs;olni pote&longs;t <lb/>in plura quadrata, ita etiam arithmeticum, vt præ&longs;ens, qui re&longs;oluitur in <lb/>quatuor quadrata arithmetica. </s> <s id="id.000915"><expan abbr="Neq;">Neque</expan> enim pote&longs;t quilibet numerus, vt opi­ <lb/>nantur ageometreti, in hunc modum di&longs;poni, &longs;ed &longs;olum ij, qui producuntur <lb/>ex multiplicatione numeri alicuius in &longs;e ip&longs;um.</s></p><p type="main"> <s id="id.000916">Per altera parte longius, intelligit numerum, qui producitur à duobus <lb/><figure id="id.009.01.048.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.048.2.jpg" place="text"/> <lb/>numeris inæqualibus inuicem multiplicatis, qualis e&longs;t <lb/>duodenarius, qui ex ductu trium in quatuor produci­ <lb/>tur, & refert figuram altera parte longiorem, &longs;iue, vt <lb/>ait Boetius longilateram, cuius vnum latus e&longs;t maius <lb/>altero, vt in appo&longs;ita figura videre licet. </s> <s id="id.000917">atque hæc <lb/>&longs;unt, quæ ex Mathematicis petenda erant, ad huius <lb/>loci intelligentiam.</s></p><p type="main"> <s id="id.000918"><arrow.to.target n="marg25"/></s></p><p type="margin"> <s id="id.000919"><margin.target id="marg25"/>25</s></p><p type="main"> <s id="id.000920">Tex. 11. <emph type="italics"/>(Per &longs;e autem, & &longs;ecundum quod ip&longs;um, idem, vt per &longs;e lineæ inest<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.049.jpg" pagenum="49"/><emph type="italics"/>punctum, & rectum; etenim &longs;ecundum quod linea, & triangulo, &longs;ecundum quod <lb/>triangulum duo recti: etenim per &longs;e triangulum duobus rectis æquale. </s> <s id="id.000921">Vniuer&longs;ale <lb/>autem e&longs;t tunc, quando in quolibet, & primo mon&longs;tratur, vt duos rectos habere, <lb/><expan abbr="neq;">neque</expan> figuræ e&longs;t vniuer&longs;ale, quamuis e&longs;t mon&longs;irare de figura, quod duos rectos habet, <lb/>&longs;ed non de qualibet figura, <expan abbr="neq;">neque</expan> vtitur qualibet figura monstrans, quadrangulum <lb/>enim figura a quidem est, non habet autem duobus rectis æquales. </s> <s id="id.000922">Aequicrus verò <lb/>babet quidem <expan abbr="quodcunq;">quodcunque</expan> duobus rectis æquales, &longs;ed non primò, &longs;ed triangulum <lb/>prius. </s> <s id="id.000923">quod igitur quoduis primum mon&longs;tratur duos rectos habens, aut <expan abbr="quodcunq;">quodcunque</expan> <lb/>aliud, huic primo ine&longs;t vniuer&longs;ale, & demonstratio de hoc vniuer&longs;aliter e&longs;t, de alijs <lb/>verò quodammodo, non per &longs;e, <expan abbr="neq;">neque</expan> de æquicrure e&longs;t vniuer&longs;aliter, &longs;ed in plus)<emph.end type="italics"/> pro <lb/>quorum intelligentia nece&longs;&longs;aria &longs;unt ea, quæ primo Priorum &longs;ecto 3. cap. 1. <lb/>&longs;crip&longs;imus. </s> <s id="id.000924">deinde memineris figuram vniuer&longs;aliorem e&longs;&longs;e triangulo, & tri­ <lb/>angulum vniuer&longs;alius æquicrure. </s> <s id="id.000925">quando ait (vt duos rectos habere) vult <lb/>dicere, habere duos angulos rectos non actu, &longs;ed potentia; quæ affectio e&longs;t <lb/>trianguli, quia, vt &longs;uperius diximus, habet tres angulos æquales duobus <lb/>rectis angulis: quæ proprietas vniuer&longs;aliter, & primò competit triangulo. <lb/></s> <s id="id.000926">non autem figuræ, quia figura e&longs;t vniuer&longs;alior. </s> <s id="id.000927"><expan abbr="neq;">neque</expan> i&longs;o&longs;celi, quia i&longs;o&longs;celes e&longs;t <lb/>re&longs;trictius triangulo. </s> <s id="id.000928">omittimus reliqua &longs;ingillatim exponere, tum quia &longs;a­ <lb/>tis clara &longs;unt, tum quia ab interpretibus benè explicantur.</s></p><p type="main"> <s id="id.000929"><arrow.to.target n="marg26"/></s></p><p type="margin"> <s id="id.000930"><margin.target id="marg26"/>26</s></p><p type="main"> <s id="id.000931">Tex. 13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem <lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro­ <lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt, <lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio <lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan> <lb/>ex 28. primi Elem. de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="id.009.01.049.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.049.1.jpg" place="text"/> <lb/>cen&longs;ui. </s> <s id="id.000932">Quando igitur duæ rectæ con&longs;titu­ <lb/>tæ fuerint, vt A B, C D, in quas alia recta, <lb/>vt G F, incidens, faciat duos angulos in­ <lb/>ternos, re&longs;pectu rectarum A B, C D, & ad <lb/>ea&longs;dem partes rectæ E F, vt &longs;unt ex parte <lb/>&longs;ini&longs;tra anguli A G H, C H G; exparte ve­ <lb/>rò dextra B G H, D H G; &longs;i <expan abbr="inquã">inquam</expan> linea E F, <lb/>fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus <lb/>rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro­ <lb/>bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. </s> <s id="id.000933">Verum, <lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tantũ">tantum</expan> æqua­ <lb/>les duobus rectis, verum etiam rectos, quo etiam modo <expan abbr="probar&etilde;tur">probarentur</expan> cædem <lb/>lineæ e&longs;&longs;e parallelæ, vt in &longs;equenti figura, cum &longs;int anguli A G I, C I G, re­ <lb/><figure id="id.009.01.049.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.049.2.jpg" place="text"/> <lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan­ <lb/>tia. </s> <s id="id.000934">Ex his facile textum in hunc modum expo­ <lb/>nemus; &longs;i quis igitur mon&longs;trauerit, quod rectæ <lb/>A B, C D, nunquam coincidunt, etiam&longs;i in in&longs;i­ <lb/>nitum producantur, &longs;eu quod &longs;unt æquidi&longs;tantes, <lb/>quando anguli prædicti interni &longs;unt duo recti, <lb/>videbitur <expan abbr="vtiq;">vtique</expan> huius e&longs;&longs;e demon&longs;tratio de vniuer&longs;ali per &longs;e, & de primo &longs;u­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.050.jpg" pagenum="50"/>biecto, vel &longs;ecundum quod ip&longs;um, eò quod probatur vniuer&longs;aliter de lineis <lb/>omnibus habentibus prædictos angulos rectos. </s> <s id="id.000935">non autem de omni, &longs;ecun­ <lb/>dum quod ip&longs;um, &longs;i quidem non competit affectio hæc, e&longs;&longs;e parallelas, li­ <lb/>neis habentibus illos angulos rectos actu; &longs;ed primò, & vniuer&longs;aliter, & &longs;e­ <lb/>cundum quod ip&longs;um competit lineis habentibus illos angulos æquales duo­ <lb/>bus rectis, <expan abbr="quomodocunq;">quomodocunque</expan> æquales &longs;int duobus rectis, &longs;iue ambo &longs;int recti, <lb/>&longs;iue vnus acutus, alter obtu&longs;us, &longs;ed tamen ambo &longs;imul æquentur duobus re­ <lb/>ctis, quales &longs;unt lineæ primæ figuræ. </s> <s id="id.000936">In tertio igitur errore, vniuer&longs;ale exi­ <lb/>&longs;tit quidem, & habet nomen, &longs;ed tamen prætermittetur, &longs;eu &longs;trictius &longs;ume­ <lb/>tur, quam oportet. </s> <s id="id.000937">alij latini, quos quidem viderim, præter Zabarellana <lb/>perperam omnino ob mathematicarum ignorantiam, exemplum i&longs;tud in­ <lb/>terpretantur.</s></p><p type="main"> <s id="id.000938"><arrow.to.target n="marg27"/></s></p><p type="margin"> <s id="id.000939"><margin.target id="marg27"/>27</s></p><p type="main"> <s id="id.000940">Ibidem <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, &longs;ecundum quod I&longs;o­ <lb/>&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> i&longs;tud e&longs;t &longs;ecundum exemplum tertij erroris. </s> <s id="id.000941">Por­ <lb/>rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i <lb/>accideret, vt ex illis tribus vna tantum &longs;pecies, v. <!-- REMOVE S-->g. <!-- REMOVE S-->I&longs;o&longs;celes in mundo re­ <lb/>periretur; <expan abbr="tunc&qacute;">tuncque</expan>; qui&longs;piam de I&longs;o&longs;cele o&longs;tenderet affectionem quampiam, <lb/>putans &longs;e <expan abbr="o&longs;t&etilde;di&longs;&longs;e">o&longs;tendi&longs;&longs;e</expan> pa&longs;&longs;ionem de proprio &longs;ubiecto, & primo, falleretur, quia <lb/>aifectio illa competeret I&longs;o&longs;celi, non vt huic &longs;peciei I&longs;o&longs;celis, &longs;ed quatenus <lb/>e&longs;t triangulum, cui primo, & per &longs;e, & &longs;ecundum quod ip&longs;um conuenit. </s> <s id="id.000942">hoc <lb/>loco di&longs;ce&longs;&longs;imus à Zabarella, qui putat i&longs;tud e&longs;&longs;e exemplum primi erroris, <lb/>cum verba textus adeo clara &longs;int, vt expo&longs;itionem illius nullo modo admit­ <lb/>tant. </s> <s id="id.000943">&longs;unt autem hæc textus verba <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o­ <lb/>&longs;celes, &longs;ecundum quod I&longs;o&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> quibus verbis manife&longs;tè <lb/>apparet Ari&longs;t. accipere pro &longs;ubiecto vniuer&longs;ali non indiuiduum vnum, vt in <lb/>primo errore contingit, &longs;ed &longs;peciem loco generis, &longs;cilicet I&longs;o&longs;celes, quod <lb/>e&longs;t &longs;pecies trianguli pro genere ip&longs;o, nimirum pro Triangulo. </s> <s id="id.000944">ait enim, &longs;i <lb/>non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, <expan abbr="&longs;ecundũ">&longs;ecundum</expan> quod I&longs;o&longs;celes: quibus verbis cla­ <lb/>rè &longs;peciem, non indiuiduum, &longs;igni&longs;icat, ex his duobus exemplis manife&longs;tus <lb/>e&longs;t tertius error, qui erat, quando erat <emph type="italics"/>(vt in parte totum)<emph.end type="italics"/> <expan abbr="quod&qacute;">quodque</expan>; illis verbis <lb/>expo&longs;uerat <emph type="italics"/>(vei contingit etiam, vt in parte totum, in quo mon&longs;tratur: ijs emm, <lb/>quæ &longs;unt in parte inerit quidem demon&longs;tratio, & erit de omni, &longs;ed tamen non erit <lb/>buius primi vniuer&longs;aliter demon&longs;tratio. </s> <s id="id.000945">dico auttm huius primi, &longs;ecundum quod <lb/>buius demonstrationem, quando &longs;it primi vniuer&longs;aliter)<emph.end type="italics"/> ide&longs;t, quando vniuer&longs;ale <lb/>&longs;ubiectum exi&longs;tit quidem, &longs;ed tamen non de ip&longs;o &longs;it demon&longs;tratio, &longs;ed de ali­ <lb/>qua parte ip&longs;ius, v. <!-- REMOVE S-->g. <!-- REMOVE S-->de &longs;pecie aliqua demon&longs;tratur aliquid, quod deberet <lb/>o&longs;tendi primò de ip&longs;o vniuer&longs;ali, cum illi primò competat.</s> </p><p type="main"> <s id="id.000946"><arrow.to.target n="marg28"/></s></p><p type="margin"> <s id="id.000947"><margin.target id="marg28"/>28</s></p><p type="main"> <s id="id.000948">Ibidem <emph type="italics"/>(Et proportionale, quod alternatim, &longs;ecundum quod numeri, & &longs;ecun­ <lb/>dum quod lineæ, & &longs;ecundum quod &longs;olida, & &longs;ecundum quod tempora: quemad­ <lb/>modum & mon&longs;trabatur aliquando &longs;eor&longs;um, contingens <expan abbr="vtiq;">vtique</expan> de omnibus vnica <lb/>demon&longs;tratione mon&longs;irari; &longs;ed quia non &longs;unt nominatum quidam omnia hæc vnum, <lb/>numeri, longitudines, tempora &longs;olida, & &longs;pecie differunt à &longs;einuicem &longs;cor&longs;um <expan abbr="ac-cipiebãtur">ac­ <lb/>cipiebantur</expan>. </s> <s id="id.000949">nunc autem vniuer &longs;aliter mon&longs;tratur, <expan abbr="neq;">neque</expan> enim &longs;ecundum quod lineæ, <lb/>aut &longs;ecundum quod numeri, inerat; &longs;ed &longs;ecundum quod boc, quod vniuer &longs;ale &longs;up­ <lb/>ponunt e&longs;&longs;e)<emph.end type="italics"/> affert exemplum &longs;ecundi erroris, quiaccidit, quando vniuer&longs;a­ <lb/>le exi&longs;tit quidem, &longs;ed tamen e&longs;t innominatum, pro cuius explicatione &longs;cien­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.051.jpg" pagenum="51"/>dum quid &longs;it alterna proportio. </s> <s id="id.000950">Alternam igitur proportionem definit Eu­ <lb/>clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad <expan abbr="anteced&etilde;tem">antecedentem</expan>, <lb/><figure id="id.009.01.051.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.051.1.jpg" place="text"/> <lb/>& con&longs;equentis ad con&longs;equentem. </s> <s id="id.000951">Explico, exponantur qua­ <lb/>tuor quantitates proportionales, v.g. <!-- REMOVE S-->vt 6. ad 3. ita &longs;int 4. ad <lb/>2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al­ <lb/>ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri­ <lb/>mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna­ <lb/>tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e­ <lb/>quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu­ <lb/>dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. </s> <s id="id.000952">quando igi­ <lb/>tur Ari&longs;t. ait, mon&longs;tramus proportionale, ide&longs;t, qua&longs;uis quatuor quantita­ <lb/>tes proportionales, habere hanc proprietatem, vt &longs;int etiam alternatim <lb/>proportionales, & non mon&longs;tramus vnica demon&longs;tratione de omni quouis <lb/>proportionali, &longs;ed &longs;eparatim de magnitudinibus in 16. quinti, de numeris <lb/>in 13. &longs;eptimi, & &longs;eor&longs;um de temporibus in a&longs;tronomia, vel phy&longs;ica; hoc <lb/>modo non o&longs;tendimus vniuer&longs;aliter de primo &longs;ubiecto, quia talis affectio <lb/>conuenit &longs;ingulis, non vt numeri, aut ma gnitudines, aut tempora &longs;unt, &longs;ed <lb/>&longs;ecundum quandam naturam illis omnibus communem, cui primò illa pa&longs;­ <lb/>&longs;io debetur; quæ quidem natura communis nomine caret, & propterea e&longs;t <lb/>cau&longs;a erroris.</s></p><p type="main"> <s id="id.000953"><arrow.to.target n="marg29"/></s></p><p type="margin"> <s id="id.000954"><margin.target id="marg29"/>29</s></p><p type="main"> <s id="id.000955"><emph type="italics"/>Nunc autem vniuer&longs;aliter demon&longs;tratur)<emph.end type="italics"/> nu&longs;quam apud Mathematicos in­ <lb/>uenio hanc demon&longs;trationem vniuer&longs;alem de illo communi omnibus præ­ <lb/>dictis, quare dicendum cum Zabarella, illud, nunc, e&longs;&longs;e intelligendum &longs;ic, <lb/>nunc autem, ide&longs;t, in præ&longs;entia autem deberet vniuer&longs;aliter demon&longs;trari, <lb/>quod tamen cum non &longs;iat, contingit nos decipi putantes vniuer&longs;aliter de­ <lb/>mon&longs;tra&longs;&longs;e. </s> <s id="id.000956">vel dicendum i&longs;tud verificari tantum de lineis, &longs;uperficiebus, & <lb/>&longs;olidis, de quibus &longs;imul in vnica natura communi, quæ e&longs;t magnitudo, de­ <lb/>mon&longs;tratur in 16. quinti vniuer&longs;aliter. </s> <s id="id.000957"><expan abbr="atq;">atque</expan> hoc modo explicatum e&longs;t exem­ <lb/>plum &longs;ecundi erroris, qui verbis illis <emph type="italics"/>(Vel &longs;it quidem, &longs;ed innominatum &longs;it in <lb/>rebus &longs;pecie differentibus)<emph.end type="italics"/> continebatur.</s></p><p type="main"> <s id="id.000958"><arrow.to.target n="marg30"/></s></p><p type="margin"> <s id="id.000959"><margin.target id="marg30"/>30</s></p><p type="main"> <s id="id.000960">Ibidem <emph type="italics"/>(Propter hoc &longs;i quis mon&longs;trauerit &longs;ingulum triangulum. </s> <s id="id.000961">demon&longs;tratio­ <lb/>ne aut vna, aut altera, quod duos rectos habet vnumquodque, <expan abbr="æquilateiũ">æquilateium</expan> &longs;eor&longs;um, <lb/>& &longs;calenum, & æquicrus: nondum nouit triangulum, quod duobus rectis, ni&longs;i &longs;o­ <lb/>phi&longs;tico modo, <expan abbr="neq;">neque</expan> vniuer&longs; aliter triangulum, <expan abbr="neq;">neque</expan> &longs;i vllum e&longs;t præter prædicta <lb/>triangulum alterum. </s> <s id="id.000962">non enim &longs;ecundum quod triangulum, <expan abbr="neq;">neque</expan> omnetriangulum, <lb/>ni&longs;i &longs;ecundum numerum, &longs;ecundum &longs;peciem autem non omne; & &longs;i nullum e&longs;t, quod <lb/>non nouit)<emph.end type="italics"/> vltimo loco ponit exemplum primi erroris, quem &longs;upra verbis il­ <lb/>lis <emph type="italics"/>(Quando vel nibil &longs;it accipere &longs;uperius, præter &longs;ingulare)<emph.end type="italics"/> expre&longs;&longs;erat, quod, <lb/>vt benè intelligamus, opus e&longs;t ea, legere, quæ libro primo Priorum &longs;ecto 3. <lb/>cap. 1. &longs;crip&longs;imus de proprietate illa trianguli, quod &longs;cilicet habet tres an­ <lb/>gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum <lb/>hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no­ <lb/>tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten­ <lb/>ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian­ <lb/>gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, & <lb/>de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de <expan abbr="vno&qacute;">vnoque</expan>; <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.052.jpg" pagenum="52"/>&longs;epatatim o&longs;tendit, aut vtens diuerfis demon&longs;trationibus, vna pro æquila­ <lb/>tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod <expan abbr="vnumquodq;">vnumquodque</expan> <lb/>illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no­ <lb/>uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia <lb/>non cogno&longs;cit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com­ <lb/>munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co­ <lb/>gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian­ <lb/>gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de <lb/>quibus &longs;eparatim <expan abbr="demõ&longs;trauit">demon&longs;trauit</expan>, & &longs;ecundum numernm, ide&longs;t de vnoquoque, <lb/>quatenus e&longs;t vnum numero. </s> <s id="id.000963">non nouit autem &longs;ecundum &longs;peciem, ideft fecun­ <lb/>dum naturam, & formam communem illis tribus indiuiduis, quæ e&longs;t natu­ <lb/>ra trianguli. </s> <s id="id.000964">hoc autem e&longs;&longs;e exemplum primi erroris manife&longs;tè conuincitnr, <lb/>tum ex verbis illis, quando nihil &longs;it &longs;uperius, præter &longs;ingulare, tum ex hu­ <lb/>ius textus verbis illis <emph type="italics"/>(Singulum triangulum)<emph.end type="italics"/> & ex illis <emph type="italics"/>(Ni&longs;i &longs;ecundum nume­ <lb/>rum)<emph.end type="italics"/> ide&longs;t, ni&longs;i de vno, quod &longs;it vnum numero. </s> <s id="id.000965">propterea nos de &longs;in gulari <lb/>triangulo omi&longs;&longs;a Zabarellæ &longs;ententia explicauimus tandem in confirma­ <lb/>tionem no&longs;træ expo&longs;itionis in hæc tria errata illud non omittendum, &longs;atius <lb/>e&longs;&longs;e dicere, Ari&longs;t. attuli&longs;&longs;e pro tribus erratis tria exempla ordine retrogra­ <lb/>do, quàm, quod facit Zabarella, primum e&longs;&longs;e pro tertio, &longs;ecundum pro pri­ <lb/>mo, tertium verò pro &longs;ecundo; eo enim modo, Ari&longs;t. confu&longs;ionem nulla ra­ <lb/>tione, imò contra omnem rationem imponimus.</s></p><p type="main"> <s id="id.000966"><arrow.to.target n="marg31"/></s></p><p type="margin"> <s id="id.000967"><margin.target id="marg31"/>31</s></p><p type="main"> <s id="id.000968">Textu 14. continet quidem quædam mathematica, &longs;ed ferè eadem cum <lb/>&longs;uperioribus, quæ quia tum ex prædictis facile intelligi po&longs;&longs;unt, tum quia <lb/>benè ab expo&longs;itoribus explicantur, ne actum agamus, prætermittimus.</s></p><p type="main"> <s id="id.000969"><arrow.to.target n="marg32"/></s></p><p type="margin"> <s id="id.000970"><margin.target id="marg32"/>32</s></p><p type="main"> <s id="id.000971">Tex. 20. <emph type="italics"/>(Ni&longs;i magnitudines numeri &longs;int)<emph.end type="italics"/> hoc e&longs;t, ni&longs;i magnitudines &longs;int di­ <lb/>fcretæ, ita vt cadant &longs;ub numernm, vt &longs;i linea quæpiam diuidatur in partes <lb/>decem, vel duodecim, tunc euadit quantitas di&longs;creta, &longs;iue numerus. </s> <s id="id.000972">& tunc <lb/>linca numerus e&longs;t. </s> <s id="id.000973">idem de &longs;uperficie, ac &longs;olido intelligendum.</s></p><p type="main"> <s id="id.000974"><arrow.to.target n="marg33"/></s></p><p type="margin"> <s id="id.000975"><margin.target id="marg33"/>33</s></p><p type="main"> <s id="id.000976">Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna <lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus, <lb/>quæ ad nos pertinent, vult Ari&longs;t. docere, quod non debet Geometra o&longs;ten­ <lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam <lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id, <lb/>quod o&longs;tenditur in 4. noni Elem. &longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s> <s id="id.000977">nonnulli latinorum <lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos <lb/>geometricos, at Euclides definit. </s> <s id="id.000978">19. &longs;eptimi, &longs;ic arithmeticum cubum de­ <lb/>finit, cubus numerus e&longs;t, qui &longs;ub tribus numeris æqualibus continetur, qua­ <lb/>lis e&longs;t. </s> <s id="id.000979">8. qui e&longs;t ad in&longs;tar cubi geometrici, & continetur&longs;ub tribus binarijs <lb/>multiplicatis inuicem, quæ multiplicatio &longs;ic in&longs;tituitur, exponuntur tres bi­ <lb/><figure id="id.009.01.052.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.052.1.jpg" place="text"/> <lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></s> <s id="id.000980">4. qui e&longs;t numerus quadratus huius figuræ, <figure id="id.009.01.052.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.052.2.jpg" place="text"/>, deinde <lb/>tertius binarius ducitur in prædictum quadratum 4. & pro­ <lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua­ <lb/>ternarij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe­ <lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.053.jpg" pagenum="53"/>& altitudo, e&longs;t 2. Similiter cubus numerus e&longs;t 27. quia &longs;it ex tribus terna­ <lb/>rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis &longs;it 9. <lb/><figure id="id.009.01.053.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.053.1.jpg" place="text"/> <lb/>qui e&longs;t quadratus. </s> <s id="id.000981">quo deinde ducto in tertium ter­ <lb/>narium, producitur 27. qui e&longs;t cubus, & refert &longs;igu­ <lb/>ram cubicam hanc. </s> <s id="id.000982">Iam verò &longs;i cubus 8. multipli­ <lb/>cet cubum 27. procreabitur 216. qui pariter cubus <lb/>e&longs;t. </s> <s id="id.000983"><expan abbr="atq;">atque</expan> hoc &longs;ibi volunt verba illa, &longs;i duo cubi cubus, <lb/>ide&longs;t, &longs;i duo numeri cubi multiplicentur mutuò, cu­ <lb/>bus alter producetur; ex quibus videas, quam in­ <lb/>eptè illi <expan abbr="interpret&etilde;tur">interpretentur</expan> hunc locum, qui dicunt, Ari­ <lb/>&longs;totilem velle dicere non pertinere ad Geometram <lb/>probare duos cubos geometricos &longs;ibi additos face­ <lb/>re alium cubum, quod erat problema Delphicum de <lb/>duplatione cubi, nondum inuentum; bis enim i&longs;ti peccant, primo in Logi­ <lb/>cam, quia &longs;ic non tran&longs;iret Geometra de genere in genus, ip&longs;ius enim e&longs;t <lb/>agere de duplatione cubi; &longs;ecundò in Mathematicas, cum nondum noue­ <lb/>rint arithmeticos cubos; & præterca ignorent duos cubos &longs;ibi additos, non <lb/>facere alium cubum. </s> <s id="id.000984">Quod præterea hoc loco intelligendi &longs;int cubi arith­ <lb/>metici certò certius con&longs;tat, ex &longs;equenti 24. textu, vbi &longs;ic dicitur <emph type="italics"/>(Veluti <lb/>Arithmetica quidem, quid impar, aut par, aut quadrangulum, aut cubus.)<emph.end type="italics"/></s></p><p type="main"> <s id="id.000985"><arrow.to.target n="marg34"/></s></p><p type="margin"> <s id="id.000986"><margin.target id="marg34"/>34</s></p><p type="main"> <s id="id.000987">Ibidem <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> alij &longs;cientiæ quod alterius, ni&longs;i <expan abbr="quæcunq;">quæcunque</expan> ita &longs;e habent inter &longs;e, <lb/>vt &longs;it alterum &longs;ub altero, vt per&longs;pectiua ad Geometriam, & harmonica ad Arith­ <lb/>meticam)<emph.end type="italics"/> excipit ab illa regula (qua prohibetur, quamuis &longs;cientiam in alie­ <lb/>nam falcem immittere) &longs;cientias &longs;ubalternatas, quæ propriè in Mathemati­ <lb/>cis reperiuntur, Per&longs;pectiua enim propriè &longs;ubalternatur Geometriæ, quia <lb/>vtitur Demon&longs;trationibus linearibus, quas applicat lineis vi&longs;ualibus, & Mu­ <lb/>&longs;ica &longs;ubalternatur Arithmeticæ, quia ab ip&longs;a mutuatur <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> nu­ <lb/>merorum, quas applicat numeris &longs;onoris. </s> <s id="id.000988">v.g. <!-- REMOVE S-->Per&longs;pectiua dicit, ea, quæ vi­ <lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden­ <lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora <lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat <lb/><figure id="id.009.01.053.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.053.2.jpg" place="text"/> <lb/>per 21. primi Elem. &longs;it enim ma­ <lb/>gnitudo vi&longs;a A B, remotior ab o­ <lb/>culo in C, po&longs;ito, & vi&longs;a propin­ <lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo­ <lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s> <s id="id.000989">Hine <lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue <lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem <lb/>optica. </s> <s id="id.000990">Exemplum &longs;ubalternationis Muficæ &longs;it, <expan abbr="con&longs;onãtia">con&longs;onantia</expan> Diapa&longs;on, quam <lb/>vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio­ <lb/>rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa­ <lb/>riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.054.jpg" pagenum="54"/><figure id="id.009.01.054.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.054.1.jpg" place="text"/> <lb/>proportionem, quam 2. ad 1. <lb/>&longs;iue duplam, ergo etiam &longs;o­ <lb/>nus totius chordæ A B, ad <expan abbr="&longs;o-nũ">&longs;o­ <lb/>num</expan> chordæ dimidiæ A C, ha­ <lb/>bebit eandem rationem, <expan abbr="nimirũ">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s> <s id="id.000991">&longs;ed &longs;onus chor­ <lb/>dæ A B, ad &longs;onum chordæ A C, con&longs;onat diapa&longs;on, &longs;eu octauam, ergo in <lb/>data chorda collocata e&longs;t con&longs;onantia diapa&longs;on, quod oportebat. </s> <s id="id.000992">vides me­ <lb/>dium e&longs;&longs;e arithmeticam, conclu&longs;ionem verò harmonicam. </s> <s id="id.000993">Aliud exemplum <lb/>Tonus, quod e&longs;t <expan abbr="interuallũ">interuallum</expan> primæ vocis, Vt, ad &longs;ecundam, Rè, in duo æqua­ <lb/>lia &longs;emitonia diuidi nequit, ratio e&longs;t Arithmetica, quia proportio &longs;uper­ <lb/>particularis in duo æqualia arithmeticè &longs;ecari nequit; at Tonus con&longs;i&longs;tit in <lb/>ratione &longs;uperparticulari, nempè in &longs;e&longs;quioctaua, ergo Tonus bifariam diui­ <lb/>di nequit. </s> <s id="id.000994">de&longs;umptum e&longs;t ex Boetio.</s></p><p type="main"> <s id="id.000995"><arrow.to.target n="marg35"/></s></p><p type="margin"> <s id="id.000996"><margin.target id="marg35"/>35</s></p><p type="main"> <s id="id.000997">Tex. 23. <emph type="italics"/>(Est autem &longs;ic mon&longs;trare, quemadmodum Bry&longs;o quadraturam, &longs;ecun­ <lb/>dum enim commune mon&longs;trant tales rationes)<emph.end type="italics"/> cum velit e&longs;tendere veram de­ <lb/>mon&longs;trationem con&longs;tare debere ex proprijs, non autem ex communibus; <lb/>primum affert exemplum demon&longs;trationis cuiu&longs;dam Bry&longs;onis, quæ ex com­ <lb/>munibus procedat, vt autem benè intelligamus, quale&longs;nam &longs;int huin&longs;modi <lb/>demon&longs;trationes, quæ per communia o&longs;tendunt, legenda prius ea &longs;unt, quæ <lb/>&longs;crip&longs;imus de quadratura circuli in pr&ecedil;dicamento relationis. </s> <s id="id.000998">Bry&longs;o itaque, <lb/>vt tradit Alexander, in hunc modum conabatur quadrare <expan abbr="circulũ">circulum</expan>. </s> <s id="id.000999">&longs;it qua­ <lb/>drandus circulus A B C D, cui circum&longs;eribatur quadratum E F G H. per <lb/>7 quarti, & alterum quadratum I L M N, eidem in&longs;cribatur per 6. quarti, <lb/>quid autem &longs;it circum&longs;cribere, & in&longs;cribere figuram circulo, ex definitione <lb/><figure id="id.009.01.054.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.054.2.jpg" place="text"/> <lb/>3. & 4. eiu&longs;dem libri petatur, quamuis <lb/>ex in&longs;pectione figuræ <expan abbr="pr&ecedil;s&etilde;tis">pr&ecedil;sentis</expan> &longs;atis per­ <lb/>cipi po&longs;&longs;it; deinde aliud <expan abbr="quadratũ">quadratum</expan> me­ <lb/>dium inter prædicta duo con&longs;tituatur, <lb/><expan abbr="&longs;it&qacute;">&longs;itque</expan>; O P Q R. </s> <s id="id.001000">Iam &longs;ic o&longs;tendebat i&longs;tud <lb/>medium quadratum e&longs;&longs;e æquale circu­ <lb/>lo propo&longs;ito. </s> <s id="id.001001"><expan abbr="Quæcunq;">Quæcunque</expan> &longs;unt, &longs;imul ma­ <lb/>iora eodem, & minora eodem, &longs;unt in­ <lb/>uicem æqualia, &longs;ed circulus, & quadra­ <lb/>tum medium, &longs;unt ambo maiora qua­ <lb/>drato in&longs;cripto, & ambo minora qua­ <lb/>drato circum&longs;cripto, ergo circulus, & <lb/>quadratum medium, &longs;unt æqualia. </s> <s id="id.001002">vte­ <lb/>batur, inquit Ari&longs;t pr&ecedil;dicto principio, <lb/>etiam numeris, lineis, temporibus, & <lb/>qualitatibus communi, <expan abbr="neq;">neque</expan> deducto ex natura circuli, aut quadrati, de qui­ <lb/>bus erat demon&longs;tratio. </s> <s id="id.001003">præterea aduertendum e&longs;t, illud e&longs;&longs;e fal&longs;um, nam &longs;ex, <lb/>& quinque, ambo &longs;unt maiores, quam quatuor, & minores, quam &longs;eptem, <lb/>& tamen non &longs;unt æquales.</s></p><p type="main"> <s id="id.001004"><arrow.to.target n="marg36"/></s></p><p type="margin"> <s id="id.001005"><margin.target id="marg36"/>36</s></p><p type="main"> <s id="id.001006">In codem textu <emph type="italics"/>(<expan abbr="Vnumquodq;">Vnumquodque</expan> autem &longs;cimus, non &longs;ecundum accidens, quando <lb/>&longs;ecundum illud cogno&longs;camus, &longs;ecundum quod ine&longs;t ex principijs illius, &longs;ecundam <lb/>quod illud; vt duobus rectis æquales, habere, cui ine&longs;t per &longs;e, quod dictum e&longs;t ex<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.055.jpg" pagenum="55"/><emph type="italics"/>principijs huius)<emph.end type="italics"/> affert nunc exemplum alterius demon&longs;trationis, quæ non <lb/>ex communibus, vt præcedens Bry&longs;onis, &longs;ed ex proprijs principijs o&longs;tendit <lb/>affectionem de &longs;ubiecto proprio. </s> <s id="id.001007">E&longs;t autem illud exemplum toties decan­ <lb/>tatum de triangulo habente tres angulos æquales duobus rectis angulis; id­ <lb/>circo operæpretium e&longs;&longs;e puto explicare demon&longs;trationem, 32. primi Eucli­ <lb/>dis, quæ i&longs;tud ex proprijs principijs demon&longs;trat, & quam hoc loco Ari&longs;to­ <lb/>teles innuit, hoc enim modo ip&longs;ius Ari&longs;t. mentem probè penetrare poteri­ <lb/><figure id="id.009.01.055.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.055.1.jpg" place="text"/> <lb/>mus. </s> <s id="id.001008">&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. <!-- KEEP S--></s> <s id="id.001009">Dico ag­ <lb/>gregatum <expan abbr="triũ">trium</expan> ip&longs;ius angulorum A, B, C, <lb/>e&longs;&longs;e æquale aggregato ex duobus angu­ <lb/>lis rectis (vt autem melius intelligas, quæ <lb/>&longs;equuntur, lege prius ea, quæ dicta &longs;unt <lb/>in lib. 1. Priorum &longs;ecto 3. cap. 1.) produ­ <lb/>catur latus B C, <expan abbr="v&longs;q;">v&longs;que</expan> in D, vt fiat angulus <lb/>externus A C D; Iam &longs;ic, quoniam <expan abbr="pro-batũ">pro­ <lb/>batum</expan> e&longs;t in 13. primi, duos angulos, quos <lb/>facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares <lb/>duobus rectis: & quia pariter in prima parte huins propo&longs;. </s> <s id="id.001010">32. probatum <lb/>e&longs;t ab Euclide duos angulos A B, e&longs;&longs;e æquales externo angulo A C D: &longs;i ter­ <lb/>tius angulus reliquus A C B, &longs;umatur bis, &longs;emel cum duobus angulis A, B, <lb/>& &longs;emel cum externo A C D, <expan abbr="add&etilde;tur">addentur</expan> æqualia æqualibus, & propterea tres <lb/>anguli A, B, A C B, &longs;imul &longs;umpti, erunt æquales duobus A C D, A C B, &longs;imul <lb/>&longs;umptis; &longs;ed his duobus &longs;unt æquales duo recti, ergo cum quæ &longs;unt æqualia <lb/>vni tertio, &longs;int etiam æqualia inuicem, erit aggregatum trium angulorum <lb/>A, B, A C B, æquale aggregato duorum rectorum; quod erat demon&longs;tran­ <lb/>dum. </s> <s id="id.001011">Medium <expan abbr="itaq;">itaque</expan> huius demon&longs;trationis, &longs;i res ad trutinam Logicam ex­ <lb/>pendatur, e&longs;t, quod partes aggregati <expan abbr="triũ">trium</expan> <expan abbr="angulorũ">angulorum</expan> A, B, A C B, &longs;unt æqua­ <lb/>les partibus aggregati <expan abbr="duorũ">duorum</expan>, & ideo <expan abbr="aggregatũ">aggregatum</expan>, aggrega to æqua­ <lb/>le e&longs;t. </s> <s id="id.001012">quod medium e&longs;t in genere cau&longs;æ materialis. </s> <s id="id.001013">quod verò partes illius <lb/>&longs;int æquales partibus huius, probatur, per dignitatem <expan abbr="illã">illam</expan>, quæ &longs;unt æqualia <lb/>vni tertio, &longs;unt etiam inter &longs;e. </s> <s id="id.001014">partes porrò aggregati trium angulorum <lb/>erant hæ, anguli A, B, vna; altera verò angulus A C B; partes verò aggre­ <lb/>gati duorum rectorum erant A C B, A C D, quibus partibus, illæ &longs;unt æqua­ <lb/>les, & ideo totum toti æquale. </s> <s id="id.001015">quod medium e&longs;t omnino intrin&longs;ecum, & ex <lb/>proprijs ip&longs;ius trianguli, &longs;iue ex proprijs angulorum ip&longs;ius, cum &longs;int ip&longs;ius <lb/>partes. </s> <s id="id.001016">quod pariter medium ex parte pa&longs;&longs;ionis, quæ demon&longs;tratur, e&longs;t ex <lb/>proprijs, cum &longs;int partes illius materiales. </s> <s id="id.001017">per materiam autem oportet <lb/>hoc loco intelligere materiam intelligibilem, ide&longs;t quantitatem à qualita­ <lb/>tibas ab&longs;tractam, & terminatam, de qua pluribus agemus infra in tractatu <lb/>de natura mathematicarum. </s> <s id="id.001018">Hinc videas eos magnopere decipi, qui pu­ <lb/>tant, hanc demon&longs;trationem e&longs;&longs;e per extrin&longs;eca, eò quod ad demon&longs;tran­ <lb/>dum producatur linea B C, in D, putantes lineam illam productam C D, <lb/>e&longs;&longs;e demon&longs;trationis medium; lineæ <expan abbr="namq;">namque</expan> huiu&longs;modi, quæ in demon&longs;tra­ <lb/>tionibus geometricis con&longs;truuntur, nunquam &longs;unt media propria demon­ <lb/>&longs;trationum, &longs;ed tantummodo a&longs;&longs;umuntur ad probandum medium iam ex­ <lb/>cogitatum e&longs;&longs;e veram cau&longs;am conclu&longs;ionis. </s> <s id="id.001019">Hinc etiam manife&longs;tè colligas <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.056.jpg" pagenum="56"/>Mathematicas facultates habere demon&longs;trationes perfecti&longs;&longs;imas, quod <lb/>ageometreti negare &longs;olent, &longs;ed audacter aiunt exempla Ari&longs;t. non e&longs;&longs;e vera: <lb/><expan abbr="neq;">neque</expan> requiri veritatem exemplorum; in <expan abbr="quorũ">quorum</expan> <expan abbr="vtroq;">vtroque</expan> peccant, nam dictum <lb/>illud v&longs;urpari &longs;olet, & debet de exemplis moralibus. </s> <s id="id.001020">at vero requiri confor­ <lb/>mitatem exemplorum cum regulis traditis, nemo &longs;anæ mentis dubitabit. <lb/></s> <s id="id.001021">Vernm i&longs;ti confundunt conformitatem cum veritate. </s> <s id="id.001022">Veritas exemplo tunc <lb/>ine&longs;t, quando illud, quod in exemplo narratur, verè extitit, vt &longs;i quis in <lb/>exemplum pudicitiæ afferret hi&longs;toriam Io&longs;ephi, <expan abbr="verũ">verum</expan> i&longs;tuà e&longs;&longs;et exemplum. <lb/></s> <s id="id.001023">quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla <lb/>&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. <!-- REMOVE S-->g. <!-- REMOVE S-->narratur ab <lb/>Ari&longs;t. de quodam filio, qui patrem crudeliter traxerat, qui po&longs;tea grandior <lb/>factus, cum filium procrea&longs;&longs;et, ab eodem pariter raptatus e&longs;t ip&longs;e, v&longs;que ad <lb/>eundem locum, quo ip&longs;e patrem &longs;uum impiè raptauerat. </s> <s id="id.001024">non e&longs;t nece&longs;&longs;e, ta­ <lb/>lem extiti&longs;&longs;e filium, <expan abbr="neq;">neque</expan> patrem. </s> <s id="id.001025">Verumtamen &longs;emper conformitas exem­ <lb/>pli cum regulis, & præceptis, quæ traduntur nece&longs;&longs;aria e&longs;t, alioquin exem­ <lb/>pla de&longs;truerent id, quod præceptio con&longs;truit, <expan abbr="illi&qacute;">illique</expan> contraria e&longs;&longs;et, quod om­ <lb/>nino ab&longs;urdum foret. </s> <s id="id.001026">non &longs;ecus, ac &longs;i quis vellet alium docere characteres <lb/>latinos, <expan abbr="illi&qacute;">illique</expan>; barbaros, quos Gothicos vocant in exemplum proponeret. </s> <s id="id.001027">re­ <lb/>quiritur igitur &longs;emper in omni exemplo conformitas cum eo, quod doce­ <lb/>tur; in moralibus tamen non &longs;emper requiritur veritas, vti diximus; Alij <lb/>verò dicunt non requiri in exemplis determinatam veritatem, &longs;ed &longs;atis e&longs;&longs;e, <lb/>&longs;i exemplum verum &longs;it &longs;ecundum opinionem aliquorum: <expan abbr="quorũ">quorum</expan> &longs;ententiam <lb/>non improbamus. </s> <s id="id.001028">Exempla igitur ab Ari&longs;t. pa&longs;&longs;im ex mathem aticis allata, <lb/>congrua, <expan abbr="conformia&qacute;">conformiaque</expan>; omninò &longs;unt ip&longs;ius doctrinæ, aliter ip&longs;um perpetuò <lb/>mentientem facimus. </s> <s id="id.001029">Po&longs;tremò illud etiam e&longs;t aduertendum, fortè Ari&longs;t. in <lb/>præ&longs;enti textu &longs;pecta&longs;&longs;e <expan abbr="nõ">non</expan> ad hanc Euclidianam demon&longs;trationem, &longs;ed po­ <lb/>tius ad Pithagoricam. </s> <s id="id.001030">Pithagorei enim eam aliter, quamuis per idem me­ <lb/>dium, &longs;cilicet à cau&longs;a materiali, demon&longs;trabant; con&longs;truebant enim aliter, <lb/><expan abbr="neq;">neque</expan> vlla vtebantur diui&longs;ione. </s> <s id="id.001031">quod dictum velim propter nonnullos, qui ab <lb/>huiu&longs;modi diui&longs;ionibus abhorrent, <expan abbr="timent&qacute;">timentque</expan>; ne demon&longs;trationis perfectio­ <lb/>ni per eas plurimum derogetur. </s> <s id="id.001032">Pithagoreorum demon&longs;trationem vide <lb/>apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro­ <lb/>clus in comm. <!-- REMOVE S-->eiu&longs;dem recitat.</s> </p><p type="main"> <s id="id.001033"><arrow.to.target n="marg37"/></s></p><p type="margin"> <s id="id.001034"><margin.target id="marg37"/>37</s></p><p type="main"> <s id="id.001035">Ibidem <emph type="italics"/>(Sed quemadmodŭ harmonica per Arithmeticam)<emph.end type="italics"/> vide &longs;upra tex. <!-- REMOVE S-->20.</s> </p><p type="main"> <s id="id.001036"><arrow.to.target n="marg38"/></s></p><p type="margin"> <s id="id.001037"><margin.target id="marg38"/>38</s></p><p type="main"> <s id="id.001038">Ibidem <emph type="italics"/>(Demon&longs;tratio autem non computatur in aliud genus; m&longs;i, vt dictum <lb/>e&longs;t geometricæ demon&longs;trationes in Per&longs;pectiuas, aut Mcchamcas, & arithmeticæ in <lb/>harmonicas)<emph.end type="italics"/> exempla &longs;ubalternationis Per&longs;pectiuæ, & Mu&longs;icæ in tex. <!-- REMOVE S-->20. at­ <lb/>tulimus; nunc Mechanicæ &longs;ubalternationis, quam hic Ari&longs;t. in&longs;inuat, exem­ <lb/>plum &longs;it illud, quod Archimedes prop. 14. primi Aequep. <!-- REMOVE S-->demon&longs;trat, ni­ <lb/>mirum centrum grauitatis omnis trianguli e&longs;&longs;e punctum illud, in quo rectæ <lb/>lineæ ab angulis trianguli ad dimidia latera oppo&longs;ita ductæ concurrunt. </s> <s id="id.001039">&longs;it <lb/>triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita <lb/>vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F. <lb/><!-- KEEP S--></s> <s id="id.001040">Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli. </s> <s id="id.001041">Quoniam enim in 13. <lb/>Aequep. <!-- REMOVE S-->probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo <lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centrũ">centrum</expan> grauitatis. <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.057.jpg" pagenum="57"/><figure id="id.009.01.057.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.057.1.jpg" place="text"/> <lb/>&longs;ed eadem ratione erit etiam in linea B E, er­ <lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;olũ">&longs;olum</expan> e&longs;t in vtra­ <lb/>que, quod erat demon&longs;trandum. </s> <s id="id.001042">ex quibus ap­ <lb/>paret, qua ratione mechanica conclu&longs;io Geo­ <lb/>metriæ &longs;ubiaceat, dum lineari di&longs;cur&longs;u ip&longs;a <lb/>demon&longs;tratio perficitur. </s> <s id="id.001043">Scias præterea cen­ <lb/>trum grauitatis e&longs;&longs;e tale punctum, ex quo &longs;i &longs;u­ <lb/>&longs;pendatur corpus triangulare vniformis cra&longs;­ <lb/>&longs;itici, manet &longs;emper horizonti parallelum, &longs;i <lb/>tamen antequam &longs;u&longs;penderetur, iacebat plano horizontis, æquidi&longs;tans; <lb/><expan abbr="neq;">neque</expan> &longs;i &longs;u&longs;pen&longs;um feratur huc illud nutat, &longs;ed &longs;emper in <expan abbr="cod&etilde;">codem</expan> &longs;itu per&longs;euerat.</s></p><p type="main"> <s id="id.001044"><arrow.to.target n="marg39"/></s></p><p type="margin"> <s id="id.001045"><margin.target id="marg39"/>39</s></p><p type="main"> <s id="id.001046">Tex. 24. <emph type="italics"/>(Veluti Arithmetica quidem quid impar, aut par; aut quadrangu­ <lb/>lum, aut cubus)<emph.end type="italics"/> cogno&longs;cas hinc certò certius quadrangulum, & cubum e&longs;&longs;e <lb/>&longs;pecies numerorum, &longs;icuti &longs;upra tex. <!-- REMOVE S-->9. & 20. explicauimus, quò nunc te vi­ <lb/>ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.</s> </p><p type="main"> <s id="id.001047"><arrow.to.target n="marg40"/></s></p><p type="margin"> <s id="id.001048"><margin.target id="marg40"/>40</s></p><p type="main"> <s id="id.001049">Ibidem <emph type="italics"/>(Geometrica verò quid irrationale, aut refrangi, aut concurrere)<emph.end type="italics"/> per <lb/>verbum, irrationale, non videtur Ari&longs;t. intellexi&longs;&longs;e proprietatem illam duo­ <lb/>rum linearum incommen&longs;urabilium longitudine, & potentia, quia v&longs;us fui&longs;­ <lb/>&longs;et verbo, <foreign lang="greek">a/rrpton.</foreign> quod apud Geometras v&longs;urpari &longs;olet in illa &longs;ignificatio­ <lb/>ne, &longs;ed v&longs;us e&longs;t verbo, <foreign lang="greek">a\logon,</foreign> quod latinè redditur improportionale.</s></p><p type="main"> <s id="id.001050"><arrow.to.target n="marg41"/></s></p><p type="margin"> <s id="id.001051"><margin.target id="marg41"/>41</s></p><p type="main"> <s id="id.001052">Per verbum <emph type="italics"/>(Refrangi)<emph.end type="italics"/> &longs;eu frangi, intelligit lineam aliquam rectam, non <lb/>in directum rendere, &longs;ed in aliquo puncto frangi, &longs;eu declinari à rectitudine, <lb/>ita vt con&longs;tituat angulum.</s></p><p type="main"> <s id="id.001053">Per verbum <emph type="italics"/>(Concurrere)<emph.end type="italics"/> intelligit, non e&longs;&longs;e parallelas, &longs;ed ad idem ali­ <lb/>quod punctum coire, &longs;i protrahantur.</s></p><p type="main"> <s id="id.001054"><arrow.to.target n="marg42"/></s></p><p type="margin"> <s id="id.001055"><margin.target id="marg42"/>42</s></p><p type="main"> <s id="id.001056">Ibidem <emph type="italics"/>(Et Astrologia &longs;imiliter)<emph.end type="italics"/> per A&longs;trologiam intelligit Ari&longs;t. non iu­ <lb/>diciariam, quamuis à recentioribus hoc nomine vocetur, &longs;ed quam hodie <lb/>dicunt A&longs;tronomiam, <expan abbr="ait&qacute;">aitque</expan>; ip&longs;am con&longs;iderare quantitatem, figuram, mo­ <lb/>tum, & locum totius Mundi, ac partium ip&longs;ius integrantium, vt &longs;unt Cœli, <lb/>& Elementa.</s></p><p type="main"> <s id="id.001057"><arrow.to.target n="marg43"/></s></p><p type="margin"> <s id="id.001058"><margin.target id="marg43"/>43</s></p><p type="main"> <s id="id.001059">Tex. 25. <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> Geometra fal&longs;a &longs;upponit, quemadmodum quidam a&longs;&longs;eruere di­ <lb/>centes, quod non oportet fal&longs;o vti: Geometram verò mentiri dicentem pedalem, non <lb/>pedalem, aut rectam de&longs;criptam, non rectam <expan abbr="exist&etilde;tem">existentem</expan>: Geometra verò nihil con­ <lb/>cludit eò, quod bæc e&longs;t linea, &longs;ed quæ per hæc e&longs;tenduntur)<emph.end type="italics"/> innuit his verbis eam <lb/>materiam intelligibilem, quæ e&longs;t &longs;ubiectum Geometriæ: eam &longs;cilicet, quæ <lb/>&longs;ub figuris Geometricis &longs;en&longs;ibilibus, & <expan abbr="plerunq;">plerunque</expan> fal&longs;is latet; nam &longs;æpè Geo­ <lb/>metra vtitur linea quadam &longs;en&longs;ibili pro recta, quæ verè nec e&longs;t linea mathe­ <lb/>matica, nec recta; &longs;upponit aliquando talem lineam e&longs;&longs;e pedalem, quæ ve­ <lb/>rè non e&longs;t pedalis: Verumtamen non mentitur, quia re&longs;picit ad veram li­ <lb/>neam mathematicam, quæ &longs;ub illa intelligitur, & quæ recta concipitur; & <lb/>quidem hæc omnia verè concipiuntur, quoniam ita e&longs;&longs;e re vera po&longs;&longs;unt.</s></p><p type="main"> <s id="id.001060"><arrow.to.target n="marg44"/></s></p><p type="margin"> <s id="id.001061"><margin.target id="marg44"/>44</s></p><p type="main"> <s id="id.001062">Tex. 28. <emph type="italics"/>(Coaltern as verò coincidere)<emph.end type="italics"/> per coalternas intelligendas e&longs;&longs;e pa­ <lb/>rallelas lineas, alias, & nunc <expan abbr="quoq;">quoque</expan> monemus.</s></p><p type="main"> <s id="id.001063"><arrow.to.target n="marg45"/></s></p><p type="margin"> <s id="id.001064"><margin.target id="marg45"/>45</s></p><p type="main"> <s id="id.001065">Tex. 29. <emph type="italics"/>(In Matbematicis verò non est &longs;imiliter paralogi&longs;mus, quoniam me­ <lb/>diŭ e&longs;t &longs;emper, quod duplex, de hoc enim omni, & hoc rur&longs;us de alio dicitur omni)<emph.end type="italics"/> <lb/>aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.058.jpg" pagenum="58"/>di&longs;ciplinis, idem tamen apud græcos <foreign lang="greek">maqhmata</foreign> &longs;unt, ac apud latinos di&longs;ci­ <lb/>plmæ; verbum autem <foreign lang="greek">maqhmata</foreign> v&longs;urpat hoc loco Ari&longs;toteles. <!-- KEEP S--></s> <s id="id.001066">Porrò non <lb/>e&longs;t in mathematicis, &longs;icut in alijs paralogi&longs;mus, quia in omni demon&longs;tra­ <lb/>tione maius extremum dicitur de omni medio, & rur&longs;us medium dicitur de <lb/>omni minori extremo, ac &longs;i diceret mathematicæ demon&longs;trationes &longs;unt in <lb/>primo modo, qui barbarè à latinis recentioribus Barbara appellatur. </s> <s id="id.001067">Hæc <lb/>e&longs;t autem pulcherrima mathematicarum commendatio, quippe præclarum <lb/>e&longs;t à laudato laudari. </s> <s id="id.001068">In mathematicis, inquit, non accidit &longs;imiliter para­ <lb/>logi&longs;mus, ide&longs;t, tam frequenter, quemadmodum in &longs;yllogi&longs;mis dialecticis, <lb/>quia modus argumentandi mathematicarum e&longs;t perfecti&longs;&longs;imus, quippe in <lb/>primo modo primæ figuræ.</s></p><p type="main"> <s id="id.001069"><arrow.to.target n="marg46"/></s></p><p type="margin"> <s id="id.001070"><margin.target id="marg46"/>46</s></p><p type="main"> <s id="id.001071">Eodem tex. <emph type="italics"/>(Contingit autem quo&longs;dam non &longs;yllogi&longs;ticè dicere, & quod ex vtri&longs;­ <lb/>que con&longs;equentia accipiunt, quemadmodum & Cæneus facit, quod ignis in multi­ <lb/>plici proportione: etenim ignis celeriter gignitur, vt ait: & hæc est proportio. </s> <s id="id.001072">&longs;ic <lb/>autem non e&longs;t &longs;yllog &longs;mus, ni&longs;i celerrimam proportio &longs;equatur multiplex: & ignem <lb/>celerrima in motu proportio)<emph.end type="italics"/> verba illa (in multiplici proportione) græcè &longs;ic <lb/>&longs;e habent, <foreign lang="greek">en th pollaplasioni analogia,</foreign> quod melius redditur latinè in mul­ <lb/>tiplici proportione, quemadmodum fecimus, quam in multiplicata, quem­ <lb/>admodum in vulgata editione. </s> <s id="id.001073">porrò quid inter multiplicem, & multipli­ <lb/>catam rationem inter&longs;it, optimè declarat no&longs;ter Clauius ad 4. definit. </s> <s id="id.001074">lib. 5. <lb/>Elem. ex quo etiam loco pauca decerpam, quæ huic loco declarando con­ <lb/>ducunt. </s> <s id="id.001075">Proportio igitur multiplex e&longs;t habitudo inter duas quantitates in­ <lb/>æquales, quarum maior continet minorem, bis, vel ter, vel quater, &c. </s> <s id="id.001076">vn­ <lb/>de proportio multiplex habet &longs;ub &longs;e genera infinita, quando enim maior <lb/>continet minorem bis, dicitur Dupla: quando ter, Tripla: quando quater, <lb/>Quadrupla: & &longs;ic in infinitum: v. <!-- REMOVE S-->g. <!-- REMOVE S-->2. ad 1. e&longs;t proportio dupla; 3. ad 1. tri­ <lb/>pla; 4. ad 1. quadrupla, &c. </s> <s id="id.001077">omnes tamen continentur &longs;ub genere multipli­ <lb/>cis rationis. </s> <s id="id.001078">porrò &longs;i qu&ecedil;piam proportio ex genere multiplici progrediatur <lb/>per plures terminos, v. <!-- REMOVE S-->g. <!-- REMOVE S-->proportio quadrupla progrediatur hoc modo, <lb/>1. 4. 16. 64. 256. &c. </s> <s id="id.001079">fit, vt &longs;ub&longs;equentes termini mirum in modum augean­ <lb/>tur. </s> <s id="id.001080">hic vides primum ip&longs;am quadruplam rationem in di&longs;po&longs;itis terminis <lb/>progredi, quia quilibet &longs;equens terminus ad præcedentem e&longs;t quadruplus. <lb/></s> <s id="id.001081">cernis etiam in paucis terminis, quinque &longs;cilicet magnum factum e&longs;&longs;e incre­ <lb/>mentum, cum <expan abbr="v&longs;q;">v&longs;que</expan> ad 256. excreuerint. </s> <s id="id.001082">Cæneus igitur dicens ignem augeri <lb/>&longs;ecundum multiplicem rationem, vnam ex prædictis intelligebat aliquam, <lb/>quia quælibet illarum magnopere cre&longs;cit, &longs;i propagetur, vt ad 10. quinti <lb/>definit. </s> <s id="id.001083">traditur: & vt paulo ante exemplo licuit per&longs;picere. </s> <s id="id.001084">argumentaba­ <lb/>tur igitur Cæneus in hunc modum; quod in multiplici ratione augetur, ce­ <lb/>lerrimè augetur: ignis celerrimè augetur, ergo ignis in multiplici ratione <lb/>augetur, quæ argumentatio vitio&longs;a e&longs;t, ex duabus quippe affirmatiuis in &longs;e­ <lb/>cunda figura procedens, vt colligitur ex verbis illis tex. <emph type="italics"/>(Ex viri&longs;que con&longs;e­ <lb/>quentia accipiunt<emph.end type="italics"/>) ex his mathematica huius locis patere &longs;atis po&longs;&longs;unt.</s></p><p type="main"> <s id="id.001085"><arrow.to.target n="marg47"/></s></p><p type="margin"> <s id="id.001086"><margin.target id="marg47"/>47</s></p><p type="main"> <s id="id.001087">Ibidem (<emph type="italics"/>Conuertuntur autem magis, quæ&longs;unt in mathematicis, quoniam nul­ <lb/>lum accidens accipiunt (m quo quidem ijs præ&longs;tăt, quæ di&longs;putationibus traduntur) <lb/>&longs;ed definitiones<emph.end type="italics"/>) Hæc e&longs;t altera mathematicarum laus, vnde earum quoque <lb/>præ&longs;tantia elucet, quia &longs;cilicet mathematicæ pro medijs vtuntur definitio­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.059.jpg" pagenum="59"/>nibus &longs;ubiecti, aut pa&longs;&longs;ionis, quæ nuilo modo &longs;unt accidentalia conclu&longs;ioni, <lb/>v. <!-- REMOVE S-->g. <!-- REMOVE S-->in prima Euclidis demon&longs;tratione per definitionem &longs;ubiecti probantur <lb/>tres lineæ e&longs;&longs;e æquales, quia nimirum &longs;int &longs;emidiametri circulorum æqua­ <lb/>lium, quæ e&longs;t ip&longs;arum definitio. </s> <s id="id.001088">& in 4. primi probantur ba&longs;is, & anguli <lb/>vnius trianguli æquales e&longs;&longs;e ba&longs;i, & angulis alterius trianguli per formalem <lb/>definitionem pa&longs;&longs;ionis, videlicet æqualitatis, quæ traditur in octauo axio­ <lb/>mate &longs;ic, quæ &longs;ibi mutuo congruunt, ea inter &longs;e &longs;unt æqualia. </s> <s id="id.001089">probat igitur <lb/>Euclides in quarta ba&longs;im, & angulos vnius trianguli e&longs;&longs;e æqualia ba&longs;i, & an­ <lb/>gulis alterius trianguli, quia o&longs;tendit, quod, &longs;i ba&longs;is illa huic ba&longs;i, & illi an­ <lb/>guli hi&longs;ce angulis &longs;uperponantur, congruunt; ex qua congruentia mutua, <lb/>quæ e&longs;t æqualitatis definitio, infert æqualitatem ip&longs;arum ba&longs;ium, necnon <lb/>angulorum. </s> <s id="id.001090">eadem deinde æqualitatis definitione totam demon&longs;trationem <lb/>concludit, &longs;cilicet totum triangulum toti triangulo æquale e&longs;&longs;e, quia vnum <lb/>alteri congruat. </s> <s id="id.001091">A&longs;tronomi <expan abbr="quoq;">quoque</expan> demon&longs;trant eclyp&longs;im de Luna, per in­ <lb/>rerpo&longs;itionem terræ inter Lunam, & Solem, quæ interpo&longs;itio e&longs;t definitio <lb/>cau&longs;alis ip&longs;ius eclyp&longs;is, &longs;cilicet pa&longs;&longs;ionis. </s> <s id="id.001092">huiu&longs;modi <expan abbr="&longs;exc&etilde;tas">&longs;excentas</expan> reperies apud <lb/>Geometras, Arithmeticos, A&longs;tronomos, <expan abbr="cæteros&qacute;">cæterosque</expan>; Mathematicas demon­ <lb/>&longs;trationes: ita vt meritò dixerit Ari&longs;t. Mathematicas alias omnes natura­ <lb/>les &longs;cientias, quæ di&longs;putabilibus rationibus traduntur ex hac parte antecel­ <lb/>lere. </s> <s id="id.001093">a&longs;&longs;umunt igitur terminos conuertibiles, quia adhibent &longs;æpè definitio­ <lb/>nes ad demon&longs;trandum. </s> <s id="id.001094">Reliqua logici expo&longs;itores declarant.</s></p><p type="main"> <s id="id.001095"><arrow.to.target n="marg48"/></s></p><p type="margin"> <s id="id.001096"><margin.target id="marg48"/>48</s></p><p type="main"> <s id="id.001097">Tex. 30. (<emph type="italics"/>Rur&longs;us quemadmodum mon&longs;trant Lunam, quod &longs;phærica &longs;it per aug­ <lb/>menta: &longs;i enim quod ita augetur, e&longs;t &longs;phæricum; augetur autem Luna; planŭ quod <lb/>&longs;phærica<emph.end type="italics"/>) Illius demon&longs;trationis, quæ ab effectu procedit, affert exemplum <lb/>ex a&longs;tronomia; A&longs;tronomi enim demon&longs;trant Lunam e&longs;&longs;e &longs;phæricam ab ef­ <lb/>fectu ip&longs;ius &longs;phæricitatis, qui e&longs;t illuminatio &longs;phærica: &longs;ic enim ratiocinan­ <lb/>tur: ea, quæ &longs;phæricè illuminantur &longs;unt &longs;phærica, Luna &longs;phæricè illumina­ <lb/>tur, ergo &longs;phærica e&longs;t: quæ argumentatio fu&longs;ius explicanda e&longs;t; quod ait, <lb/>quod ita augetur, ide&longs;t, &longs;phæricè, e&longs;t &longs;phæricum, ide&longs;t, quia lumen nouæ Lu­ <lb/>næ augetur &longs;phæricè, hoc e&longs;t, ad eum modum, quo quæuis &longs;phæra obiecta <lb/>corpori lumino&longs;o &longs;olet illuminari. </s> <s id="id.001098">illuminatio porrò Lunæ in &longs;e &longs;emper e&longs;t <lb/>eadem, quia &longs;emper dimidium Lunæ quod Solem a&longs;picit, illuminatur; dici­ <lb/>tur tamen augeri re&longs;pectu oculi no&longs;tri, quia &longs;cilicet initio facto à nouilunio <lb/>pars illuminata incipit quotidie magis vergere ad oculum no&longs;trum, ita vt <lb/>in dies maiorem, ac maiorem illuminationem videamus, donec opponatur <lb/>Soli, in qua oppo&longs;itione totum ferè Lunæ <expan abbr="illuminatũ">illuminatum</expan> con&longs;picitur. </s> <s id="id.001099">Vt autem <lb/>huius illuminationis non iniucundam f cias experientiam; cape &longs;phæram <lb/>quampiam &longs;olidam manu, cum qua recede ad medium cubiculi, & pone lu­ <lb/>men &longs;eor&longs;um ad partem aliquam: deinde brachio exten&longs;o oppone &longs;phæram <lb/>lumini, quo &longs;itu nihil de illuminatione videbis, quamuis dimidium ferè il­ <lb/>lius illuminetur. </s> <s id="id.001100">po&longs;tea conuerte te ip&longs;um ibidem paulatim, ita vt aliquid <lb/>illuminationis oculo tuo appareat; & videbis partem illam illuminationis, <lb/>falcatæ, &longs;eu nouæ Lunæ &longs;imilem. </s> <s id="id.001101">Deinde adhuc magis te conuerte, & cer­ <lb/>nes illuminationem dimidiatæ Lunæ &longs;imilem: verte adhuc te ip&longs;um donec <lb/>&longs;it &longs;phæra ita lumini oppo&longs;ita, vt inter ip&longs;am, & lumen oculus tuus &longs;it me­ <lb/>dius; apparebit tunc tota illuminatio, quæ erit in&longs;tar plenilunij. </s> <s id="id.001102">perge ad­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.060.jpg" pagenum="60"/>huc te ip&longs;um conuertere, & videbis paulatim lumen oculo tuo decre&longs;cere <lb/>non aliter ac in Luna &longs;ene&longs;cente. </s> <s id="id.001103"><expan abbr="atq;">atque</expan> hoe e&longs;t &longs;phæricè illuminari, fierique <lb/>&longs;phærica illuminationis augmenta. </s> <s id="id.001104">cum ergo videamus Lunam eo modo lu­ <lb/>mine augeri, quo &longs;phæra, hinc ip&longs;am <expan abbr="quoq;">quoque</expan> &longs;phæricam-e&longs;&longs;e argumentamur.</s></p><p type="main"> <s id="id.001105"><arrow.to.target n="marg49"/></s></p><p type="margin"> <s id="id.001106"><margin.target id="marg49"/>49</s></p><p type="main"> <s id="id.001107">Po&longs;t nonnulla (<emph type="italics"/>Vt Per&longs;pectiua ad Geometriam, & Mechanica ad Stereome­ <lb/>tricam, & Harmonica ad Arithmeticam, vt Apparentia ad A&longs;trologicam<emph.end type="italics"/>) &longs;upra <lb/>tex. <!-- REMOVE S-->20. exempla &longs;ubalternationum Per&longs;pectiuæ, & Mechanicæ cum Geo­ <lb/>metria &longs;unt allata. </s> <s id="id.001108">hic primo notandum Stereometriam non ef&longs;e &longs;cientiam <lb/>di&longs;tinctam à Geometria, ni&longs;i &longs;icuti partem à toto: nam cum Geometria <lb/>con&longs;ideret quantitatem, &longs;ecundum tres dimen&longs;iones, longitudinem, latitu­ <lb/>dinem, & profunditatem, oritur triplex illius diui&longs;ro, de lineis, de &longs;uperfi­ <lb/>ciebus, de &longs;olidis. </s> <s id="id.001109">pars igitur, quæ de &longs;olidis tractat, <expan abbr="pattim&qacute;">pattimque</expan>; continetur <lb/>11. 12. 13. 14. & 15. Euclidis, partim aliorum Geometrarum libris, vt li­ <lb/>bro Archim. <!-- REMOVE S-->de Sphæra, & Cyl. <!-- REMOVE S-->& &longs;imilibus, dicitur Stereometria à græco <lb/><foreign lang="greek">steoeov,</foreign> ide&longs;t &longs;olidum. </s> <s id="id.001110">Porrò cur malit Ari&longs;t. Mechanicam &longs;ubalternari Ste­ <lb/>reometriæ, quam toti Geometriæ, qua tamen, vt videre e&longs;t apud Archime­ <lb/>dem, innititur, fortè ea ratio e&longs;t, quia Mechanica præcipuè con&longs;iderat ma­ <lb/>chinas, quæ corpora &longs;unt, & propterea præcipuè, & primò debet Stereome­ <lb/>triæ, quæ corpora pariter contemplatur, &longs;ubalternari. </s> <s id="id.001111">Quod ait Apparen­ <lb/>tia ad A&longs;irol. <!-- KEEP S--></s> <s id="id.001112">inteiligit per Apparentia vulgarem quandam Nautarum, & <lb/>Agricolarum a&longs;tronomiam, quæ quodammodo &longs;ubaiternatur, & pendet ex <lb/>&longs;cientia A&longs;trologiæ; indiget enim cognitione ortus, & motus a&longs;trorum, <lb/>præ&longs;ertim Lunæ, Hyadum, Pleiadum, & Canis. </s> <s id="id.001113">Reliqua <expan abbr="v&longs;q;">v&longs;que</expan> ad &longs;inem ca­ <lb/>pitis optimè à Zabarella explicantur, <expan abbr="neq;">neque</expan> ad nos pertinet, cum de Mathe­ <lb/>maticis agant, quatenus ad Logicum &longs;pectant.</s></p><p type="main"> <s id="id.001114"><arrow.to.target n="marg50"/></s></p><p type="margin"> <s id="id.001115"><margin.target id="marg50"/>50</s></p><p type="main"> <s id="id.001116">Po&longs;t nonnulla (<emph type="italics"/>Hic enim ip&longs;um quidem quod &longs;en&longs;itiuorum e&longs;t &longs;cire, ip&longs;um ve­ <lb/>rò Propter quid Mathematicorum; hi <expan abbr="namq;">namque</expan> habent cau&longs;arum demon&longs;trationes, <lb/>&c.<emph.end type="italics"/>) &longs;en&longs;us e&longs;t in &longs;ubalternatis, & dependentibus di&longs;ciplinis, quas &longs;en&longs;itiuas <lb/>appellat, quia de rebus &longs;en&longs;ibilibus &longs;unt, vt in Per&longs;pectiua de obiectis vifibi­ <lb/>libus, & in Mu&longs;ica de &longs;onis cogno&longs;citur Quod, ide&longs;t effectus: cuius effectus <lb/>cau&longs;a, &longs;eu Propter quid &longs;citur auxilio Mathematicarum, ide&longs;t, traditur à <lb/>&longs;cientijs &longs;ubalternantibus. </s> <s id="id.001117">v. <!-- REMOVE S-->g. <!-- REMOVE S-->alicuius effectus in Per&longs;pectiua cau&longs;a inqui­ <lb/>ritur, & inuenitur ope Geometriæ, cuiilla &longs;ubiacet. </s> <s id="id.001118">Hic obiter notandum, <lb/>Ari&longs;t. fateri manife&longs;tè Mathematicas &longs;ubalternatas, &longs;eu medias o&longs;tendere <lb/>per cau&longs;as, quas &longs;ubalternantium ope perue&longs;tigant.</s></p><p type="main"> <s id="id.001119"><arrow.to.target n="marg51"/></s></p><p type="margin"> <s id="id.001120"><margin.target id="marg51"/>51</s></p><p type="main"> <s id="id.001121">Et po&longs;tea (<emph type="italics"/>Se habet autem & ad Per&longs;pectiuam, vt hæc ad Geometriam, alia ad <lb/>hanc, vt quoæ e&longs;t de Iride ip&longs;um enim quod Naturalis e&longs;t &longs;cire, ip&longs;um vcrò Prop­ <lb/>ter quid Per&longs;pectiui<emph.end type="italics"/>) &longs;icut &longs;e habet, inquit, <expan abbr="&longs;ci&etilde;tià">&longs;cientià</expan> Naturalis de Iride ad Per­ <lb/>&longs;pectiuam, ita Per&longs;pectiua ad Geomettiam. </s> <s id="id.001122">qua verò ratione cau&longs;a Iridis <lb/>pertineat ad opticam, <expan abbr="atq;">atque</expan> hine tandem ad Geometriam, optimè patebit <lb/>in Meteoris, cum ip&longs;ius demon&longs;trationem afferemus.</s></p><p type="main"> <s id="id.001123"><arrow.to.target n="marg52"/></s></p><p type="margin"> <s id="id.001124"><margin.target id="marg52"/>52</s></p><p type="main"> <s id="id.001125">Tex. 37. (<emph type="italics"/>Vt æquicruri, & Scaleno hoc, quod e&longs;t duobus rectis æquales habere <lb/>&longs;ecandum commune aliquod ine&longs;t<emph.end type="italics"/>) quid &longs;it habcre tres æquales duobus rectis <lb/>&longs;atis explicatum e&longs;t lib. r. </s> <s id="id.001126">Priorum &longs;ecto 3. cap. r. </s> <s id="id.001127">nunc igitur paraphra&longs;im <lb/>&longs;olum huius loci dabo. </s> <s id="id.001128">Triangnlo I&longs;o&longs;celi, & Scaleno connenit pa&longs;&longs;io illa, <lb/>habere tres angulos æquales duobus rectis angulis &longs;ecundum aliquod com­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.061.jpg" pagenum="61"/>mune, quia illis competit, quatenus ambo &longs;unt figura quædam, ide&longs;t, qua­ <lb/>tenus <expan abbr="vtrumq;">vtrumque</expan> illorum triangulum e&longs;t; triangulo <expan abbr="namq;">namque</expan> omni primo com­ <lb/>petit habere tres angulos æquales duobus rectis.</s></p><p type="main"> <s id="id.001129"><arrow.to.target n="marg53"/></s></p><p type="margin"> <s id="id.001130"><margin.target id="marg53"/>53</s></p><p type="main"> <s id="id.001131">Tex. 38. (<emph type="italics"/>Et quemadmodum in alijs principium &longs;implex, boc autem non idem <lb/>vbique, &longs;ed in pondere quidem mina, in cătu verò die&longs;is<emph.end type="italics"/>) Die&longs;is apud Muficos e&longs;t <lb/>pars Toni. </s> <s id="id.001132">Tonus autem e&longs;t interuallum duarum vocum, quale e&longs;t inter pri­ <lb/>mam vocem, Vt, & &longs;ecundam Rè, vt modo loquuntur. </s> <s id="id.001133">i&longs;tud interuallum <lb/>diuidunt Mu&longs;ici primum in &longs;emitonia, non tamen æqualia, &longs;ed vnum maius <lb/>altero. </s> <s id="id.001134">minus iterum in duas partes æquales &longs;ubdiuidunt, quarum <expan abbr="vtramq;">vtramque</expan> <lb/>veteres harmonici die&longs;im dixerunt. </s> <s id="id.001135">& h&ecedil;c die&longs;is e&longs;t minima vox ab eis con­ <lb/>&longs;iderata; & quæ prima cadit &longs;ub &longs;en&longs;um; & propterea veluti &longs;implex prin­ <lb/>cipium, & clementum, ex quo alia maiora interualla conftent; & in quod <lb/>re&longs;oluuntur. <foreign lang="greek">die/ois</foreign> porrò græcè valet inter alia, diui&longs;ionem. </s> <s id="id.001136">igitur interual­ <lb/>lum i&longs;tud minimum dictum e&longs;t die&longs;is, quod &longs;it quædam diui&longs;io, &longs;eu &longs;egmen­ <lb/>turn Toni (<emph type="italics"/>Quemadmodum in pondere mina<emph.end type="italics"/>) qui de ponderibus antiquis tra­ <lb/>ctant, a&longs;&longs;erunt, Minam fui&longs;&longs;e maiorem libra per &longs;emunciam, æquipondera­ <lb/>bat enim centum drachmis: quæ refragantur huic loco. </s> <s id="id.001137">&longs;ed fortè <expan abbr="dic&etilde;dum">dicendum</expan>, <lb/>Ari&longs;t. con&longs;idera&longs;&longs;e, Minam re&longs;pectu Talenti, re&longs;pectu enim illius dici pote&longs;t <lb/>principium, cum &longs;ex millia minarum in Attico talento continerentur.</s></p><figure id="id.009.01.061.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.061.1.jpg" place="text"/><p type="main"> <s id="id.001138"><arrow.to.target n="marg54"/></s></p><p type="margin"> <s id="id.001139"><margin.target id="marg54"/>54</s></p><p type="main"> <s id="id.001140">Tex. 39. <emph type="italics"/>(Si enim quod duobus rectis ine&longs;t, non in <lb/>quantum æquicrus, &longs;ed in quantum triangulus, no­ <lb/>&longs;cens, &c.)<emph.end type="italics"/> ide&longs;t, &longs;i enim qui cogno&longs;cit, quod ha­ <lb/>bere tres angulos æquales duobus rectis conuenit <lb/>æquicruri, non quatenus æquicrus e&longs;t, &longs;ed quate­ <lb/>nus triangulus e&longs;t, &c. </s> <s id="id.001141">quid &longs;it habere tres æqua­ <lb/>les duobus rectis, &c. </s> <s id="id.001142">fusè explicatum e&longs;t in lib. 1. <lb/>Priorum &longs;ecto 3. cap. 1. quò te nunc mitto.</s></p><p type="main"> <s id="id.001143"><arrow.to.target n="marg55"/></s></p><p type="margin"> <s id="id.001144"><margin.target id="marg55"/>55</s></p><p type="main"> <s id="id.001145">Po&longs;t pauca <emph type="italics"/>(Ine&longs;t omni triangulo hoc quod est <lb/>duos, &c.)<emph.end type="italics"/> ide&longs;t, hæc proprietas, quæ e&longs;t habere <lb/>duos angulos rectos non actu, &longs;ed per æquiualen­ <lb/>tiam trium angulorum trianguli. </s> <s id="id.001146">Vide quæ im­ <lb/>mediatè &longs;upra de hac re dixi, & quò te remi&longs;r.</s></p><p type="main"> <s id="id.001147"><arrow.to.target n="marg56"/></s></p><p type="margin"> <s id="id.001148"><margin.target id="marg56"/>56</s></p><p type="main"> <s id="id.001149">Eodem tex <emph type="italics"/>(Quando igitur cognofcimes, quod­ <lb/>quatuor exteriores &longs;unt æquales, quoniam I&longs;o&longs;celes, <lb/>adhuc defseit, propier quid I&longs;o&longs;celes? </s> <s id="id.001150">quoniain trian­ <lb/>gulus: & hoc quoniam figura rectilinea, &c.)<emph.end type="italics"/> exem­ <lb/>plo geometrico vult o&longs;tendere demon&longs;trationem <lb/>vniuer&longs;alem e&longs;&longs;e particulari præ&longs;tantiorem: e&longs;t <lb/>autem exemplum de pulcherrima, <expan abbr="atq;">atque</expan> admira­ <lb/>bili proprietate, quæ omnibus figuris rectilineis <lb/>conuenit, e&longs;t <expan abbr="&qacute;">que</expan>; huiu&longs;modi: Omnis figuræ rectili­ <lb/>neæ anguli externi omnes &longs;imul &longs;umpti, &longs;unt æqua <lb/>les quatuor rectis angulis, quæ affectio demon­ <lb/>&longs;tratur in &longs;cholio 32. primi Elem. dicuntur autern <lb/>anguli externi, qui productis lateribus fiunt, vt in <lb/>triangulo pra&longs;enti anguli externi &longs;unt, B D C, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.062.jpg" pagenum="62"/>D F E, F B A, ita vt quælibet figura tot angulos externos &longs;ortiatur, quot <lb/>habet latera; cum exproductis lateribus oriantur. </s> <s id="id.001151">Vt autem propo&longs;itio ve­ <lb/>rificetur, &longs;ingula latera ordinatim &longs;unt producenda, hoc e&longs;t, ver&longs;us eandem <lb/>partem, vt in figuris appo&longs;itis vides. </s> <s id="id.001152">Quæuis igitur figura rectilinea, &longs;iue <lb/>trilatera &longs;it, &longs;iue quadrilatera, vel etiam millelatera, & proinde mille quo­ <lb/>que angulos externos habeat, hanc tamen mirabilem proprietatem (quod <lb/>vix credi pote&longs;t) po&longs;&longs;idet, vt omnes illi anguli externi &longs;imul &longs;int æquales <lb/>quatuor rectis angulis. </s> <s id="id.001153">vnde tres externi anguli trianguli, & quatuor exter­ <lb/>ni quadranguli, & quinque externi <expan abbr="p&etilde;tagoni">pentagoni</expan>, &c. </s> <s id="id.001154">&longs;unt æquales quatuor tan­ <lb/>tum rectis, nec aliter res &longs;e habet in figura millelatera. </s> <s id="id.001155">Ex quo fit, vt an­ <lb/>guli externi cuiu &longs;uis figuræ &longs;int æquales angulis omnibus externis alterius <lb/>cuiu&longs;libet figuræ. </s> <s id="id.001156">Ari&longs;t. igitur inquit, quando cogno&longs;cimus, quod quatuor <lb/>angulis rectis &longs;unt æquales exteriores omnes anguli alicuius figuræ, quo­ <lb/>niam figura illa e&longs;t triangulum &longs;calenum, adhuc talis cognitio e&longs;t defecti­ <lb/>ua, quia non illi competit illa pa&longs;&longs;io, quia &longs;it triangulum &longs;calenum, neque <lb/>competit &longs;caleno, quia &longs;it triangulum; &longs;ed his omnibus competit, quia &longs;unt <lb/>figuræ rectilineæ, cui hæc proprietas ine&longs;t primo, & vniuer&longs;aliter: qui igi­ <lb/>tur &longs;cit, &longs;calenum habere prædictam affectionem, ex eo, quod &longs;it figura re­ <lb/>ctilinea, perfectius &longs;cit, quia nihil amplius quæri pote&longs;t, quia illa figura re­ <lb/>ctilinea illud vniuer&longs;ale e&longs;t, cui primo competit; reliquis autem per illam. <lb/></s> <s id="id.001157">qui igitur vniuer&longs;ale &longs;cit, perfectius &longs;cit; quod volebat Ari&longs;t. demon&longs;trare.</s></p><p type="main"> <s id="id.001158"><arrow.to.target n="marg57"/></s></p><p type="margin"> <s id="id.001159"><margin.target id="marg57"/>57</s></p><p type="main"> <s id="id.001160">Eodem tex. <emph type="italics"/>(Vt &longs;i quis nouit, quod omnis triangulus habet tres duobus rectis <lb/>æquales)<emph.end type="italics"/> nihil &longs;requentius. </s> <s id="id.001161">vide &longs;upra lib. 1. Priorum &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.001162"><arrow.to.target n="marg58"/></s></p><p type="margin"> <s id="id.001163"><margin.target id="marg58"/>58</s></p><p type="main"> <s id="id.001164">Tex. 43. <emph type="italics"/>(Sed planum, quod et&longs;i e&longs;&longs;et &longs;entire triangulum, quod duobus rectis <lb/>æquales habet angulos)<emph.end type="italics"/> vide &longs;upra lib. 1. Priorum &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.001165"><arrow.to.target n="marg59"/></s></p><p type="margin"> <s id="id.001166"><margin.target id="marg59"/>59</s></p><p type="main"> <s id="id.001167">Po&longs;t pauca <emph type="italics"/>(Quare & &longs;i &longs;upra Lunam e&longs;&longs;emus, & videremus obiectam terram, <lb/>non <expan abbr="vtiq;">vtique</expan> &longs;ciremus cau&longs;am eclyp&longs;is)<emph.end type="italics"/> loquitur de defectu Lunæ, qui fit, quando <lb/>terra inter Lunam, & Solem po&longs;ita, impedit, ne lumen Solis feratur in Lu­ <lb/>nam, &longs;ed efficit, vt vmbra ip&longs;ius terræ eam contegat.</s></p><p type="main"> <s id="id.001168"><arrow.to.target n="marg60"/></s></p><p type="margin"> <s id="id.001169"><margin.target id="marg60"/>60</s></p><figure id="id.009.01.062.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.062.1.jpg" place="text"/><p type="main"> <s id="id.001170">Et paulo po&longs;t <emph type="italics"/>(Qutmadmodŭ &longs;t vi­ <lb/>trum perforatum videremus, & lumen <lb/>permeans, planum vtique e&longs;&longs;et propter <lb/>quid comburit)<emph.end type="italics"/> Ioquitur de ea com­ <lb/>bu&longs;tione, cuæ fit per refractionem <lb/>media &longs;phæra vitrea. </s> <s id="id.001171">de qua Vitel­ <lb/>lio propo&longs;. </s> <s id="id.001172">48. decimi libri; non au­ <lb/>tem de ea, quæ fit per reflexionem <lb/>ex &longs;peculo concauo quando combu­ <lb/>&longs;tio fit per refractionem, cau&longs;atur à <lb/>radijs Solis vitrum permeantibus, <lb/>in quo ita franguntur, vt egredien­ <lb/>tes è vitro &longs;imul vniantur, ex qua <lb/>vnione ita calor intenditur, vt ibi <lb/>comburat. </s> <s id="id.001173">vt in appo&longs;ita figura cer­ <lb/>nere facile e&longs;t; in qua radij à Sole <lb/>manentes, &longs;phæram vitream perua­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.063.jpg" pagenum="63"/>dunt, <expan abbr="atq;">atque</expan> in exitu ita refraguntur, vt ad A, punctum coaceruati, ibi po&longs;­ <lb/>&longs;int, &longs;i quid combu&longs;tibile occurrat, comburere. </s> <s id="id.001174">Si igitur, inquit Ari&longs;t. vide­ <lb/>remus illos radios &longs;ic permeare, & refrangi, planum <expan abbr="vtiq;">vtique</expan> nobis e&longs;&longs;et pro­ <lb/>pter quid incendant. <lb/><arrow.to.target n="marg61"/></s></p><p type="margin"> <s id="id.001175"><margin.target id="marg61"/>61</s></p><p type="main"> <s id="id.001176">Ad finem tex. <!-- REMOVE S-->43. <emph type="italics"/>(Principia enim duplicia &longs;unt, ex quibus, & circa quod: <lb/>quæ quidem igitur, ex quibus, communia &longs;unt: quæ autem circa quod propria, vt <lb/>numerus, magnitudo)<emph.end type="italics"/> nonnulli codices corruptè legunt (vt numerus magni­ <lb/>tudine) &longs;ed ex græco tex. <!-- REMOVE S-->corrigendi &longs;unt, vti fecimus. </s> <s id="id.001177">Cæterum per prin­ <lb/>cipia, ex quibus intelligit Dignitates, quia ex illis di&longs;currimus. </s> <s id="id.001178">per princi­ <lb/>pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli­ <lb/>dem, explicatur &longs;ubiectum, circa quod &longs;cientia ver&longs;atur; vt in definitioni­ <lb/>bus primi Elem. docemur, quid &longs;it linea, quid triangulum, quid circulus, <lb/>quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe­ <lb/>culatur. </s> <s id="id.001179">In &longs;eptimo verò traduntur definitiones numerorum, quid &longs;it nu­ <lb/>merus, quid impar, quid compo&longs;itus, quadratus, cubus, & reliquæ nume­ <lb/>rorum &longs;pecies, quæ &longs;unt materia &longs;eptimi, octaui, & noni, in quibus de Arith­ <lb/>metica tractatur.</s></p><p type="main"> <s id="id.001180"><arrow.to.target n="marg62"/></s></p><p type="margin"> <s id="id.001181"><margin.target id="marg62"/>62</s></p><p type="main"> <s id="id.001182">Tex. 44. <emph type="italics"/>(Commen&longs;urabilem namq e&longs;&longs;e diametrum verè opinari, ab&longs;urdum e&longs;t)<emph.end type="italics"/> <lb/>vide, quæ de <expan abbr="comm&etilde;&longs;urabilitate">commen&longs;urabilitate</expan> diametri quadrati cum latere expo&longs;uimus <lb/>lib. 1. Priorum &longs;ecto 1. cap. 23. ait igitur Ari&longs;t. ab&longs;urdum e&longs;&longs;e opinari dia­ <lb/>metrum e&longs;&longs;e commen&longs;urabilem co&longs;tæ, &longs;eu lateri eiu&longs;dem quadrati, reli­ <lb/>qua &longs;unt Logica.</s></p></chap><chap><p type="head"> <s id="id.001183"><emph type="italics"/>Ex Secundo Posteriorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001184"><arrow.to.target n="marg63"/></s></p><p type="margin"> <s id="id.001185"><margin.target id="marg63"/>63</s></p><p type="main"> <s id="id.001186">Tex. 1. <emph type="italics"/>(Dico autem &longs;impliciter quidem &longs;ubiectum, vt Lunam, aut ter­ <lb/>ram, aut Solem, aut triangulum; aliquid verò defectum, æqualitatem, <lb/>inæqualitatem. </s> <s id="id.001187">&longs;i in medio, aut non)<emph.end type="italics"/> Zabarella locum hunc, etiam <lb/>quatenus ad Mathematicum attinet, optimè declarat. </s> <s id="id.001188">In quæ­ <lb/>&longs;tionibus, & demon&longs;trationibus duo &longs;unt, &longs;ubiectum, & <expan abbr="prædicatũ">prædicatum</expan>, <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> <lb/>cau&longs;æ exi&longs;tunt, & quæruntur: v. <!-- REMOVE S-->g. <!-- REMOVE S-->Luna, terra, Sol, & triangulum &longs;unt &longs;u­ <lb/>biectum in demon&longs;tratione, quorum prædicata &longs;unt, Lunæ quidem, & So­ <lb/>lis, eclyp&longs;is. </s> <s id="id.001189">terræ autem e&longs;&longs;e in medio mundi, quod ab A&longs;tronomis ratione <lb/>ab eclyp&longs;ibus de&longs;umpta, euidentius, quam ab alio quoquam demon&longs;tratur, <lb/>vt patet ex tractatu de &longs;phœra. </s> <s id="id.001190">in quo Zabarella non probatur, qui &longs;olum <lb/>ait, terram e&longs;&longs;e in medio mundi, à Phy&longs;icis demon&longs;trari. </s> <s id="id.001191"><expan abbr="triãgulum">triangulum</expan> autem, <lb/>&longs;eu angulorum ip&longs;ius <expan abbr="prædicatũ">prædicatum</expan> e&longs;t æqualitas, & inæqualitas: vt cum in 32. <lb/>primi Elem. demon&longs;trat Euclides, omne triangulum habcre tres angulos <lb/>æquales duobus rectis.</s></p><p type="main"> <s id="id.001192"><arrow.to.target n="marg64"/></s></p><p type="margin"> <s id="id.001193"><margin.target id="marg64"/>64</s></p><p type="main"> <s id="id.001194">Ibidem <emph type="italics"/>(Quid e&longs;t con&longs;onantia? </s> <s id="id.001195">ratio numerorum in acuto, & graui, &c)<emph.end type="italics"/> tan­ <lb/>git breuiter Ari&longs;t. cau&longs;am formalem con&longs;onantiæ, & con&longs;equenter defini­ <lb/>tionem ip&longs;ius. </s> <s id="id.001196">definiunt igitur Mu&longs;ici con&longs;onantiam hoc modo; Con&longs;onan­ <lb/>tia e&longs;t compo&longs;itio &longs;oni grauis, & acuti, quæ &longs;uauiter auribus accidit; & quo­ <lb/>rum &longs;onorum proportio ad inuicem &longs;it &longs;icuti proportio numerorum, qui <lb/>quaternario includuntur: vt e&longs;t proportio 2. ad 1. vel 3. ad 1. vel 4. ad 1. <lb/>vel 3. ad 2. vel 4. ad 3. <expan abbr="Quotie&longs;eunq;">Quotie&longs;eunque</expan> igitur duo &longs;oni habuerin quampiam <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.064.jpg" pagenum="64"/>ex <expan abbr="quinq;">quinque</expan> prædictis proportionibus, &longs;i &longs;imul coaluerint, ita vt ex eis vnue <lb/>tantum &longs;onus efficiatur; &longs;onus ille erit concordans, & auribus gratus. </s> <s id="id.001197"><expan abbr="atq;">atque</expan> <lb/>hæc e&longs;t &longs;ententia pri&longs;corum præ&longs;ertim Pythagoreorum, qui propterea di­ <lb/>cebantnon licere Mu&longs;ico vltra quaternarium pertran&longs;ire, eò quod &longs;olæ pro­ <lb/>portiones, vt diximus, numerorum quaternario contentorum, concordem, <lb/>ac con&longs;onantem concentum efficere poterant: quod vt adhuc melius per­ <lb/><figure id="id.009.01.064.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.064.1.jpg" place="text"/> <lb/>cipiamus, accipe exemplum. </s> <s id="id.001198">Sint duæ chordæ <lb/>A, & B, æqualis cra&longs;&longs;itici, & æquè ten&longs;æ. </s> <s id="id.001199">qua­ <lb/>rum A, dupla &longs;it ip&longs;ius B, quia igitur corpora <lb/>&longs;onantia &longs;unt in dupla proportione, erunt pa­ <lb/>riter corum &longs;oni in ratione dupla (vt patet ex <lb/>principijs harmonicæ) hoc e&longs;t, <expan abbr="eorũ">eorum</expan> &longs;oni erunt, <lb/>vt 2. ad 1. quia &longs;cilicet &longs;onus maioris chordæ A, erit duplus ad &longs;onum mi­ <lb/>noris chordæ B. hoc e&longs;t, erit, vt 2. ad 1. & propterea, &longs;i &longs;imul ambæ chordæ <lb/>pul&longs;entur, &longs;onus, quem ex duobus mixtum edent, con&longs;onans, <expan abbr="atq;">atque</expan> grati&longs;&longs;i­ <lb/>mus auribus no&longs;tris perueniet. </s> <s id="id.001200">huiu&longs;modi porrò con&longs;onantia, quæ e&longs;t in <lb/>proportione dupla, <expan abbr="quæ&qacute;">quæque</expan> omnium &longs;uaui&longs;&longs;ima e&longs;t, à græcis dicebatur Dia­ <lb/>pafon. </s> <s id="id.001201"><expan abbr="atq;">atque</expan> hæc in præ&longs;entia &longs;ufficiant, cum plura de his ad &longs;ectionem pro­ <lb/>blematum 19. quæ tota e&longs;t de Mu&longs;ica, dicenda &longs;int.</s></p><p type="main"> <s id="id.001202"><arrow.to.target n="marg65"/></s></p><p type="margin"> <s id="id.001203"><margin.target id="marg65"/>65</s></p><p type="main"> <s id="id.001204">Tex. 2. <emph type="italics"/>(Vt quod omnis triangulus duobus rectis æquales babet)<emph.end type="italics"/> vide anno­ <lb/>tata lib. 1. Priorum &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.001205"><arrow.to.target n="marg66"/></s></p><p type="margin"> <s id="id.001206"><margin.target id="marg66"/>66</s></p><p type="main"> <s id="id.001207">Eodem tex. <emph type="italics"/>(Definitiones verò apparent omnes &longs;upponentes, & accipientes <lb/>ip&longs;um quid e&longs;t, vt Mathematicæ, quid vnitas, quid par, & impar)<emph.end type="italics"/> alludit ad de­ <lb/>finitiones 7. Elem. vbi agitur de numeris. </s> <s id="id.001208">Quæ verò hoc loco de principijs <lb/>dicuntur, luculenti&longs;&longs;imè patent con&longs;ideranti definitiones, & axiomata, quæ <lb/>Mathematicis demon&longs;trationibus in omnibus ferè libris præmittuntur; ex <lb/>quibus &longs;tatim demon&longs;trationes deriuantur.</s></p><p type="main"> <s id="id.001209"><arrow.to.target n="marg67"/></s></p><p type="margin"> <s id="id.001210"><margin.target id="marg67"/>67</s></p><p type="main"> <s id="id.001211">Et paulo po&longs;t <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> <expan abbr="vtiq;">vtique</expan> de plano figura, non enim e&longs;t planum figura, <expan abbr="neq;">neque</expan> fi­ <lb/>gura planum)<emph.end type="italics"/> alludit ad definitiones planarum figurarum, qualis e&longs;t circu­ <lb/>lus, cuius definitio e&longs;t inter definitiones primi Elem. 15. & e&longs;t huiu&longs;modi: <lb/>circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap­ <lb/>pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca­ <lb/>dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione <lb/>non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb/>&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur; <expan abbr="neq;">neque</expan> figura e&longs;t <lb/>plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam­ <lb/>plurimæ.</s></p><p type="main"> <s id="id.001212"><arrow.to.target n="marg68"/></s></p><p type="margin"> <s id="id.001213"><margin.target id="marg68"/>68</s></p><p type="main"> <s id="id.001214">Ibidem <emph type="italics"/>(Quoniam mon&longs;tratum e&longs;t I&longs;o&longs;celes habere tres angulos æquales duo­ <lb/>bus rectis, &longs;i id de omni triangulo mon&longs;tratum &longs;it)<emph.end type="italics"/> ex dictis lib. 1. Priorum &longs;ecto <lb/>3. cap. 1. petatur huius loci declaratio.</s></p><p type="main"> <s id="id.001215"><arrow.to.target n="marg69"/></s></p><p type="margin"> <s id="id.001216"><margin.target id="marg69"/>69</s></p><p type="main"> <s id="id.001217">Tex. 7. <emph type="italics"/>(Quid enim &longs;ignificat triangulum, accipit Geometra)<emph.end type="italics"/> vt manife&longs;tum <lb/>e&longs;t in 20. dednitione primi Elem.<!-- KEEP S--></s></p><p type="main"> <s id="id.001218"><arrow.to.target n="marg70"/></s></p><p type="margin"> <s id="id.001219"><margin.target id="marg70"/>70</s></p><p type="main"> <s id="id.001220">Ibidem <emph type="italics"/>(Quod autem &longs;it, monstrat)<emph.end type="italics"/> vt per&longs;picuum e&longs;t in prima <expan abbr="demõ&longs;tra-tione">demon&longs;tra­ <lb/>tione</expan> primi Elem. vbi triangulum æquilaterum con&longs;truit, & po&longs;tea probat <lb/>illud e&longs;&longs;e triangulum æquilaterum. </s> <s id="id.001221">Certum tamen e&longs;t, Geometram luppo­ <lb/>nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.065.jpg" pagenum="65"/>quod tamen non ob&longs;tat, quominus probare po&longs;&longs;it, aliquando po&longs;&longs;e <expan abbr="cõ&longs;trni">con&longs;trni</expan>, <lb/>& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione, <lb/>Euclidis.<!-- KEEP S--></s></p><p type="main"> <s id="id.001222"><arrow.to.target n="marg71"/></s></p><p type="margin"> <s id="id.001223"><margin.target id="marg71"/>71</s></p><p type="main"> <s id="id.001224">Tex. 11. <emph type="italics"/>(Manife&longs;tum autem, & &longs;ic, propter quid e&longs;t rectus in &longs;emicirculo)<emph.end type="italics"/> <lb/>affert exemplum demon&longs;trationis per cau&longs;am materialem, <expan abbr="id&qacute;">idque</expan>; vti &longs;olet ex <lb/>Mathematicis petitum, e&longs;t enim apud Euclidem 31. demon&longs;tratio 3. Elem. <lb/>vbi ip&longs;e o&longs;tendit angulum in &longs;emicirculo e&longs;&longs;e rectum. </s> <s id="id.001225">Vbi aduertendum e&longs;t <lb/>propo&longs;itionem hanc 31. ab Euclide demon&longs;trari duobus modis; ex quibus <lb/>&longs;ecundum innuit hoc loco Ari&longs;t. cui a&longs;cripta e&longs;t figura &longs;imilis huic no&longs;træ; <lb/>in editione Clauiana. </s> <s id="id.001226">quod fortè non benè aduertens Iacobus Zabarella, <lb/>alioquin in his &longs;atis oculatus incidit in errorem, dicens, &longs;e nullo pacto vi­ <lb/>dere medium Euclidianæ demon&longs;trationis e&longs;&longs;e cau&longs;am materialem; quod <lb/>tamen nos mox aperiemus. </s> <s id="id.001227">per angulum in &longs;emicirculo intelligas eum, qui <lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum <lb/><figure id="id.009.01.065.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.065.1.jpg" place="text"/> <lb/>circumferentiæ coeuntibus, vt in figura <lb/>præ&longs;enti vides lineas A C, B C, ad C, pun­ <lb/>ctum conuenire, <expan abbr="ibi&qacute;">ibique</expan>; facere angulum, <lb/>A C B, qui dicitur angulus in &longs;emicircu­ <lb/>lo, quia de&longs;criptus e&longs;t in &longs;emicirculo A­ <lb/>C B. <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;anè mirabilis hæc &longs;emicirculi <lb/>proprietas, cum <expan abbr="vbicunq;">vbicunque</expan> punctum C, in <lb/>periphæria &longs;umptum fuerit, &longs;emper ta­ <lb/>men angulus A C B, fiat rectus. </s> <s id="id.001228">quod Euclides eodem pror&longs;us medio, quod <lb/>Ari&longs;t. hic innuit, hoc modo demon&longs;trat. </s> <s id="id.001229">ducta enim recta D C, à centro D, <lb/>ad punctum C, exurgunt duo l&longs;o&longs;celia triangula A D C, C D B, ergo per <lb/>5. primi, anguli D C A, D A C, &longs;unt æquales: pariter anguli D C B, D B C, <lb/>æquales &longs;unt. </s> <s id="id.001230">& quia per 32. primi, anguli D A C, D C A, &longs;imul &longs;unt æqua­ <lb/>les angulo externo C D B, & inter &longs;e æquales, erit angulus A C D, dimidium <lb/>anguli C D B. eadem ratione probatur angulus D C B, e&longs;&longs;e dimidium an­ <lb/>guli C D A. ergo totus angulus A C B, dimidium erit duorum angulorum <lb/>A D C, C D B, qui per 13. primi, &longs;unt vel recti, vel duobus rectis <expan abbr="æquiual&etilde;t">æquiualent</expan>. <lb/></s> <s id="id.001231">Sequitur igitur, angulum A C B, in &longs;emicirculo e&longs;&longs;e dimidium duorum re­ <lb/>ctorum; & quia omnes recti &longs;unt æquales, &longs;equitur dimidium duorum re­ <lb/>ctorum, nihil aliud e&longs;&longs;e, quam vnum rectum angulum, ergo angulus in &longs;e­ <lb/>micirculo, cum &longs;it &longs;emi&longs;&longs;is duorum <expan abbr="rectorũ">rectorum</expan>, erit vnus rectus angules; quod <lb/>erat probandum. </s> <s id="id.001232">ex quibus vides medium illud, quod Ari&longs;t. a&longs;&longs;ump&longs;it, e&longs;&longs;e <lb/>omnino idem cum eo, quo Euclides vtitur, &longs;cilicet, e&longs;&longs;e dimidium duorum <lb/>rectorum, & propterea e&longs;&longs;e rectum: quod etiam medium in toto demon­ <lb/>&longs;trationis decur&longs;u e&longs;t vltimum, & principale, quod proximè conclu&longs;ionem <lb/>attingit, & propterea dici meretur e&longs;&longs;e medium huius demon&longs;trationis. <lb/></s> <s id="id.001233">Cæterum, quod medium i&longs;tud &longs;it in genere cau&longs;æ materialis, patet ex co, <lb/>quod e&longs;t, e&longs;&longs;e dimidium; nam e&longs;&longs;e dimidium, vel e&longs;&longs;e tertiam partem, & &longs;i­ <lb/>milia, nihil aliud e&longs;t, quam e&longs;&longs;e partem; e&longs;&longs;e autem partem e&longs;t e&longs;&longs;e materiam <lb/>totius, etiam ex &longs;ententia ip&longs;ius Ari&longs;t. ex hac præterea materia conflatur <lb/>definitio minoris extremi, vel &longs;ubiecti; dum dicitur, angulus in &longs;emicircu­ <lb/>lo e&longs;t dimidium duorum rectorum. </s> <s id="id.001234">&longs;yllogi&longs;mus enim reducitur tandem ad <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.066.jpg" pagenum="66"/>hanc formam, dimidium duorum rectorum e&longs;t rectus, angulus in &longs;emicir­ <lb/>culo e&longs;t dimidium duorum <expan abbr="rectorũ">rectorum</expan>, ergo angulus in &longs;emicirculo e&longs;t rectus. <lb/></s> <s id="id.001235">vides in minori propo&longs;itione contineri definitionem &longs;ubiecti materialem? <lb/></s> <s id="id.001236">adeò vt hæc &longs;it demon&longs;tratio omnibus numeris ab&longs;oluta per cau&longs;am mate­ <lb/>rialem, vt benè &longs;entit Ari&longs;t. <!-- KEEP S--></s> <s id="id.001237">Reliqua ad logicum pertinent, etiam&longs;i per cha­ <lb/>racteres more mathematicorum exponantur.</s></p><p type="main"> <s id="id.001238"><arrow.to.target n="marg72"/></s></p><p type="margin"> <s id="id.001239"><margin.target id="marg72"/>72</s></p><p type="main"> <s id="id.001240">Tex. 24. <emph type="italics"/>(Vt propter quid re&longs;onat? </s> <s id="id.001241">aut propter quid apparet? </s> <s id="id.001242">aut propter quid <lb/>Iris? <!-- KEEP S--></s> <s id="id.001243">omnia enim hær idem problemata &longs;unt genere, omnia enim &longs;unt refractio, &longs;ed <lb/>&longs;pecie altera)<emph.end type="italics"/> propter quid re&longs;onat? </s> <s id="id.001244">&longs;cilicet echo; propter quid apparet? <lb/></s> <s id="id.001245">&longs;cilicet imago in &longs;peculo. </s> <s id="id.001246">dicit cau&longs;am echo, imaginis in &longs;peculo, & iridis <lb/>in nubibus e&longs;&longs;e eandem; nimirum refractionem; quamuis tres illæ refractio­ <lb/>nes, &longs;eu; vt melius loquamur, reflexiones differant &longs;pecie ab inuicem, illa <lb/>enim e&longs;t repercu&longs;&longs;io vocis; hæc reflexio &longs;peciei vi&longs;ibilis ex corpore ter&longs;o; <lb/>i&longs;ta <expan abbr="deniq;">denique</expan> radiorum Solis ex nube rorida in &longs;tato angulo repercu&longs;&longs;us. </s> <s id="id.001247">qua <lb/>ratione autem i&longs;ta omnia fiant, longum e&longs;&longs;et exponere, & ab intelligentia <lb/>huius loci fortè alienum. </s> <s id="id.001248">Illud tamen non prætereundum, quod &longs;i propriè <lb/>cum Per&longs;pectiuis loqui velimus, dicendum e&longs;&longs;e, omnia illa e&longs;&longs;e reflexionem, <lb/>non refractionem. </s> <s id="id.001249">nam reflexio e&longs;t, quando linea vi&longs;ualis, per quam fertur <lb/>&longs;péecies in aliquod corpus ter&longs;um, impingit, ex quo deinde ad oculos refle­ <lb/>ctitur. </s> <s id="id.001250">refractio tunc e&longs;t, quando &longs;pecies obiectivi&longs;ibilis tran&longs;it per media <lb/>diuer&longs;æ cra&longs;&longs;itiei., vt quando &longs;pecies lapilli per aquam primùm, deinde per <lb/>æerem means ad oculum peruenit; tunc enim linea, per quam &longs;pecies pro­ <lb/>greditur, frangitur in confinio aquæ, & aeris, ita vt &longs;pecies non per vnicam <lb/>lineam rectam, &longs;ed per fractam, &longs;eu refractam in confinio illo, oculis tan­ <lb/>dem accidat.</s></p><p type="main"> <s id="id.001251">In fine textus <emph type="italics"/>(Quoniam Luna deficit)<emph.end type="italics"/> non intelligit defectum illum, qui <lb/>eclyp&longs;is appellatur, &longs;ed ilium, quo paulatim lumen Lunæ minus oculis no­ <lb/>&longs;tris apparet: decre&longs;cente enim Luna &longs;olent humida augeri.</s></p><p type="main"> <s id="id.001252"><arrow.to.target n="marg73"/></s></p><p type="margin"> <s id="id.001253"><margin.target id="marg73"/>73</s></p><p type="main"> <s id="id.001254">Tex. 25. <emph type="italics"/>(Vt propter quid, & permutatim proportionale? </s> <s id="id.001255">& c.<emph.end type="italics"/>) quod quan­ <lb/>titates, quæ &longs;unt proportionales, &longs;int etiam alternatim, &longs;eu permutatim <lb/>proportionales explicatum e&longs;t ad tex. <!-- REMOVE S-->13. primi Po&longs;ter. quæ etiam nece&longs;&longs;a­ <lb/>ria &longs;unt ad hunc locum benè intelligendum. </s> <s id="id.001256">Illud autem commune propter <lb/>quod ea, quæ &longs;unt proportionalia, &longs;int etiam permutatim proportionalia, <lb/>e&longs;t quoddam innominatum, de quo ibi dictum e&longs;t, quod cum conueniat li­ <lb/>neis, & numeris, & tamen &longs;eparatim de vtri&longs;que illa pa&longs;&longs;io demon&longs;tretur, <lb/>quærit cuinam primò, & per &longs;e conueniat hæc pa&longs;&longs;io, e&longs;&longs;e permutatim pro­ <lb/>portionale; &longs;cilicet quidnam &longs;it illud innominatum; in quo deinde commu­ <lb/>nicent lineæ, & numeri, vt inde habeant e&longs;&longs;e etiam permutatim propor­ <lb/>tionalia.</s></p><p type="main"> <s id="id.001257"><arrow.to.target n="marg74"/></s></p><p type="margin"> <s id="id.001258"><margin.target id="marg74"/>74</s></p><p type="main"> <s id="id.001259">Ibidem (<emph type="italics"/>Hic quidem forta&longs;&longs;e proportionaliter habere latera, & angulos<emph.end type="italics"/>) vult <lb/>indicare, in quonam con&longs;i&longs;tat &longs;imilitudo inter duas-figuras rectilineas geo­ <lb/>metricas, quam &longs;imilitudinem Euclides definit. </s> <s id="id.001260">1. &longs;exti, &longs;ic explicat: &longs;imi­ <lb/>les figuræ rectilíneæ &longs;unt; quæ & angulos &longs;ingulos, &longs;ingulis angulis æquales <lb/>habent, <expan abbr="atq;">atque</expan> etiam latera, quæ circa angulos æquales &longs;unt proportionalia. <lb/></s> <s id="id.001261">vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angulũ">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.067.jpg" pagenum="67"/><figure id="id.009.01.067.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.067.1.jpg" place="text"/> <lb/>circa angulos æquales, v. <!-- REMOVE S-->g. <!-- REMOVE S-->cirea an­ <lb/>gulos A, & D, habeant proportiona­ <lb/>lia, hoc e&longs;t, vt latus A B, ad latus A C; <lb/>ita &longs;it latus D E, ad latus D F; & &longs;ic de <lb/>lateribus alijs circa reliquos angulos <lb/>æquales; erunt tunc prædicta duo tri­ <lb/>angula fimilia.</s> </p><p type="main"> <s id="id.001262"><arrow.to.target n="marg75"/></s></p><p type="margin"> <s id="id.001263"><margin.target id="marg75"/>75</s></p><p type="main"> <s id="id.001264">Ibidem (<emph type="italics"/>Vt extrin&longs;ecos æquales e&longs;&longs;e<emph.end type="italics"/>) ide&longs;t extrin&longs;ecos angulos cuiu&longs;uis fi­ <lb/>guræ rectilineæ æquales e&longs;&longs;e quatuor rectis angulis: vide quæ &longs;crip&longs;imus de <lb/>hac re ad tex. <!-- REMOVE S-->39. &longs;ecundi Po&longs;ter. quæ huic pariter loco &longs;atisfaciunt.</s> </p><p type="head"> <s id="id.001265"><emph type="italics"/>EX TOPICIS.<emph.end type="italics"/></s></p><p type="head"> <s id="id.001266"><emph type="italics"/>Ex Primo Libro.<emph.end type="italics"/><!-- KEEP S--></s></p></chap><chap><p type="main"> <s id="id.001267"><arrow.to.target n="marg76"/></s></p><p type="margin"> <s id="id.001268"><margin.target id="marg76"/>76</s></p><p type="main"> <s id="id.001269">Cap. 13. (<emph type="italics"/>Con&longs;iderare, quod diameter est co&longs;tæ incommen&longs;urabilis<emph.end type="italics"/>) vide <lb/>quæ de hac re &longs;crip&longs;i lib. 1. Priorum &longs;ecto 1. cap. 23.</s></p><p type="main"> <s id="id.001270"><arrow.to.target n="marg77"/></s></p><p type="margin"> <s id="id.001271"><margin.target id="marg77"/>77</s></p><p type="main"> <s id="id.001272">Eodem cap. (<emph type="italics"/>Similiter autem & acutum; non enim idem &longs;impliciter <lb/>in omnibus dicitur: nam vox acuta quidem velox (&longs;icut dicunt, qui &longs;e­ <lb/>cundum numeros harmonici &longs;unt) angulus autem acutus, qui minor e&longs;trecto; gla­ <lb/>dius verò, qui e&longs;t anguli acuti<emph.end type="italics"/>) affert tres &longs;pecies acuti, aliud dicens e&longs;&longs;e acu­ <lb/>tum, quod e&longs;t in voce acuta; aliud, quod e&longs;t in angulo acuto: aliud denique, <lb/>quod e&longs;t in gladio acuto horum enim trium acumen diuer&longs;o modo &longs;e habet. <lb/></s> <s id="id.001273">nam acumen vocis, & &longs;oni ex celeritate motus, qua aer percu&longs;&longs;us impelli­ <lb/>tur; grauitatem autem ex tarditate oriri tradiderunt antiqui Mu&longs;ici om­ <lb/>nes: quamuis non ex &longs;ola celeritate, & tarditate, &longs;ed ex alijs etiam cau&longs;is <lb/>oriri po&longs;&longs;e voluerint. </s> <s id="id.001274">Primus <expan abbr="omniũ">omnium</expan> Architas Tarentinus, vt e&longs;t apud Por­ <lb/>phirium in harmonicis Ptolæmei, & Zarlinum pag. </s> <s id="id.001275">58. complem. </s> <s id="id.001276">mu&longs;ica­ <lb/>lium, ait, &longs;i virga celerius feriat aerem, gigni motum celeriorem in aere, <lb/><expan abbr="atq;">atque</expan> hinc &longs;onum acutiorem reddi, experientia con&longs;tat: &longs;i autem eadem vir­ <lb/>ga tardius aerem feriat, gigni motum in aere tardiorem, ex quo etiam &longs;o­ <lb/>num grauem, vt experientia docet. </s> <s id="id.001277">Ptolæmeus deinde lib. 1. cap. 3. Harm. <lb/><!-- REMOVE S-->cum ex alijs, tum ex celeritate oriri &longs;onum acutum, grauem verò ex tardi­ <lb/>tate a&longs;&longs;erit; vt &longs;i chorda eadem parum inten&longs;a pul&longs;etur, tardius aerem ver­ <lb/>berat, & ideo grauiorem &longs;onum efficit: &longs;i autem magis intendatur, validius <lb/>aerem pul&longs;abit, & proinde citiorem motum illi imprimet, & propterea <lb/>acutiorem &longs;onum reddet. </s> <s id="id.001278">hæc ille. </s> <s id="id.001279">videmus etiam, quod cannæ organo­ <lb/>rum maiores cum plus aeris moucant, & idcirco tardius, &longs;onum grauiorem <lb/>emittunt, quàm cannæ graciliores, quæ quia parum aeris cient, & ideo ce­ <lb/>lerius, &longs;onum acutum edunt. </s> <s id="id.001280">ab hac &longs;ententia po&longs;teriores Mu&longs;ici non recel­ <lb/>&longs;erunt, vt videre e&longs;t apud Zarlinum.<!-- KEEP S--></s></p><p type="main"> <s id="id.001281">In quo po&longs;tea con&longs;i&longs;tat ratio acnti anguli, explicat <expan abbr="induc&etilde;s">inducens</expan> definitionem <lb/>ip&longs;ius, quæ e&longs;t inter de&longs;initiones primi Elem. huiu&longs;modi, Angulus acutus <lb/>e&longs;t, qui minor recto e&longs;t. </s> <s id="id.001282">Demum explicat, cur nam gladius dicatur acutus, <lb/>quia nimirum habet angulum acutum &longs;uperficialem, ide&longs;t, quem duæ &longs;uper­ <lb/>ficies &longs;imul in acie gladij concurrentes efficiunt.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.068.jpg" pagenum="68"/><p type="main"> <s id="id.001283"><arrow.to.target n="marg78"/></s></p><p type="margin"> <s id="id.001284"><margin.target id="marg78"/>78</s></p><p type="main"> <s id="id.001285">Eodem cap. (<emph type="italics"/>Rur&longs;um &longs;i eorundem; quæ &longs;unt &longs;ub eodem nomine diuer&longs;æ d ffe­ <lb/>rentiæ &longs;unt; vt coloris, qui e&longs;t in corporibus, & in melodijs<emph.end type="italics"/>) veteres Mu&longs;ici can­ <lb/>tilenas omnes ad tria genera reuocarunt, videlicet Enharmonicum, Chro­ <lb/>maticum, & Diatonicum; quæ diftinguebantur inuicem ex varia diui&longs;ione <lb/>interuallorum, ex quibus ip&longs;orum Monochordia conftabant: &longs;iue ex varijs <lb/>vocum interuallis, v. <!-- REMOVE S-->g. <!-- REMOVE S-->quia in vno continebantur plures toni, vt in Diato­ <lb/>nico; in alio plures die&longs;es, vt in Enharmonico; in tertio verò plura &longs;emito­ <lb/>nia, vt in Chromatico: quæ vox deducitur à chroma, græco, quod latinis <lb/>e&longs;t color; quare Chromaticum latinè redditur coloratum. </s> <s id="id.001286">Hic e&longs;t igitur <lb/>color ille, quem hic Ari&longs;t. innuit. </s> <s id="id.001287">quod genus for&longs;itan à calore denomina­ <lb/>batur, quòd ip&longs;ius notæ mu&longs;icales e&longs;&longs;ent coloratæ, vt hoc modo ab alijs ge­ <lb/>neribus digno&longs;ceretur. quam con&longs;uetudinem exi&longs;timat Zarlinus cap. 46. &longs;e­ <lb/>cundæ partis, etiam no&longs;tra tempe&longs;tate aliquo modo per&longs;euerare, cum vi­ <lb/>deamus in organis, & alijs huiu&longs;modi in&longs;trumentis, quæ pinnas, vulgò ta­ <lb/>&longs;tos, habent; illas inquam pinnas, quæ chromaticis interuallis deputatæ <lb/>&longs;unt, colore nigro tinctas e&longs;&longs;e.</s></p></chap><chap><p type="head"> <s id="id.001288"><emph type="italics"/>Libro Quarto.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001289"><arrow.to.target n="marg79"/></s></p><p type="margin"> <s id="id.001290"><margin.target id="marg79"/>79</s></p><p type="main"> <s id="id.001291">Cap. 1. loco 10. (<emph type="italics"/>Si quis in&longs;ecabiles ponens lineas<emph.end type="italics"/>) nonnulli antiquorum <lb/>Philo&longs;ophorum putarunt omnia ex indiui&longs;ibilibus componi, vt Demo­ <lb/>critus, & Leucippus, & propterea dixerunt, etiam lineas con&longs;tare ex lineis <lb/>quibu&longs;dam ade ò paruis, quæ omnino e&longs;ient in&longs;ecabiles, &longs;eu indiui&longs;ibiles: de <lb/>quibus plura in libello de line is in&longs;ecabilibus.</s> </p></chap><chap><p type="head"> <s id="id.001292"><emph type="italics"/>Libro Sexio.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001293"><arrow.to.target n="marg80"/></s></p><p type="margin"> <s id="id.001294"><margin.target id="marg80"/>80</s></p><p type="main"> <s id="id.001295">Cap. 2. loco 32. (<emph type="italics"/>Vt qui lineam definiunt longitudinem &longs;ine latitudine e&longs;&longs;e<emph.end type="italics"/>) <lb/>&longs;uppenimus lectorem inteil exi&longs;&longs;e definitiones &longs;altem primi Elem. in­ <lb/>ter quas definitio lineæ e&longs;t &longs;ecunda, <expan abbr="cadem&qacute;">cademque</expan>; cum hac Ari&longs;totelis.<!-- KEEP S--></s></p></chap><chap><p type="head"> <s id="id.001296"><emph type="italics"/>Libro Octauo.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001297"><arrow.to.target n="marg81"/></s></p><p type="margin"> <s id="id.001298"><margin.target id="marg81"/>81</s></p><p type="main"> <s id="id.001299">Cap. 2. loco 41. (<emph type="italics"/>Videntur dutem in di&longs;ciplinis, &longs;eu Mathematicis quædam <lb/>ob de&longs;initionis defectum non facile de&longs;cribi; vt & quoniam, quæ ad latus &longs;e­ <lb/>cat planum linea, &longs;imiliter diuidit & lineam, & locum: definitione autem dicta, <lb/>&longs;tatim manife&longs;tum e&longs;t, quod dicitur, nam eandem ablationem habent loca, & linea, <lb/>&longs;ive latus planæ figuræ, est autem definitio eiu&longs;dem proportionis hæc<emph.end type="italics"/>) mendosè <lb/>lugitur à nonnullis (<emph type="italics"/>E&longs;t distem de&longs;initio eiu&longs;dem orationis hæc<emph.end type="italics"/>) quos puto de­ <lb/>ceptos ab æquiuoco <foreign lang="greek">lsgous</foreign> quod & orationem, & rationem, &longs;iue proportio­ <lb/>nem &longs;ignificat: hic autem &longs;ignificare proportionem res &longs;ubrecta &longs;atis mani­ <lb/>fe&longs;tat. </s> <s id="id.001300">Notandum po&longs;tea cum Alexandro (quod & &longs;uperius alias commo­ <lb/>nui in cap. de Priori, & alibi) per verbum (De&longs;cribi) fignificari hoc loco <lb/>geometricè demon&longs;trare, quoniam Geometræ <expan abbr="nõ">non</expan> ni&longs;i adhibit is de&longs;criptio­ <lb/>nibus, &longs;eu figutis demon&longs;trant. </s> <s id="id.001301">Vult autem Ari&longs;t. exemplo mathematico <lb/>o&longs;tendere, difficile e&longs;&longs;e di&longs;putare, aut <expan abbr="argum&etilde;tari">argumentari</expan>, ni&longs;i prius rectè a&longs;&longs;ignetur <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.069.jpg" pagenum="69"/>definitio illius rei, de qua di&longs;&longs;eritur. </s> <s id="id.001302">Porrò exemplum mathematicum hic <lb/>allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro­ <lb/>po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e­ <lb/>cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t <lb/><figure id="id.009.01.069.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.069.1.jpg" place="text"/> <lb/>in eadem proportione, vt in triangulo A B C, <lb/>linea D E, parallela ba&longs;i B C, &longs;ecat latera A B, <lb/>& A C, in punctis D, & E, in eadem ratione, <lb/>in qua etiam fecat totum triangulum, ita vt <lb/>eadem &longs;it proportio lineæ A D, ad D B, & lineæ <lb/>A E, ad E C, quæ e&longs;t partium totalis trianguli <lb/>A B C, &longs;eilicet quæ e&longs;t partis A D E, ad partem <lb/>E D C, fiue ad partem D E B. quod con&longs;tat ex <lb/>&longs;ecunda 6. Elem. <!-- KEEP S--></s> <s id="id.001303">Inquit ergo Ari&longs;t. <!-- KEEP S--></s> <s id="id.001304">Si quis <lb/>vellet hoc demon&longs;trare nondum præmi&longs;&longs;a defi­ <lb/>nitione eorum, quæ habent eandem rationem, &longs;iue nondum definitione al­ <lb/>lata quantitatum proportionalium, hic difficile id valeret o&longs;tendere: at ve­ <lb/>rò allata prins definitione quantitatum proportionalium facile demon&longs;tra­ <lb/>bit. </s> <s id="id.001305">Subdit verò Ari&longs;t. dictam definitionem, dicens, tunc quantitates e&longs;&longs;e <lb/>proportionales, quando habent eandem ablationem, ide&longs;t, eandem diui&longs;io­ <lb/>nem, ide&longs;t, eadem diui&longs;io ne tantum proportionaliter de vna, quantum de <lb/>altera magnitudine re&longs;ecatur: Quemadmodum etiam Euclides loco cita­ <lb/>to probat, latera illius trianguli, & &longs;uperficiem e&longs;&longs;e &longs;imiliter diui&longs;a, ex quo <lb/>&longs;equitur e&longs;&longs;e proportionalia. </s> <s id="id.001306">Porrò Euclides definit. </s> <s id="id.001307">&longs;eptima 5. paulo ali­ <lb/>ter definit quantitates proportionales e&longs;&longs;e illas, quæ eandem habent ratio­ <lb/>nem, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;it, vt prima ad &longs;ecundam, ita tertia ad quartam. </s> <s id="id.001308">ex quibus <lb/>quoad Mathematicas &longs;pectat, huic loco &longs;atisfactum &longs;it.</s></p><p type="main"> <s id="id.001309"><arrow.to.target n="marg82"/></s></p><p type="margin"> <s id="id.001310"><margin.target id="marg82"/>82</s></p><p type="main"> <s id="id.001311">Cap. 4. loco 86. <emph type="italics"/>(Tentandum autem, & ea, in quæ &longs;æpi&longs;&longs;imè incidunt di&longs;puta­ <lb/>tiones, tenere, nam quemadmodum in Geometria ante opus e&longs;t circa elementa exer­ <lb/>citatum e&longs;&longs;e, & in numeris circa capitales promptè &longs;e habere, & multum refert ad <lb/>boc, & alium numerum cogno&longs;cere multiplicatum)<emph.end type="italics"/> Elementa vocabant antiqui <lb/>demon&longs;trationes faciliores, & &longs;impliciores, quales propriè &longs;unt omnes, quæ <lb/>&longs;ex prioribus libris Euclidianis continentur: ex illis enim tanquam ex ele­ <lb/>mentis ab&longs;tru&longs;iores, & difficiliores demon&longs;trationes deducebant. </s> <s id="id.001312"><expan abbr="atq;">atque</expan> hæc <lb/>e&longs;t ratio, cur Euclides &longs;uos libros elementa nuncupauerit. </s> <s id="id.001313">ait igitur curan­ <lb/>dum e&longs;&longs;e horum elementorum cognitionem in promptu habere, quia fre­ <lb/>quens de ip&longs;is incidit di&longs;putatio. </s> <s id="id.001314">Per capitales numeros intelligo &longs;implices <lb/>ab vnitate, <expan abbr="v&longs;q;">v&longs;que</expan> ad nouem inclu&longs;iuè. </s> <s id="id.001315">& quando ait, alium numerum cogno­ <lb/>&longs;cere multiplicatum, &longs;ignificat vtile valdè e&longs;&longs;e ad quotidianum v&longs;um <lb/>cogno&longs;cere, quemnam numerum producant numeri capitales, <lb/>&longs;i ad inuicem multiplicentur, quamuis huiu&longs;modi co­ <lb/>gnitio facilis, ac leuis &longs;it: qua de cau&longs;a vide­ <lb/>mus v&longs;que in hanc diem pueros diu in <lb/>Abaco memoriter perdi&longs;cen­ <lb/>do detineri.</s></p></chap><chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.070.jpg" pagenum="70"/><p type="head"> <s id="id.001316"><emph type="italics"/>Ex Primo Elenchorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001317"><arrow.to.target n="marg83"/></s></p><p type="margin"> <s id="id.001318"><margin.target id="marg83"/>83</s></p><p type="main"> <s id="id.001319">Cap. 10. <emph type="italics"/>(Nam p&longs;eudograpbiæ non contentio&longs;æ (&longs;ecundum enim ea, quæ <lb/>&longs;ub arte &longs;unt, captio&longs;æ &longs;unt ratiocinationes) <expan abbr="neq;">neque</expan> &longs;i aliqua e&longs;t p&longs;eudogra­ <lb/>phia circa verum, vt Hippocratis quadratura, quæ per lunulas, &longs;ed, vt <lb/>Bry&longs;&longs;o quadrauit circulum; & tamet&longs;i quadretur circulus, quia tamen <lb/>non &longs;ecundum rem, ideo &longs;ophi&longs;ticus)<emph.end type="italics"/> qua ratione Hippocrates orbi quadrum <lb/>exhibere æquale tentauerit, explicatum e&longs;t abundè in 2. Priorum cap. 31. <lb/>& quo itidem modo Bry&longs;&longs;o lib. 1. Po&longs;ter. tex. <!-- REMOVE S-->23. <expan abbr="&longs;olũmodo">&longs;olummodo</expan> id hoc loco no­ <lb/>tandum per p&longs;eudographiam intelligere, vt apertè etiam inferius explicat, <lb/>Geometricam demon&longs;trationem fallacem, eò quod demon&longs;trationes geo­ <lb/>metricæ fiant adhibitis de&longs;criptionibus, &longs;eu figurationibus: p&longs;endographia <lb/>autem latinè idem e&longs;t, ac fal&longs;a de&longs;criptio; quemadmodum è contrariò, &longs;i­ <lb/>cuti &longs;upra in Topicis, & alibi ob&longs;eruaui, per de&longs;cribere intelligit geometri­ <lb/>cè demon&longs;trare, & per de&longs;criptiones intelligit demon&longs;trationes geometri­ <lb/>cas. </s> <s id="id.001320">Qua ratione item Hippocrates ex ijs, quæ &longs;ub arte Geometriæ &longs;unt, <lb/>procederet ibi dictum e&longs;t, propter quod non e&longs;t contentio&longs;a, quamuis fallax <lb/>ip&longs;ius demon&longs;tratio: appellat enim Ari&longs;t illas demon&longs;trationes contentio­ <lb/>&longs;as, quæ non procedunt ex proprijs illius &longs;cientiæ, in qua fiunt, &longs;ed ex com­ <lb/>munibus alijs &longs;cientijs: captio&longs;as verò, & &longs;ophi&longs;ticas, quæ ex proprijs &longs;cien­ <lb/>tiæ, in qua fiunt, decipiunt. </s> <s id="id.001321">At verò demon&longs;tratio, &longs;eu p&longs;eudographia Bry&longs;­ <lb/>&longs;onis erat contentio&longs;a, quia ex communibus, & extra Geometriam petitis <lb/>argumentabatur: quemadmodum ibi explicatum e&longs;t.</s></p><p type="main"> <s id="id.001322"><arrow.to.target n="marg84"/></s></p><p type="margin"> <s id="id.001323"><margin.target id="marg84"/>84</s></p><p type="main"> <s id="id.001324">Eodem cap. <emph type="italics"/>(Quadratura per lunulas non contentio&longs;a)<emph.end type="italics"/> inquit Hippocratis <lb/>tetragoni&longs;mum, de quo in 2. Priorum, quæ non contentio&longs;a dicitur, quia ex <lb/>proprijs Geometriæ deducebatur.</s></p><p type="main"> <s id="id.001325"><arrow.to.target n="marg85"/></s></p><p type="margin"> <s id="id.001326"><margin.target id="marg85"/>85</s></p><p type="main"> <s id="id.001327">Ibidem <emph type="italics"/>(Bry&longs;&longs;onis autem contentio&longs;a: & illam quidem non e&longs;t transferre, ni&longs;i <lb/>ad Geometriam &longs;olum; eo quod ex proprijs &longs;it principijs)<emph.end type="italics"/> <expan abbr="quãdo">quando</expan> ait <emph type="italics"/>(& illam qui­ <lb/>dem)<emph.end type="italics"/> intelligit quadrationem Hippocratis. </s> <s id="id.001328">vide 2. Prior cap. 31. & quæ pau­ <lb/>lo ante in præcedentibus locis diximus.</s></p><p type="main"> <s id="id.001329"><arrow.to.target n="marg86"/></s></p><p type="margin"> <s id="id.001330"><margin.target id="marg86"/>86</s></p><p type="main"> <s id="id.001331">Ibidem <emph type="italics"/>(Hanc autem ad plures)<emph.end type="italics"/> intelligit tetragoni&longs;mum Bry&longs;&longs;onis, qui <lb/>per communia deducebatur. </s> <s id="id.001332">lege &longs;uperius dicta in præcedentibus locis hu­ <lb/>ius capituli.</s></p><p type="main"> <s id="id.001333"><arrow.to.target n="marg87"/></s></p><p type="margin"> <s id="id.001334"><margin.target id="marg87"/>87</s></p><figure id="id.009.01.070.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.070.1.jpg" place="text"/><p type="main"> <s id="id.001335">Ad &longs;inem cap. <emph type="italics"/>(Aut vt Antiphon quadra­ <lb/>uit)<emph.end type="italics"/> &longs;imile peccatum pecca&longs;&longs;e Antiphon­ <lb/>tem in orbe quadrando, ac Hippocratem, <lb/>Ari&longs;t. his verbis videtur &longs;ignificare, ide&longs;t, <lb/>ip&longs;um, quamuis ex proprijs Geometriæ, <lb/>fal&longs;is tamen ratiocinatum e&longs;&longs;e. </s> <s id="id.001336">Cæterum <lb/>Antiphontem in hunc modum orbem ad <lb/>quadrum redigere tenta&longs;&longs;e, tradit Simpli­ <lb/>cius. </s> <s id="id.001337">circulo quadrando in&longs;cribebat pri­ <lb/>mò quadratum A B C D. deinde in &longs;ingu­ <lb/>lis quatuor &longs;egmentis in&longs;cribebat totidem <lb/>trigona æquilatera, vt patet in ad&longs;cripta <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.071.jpg" pagenum="71"/>figura. </s> <s id="id.001338">po&longs;tea &longs;uper &longs;ingula latera horum triangulorum in reliquis &longs;egmen­ <lb/>tis in&longs;cribebat adhuc triangula &longs;imilia triangulo A I E. alia in&longs;uper trigona <lb/>&longs;uper latera i&longs;torum con&longs;tituebat, donec ambitus figuræ illius multilateræ <lb/>in circulo delinearæ, circumferentiæ circuli aptaretur. </s> <s id="id.001339">quod fieri po&longs;&longs;e ille <lb/>falsò contra Geometriæ principia a&longs;&longs;umebat; e&longs;t enim principium Geome­ <lb/>tricum continuum e&longs;&longs;e diui&longs;ibile in infinitum, <expan abbr="neq;">neque</expan> per diui&longs;ionem ab&longs;umi <lb/>po&longs;&longs;e; cui principio aduer&longs;atur, dum putat &longs;e con&longs;umpturum rotum circu­ <lb/>lum, diuidendo illud in triangula &longs;emper minora; vel quia putat, lineam <lb/>curuam con&longs;tare ex minimis lineis rectis. </s> <s id="id.001340">Similiter igitur <expan abbr="atq;">atque</expan> Hippocra­ <lb/>res errauit, qúi pariter in Geometria fallebatur: Antiphon quidem contra <lb/>principia illius: Hippocrates verò a&longs;&longs;umens fal&longs;i quidpiam in Geometria. <lb/><!-- KEEP S--></s> <s id="id.001341">At Bry&longs;&longs;o, eo quod per communia alijs &longs;cientijs deduceret ratiocinatio­ <lb/>nem propterea p&longs;eudographia Antiphontis non litigio&longs;a quidem, &longs;ed <lb/>tamen fallax extitit, non enim per communia alijs &longs;cientijs <lb/>procedat; vnde nec transferri poterat ip&longs;ius fal&longs;a de­ <lb/>&longs;criptio, &longs;eu demon&longs;tratio extra Geometriæ li­ <lb/>mites, quod cau&longs;a e&longs;t contentionis.</s></p><p type="head"> <s id="id.001342"><emph type="italics"/>Logicorum locorum finis.<emph.end type="italics"/></s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.072.jpg" pagenum="72"/><chap><p type="head"> <s id="id.001343">EX PRIMO LIBRO <lb/>PHYSICORVM. <lb/><arrow.to.target n="marg88"/></s></p><p type="margin"> <s id="id.001344"><margin.target id="marg88"/>88</s></p><p type="main"> <s id="id.001345">Tex. 11. <emph type="italics"/>(Simul autem <expan abbr="neq;">neque</expan> conuenit omnia &longs;oluere', &longs;ed <expan abbr="quæcunq;">quæcunque</expan> ex <lb/>principijs aliquis demon&longs;trans <expan abbr="m&etilde;titur">mentitur</expan>; <expan abbr="quæcunq;">quæcunque</expan> verò non, minimè: <lb/>vt tetragoni&longs;mum, eum quidem, qui per &longs;ectiones Geometrici est di&longs;­ <lb/>&longs;oluere: illum autem, qui Antiphontis non Geometrici e&longs;t<emph.end type="italics"/>) Tetrago­ <lb/>ni&longs;mum, &longs;eu circuli quadraturam per &longs;ectiones, e&longs;&longs;e illam Hip­ <lb/>pocratis Chij exi&longs;timant græci expo&longs;itores, qui per lunulas, quas Ari&longs;t. &longs;e­ <lb/>ctiones appellat, orbem quadrare tentabat. </s> <s id="id.001346">Eius den on&longs;trationem expli­ <lb/>caui ad cap. 31. de Abductione in 2. Priorum, quam inibi videas. </s> <s id="id.001347">hoc &longs;olum <lb/>hic notandum pertinere ad Geometram, ip&longs;am refellere, quia ex fal&longs;a qua­ <lb/>dam præmi&longs;&longs;a ex Geometria de&longs;umpta, ratiocinabatur, idcirco debet (in­ <lb/>quit Ari&longs;t.) Geometra illius deceptionem inuenire. </s> <s id="id.001348">Tetragoni&longs;mum autem <lb/>Antiphontis non e&longs;t Geometræ <expan abbr="cõfutare">confutare</expan>, quia aduer&longs;abatur principijs Geo­ <lb/>metriæ, &longs;upponebat enim circuli circumferentiam ex indiuiduis, <expan abbr="minimis&qacute;">minimisque</expan>; <lb/>lineis rectis componi: cuius fal&longs;am demon&longs;trationem explicatam inuenies <lb/>ad cap. 10. primi Elench. <!-- REMOVE S-->po&longs;&longs;umus addere tertiam rationem quia &longs;cilicet <lb/>Hippocrates non procedebat per communia alijs &longs;cientijs, vt videre e&longs;t ad <lb/>tex. <!-- REMOVE S-->23. primi Po&longs;ter. cap. 8. vbi ip&longs;ius p&longs;eudographiam expo&longs;ui. Quemad­ <lb/>modum igitur Geometra di&longs;&longs;oluit fal&longs;as tantummodo rationes eas, quæ &longs;er­ <lb/>uatis Geometricis principijs procedunt; non autem eas, quæ Geometriæ <lb/>principia conuellunt: ita Phy&longs;ico non incumbit <expan abbr="cõtra">contra</expan> Parmenidem, ac Me­ <lb/>li&longs;&longs;um naturæ principia de&longs;truentes di&longs;ceptare, aut fallaces eorum rationes <lb/>coarguere. </s> <s id="id.001349">Hoc volebat Ari&longs;toteles inferre.</s></p></chap><chap><p type="head"> <s id="id.001350"><emph type="italics"/>Ex Secundo Phy&longs;icorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001351"><arrow.to.target n="marg89"/></s></p><p type="margin"> <s id="id.001352"><margin.target id="marg89"/>89</s></p><p type="main"> <s id="id.001353">Tex. 20. (<emph type="italics"/>Geometria enim de phy&longs;ica linea con&longs;iderat, &longs;ed non quatenus <lb/>e&longs;t phy&longs;ici: Per&longs;pectiua autem mathematicam quidem lineam, &longs;ed non <lb/>quatenus phy&longs;ica e&longs;t<emph.end type="italics"/>) quamuis textus hic non pertineat ad Mathe­ <lb/>maticum, libuit tamen illum in ordinem no&longs;trum recen&longs;ere, ope­ <lb/>ræpretium etenim e&longs;t ea, quæ in ip&longs;o continentur à nonnullis recentioribus <lb/>rectè intelligi, vt ab his moniti, ab inani quadam optices impugnatione ab­ <lb/>&longs;tineant, ac tandem ex Ari&longs;t. lineas illas vi&longs;uales quas ip&longs;i de medio tollunt, <lb/>per&longs;picuè videant. </s> <s id="id.001354">cætera, quæ in præcedentibus locis Ari&longs;t. de Natura Ma­ <lb/>thematicarum habet, &longs;unt præter no&longs;trum in&longs;titutum.</s></p><p type="main"> <s id="id.001355"><arrow.to.target n="marg90"/></s></p><p type="margin"> <s id="id.001356"><margin.target id="marg90"/>90</s></p><p type="main"> <s id="id.001357">Tex. 28. (<emph type="italics"/>Alio autem modo, &longs;orma, & exemplum: bæc autem e&longs;t ratio ip&longs;ius, <lb/>quod quid erat e&longs;&longs;e, & huius genera, vt ip&longs;ius diapa&longs;on duo ad vnum, & omnino <lb/>numerus, & partes, quæ in ratione &longs;unt<emph.end type="italics"/>) vt benè intelligas, quod in præ&longs;enti <lb/>textu <expan abbr="mathematicũ">mathematicum</expan> e&longs;t, con&longs;ule prius, quæ &longs;crip&longs;i ad tex. <!-- REMOVE S-->1. cap. primi 2. Po­ <lb/>&longs;ter. <!-- REMOVE S-->&longs;uper verba illa (<emph type="italics"/>Quid e&longs;t con&longs;onantia?<emph.end type="italics"/>) vbi per&longs;picuè videbis, cur <expan abbr="con-&longs;onãtiæ">con­ <lb/>&longs;onantiæ</expan>, quæ dicitur Diapa&longs;on, e&longs;&longs;entia, & definitio &longs;it ip&longs;a proportio dupla, <lb/>quæ &longs;ub his num. </s> <s id="id.001358">2.1. continetur: quibus per&longs;pectis facilis erit phy&longs;ico totius <lb/>loci intelligentia.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.073.jpg" pagenum="73"/><p type="main"> <s id="id.001359"><arrow.to.target n="marg91"/></s></p><p type="margin"> <s id="id.001360"><margin.target id="marg91"/>91</s></p><p type="main"> <s id="id.001361">Tex. 68. (<emph type="italics"/>Aut enim ad ip&longs;um quid e&longs;t, reducitur ip&longs;um propter quid in immo­ <lb/>bihbus, vt in Mathematicis, ad definitionem enim recti, aut commen&longs;urabilis, aut <lb/>alius cuiu&longs;piam reducitur vltimum<emph.end type="italics"/>) ex his manife&longs;tè videas Mathematicas <expan abbr="de-mõ&longs;trare">de­ <lb/>mon&longs;trare</expan> per cau&longs;am formalem, cum cau&longs;am ip&longs;am ad ip&longs;um quid e&longs;t, ide&longs;t, <lb/>ad definitionem reducant. </s> <s id="id.001362">quorum exempla in logicis ex Mathematicis at­ <lb/>tuli: &longs;ed etiam &longs;equentis loci exemplum de triangulo idem apertè manife­ <lb/>&longs;tat; in quo probat duos angulos A C B, A C D, e&longs;&longs;e rectos, ex definitione <lb/>ip&longs;orum, &longs;iue ex definitione lineæ perpendicularis A C, quod idem e&longs;t.</s></p><p type="main"> <s id="id.001363"><arrow.to.target n="marg92"/></s></p><p type="margin"> <s id="id.001364"><margin.target id="marg92"/>92</s></p><p type="main"> <s id="id.001365">Tex 89. (<emph type="italics"/>E&longs;t autem nece&longs;&longs;arium in Mathematicis, & in his, quæ &longs;ecundum <lb/>natur am fiunt qua&longs;i eodem modo; quoniam enim hocrectum e&longs;t, nece&longs;&longs;e e&longs;t, trian­ <lb/>gulum trcs angulos habere æquales duobus rectis; &longs;ed non, &longs;i hoc, illud; &longs;ed &longs;i hoc <lb/>non e&longs;t, <expan abbr="neq;">neque</expan> rectum e&longs;t.<emph.end type="italics"/>) cum animaduerterim non parum e&longs;&longs;e di&longs;&longs;en&longs;ionis, & <lb/>difficultatis in exemplo hoc mathematico explicando, ita vt recentiores <lb/>quidam textum <expan abbr="hũc">hunc</expan> pro arbitratu &longs;uo perperam latinè verterint: ideò pri­ <lb/>mum ex græcis codicibus interpretationem hanc veram attuli. </s> <s id="id.001366">deinde, quia <lb/>etiam græci in exemplo mathematico enodando, vel malè, vt Simplicius; <lb/>vel ob&longs;curè nimis, vt reliqui; Latini verò vel nihil, vel peius multò loquun­ <lb/>tur, ideò &longs;ic ego exponendum cen&longs;ui. </s> <s id="id.001367">cum velit Ari&longs;t. o&longs;tendere nece&longs;&longs;ita­ <lb/>tem, quæ in &longs;cientijs inter præmi&longs;&longs;as, &longs;cu medium, & conclu&longs;ionem reperi­ <lb/>tur, affert exemplum illud mathematicum &longs;ibi familiare, demon&longs;trationem <lb/>&longs;cilicet illam, qua o&longs;tenditur, omne triangulum habere tres angulos æqua­ <lb/>les duobus rectis angulis, cuius fu&longs;i&longs;&longs;imam explicationem inuenies &longs;upra in <lb/>primo Priorum, &longs;ecto 3. cap. 1. quam nece&longs;&longs;e e&longs;t, con&longs;ulas. </s> <s id="id.001368">pro medio autem <lb/>huius pa&longs;&longs;ionis accipit lineam perpendicularem, quam innuit verbis illis <lb/><emph type="italics"/>(quoniam enim hoc rectum e&longs;t<emph.end type="italics"/>) vt in figura &longs;it triangulum A B C, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vt latus <lb/><figure id="id.009.01.073.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.073.1.jpg" place="text"/> <lb/>A C, &longs;it perpendiculare <expan abbr="cũ">cum</expan> latere B C, & pro­ <lb/>ducatur B C, in D; tunc triangulum A B C, <lb/>habere tres angulos, A, B, & A C B, æquales <lb/>duobus rectis planum erit: nam <expan abbr="cũ">cum</expan> latus A C, <lb/>&longs;it perpendiculare (quod Ari&longs;t. dicit, cum <expan abbr="re-ctũ">re­ <lb/>ctum</expan> hoc &longs;it) erunt duo anguli deinceps A C B, <lb/>A C D, recti, ex definitione lineæ perpendicu­ <lb/>laris, cum ergo duo anguli A, & B, externo, <expan abbr="recto&qacute;">rectoque</expan>; A C D, &longs;int æquales per <lb/>32. primi, & reliquus angulus A C B, communis, ide&longs;t, &longs;it angulus triangu­ <lb/>li, & angulus vnus lineæ perpendicularis, & ideò rectus; manife&longs;tè apparet, <lb/>tres angulos A, B, A C B, e&longs;&longs;e æquales nece&longs;&longs;ariò duobus rectis, ex po&longs;itio­ <lb/>ne illius recti, &longs;iue lateris perpendicularis, quia ex verò, verum nece&longs;&longs;ariò <lb/>&longs;equitur; non tamen po&longs;ita hac pa&longs;&longs;ione, &longs;iue conclu&longs;ione, habere &longs;cilicet <lb/>tres angulos æquales duobus rectis, nece&longs;&longs;ariò &longs;equitur illud e&longs;&longs;e rectum, <lb/>idelt latus illud A C, e&longs;&longs;e perpendiculare ad latus B C, quia verum <lb/>&longs;equi pote&longs;t ex verò, & falsò. </s> <s id="id.001369">valebit tamen hæc con&longs;e quen­ <lb/>tia, &longs;i triangulum non habet hanc proprietatem, ne­ <lb/>que illud rectum e&longs;t, ide&longs;t, <expan abbr="neq;">neque</expan> latus prædi­ <lb/>ctum crit <expan abbr="perp&etilde;diculare">perpendiculare</expan>, quia fal&longs;um <lb/>non, ni&longs;i exfal&longs;o &longs;equitur.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.074.jpg" pagenum="74"/><chap><p type="head"> <s id="id.001370"><emph type="italics"/>Ex Tertio Phy&longs;icorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001371"><arrow.to.target n="marg93"/></s></p><p type="margin"> <s id="id.001372"><margin.target id="marg93"/>93</s></p><p type="main"> <s id="id.001373">Tex. 26. <emph type="italics"/>(Et hi quidem infinitum e&longs;&longs;e par; hoc enim compræhen&longs;um, & <lb/>ab impari terminatum tribuit ijs, quæ &longs;unt, infinitatem. </s> <s id="id.001374">&longs;ignum autem <lb/>huius id e&longs;&longs;e, quod contingit in numcris, circumpo&longs;it is enim Gnomoni­ <lb/>bus circa vnum, & &longs;eor&longs;um, aliquando quidcm &longs;emper aliam fieri &longs;pe­ <lb/>ciem, aliquando autem vnam)<emph.end type="italics"/> vt melius percipiantur ea, quæ &longs;equuntur, lege <lb/>prius, quæ in cap. de Motu in po&longs;t prædicamentis &longs;crip&longs;i de Gnomone, ad <lb/>&longs;imilitudinem enim Gnomonis illius Geometrici, inueniuntur etiam in nu­ <lb/>meris Gnomones Arithmetici. </s> <s id="id.001375">Pythagorici enim (à quibus i&longs;ta mutuatus <lb/>e&longs;t Ari&longs;t. numeros impares &longs;olos appellabant Gnomones, eò quod in for­ <lb/>mam normæ æquilateræ, &longs;iue Gnomonis con&longs;titui po&longs;&longs;int, vt patet in his <lb/><figure id="id.009.01.074.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.074.1.jpg" place="text"/> <lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de <lb/>reliquis imparibus. </s> <s id="id.001376">pares autem numeri, quia ne­ <lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum <lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la­ <lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di­ <lb/>&longs;ponatur &longs;ic <figure id="id.009.01.074.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.074.2.jpg" place="text"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con­ <lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="id.009.01.074.3.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.074.3.jpg" place="text"/> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s> <s id="id.001377">Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e <lb/>infinitum ip&longs;um, quia videbant ip&longs;um e&longs;&longs;e cau&longs;am perpetuæ diui&longs;ionis, cum <lb/>quælibet res quanta &longs;it diui&longs;ibilis bifariam, ide&longs;t in duo &longs;ecundum numerum <lb/>parem, & &longs;ubdiui&longs;ibilis po&longs;tea bifariam, & &longs;ic in infinitum, vt de linea pro­ <lb/>blematicè probatur in 10. primi Elem. quamuis theorematicè &longs;it axioma. <lb/></s> <s id="id.001378">hunc porrò numerum parem dicebant terminatum e&longs;&longs;e ab impari, quia ori­ <lb/>tur ex diui&longs;ione cuiu&longs;uis rei, quæ vna &longs;it, &longs;umentes vnitatem pro impari. <lb/></s> <s id="id.001379">&longs;ignum præterea huius finitatis ab impari, & infinitatis à pari numero pro­ <lb/>cedentis, aiunt e&longs;&longs;e Gnomones, numeros &longs;cilicet impares: Gnomones enim, <lb/>ide&longs;t impares numeri vnitati additi, producunt eandem perpetuò numero­ <lb/>rum formam, videlicet quadratum: at verò è contrariò numeri pares vni­ <lb/>tati additi, conflant perpetuò varias numerorum formas: quapropter vi­ <lb/>dentur numeri impares e&longs;&longs;e finitatis cau&longs;a; &longs;icut pares exaduersò infinitatis <lb/>principium. </s> <s id="id.001380">quæ vt melius intelligas, declaranda e&longs;t 26. propo&longs;. </s> <s id="id.001381">7. Arith­ <lb/>metices lordani, vbi i&longs;tud idem demon&longs;trat, quæ e&longs;t hæc. </s> <s id="id.001382">&longs;it vnitas, & &longs;uo or­ <lb/>dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c. <lb/><figure id="id.009.01.074.4.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.074.4.jpg" place="text"/> <lb/>&longs;i igitur vnitati addatur ternarius in Gnomo­ <lb/>nis modum, vt vides in prima figura, produ­ <lb/>cetur quaternarius numerus, qui e&longs;t numerus <lb/>quadratus (quid &longs;it quadratus numerus expli­ <lb/>caui in Logicis tex. <!-- REMOVE S-->9. primi Po&longs;ter.) etfi huic <lb/>quaternario addatur &longs;equens impar, qui e&longs;t <lb/>quinarius in modum Gnomonis, vt in &longs;ecunda <lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s> <s id="id.001383">et&longs;i huic &longs;imiliter <lb/>addatur &longs;e quens impar, nimirum &longs;eptenarius, conflabitur &longs;edenarius, qui <lb/>numerus pariter quadratus e&longs;t, vt in tertia figura, & hoc modo, &longs;i in infini­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.075.jpg" pagenum="75"/>tum procedatur, numeri &longs;emper quadrati progignentur. </s> <s id="id.001384">Vides igitur, qui <lb/>ratione Gnomonum, &longs;iue imparium additione fiat &longs;emper eadem &longs;pecies, <lb/>&longs;cilicet quadratus numerus, quod &longs;ignum e&longs;t, inquiunt, imparem numerum <lb/>non infinitatis, &longs;ed finitatis e&longs;&longs;e auctorem. </s> <s id="id.001385">Po&longs;t prædictam 26. propo&longs;itio­ <lb/>nem Iotdani, &longs;unt aliquot propo&longs;itiones, quarum &longs;umma hæc e&longs;t: &longs;i pares <lb/>numeri ab vnitate coaceruentur; coaceruati erunt &longs;emper variæ formæ nu­ <lb/>merorum. </s> <s id="id.001386">quæ &longs;ic explicantur: &longs;int ab vnitate pares di&longs;po&longs;iti ordinatim <lb/>hoc modo, 1. 2. 4. 6. &c. </s> <s id="id.001387">&longs;i igitur vnitati binarius coaceruetur, fit numerus <lb/><figure id="id.009.01.075.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.075.1.jpg" place="text"/> <lb/>triangularis, vt in prima figura. </s> <s id="id.001388">&longs;i huic ternario <lb/>coaceruetur &longs;equens par, fiet altera &longs;pecies, ni­ <lb/>mirum hexagonus numerus, vt in &longs;ecunda figu­ <lb/>ra. </s> <s id="id.001389">cui &longs;i &longs;equens addatur par, &longs;cilicet &longs;enarius, <lb/>fiet iterum noua numeri forma, v. <!-- REMOVE S-->g. <!-- KEEP S--></s> <s id="id.001390">dodecago­ <lb/>nus, vt in tertia figura. </s> <s id="id.001391">& &longs;ic &longs;emper in infinitum nouæ ac variæ numerorum <lb/>formæ ex hac additione parium prouenient, quod argumento e&longs;t numerum <lb/>parem infiniti naturam &longs;apere. </s> <s id="id.001392">Porrò reperiri numeros triangulares, pen­ <lb/>tagonos, & &longs;imiles, con&longs;tat ex Arithmetica Nicomachi, Boetij, & Iordani, <lb/>citati in definitionibus 7. &longs;uæ Arithmeticæ, atque ex tractatu Diophantis <lb/>Alex. de numeris rectangulis. </s> <s id="id.001393"><expan abbr="atq;">atque</expan> ex his locus hic &longs;atis clarus redditur.</s></p><p type="main"> <s id="id.001394"><arrow.to.target n="marg94"/></s></p><p type="margin"> <s id="id.001395"><margin.target id="marg94"/>94</s></p><p type="main"> <s id="id.001396">Tex. 31. <emph type="italics"/>(Vtuntur etiam Mathematici infinito)<emph.end type="italics"/> <expan abbr="aliquãdo">aliquando</expan> Mathematici du­ <lb/>cunt lineas quantumuis longas, &longs;eu indefinitæ longitudinis, quas etiam in­ <lb/>finitas appellant: & hoc modo vtuntur infinito, vt infra tex. <!-- REMOVE S-->71. ip&longs;e Ari&longs;t. <lb/>exponit. </s> <s id="id.001397">alio præterea modo vtuntur infinito, vt quando &longs;upponunt data <lb/>quauis quantitate po&longs;&longs;e &longs;umi maiorem, vel etiam minorem in infinitum, vt <lb/>patet ex 6. po&longs;tulato primi Elem. editionis Clauianæ. </s> <s id="id.001398">numerum <expan abbr="quoq;">quoque</expan> au­ <lb/>geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem. vel demum <lb/>quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur <lb/>po&longs;&longs;e &longs;ub diuidi in <expan abbr="infinitũ">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinitũ">infinitum</expan> in v&longs;u e&longs;t.</s></p><p type="main"> <s id="id.001399"><arrow.to.target n="marg95"/></s></p><p type="margin"> <s id="id.001400"><margin.target id="marg95"/>95</s></p><p type="main"> <s id="id.001401">Tex. 68. & 69. plura de magnitudine, & numero continent; &longs;ed quæ non <lb/>indigeant opera no&longs;tra.</s></p><p type="main"> <s id="id.001402"><arrow.to.target n="marg96"/></s></p><p type="margin"> <s id="id.001403"><margin.target id="marg96"/>96</s></p><p type="main"> <s id="id.001404">Tex. 71. <emph type="italics"/>(Non remouet autem ratio Mathematicos à contemplatione auferens <lb/>&longs;ic e&longs;&longs;e infinitum, vt actu &longs;it ver&longs;us augmentum, vt intran&longs;ibile, <expan abbr="ncq;">ncque</expan> enim nunc in­ <lb/>digent infinito, <expan abbr="neq;">neque</expan> vtuntur, &longs;ed &longs;olum e&longs;&longs;e <expan abbr="quantumcunqu;">quantumcunque</expan> velint finitam)<emph.end type="italics"/> ratio <lb/>phy&longs;ica tollens infinitum actu, non e&longs;t Mathematicis impedimento, quia ip&longs;i <lb/>non vtuntur infinito actu; quam enim ip&longs;i ducunt lineam infinitam, non e&longs;t <lb/>verè infinita, &longs;ed indefinita, eam enim quantumlibet magnam producunt, vt <lb/>po&longs;&longs;it ad demon&longs;trandum &longs;ufficere.</s></p></chap><chap><p type="head"> <s id="id.001405"><emph type="italics"/>Ex Quarto Phy&longs;icorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001406"><arrow.to.target n="marg97"/></s></p><p type="margin"> <s id="id.001407"><margin.target id="marg97"/>97</s></p><p type="main"> <s id="id.001408">Tex. 120. ter in hoc textu meminit commen&longs;urabilitatis, & incommen­ <lb/>&longs;urabilitatis, quæ e&longs;t diametri ad co&longs;tam: cuius explicationem vide <lb/>primo Priorum, &longs;ecto primo, cap. 23.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.076.jpg" pagenum="76"/><chap><p type="head"> <s id="id.001409"><emph type="italics"/>Ex Quinto Phy&longs;icorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001410"><arrow.to.target n="marg98"/></s></p><p type="margin"> <s id="id.001411"><margin.target id="marg98"/>98</s></p><p type="main"> <s id="id.001412">Tex. 6. <emph type="italics"/>(Vt media grauis ad vltimam, & acuta ad primam)<emph.end type="italics"/> alludit ad or­ <lb/>dinem chordarum in mu&longs;icis in&longs;trumentis, vbi media chorda edit &longs;o­ <lb/>num, re&longs;pectu quidem vltimæ, & &longs;upremæ chordæ grauem: re&longs;pectu verò <lb/>primæ, & infimæ acutum.</s></p></chap><chap><p type="head"> <s id="id.001413"><emph type="italics"/>Ex Octauo Phy&longs;icorum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001414"><arrow.to.target n="marg99"/></s></p><p type="margin"> <s id="id.001415"><margin.target id="marg99"/>99</s></p><p type="main"> <s id="id.001416">Tex. 15. <emph type="italics"/>(Etenim triangulus habet tres angulos æquales duobus rectis angulis)<emph.end type="italics"/> <lb/>lib. 1. Priorum, &longs;ecto 3. cap. 1. huius rei explicationem reperies.</s></p></chap><chap><p type="head"> <s id="id.001417"><emph type="italics"/>EX PRIMO DE COELO.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001418"><arrow.to.target n="marg100"/></s></p><p type="margin"> <s id="id.001419"><margin.target id="marg100"/>100</s></p><p type="main"> <s id="id.001420">Tex. 33. <emph type="italics"/>(Vt &longs;i quis minimam quădam e&longs;&longs;e dicat magnitudinem, hic enim <lb/>minimum introducens, maxima <expan abbr="vbiq;">vbique</expan> amoueret mathematicorŭ)<emph.end type="italics"/> ide&longs;t, <lb/>&longs;i quis, vt Democritus po&longs;uerit in magnitudinibus e&longs;&longs;e minima, <lb/>&longs;eu indiui&longs;ibilia, ex quibus entia mathematica componerentur, <lb/>hic euerteret maxima mathematicorum, ide&longs;t maxime ip&longs;orum demon&longs;tra­ <lb/>tiones, atque etiam effata euerterentur: v. <!-- REMOVE S-->g. <!-- REMOVE S-->10. primi Elem. quæ docet <lb/>quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con­ <lb/>&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. </s> <s id="id.001421">pa­ <lb/>riter totus ferè decimus liber Elem. deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da­ <lb/>rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ e&longs;&longs;ent lineæ incom­ <lb/>men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen­ <lb/>&longs;urarentur. </s> <s id="id.001422">po&longs;tulatum <expan abbr="quoq;">quoque</expan> illud, qualibet data magnitudine &longs;umi po&longs;&longs;e <lb/>minorem pror&longs;us irritum redderetur, quia data atomo, illa minor accipi <lb/>non po&longs;&longs;et.</s></p><p type="main"> <s id="id.001423"><arrow.to.target n="marg101"/></s></p><p type="margin"> <s id="id.001424"><margin.target id="marg101"/>101</s></p><p type="main"> <s id="id.001425">Tex. 36. <emph type="italics"/>(Sit <expan abbr="itaq;">itaque</expan> linea, in qua A G E, infinita ad partes E; & alia vtrinque <lb/>infinita, in qua <foreign lang="greek">b</foreign> B; &longs;i <expan abbr="itaq;">itaque</expan> de&longs;cribat circulum linea A G E, cir ca centrum G, fe-<emph.end type="italics"/> <lb/><figure id="id.009.01.076.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.076.1.jpg" place="text"/> <lb/><emph type="italics"/>retur circulariter linea A G E, &longs;ecans ali­ <lb/>quando lineam <foreign lang="greek">b</foreign> B, tempore finito; totum <lb/>enim tempus, in quo circulariter latum <lb/>e&longs;t Cœlum finitum e&longs;t, & ablatum igitur, <lb/>quo &longs;ecans ferebatur; erit igitur aliqued <lb/>prmcipium, quo primum linea A G E, li­ <lb/>neam <foreign lang="greek">b</foreign> B, &longs;ecuit. </s> <s id="id.001426">&longs;ed impo&longs;&longs;ibile est; non <lb/>est igitur circulariter verti <expan abbr="infinitũ">infinitum</expan>, quare <lb/><expan abbr="neq;">neque</expan> mundum, &longs;i e&longs;&longs;et infinitus)<emph.end type="italics"/> quamuis <lb/>textus hic parum &longs;it mathematicus, <lb/>quia tamen &longs;upponit figuram mathe­ <lb/>maticam, quæ in codicibus pariter, ac <lb/>commentarijs de&longs;ideratur, illam pla­ <lb/>cuit apponere. </s> <s id="id.001427">in qua quidem, quamuis duæ lineæ infinitæ &longs;upponantur, vna <lb/>ad alteram <expan abbr="tãtum">tantum</expan> partem in qua E: altera verò ad <expan abbr="vtramq;">vtramque</expan> partem <foreign lang="greek">b,</foreign> & B, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.077.jpg" pagenum="77"/>non potuerunt tamen de&longs;cribi, ni&longs;i finitæ; appo&longs;itæ idcircò &longs;unt ad partes <lb/>illas, ad quas deberent e&longs;&longs;e infinitæ lineolæ quædam infinitatem indicantes. <lb/></s> <s id="id.001428">debemus po&longs;tea, vt mentem Ari&longs;t. percipiamus concipere lineam A G E, <lb/>moueri circulariter facto centro in G. quæ quia infinita &longs;upponitur ad par­ <lb/>tem E, &longs;ecabit nece&longs;&longs;ariò alteram <expan abbr="vtrinq;">vtrinque</expan> infinitam <foreign lang="greek">b</foreign> B, <expan abbr="illam&qacute;">illamque</expan>; nece&longs;&longs;ariò <lb/>finito tempore percurret, finito enim tempore tota mundi circulatio per­ <lb/>agitur, &longs;patio videlicet viginti quatuor horarum. </s> <s id="id.001429">ex quo Ari&longs;t. infert mun­ <lb/>dum non po&longs;&longs;e e&longs;&longs;e infinitæ magnitudinis; quia &longs;i mundus e&longs;&longs;et infinitus; &. <lb/></s> <s id="id.001430">duæ lineæ infinitæ, quales &longs;unt prædictæ in ip&longs;o, <expan abbr="atq;">atque</expan> cum ip&longs;o moueri alte­ <lb/>ra earum A E, intelligatur, alteram <foreign lang="greek">b</foreign> B, manentem in tempore finito, ide&longs;t, <lb/>in diurna conuer&longs;ione pertran&longs;ibit: fieri autem nequit, vt infinita magni­ <lb/>tudo finito tempore percurratur; quare dicendum e&longs;t, mundum e&longs;&longs;e finita <lb/>magnitudine præditum.</s></p><p type="main"> <s id="id.001431"><arrow.to.target n="marg102"/></s></p><p type="margin"> <s id="id.001432"><margin.target id="marg102"/>102</s></p><p type="main"> <s id="id.001433">Tex. 48. <emph type="italics"/>(Nihil autem refert grauitates, commen&longs;urabiles &longs;int, an incommen­ <lb/>&longs;ur abiles)<emph.end type="italics"/> quidnam &longs;it commen&longs;urabilitas, & incommen&longs;urabilitas, expli­ <lb/>catum e&longs;t lib. 1. Priorum, &longs;ecto 1. cap. 23.</s></p><p type="main"> <s id="id.001434"><arrow.to.target n="marg103"/></s></p><p type="margin"> <s id="id.001435"><margin.target id="marg103"/>103</s></p><p type="main"> <s id="id.001436">Tex. 119. <emph type="italics"/>(Est autem impo&longs;&longs;ibile, & po&longs;&longs;ibile; fal&longs;um, & verum, ex &longs;uppo&longs;itio­ <lb/>ne quidem, dico autem, vt triangulum impo&longs;&longs;ibile e&longs;t duos rectos habere, &longs;i hæc)<emph.end type="italics"/> <lb/>ide&longs;t, &longs;i &longs;upponantur fal&longs;a quædam, quæ &longs;upponi po&longs;&longs;unt, &longs;equetur impo&longs;&longs;i­ <lb/>bile e&longs;&longs;e triangulum habere tres angulos æquales duobus rectis angulis, vi­ <lb/>de, quæ &longs;crip&longs;i lib. 1. Priorum, &longs;ecto 3. cap. 1. de hoc, quod e&longs;t, habere tres <lb/>angulos æquales duobus rectis. </s> <s id="id.001437">v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i in triangulo pag. </s> <s id="id.001438">73. producto late­ <lb/>re A C, in D. &longs;i &longs;upponatur externus angulus B C D, non e&longs;&longs;e æqualis duobus <lb/>internis, & oppofitis A, & B, nunquam poterimus eo modo, quo Euclides, <lb/>demon&longs;trare pa&longs;&longs;ionem prædictam de triangulo A B C. huiu&longs;modi impo&longs;&longs;i­ <lb/>bile, cuius oppo&longs;itum non &longs;olum po&longs;&longs;ibile, &longs;ed etiam nece&longs;&longs;arium e&longs;t, vocat <lb/>Ari&longs;t. impo&longs;&longs;ibile ex &longs;uppo&longs;itione, quia &longs;cilicet impo&longs;&longs;ibile euadit ex quo­ <lb/>dam fal&longs;o &longs;uo &longs;uppo&longs;ito, vt in allato exemplo, triangulum habere tres an­ <lb/>gulos æquales duobus rectis, quamuis nece&longs;&longs;arium &longs;it, tamen ex fal&longs;a &longs;up­ <lb/>po&longs;itione, impo&longs;&longs;ibile oua&longs;it.</s></p><p type="main"> <s id="id.001439"><arrow.to.target n="marg104"/></s></p><p type="margin"> <s id="id.001440"><margin.target id="marg104"/>104</s></p><p type="main"> <s id="id.001441">Ibidem <emph type="italics"/>(Et diameter commen&longs;urabilis est co&longs;tæ, &longs;i bæc)<emph.end type="italics"/> vide primo Priorum, <lb/>&longs;ecto 3. cap. 23. hoc &longs;olum nunc addendum <emph type="italics"/>(Si hæc)<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i &longs;upponamus li­ <lb/>neas e&longs;&longs;e compo&longs;itas ex indiui&longs;ibilibus, con&longs;ectarium erit diametrum e&longs;&longs;e <lb/>commen&longs;urabilem co&longs;tæ, quia indiui&longs;ibile illud, ex quo vtraque linea con­ <lb/>&longs;tat, erit <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> men&longs;ura communis.</s> </p></chap><chap><p type="head"> <s id="id.001442"><emph type="italics"/>Ex Secundo de Cælo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.001443"><arrow.to.target n="marg105"/></s></p><p type="margin"> <s id="id.001444"><margin.target id="marg105"/>105</s></p><p type="main"> <s id="id.001445">Tex. 24. <emph type="italics"/>(Amplius qui &longs;olida diuidunt in plana, <expan abbr="atq;">atque</expan> ex planis corpora <lb/>generant, his te&longs;tes fui&longs;&longs;e videntur: &longs;olam enim figurarum &longs;olidarum <lb/>&longs;phæram non diuidunt, vt non plures &longs;uperficies. </s> <s id="id.001446">quam vnam <expan abbr="hab&etilde;um">habenum</expan>. <lb/></s> <s id="id.001447">diui&longs;io enim in plana non perinde e&longs;&longs;icitur, vt qui&longs;piam <expan abbr="diuid&etilde;s">diuidens</expan> in par­ <lb/>tes diuidat totum, &longs;ed vt in &longs;pecie diuer&longs;a: patct igitur &longs;phæram e&longs;&longs;e &longs;olidarum <lb/>primam)<emph.end type="italics"/> qui &longs;olida diuidunt in plana, ca diuidunt <expan abbr="&longs;ecũdum">&longs;ecundum</expan> numerum &longs;uper­ <lb/>&longs;icierum, quibus ambiuntur, v. <!-- REMOVE S-->g. <!-- REMOVE S-->diuidunt cubum in &longs;ex &longs;uperficies, quia <lb/>cubus &longs;ex quadratis planis &longs;uperficiebus continetur: qua ratione nequcunt <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.078.jpg" pagenum="78"/>&longs;ohæram in plana vlla re&longs;oluere, <expan abbr="neq;">neque</expan> in alias plures &longs;uperficies, quia &longs;phæ­ <lb/>ra ambitur vnica tantum &longs;uperficie &longs;phærica. </s> <s id="id.001448">quando verò ex planis corpo­ <lb/>ra generant, vt facit Plato in Timæo, accipíunt primò triangulum æquila­ <lb/>terum, & ex quatuor triangulis æquilateris &longs;imul compactis conficiunt py­ <lb/>ramidem; & hoc modo alia &longs;olida à pluribus &longs;uperficiebus ambita con&longs;ti­ <lb/>tuunt: verum hac ratione nullo modo po&longs;&longs;unt &longs;phæram componere, quia <lb/>vnica tantum, <expan abbr="ea&qacute;">eaque</expan>; &longs;phærica &longs;uperficie compræhenditur: <expan abbr="atq;">atque</expan> hoc pacto i&longs;ti <lb/>diuidentes, & componentes corpora fidem faciunt, &longs;phæram, cum ex nullis <lb/>componatur, &longs;olidorum e&longs;&longs;e primam.</s></p><p type="main"> <s id="id.001449"><arrow.to.target n="marg106"/></s></p><p type="margin"> <s id="id.001450"><margin.target id="marg106"/>106</s></p><p type="main"> <s id="id.001451">Tex. 25. <emph type="italics"/>(Est autem, & &longs;ecundum numerorum ordinem a&longs;&longs;ignantibus, &longs;ic po­ <lb/>nentibus rationabili&longs;&longs;imam, circulum quidem &longs;ecundum vnum; triangulum autem <lb/>&longs;ecundum dualitatem, quoniam duo recti. </s> <s id="id.001452">&longs;i autem &longs;ecundum triangulum, vnum. <lb/></s> <s id="id.001453">circulus non erit figura)<emph.end type="italics"/> In ordine figurarum conueniens e&longs;t, inquit, primam <lb/>facere circulum propter &longs;implici&longs;simam ip&longs;ius naturam, cum vnica, ac per­ <lb/>fecta circulari linea comprehendatur: <expan abbr="Triangulũ">Triangulum</expan> verò &longs;ecundam, quoniam <lb/>duo anguli recti, ide&longs;t, quia triangulum habet tres angulos æquales duobus <lb/>rectis angulis; quod fusè explicatnm e&longs;t lib. 1. Priorum, &longs;ecto 3. cap. 1. De­ <lb/>mum &longs;i primum locum dederimus triangulo, nullus alius remanet pro cir­ <lb/>culo, quod e&longs;t inconueniens, ergo circulus prima figura erit.</s></p><p type="main"> <s id="id.001454"><arrow.to.target n="marg107"/></s></p><p type="margin"> <s id="id.001455"><margin.target id="marg107"/>107</s></p><p type="main"> <s id="id.001456">Tex. 31. <emph type="italics"/>(At verò, quod aquæ &longs;uperficies talis &longs;it, manife&longs;tum e&longs;t hac &longs;uppo&longs;i­ <lb/>tione &longs;umpta, quod apta natura e&longs;t &longs;emper confluere aqua ad magis concauum: ma­ <lb/>gis autem concauum e&longs;t, quod centro propinquius est. </s> <s id="id.001457">ducantur ergo ex centro A,<emph.end type="italics"/> <lb/><figure id="id.009.01.078.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.078.1.jpg" place="text"/> <lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C, <lb/>ducta igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis, <lb/>quæ ex centro. </s> <s id="id.001458">magis igitur concauus locus e&longs;t, quare <lb/>influet aqua, donec <expan abbr="vtiq;">vtique</expan> æquetur. </s> <s id="id.001459">æqualis e&longs;t autem eis, <lb/>quæ ex centro linea A E, quare nece&longs;&longs;e e&longs;t apud eas, quæ <lb/>ex centro, e&longs;&longs;e aquam, tunc enim quie&longs;cet. </s> <s id="id.001460">linea autem, <lb/>quæ eas, quæ ex centro tangit, circularis e&longs;t, &longs;phærica <lb/>igitur aquæ &longs;uperficies e&longs;t, in qua B E C.)<emph.end type="italics"/> toto hoc <lb/>textu lineari demon&longs;tratione probat aquæ manen­ <lb/>tis &longs;uperficiem e&longs;&longs;e &longs;phæricam: quæ demon&longs;tratio <lb/>per&longs;picua euadit, &longs;i &longs;igura, quæ in codicibus tam <lb/>græcis, quam latinis, <expan abbr="atq;">atque</expan> etiam in commentarijs de&longs;ideratur, quemadmo­ <lb/>dum fecimus, re&longs;tituatur. </s> <s id="id.001461">&longs;it igitur in præcedenti figura A, centrum mundi, <lb/>ex quo educantur duæ rectæ lineæ æquales A B, A C, quæ deinde alia recta <lb/>B C, coniungantur. </s> <s id="id.001462">educatur <expan abbr="quoq;">quoque</expan> recta alia ex centro A, quæ pertingat <lb/>ad B C, quæ ba&longs;is e&longs;t trianguli B A C, & producatur vlterius quantumlibet <lb/>in E. intelligatur demum circumferentia tran&longs;ire per puncta B, & C, quia <lb/>illæ duæ lineæ A B, A C, &longs;unt æquales, quæ circumferentia alteram A D, quæ <lb/>fuit protracta, &longs;ecet in E. <!-- KEEP S--></s> <s id="id.001463">Iam &longs;ic argumentatur: aqua natura &longs;ua &longs;emper <lb/>de&longs;luit ad locum magis concauum, ide&longs;t, ad loca centro A, terræ propin­ <lb/>quiora, quale e&longs;&longs;et in figura locus D, re&longs;pectu locorum B, & C, quia A D, <lb/>linea minor e&longs;t ijs, quæ ex centro eductæ &longs;unt A B, A C. quapropter aqua <lb/>debet de&longs;luere ex B, ad D, vel ex C, ad idem D, donec pertingat ad E. qui <lb/>locus non e&longs;t decliuior punctis B, & C. quare cum loca B, E, C, quæ &longs;unt ex­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.079.jpg" pagenum="79"/>trema linearum, &longs;int æquè decliuia, nece&longs;&longs;e e&longs;t aquæ &longs;uperficiem apud ip&longs;a <lb/>con&longs;i&longs;tere, tunc enim debet quie&longs;cere, aliter nunquam quie&longs;ceret; &longs;ed vide­ <lb/>mus aquam manentem, & quietam, ergo quie&longs;cit circa puncta B, E, C, à <lb/>centro terræ æquidi&longs;tantia, per quæ tran&longs;it linea circularis coniungens illa; <lb/>et&longs;i &longs;uperficies per eiu&longs;modi loca pertran&longs;iret, e&longs;&longs;et &longs;phærica: &longs;ed &longs;uper&longs;i­ <lb/>cies aquæ tran&longs;it per talia loca, ergo &longs;phærica e&longs;t. </s> <s id="id.001464">Huius etiam habes acu­ <lb/>ti&longs;&longs;imam Archimedis demon&longs;trationem initio libelli de ijs, quæ vehuntur <lb/>in aqua, quam in &longs;uam &longs;phæram retulit Clauius.<!-- KEEP S--></s></p><p type="main"> <s id="id.001465"><arrow.to.target n="marg108"/></s></p><p type="margin"> <s id="id.001466"><margin.target id="marg108"/>108</s></p><p type="main"> <s id="id.001467">Tex. 46. <emph type="italics"/>(Reliquum e&longs;t orbes quidem moueri, stellas verò quie&longs;cere, & infixas <lb/>ip&longs;is orbibus ferri; &longs;olum enim &longs;ic nullum ab&longs;urdum accidit. </s> <s id="id.001468">celeriorem enim e&longs;&longs;e <lb/>maioris circuli velocitatem, rationabile e&longs;t circa idem centrum infixis: vt enim in <lb/>alijs maius corpus velocius fertur propria latione, &longs;ic, & in circularibus: maius <lb/>enim e&longs;t eorum, quæ auferuntur ab eis, quæ ex centro, maioris circuli &longs;egmentum)<emph.end type="italics"/> <lb/>ex intellectione vltimæ periodi textus totius intelligentia pendet: &longs;it igitur <lb/><figure id="id.009.01.079.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.079.1.jpg" place="text"/> <lb/>figura præ&longs;ens, in qua cum &longs;int duo circuli concen­ <lb/>trici, vnus altero maior, <expan abbr="eductæ&qacute;">eductæque</expan>; &longs;int ex <expan abbr="c&etilde;tro">centro</expan> duæ <lb/>&longs;emidiametri A D, A E, quæ <expan abbr="vtrunq;">vtrunque</expan> circulum &longs;e­ <lb/>cant, apparet maius e&longs;&longs;e <expan abbr="&longs;egmentũ">&longs;egmentum</expan> D E, quod è ma­ <lb/>iori circulo &longs;emidiametri ex <expan abbr="c&etilde;tro">centro</expan> eductæ auferunt, <lb/>quam &longs;egmentum B C, minoris circuli, quod ei&longs;dem <lb/>&longs;emidia metris intercipitur. </s> <s id="id.001469">Verumtamen &longs;i circuli <lb/>ambo &longs;imul moueantur, maior circulus æquali tem­ <lb/>pore maius illud &longs;patium D E, & minor minus B C, <lb/>pertran&longs;ibit: idem igitur de cœle&longs;tibus orbibus di­ <lb/>cendum, qui quamuis omnes diurnum &longs;imul motum <lb/>ab&longs;oluunt, maiores tamen celerius conuertuntur: quo fit, vt &longs;tellæ maiori­ <lb/>bus circulis infixæ, <expan abbr="atq;">atque</expan> delatæ, maiori celeritate &longs;uos cur&longs;us peragant, ne­ <lb/>que oportet eas, dum mouentur cœlum di&longs;&longs;ecare, quod accideret, &longs;i pro­ <lb/>prio motu veluti pri&longs;ces per aquam progrederentur.</s></p><p type="main"> <s id="id.001470">Hæc quidem Ari&longs;t. con&longs;entanea ob&longs;eruationibus veterum A&longs;tronomo­ <lb/>rum; at verò illis no&longs;træ ætatis ob&longs;eruationes repugnant; præ&longs;ertim illæ, <lb/>quæ fiunt circa &longs;tellas errantes: ex quibus fatendum e&longs;&longs;e videtur, Cœlum, <lb/>qua parte Planetas continet, liquidum e&longs;&longs;e, ac per illud Planetas proprio <lb/>motu, ceu pi&longs;ces in aqua progredi. </s> <s id="id.001471">Tycho <expan abbr="namq;">namque</expan> Brahe, <expan abbr="alij&qacute;">alijque</expan>; plures exactè <lb/>demon&longs;trant Cometas in regione Planetarum e&longs;&longs;e, <expan abbr="eos&qacute;">eosque</expan>; motu quodam in <lb/>tran&longs;uer&longs;um moueri, quo nece&longs;&longs;ario C&ecedil;lú deberent perforare; ijdem o&longs;ten­ <lb/>dunt nonnullos Planetas, Martem præ&longs;ertim, ac Venerem modo &longs;upra So­ <lb/>lem, modo infra a&longs;cendere, & de&longs;cendere. </s> <s id="id.001472">Idem patet ex ob&longs;eruatione no­ <lb/>ua per nouum Tele&longs;copij i <expan abbr="&longs;trum&etilde;tum">&longs;trumentum</expan> in Venere facta, quæ lunulata <expan abbr="vtrinq;">vtrinque</expan> <lb/>à Sole apparet: quando nimirum e&longs;t in imo epicyclo. </s> <s id="id.001473"><expan abbr="iterum&qacute;">iterumque</expan>; rotunda ve­ <lb/>luti Luna plena, cum in &longs;ummo epicyclo ver&longs;atur: quæ minimè apparerent, <lb/>ni&longs;i &longs;upra, ac infra Solem circumiret. </s> <s id="id.001474">His rationibus conantur ip&longs;i proba­ <lb/>re Cœlum e&longs;&longs;e liquidum; <expan abbr="atq;">atque</expan> in eo Planetas, veluti aues in aere, permearc: <lb/>quarum &longs;olutio mihi nulla occurrit, alijs forta&longs;&longs;is occurrct.</s></p><p type="main"> <s id="id.001475"><arrow.to.target n="marg109"/></s></p><p type="margin"> <s id="id.001476"><margin.target id="marg109"/>109</s></p><p type="main"> <s id="id.001477">Tex. 57. <emph type="italics"/>(De ordine autem ip&longs;orum, quo quidem modo &longs;ingula di&longs;ponantur, vt <lb/>quædam &longs;int priora, quædam posteriora, & quomodo &longs;patijs &longs;e ă<expan abbr="habeãt">habeant</expan> ad inuicem,<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.080.jpg" pagenum="80"/><emph type="italics"/>ex ijs circa A&longs;trologiam, con&longs;ideretur: dicitur enim &longs;ufficienter)<emph.end type="italics"/> &longs;umit hoc loco <lb/>A&longs;trologiam, pro A&longs;tronomia, &longs;i iuxta recentiores loqui velimus. </s> <s id="id.001478">Dicit igi­ <lb/>tur ordinem cœlorum, ac &longs;yderum, item &longs;itum, & proportiones magnitu­ <lb/>dinum corundem, cum per naturalis &longs;cientiæ princip ia &longs;ciri nequeant, ex <lb/>rationibus A&longs;tronomorum petenda e&longs;&longs;e, apud quos i&longs;ta &longs;ufficienter <expan abbr="demon-&longs;tr&etilde;tur">demon­ <lb/>&longs;trentur</expan>. </s> <s id="id.001479">& meritò quidem hæc dicuntur; po&longs;teriores enim ab Ari&longs;t. ordines, <lb/>&longs;itus, ac magnitudines tam cœlorum, quam &longs;yderum firmis rationibus, <expan abbr="atq;">atque</expan> <lb/>inuentu peracutis demon&longs;trarunt. </s> <s id="id.001480">quorum princeps fuit ptolæmeus; no&longs;tra <lb/>tamen ætate Tycho Brahe, qui certis ob&longs;eruationibus, quas maximo labo­ <lb/>re, ac &longs;umptu exantlauit, in nonnullis à Ptolæmeo, ac reliquis di&longs;&longs;entjt: &longs;tan­ <lb/>dum autem e&longs;&longs;e recentioribus ob&longs;eruationibus apud A&longs;tronomiæ peritos in <lb/>confe&longs;&longs;o e&longs;t.</s></p><p type="main"> <s id="id.001481"><arrow.to.target n="marg110"/></s></p><p type="margin"> <s id="id.001482"><margin.target id="marg110"/>110</s></p><p type="main"> <s id="id.001483">Tex. <emph type="italics"/>(Luna autem o&longs;tenditur per ea, quæ circa vi&longs;um, quod &longs;phærica &longs;it: non <lb/>enim <expan abbr="vtiq;">vtique</expan> fieret accre&longs;cens, & decre&longs;cens, plurimŭ quidem alter a ex parte curua, <lb/>altera concaua, aut <expan abbr="vtrmq;">vtrmque</expan> curua, &longs;emel autem bipartita)<emph.end type="italics"/> ait per ea, quæ circa <lb/>vi&longs;um, ide&longs;t per opticem probari Lunam e&longs;&longs;e &longs;phæricam: &longs;ed con&longs;ule, quæ <lb/>primo Po&longs;ter. tex. <!-- REMOVE S-->3. de hac re &longs;crip&longs;i, & plenam etiam huius loci intelligen­ <lb/>tiam a&longs;&longs;equeris, præ&longs;ertim &longs;i experimentum ibi traditum inieris.</s> </p><p type="main"> <s id="id.001484"><arrow.to.target n="marg111"/></s></p><p type="margin"> <s id="id.001485"><margin.target id="marg111"/>111</s></p><p type="main"> <s id="id.001486">Ibidem <emph type="italics"/>(Et rur&longs;us per Astrologica, quia <expan abbr="vtiq;">vtique</expan> non e&longs;&longs;ent &longs;olis eclyp&longs;es lunulæ <lb/>&longs;peciem præ&longs;eferentes. </s> <s id="id.001487">Quare &longs;i vnum est tale, palam e&longs;t, quod & alia <expan abbr="vtiq;">vtique</expan> erunt <lb/>talia)<emph.end type="italics"/> &longs;icuti <expan abbr="præced&etilde;s">præcedens</expan> &longs;phæricitatis Lunæ ratio ex Per&longs;pectiua de&longs;umpta e&longs;t, <lb/>ita præ&longs;ens ex A&longs;tronomia, ex eò enim, quod eclyp&longs;is Solis habeat figuram <lb/>lunulæ, ide&longs;t, &longs;i in&longs;tar Lunæ falcatæ, probant A&longs;tronomi Lunam e&longs;&longs;e &longs;phæri­ <lb/>cam. </s> <s id="id.001488">intellige tamen partem illam Solis, quæ non eclyp&longs;atur, habere figu­ <lb/>ram lunulæ, pars enim à Luna obumbrata non videtur, et&longs;i videretur oua­ <lb/>lem quandam &longs;peciem, præ&longs;eferret: pars igitur, illa e&longs;t corniculata, quia <lb/><figure id="id.009.01.080.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.080.1.jpg" place="text"/> <lb/>cum Solis defectio ex interpo&longs;itione Lunæ inter nos, & <lb/>Solem contingat, & Luna &longs;it &longs;phærica, nece&longs;&longs;ariò &longs;phæ­ <lb/>ricè, & circulariter Solem obumbrabit; quare pars illa <lb/>non obumbrata remanet falcata, & corniculata, vt in <lb/>præ&longs;enti figura vidcre e&longs;t; vbi cernis, Lunam Solem or­ <lb/>biculariter offu&longs;care in linea A D C, partem Solis de­ <lb/>tectam <expan abbr="contentã">contentam</expan> lineis curuis A B C D, e&longs;&longs;e lunularem, <lb/>& falcatam; cum ergo in hunc modum fiat Solis deli­ <lb/>quium, &longs;ignum certum e&longs;t, Lunam e&longs;&longs;e &longs;phæricam.</s></p><p type="main"> <s id="id.001489"><arrow.to.target n="marg112"/></s></p><p type="margin"> <s id="id.001490"><margin.target id="marg112"/>112</s></p><p type="main"> <s id="id.001491">Tex. 107. <emph type="italics"/>(Quod autem dubitatur, hoc e&longs;t; videre autem non e&longs;t difficile, &longs;i pa­ <lb/>rum con&longs;iderauerimus, & di&longs;tinxerimus, quonam modo cen&longs;eamus quantamuis ma­ <lb/>gnitudinem grauem ad medium ferri. </s> <s id="id.001492">manife&longs;tum enim e&longs;t, quod non quou&longs;que ex­ <lb/>tremum tangat ip&longs;um centrum; &longs;ed maior pars vincat, oportet, <expan abbr="quou&longs;q;">quou&longs;que</expan> &longs;uo medio <lb/>ip&longs;um medium compræhendat; <expan abbr="hucn&longs;q;">hucn&longs;que</expan> enim habet propen&longs;ioncm)<emph.end type="italics"/> &longs;en&longs;us Ari&longs;to­ <lb/>telis e&longs;t, debere nos exi&longs;timare, quod &longs;i quæpiam grauis magnitudo de&longs;cen­ <lb/>dat ad centrum mundi, eam non perman&longs;uram, &longs;latim ac ip&longs;ius extremum <lb/>centrum mundi attigent; &longs;ed cò <expan abbr="v&longs;q;">v&longs;que</expan> de&longs;cen&longs;uram, <expan abbr="quou&longs;q;">quou&longs;que</expan> ip&longs;ius medium, <lb/>mundi medium, &longs;iue centrum a&longs;&longs;equutum &longs;it; maior enim ip&longs;ius pars, in qua <lb/>&longs;cilicet medium e&longs;t, minorem partem propellit, donec vtrinque à centro <lb/>mundi æquè emineat; omne enim graue <expan abbr="hucu&longs;q;">hucu&longs;que</expan> habet propen&longs;ionem, &longs;iue <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.081.jpg" pagenum="81"/><expan abbr="hucu&longs;q;">hucu&longs;que</expan> grauitat, v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;i lapis illuc de&longs;cenderet, non quie&longs;ceret &longs;tatim ac <lb/>prima ip&longs;ius pars ad mundi centrum pertingeret, &longs;ed reliquæ ip&longs;ius partes <lb/>adhuc grauitarent, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; vlterius primam partem impellerent, donec lapi­ <lb/>áis medium, mundi medio congrueret: quo facto lapis quie&longs;ceret. </s> <s id="id.001493">quæ num <lb/>vera &longs;int, vt intelligamus, oportet prius præmittere, iuxta Mathematicos <lb/>duplex e&longs;&longs;e medium, &longs;iue centrum cuiu&longs;uis magnitudinis: aliud enim e&longs;t <lb/>centrum molis, aliud e&longs;t centrum grauitatis. </s> <s id="id.001494">centrum molis e&longs;t illud pun­ <lb/>ctum, à quo extrema æquidi&longs;tant: centrum grauitatis e&longs;t punctum illud, à <lb/>quo extrema æque ponderant, &longs;iue à quo graue &longs;u&longs;pen&longs;um æquè ponderat, <lb/>&longs;iue in æquilibrio manet. </s> <s id="id.001495">Porrò in corporibus regularibus, &longs;i vnifo mia &longs;int <lb/>idem, & vnum &longs;unt centrum molis, ac centrum grauitatis: vt in &longs;phæra <lb/>plumbea, idem crit <expan abbr="vtrumq;">vtrumque</expan> centrum: &longs;i verò difformia &longs;int in grauitate, <lb/>vt in &longs;phæra partim plumbea, partim lignea, diuer&longs;um erit centrum molis, <lb/>à centro grauitatis; illud enim erit in medio &longs;phæræ; centrum verò graui­ <lb/>tatis in parte plumbea exi&longs;tet. </s> <s id="id.001496">In corporibus deinde irregularibus, etiam&longs;i <lb/>&longs;int vniformis ponderis, aliud tamen e&longs;&longs;e pote&longs;t centrum molis à <expan abbr="c&etilde;tro">centro</expan> gra­ <lb/>uitatis, vt in corpore oblongo, cuius alterum extremum &longs;it reliquis parti­ <lb/>bus multò maius, vti e&longs;t claua: vbi centrum molis erit in medio longitudi­ <lb/>nis clauæ; centrum verò grauitatis, erit propinquius capiti clauæ. </s> <s id="id.001497">quando <lb/>igitur Ari&longs;t. ait, graue de&longs;cen&longs;urum, donec ip&longs;ius medium, &longs;iue centrum, <lb/>mundi centrum attingat; benè dicit, &longs;i de medio grauitatis intelligat; ma­ <lb/>lè autem &longs;i de medio molis. </s> <s id="id.001498">quia grauia omnia ratione centri grauitatis <lb/>ponderant, <expan abbr="neq;">neque</expan> manent; ni&longs;i ip&longs;um maneat: quare ni&longs;i ip&longs;um <expan abbr="attingãt">attingant</expan> cen­ <lb/>trum mundi &longs;emper grauitabunt, & mouebuntur. </s> <s id="id.001499">Verum enim verò ex an­ <lb/>tiquorum monumentis manife&longs;tum e&longs;t, Archimedem, qui multò po&longs;t Ari­ <lb/>&longs;totelem floruit, primum omnium de centro grauitatis e&longs;&longs;e philo&longs;ophatum, <lb/>qua ratione dicendum e&longs;&longs;et, Ari&longs;totelem de centro, molis loquutum e&longs;&longs;e, <lb/>& perinde non <expan abbr="v&longs;quequaq;">v&longs;quequaque</expan> verè.</s></p><p type="main"> <s id="id.001500"><arrow.to.target n="marg113"/></s></p><p type="margin"> <s id="id.001501"><margin.target id="marg113"/>113</s></p><p type="main"> <s id="id.001502">Tex. 109. <emph type="italics"/>(Præterea <expan abbr="quoq;">quoque</expan> & per ta, quæ apparent &longs;ecundum &longs;en&longs;um, neque <lb/>enim Lunæ eclyp&longs;es tales <expan abbr="haber&etilde;t">haberent</expan> deci&longs;iones; nunc enim in ijs, quæ &longs;ecundum men­ <lb/>&longs;em fiunt, figurationibus, omnes accipit diui&longs;iones: etenim recta fit, & vtrinque <lb/>curua, & concaua)<emph.end type="italics"/> probat terram e&longs;&longs;e &longs;phæricam ratione a&longs;tronomica, ex <lb/>Lunæ eclyp&longs;ibus de&longs;umpta: nam ni&longs;i terra e&longs;&longs;et rotunda, nunquam Luna in <lb/>eclyp&longs;i haberet tales deci&longs;iones, ide&longs;t non haberet falcatas, aut lunulatas <lb/>partes illas, quæ in eclyp&longs;i ob&longs;curantur, & quafi à Luna re&longs;ecantur. </s> <s id="id.001503">quam­ <lb/>uis enim &longs;ingulis men&longs;ibus Luna terminetur modo linea concaua, vt quan­ <lb/>do noua e&longs;t; modo recta, vt quando diuidua e&longs;t: modo vtrinque curua, vt <lb/>cum à diuiduæad plenilunium tendit. </s> <s id="id.001504">quod fu&longs;ius primo Po&longs;ter. tex. <!-- REMOVE S-->30. ex­ <lb/>po&longs;ui. </s> <s id="id.001505">in eclyp&longs;ibus tamen &longs;emper curuam habet lineam illam, quæ partem <lb/>ec'yp&longs;atam de&longs;init; vt paulo po&longs;t explicabo. </s> <s id="id.001506">Vide precedentem textum 59. <lb/>& ca, quæ ibi annotaui, <expan abbr="quæq;">quæque</expan> tibi propo&longs;ui, & plenam huius loci intelligen­ <lb/>tiam a&longs;&longs;equeris. </s> <s id="id.001507">vide etiam, quæ mox &longs;ubdam circa huius loci reliquum.</s></p><p type="main"> <s id="id.001508"><arrow.to.target n="marg114"/></s></p><p type="margin"> <s id="id.001509"><margin.target id="marg114"/>114</s></p><p type="main"> <s id="id.001510">Ibidem <emph type="italics"/>(Circa autem eclyp&longs;es, &longs;emper curuam habet terminătem lmeam: qua­ <lb/>re qaon'am eclyp&longs;im palitur propter terræ obiectionem, terræ <expan abbr="circumfer&etilde;tia">circumferentia</expan> &longs;phæ­ <lb/>rica exi&longs;tens, figuræ cau&longs;a erit)<emph.end type="italics"/> probat rotunditatem terræ ab eclyp&longs;i lunari, <lb/>ex eo, quod Luna &longs;phæricè eclyp&longs;etur, quod innuitur illis verbis, &longs;emper <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.082.jpg" pagenum="82"/>curuam hzbet terminantem lineam, linea &longs;cilicet, quæ terminat partem <lb/>eclyp&longs;atam à non eclyp&longs;ata, &longs;emper apparet circularis; cum autem hæc li­ <lb/>nea &longs;it terminus vmbræ terræ, quæ lumen obumbrat, &longs;ignum <expan abbr="manife&longs;tũ">manife&longs;tum</expan> e&longs;t <lb/>vmbram ip&longs;am e&longs;&longs;e rotundam; nam cum Luna deficiat propter terræ obie­ <lb/>ctionem inter ip&longs;am, & Solem, ita, vt vmbra terræ protendatur <expan abbr="v&longs;q;">v&longs;que</expan> ad Lu­ <lb/>nam, <expan abbr="eam&qacute;">eamque</expan>; in omni eclyp&longs;atione, &longs;iue eclyp&longs;is &longs;it &longs;upra terram, &longs;iue infra, <lb/>ad quamlibet <expan abbr="deniq;">denique</expan> partem terræ fiat, orbiculariter eam contegit, &longs;ignum <lb/>per&longs;picuum e&longs;t terram proijcere quoquouer&longs;us vmbram rotundam, quæ vt <lb/>in &longs;phæra o&longs;tenditur, e&longs;t rotunda ad modum coni; cum ergo vmbra terræ <lb/>ex quauis parte proijciatur, &longs;it rotunda, certò certius colligitur, <expan abbr="terram&qacute;">terramque</expan>; <lb/><expan abbr="quoq;">quoque</expan> ip&longs;am rotunda &longs;igura præditam e&longs;&longs;e. </s> <s id="id.001511">hanc eandem rationem, &longs;i libue­ <lb/>rit, fu&longs;ius pertractatam videre poteris apud P. <!-- REMOVE S-->Clauium in &longs;phæra.</s> </p><p type="main"> <s id="id.001512"><arrow.to.target n="marg115"/></s></p><p type="margin"> <s id="id.001513"><margin.target id="marg115"/>115</s></p><p type="main"> <s id="id.001514">Tex. <emph type="italics"/>(Præterea per astrorum apparentiam, non &longs;olum manife&longs;ium e&longs;t, quod re­ <lb/>tunda, &longs;ed & quod magnitudine non magna &longs;it; paruo enim facto ncbis tran&longs;itu ad <lb/>meridiem, & Vr&longs;am, manifa&longs;tè fit alter horizon circulus, ita vt a&longs;tra, quæ &longs;uper <lb/>caput, magnam habcant mutationem, & non eadem appareant, & ad Vr&longs;am, & ad <lb/>meridiem tran&longs;euntibus, quædam enim in Acgypto quidem stellæ <expan abbr="vid&etilde;tur">videntur</expan>, & cir­ <lb/>ca Cyprum, in ijs autem, quæ ad Vr&longs;am vergunt regionibus, non <expan abbr="via&etilde;tur">viaentur</expan>. </s> <s id="id.001515">& a&longs;tro­ <lb/>rum ea, quæ &longs;emper in ijs, quæ ad Vr&longs;am vergunt, apparent, in illis locis occidunt. <lb/></s> <s id="id.001516">Quare non &longs;olum ex his manife&longs;tum e&longs;t rotundam e&longs;&longs;e figuram terræ, &longs;ed & &longs;phæræ <lb/>non magnæ: non enim tam celeriter in&longs;igne quippiam faceret, tran&longs;latis nobis adeò <lb/>parum)<emph.end type="italics"/> hic textus ei, qui &longs;phæram mundi audiuerit perfacilis e&longs;t: propte­ <lb/>rea eum breuiter &longs;ic paraphra&longs;ticè exponam. </s> <s id="id.001517">Terram e&longs;&longs;e rotundam, <expan abbr="atq;">atque</expan> <lb/>re&longs;pectu cœle&longs;tium corporum non magnam, &longs;ignum e&longs;t, quod facto à nobis <lb/>paruo itinere &longs;iue ad meridionalem plagam, &longs;iue ad <expan abbr="&longs;ept&etilde;trionalem">&longs;eptentrionalem</expan> (quam <lb/>Vr&longs;am dicit) magnopere mutatur horizon: quod apparet primo ex varia­ <lb/>tionc a&longs;trorum, nam quæ in primo loco &longs;upra no&longs;trum verticem <expan abbr="trã&longs;ibant">tran&longs;ibant</expan>, <lb/>in &longs;ecundo loco non amplius, &longs;ed alia, <expan abbr="atq;">atque</expan> alia valde ab inuicem &longs;eiuncta <lb/><figure id="id.009.01.082.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.082.1.jpg" place="text"/> <lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s> <s id="id.001518">&longs;it in <lb/>præ&longs;enti figura terra, vbi A, in qua facta parua <lb/>mutatione ex loco F, in locum G, fieret magna <lb/>mutatio <expan abbr="a&longs;trorũ">a&longs;trorum</expan> ver&longs;icalium B, in C, quæ mul­ <lb/>tum ab inuicem di&longs;tant. </s> <s id="id.001519">&longs;i autem terra e&longs;&longs;et <lb/>maior, v. <!-- REMOVE S-->g. <!-- REMOVE S-->circulus medius, tunc facta maio­ <lb/>ri mutatione ex D, in E, fieret eadem a&longs;trorum <lb/>variatio ex B, in C; &longs;ed cum nos experiamur <lb/>&longs;ieri magnam a&longs;trorum mutationem, ex parua <lb/>locorum intercapedine, &longs;ignum e&longs;t magnope­ <lb/>re mutari horizontem, ac proinde terram e&longs;&longs;e <lb/>rotundam, ac re&longs;pectu cœle&longs;tium corporum <lb/>paruam. </s> <s id="id.001520">aliud præterea &longs;ignum hums horizontis permutationis e&longs;t, quod <lb/>&longs;tellæ, quæ in priori loco &longs;upra horizontem apparebant, mutato paululum <lb/>loco ad alterutram plagam, &longs;tatim ab&longs;conduntur; aliæ verò nouæ <expan abbr="appar&etilde;t">apparent</expan> <lb/>vt in Acgypto, & Cypro, &longs;tella, quæ dicitur Canobus &longs;upra horizontem <lb/>a&longs;cendit; quæ &longs;i paululum Vr&longs;am, &longs;eu &longs;eptentrionem ambulaueris, &longs;tatim <lb/>latitabit. </s> <s id="id.001521">Demum ciu&longs;dem citæ mutationis &longs;initoris indicium etiam &longs;it, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.083.jpg" pagenum="83"/>quod regiones &longs;eptentrionales incolentibus plurima &longs;unt a&longs;tra, quæ nun­ <lb/>quam occidunt, quamuis horizontem leuiter per&longs;iringant, quæ tamen Cy­ <lb/>prijs, <expan abbr="atq;">atque</expan> Aegyptijs oriuntur, <expan abbr="atq;">atque</expan> occidunt. </s> <s id="id.001522">ex quibus & rotunditas, & <lb/>paruitas terræ colligi pote&longs;t. </s> <s id="id.001523">has ea&longs;dem rationes fu&longs;ius explicatas repe­ <lb/>ries apud P. <!-- REMOVE S-->Clauium in &longs;phæra.</s> </p><p type="main"> <s id="id.001524"><arrow.to.target n="marg116"/></s></p><p type="margin"> <s id="id.001525"><margin.target id="marg116"/>116</s></p><p type="main"> <s id="id.001526">Tex. 111. <emph type="italics"/>(Quapropter existimantes eum, qui circa Herculcas columnas e&longs;t lo­ <lb/>cum coniungi ei, qui circa Indiam, & boc modo mare vnum e&longs;&longs;e, nen admcdum <lb/>incredibilia exi&longs;timare videntur &c.)<emph.end type="italics"/> exi&longs;timatores ho&longs;ce non perperam exi­ <lb/>&longs;tima&longs;&longs;e apertè <expan abbr="cõuincunt">conuincunt</expan> Chri&longs;tophori Columbi, Argonautarum principis <lb/>nauigationes; quibus nouus orbis repertus e&longs;t, qui inter columnas Hercu­ <lb/>lis, <expan abbr="atq;">atque</expan> orientalem Indiam totus vna <expan abbr="cũ">cum</expan> mari Oceano Atlantico interiacet.</s></p><p type="main"> <s id="id.001527"><arrow.to.target n="marg117"/></s></p><p type="margin"> <s id="id.001528"><margin.target id="marg117"/>117</s></p><p type="main"> <s id="id.001529">Tex. 112. <emph type="italics"/>(Matbematicorum etiam, qui circum ferentiæ magnitudinemratio­ <lb/>cinari tentant, ad 400. dicunt ftadiorum millia, &c.)<emph.end type="italics"/> quam &longs;ubtilibus rationi­ <lb/>bus inue&longs;tigauerint A&longs;tronomi quantitatem terræ, optimè, ac dilucidè ex­ <lb/>ponitur à P. <!-- REMOVE S-->Clauio in &longs;phæra: quem &longs;i libet, con&longs;ule, ne inani labore opu­ <lb/>&longs;culum i&longs;tud exere&longs;cat.</s> </p></chap><chap><p type="head"> <s id="id.001530"><emph type="italics"/>Ex Tertio de Cœlo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.001531"><arrow.to.target n="marg118"/></s></p><p type="margin"> <s id="id.001532"><margin.target id="marg118"/>118</s></p><p type="main"> <s id="id.001533">Tex. 40. <emph type="italics"/>(Figuræ autem omnes componuntur ex pyramidibus: rectilinea <lb/>quidem ex rectilmeis: fphæra verò ex octo partibus componitur)<emph.end type="italics"/> Ale­ <lb/>xander exiftimat, Ari&longs;totelem dicere &longs;phæram con&longs;tare ex octo <lb/>partibus illis, quæ de&longs;ignantur per tres circulos, quorum duo &longs;e­ <lb/>cant &longs;e mucuò ad angulos rectos, vt in &longs;phæra mundi faciunt duo coluri; <lb/>tertius verò medios illos diuidit æquidi&longs;tanter à &longs;ectionibus <expan abbr="illorũ">illorum</expan> mutuis, <lb/>quemadmodum æquator in &longs;phæra mundi &longs;ecat duos coluros. </s> <s id="id.001534">ex quibus &longs;e­ <lb/>ctionibus tota &longs;phæra in octo partes diuiditur, quibus &longs;phæram componi <lb/>vult Ari&longs;toteles. <!-- KEEP S--></s> <s id="id.001535">aduerte tamen hanc &longs;phæræ compo&longs;itionem nullo modo <lb/>habere partes actu, cum &longs;phæra &longs;it vnica &longs;implici &longs;uperficie terminata; &longs;ed <lb/>quæ tantum &longs;int à prædictis imaginatis circulis de&longs;ignatæ: at verò aliæ fi­ <lb/>guræ, quæ pluribus planis terminantur, vt cubus, octaedrum, & &longs;imilia, quæ <lb/>Ari&longs;t. vocat rectiliheas, quia terminantur &longs;uperficiebus rectilineis actu di­ <lb/>&longs;tinctis ab inuicem ex natura &longs;ua, non per no&longs;tram de&longs;ignationem, ideò re­ <lb/>ctè dicuntur componi ex pyramidibus, v. <!-- REMOVE S-->g. <!-- REMOVE S-->dicimus cubum componi ex &longs;ex <lb/>pyramidibus, quia cum habeat &longs;ex ba&longs;es, cogitamus &longs;upra <expan abbr="vnamquamq;">vnamquamque</expan> il­ <lb/>larum &longs;ingulas pyramides erigi, quarum omnium vertices ad idem punctum <lb/>medium intra cubum imaginatum coeant. </s> <s id="id.001536">& &longs;ic de reliquis &longs;olidis. </s> <s id="id.001537">quæ qua <lb/>ratione re&longs;oluantur in plures pyramides, con&longs;tat ex 10. 11. 12. & 13. Ele­ <lb/>mentorum Euclidis, at verò in &longs;phæra nullum reale compo&longs;itionis, aut di­ <lb/>ui&longs;ionis fundamentum reperitur.</s></p><p type="main"> <s id="id.001538"><arrow.to.target n="marg119"/></s></p><p type="margin"> <s id="id.001539"><margin.target id="marg119"/>119</s></p><p type="main"> <s id="id.001540">Tex. <emph type="italics"/>(Ad hæc nece&longs;&longs;e e&longs;t non omne corpus e&longs;&longs;e diui&longs;ibile dicere, &longs;ed repugnare <lb/>certi&longs;&longs;i nis &longs;cientijs; nam Mathematicæ ip&longs;um quideæ intelligibile, accipiunt diui­ <lb/>&longs;ibile)<emph.end type="italics"/> ip&longs;um intelligibile, ide&longs;t, quantitatem ab&longs;tractam tam continuam, <lb/>quam di&longs;eretam, quam &longs;tatuunt Philo&longs;ophi e&longs;&longs;e &longs;ubiectam materiam ma­ <lb/>thematicarum. </s> <s id="id.001541">quam ideo appellant intelligibilem, quia cum &longs;it ab&longs;tracta <lb/>per intellectum à &longs;en&longs;ibilibus affectionibus, re&longs;tat vt &longs;it tantummodo intel­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.084.jpg" pagenum="84"/>lectu perceptibilis. </s> <s id="id.001542">Hanc eandem &longs;upponunt e&longs;&longs;e diui&longs;ibilem in infinitum, <lb/>vt &longs;upra 3. Phy&longs;. textu 31. dictum e&longs;t.</s></p><p type="main"> <s id="id.001543"><arrow.to.target n="marg120"/></s></p><p type="margin"> <s id="id.001544"><margin.target id="marg120"/>120</s></p><p type="main"> <s id="id.001545">Tex. 66. <emph type="italics"/>(Ommnò autem eniti &longs;implicibus corporibus figur as tribuere irratio­ <lb/>nabile e&longs;t. </s> <s id="id.001546">primò quidem, quia accidit non repleri totum; nam in planis tres figuræ <lb/>videntur implere locum, Triangulus, Quadratum, & Sexangulus)<emph.end type="italics"/> per &longs;implicia <lb/>corpora intelligit quatuor elementa. </s> <s id="id.001547">Vult enim probare quatuor elemen­ <lb/>ta non habere figuras illas mathematicas, quas illis Plato tribuebat, vt au­ <lb/>tem Ari&longs;t. rationem probè percipiamus, &longs;ciendum, quod implere totum, <lb/>&longs;iue locum, illæ figuræ dicuntur, quæ &longs;imul &longs;uis angulis in plano quopiam ad <lb/>vnum, <expan abbr="atq;">atque</expan> idem punctum vnitæ locum illum totum, qui cirea punctum il­ <lb/>lud con&longs;i&longs;tit, <expan abbr="cõtegunt">contegunt</expan>, ita vt nihil vacui inter ip&longs;as relinquatur. </s> <s id="id.001548">tales &longs;unt, <lb/>quibus fieri po&longs;&longs;unt pauimenta, oportet enim, vt &longs;imul vnitæ nihil vacui in <lb/>pauimento relinquant. </s> <s id="id.001549">huiu&longs;modi &longs;unt triangula æquilatera (de his enim <lb/>intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia; <lb/><figure id="id.009.01.084.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.084.1.jpg" place="text"/> <lb/>nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui­ <lb/>re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t, <lb/>quia omnes anguli circa idem punctum (y. </s> <s id="id.001550">g. <!-- REMOVE S-->A, in hac <lb/>figura) in plano, quotquot fuerint con&longs;tituti, &longs;unt æqua­ <lb/>les quatuor rectis, ex coroll. </s> <s id="id.001551">&longs;ecundo 15. primi Elemen­ <lb/>ti: cum igitur &longs;ex anguli, trianguli æquilateri <expan abbr="æquiualeãt">æquiualeant</expan> <lb/>quatuor rectis angulis, con&longs;tituti omnes circa punctum <lb/>A, totum locum circa illud implere po&longs;&longs;unt. </s> <s id="id.001552">Quadratum etiam replere lo­ <lb/><figure id="id.009.01.084.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.084.2.jpg" place="text"/> <lb/>cum manife&longs;tum e&longs;t, cum enim ip&longs;ius anguli &longs;intrecti, &longs;i <lb/>quatuor quadrata ad idem punctum A, copulentur, vt in <lb/>figura apparet, replebunt eadem de cau&longs;a vacuum.</s></p><p type="main"> <s id="id.001553">Hexagonum quoque regulare, ide&longs;t æquilaterum, & <lb/>æquiangulum idem præ&longs;tare pote&longs;t; cum enim tres angu­ <lb/>li ip&longs;ius æquiualeant quatuor rectis, &longs;i tria hexagona ad <lb/>idem punctum A, vt in &longs;igura adaptentur, nece&longs;&longs;ariò ni­ <lb/>hil vacui inter ip&longs;a relinquetur, vt in figura hac o&longs;tenditur. </s> <s id="id.001554">præter has tres <lb/><figure id="id.009.01.084.3.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.084.3.jpg" place="text"/> <lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol­ <lb/>&longs;it. </s> <s id="id.001555">cuius demon&longs;trationem perfectam videre pote­ <lb/>ris in fine commentarij P. <!-- REMOVE S-->Clauij &longs;uper 4. Elem. nos <lb/>ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi­ <lb/>ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um <lb/>Ari&longs;totelis patefaciunt. </s> <s id="id.001556">Aliæ porrò figuræ replen­ <lb/>tes locum planum, quibus aliquando Architectores <lb/>vtuntur, vel &longs;unt irregulares, vel ad prædictas redu­ <lb/>ci po&longs;&longs;unt. </s> <s id="id.001557">cum igitur tres tantum ex figuris planis <lb/>totum repleant, hæ &longs;olæ poterunt elementis attri­ <lb/>bui, ac propterea non &longs;ufficient, ni&longs;i pro tribus elementis. </s> <s id="id.001558">quare quartum <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> figura relinquetur; quod e&longs;t ab&longs;urdum.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.085.jpg" pagenum="85"/><p type="head"> <s id="id.001559"><emph type="italics"/>Admirabilis quædam A&pgrave;um industria.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001560">Cæterum occa&longs;ione harum figurarum illud hoc loco apponere vi­ <lb/>&longs;um e&longs;t, quod Pappus <expan abbr="Alexãdrinus">Alexandrinus</expan> initio quinti libri collectionum <lb/>mathematicarum &longs;cribit, De admirabili Apum indu&longs;tria, atque <lb/>prudentia in con&longs;truendo &longs;uas cellulas figura hexagona regulari. <lb/></s> <s id="id.001561">cum enim vellent omne vacuum excludere, & præterea capaci&longs;&longs;imam <expan abbr="om-niũ">om­ <lb/>nium</expan> figuram habere, hexagonam accepere, quæ inter prædictas tres vtrum­ <lb/>que præ&longs;tat, nam & inane omne excludit, & illarum trium capaci&longs;&longs;ima e&longs;t, <lb/>cum magis ad circularem figuram accedat: vt patet ex tractatu de figuris <lb/>I&longs;operimetris, qui e&longs;t apud Clauium in &longs;phæra, necnonin Geometria pra­ <lb/>ctica. </s> <s id="id.001562">hoc ideò libentius recen&longs;ui, quia animaduerti naturales hi&longs;toriogra­ <lb/>phos omnes latere, vel ip&longs;um Aldobrandum no&longs;trum, qui quamuis indu­ <lb/>&longs;trio&longs;æ Apis in&longs;tar omnia delibauerit, i&longs;tud tamen de Apibus artificium tan­ <lb/>ta &longs;apientia plenum, ne&longs;cio quo modo prætermi&longs;it.</s></p><p type="main"> <s id="id.001563"><arrow.to.target n="marg121"/></s></p><p type="margin"> <s id="id.001564"><margin.target id="marg121"/>121</s></p><p type="main"> <s id="id.001565">Ibidem <emph type="italics"/>(In folidis verò duæ &longs;olum pyramis, & cubus)<emph.end type="italics"/> ide&longs;t replent locum <lb/>&longs;olidum. </s> <s id="id.001566">nullum reperi, qui in hoc loco explicando non errauerit; nam Græ­ <lb/>ci, qui alioqui &longs;olent mathematica probè intelligere, hic omnes lap&longs;i &longs;unt, <lb/><expan abbr="&longs;ecum&qacute;">&longs;ecumque</expan>; & Arabes, & Latinos in <expan abbr="eãdem">eandem</expan> foueam &longs;upra &longs;e mi&longs;erè traxerunt. <lb/></s> <s id="id.001567">communis ferè error omnium fuit, pyramides plures &longs;imul compactas po&longs;­ <lb/>&longs;e replere &longs;olidum locum. </s> <s id="id.001568">quod vt melius intelligamus, &longs;ciendum e&longs;t, reple­ <lb/>re locum <expan abbr="&longs;olidũ">&longs;olidum</expan> nihil aliud e&longs;&longs;e, quam &longs;i plura corpora &longs;olida &longs;imul ad idem <lb/>punctum coaptata, ita con&longs;tipentur, vt totum &longs;patium, quod e&longs;t circa pun­ <lb/>ctum illud omninò occupent, hoc e&longs;t, nihil vacui inter ip&longs;a relinquatur: &longs;i­ <lb/>cut enim prædictæ tres &longs;iguræ planæ, de quibus paulò ante, replent locum <lb/>planum, ide&longs;t &longs;uper&longs;iciem; ita cubi replent &longs;olidum, ide&longs;t &longs;oliditatem &longs;imul <lb/>vniti con&longs;tituunt, ita vt &longs;i octo cubi &longs;imul ad idem punctum <expan abbr="coapt&etilde;tur">coaptentur</expan>, con­ <lb/>&longs;tituant corpus &longs;olidum ex octo illius con&longs;latum, <expan abbr="nihil&qacute;">nihilque</expan>; inane inter ip&longs;os <lb/>cubos relinquatur. </s> <s id="id.001569">& &longs;icuti planæ illæ figuræ erant conficiendis pauimentis <lb/>aptæ, ita &longs;olidæ hæ muris, qui corpora &longs;unt &longs;olida, <expan abbr="con&longs;tru&etilde;dis">con&longs;truendis</expan> idonea &longs;unt. <lb/></s> <s id="id.001570"><expan abbr="Notã">Notam</expan> dum præterea, quod per pyramidem debemus intelligere pyramidem <lb/>regularem, quæ dicitur etiam Tetraedrum, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;ecunda inter <expan abbr="quinq;">quinque</expan> cor­ <lb/>pora regularia rectilinea, quæ alias Platonica corpora dicuntur. </s> <s id="id.001571"><expan abbr="eorum&qacute;">eorumque</expan>; <lb/>defraitiones &longs;unt in 11. Elem. <!-- KEEP S--></s> <s id="id.001572">Tetraedrum autem &longs;ic definitur, e&longs;t figura &longs;o­ <lb/>lida &longs;ub quatuor triangulis æquilateris, <expan abbr="atq;">atque</expan> inuicem æqualious contenta: <lb/>de hac inquam e&longs;t &longs;ermo. </s> <s id="id.001573">quia &longs;i liceret intelligere de irregularibus figuris, <lb/>infinitæ reperir entar figuræ tam planæ, quam &longs;olidæ, quæ vtrumque locum <lb/>complerent. </s> <s id="id.001574">Aduertendum tandem Ari&longs;t. videri loqui de repletione loci <lb/>&longs;olidi, quia tran&longs;it à planïs figuris ad &longs;olidas. </s> <s id="id.001575">& quia &longs;i hæ duæ pyramis, & <lb/>cubus replent locum &longs;olummedo &longs;ecundum &longs;uas &longs;uperficies, quæ &longs;unt trian­ <lb/>gulum, & quadratum, iam de his cum proximè ante dixi&longs;&longs;et, quid opus fui&longs;­ <lb/>&longs;et idem po&longs;t modum repetere. </s> <s id="id.001576">ad hæc &longs;i in medium &longs;olida hæc duo profert, <lb/><expan abbr="ait&qacute;">aitque</expan>; ip&longs;a replere locum, intelligens, planum, profectò non loquitur forma­ <lb/>liter, ide&longs;t de ip&longs;is, vt &longs;oh da &longs;unt. </s> <s id="id.001577">Quare Ari&longs;t. videretur &longs;ibi non con&longs;tare, <lb/>vel perperam exi&longs;tima&longs;&longs;e plura Tetraedra complere &longs;oliditatem. </s> <s id="id.001578">deceptus <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.086.jpg" pagenum="86"/>fortè fuit Ari&longs;t. cò quod videret Ico&longs;aedrum con&longs;tare ex viginti pyramidi­ <lb/>bus, verùm illæ non &longs;unt regulares, ide&longs;t <expan abbr="nõ">non</expan> &longs;unt Tetraedra, vt po&longs;tea o&longs;ten­ <lb/>dam. </s> <s id="id.001579">Verum quidem e&longs;t octo cubos &longs;imul adactos &longs;oliditatem conficere, <lb/>quia ad id nece&longs;&longs;arij &longs;unt octo anguli &longs;olidi, quos octo cubi præbere po&longs;&longs;unt, <lb/>cum anguli ip&longs;orum &longs;int recti, & &longs;olidi. </s> <s id="id.001580">Verum enim verò plures pyramides <lb/>regulares, &longs;iue plura Tetraedra non po&longs;&longs;e replere vacuum, <expan abbr="&longs;olidum&qacute;">&longs;olidumque</expan>; con­ <lb/>&longs;tituere, ex eo patet, quia &longs;i id præ&longs;tarent, conflarent nece&longs;&longs;ariò, vel vnum <lb/>ex <expan abbr="quinq;">quinque</expan> corporibus regularibus, de quibus in 13. Elemen. <!-- REMOVE S-->vel aliud quod­ <lb/>piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem. non dantur, ni&longs;i illa. <lb/></s> <s id="id.001581">quinque; <expan abbr="neq;">neque</expan> vllum ex illis, quia diameter huiu&longs;modi corporis, quod com­ <lb/>poneretur ex illis pyramidibus, e&longs;&longs;et dupla lateris eiu&longs;dem, vt patet, quia <lb/>pyramides illæ omnes concurrerent ad centrum &longs;phæræ illas omnes com­ <lb/>plectentis, quare latus vnius pyramidis à &longs;uperficie &longs;phæræ incipiens de&longs;i­ <lb/>neret in centrum, ergo latus i&longs;tud e&longs;&longs;et &longs;emidiameter, quapropter tota dia­ <lb/>meter illius &longs;ph&ecedil;ræ, & con&longs;equenter huius corporis in illa in&longs;cripti, e&longs;&longs;et du­ <lb/>pla lateris eiu&longs;dem figuræ &longs;olidæ in&longs;criptæ, &longs;ed nullo talis proportio diame­ <lb/>tri alicuius ex illis <expan abbr="quinq;">quinque</expan> &longs;olidis regularibus ad latus eiu&longs;dem reperitur, <lb/>quæ &longs;it nimirum dupla, vt patet ex vltimis demon&longs;trationibus 13. Elem. ini­ <lb/>tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla <lb/>inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani­ <lb/>fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo­ <lb/>do replere locum &longs;olidum. </s> <s id="id.001582">cum igitur animaduerterem, &longs;en&longs;um Ari&longs;t. nullo <lb/>modo po&longs;&longs;e verificari de repletione &longs;olidi per plura Tetraedra, & omnes <lb/>tamen commentatores auctoritate Ari&longs;t. decepti pro ip&longs;o &longs;tarent, dubius, <lb/><expan abbr="anceps&qacute;">ancepsque</expan>; diu hæ&longs;i, neque quid quam mea Minerua a&longs;&longs;erere au&longs;us &longs;um, &longs;ed P. <lb/><!-- REMOVE S-->Clauium præceptorem meum per literas con&longs;ului, qui in hunc modum hu­ <lb/>mani&longs;&longs;imè re&longs;pondit; cubus implet locum quater &longs;umptus, ad idem enim <lb/>punctum quatuor cubi coaptantur: &longs;ic etiam pyramis &longs;exies &longs;umpta, &longs;eu &longs;ex <lb/>pyramides ad idem punctum iunctæratione &longs;ub&longs;tantium <expan abbr="triangulorũ">triangulorum</expan> æqui­ <lb/>laterorum. </s> <s id="id.001583">Verum hac ratione non videntur implere locum lolidum, fa­ <lb/>teor; &longs;ed tamen Ari&longs;t. in co tex. <!-- REMOVE S-->non loquitur de repletione loci &longs;olidi. </s> <s id="id.001584">hæc <lb/>ip&longs;e. </s> <s id="id.001585">&longs;i igitur libeat Ari&longs;totelem, quod fortè Clauius intendebat defendere, <lb/>dicendum e&longs;t cum eo Ari&longs;t non loqui de repletione loci &longs;olidi: <expan abbr="neq;">neque</expan> loqui <lb/>de cubo, & Tetraedro, quatenus &longs;unt corpora, &longs;ed quatenus habent &longs;uper­ <lb/>ficies, cubus quidem &longs;ex quadratas, Tetraedrum autem quatuor æquilate­ <lb/>ras &longs;uperficies, quæ duæ figuræ, vt &longs;upra in hoc textu vidimus, replent lo­ <lb/>cum: <expan abbr="atq;">atque</expan> hoc modo facimus Ari&longs;totelem non formaliter loquentem. </s> <s id="id.001586">ex­ <lb/>aduersò ne videamur magis Ari&longs;t. quam veritatem &longs;equi, videtur dicen­ <lb/>dum, Ari&longs;totilem formaliter locutum e&longs;&longs;e, & vt patet ex rationibus &longs;upra <lb/>allatis de repletione &longs;olidi e&longs;&longs;e intelligendum, vt etiam intellexerunt omnes <lb/>huius loci expo&longs;itores; Verumtamen ip&longs;um erra&longs;&longs;e, dum plures pyramides <lb/>replere &longs;olidum exi&longs;timauit. </s> <s id="id.001587">Vtrumuis dixerimus, non tamen Ari&longs;t. ab om­ <lb/>ni crrore vindicabimus. </s> <s id="id.001588">Hoc tamen certum e&longs;t, ex prædictis, Græcos om­ <lb/>nes pariter, ac Latinos, illos &longs;equentes, lapos e&longs;&longs;e, a&longs;&longs;erentes duodecim py­ <lb/>ramides complere &longs;olidum locum, <expan abbr="atq;">atque</expan> Dodecaedrum con&longs;tituere; nam py­ <lb/>ramides Dodecaedron con&longs;tituentes non &longs;unt regulares, ide&longs;t, non &longs;unt Te­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.087.jpg" pagenum="87"/>traedra (de quibus tamen Ari&longs;t. loquitur) vt patet ex &longs;upradictis. </s> <s id="id.001589">Indul­ <lb/>geas Lector, &longs;i hoc loco nece&longs;&longs;e fuit in Geometriæ penetralia ingredi: ope­ <lb/>ræpretium enim e&longs;t aliquando ip&longs;is Mathematicis &longs;atisfacere. </s> <s id="id.001590">tu verò, &longs;i <lb/>adeo es mathematicis imbutus, con&longs;ule po&longs;tremas demon&longs;tra. </s> <s id="id.001591">13. Elem. & <lb/>præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima, <lb/><expan abbr="atq;">atque</expan> huc &longs;pectantia reperies ex his omnibus Mathematica, quæ no&longs;træ &longs;unt <lb/>partes, per&longs;picuè &longs;atis expo&longs;uimus.</s></p><p type="main"> <s id="id.001592">Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap. 38. &longs;pecu­ <lb/>lationem 10. Benedicti de placitis Ari&longs;t. <!-- REMOVE S--><expan abbr="reperi&qacute;">reperique</expan>; ab eo vno Ari&longs;t. hoc loco <lb/>erroris notari, dum a&longs;&longs;eruit duodecim pyramides replere <expan abbr="locũ">locum</expan> corporeum, <lb/>ide&longs;t, vt exponit ip&longs;e, &longs;ex pyramides &longs;uper hexagonam aliquam figuram <lb/>&longs;uperficialem, & &longs;ex &longs;ub eadem, id præ&longs;tarent, cum potius maius vacuum <lb/>remaneat ad quamlibet partium &longs;upra, & infra, quam plenum. </s> <s id="id.001593">hæc ip&longs;e. </s> <s id="id.001594">&longs;ed <lb/>expo&longs;itio i&longs;ta puerili, ne dum Ari&longs;t. ingenio pror&longs;us indigna e&longs;t: vt propte­ <lb/>rea exi&longs;timem ca&longs;u potius eum Ari&longs;t. rectè reprehendi&longs;&longs;e, quam ex certa <lb/>&longs;cientia, cum illius erratum maiori errato conetur corrigere. </s> <s id="id.001595">Incidi po­ <lb/>&longs;tremò in Indicem librorum, quem Maurolyius &longs;uæ Co&longs;mographiæ præpo­ <lb/>nit, vbi &longs;ic ait: Demon&longs;tramus autem in libello de figuris planis, <expan abbr="&longs;olidis&qacute;">&longs;olidisque</expan>; <lb/>locum replentibus, cubos per &longs;e, pyramides verò cum octacdris compactas <lb/>dumtaxat implere locum, qua in re Auerroem erra&longs;&longs;e pueriliter manife&longs;tum <lb/>erit. </s> <s id="id.001596">Vides igitur tanti viri auctoritate confirmari no&longs;tram &longs;ententiam, py­ <lb/>ramides videlicet per &longs;e, non replere vacuum. </s> <s id="id.001597">cum igitur con&longs;tet vnam tan­ <lb/>tum ex figuris &longs;olidis, &longs;iue etiam dicas, vt perperam Ari&longs;t. & alij plures exi­ <lb/>&longs;timarunt, replere totum &longs;olidum; nulla ratione poterunt <expan abbr="elem&etilde;ta">elementa</expan> quatuor, <lb/>quatuor diuer&longs;is figuris indui, &longs;ed vnum tantummodo, quare reliqua <expan abbr="ab&longs;q;">ab&longs;que</expan> <lb/>figura remanere nece&longs;&longs;e e&longs;&longs;et: quod e&longs;t omnino inconueniens.</s></p><p type="main"> <s id="id.001598"><arrow.to.target n="marg122"/></s></p><p type="margin"> <s id="id.001599"><margin.target id="marg122"/>122</s></p><p type="main"> <s id="id.001600">Tex. 71 <emph type="italics"/>(Deinde &longs;i terra e&longs;t cubus &c.)<emph.end type="italics"/> lege definitiones 11. Elem. quæ &longs;unt <lb/>admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium, <lb/>quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha­ <lb/>bes in &longs;phæra Clau. <!-- KEEP S--></s> <s id="id.001601">Simpl. <!-- REMOVE S-->etiam hoc loco &longs;atisfacit.</s> </p></chap><chap><p type="head"> <s id="id.001602"><emph type="italics"/>Ex Quarto de Cœlo.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.001603"><arrow.to.target n="marg123"/></s></p><p type="margin"> <s id="id.001604"><margin.target id="marg123"/>123</s></p><p type="main"> <s id="id.001605">Tex. 33. <emph type="italics"/>(Deinde ad &longs;imiles videtur angulos ignis quidem &longs;ur&longs;um ferri, <lb/>terra autem deor&longs;um, & omninò quod grauitatem babet, quare nece&longs;&longs;e <lb/>est ferri ad medium. </s> <s id="id.001606">boc autem vtrum accidit ad ip&longs;um tcrræ medium, <lb/>an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)<emph.end type="italics"/> cum vellet <lb/><figure id="id.009.01.087.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.087.1.jpg" place="text"/> <lb/>probare Ari&longs;toteles dari <expan abbr="pũctum">punctum</expan> quoddam in medio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;cendat; vtitur, præter alias, etiam <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;­ <lb/>modi. </s> <s id="id.001607">videmus ignem, & cætera l&ecedil;uia a&longs;cendere à <lb/>terra &longs;ur&longs;um ad angulos æquales; &longs;imiliter videmus <lb/>terram, & c&ecedil;tera grauia de&longs;cendere ad terram dcor­ <lb/>&longs;um ad angulos æquales, quod &longs;ignum e&longs;t omnia i&longs;ta <lb/>idem mundi medium re&longs;picere: v.g. <!-- REMOVE S-->&longs;it terra in &longs;igu­ <lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine cen­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.088.jpg" pagenum="88"/>trum A. via, qua a&longs;cendit ignis &longs;it in linea A C B, quæ facit angulos in &longs;u­ <lb/>perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per <lb/>candem lineam <expan abbr="faci&etilde;s">faciens</expan> eo&longs;dem angulos æquales de&longs;cendit. </s> <s id="id.001608">linea autem, quæ <lb/>facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu­ <lb/>ra, & probari pote&longs;t geometricè ex primis tertij Elem. ex quibus patet tam <lb/>læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A, <lb/>&longs;phæræ. </s> <s id="id.001609">Vtrum autem i&longs;tud centrum &longs;it idem cum <expan abbr="c&etilde;tro">centro</expan> totius mundi, alius, <lb/>inquit, e&longs;t &longs;ermo, hoc e&longs;t, ad a&longs;tronomum pertinet. </s> <s id="id.001610">vide igitur hac de re <lb/>pulchram de&longs;&longs;ertationem apud Clauium in &longs;phæra: qui probat euidenter <lb/>e&longs;&longs;e vnum, & idem.</s></p><p type="main"> <s id="id.001611"><arrow.to.target n="marg124"/></s></p><p type="margin"> <s id="id.001612"><margin.target id="marg124"/>124<!-- KEEP S--></s></p><p type="main"> <s id="id.001613"><emph type="italics"/>Hoc loco de&longs;ideratur commentarius in cap. vlt. </s> <s id="id.001614">de Cœlo. <!-- KEEP S--></s> <s id="id.001615">cuius loco ìn-<emph.end type="italics"/></s></p><p type="main"> <s id="id.001616"><arrow.to.target n="marg125"/> <lb/><arrow.to.target n="marg126"/> <lb/><emph type="italics"/>terim Lector adeat Di&longs;cur&longs;um Italicum Galilæi Galilæi, de his,<emph.end type="italics"/> <lb/><arrow.to.target n="marg127"/> <lb/><emph type="italics"/>quæ in aqua mouentur, ac natant: ubi propè finem, plura in hu-<emph.end type="italics"/> <lb/><arrow.to.target n="marg128"/> <lb/><emph type="italics"/>ius capitis explicationem affert.<emph.end type="italics"/> <lb/><arrow.to.target n="marg129"/></s></p><p type="margin"> <s id="id.001617"><margin.target id="marg125"/>125</s></p><p type="margin"> <s id="id.001618"><margin.target id="marg126"/>126</s></p><p type="margin"> <s id="id.001619"><margin.target id="marg127"/>127</s></p><p type="margin"> <s id="id.001620"><margin.target id="marg128"/>128</s></p><p type="margin"> <s id="id.001621"><margin.target id="marg129"/>129</s></p></chap><chap><p type="head"> <s id="id.001622"><emph type="italics"/>Ex Lib. 2. de Generatione, & Corruptione.<emph.end type="italics"/> <lb/><arrow.to.target n="marg130"/><!-- KEEP S--></s></p><p type="margin"> <s id="id.001623"><margin.target id="marg130"/>130</s></p><p type="main"> <s id="id.001624">Tex. 56. <emph type="italics"/>(ldeoqué non prima latio cau&longs;a Generationis, & Corruptionis e&longs;t, <lb/>&longs;ed quæ &longs;ecundum obliquum circulum, in hac enim & continuum vnum <lb/>e&longs;t & moueri duobus motibus)<emph.end type="italics"/> per primam lationem intelligit mo­ <lb/>tum primi mobilis, qul &longs;it &longs;uper polis mundi, quo Stellæ omnes <lb/>ab oriente in occidentem rectà feruntur. </s> <s id="id.001625">per obliquum verò circulum in­ <lb/>telligit Zodiacum, qui obliquus e&longs;t, quia poli eius &longs;unt alij à polis mundi, & <lb/>quia non tendit rectà ab ortu ad occa&longs;um, &longs;ed in &longs;phæra mundi tran&longs;uer­ <lb/>&longs;us e&longs;t, & deflectit à &longs;eptentrione in meridiem, quamuis non rectà, vt in <lb/>&longs;phæra explicari &longs;olet. </s> <s id="id.001626">motus ergo Planetarum, qui fit &longs;ecundum hunc cir­ <lb/>culum, & ip&longs;e obliquus, & tran&longs;uer&longs;us codem modo erit; ferrentur que per <lb/>eum à Borea ad Au&longs;trum, & è conuer&longs;o; ex quo acce&longs;&longs;u, & rece&longs;&longs;u efficiunt <lb/>æ&longs;tatem, & hyemem, item generationes, & corruptiones. </s> <s id="id.001627">Sol porrò, & pla­ <lb/>netæ, qui motibus proprijs hunc circulum peragunt, dicuntur moueri duo­ <lb/>bus motibus, & quidem contrarijs: quoniam dum Sol. </s> <s id="id.001628">v. <!-- REMOVE S-->g. <!-- REMOVE S-->per Zodiacum <lb/>graditur motu proprio, interim etiam à primo mobili fertur ab ortu in oc­ <lb/>ca&longs;um: ex quibus duobus motibus fit vnus tantum Solis motus &longs;piralis, qui <lb/>mixtus e&longs;t, ide&longs;t, qui fit à duobus motoribus; vnde re vera Sol non mouetur <lb/>duobus motibus contrarijs re ip&longs;a di&longs;tinctis; hoc enim impo&longs;&longs;ibile e&longs;t: &longs;ed <lb/>motu mixto ex duobus, qui &longs;piralis e&longs;t, circa mundum de&longs;cribens &longs;piras ab <lb/>vno tropico ad alterum: qui, vt dixi, cau&longs;atur à duobus motoribus, qui &longs;unt <lb/>Sol ip&longs;e, mouens &longs;e ip&longs;um per Zodiacum: & primum mobile mouens in&longs;u­ <lb/>per ip&longs;um Solem, & Zodiacum ab ortu in occa&longs;um circa mundum.</s> </p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.089.jpg" pagenum="89"/><chap><p type="head"> <s id="id.001629"><emph type="italics"/>EX PRIMO METEORORVM.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001630"><arrow.to.target n="marg131"/></s></p><p type="margin"> <s id="id.001631"><margin.target id="marg131"/>131</s></p><p type="main"> <s id="id.001632">Svmma 1. cap. 3. <emph type="italics"/>(Moles autem terræ quanta &longs;it ad ambientes magnitudi­ <lb/>nes, non immamfestum, iam enim vi&longs;um est per a&longs;trologica theoremata, <lb/>quod multò etiam quibu&longs;dam a&longs;tris est minor)<emph.end type="italics"/> Quantitas terræ non &longs;o­ <lb/>lum ab&longs;olutè con&longs;iderata, ab A&longs;tronomis explorata habetur, vt vi­ <lb/>dere e&longs;t in &longs;phæra Clauij; &longs;ed etiam re&longs;pectiuè con&longs;iderata, ide&longs;t re&longs;pectu <lb/>aliorum elementorum, & ip&longs;orum etiam a&longs;trorum; cuius demon&longs;trationes <lb/>&longs;unt partim in libello Ari&longs;tarchi Samij, de magnitudine, & di&longs;tantia Solis, <lb/>& Lunæ, partim apud Ptolæmeum in magna Syntaxi, &longs;iue Almage&longs;to: par­ <lb/>tim apud Albategnium de &longs;cientia &longs;tellarum: partim demum apud Ticho­ <lb/>nem Brahe. </s> <s id="id.001633">Porrò facile e&longs;t demon&longs;trare Solem e&longs;&longs;e terra multò maiorem, <lb/>terram verò maiorem Luna, <expan abbr="id&qacute;">idque</expan>; ex eclyp&longs;i lunari, cuius imaginem habes <lb/>in figura &longs;equenti; vbi vmbra terræ e&longs;t D B E, in quam Luna nigricans im­ <lb/>mergitur, ac lumine deficit, reliqua cognitu &longs;unt facilia: quia igitur A&longs;tro­ <lb/>nomi ob&longs;eruarunt vmbram terræ paulò &longs;upra Lunam pertingere, cum &longs;upe­ <lb/>riora a&longs;tra non adeat, hinc collegerunt eam nece&longs;&longs;ariò e&longs;&longs;e acuminatam, &longs;eu <lb/>conicam, vt figura refert. </s> <s id="id.001634">Cum ergo terra vmbram proijciat turbinatam, <lb/>nece&longs;&longs;ariò corpus Solis, quod ip&longs;am illuminat, eadem maior erit: quoti­ <lb/>diana enim experientia docemur, corpore illuminante exi&longs;tente maiore <lb/>quà &longs;it illuminatum, vmbram proijci fa&longs;tigiatam: cum deinde Solem val­ <lb/>de a terra di&longs;tare certum &longs;it, optimè infertur, eum re&longs;pectu terræ e&longs;&longs;e maxi­ <lb/>mum: quanto enim duæ lineæ, &longs;iue radij B A, B C. à terra ad partes Solis <lb/><figure id="id.009.01.089.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.089.1.jpg" place="text"/> <lb/>magis elongantur, tan­ <lb/>to maius corpus <expan abbr="illu-minãs">illu­ <lb/>minans</expan> intercipiunt. </s> <s id="id.001635">ha­ <lb/>ctenus de magnitudine <lb/>terræ ad Solem. </s> <s id="id.001636">Cum <lb/>verò Luna eclyp&longs;atio­ <lb/>nis tempore, aliquan­ <lb/>do non &longs;olum tota in <lb/>vmbræ vertice lateat, <lb/>verùm etiam <expan abbr="aliquãdo">aliquando</expan> <lb/>moram trahat, euidens <lb/>e&longs;t, eam e&longs;&longs;e multò mi­ <lb/>norem illa vmbræ par­ <lb/>te, in quam immergi­ <lb/>tur; quæ pars cum &longs;it <lb/>conicæ vmbræ media, <lb/>crit multò gracilior <lb/>quàm &longs;it ip&longs;a terra. <lb/></s> <s id="id.001637">Ex quo manife&longs;tè apparet, Lunam, quæ illa vmbra minor e&longs;t, e&longs;&longs;e à fortio­ <lb/>ri multò minorem ip&longs;a terre&longs;tri mole. </s> <s id="id.001638">Atque hæc de comparatione terræ <lb/>ad Lunam. <!-- KEEP S--></s> <s id="id.001639">harum rerum demon&longs;trationes exactiores pertractare non e&longs;t <lb/>huius loci.</s></p><p type="main"> <s id="id.001640"><arrow.to.target n="marg132"/></s></p><p type="margin"> <s id="id.001641"><margin.target id="marg132"/>132</s></p><p type="main"> <s id="id.001642">Eodem cap. <emph type="italics"/>(Con&longs;iderautes vtique, quæ nunc c&longs;tenduntur per Mathematica<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.090.jpg" pagenum="90"/><emph type="italics"/>&longs;ufficienter, fortè vtique de&longs;isterent ab hac puerili opinione; valde enim &longs;implex <lb/>e&longs;t putare <expan abbr="vnumquodq;">vnumquodque</expan> eorum quæ feruntur e&longs;&longs;e paruum magnitudinibus, quia vi­ <lb/>detur a&longs;picientibus, binc nobis &longs;ic)<emph.end type="italics"/> vtinam i&longs;ta, necnon alia his &longs;imilia, quæ <lb/>pa&longs;&longs;im apud Ari&longs;t. occurrunt, <expan abbr="pleriq;">plerique</expan> no&longs;træ ætatis con&longs;iderarent, qui nulla <lb/>ratione probari po&longs;&longs;e exi&longs;timant, Solem, v. <!-- REMOVE S-->g. <!-- REMOVE S-->terra e&longs;&longs;e centies &longs;exagies &longs;e­ <lb/>xies maiorem; &longs;ed etiam, quod peius e&longs;t, negant e&longs;&longs;e maiorem; ad demon­ <lb/>&longs;trationes autem a&longs;tronomicas dicunt &longs;e exi&longs;timare eas e&longs;&longs;e fallaces; at que <lb/>impo&longs;libile e&longs;&longs;e nos res adeo à nobis di&longs;taptes &longs;ufficienter perue&longs;tigare: <lb/>quanto &longs;apientius, ac prudentius eorum Magi&longs;ter Ari&longs;t. alibi &longs;æpius, &longs;ed hoc <lb/>præcipuè loco; quippe qui Mathematicis &longs;ufficienter excultus erat; quibus <lb/>i&longs;ti de&longs;tituti, nullo vnquam modo ve&longs;tigia præceptoris a&longs;&longs;equi poterunt.</s> </p><p type="main"> <s id="id.001643"><arrow.to.target n="marg133"/></s></p><p type="margin"> <s id="id.001644"><margin.target id="marg133"/>133</s></p><p type="main"> <s id="id.001645">Summa 1. cap. 4. <emph type="italics"/>(Quæ igitur astrorum e&longs;t, velox quidem; longè autem: quæ <lb/>verò Lunæ deor&longs;um quidem, tarda autem: quæautem Solis ambo hæc babet &longs;uffi­ <lb/>cienter)<emph.end type="italics"/> quæ igitur a&longs;trorum, ide&longs;t latio a&longs;trorum e&longs;t velox, &longs;ed procul à ter­ <lb/>ra; Lunæ verò latio terræ quidem proxima, tarda tamen: at verò Solis la­ <lb/>tio medio modo &longs;e habet inter vtrumque, ide&longs;t, quia <expan abbr="neq;">neque</expan> nimis vt a&longs;tra di­ <lb/>&longs;tat, <expan abbr="neq;">neque</expan> tardè &longs;icut Luna circunfertur. </s> <s id="id.001646">exi&longs;timo Ari&longs;t. loqui de motu diur­ <lb/>no, quia &longs;ecundum hunc a&longs;tra inerrantia &longs;unt Sole citatiora, Sol verò ip&longs;a <lb/>Luna citior. </s> <s id="id.001647">Verumenimuerò illud non prætereundum, quod plurium inua­ <lb/>luerit opinio exi&longs;timantium Ari&longs;t. his verbis, Solem &longs;upra Lunam proximè <lb/>colloca&longs;&longs;e; quod tamen ex ip&longs;is nullo pacto deduci pote&longs;t; &longs;ed &longs;olummodo <lb/>ip&longs;um &longs;upra Lunam colloca&longs;&longs;e. </s> <s id="id.001648">quod &longs;i ita &longs;en&longs;i&longs;&longs;et venia dignus haberetur, <lb/>cum tunc temporis nondum fortè adinuentæ e&longs;&longs;ent demon&longs;trationes illæ <lb/>a&longs;tronomicæ, quibus ordo Planetarum certi&longs;&longs;imè con&longs;tat, <expan abbr="Sol&qacute;">Solque</expan>; medius in­ <lb/>ter Planetas collocatur. </s> <s id="id.001649">At verò nulla ratione ferendi &longs;unt <expan abbr="quicunq;">quicunque</expan> no&longs;tra <lb/>hac tempe&longs;tate non &longs;olum Ari&longs;t. ita &longs;en&longs;i&longs;&longs;e, &longs;ed etiam contra firmi&longs;&longs;imas <lb/>aftronomorum demon&longs;trationes, quibus adeò Ari&longs;t. deferebat, vnica, vt pu­ <lb/>tant ip&longs;ius auctoritate fulti, Solem &longs;ecundum à Luna locum occupare om­ <lb/>ni ope defendunt.</s></p><p type="main"> <s id="id.001650"><arrow.to.target n="marg134"/></s></p><p type="margin"> <s id="id.001651"><margin.target id="marg134"/>134</s></p><p type="main"> <s id="id.001652">Summa 2. cap. 3. <emph type="italics"/>(Quod accidit circa Mercurij stellam, quia enim modicum <lb/><expan abbr="&longs;upera&longs;c&etilde;dis">&longs;upera&longs;cendis</expan>, &longs;æpè non apparet, it a vt po&longs;t tempus multum appareat)<emph.end type="italics"/> quod Mer­ <lb/>curius non ni&longs;i rarò con&longs;pici po&longs;&longs;it, cau&longs;a e&longs;t, quia parum à Sole elongatur, <lb/>&longs;iue ip&longs;um antecedat, &longs;iue &longs;ub&longs;equatur. </s> <s id="id.001653">ex quo fit, vt diu ferè &longs;imul cum So­ <lb/>le cit cumferatur, & propterea &longs;iue oriatur, &longs;iue occidat, parum &longs;upra ho­ <lb/>rizontem eleuatus apparere pote&longs;t, quod Ari&longs;t. ait modicum <expan abbr="&longs;upera&longs;c&etilde;dit">&longs;upera&longs;cendit</expan>. <lb/></s> <s id="id.001654">vnde fit tum propter nimiam Solis vicinitatem, cuius lumine tegitur; tum <lb/>propter vapores, qui horizonti vt plurimum incumbunt, vt rarò, & po&longs;t ma­ <lb/>gna temporis interualla con&longs;piciatur. </s> <s id="id.001655">non me fugit hæc omnia ab a&longs;trono­ <lb/>mis per epiciclum excu&longs;ari; &longs;ed ego mediocritati eorum, in quorum gra­ <lb/>tiam hæc &longs;cribo, con&longs;ultum volo.</s></p><p type="main"> <s id="id.001656"><arrow.to.target n="marg135"/></s></p><p type="margin"> <s id="id.001657"><margin.target id="marg135"/>135</s></p><p type="main"> <s id="id.001658">Eodem cap. <emph type="italics"/>(Ad au&longs;trum autem quando feratur, copiam quidem habere talís <lb/>humiditatis, &longs;ed quia parua e&longs;t &longs;ictio circuli, quæ &longs;uper terram, quæ autem deor­ <lb/>&longs;um multiplex, non po&longs;&longs;e vi&longs;um hominum fractum ferri ad Solem, <expan abbr="neq;">neque</expan> ip&longs;i tropico <lb/>au&longs;trino appropinquanti; <expan abbr="neq;">neque</expan> in æ&longs;tiuis ver&longs;iombus exi&longs;tente Sole. <!-- KEEP S--></s> <s id="id.001659">quapropter in <lb/>lis quidem locis neque fieri cometem ip&longs;um. </s> <s id="id.001660">quando verò ad Boream &longs;ubdefecerit, <lb/>accipere comam, quia magna e&longs;t circun&longs;erentia, quæ e&longs;t &longs;upra horizontem; quæ au-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.091.jpg" pagenum="91"/><emph type="italics"/>tem e&longs;t &longs;ubtus, pars circuli parua; facilè enim vi&longs;um hominum pertingere tunc ad <lb/>Solem)<emph.end type="italics"/> cur cometa in regione au&longs;trali vltra Solis, <expan abbr="anni&qacute;">annique</expan>; vias con&longs;titutus <lb/>non appareret, cau&longs;am referebat Hippocrates paruitatem circuli, qucm <lb/>motu diurno cometa de&longs;cribebat, ob quam adeò parum &longs;upra horizontem <lb/>attolleretur, vt <expan abbr="nõ">non</expan> po&longs;&longs;et vi&longs;us no&longs;ter ab ip&longs;o ad Solem reflecti; quod &longs;ecun­ <lb/>dum ip&longs;um erat nece&longs;&longs;arium ad cometarum apparitionem. </s> <s id="id.001661">I oquitur igitur <lb/>Hippocrates de circulis, quos diurna conuer&longs;ione cometes circumducir, qui <lb/>omninò &longs;imiles &longs;unt ijs, quos etiam Sol, <expan abbr="reliqua&qacute;">reliquaque</expan>; a&longs;tra eodem motu de &longs;i­ <lb/>gnant. </s> <s id="id.001662">qui quidem omnes in no&longs;tr a &longs;phæra obliqua ita &longs;e habent, vt ij, qui <lb/>&longs;unt vltra æquatorem ad Capricorm tropicum, minus &longs;upra horizontem <lb/>extent, quàm infra de primantur, & tanto minus, quanto magis ab æquato­ <lb/>re in auftrum recedunt: contra verò faciunt, qui citra æquatorem ad Can­ <lb/>cri conuer&longs;ionem co&longs;&longs;ocantur, quanto enim magis ab æquatore in boream <lb/>remouentur, tantò eorum &longs;ectio, quæ e&longs;t &longs;upra horizontem, maior e&longs;t ea, <lb/>quæ infra horizontem latet. </s> <s id="id.001663">quæ quidem omnia clara &longs;unt adhibita &longs;phæra <lb/>materiali, quam &longs;i ad tuam poli eleuationem accommodaueris, illicò vi­ <lb/>debis tropici, Cancri &longs;ectionem, quæ e&longs;t &longs;upra horizontem multo maiorem <lb/>ea, quæ e&longs;t infra. </s> <s id="id.001664">oppo&longs;itum verò in altero Capricorni tropico, cuius mini­ <lb/>mam portionem &longs;upra, maximam verò infra horizontem exi&longs;tere videbis. <lb/></s> <s id="id.001665">Idem proportionaliter imaginari debes de circulis, quos cometa tam vltra <lb/>Capricornum, quàm citra Cancrum delineat; nam eorum, qui &longs;unt vltra <lb/>Capricornum ad au&longs;trum minores adhuc &longs;ectiones &longs;upra horizontem exi­ <lb/>&longs;terent, quàm opus &longs;it ad cometen &longs;pectandum. </s> <s id="id.001666"><expan abbr="At&qacute;ue">Atque</expan> hæc cau&longs;a e&longs;t ex &longs;en­ <lb/>tentia Hippocr. <!-- REMOVE S-->cur in illa au&longs;trali plaga <expan abbr="nũquam">nunquam</expan> cometes effulgeat. </s> <s id="id.001667">è con­ <lb/>trario autem, quia ad boream &longs;ectiones illæ maximæ &longs;unt, <expan abbr="aptæ&qacute;">aptæque</expan>; ad refra­ <lb/>ctionem vi&longs;us no&longs;tri v&longs;que ad Solem, idcircò in hac mundi parte cometas <lb/>con&longs;picere &longs;olemus. </s> <s id="id.001668">Reliqua Vicomercatus, <expan abbr="atq;">atque</expan> Alexand. <!-- REMOVE S-->optimè expli­ <lb/>cant, quos tu con&longs;ule, ne actum agatur.</s> </p></chap><chap><p type="head"> <s id="id.001669"><emph type="italics"/>In cap. 4. &longs;ummæ 2. lib. 1. Meteor. <!-- REMOVE S-->de Cometis.<emph.end type="italics"/><!-- KEEP S--></s> </p><p type="main"> <s id="id.001670"><arrow.to.target n="marg136"/></s></p><p type="margin"> <s id="id.001671"><margin.target id="marg136"/>136</s></p><p type="main"> <s id="id.001672">In præ&longs;enti cap. <!-- REMOVE S-->Ari&longs;t. &longs;uam de Cometis &longs;ententiam exponit: Come­ <lb/>tam nimirum infra Lunam in elementari mundo procreari, & ignitum <lb/>quoddam Meteoron, ex lenta, pingui, <expan abbr="&longs;icca&qacute;">&longs;iccaque</expan>; materia à terra in &longs;u­ <lb/>premam aeris regionem attracta, exi&longs;tere; <expan abbr="ibi&qacute;">ibique</expan>; rapti aeris calore, <lb/>vel elementi ignis (quod illic e&longs;&longs;e putat) vicinitate, vel etiam vi a&longs;trorum <lb/>incendi, <expan abbr="atq;">atque</expan> impelli. </s> <s id="id.001673">Hanć porrò opinionem & &longs;i probabilibus tantum ra­ <lb/>tionibus confirmatam vulgò tamen <expan abbr="v&longs;q;">v&longs;que</expan> ad hanc diem receptam, cum fal­ <lb/>&longs;am e&longs;&longs;e a&longs;tronomi exi&longs;timent, non erit abs re rationes eas ex &longs;ecundo pro­ <lb/>gymn. </s> <s id="id.001674">Tichonis volumine, de&longs;umptas hic breuiter referre, quibus a&longs;trono­ <lb/>mus ille eos &longs;upra Lunam in ætherea regione collocauit: quas quidem ra­ <lb/>tiones ille ex diuturnis ob&longs;eruationibus per exqui&longs;ita organa factis adinue­ <lb/>nit: ea&longs;que Mathematicis linearum, ac numerorum demon&longs;trationibus <lb/>explicauit.</s></p><p type="main"> <s id="id.001675">Prima. <!-- KEEP S--></s> <s id="id.001676">&longs;ed vt ab auctoritate, in quam obiter incidimus <expan abbr="initiũ">initium</expan> faciamus, <lb/>non e&longs;t exi&longs;timandum nonnuilos &longs;olum ex recentioribus id con&longs;tanter a&longs;&longs;e­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.092.jpg" pagenum="92"/>uera&longs;&longs;e, &longs;ed &longs;uperiori etiam ætate id ip&longs;um Hieron. </s> <s id="id.001677">Cardan. <!-- REMOVE S-->libro de &longs;ubtili­ <lb/>tate conatus e&longs;t, <expan abbr="neq;">neque</expan> irrito conatu, demon&longs;trare; qui præterea idem cum <lb/>&longs;e ip &longs;o &longs;en&longs;i&longs;&longs;e ait Albumazar. </s> <s id="id.001678">quibus etiam ex antiquis Seneca annumeran­ <lb/>dus e&longs;t. </s> <s id="id.001679">pr&ecedil;dicti autem recentiores omnes varijs demon&longs;trationibus ex ac­ <lb/>curata ob&longs;eruatione erutis illud certò certius con&longs;irmare contendunt: <expan abbr="id&qacute;">idque</expan>; <lb/>non in vno dumtaxat, &longs;ed in <expan abbr="quinq;">quinque</expan> cometis; quorum demon&longs;trationes apud <lb/>Tychonem partim in progymn. </s> <s id="id.001680">partim in epi&longs;t. <!-- REMOVE S-->fu&longs;ius explicatas reperies.</s> </p><p type="main"> <s id="id.001681">2. Quarum poti&longs;&longs;ima illa e&longs;t, quæ ex parallaxi, &longs;eu a&longs;pectus diuer&longs;itate <lb/>de&longs;umitur, certi&longs;&longs;imum enim e&longs;t lumen illud e&longs;&longs;e altero &longs;ublimius, quod mi­ <lb/>norem exhibet parallaxim: expertos autem &longs;e e&longs;&longs;e hi omnes, affirmant ho­ <lb/>&longs;ce quinque cometas multò minorem pati parallaxim, quam Lunam; imò <lb/>quempiam minorem, quàm Sol ip&longs;e patiatur, quo po&longs;ito manife&longs;tè conuin­ <lb/>ceretur eos omnes &longs;upra Lunam in ætherea regione efful&longs;i&longs;&longs;e.</s></p><p type="main"> <s id="id.001682">3. Ratio, qua etiam ante nouas ob&longs;eruationes vti &longs;olebant, de&longs;umitur <lb/>ex motu cometæ diurno, quo &longs;cilicet oritur, & occidit, quemadmodum cæ­ <lb/>tera &longs;ydera, hoc e&longs;t &longs;patio 24. horarum diurnam conuer&longs;ionem circa totam <lb/>terram ab&longs;oluit. </s> <s id="id.001683">&longs;i igitur comete e&longs;&longs;et in &longs;ublimiori aeris regione, vbi cæte­ <lb/>ra ignita meteora collocantur, <expan abbr="moueretur&qacute;">mouereturque</expan>; diurno motu circa terram, &longs;e­ <lb/>queretur nece&longs;&longs;ariò eum tanta velocitate videri à nobis circumferri, vt po­ <lb/>tius fulgor quidam, &longs;eu radius pertran&longs;iens ab oriente in occidentem appa­ <lb/>reret, quam &longs;tella qu&ecedil;dam: <expan abbr="id&qacute;">idque</expan>; propter propinquitatem; a&longs;tra enim ob ni­ <lb/>miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.</s></p><figure id="id.009.01.092.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.092.1.jpg" place="text"/><p type="main"> <s id="id.001684">Quod melius ex &longs;equenti figura <lb/><expan abbr="cõuincitur">conuincitur</expan>, vbi circulus interior e&longs;t <lb/>terra, cuius &longs;emidiameter A B. cir­ <lb/>culus verò exterior e&longs;t cometæ gy­ <lb/>rus, quem ip&longs;e &longs;patio 24. horarum <lb/>percurrit, qui &longs;ecundum veram pro­ <lb/>portionem deberet adhuc ip&longs;i terræ <lb/>propinquior, ac proinde minor e&longs;&longs;e, <lb/>iuxta aeris &longs;upremam partem. </s> <s id="id.001685">hori­ <lb/>zon e&longs;t recta D C, tangens terram in <lb/>B, vbi e&longs;t oculus no&longs;ter, qui nihil in­ <lb/>fra ip&longs;am D C, videre pote&longs;t; quare <lb/>&longs;i cometa 24. horarum totum gyrum <lb/>D C E, percurrit, non videbitur, ni&longs;i <lb/>quando percurret portionem D C, <lb/>&longs;upra horizontem; quæ quidem por­ <lb/>tio, <expan abbr="neq;">neque</expan> &longs;emihoræ re&longs;ponderet, &longs;i &longs;i­ <lb/>gura iuxta veram proportionem con&longs;trueretur. </s> <s id="id.001686">experientia tamen con&longs;tat, <lb/>cometas videri &longs;upra horizontem tot horis, quot &longs;tellæ fixæ, &longs;ub quibus mo­ <lb/>uentur: non ergo e&longs;t in &longs;upremo aere. </s> <s id="id.001687">Quod &longs;i &longs;iat figura, in qua exterior <lb/>cometæ ambitus adeò magnus &longs;it, vt ip&longs;ius portio D C, &longs;upra horizontem <lb/>exi&longs;tens, re&longs;pondeat tempori, quo cometa &longs;upra no&longs;trum pariter horizon­ <lb/>tem &longs;pectatur, ea figura terræ &longs;emidiametrum A B. toties multiplicabit, vt <lb/>ip&longs;i Lunæ circuitui proximè accedat.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.093.jpg" pagenum="93"/><p type="main"> <s id="id.001688">Præterea aiunt, quis &longs;anæ mentis dixerit, Meteoron vlium ex materia <lb/>vaga, ac fluxa con&longs;tans, po&longs;&longs;e tanta pernicitate moueri, vt diurnam con­ <lb/>uer&longs;ionem ab&longs;oluat? </s> <s id="id.001689">vnde illi motus i&longs;te? </s> <s id="id.001690">præ&longs;ertim cum videamus cætera <lb/>ignita meteora e&longs;&longs;e ad modum temporanea, <expan abbr="atq;">atque</expan> euanida.</s></p><p type="main"> <s id="id.001691">4. Comprobationem nobis &longs;uppeditant ex via, &longs;eu ductus circuli, quem <lb/>toto durationis tempore proprio cur&longs;u de&longs;ignarunt: prædicti <expan abbr="namq;">namque</expan> quin­ <lb/>que cometæ motu &longs;ibi proprio, quo ab occidente non omninò orientem <lb/>ver&longs;us, &longs;ed ad aquilonem deflectentes ab initio &longs;uæ apparitionis, <expan abbr="v&longs;q;">v&longs;que</expan> ad vl­ <lb/>timum fiuem exqui&longs;iti&longs;&longs;imè portionem circuli maximi in c&ecedil;lo de&longs;ignarunt; <lb/>non aiiter quàm Sol proprio motu per eclypticam in cœlo mundi &longs;phæram <lb/>in duo æqualia diuidentem de&longs;cribit. </s> <s id="id.001692">necnon aliter ac Luna &longs;uum iter per <lb/>circulum maximum cœlum bifariam diuidentem perficit. </s> <s id="id.001693">quapropter co­ <lb/>metas ho&longs;ce <expan abbr="nõ">non</expan> minus quam Sol, vel Luna in ip&longs;o æthere &longs;patiatos e&longs;&longs;e con­ <lb/>tendunt. </s> <s id="id.001694">qui enim, aiunt, fieri potui&longs;&longs;et, &longs;i in mundo elementari flagra&longs;&longs;ent, <lb/>vt tam regulari, <expan abbr="atq;">atque</expan> con&longs;tanti ductu circuli maximi portionem tam exactè <lb/>delinea&longs;&longs;ent, quam quidem inter elementa vagum, <expan abbr="atq;">atque</expan> in&longs;tabilem pro ma­ <lb/>teriæ in&longs;tabilitate exercere debui&longs;&longs;ent?</s></p><p type="main"> <s id="id.001695">5. Adde, quod in maximo hoc circulo de&longs;cribendo, etiam &longs;i inæquali ve­ <lb/>locitate vi&longs;i &longs;int moueri, inæqualitatem tamen illam regularem <expan abbr="vbiq;">vbique</expan> &longs;em­ <lb/>per &longs;eruauerunt, in principio quidem velociores, deinde &longs;ucce&longs;&longs;iuè, & pro­ <lb/>portionaliter velocitatem illam &longs;imili analogia &longs;emper &longs;eruata <expan abbr="inhibuerũt">inhibuerunt</expan>, <lb/>nullo igitur pacto inordinatam inæqualitatem, qua à tardiore motu &longs;ubito <lb/>in celeriorem, & rur&longs;us &longs;tatim ab hoc in <expan abbr="illũ">illum</expan> pro&longs;ilirent exhibuerunt: prout <lb/>omnia Meteora, quæ in mundi parte elementari ex flammanti materia ge­ <lb/>nerantur, talem di&longs;parem, <expan abbr="atq;">atque</expan> incon&longs;tantem motum obtinere cernuntur.</s></p><p type="main"> <s id="id.001696">6. Argumento præterea e&longs;t cometas ho&longs;ce minimè elementares fui&longs;&longs;e, <lb/>quod hic eorum proprius motus, quo maximo illo tramite ferebantur, nua­ <lb/>quam tantus fuit, vt proprium Lunæ motum, vel tardi&longs;&longs;imum adæquauerit, <lb/>quæ quidem cum lenti&longs;&longs;ima e&longs;t plus denis gradibus vna die promouetur; <lb/>cum tamen cometæ initio cum veloci&longs;&longs;imi &longs;unt non multum vltra quinos <lb/>gradus diurno motu progre&longs;&longs;i &longs;int, vt ob id longè &longs;upra Lunam cur&longs;um &longs;uum <lb/>ab&longs;olui&longs;&longs;e manife&longs;tè comprobari po&longs;&longs;it: quo enim &longs;ydera magis à terra at­ <lb/>tolluntur, <expan abbr="octauæ&qacute;">octauæque</expan>; &longs;phæræ propius accedunt, eò tardioribus proprijs la­ <lb/>tionibus proferuntur: ita vt &longs;teilæ i&longs;tæ cœlo ad&longs;cititiæ &longs;upra Lunam admo­ <lb/>dum euehendæ videantur. </s> <s id="id.001697">Quod &longs;i in &longs;uprema aeris regione con&longs;lagrarent, <lb/>qua nam ratione vnà cum toto cœlo diurnam conuer&longs;ionem ab&longs;olui&longs;&longs;ent: <lb/><expan abbr="neq;">neque</expan> enim putandum e&longs;t &longs;upremum hunc aeris limbum eadem pernecitate, <lb/>qua cœle&longs;tes orbes, verum minori admodum imò tardi&longs;&longs;imè à diurno mo­ <lb/>tu, &longs;i tamen eo rapitur circumduci.</s></p><p type="main"> <s id="id.001698">7. Tandem argumentum ex ip&longs;orum duratione de&longs;umatur. </s> <s id="id.001699">cætera nam­ <lb/>que meteora &longs;tatim <expan abbr="atq;">atque</expan> apparuerint, veluti temporanea pror&longs;us, <expan abbr="atq;">atque</expan> eua­ <lb/>nida extinguuntur: At verò cometæ ad men&longs;em aliquando integrum per­ <lb/>&longs;euerant. </s> <s id="id.001700">quì igitur fieri potuerit, vt in hac corruptibili <expan abbr="mũdi">mundi</expan> parte ex ma­ <lb/>teria adeò &longs;luxa, & vaga, quam illis Ari&longs;teteles &longs;upponit, tandiu perdura­ <lb/>re potui&longs;&longs;ent.</s></p><p type="main"> <s id="id.001701"><expan abbr="Atq;">Atque</expan> hæ &longs;unt rationes, quibus plurimi a&longs;tronomorum recentiorum, co­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.094.jpg" pagenum="94"/>metas ho&longs;ce motum æthereæ regioni conformem, contrà quam Ari&longs;t. opi­ <lb/>natus e&longs;t, obtinui&longs;&longs;e, munifeftum e&longs;&longs;e volunt; ac proinde eorum locum, & <lb/>cur&longs;um in cœle&longs;ti mundi parte extiti&longs;&longs;e, &longs;e comproba&longs;&longs;e exi&longs;timant: qua de <lb/>re prudentis Lectoris e&longs;to iudicium: <expan abbr="neq;">neque</expan> enim, vt ille cecinit, no&longs;trum e&longs;t, <lb/>tantas componere lites.</s></p><p type="main"> <s id="id.001702">Verumenimuerò Peripatetica omnis &longs;chola reclamat; Cœlum e&longs;t inge­ <lb/>nerabile, & incorruptibile, mhil igitur noui cœlo pote&longs;t accidere. </s> <s id="id.001703">&longs;ed age <lb/>re&longs;pondent, nonne omnium a&longs;tronomorum con&longs;en&longs;u &longs;tellæ tres nouæ no&longs;tro <lb/>hoc &longs;æculo in cœlo toti mundo con&longs;picuæ illuxerunt? </s> <s id="id.001704"><expan abbr="eas&qacute;">easque</expan>; in octaua &longs;phæ­ <lb/>ra re&longs;edi&longs;&longs;e conftans e&longs;t omnium a&longs;&longs;ertio? </s> <s id="id.001705">quarum prior anno 1572. in con­ <lb/>&longs;tellatione Ca&longs;&longs;iopeæ apparuit. </s> <s id="id.001706">Secunda anno 1600. in Cygno, quæ nec dum <lb/>extinguitur. </s> <s id="id.001707">Tertia anno 1604. inter Sagittarij &longs;tellas vi&longs;a e&longs;t, de quibus vi­ <lb/>de P. <!-- REMOVE S-->Clauium in &longs;phæra breuiter de illis tractantem: aut &longs;i mauis, & vacat, <lb/>vide quoad primam primum volumen progymna&longs;matum Tychonis Brahe, <lb/>vbi etiam aliorum a&longs;tronomorum de eadem certi&longs;&longs;imas commentationes <lb/>reperies. </s> <s id="id.001708">con&longs;ule etiam de reliquis duabus Ioannis Kepleri Cæ&longs;areæ Maie­ <lb/>&longs;tatis Mathematici commentaria; & coactus libenter fateberis noui ali­ <lb/>quid cœlo aduenire po&longs;&longs;e.</s></p><p type="main"> <s id="id.001709">Po&longs;tremò tandem po&longs;&longs;et qui&longs;piam in hunc <expan abbr="modũ">modum</expan> opponere: etiam &longs;i con­ <lb/>&longs;tet <expan abbr="quinq;">quinque</expan> cometas c&ecedil;lo oberra&longs;&longs;e, non propterea dicemus reliquos omnes <lb/>e&longs;&longs;e pariter cœle&longs;tes, <expan abbr="nullum&qacute;">nullumque</expan>; proinde &longs;ublunarem. </s> <s id="id.001710">Huic memorati A&longs;tro­ <lb/>nomi &longs;ic re&longs;ponderent; id quidem mathematica, & infallibili ratione non <lb/>colligi, imò aliquot parum infra Lunam extiti&longs;&longs;e, non omninò negandum <lb/>videri: at verò in &longs;uperiori aeris plaga, in tam fluxa, ac in&longs;tabili mundi par­ <lb/>te, cometas vnquam efful&longs;i&longs;&longs;e, nemo &longs;ibi ob allatas rationes meritò per&longs;ua­ <lb/>dere po&longs;&longs;e.</s></p><p type="main"> <s id="id.001711"><arrow.to.target n="marg137"/></s></p><p type="margin"> <s id="id.001712"><margin.target id="marg137"/>137</s></p><p type="main"> <s id="id.001713">Summæ 2. cap. 5. <emph type="italics"/>(Ad hæc autem &longs;i quemadmodum o&longs;tenditur in ijs, quæ cir­ <lb/>ca Astrologiam &longs;peculationibus, Solis magnitudo maior e&longs;t quàm terræ; & diftax­ <lb/>tia multò maior a&longs;trorum ad terram quàm So is; &longs;icut Solis ad terram quàm Lu­ <lb/>næ; non <expan abbr="vtiq;">vtique</expan> longè alicubi à terra conus, qui à Sole, conijciet radios, <expan abbr="neq;">neque</expan> vtique <lb/>vmbraterræ, quæ vocatur nox, erit apud astra; &longs;ed nece&longs;&longs;e Solem omnia a&longs;tra cir­ <lb/>cun&longs;picere, & nulli ip&longs;orum terram ob&longs;istere)<emph.end type="italics"/> ex dictis &longs;umma 1. cap. 3. huius, <lb/>& ex figura ibi de&longs;cripta, facilè e&longs;t intelligere præ&longs;entem locum; nam cum <lb/>Sol &longs;it multò maior terra, vt ibi probatur, ac minus di&longs;ter à terra quàm fixæ <lb/>&longs;tellæ, magis tamen quàm Luna, vt patet ex &longs;olari eclyp&longs;i, &longs;equitur nece&longs;&longs;a­ <lb/>riò vmbram terræ, quæ nox e&longs;t ip&longs;a, effici turbinatam, & valdè procul à ter­ <lb/>ra acumen coni vmbræ a&longs;cendet, &longs;ed paulò &longs;upra Lunam conus hic vmbræ <lb/>permittet radios Solis &longs;e ip&longs;um ambientes iterum &longs;imul committi, quod il­ <lb/>lis verbis <emph type="italics"/>(Conijciet radios)<emph.end type="italics"/> ide&longs;t committet radios expre&longs;&longs;it Ari&longs;t. cum igi­ <lb/>tur vmbra apud Lunam &longs;it &longs;atis gracilis, breui &longs;upra Lunam de&longs;inet, neque <lb/>vllo pacto ad affixa &longs;ydera protendetur, <expan abbr="neq;">neque</expan> illis renebras offundet. </s> <s id="id.001714">quod <lb/>etiam experientia confirmat, cum nunquam a&longs;tra illa, quæ Soli opponuntur, <lb/><expan abbr="quæ&qacute;">quæque</expan>; vertex vmb æ collimat, vllam <expan abbr="patiãtur">patiantur</expan> eclyp&longs;im. </s> <s id="id.001715">quare &longs;ine vllo ter­ <lb/>ræ impedimento Sol pote&longs;t af&longs;ixa omuia &longs;ydera perlu&longs;lrare. </s> <s id="id.001716">Exactiores ha­ <lb/>rum rerum demon&longs;trationes &longs;unt alterius loci.</s></p><p type="main"> <s id="id.001717"><arrow.to.target n="marg138"/></s></p><p type="margin"> <s id="id.001718"><margin.target id="marg138"/>138</s></p><p type="main"> <s id="id.001719">Eodem cap. <emph type="italics"/>(Amplius autem e&longs;t tertia quædam opinio de ip&longs;o, dicunt enim<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.095.jpg" pagenum="95"/><emph type="italics"/>quidam lac e&longs;&longs;e reflexionem no&longs;tri vi&longs;us ad Solem; &longs;icut & &longs;tellam comatam; im­ <lb/>po&longs;&longs;ibile autem e&longs;t & hoc, &longs;i enim videns quieuerit & &longs;peculum, & quod videtur <lb/>omne in eodem puncto &longs;peculi eadem apparebit <expan abbr="vtiq;">vtique</expan> pars imaginis, &longs;i autem mo­ <lb/>ueatur &longs;peculum, & quod videtur, in eadem quidem di&longs;tantia ad videns, & quie­ <lb/>&longs;cens; ad inuicem autem <expan abbr="neq;">neque</expan> æquè velociter, <expan abbr="neq;">neque</expan> in eadem &longs;emper di&longs;tantia im­ <lb/>po&longs;&longs;ibile eandem imaginem in eadem e&longs;&longs;e parte &longs;peculi. </s> <s id="id.001720">Quæ autem in lactis circu­ <lb/>lo feruntur a&longs;tra, & Sol, ad quem fit reflexio, mouentur manentibus nobis, & &longs;i­ <lb/>militer, & æqualiter ad nos di&longs;tantia; à &longs;e ip&longs;is autem non æqualiter: aliquando <lb/>enim medijs noctibus Delphin oritur, aliquando verò diluculo. </s> <s id="id.001721">partes autem lactis <lb/>eædem manent in vnoquoque; atqui non oportebat, &longs;i erat imago, &longs;ed non in ei&longs;dem <lb/>adhuc e&longs;&longs;et hæc pa&longs;&longs;io locis)<emph.end type="italics"/> in his Ari&longs;t. confutat opinionem dicentium Gala­ <lb/>xiam apparere per quandam reflexionem vi&longs;us no&longs;tri ab illa parte c&ecedil;li, ceu, <lb/>ex quodam &longs;peculo ad Solem: probat autem hoc e&longs;&longs;e impo &longs;&longs;ibile ratione <lb/>de&longs;umpta ex parte Optices, quæ dicitur Catoptrica, &longs;iue &longs;pecularia, quia <lb/>tractat de vi&longs;ione reflexa, quæ fit mediante &longs;peculo, quam quidem rationem <lb/>&longs;i vellem mathematicè explicare, longa nimis, ac præter in&longs;titutum fieret <lb/>tractatio. </s> <s id="id.001722">Pauca tamen addam, quæ Ari&longs;totelis <expan abbr="&longs;ent&etilde;tiam">&longs;ententiam</expan> &longs;atis per&longs;picuam <lb/>reddant. </s> <s id="id.001723">&longs;i igitur inquit, Galaxia nihil aliud e&longs;&longs;et quàm reflexio no&longs;tri vi&longs;us <lb/>ex illa cœli parte, in qua ip&longs;a apparet tanquam ex &longs;peculo ad Solem, ita vt <lb/>nihil aliud ip&longs;a e&longs;&longs;et, quàm Sol vi&longs;us per reflexionem exilla cœli parte tan­ <lb/>quam &longs;peculo; &longs;equeretur eam non &longs;emper in eadem cœli parte apparere, <lb/>&longs;ed modo in vna, modo in alia, ita vt &longs;patio vnius anni totum cœlum perua­ <lb/>garetur: quod tamen non accidit. </s> <s id="id.001724">quod autem illud con&longs;equatur manife­ <lb/>&longs;tum e&longs;&longs;e pote&longs;t ex ob&longs;eruatione eorum, quæ ex &longs;peculis videntur: tunc enim <lb/>res per &longs;pe culum vi&longs;a in eadem &longs;peculi parte apparet, quando & videns, & <lb/>&longs;peculum, & obiectum immota manent: quod &longs;i & &longs;peculum, & obiectum ad <lb/>inuicem accedant, vel recedant, &longs;eruata tamen eadem ab in&longs;pectore di&longs;tan­ <lb/>tia, nullo modo fieri pote&longs;t, vt eadem imago, in eadem &longs;peculi parte &longs;pe­ <lb/>ctanti videatur, ni&longs;i obiectum &longs;peculo per eandem lineam accedat, &longs;ecun­ <lb/>dum quam illi incidebat. </s> <s id="id.001725">At verò partibus illis lactei circuli, &longs;iue a&longs;tris, quæ <lb/>in eo fulgent, Sol perpetuò accedit, vel recedit, <expan abbr="neq;">neque</expan> per lineam incidentiæ <lb/><expan abbr="eãdem">eandem</expan>, &longs;eruata tamen eadem à nobis di&longs;tatia, quod quidem inde patet, quia <lb/>Delphini con&longs;tellatio, qui in ip&longs;o ferè lacte exi&longs;tit, <expan abbr="aliquãdo">aliquando</expan> medijs noctibus, <lb/>aliquando verò mane, aliquando etiam ve&longs;peri oritur; quod inde accidit, <lb/>quia illi Sol modò appropinquat, modò coniungitur, modò ab eo recedit, <lb/>quare nece&longs;&longs;e e&longs;&longs;et, vt lacteus orbis, non &longs;emper in ij&longs;dem locis, &longs;ed perpe­ <lb/>tuò in alijs, <expan abbr="atq;">atque</expan> alijs cernerecur, cuius tamen contrarium videmus. </s> <s id="id.001726">ex qui­ <lb/>bus con&longs;tat fal&longs;am omninò e&longs;&longs;e eorum &longs;ententiam, qui Galaxiam per huiu&longs;­ <lb/>modi re&longs;lexionem fieri opinabantur. </s> <s id="id.001727">Quæ dicta &longs;unt de &longs;peculo, & obiecto <lb/>&longs;atius e&longs;t a&longs;&longs;umpto aliquo &longs;peculo experiri, quàm ea pluribus ob&longs;curare: qua <lb/>etiam experientia Ari&longs;t. ratio confirmabiaur.</s></p><p type="main"> <s id="id.001728"><arrow.to.target n="marg139"/></s></p><p type="margin"> <s id="id.001729"><margin.target id="marg139"/>139</s></p><p type="main"> <s id="id.001730">Ibidem <emph type="italics"/>(Quæ autem in lactis cir culo feruntur astra, & Sol, ad quem fit refle­ <lb/>xio, mouentur mancntibus nobis, & &longs;imiliter, & æqualiter ad nos di&longs;t antia à &longs;e <lb/>ip&longs;is autem non æqualiter)<emph.end type="italics"/> quæ hic ab Ari&longs;totele dicuntur <expan abbr="nõ">non</expan> &longs;unt v&longs;quequae; <lb/>vera propter apogæum, ac porigæum Solis, quæ quidem duo ab omnibus <lb/>a&longs;tronomis a&longs;&longs;eruatur: quando igitur Sol e&longs;t in apogæo, maiori multo in­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.096.jpg" pagenum="96"/>teruallo di&longs;tat à nobis, quàm quando e&longs;t in perigæo, interuallum enim illud <lb/>con&longs;tat diametris terræ duobus, & quadraginta, hoc e&longs;t milliarijs 208000. <lb/>ferè, ide&longs;t octonis millibus &longs;upra ducenta millia. </s> <s id="id.001731">quæ differentia facit vt Sol <lb/>manife&longs;tè appareat nobis minor apogæus, quàm perigæus. </s> <s id="id.001732">Sol præterea &longs;i­ <lb/>militer ip&longs;is inerrantibus &longs;tellis fit tantumdem modo remotior, modo pro­ <lb/>pinquior: &longs;ed fortè Ari&longs;t. i&longs;ta non occurrerunt, vel tunc temporis nondum <lb/>per&longs;pecta erant.</s></p><p type="main"> <s id="id.001733"><arrow.to.target n="marg140"/></s></p><p type="margin"> <s id="id.001734"><margin.target id="marg140"/>140</s></p><p type="main"> <s id="id.001735">Ibidem <emph type="italics"/>(Aliquando enim medijs noctibus Delphin oritur)<emph.end type="italics"/> vt probet, Gala­ <lb/>xiam non &longs;emper &longs;eruare à Sole di&longs;tantiam eandem, accipit tanquam huius <lb/>rei &longs;ignum, manife&longs;tum, quod Delphini con&longs;tellatio aliquando medijs no­ <lb/>ctibus oriatur &longs;upra horizontem, aliquando verò diluculo; non ideò tamen <lb/>putes hanc rationem &longs;upponere Delphinum e&longs;&longs;e in ip&longs;o lacteo circulo, quod <lb/>tamen verum non e&longs;t, non enim e&longs;t in Galaxia, &longs;ed tamen illi proximus, vt <lb/>noctu videre e&longs;t in cœlo, vel etiam &longs;i mauis in globo a&longs;tronomico: non ta­ <lb/>men ob id Ari&longs;t. ratio minus valida redditur, cum Delphinus &longs;emper Gala­ <lb/>xiæ eodem modo &longs;it proximus, <expan abbr="eo&qacute;">eoque</expan>; moto, ip&longs;a pariter moueatur.</s></p><p type="main"> <s id="id.001736"><arrow.to.target n="marg141"/></s></p><p type="margin"> <s id="id.001737"><margin.target id="marg141"/>141</s></p><p type="main"> <s id="id.001738">Summæ 2. cap. 6. Sunt qui velint Ari&longs;t. <!-- REMOVE S-->Galaxiam nihil aliud e&longs;&longs;e, quàm <lb/>quandam refractionem lucis &longs;tellarum illarum, quæ &longs;unt in ætherea Gala­ <lb/>xia, quæ inquam refractio fiat circa &longs;upremam aeris regionem ex occur&longs;u <lb/>exhalationum, quæ ibi perpetuò con&longs;eruantur, & vi earumdem &longs;tellarum <lb/>&longs;ur&longs;um &longs;emper attrahuntur, quæ refractio fiat ad eum modum, quo halo cir­ <lb/>ca Solem, & Lunam. <!-- KEEP S--></s> <s id="id.001739">& quemadmodum halo, &longs;iue area omnibus <expan abbr="vndecunq;">vndecunque</expan> <lb/>a&longs;picientibus &longs;emper videntur in eodem cœli loco, hoc e&longs;t è regione Solis, <lb/>vel Lunæ; &longs;imiliter Galaxia in aere omnibus <expan abbr="vndecunq;">vndecunque</expan> intuentibus appa­ <lb/>reat in eadem cœli parte, ide&longs;t ex aduersò eorumdem &longs;yderum, quæ cœle­ <lb/>&longs;tem lacteam viam conficiunt. </s> <s id="id.001740">Porrò qui &longs;ic mentem Ari&longs;t. exponunt, nul­ <lb/>lo modo po&longs;&longs;unt à Mathematicis redargui per rationem de&longs;umptam à di­ <lb/>uer&longs;itate a&longs;pectus (quam po&longs;tea explicabo) quamuis phy&longs;icis rationibus re­ <lb/>fellantur. </s> <s id="id.001741">Alij &longs;unt, quorum &longs;ententia magis videtur <expan abbr="improbãda">improbanda</expan>, cò quod <lb/>Ari&longs;t. &longs;ummum Philo&longs;ophum pueriliter in a&longs;tronomia lap&longs;um fateri cogan­ <lb/>tur. </s> <s id="id.001742">Exi&longs;timant hi Galaxiam hanc Ari&longs;totelicam nihil aliud e&longs;&longs;e, quàm ip­ <lb/>&longs;as tenues exhalationes in aere &longs;ubuectas, directèque infra &longs;tellas illas la­ <lb/>cteum circulum in cœlo con&longs;tituentes nobis obiectas. </s> <s id="id.001743">qui præter innumera, <lb/>ac magna ab&longs;urda è naturali Philo&longs;ophia petita, vnum maximum ex A&longs;tro­ <lb/>nomia, nempè ex diuer&longs;itate a&longs;pectus de&longs;umptum, nullo modo vitare po&longs;­ <lb/>&longs;unt; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; huiu&longs;modi, quia &longs;i lacteus hic circulus e&longs;&longs;et in aere, non ab om­ <lb/>nibus, <expan abbr="neq;">neque</expan> ex omni terræ loco per eadem &longs;ydera commeare cerneretur, &longs;ed <lb/>è diuer&longs;is, & præcipuè ab inuicem valde di&longs;&longs;itis, circa diuer&longs;a a&longs;tra &longs;e &longs;e ocu­ <lb/>lis no&longs;tris obijceret: at te&longs;timonio &longs;en&longs;us con&longs;tat, Galaxiam &longs;emper in eo­ <lb/>dem loco; <expan abbr="eadem&qacute;">eademque</expan>; à &longs;yderibus fixis di&longs;tantia albicare, ergò nullo modo <lb/>viam hanc in aere qua&longs;i pendulam fabricare debemus. </s> <s id="id.001744">rationem hanc di­ <lb/>uer&longs;itatis a&longs;pectus a&longs;tronomicè magis explicatam reperies apud Clauium <lb/>in &longs;phæra. </s> <s id="id.001745">Porrò hæc ratio quamuis adeo certa, ac no&longs;tra tempe&longs;tate vul­ <lb/>gata, parum tamen à nonnullis de rebus Meteorologicis commentaria con­ <lb/>farcinantibus intellecta, minimè eos ab&longs;terrere potuit, quin prædictam opi­ <lb/>nionem, non &longs;olum Ari&longs;toteli imponerent, verum etiam ip&longs;i <expan abbr="tãquam">tanquam</expan> veram <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.097.jpg" pagenum="97"/>a&longs;truerent: huiu&longs;modi patiuntur incommoda, qui <expan abbr="ab&longs;q;">ab&longs;que</expan> Mathematicarum <lb/>auxilio Philo&longs;ophiam aggrediuntur.</s></p><p type="main"> <s id="id.001746"><arrow.to.target n="marg142"/></s></p><p type="margin"> <s id="id.001747"><margin.target id="marg142"/>142</s></p><p type="main"> <s id="id.001748">Eodem cap. <emph type="italics"/>(Ad hæc autem locus plenus e&longs;t a&longs;tris maximis, & fulgidi&longs;&longs;imis, <lb/>& adhuc &longs;par&longs;is vocatis)<emph.end type="italics"/> non &longs;olum viam hanc lacteam a&longs;tris plurimis refer­ <lb/>ti&longs;&longs;imam e&longs;&longs;e videmus, &longs;ed præterea eandem &longs;tellarum admodum feracem <lb/>appellare licebit, &longs;i quidem &longs;tellæ omnes illæ nouæ, quæ no&longs;tra tempe&longs;tate <lb/>apparuerunt, omnes in hac via exortæ &longs;unt. </s> <s id="id.001749">prima enim anno 1572. efful&longs;it <lb/>in Ca&longs;&longs;iopea; altera anno 1600. in Cygno. </s> <s id="id.001750">tertia demum anno 1604. in Sa­ <lb/>gittario, quæ omnes con&longs;tellationes intra lacteum circulum continentur. <lb/></s> <s id="id.001751">Veri&longs;&longs;imum præterea e&longs;&longs;e hoc idem confirmatur in&longs;trumenti illius mirabi­ <lb/>lis auxilio, quod &longs;uperiori anno in Belgio excogitatum, & po&longs;tea in Italia <lb/>à Galilæo perfectius <expan abbr="redditũ">redditum</expan> e&longs;t, <expan abbr="quod&qacute;">quodque</expan>; ip&longs;e primum Italicè Cannocchiale, <lb/>Latinè verò, & quidem aptè à Græcis mutuato vocabulo alius Tele&longs;copium <lb/>appellauit: hoc inquam &longs;pecillo adhibito per&longs;picuum &longs;tatim fit non &longs;olum <lb/>in via lactea innumeras &longs;tellas contineri, verum quid ip&longs;a &longs;it, certò certius <lb/>con&longs;tat; &longs;ed &longs;atius e&longs;t ip&longs;ius Galilæi verba ex Nuncio &longs;ydereo referre: Quod <lb/>tertio inquit, loco à nobis fuit ob&longs;eruatum e&longs;t ip&longs;iu&longs;met lactei circuli e&longs;&longs;en­ <lb/>tia, &longs;en materies, quam Tele&longs;copij beneficio adeò ad &longs;en&longs;um licet intueri, <lb/>vt & altercationes omnes, quæ per tot &longs;æcula Philo&longs;ophos excruciarunt ab <lb/>oculata certitudine <expan abbr="dirimãtur">dirimantur</expan>, <expan abbr="nos&qacute;">nosque</expan>; à verbo&longs;is di&longs;putationibus liberemur: <lb/>e&longs;t enim Galaxia nihil aliud, quàm innumerarum &longs;tellarum coaceruatim <lb/>con&longs;itarum congeries, in <expan abbr="quãcunq;">quancunque</expan> enim regionem illius &longs;pecillum dirigas, <lb/>&longs;tatim &longs;tellarum ingens fre quentia &longs;e &longs;e in con&longs;pectum profert, <expan abbr="quarũ">quarum</expan> com­ <lb/>plures &longs;atis magnæ, ac valdè con&longs;picuæ videntur; &longs;ed exiguarum multitudo <lb/>pror&longs;us inexplorabilis e&longs;t. </s> <s id="id.001752">hæc ille.</s></p><p type="main"> <s id="id.001753"><arrow.to.target n="marg143"/></s></p><p type="margin"> <s id="id.001754"><margin.target id="marg143"/>143</s></p><p type="main"> <s id="id.001755">Eodem cap. <emph type="italics"/>(Con&longs;ideretur autem & circulus, & quæ &longs;unt in ip&longs;o a&longs;tra ex de­ <lb/>&longs;criptione)<emph.end type="italics"/> id e&longs;t, con&longs;ideretur Galaxia, & a&longs;tra ip&longs;ius in&longs;piciantur diligenter <lb/>ex de&longs;criptione alicuius Globi a&longs;tronomici, in quo &longs;olent A&longs;tronomi omnes <lb/>con&longs;tellationes, ac &longs;tellas &longs;uis locis reddere, <expan abbr="atq;">atque</expan> etiam lacteum ip&longs;um cir­ <lb/>culum graphicè effingere. </s> <s id="id.001756">huiu&longs;modi globum veteres &longs;ph&ecedil;ram Aratæam di­ <lb/>cebant ab Arato Poeta græco, qui <expan abbr="cõ&longs;tellationes">con&longs;tellationes</expan> omnes carmine pro&longs;e quu­ <lb/>tus e&longs;t, ac proinde globum hunc ordine expo&longs;uit:</s></p><p type="main"> <s id="id.001757"><arrow.to.target n="marg144"/></s></p><p type="margin"> <s id="id.001758"><margin.target id="marg144"/>144</s></p><p type="main"> <s id="id.001759">Eodem cap. <emph type="italics"/>(Spar&longs;a autem vocata)<emph.end type="italics"/> putò &longs;par&longs;a hæc &longs;ydera illa e&longs;&longs;e, quæ <lb/>recentiores informia appellant, eò quod ad aliorum a&longs;teri&longs;morum formas <lb/>minimè reuocentur.</s></p><p type="main"> <s id="id.001760"><arrow.to.target n="marg145"/></s></p><p type="margin"> <s id="id.001761"><margin.target id="marg145"/>145</s></p><p type="main"> <s id="id.001762">Summa 4. cap. 1. <emph type="italics"/>(In A&longs;ia igitur plurimi ex Parna&longs;&longs;o vocato monte videntur <lb/>&longs;tuentes)<emph.end type="italics"/> rectè dubitat Alexander, qua ratione mons Parna&longs;&longs;us ab Ari&longs;t. po­ <lb/>natur in A&longs;ia, cum certò certius con&longs;tet, ip&longs;um in Græcia Europæ regione <lb/>&longs;itum e&longs;&longs;e. </s> <s id="id.001763">fortè legendum e&longs;t, vt vult Vicomercatus, ex Paropame&longs;&longs;o, non <lb/>autem ex Parna&longs;&longs;o, quamuis Græci codices aduer&longs;entur; Paropame&longs;&longs;um <lb/><expan abbr="namq;">namque</expan> Plinius, & Strabo in A&longs;ia collocant, <expan abbr="volunt&qacute;">voluntque</expan>; ip&longs;um e&longs;&longs;e iugum quod­ <lb/>dam montis Cauca&longs;i: Cauca&longs;um autem &longs;upra Pontum orifi, & <expan abbr="v&longs;q;">v&longs;que</expan> ad Hir­ <lb/>canum, & vltra mare per totam A&longs;iam &longs;e proferre, tradunt veteres Geo­ <lb/>graphi. </s> <s id="id.001764">vide The&longs;aurum geographicum Abrahami Ortelij. </s> <s id="id.001765">Strabo lib. 15. <lb/>&longs;ic: Indiam à &longs;eptentrione Tauri extrema terminant, ab Ariana v&longs;que in <lb/>orientale mare, quæ extrema indigenæ particulatim nominant Poropami&longs;­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.098.jpg" pagenum="98"/>&longs;um, Emodum, Imauum, & alijs nominibus: Macedones verò Cauca&longs;um <lb/>vocant.</s></p><p type="main"> <s id="id.001766"><arrow.to.target n="marg146"/></s></p><p type="margin"> <s id="id.001767"><margin.target id="marg146"/>146</s></p><p type="main"> <s id="id.001768">Ibidem <emph type="italics"/>(Apparet mare, quod e&longs;t extra)<emph.end type="italics"/> intelligit illud mare <expan abbr="Oceanũ">Oceanum</expan>, quod <lb/>Arabiam, ac Per&longs;iam alluit, <expan abbr="Indico&qacute;">Indicoque</expan>; Oceano committitur: <expan abbr="quod&qacute;">quodque</expan>; à pri­ <lb/>&longs;cis Geographis Rubrum mare appellatur, cuius alterum Rubrum mare, <lb/>quod inter Africam, & Arabiam &longs;e in&longs;inuat, e&longs;t quidam &longs;inus, quem nunc <lb/>communiter omnes Rubrum mare appellant. </s> <s id="id.001769">de illo inquam meritò intel­ <lb/>ligit Alexander, non de hoc Aegyptiaco, cum ex a&longs;pectu illius à monte Pa­ <lb/>ropame&longs;&longs;o, &longs;equatur ip&longs;um e&longs;&longs;e editi&longs;&longs;imum, quod non &longs;equeretur ex altero <lb/>ob illius propinquitatem. </s> <s id="id.001770">Dixit autem mare, quod e&longs;t extra, ide&longs;t extra <lb/>terram habitatam, ad di&longs;tinctionem maris Mediterranei, quod e&longs;t intra <lb/>terram habitatam, ac propterea Mediterraneum dictum e&longs;t.</s></p><p type="main"> <s id="id.001771"><arrow.to.target n="marg147"/></s></p><p type="margin"> <s id="id.001772"><margin.target id="marg147"/>147</s></p><p type="main"> <s id="id.001773">Ibidem <emph type="italics"/>(Ex hoc igitur fluunt & alij fluuij, & Bactrus, & Choa&longs;pes, & Ara­ <lb/>xes. </s> <s id="id.001774">ab hoc autem ab&longs;cinditur Tanais pars exi&longs;tens in Meotidem paludem fluit au­ <lb/>tem, & Indus ex ip&longs;o, omnium fluuiorum fluxio maxima)<emph.end type="italics"/> hæc omnia &longs;unt fal&longs;a, <lb/>& impo&longs;&longs;ibilia; nam cum Bactrus Bactrianam regionem irriget, quæ e&longs;t vl­ <lb/>tra Per&longs;iam, Choa&longs;pes verò Per&longs;iam ip&longs;am, Indus <expan abbr="deniq;">denique</expan> in India oriatur: <lb/>quì fieri pote&longs;t, vt in Regionibus adeò inuicem di&longs;&longs;itis orti fluuij ab eodem <lb/><expan abbr="quoq;">quoque</expan> Paropame&longs;&longs;o monte ortum ducant. </s> <s id="id.001775">nec minus fal&longs;um e&longs;t illud de Ta­ <lb/>nai, quod &longs;it qua&longs;i ip&longs;ius Araxis ramus quidam, Tanais enim ex Riphæis <lb/><expan abbr="mõtibus">montibus</expan> Scythiæ delabitur in Meotidem paludem longè longius ab Araxi. <lb/><!-- KEEP S--></s> <s id="id.001776"><expan abbr="eum&qacute;">eumque</expan>; terminum inter Europam, & A&longs;iam Geographi con&longs;tituunt, vnde <lb/>Diony&longs;ius Afer &longs;ic cecinit:</s></p><p type="main"> <s id="id.001777"><emph type="italics"/>Europam, <expan abbr="atq;">atque</expan> A&longs;iam Tanais di&longs;terminat amnis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001778">verùm huiu&longs;modi errata Ari&longs;t. <!-- REMOVE S--><expan abbr="atq;">atque</expan> adeò Geographis illius temporis con­ <lb/>donanda &longs;unt, cum nondum Geographia &longs;atis exculta e&longs;&longs;et.</s> </p><p type="head"> <s id="id.001779"><emph type="italics"/>De altitudine montis Cauca&longs;i.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001780"><arrow.to.target n="marg148"/></s></p><p type="margin"> <s id="id.001781"><margin.target id="marg148"/>148</s></p><p type="main"> <s id="id.001782">Eod. <!-- REMOVE S-->cap. <emph type="italics"/>(Cauca&longs;us autem maximus mons e&longs;t eorum qui ad ori<expan abbr="&etilde;tem">entem</expan> æ&longs;tiua­ <lb/>lem, & multitudine, & altitudine &longs;igna autem altitudinis quidem, quia <lb/>videtur & à vocatis Profundis, & à nauigantibus in Stagnum in&longs;uper il­ <lb/>lu&longs;trantur à Sole ip&longs;ius &longs;ummitates, v&longs;que ad tertiam partem nocte, & ab <lb/>aurora, & iterum a ve&longs;pera)<emph.end type="italics"/> Cauca&longs;us mons &longs;itus e&longs;t inter mare Euxinum, & <lb/>Ca&longs;pium, &longs;upra Cholchidem, & Iberiam regiones, vbi polus eleuatur 47. <lb/>circiter grad. <!-- REMOVE S-->ac re&longs;pectu Græciæ, & maris Euxini vergit ad eam mundi pla­ <lb/>gam, vnde illis æ&longs;tiuo tempore Sol oritur. </s> <s id="id.001783">ait Ari&longs;t. eum e&longs;&longs;e omnium mon­ <lb/>tium illius plagæ alti&longs;&longs;imum, quod probat primò, quia admodum à longè <lb/>cernitur, <expan abbr="nimirũ">nimirum</expan> ab illo Euxini loco, qui Profunda vocatur, eò quòd à Nau­ <lb/>tis nu&longs;quam ibi fundus reperiatur. </s> <s id="id.001784">& præterca à Nauigantibus in Stagnum, <lb/>&longs;iue in Meotidem paludem, quæ quidem loca minimùm di&longs;tant a Cauca&longs;o <lb/>560. milliaribus. </s> <s id="id.001785">Secundò, probat il ius altitudinem ex eo, quòd &longs;ummi­ <lb/>tates ip&longs;ius <expan abbr="v&longs;q;">v&longs;que</expan> ad tertiam partem nocte, & ve&longs;peri à Sole illu&longs;trentur. </s> <s id="id.001786">Lo­ <lb/>cum hunc fusè pertractat eruditi&longs;&longs;imus Iacobus Mazonius &longs;ectione 3. & 4. <lb/>de Comparatione Platonis, & Ari&longs;t. quo in opere plurima habet ex Mathe­ <lb/>maticis de&longs;umpta, quibus naturalem Philo&longs;ophiam mirificè illu&longs;trat, <expan abbr="mani-fe&longs;tum&qacute;">mani­ <lb/>fe&longs;tumque</expan>; reddit, quàm nece&longs;&longs;ariæ &longs;int Mathematicæ ad philo&longs;ophicæ veri­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.099.jpg" pagenum="99"/>tatis in&longs;pectionem. </s> <s id="id.001787">Is igitur &longs;ect. </s> <s id="id.001788">3. cap. 5. de hoc Ari&longs;t. loco &longs;ie loquitur: <lb/>hic locus diligenter expendendus videtur tum quia difficillimus e&longs;t, <expan abbr="tũ">tum</expan> quia <lb/>multis an&longs;am dedit reprehendendi Ari&longs;t. tanquam puerilia effutientem. </s> <s id="id.001789">tex­ <lb/>tus <expan abbr="itaq;">itaque</expan> Ari&longs;t. duplicem habet &longs;en&longs;um; alter à quo non abhorret <expan abbr="Alexãder">Alexander</expan>; <lb/>vt tertia illa pars ad montem referatur, qua&longs;i dicat, quod antequam Sol ima <lb/>montis illu&longs;tret, illuminat illius cacumen <expan abbr="v&longs;q;">v&longs;que</expan> ad tertiam montis partem: <lb/>&longs;ed hæc Mazonij expo&longs;itio nulla e&longs;t, cuiu&longs;libet enim montis etiam medio­ <lb/>cris altitudinis Sol illu&longs;trat non &longs;olum tertiam partem, &longs;ed & dimidium, & <lb/>duas tertias, & ferè totum, antequam ad planam illius ba&longs;im de&longs;cendat. <lb/></s> <s id="id.001790">Ego &longs;ic exponendum cen&longs;eo, vt Ari&longs;t. dicat, mane, ide&longs;t initio Crepu&longs;culi <lb/>matutini, & ve&longs;pere, ide&longs;t, in fine Crepu&longs;culi ve&longs;pertini ip&longs;ius tertiam par­ <lb/>tem illuminatam con&longs;pici ab ijs, quorum horizonti tunc incipit, vel de&longs;init <lb/>Crepu&longs;culum; ex quibus illi nece&longs;&longs;ariò re&longs;pectu Cauca&longs;i &longs;unt occidentales, <lb/>quì manè hoc vident, vti &longs;unt ij, qui in Euxino, &longs;eu Ponto, & Meotide naui­ <lb/>gant, vel loca proxima inhabitant: illi verò, qui in fine Crepu&longs;culi ve&longs;per­ <lb/>tini hoc cernunt, nece&longs;&longs;ariò re&longs;pectu Cauca&longs;i erunt orientales. </s> <s id="id.001791">Alter huius <lb/>loci &longs;en&longs;us e&longs;t, ait Mazonius, vt non de tertia montis parte, &longs;ed de tertia <lb/>noctis portione loquatur, ita vt manè. </s> <s id="id.001792">v. <!-- REMOVE S-->g. <!-- REMOVE S-->initio tertiæ, & vltimæ noctis <lb/>parte, cacumen Cauca&longs;i illuminetur. </s> <s id="id.001793">hæc ille. </s> <s id="id.001794">vbi animaduertendum expo­ <lb/>&longs;itionem <expan abbr="hãc">hanc</expan> parùm differre à no&longs;tra modò allata, cùm <expan abbr="vtraq;">vtraque</expan> in idem tem­ <lb/>pus recidat; nam &longs;i dixerimus initio Crepu&longs;culi matutini illuminari ter­ <lb/>tiam partem Cauca&longs;i, tempus hoc coincidit cum initio tertiæ partis noctis, <lb/>quantitas enim Crepu&longs;culi in poli eleuatione 47. grad. <!-- REMOVE S-->qualem habet Cau­ <lb/>ca&longs;us, per totam æ&longs;tatem tres horas plus minus continet, vt patet ex tabu­ <lb/>la quantitatis Crepu&longs;culi, quæ e&longs;t apud Nonium, & apud Clauium in &longs;phæ­ <lb/>ra vltimæ editionis; quæ quantitas reperiri geometrico calculo pote&longs;t, vt <lb/>docent Nonius, Clauius, & Maginus lib. 10. primi mob. </s> <s id="id.001795">quod quidem trium <lb/>circiter horarum tempus e&longs;t tertia ferè noctis pars in ijs regionibus, quibus <lb/>polus eleuatur 47. grad. <!-- REMOVE S-->&longs;iue ergo dicamus id contingere initio Crepu&longs;culi, <lb/>&longs;iue initio tertiæ partis noctis, erit idem tempus, trium &longs;cilicet horarum. <lb/></s> <s id="id.001796">&longs;i ergo, inquit Mazonius, &longs;equamur priorem declarationem, nece&longs;&longs;arium <lb/>e&longs;t dicere, quod ea tertia pars montis, quæ initio auroræ Solis lumine per­ <lb/>funditur, &longs;it ea montis altitudo, qua ip&longs;e exuperat illam aeris regionem, <lb/>vnde Crepu&longs;culum incipit apparere. </s> <s id="id.001797">quo po&longs;ito aptè, ac &longs;agaciter altitudi­ <lb/>nem Cauca&longs;i inue&longs;tigat hoc pacto. </s> <s id="id.001798">præmittit autem &longs;eptem propo&longs;itiones <lb/>apud Mathematicos manife&longs;tas, quas ego mi&longs;&longs;as facio cum non mihi nece&longs;­ <lb/>&longs;ariæ videantur. </s> <s id="id.001799">po&longs;tea &longs;ic di&longs;currit; His ergo ita &longs;e habentibus, dico nos in­ <lb/>uenire po&longs;&longs;e viam, qua &longs;altem rudi Minerua, montis altitudinem comper­ <lb/>tam habeamus. </s> <s id="id.001800">&longs;i enim in principio Crepu&longs;culi v. <!-- REMOVE S-->g. <!-- REMOVE S-->matutini (ita enim, vt <lb/>&longs;upra annotaui intelligendus e&longs;t Ari&longs;t.) illuminatur tertia pars, nece&longs;&longs;arium <lb/>vidctur tertiam illam partem &longs;upra cam regionem collocari, ex qua Cre­ <lb/>pu&longs;culum in planitie apparere incipit, &longs;ed illa regio ex Alhazino, & Vitell. <lb/><!-- REMOVE S-->de Crepu&longs;culis milliaribus 52. à terra recedit, ergo duæ tertiæ montis par­ <lb/>tes, quæ Solem initio auroræ non vident, &longs;unt 52. milliaria ad perpendicu­ <lb/>lum, & tertia alia pars illuminata e&longs;t ad perpendiculum 26. milliaria: ita <lb/>vt totius montis altitudo perpendicularis &longs;it 78. mill. <!-- REMOVE S-->&longs;ed papè in quos acu­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.100.jpg" pagenum="100"/>leos imprudens me conieci? </s> <s id="id.001801">rident enim hoc Ari&longs;t. dictum Mathematici, <lb/>putant enim eum pueriliter lap&longs;um e&longs;&longs;e. </s> <s id="id.001802">Cæterum ego pro præceptoris tu­ <lb/>rela, dico eum &longs;equutum e&longs;&longs;e famam. </s> <s id="id.001803">hæc Mazonius, quorum nonnulla in­ <lb/>digent con&longs;ideratione cuiu&longs;modi, &longs;unt illa, quando dicit, nece&longs;&longs;arium vi­ <lb/>detur, quod ea pars &longs;upra eam regionem attollatur, vnde Crepu&longs;culum in <lb/>planitie apparere ineipit. </s> <s id="id.001804">videtur enim his verbis velle dicere, quod quan­ <lb/>do habitantibus planitiem, quæ e&longs;t ad pedem montis Cauca&longs;i, vel horizon­ <lb/>tem eiu&longs;dem, incipit Crepu&longs;culum, ij&longs;dem etiam tunc tertia montis pars <lb/>appareat illuminata; in quo &longs;en&longs;u errat po&longs;tea in colligenda montis altitu­ <lb/>dine, quamuis enim verum e&longs;&longs;et partem illuminatam eminere totam &longs;upra <lb/>52. milliaria, non tamen &longs;equitur ip&longs;am &longs;olam eminere, &longs;ed alia etiam pars <lb/>eminere pote&longs;t, quod &longs;ic geometricè demon&longs;trabo. </s> <s id="id.001805">de&longs;cribatur enim figura <lb/><figure id="id.009.01.100.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.100.1.jpg" place="text"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.101.jpg" pagenum="101"/>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio, <lb/>& Clauius, in qua terræ globus e&longs;t F L G E, regiò vaporum, & exhalatio­ <lb/>num M X N T. horizon a&longs;tronomicus O P. phy&longs;icus Q R, tangens terram <lb/>in puncto F, vbi etiam ponendus e&longs;t huius horizontis habitator, vnà cum. <lb/></s> <s id="id.001806">Cauca&longs;o F V. <!-- KEEP S--></s> <s id="id.001807">Sol A B C, qui initio Crepu&longs;culi infra horizontem O P, depri­ <lb/>mitur gr. <!-- REMOVE S-->18. vti ab A&longs;tronomis compertum e&longs;t, hoc e&longs;t, arcum D P, e&longs;&longs;e <lb/>grad. <!-- REMOVE S-->18. radius autem C I K, tangens terram, incipit illuminare halitus, <lb/>qui &longs;unt ad K, in extremo horizonte &longs;en&longs;ibili F K. quique po&longs;&longs;unt videri ab <lb/>oculo in F, ide&longs;t ab huius horizontis habitatore. </s> <s id="id.001808">Cæterùm prædicti autho­ <lb/>res po&longs;t longam ratiocinationem ex calculo planorum <expan abbr="triangulorũ">triangulorum</expan> tandem <lb/>o&longs;tendunt in triangulo H F K, latus H K, continere milliaria 3631. ex quo <lb/>detracta H L, &longs;emidiametro terræ, quæ e&longs;t milliar, 3579. reliqua L K, &longs;um­ <lb/>ma halitunm eleuatio relinquatur 52. milliar. </s> <s id="id.001809">quibus ab ip&longs;is demon&longs;tra­ <lb/>tis, &longs;i H F, terræ &longs;emidiameter, quæ continet milliar. </s> <s id="id.001810">3579. ponatur &longs;inus <lb/>totus 100000. & latus F K, ponatur tangens anguli ad H, quem pr&ecedil;dicti au­ <lb/>thores probant e&longs;&longs;e grad. <!-- REMOVE S-->8. 54. erit F K, tangens partium 15659. fiat igi­ <lb/>tur per 2. pro. </s> <s id="id.001811">trjang. </s> <s id="id.001812">rectil. </s> <s id="id.001813">Clauij; <lb/><arrow.to.target n="table4"/></s></p><table><table.target id="table4"/><row><cell>vt H F, &longs;inus totus,</cell><cell>ad milliar.</cell><cell>ita tangens F K,</cell><cell>ad milliar.</cell></row><row><cell>100000.</cell><cell>3579.</cell><cell>15659.</cell><cell>560.</cell></row></table><p type="main"> <s id="id.001814">& inueniemus per auream regulam latus F K, continere milliar. </s> <s id="id.001815">560. quan­ <lb/>ta &longs;cilicet e&longs;t di&longs;tantia ab oculo no&longs;tro ad exhalationes Crepu&longs;culi initium <lb/>efficientes. </s> <s id="id.001816">Con&longs;ideremus iam triangulum F K V, vt ip&longs;ius latus F V, quæ <lb/>e&longs;t Cauca&longs;i altitudo, in milliaribus innote&longs;cat. </s> <s id="id.001817">iam ip&longs;ius latus F K, inno­ <lb/>tuit, angulus verò ad F, e&longs;t rectus; at angulus ad K, &longs;ic manife&longs;tabitur; in <lb/>quadrilatero F K I H, quatuor anguli &longs;unt æquales 4. rectis ex 32. primi. </s> <s id="id.001818">duo <lb/>autem F, & I, &longs;unt recti ex 18. 3. ergo reliqui duo H, & K, æquales erunt duo­ <lb/>bus rectis, quorum alter H, e&longs;t gr. <!-- REMOVE S-->17. 48. vt præditi Mathematici <expan abbr="o&longs;t&etilde;dunt">o&longs;tendunt</expan>, <lb/>reliquus igitur ad K, erit gr. <!-- REMOVE S-->162. 12. vt compleat duos rectos. </s> <s id="id.001819">qui &longs;i detra­ <lb/>hatur à duobus rectis, qui &longs;unt deinceps ad lineam F K, reliquus angulus <lb/>F K V, erit gr. <!-- REMOVE S-->17. 48. &longs;i ergo latus F K, notum ponatur &longs;inus totus 100000. <lb/>latus verò F V, tangens anguli noti, erit ip&longs;a 32100. fiat igitur, <lb/><arrow.to.target n="table5"/></s> </p><table><table.target id="table5"/><row><cell>vt F K, &longs;inus totus,</cell><cell>ad milliar.</cell><cell>ita F V, tangens</cell><cell>ad milliar.</cell></row><row><cell>100000.</cell><cell>560.</cell><cell>32100.</cell><cell>180.</cell></row></table><p type="main"> <s id="id.001820"><expan abbr="inueniemus&qacute;">inueniemusque</expan>; latus F V, continere milliar. </s> <s id="id.001821">180. cuius pars F X, quæ e&longs;t in­ <lb/>fra habituum altitudinem continet milliar. </s> <s id="id.001822">52. quibus detractis ex 180. re­ <lb/>manent 128. pro tota X V, quæ tota e&longs;t &longs;upra vapores, nondum tamen illu­ <lb/>minata. </s> <s id="id.001823">vnde patet Mazonium erra&longs;&longs;e in colligenda hoc modo Cauca&longs;i al­ <lb/>titudine, ex prima Crepu&longs;culi illuminatione in horizonte Cauca&longs;i facta, <lb/>cum ex præmi&longs;&longs;o calculo con&longs;tet partem montis F V, totam tunc temporis <lb/>e&longs;&longs;e tenebro&longs;am, quamuis &longs;uperet multò regionem vaporum, contrà quàm <lb/>ip&longs;e putabat, &longs;uperat enim eam milliar. </s> <s id="id.001824">128. quare duæ tertiæ montis erunt <lb/>non 52. mill. <!-- REMOVE S-->vt ip&longs;e ait, &longs;ed mill. <!-- REMOVE S-->180. & proinde tota altitudo erit mill. <!-- REMOVE S-->270. <lb/>quod &longs;anè ridiculum e&longs;t, cum nullius montis altitudo &longs;e&longs;quimilliare tran­ <lb/>&longs;cendat. </s> <s id="id.001825">Quod &longs;i &longs;equamur alteram expo&longs;itionem, vt nimirum Ari&longs;tor. <!-- REMOVE S-->lo­ <lb/>quatur non de tertia montis parte, &longs;ed noctis, ita vt dicat, circa initium <lb/>tertiæ partis noctis apicem montis illu&longs;trari, altitudo eius erit tantum­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.102.jpg" pagenum="102"/>modo 180. quot continet latus F V. vt vidimus, quæ quamuis illa minor &longs;it, <lb/>adhuc tamen ab&longs;urda e&longs;t.</s> </p><p type="main"> <s id="id.001826">Si verò dixerimus Ari&longs;t. intelligere hæc omnia, non re&longs;pectu horizontis <lb/>Cauca&longs;i, &longs;ed alterius, cuius habitator in principio &longs;ui Crepu&longs;culi tertiam <lb/>Cauca&longs;i partem iam illu&longs;tratam videat, vti accideret &longs;i Cauca&longs;us &longs;tatuere­ <lb/>tur in L K, vbi incipit Crepu&longs;culum habitanti in F. tunc e&longs;&longs;et altitudo tanta, <lb/>quanta colligit Mazonius, &longs;i tamen Ari&longs;t. intelligatur de tertia montis par­ <lb/>te; e&longs;t enim L K, altitudo habituum 52. mill. <!-- REMOVE S-->& duæ tertiæ montis, quare <lb/>totus mons erit 78. &longs;i autem intelligatur circa tertiam noctis partem, mon­ <lb/>tis apicem illuminatum videri ab habitatore F, &longs;ic altitudo eins erit tan­ <lb/>tummodo 52. mill. <!-- REMOVE S-->quæ tamen adhuc omnem veritatem nimium &longs;uperat. <lb/></s> <s id="id.001827">Cum ergo hinc inde &longs;equantur ab&longs;urda, putat Mazonium excu&longs;andum e&longs;&longs;e <lb/>Ari&longs;tot. dicendo eum &longs;equutum e&longs;&longs;e famam, <expan abbr="loquutum&qacute;">loquutumque</expan>; e&longs;&longs;e populariter. <lb/></s> <s id="id.001828">Verumenimuerò &longs;apientiores iudicent num rectè philo&longs;ophus, cuius e&longs;t re­ <lb/>condita, <expan abbr="atq;">atque</expan> abdita docere, excu&longs;etur, &longs;i dicatur, eum, popularem famam <lb/>&longs;equutum e&longs;&longs;e.</s></p><p type="main"> <s id="id.001829">Tandem monendus mihi Lector e&longs;t, in demon&longs;tratione Magini, quæ e&longs;t <lb/>apud Mazonium &longs;ect. </s> <s id="id.001830">4. citati operis; a&longs;&longs;umi radium Solis tangentem terræ <lb/>globum, qui cum horizonte faciat angulum gr. <!-- REMOVE S-->18. quod fal&longs;um e&longs;t, &longs;olus <lb/>enim radius centralis, qui à centro Solis ad centrum terræ ducitur talem <lb/>facit angulum, <expan abbr="atq;">atque</expan> hac de cau&longs;a ip&longs;e colligit altitudinem no&longs;tra maiorem; <lb/>no&longs;tra e&longs;t 270. mill. <!-- REMOVE S-->&longs;ua verò 276. vbi etiam, &longs;icut & nos a&longs;&longs;umit horizon­ <lb/>tem Cauca&longs;i.</s> </p><p type="main"> <s id="id.001831">Aduertendum tandem Mazonium admodum aduer&longs;antia loquutum e&longs;&longs;e, <lb/>&longs;ect. </s> <s id="id.001832">enim 3. demon&longs;tratinè concludit altitudinem 76. mill. <!-- REMOVE S-->&longs;ect. </s> <s id="id.001833">verò 4. &longs;i­ <lb/>mul <expan abbr="cũ">cum</expan> Magino demon&longs;tratiuè pariter colligit altitudinem eiu&longs;dem 276. m. <lb/></s> <s id="id.001834">quæ nimis ab innicem di&longs;crepant, cum tamen <expan abbr="vtrobiq;">vtrobique</expan> demon&longs;tret, & ve­ <lb/>ritas &longs;it vna. </s> <s id="id.001835">At verò cau&longs;a huius di&longs;crepantiæ e&longs;t, quòd &longs;ect. </s> <s id="id.001836">3. accipit Cre­ <lb/>pu&longs;culum non horizontis Cauca&longs;i, &longs;ed illius, in cuius extremitate orientali, <lb/>vbi incipit Crepu&longs;culum, Cauca&longs;us &longs;itus &longs;it, <expan abbr="di&longs;tet&qacute;">di&longs;tetque</expan>; ab habitatore 560. m. <lb/></s> <s id="id.001837">vt &longs;upra o&longs;tendimus. </s> <s id="id.001838">&longs;ect. </s> <s id="id.001839">verò 4. accipit horizontem ip&longs;ius Cauca&longs;i, vt ex <lb/>figura illic de&longs;cripta videre e&longs;t. </s> <s id="id.001840">ex hac igitur horizontum varia &longs;uppo&longs;itio­ <lb/>ne, varia etiam altitudo colligitur, quamuis <expan abbr="vtrobiq;">vtrobique</expan> ex <expan abbr="vtraq;">vtraque</expan> &longs;uppo&longs;itio­ <lb/>ne <expan abbr="vtramq;">vtramque</expan> altitudinem rectè concludat. </s> <s id="id.001841"><expan abbr="Atq;">Atque</expan> hæc de Cauca&longs;o &longs;ufficiant.</s></p><p type="main"> <s id="id.001842"><arrow.to.target n="marg149"/></s></p><p type="margin"> <s id="id.001843"><margin.target id="marg149"/>149</s></p><p type="main"> <s id="id.001844">Eodem cap. <emph type="italics"/>(Ex Pyreneo autem, hic autem est mons ad occidentem æquino­ <lb/>ctiaiem in Gallia, flaunt l&longs;ter, & Tarte&longs;&longs;us, iste quidem extra columnas, I&longs;ter au­ <lb/>tem per totam Europam in Pontum Euxmum)<emph.end type="italics"/> Ari&longs;t. fortè &longs;equutus e&longs;t Herodo­ <lb/>tum, qui falsò tradit I&longs;trum, &longs;ine Dannbium ex Pyreneis de&longs;luere, nam Iu­ <lb/>ce clarius con&longs;tat ip&longs;um ex ijs Alpibus, quæ Heluetiorum montes dicuntur, <lb/>propè Ba&longs;ileam ex Adula monte ortum ducere. </s> <s id="id.001845"><expan abbr="neq;">neque</expan> verum e&longs;t Tarte&longs;&longs;um, <lb/>quem & Bœtim alij nominant ex Pyreneis de&longs;cendere. </s> <s id="id.001846">Tarte&longs;&longs;um hunc Ma­ <lb/>ginus putat e&longs;&longs;e Tagum, cui fauet vocabulorum quali&longs;cunque &longs;imilitudo. <lb/></s> <s id="id.001847">extra tamen columnas Herculis qui&longs;quis &longs;it in Oceanum occidentale illa­ <lb/>bitur. </s> <s id="id.001848">Igno&longs;cenda &longs;unt i&longs;ta Ari&longs;t. tunc enim Geographia <expan abbr="nondũ">nondum</expan> adoleuerat.</s></p><p type="main"> <s id="id.001849"><arrow.to.target n="marg150"/></s></p><p type="margin"> <s id="id.001850"><margin.target id="marg150"/>150</s></p><p type="main"> <s id="id.001851">Ad finem eiu&longs;dem cap. <emph type="italics"/>(Et circa Ligu&longs;ticam non minor Rhodano ab&longs;orbetur <lb/>quidam fluuius, & iterum egreditur &longs;ecundum alium locum)<emph.end type="italics"/> <expan abbr="incompertũ">incompertum</expan> & hoc <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.103.jpg" pagenum="103"/>Ari&longs;t. vt &longs;uperiora, ob Geographiæ illius &longs;eculi imperfectionem, nu&longs;quam <lb/>enim in tota Liguria quidpiam tale reperitur.</s></p><p type="head"> <s id="id.001852"><emph type="italics"/>De Terræ rotunditate.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001853"><arrow.to.target n="marg151"/></s></p><p type="margin"> <s id="id.001854"><margin.target id="marg151"/>151</s></p><p type="main"> <s id="id.001855">Svmma 4. cap. 2. quod e&longs;t de permutatione, & vici&longs;&longs;itudine aquarum, <lb/>& continentis. </s> <s id="id.001856">Pergratum Lectori fore exi&longs;timaui, nec alienum ab <lb/>in&longs;tituto, &longs;i occa&longs;ione huius permutationis maris, ac terræ, rem ex­ <lb/>po&longs;uero &longs;citu digniffimam, quam pridem ob&longs;eruare cœpi, ac in dies <lb/>ob&longs;eruo, præ&longs;ertim cum nullus præteritorum &longs;criptorum, quod &longs;ciam, eam <lb/>literis mandauerit: Terræ &longs;cilicet totius molem paulatim reduci ad perfe­ <lb/>ctam &longs;phæricitatem, ita vt aliquando nece&longs;&longs;e &longs;it futurum ip&longs;am à mari inun­ <lb/>dari, <expan abbr="atq;">atque</expan> omninò inhabitabilem reddi. </s> <s id="id.001857">Prrmum igitur illud ex &longs;acris lite­ <lb/>ris &longs;tatuendum, orbem terræ in &longs;uo primordio fui&longs;&longs;e ab opifice rerum om­ <lb/>nium, figura &longs;phærica donatum, hoc e&longs;t <expan abbr="ab&longs;q;">ab&longs;que</expan> montium eminentijs, atque <lb/>vallium depre&longs;&longs;ionibus. </s> <s id="id.001858">quod patet ex eo, quia <expan abbr="tũc">tunc</expan> tota Mari obtegebatur, <lb/>ita vt minimè apta e&longs;&longs;et animantibus ad inhabitandum. </s> <s id="id.001859">redditam verò ha­ <lb/>bitabilem, cum ip&longs;ius conditor <expan abbr="quãdam">quandam</expan> ip&longs;ius partem humiliorem, & quan­ <lb/>dam eminentiorem effeci&longs;&longs;et; transferendo nimirum maximam terræ por­ <lb/>tionem ex vno loco in alium, vnde illic maris concauitas, i&longs;tic verò mon­ <lb/>tium &longs;ublimitas emer&longs;it. </s> <s id="id.001860">quo facto aquæ omnes in loca illa decliuiora &longs;ua <lb/>&longs;pontè rece&longs;&longs;erunt, quæ aquarum congregatio Mare appellatum e&longs;t. </s> <s id="id.001861">Hine <lb/>nonnulli auctores graui&longs;&longs;imi a&longs;&longs;erere non dubitarunt, montes <expan abbr="cõflatos">conflatos</expan> fui&longs;­ <lb/>&longs;e ex terra illa, quæ locum illum occupabat, quem po&longs;tea maria inua&longs;erunt. <lb/></s> <s id="id.001862">quæ cum ita &longs;int. </s> <s id="id.001863">&longs;equitur terram <expan abbr="nũc">nunc</expan> e&longs;&longs;e extra naturalem &longs;uam figuram, & <lb/>propterea in quodam &longs;tatu violento, <expan abbr="viol&etilde;tum">violentum</expan> autem <expan abbr="nullũ">nullum</expan> <expan abbr="perpetuũ">perpetuum</expan>. </s> <s id="id.001864">præ­ <lb/>terea cum terra &longs;it grauior quàm aqua, nulla ratione deberent terræ partes <lb/>&longs;uperiores a quæ &longs;uperficiem &longs;uperare, cuius tamen <expan abbr="contrariũ">contrarium</expan> accidit, nam <lb/>&longs;uperficies ip&longs;a terræ, & multò magis <expan abbr="mõtana">montana</expan> loca &longs;uperficiem maris cuiu&longs;­ <lb/>uis non parum &longs;uperant; quæ altera violentia terræ, & aquæ ine&longs;t, & ideò <lb/>minimè mirum e&longs;t, imò <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> naturæ valdè conueniens terram redire ad <lb/>pri&longs;tinam, ac primigeniam figuram, ex qua con&longs;ectarium erit aquam <expan abbr="quoq;">quoque</expan> <lb/>&longs;uam pariter illam &longs;ibi primæuam recuperaturam e&longs;&longs;e figuram. </s> <s id="id.001865">cau&longs;am au­ <lb/>tem re&longs;tauratricem huius terrenæ <expan abbr="rotũditatis">rotunditatis</expan> e&longs;&longs;e aquas tum pluuiales, tum <lb/>fluuiales iamdiù ob&longs;eruauimus, vt ex &longs;equentibus ob&longs;eruationibus patebit.</s></p><p type="main"> <s id="id.001866">Primò, videmus flumina quotidie montium radices corrodere, ac qua&longs;i <lb/>&longs;uffodere, ita vt pa&longs;&longs;im ex hoc, vel illo monte magnas faciant ruinas, ac pr&ecedil;­ <lb/>cipitia, <expan abbr="atq;">atque</expan> hiac inde prærupti appareant montes, vt meritò legamus apud <lb/>Iob cap. 14. allunione paulatim terra con&longs;umitur. </s> <s id="id.001867">humum porrò illam ex <lb/>montibus delap&longs;am &longs;emper ad loca humiliora fluuij &longs;ecum detrahunt. </s> <s id="id.001868">Ex <lb/>continua etiam hac inter montes corro&longs;ione facta manife&longs;tè apparet, flumi­ <lb/>num alueos in montanis modò e&longs;&longs;e humiliores quàm olim, quamuis contra­ <lb/>rium accidat alueis &longs;luuiorum per plana decurrentium, qui modò altiores <lb/>&longs;unt <expan abbr="quã">quam</expan> exordio mundi, vt paulò po&longs;t <expan abbr="o&longs;t&etilde;dam">o&longs;tendam</expan>. </s> <s id="id.001869">Illud autem liquidò apparet <lb/>ex &longs;ignis, &longs;eu &longs;ymbolis, &longs;eu ex &longs;imilitudine terræ, aut lapidis, quæ in alti&longs;&longs;imis <lb/>fluminum ripis hinc inde pa&longs;&longs;im <expan abbr="vid&etilde;tur">videntur</expan>, quæ indicio &longs;unt montes illos iam <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.104.jpg" pagenum="104"/>olim fui&longs;&longs;e continuos, <expan abbr="atq;">atque</expan> vnam, <expan abbr="eandem&qacute;">eandemque</expan>; terram <expan abbr="contin&etilde;tem">continentem</expan>, antequam <lb/>flumen eos ab inuicem &longs;epararet; <expan abbr="flumen&qacute;">flumenque</expan>; ip&longs;um olim altius, vbi &longs;unt &longs;igna <lb/>illa ambula&longs;&longs;e; quemadmodum in Pyramo Ciliciæ amne ob&longs;eruauit Strabo, <lb/>dum libro 12. de illius ripis hæc tradit, mira præterea e&longs;t montis cæ&longs;ura, <lb/>per quam alueus ducitur; nam quemadmodum in petris per medium &longs;ci&longs;&longs;is <lb/>contingit, alterius partis depre&longs;&longs;ioribus ita conuenire alterius partis emi­ <lb/>nentias, vt coniungi po&longs;&longs;int: &longs;ic videre e&longs;t imminentes flumini petras vtrin­ <lb/>que ferè <expan abbr="v&longs;q;">v&longs;que</expan> ad montis &longs;umma pertendentes duorum, triumuè iugerum <lb/>&longs;patio concauitates qua&longs;dam eminentijs oppo&longs;itas habere. </s> <s id="id.001870">hæc Strabo de <lb/>vno, quod nos in pluribus ob&longs;eruauimus. </s> <s id="id.001871">Pr&ecedil;terea videmus quotidie pluuias <lb/>aquas, idem quantum po&longs;&longs;unt efficere, &longs;uperficies montium, eorum maxi­ <lb/>mè, qui coluntur, perpetuò ab&longs;umentes, <expan abbr="atq;">atque</expan> ad loca conuallium deducen­ <lb/>tes. </s> <s id="id.001872">hinc videre e&longs;t, montes cæteris duriores, vt &longs;unt lapido&longs;i, cæteris altio­ <lb/>res reman&longs;i&longs;&longs;e; quippe qui magis & pluuijs, & fluuialibus aquis &longs;ua duritie <lb/>ob&longs;titerunt. </s> <s id="id.001873">idem montani incolæ omnes confirmant, qui omnes aiunt &longs;ibi <lb/>hanc montium demolitionem iampridem innotui&longs;&longs;e, ex eo quod nonnulli <lb/>montes olim &longs;ibi impedimento erant, ne arcem, turremuè in vlteriore mon­ <lb/>te &longs;itam con&longs;picerent, quam deinde plures po&longs;t annos intermedio monte <lb/>depre&longs;&longs;o, commodè videbant. </s> <s id="id.001874">Ad hæc; antiqua in montium verticibus con­ <lb/>&longs;tituta ædeficia, propterea intercidunt, quia terra hinc, & inde ab aquis <lb/>paulatim confumpta, <expan abbr="deor&longs;um&qacute;">deor&longs;umque</expan>; delap&longs;a, fundamenta ip&longs;orum nuda primò <lb/>relinquit; deindé terra etiam ip&longs;a, qua fundamenta innitebatur &longs;en&longs;im de­ <lb/>lap&longs;a, ip&longs;a <expan abbr="quoq;">quoque</expan> fundamenta vnà cum toto ædeficio nece&longs;&longs;e e&longs;t collabi, hu­ <lb/>ius &longs;igna infinita propemodum videri po&longs;&longs;unt; vnum tamen, quod toti orbi <lb/>confpicuum e&longs;t, non ommittam; Capitolium videlicet Romanum, cuius <lb/>modo fundamenta tota extant, quæ olim altè &longs;ub terram de&longs;cendebant. </s> <s id="id.001875">vi­ <lb/>de pulcherrimam hac de re tractationem apud Georgium Agricolam lib. 3. <lb/>cap. 1. qui amplius addit illud, quod & mihi maximè probatur; flumina ni­ <lb/>mirum producere montes, <expan abbr="colles&qacute;">collesque</expan>; hoc modo; vult enim initio mundi non <lb/>extiti&longs;&longs;e tot particulares montes ab inuicem di&longs;cretos, &longs;ed fui&longs;&longs;e perpetua <lb/>quædam terræ iuga eminentia quidem, &longs;ed non tot vallibus di&longs;&longs;ecta: v. <!-- REMOVE S-->g. <lb/><!-- REMOVE S-->mons no&longs;ter Apenninus erat iugum, &longs;iue dor&longs;um quoddam terræ eminens <lb/>quidem, &longs;ed nullis vallibus in tot particulares colles, aut montes di&longs;&longs;ectum; <lb/>&longs;ed po&longs;tquam flumina à &longs;ummitate ip&longs;ius deor&longs;um fluere cœperunt; paula­ <lb/>tim corrodentes humum in dies magis, ac magis effecerunt valles, <expan abbr="atq;">atque</expan> hac <lb/>ratione in colles, <expan abbr="montes&qacute;">montesque</expan>; plurimes totus Apenninus diui&longs;us e&longs;t. </s> <s id="id.001876">hæc de <lb/>montibus &longs;ufficiant, nunc ad plana de&longs;cendamus.</s></p><p type="main"> <s id="id.001877">Contrarium igitur omninò accidere videmus in planis, quoniam eædem <lb/>aquæ, quæ ex montibus quotidie terram &longs;ecum deducunt, eam ad humilio­ <lb/>ra loca, vt &longs;unt plana, & campe&longs;tria, &longs;iue ibi &longs;int maria, &longs;iue arida, compor­ <lb/>tant, <expan abbr="eam&qacute;">eamque</expan>; ibidem deponunt. </s> <s id="id.001878">hinc videmus antiqua ædeficia in planis locis <lb/>ex&longs;tructa, e&longs;&longs;e iam penè tota &longs;epulta, contra quam in montanis, cuius exem­ <lb/>plum habes etiam Romæ propè ip&longs;um Capitolium, in Arcu triumphali Sep­ <lb/>timij, qui iam ferè totus ruino&longs;a vndique terra obruitur. </s> <s id="id.001879">&longs;ic Pantheon. </s> <s id="id.001880">&longs;ic <lb/>etiam templa Epi&longs;copalia, quæ <expan abbr="plerunq;">plerunque</expan> &longs;atis peruetu&longs;ta &longs;unt, admodum <lb/>infra terram con&longs;piciuntur. </s> <s id="id.001881">Idem affirmant cœmentarij, & architectores <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.105.jpg" pagenum="105"/>omnes, quibus <expan abbr="vbiq;">vbique</expan> terrarum, dum in planis ædeficiorum fundamenta ex­ <lb/>canant, occurrit primò terra quædam, quam ip&longs;i motam appellant, quæ li­ <lb/>gnis, ruderibus, ferramentis, numi&longs;matis, &longs;epulturis, <expan abbr="varijs&qacute;">varijsque</expan>; rebus per­ <lb/>mixta e&longs;t; qua eruta, reperitur terra alia, quam nunquam fui&longs;&longs;e motam, ap­ <lb/>paret, ex eo quod &longs;olida, ac benè compacta &longs;it, neque vllis externis rebus, <lb/>præ&longs;ertim artificiatis admixta, terra illa, quam motam dicunt, variam va­ <lb/>rijs in locis &longs;ortita e&longs;t altitudinem, prout aquæ plurimum, vel minimum <lb/>montanæ terræ huc, vel illuc comportarunt: alicubi vt hic Parmæ erit &longs;ex <lb/>vlnarum, alibi viginti, vt Mutinæ; alibi triginta, vt Romæ, nonnullis in lo­ <lb/>cis. </s> <s id="id.001882">Comprobatur tandem hæc no&longs;tra ob&longs;eruatio ex arte illa, qua per ea&longs;­ <lb/>dem fluuiales aquas &longs;olent, tam loca depre&longs;&longs;iora per aggerationem paula­ <lb/>tim replere, <expan abbr="atq;">atque</expan> eleuare: quàm etiam altiora per aquarum earumdem cor­ <lb/>ro&longs;ionem deprimere. </s> <s id="id.001883">qua in arte exercitati&longs;&longs;imum P. <!-- REMOVE S-->Augu&longs;tinum Spernac­ <lb/>ciatum no&longs;træ Societatis videmus modo de mandato Summi Pontificis Pa­ <lb/>dum, ac Renum Bononien&longs;em ob aggerationem &longs;tagnantes in mari emitte­ <lb/>re; cui totus hic no&longs;ter di&longs;cur&longs;us maximè probatur. </s> <s id="id.001884">Ex quibus omnibus &longs;e­ <lb/>quitur &longs;uperficiem terræ tam montium, quam planorum quotidie variari. <lb/></s> <s id="id.001885">illam nimirum deprimi, hanc attolli. </s> <s id="id.001886">vnde aliud maximum notandum &longs;e­ <lb/>quitur, videlicet hac tempe&longs;tate non e&longs;&longs;e eandem agrorum &longs;uperficiem, quæ <lb/>erat antiquitus, cum in montanis agris &longs;it multò humilior, in campe&longs;tribus <lb/>verò altior, quàm antiqua illa, ac primigenia; quapropter mirum videri <lb/>non debet, &longs;i quorumdam locorum adeò immutata natura e&longs;t, vt quæ olim <lb/>genero&longs;a vina ferebant, vel quouis alio e&longs;&longs;ent prædita munere, adeò dege­ <lb/>nerauerint, vt & vina, & alia nullius modò valoris, vel in parua copia pro­ <lb/>ferant. </s> <s id="id.001887">Quod verò ad marium aggerationem &longs;pectat, dicimus ij&longs;dem aquis <lb/>magnam arenarum copiam perpetuò impertantibus, fieri aggerationem, <lb/>hoc e&longs;t littora quotidie magis cre&longs;cere, &longs;eu in mare ingredi, & con&longs;equen­ <lb/>ter mare recedere. </s> <s id="id.001888">quod primò Ari&longs;t. te&longs;timonio in hoc cap. comprobatur, <lb/>cum quo pariter &longs;entiunt veteres Geographi, & Hi&longs;torici omnes. </s> <s id="id.001889">Ari&longs;t. igi­ <lb/>tur in comprobationem huius adducit primò magnam Aegypti aggeratio­ <lb/>nem; pars enim illa Aegypti, quæ Delta, <expan abbr="Nili&qacute;">Nilique</expan>; donum appellatur ab He­ <lb/>rodoto, ex arenis, & limo, ex Aethyopiæ montibus &longs;imul cum Nilo in mare <lb/>delabentibus, e&longs;t conflata, <expan abbr="atq;">atque</expan> antiquo littori addita, cui locum paulatim <lb/>mare ce&longs;&longs;it; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; propterea donum Nili appellata, quod ab ip&longs;o illuc are­ <lb/>nas importante &longs;it facta. </s> <s id="id.001890">&longs;ecundum, Ari&longs;t. exemplum e&longs;t Ammonia Regio, <lb/>cuius humiliora loca. </s> <s id="id.001891">f. </s> <s id="id.001892">maritima, palam e&longs;t, inquit, quod aggeratione facta, <lb/>fiunt &longs;tagna, & continens: &longs;uccedente autem tempore, &longs;tagnans aqua ob <lb/>nouam aggerationem de&longs;iccata e&longs;t, & iam annihilata. </s> <s id="id.001893">tertium e&longs;t Meotidis <lb/>Paludis; At verò, ait, & quæ &longs;unt circa Meotidem Paludem creuerunt allu­ <lb/>uione fluuiorum tantum, vt multò minores magnitudine naues, nunc innare <lb/>po&longs;&longs;int, quàm anno ab hinc &longs;exage&longs;imo. </s> <s id="id.001894">quare ex hoc facilè e&longs;t ratiocinari, <lb/>quod & primò, vt multa &longs;tagnorum, ita & hoc opus e&longs;t fluuiorum, & tan­ <lb/>dem nece&longs;&longs;e e&longs;t totum fieri &longs;iccum. </s> <s id="id.001895"><expan abbr="quartũ">quartum</expan> e&longs;t illi Bo&longs;phorus Tracius; quod <lb/>vnà cum præcedentibus &longs;atius e&longs;t apud ip&longs;um, vel potius apud eius expo&longs;i­ <lb/>torem Vicomercatum videre, vt breuitati con&longs;ulatur. </s> <s id="id.001896">Accedit & Plinij te­ <lb/>&longs;timonium, qui tradit multas terras na&longs;ci, non &longs;olum fluminum inuectu, &longs;ed <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.106.jpg" pagenum="106"/>etiam marium rece&longs;&longs;u; &longs;ic mare ab Ambraciæ portu 10. millia pa&longs;&longs;uum; ab <lb/>Athenarum verò <expan abbr="quinq;">quinque</expan> millia, & alijs in locis plus minu&longs;uè rece&longs;&longs;i&longs;&longs;e &longs;cri­ <lb/>bit. </s> <s id="id.001897">Huc facit locus quidam Strabonis ex lib. 12. de Pyramo Ciliciæ fluuio: <lb/>&longs;ic; montes verò egre&longs;&longs;us tantum limum in mare deducit, partim ex Ca­ <lb/>taonia, partim ex Ciliciæ campis, vt huiu&longs;modi de co oraculum feratur;</s></p><p type="main"> <s id="id.001898"><emph type="italics"/>Tempus erit rapidis olim cum Pyramus vndis <lb/>In &longs;acram veniet conge&longs;io litore, Cyprum:<emph.end type="italics"/></s></p><p type="main"> <s id="id.001899">hic enim fluuius è regione Cypri in&longs;ulæ in mari influit, &c. </s> <s id="id.001900">hæc Strabo.</s></p><p type="main"> <s id="id.001901">Verùm recentiora non de&longs;unt exempla. </s> <s id="id.001902">Rauenna olim erat in extremo <lb/>littore &longs;ita, nunc paulatim aggeratione aucto litore, mare multum ab ea <lb/>rece&longs;&longs;it. </s> <s id="id.001903">Patauium pariter, vt fertur mare alluebat, quod modo 25. pa&longs;&longs;uum <lb/>millibus ab eo di&longs;tat. </s> <s id="id.001904">Aæ&longs;tuarium ip&longs;um Venetum, ob arenas à varijs &longs;lu­ <lb/>minibus in ip&longs;um immi&longs;&longs;as adeò fundum extulit, vt vix amplius nauigatio­ <lb/>ni &longs;it aptum, <expan abbr="periculam&qacute;">periculamque</expan>; &longs;it ne Venetiarum mirabilis locus, ex maritimo <lb/>fiat terre&longs;tris. </s> <s id="id.001905">demum exemplum &longs;it Bononien&longs;ium Renus, qui quamuis exi­ <lb/>guus &longs;it torrens, paucis tamen annis Padum ip&longs;um, in quem immi&longs;&longs;us fue­ <lb/>rat arena ita repleuit, vt & &longs;ibi, & Pado magno vicinorum agrorum damno <lb/>viam in mare ob&longs;truxerit. </s> <s id="id.001906">Cum igitur mare ob hanc adaggerationem co­ <lb/>gatur &longs;e quotidie magis recipere, <expan abbr="fiat&qacute;">fiatque</expan>; propterea alueus ip&longs;ius angu&longs;tior, <lb/><expan abbr="atq;">atque</expan> clatior, nece&longs;&longs;e e&longs;t etiam ip&longs;am quoque maris aquam quotidie magis <lb/>coangu&longs;tari, <expan abbr="atq;">atque</expan> attolli, & aliquando futurum, vt exundare incipiat. </s> <s id="id.001907">quod <lb/>iam <expan abbr="pleri&longs;q;">pleri&longs;que</expan> in locis accidit, vt in littore Baltico, Danico, & Hollandico, <lb/>quibus in locis &longs;unt hac tempe&longs;tate extructi prælongi, ac præalti aggeres <lb/>contra maritimas innundationes: quibus antiquitus minimè fui&longs;&longs;e opus hi­ <lb/>&longs;toricorum, ac <expan abbr="Geographorũ">Geographorum</expan> &longs;ilentium comprobat. </s> <s id="id.001908">Hoc igitur modo ter­ <lb/>ra, qua montes, <expan abbr="colles&qacute;">collesque</expan>; con&longs;tant paulatim ab aquis in maris concauitates <lb/>deportata, cau&longs;a e&longs;t, vt mare &longs;en&longs;im modo hac, modo illac, terræ &longs;uperfi­ <lb/>ciei &longs;uperfundatur, <expan abbr="terra&qacute;">terraque</expan>; iterum, quemadmodum exordio mundi inhabi­ <lb/>tabilis reddatur: quod tunc maximè accidct cum aquæ tam fluuiales, quàm <lb/>pluuiæ, &longs;uper faciem terræ perpetuò di&longs;currentes, totam illam montanam <lb/>terram in pri&longs;tinum locum, vbi ab initio fuerat, <expan abbr="vnde&qacute;">vndeque</expan>; &longs;ublata fuit, re&longs;ti­ <lb/>tuerint; tunc terra erit iterum rotunda, & &longs;phærica, hoc e&longs;t &longs;uæ primigeniæ <lb/>iterum figuræ re&longs;tituetur: quapropter mare etiam rur&longs;us &longs;icut initio mundi <lb/>totam terræ faciem <expan abbr="circumquaq;">circumquaque</expan> innundabit, quod probare volebam.</s></p><p type="main"> <s id="id.001909"><emph type="italics"/>Tantum æui mutare potest longæua vetu&longs;tas.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001910">Hinc nonnulla colligi po&longs;&longs;unt non minus notatu, ac &longs;citu, quàm præceden­ <lb/>tia digni&longs;&longs;ima, quibus Ethnicorum Philo&longs;ophorum error redarguatur, &longs;ides <lb/>verò no&longs;tra magis roboretur: mundum nimirum ab æterno neutiquam ex­ <lb/>titi&longs;&longs;e, vel &longs;altem terram ab æterno non fui&longs;&longs;e hac figura præditam, qua nunc <lb/>videmus, ncc mundum perpetuò duraturum. </s> <s id="id.001911">nam &longs;i hæc montuo&longs;a illi figu­ <lb/>ra ab æterno ine&longs;&longs;et, iampridem tota illa montium tubero&longs;itas fui&longs;&longs;et ab <lb/>aquis exæ&longs;a, & con&longs;umpta: <expan abbr="neq;">neque</expan> æterna erit, quia &longs;ucce&longs;&longs;u temporis, vt pro­ <lb/>bauimus, reducetur ad rotunditatem, <expan abbr="atq;">atque</expan> à mari innun dabitur, & idcircò <lb/>inhabitabilis, vnde nece&longs;&longs;ariò mortalium genus interibit. </s> <s id="id.001912">Quapropter ni&longs;i <lb/>igne illo, quem &longs;acræ literæ innuunt catacly&longs;mus ille præueniatur, aqua <lb/>mundus interiturus e&longs;&longs;et. </s> <s id="id.001913">&longs;ed de his hactenus.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.107.jpg" pagenum="107"/><p type="main"> <s id="id.001914">Quoad magnum illud Diluuium, quod Ari&longs;t. hoc capite exi&longs;timat po&longs;t <lb/>multa &longs;ecula reuolui, hoc veritati e&longs;&longs;e con&longs;entaneum argumento &longs;unt, ac <lb/>pariter admirationi varia <expan abbr="cõchiliorum">conchiliorum</expan> genera, quæ tùm in Apennino mon­ <lb/>te, tùm in Alpibus ob&longs;eruaui; <expan abbr="Ìdem&qacute;">Ìdemque</expan>; in alijs mundi partibus inueniri pu­ <lb/>to; præ&longs;ertim in tam immen&longs;a copia, <expan abbr="atq;">atque</expan> intra vi&longs;cera montium colloca­ <lb/>ta, quæ nulla vis humana illuc contuli&longs;&longs;er, ni&longs;i temporibus catacly&longs;mi ebul­ <lb/>lientibus aquis maris &longs;uper terram facta fui&longs;&longs;et hæc varia rerum maritima­ <lb/>rum cum terre&longs;tribus commixtio: quæ quidem optimè ex Pomponio Mela <lb/>comprobantur, qui libro 1. de Numidia &longs;ic narrat: interius, & longè &longs;atis <lb/>à litore, &longs;i fides res capit, mirum admodum, &longs;pinæ pi&longs;cium, <expan abbr="Muricũ">Muricum</expan>, <expan abbr="O&longs;treo-rum&qacute;">O&longs;treo­ <lb/>rumque</expan>; fragmenta, &longs;axi atritu, vti &longs;olent fluctibus, & non differentia mari­ <lb/>nis, infixæ cautibus anchoræ, <expan abbr="alia&qacute;">aliaque</expan>; huiu&longs;modi &longs;igna, & ve&longs;tigia effu&longs;i olim <lb/><expan abbr="v&longs;q;">v&longs;que</expan> ad ea loca pelagi, in campis nihil alentibus e&longs;&longs;e inuenirique narrantur. <lb/></s> <s id="id.001915">neque locus ille Ouid. <!-- REMOVE S-->Met. <!-- REMOVE S-->15. extra rem:</s> </p><p type="main"> <s id="id.001916"><emph type="italics"/>Vidi ego, quod fuerat olim &longs;olidi&longs;&longs;ima petra <lb/>E&longs;&longs;e fretum, vidi fact as ex æquore terras: <lb/>Et procul à Pelago conchæ iacuere marinæ, <lb/>Et vetus inuenta e&longs;t in montibus anchora &longs;ummis.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001917">Nos autem Chri&longs;tiani ad Noemi Diluuium i&longs;ta referre debemus.</s></p></chap><chap><p type="head"> <s id="id.001918"><emph type="italics"/>Ex Secundo Meteororum.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.001919"><arrow.to.target n="marg152"/></s></p><p type="margin"> <s id="id.001920"><margin.target id="marg152"/>152</s></p><p type="main"> <s id="id.001921">Cap. 1. ait multa e&longs;&longs;e maria, quæ ad inuicem non communicant. <lb/></s> <s id="id.001922">Eorum rubrum mare vnum e&longs;&longs;e; quod cum Oceano <expan abbr="Atlãtico">Atlantico</expan>, qui <lb/>e&longs;t extra Herculeum fretum ad occidentem parum videtur com­ <lb/>mi&longs;ceri &longs;iue Ari&longs;t. pro Rubro mari intelligat Oceanum illum, qui <lb/>Arabiam, ac Per&longs;iam alluit, &longs;iue illius &longs;inum, qui Arabiam, <expan abbr="atq;">atque</expan> Aethiopiam <lb/>interluit, fal&longs;um e&longs;t ip&longs;um parum communicare cum occidentali Oceano, <lb/>vt quotidianis Lu&longs;itanorum nauigationibus ad Indos patet. </s> <s id="id.001923">&longs;ed meritò hoc <lb/>Ari&longs;tot. condonandum, cum tunc temporis nondum tota Africa e&longs;&longs;et certò <lb/>circumlu&longs;trata, <expan abbr="neq;">neque</expan> iter ab Hi&longs;pania ad Indos maritimum, adeo nunc fre­ <lb/>quens, patefactum e&longs;&longs;et.</s> </p><p type="main"> <s id="id.001924"><arrow.to.target n="marg153"/></s></p><p type="margin"> <s id="id.001925"><margin.target id="marg153"/>153</s></p><p type="main"> <s id="id.001926">Summæ 2. cap. 2. <emph type="italics"/>(Quapropter & circa Orionis orturm maximè fit tranquilli­ <lb/>tas)<emph.end type="italics"/> quando Medici, Philo&longs;ophi, Poetæ, ac reliqui auctores loquuntur de <lb/>ortu a&longs;trorum fixorum, aut con&longs;tellationum, quæ &longs;unt in firmamento, vti <lb/>e&longs;t Orion (& Canis, de quo po&longs;tea) intelligunt &longs;emper de ortu ip&longs;orum, qui <lb/>fit matutino tempore, quando &longs;cilicet vel &longs;imul cum Sole, vel paulò ante <lb/>Solem emergunt, ita vt videantur à nobis; qui ortus dicitur Co&longs;micus, tunc <lb/>propriè, quando &longs;imul a&longs;trum cum Sole oritur; quando autem incipit appa­ <lb/>rere manc ante Solem, dicitur ortus Heliacus. </s> <s id="id.001927">i. </s> <s id="id.001928">&longs;olaris, quia oritur quodam­ <lb/>modo ex radijs Solis, &longs;ub quibus antea latebat. </s> <s id="id.001929">A&longs;tra verò incrrantia, & <lb/>planetæ Sole tardiores oriuntur <expan abbr="vtroq;">vtroque</expan> modo. </s> <s id="id.001930">nam cùm ip&longs;a Sol, quippe il­ <lb/>lis velocior primum a&longs;&longs;equitur, ea &longs;uo lumine obtegit, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; hic occa&longs;us eo­ <lb/>rum heliacus: cum verò eadem præterierit, ac po&longs;t &longs;e reliquerit fit, vt mo­ <lb/>tu diurno toto cœlo conuer&longs;o, mane ante Solem effulgeant, &longs;iue heliacè <lb/>oriantur: & cum quotidie magis Sol ab illis recedat, ip&longs;aque magis à Sole <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.108.jpg" pagenum="108"/>elongentur, fit, vt quotidie magis ortum Solis anticipent, & citius mane au­ <lb/>te Solem videantur. </s> <s id="id.001931"><expan abbr="&longs;ic&qacute;">&longs;icque</expan>; tanto in dies citius, vt deinde media etiam nocte <lb/>oriantur; tum ante mediam noctem po&longs;tea paulò ante occa&longs;um Solis. <!-- KEEP S--></s> <s id="id.001932">de­ <lb/>mum cum fuerint Soli oppo&longs;ita, occidente Sole oriantur, qui ortus dicitur <lb/>Ve&longs;pertinus, vel Acronicus. </s> <s id="id.001933">po&longs;tea oriuntur &longs;emper in die ante Solis occa­ <lb/>&longs;um, donec Sol ip&longs;a iterum a&longs;&longs;equatur, <expan abbr="ea&qacute;">eaque</expan>; radijs &longs;uis offu&longs;cet, quod e&longs;t he­ <lb/>liacè occidere; & mox cum ip&longs;o Sole occumbant, quod Acronicè e&longs;t occi­ <lb/>dere. </s> <s id="id.001934">Totum porrò illud tempus, quo per diem oriuntur, non eorum ortui, <lb/>&longs;ed occa&longs;ui deputatur, eò quod non cernuntur oriri, vt &longs;equenti loco expli­ <lb/>cabitur. </s> <s id="id.001935">Quæ omnia adhibito Globo a&longs;tronomico, in quo con&longs;tellationes <lb/>omnes depictæ &longs;unt, <expan abbr="eo&qacute;">eoque</expan>; ad tui poli eleuationem con&longs;tituto, appo&longs;itoque <lb/>Sole &longs;uo loco in Zodiaco, qui paulatim per Zodiacum orientem ver&longs;us gra­ <lb/>diatur, & interim diurno motu globus conuertatur, ad &longs;en&longs;um manife &longs;ta <lb/>apparebunt. </s> <s id="id.001936">In &longs;umma auctores intelligunt de ortu, qui mane fit ante So­ <lb/>lem, quia tunc primum po&longs;t diuturnas latebras incipit apparere. </s> <s id="id.001937"><expan abbr="nõ">non</expan> autem <lb/>intelligunt de ortu Acronico, quia ante hunc ortum videbatur noctu, <expan abbr="itaq;">itaque</expan> <lb/>ortu Acronico non fit noua apparitio; ideo de hoc non intelligunt. </s> <s id="id.001938">fit au­ <lb/>tem ortus hic Orionis, heliacus, & matutinus, de quo Ari&longs;t. hoc loco, & alij <lb/>auctores, no&longs;tra hac tempe&longs;tate paulò ante Solis ingre&longs;&longs;um in Cancrum, &longs;i­ <lb/>ue ante &longs;ol&longs;titium æ&longs;tiuum circa 22. Iunij.<!-- REMOVE S--><arrow.to.target n="marg154"/></s></p><p type="margin"> <s id="id.001939"><margin.target id="marg154"/>154</s></p><p type="main"> <s id="id.001940">Eodem cap. <emph type="italics"/>(Incertus autem, & mole&longs;tus Orion e&longs;&longs;e videtur & occumbens, & <lb/>oriens, quia in tran&longs;mutatione temporis accidit occa&longs;us, & ortus, a&longs;tate, aut hye­ <lb/>me, & propter magnitudinem a&longs;tri dierum &longs;it aliqua pluralitas)<emph.end type="italics"/> hoc loco Vico­ <lb/>mercatus ex &longs;ententia a&longs;tronomorum occa&longs;um Orionis fieri autumni tem­ <lb/>pore, Sole Scorpionem ob&longs;idente docet, quod & verba Ari&longs;t. clarè &longs;ignifi­ <lb/>cant, cum dicat ortum ip&longs;ius fieri æ&longs;tate; in tran&longs;mutatione verò temporis, <lb/>videlicet in autumno fieri occa&longs;um. </s> <s id="id.001941">Porrò occa&longs;us hic fieri incipit primum <lb/>mane oriente Sole, <expan abbr="dicitur&qacute;">diciturque</expan>; occa&longs;us co&longs;micus, quia dum Sol e&longs;t in oriente, <lb/>Orion e&longs;t in occidente, & infra orizontem cadit: deinde paulò ante Solis or­ <lb/>tum, &longs;ed tamen nocturno tempore, ita vt occa&longs;us eius videri po&longs;&longs;it, donec <lb/>occidat parum po&longs;t Solis occa&longs;um, & tandem cum Sole ip&longs;o heliacè euane­ <lb/>&longs;cat. </s> <s id="id.001942">Scriptores autem ferè &longs;emper cum loquuntur de occa&longs;u inerrantium <lb/>&longs;yderum, de eo, qui noctu videatur, intelligunt: &longs;icuti ortum intelligunt <lb/>eum, qui noctu fit, <expan abbr="noctu&qacute;">noctuque</expan>; videtur. </s> <s id="id.001943">affixa <expan abbr="namq;">namque</expan> &longs;ydera per fex fermè men­ <lb/>&longs;es noctu oriuntur, <expan abbr="oriri&qacute;">oririque</expan>; ea con&longs;picimus, & propterea totum illud tem­ <lb/>pus, ortui ip&longs;orum deputamus: Reliquum verò <expan abbr="t&etilde;pus">tempus</expan>, quo per diem oriun­ <lb/>tur, & idcircò ortus illorum minimè apparet, nulla ratione ortui debuit <lb/>a&longs;cribi: totum verò tempus, quo noctu occidunt, & occidere cernuntur, oc­ <lb/>ca&longs;ui illorum meritò attribuitur. </s> <s id="id.001944">& <expan abbr="quemadmodũ">quemadmodum</expan> temporis illius initium, <lb/>quo primo de nocte apparere incipiunt, dicitur ab&longs;olutè ortus cuiu&longs;uis &longs;y­ <lb/>deris; &longs;ic etiam initium temporis illius, quo primum per noctem ea occide­ <lb/>re videmus, &longs;impliciter occa&longs;um appellamus.</s></p><p type="main"> <s id="id.001945"><arrow.to.target n="marg155"/></s></p><p type="margin"> <s id="id.001946"><margin.target id="marg155"/>155</s></p><p type="main"> <s id="id.001947">Eodem cap. <emph type="italics"/>(Ete&longs;iæ autem flant post ver&longs;iones, & Canis ortum)<emph.end type="italics"/> per ver&longs;io­ <lb/>nes intelligit tropicos, quod & tropici etymon confirmat, <expan abbr="cũ">cum</expan> tropicus idem <lb/>valeat, ac conuer&longs;iuus. </s> <s id="id.001948">circa Canis ortum eadem &longs;unt notanda, quæ &longs;upra <lb/>de ortu Orionis annotaui; intelligit enim eum Canis ortum, qui mane fiat <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.109.jpg" pagenum="109"/>primum paulò ante Solis ortum, cum &longs;cilicet incipit apparere.</s></p><p type="main"> <s id="id.001949">Cum porrò in c&ecedil;lo &longs;it Canis maior, & Canis minor, qui & Procyon, ide&longs;t <lb/>Anticanis dicitur, exi&longs;timo Canem maiorem e&longs;&longs;e eum, qui vulgò Canicula <lb/>nominatur, <expan abbr="&longs;olet&qacute;">&longs;oletque</expan>; vehementes, ac noxios calores excitare. </s> <s id="id.001950">de quo etiam <lb/>putò Ari&longs;t. intelligere. </s> <s id="id.001951">eius porrò ortus in no&longs;tra poli eleuatione quadra­ <lb/>ginta quinque graduum, circa diem tertium Augu&longs;ti contingit, Sole autem <lb/>10. gradum Leonis occupante. </s> <s id="id.001952">Ex Magini tabulis ante ephemerides.</s></p><p type="main"> <s id="id.001953"><arrow.to.target n="marg156"/></s></p><p type="margin"> <s id="id.001954"><margin.target id="marg156"/>156</s></p><p type="main"> <s id="id.001955">Eodem cap. <emph type="italics"/>(Duobus enim exi&longs;tentibus &longs;egmentis habitabilis regionis: vno <lb/>quidem ad &longs;uperiorem polum, qui no&longs;ter e&longs;t; altero ad alterum, & ad meridiem: <lb/><expan abbr="ea&qacute;">eaque</expan>, tympani &longs;peciem habeant, talem enim figuram terræ excidunt ex centro ip&longs;ius <lb/>ductæ lineæ, & faciunt duos conos, bunc quidem habentem ba&longs;im tropicum, alte­ <lb/>rum autem habentem ba&longs;im circulum &longs;emper manifestum, verticem autem in me­ <lb/>dio terræ. </s> <s id="id.001956">eodem autem modo ad inferiorem polum alij duo coni terræ &longs;egmenta fa­ <lb/>ciunt)<emph.end type="italics"/> vt benè duas ha&longs;ce terræ portiones, quas &longs;olas habitabiles putat Ari­ <lb/>&longs;tot. concipias, <expan abbr="reliquaq;">reliquaque</expan> huius loci intelligas, in&longs;pice &longs;equentem figuram. <lb/><figure id="id.009.01.109.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.109.1.jpg" place="text"/> <lb/>Maior circulus &longs;it cœlum, in quo polus L, articus; M, antarticus, ille eleua­ <lb/>tus &longs;upra no&longs;trum horizontem S N, 45. gradibus, i&longs;te verò totidem infra <lb/>depre&longs;&longs;us. </s> <s id="id.001957"><expan abbr="&longs;int&qacute;">&longs;intque</expan>; diametri circuli &longs;emper apparentium maximi S R, necnon <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.110.jpg" pagenum="110"/>diametri &longs;emper occultorum maximi Y N: tropicorum item T Q, Cancri, <lb/>X O, Capricorni, vt vides in figura. </s> <s id="id.001958">Terra &longs;it A B C H G F E D Z K. à cu­ <lb/>ius centro Z, educantur primo duæ lineæ rectæ Z R, Z S. ad circulum &longs;em­ <lb/>per apparentium maximum, quæ in terra tran&longs;eant per puncta B, K. & iun­ <lb/>gatur linea B K: iam vides conum S R Z, cuius ba&longs;is e&longs;t circulus &longs;emper ap­ <lb/>parens S R, vertex autem Z, in centro terræ, vt ait Ari&longs;tot. <!-- KEEP S--></s> <s id="id.001959">educantur nunc <lb/>duæ aliæ rectæ ad tropicum Cancri Z T, Z Q, quæ in terra faciant puncta <lb/>I, C, <expan abbr="iungatur&qacute;">iungaturque</expan>; recta I C; hic pariter vides conum alterum T Q Z, cuius ba­ <lb/>&longs;is e&longs;t circulus Cancri, vertex verò centrum terræ Z. con&longs;idera iam figuram <lb/>B K I C, inter duas rectas B K, I C, & duos circuli terræ arcus contentam; <lb/>hanc Ari&longs;t. appellat tympanum vnum terræ habitabile, quod e&longs;t ad Vr&longs;am, <lb/>ide&longs;t in &longs;eptentrionali plaga, in qua &longs;umus nos: quæ quidem portio &longs;i con&longs;i­ <lb/>deretur vt &longs;olida, & à reliqua terra præci&longs;a, erit corpus rotundum, <expan abbr="vtrinq;">vtrinque</expan> <lb/>tamen duobus planis circulis ad in&longs;tar tympani terminatum: Ductis dein­ <lb/>de &longs;imiliter alijs quattuor lineis à centro Z, ver&longs;us polum antarticum fit al­ <lb/>terum tympanum H D E G, au&longs;tralis terræ habitabilis, vt in figura manife­ <lb/>&longs;tum e&longs;t. </s> <s id="id.001960">fui&longs;&longs;e autem huiu&longs;modi habitabilis terræ &longs;egmenta figuræ tympa­ <lb/>ni &longs;imilia, optimè declarant veteres figuræ geographicæ Ptol&ecedil;mei, & patet <lb/>etiam ex longitudine, & latitudine, vt benè ait Ari&longs;t. quas Geographi por­ <lb/>tioni terræ habitabili attribuebant, longitudinem enim dixerunt eius di­ <lb/>men&longs;ionem ab occa&longs;u ad ortum: latitudinem autem à &longs;eptentrione in meri­ <lb/>diem, eò quòd illa multò hac longior e&longs;&longs;et. </s> <s id="id.001961">Ex quibus apparet habitatam <lb/>fui&longs;&longs;e veluti Zonam, terram ab occa&longs;u ad ortum præcingentem. </s> <s id="id.001962">quæ Zona <lb/>&longs;i &longs;umatur cum &longs;oliditate, quam ambit, ab Ari&longs;t. tympano a&longs;&longs;imilatur.</s></p><p type="main"> <s id="id.001963"><arrow.to.target n="marg157"/></s></p><p type="margin"> <s id="id.001964"><margin.target id="marg157"/>157</s></p><p type="main"> <s id="id.001965">Eodem cap. <emph type="italics"/>(Hæ autem habitari &longs;olæ po&longs;&longs;ibiles: & <expan abbr="neq;">neque</expan> vltra ver&longs;iones; vm­ <lb/>bra enim non <expan abbr="vtiq;">vtique</expan> e&longs;&longs;et ad Vr&longs;am: nunc autem inhabitabilia prius fiunt loca, quàm <lb/>&longs;ubdeficiat, aut permutetur vmbra ad meridiem. </s> <s id="id.001966">Quæ autem &longs;ub Vr&longs;a, è frigore <lb/>inhabitabilia)<emph.end type="italics"/> quod ait vltra ver&longs;iones, ide&longs;t intra tropicos in ip&longs;a &longs;cilicet <lb/>Zona torrida, non po&longs;&longs;e habitari, fal&longs;um e&longs;&longs;e o&longs;tendunt plurimæ regiones <lb/>tam veteris, quam noui orbis, &longs;uperiori &longs;eculo patefactæ, in quibus magna <lb/>in amœnitate, ac fertilitate, <expan abbr="&longs;ummis&qacute;">&longs;ummisque</expan>; delicijs viuitur. </s> <s id="id.001967">Quoad vmbram il­ <lb/>lam, intellige meridianam. </s> <s id="id.001968">i. </s> <s id="id.001969">quam Sole circa meridiem exi&longs;tente, nos qui <lb/>Boreales &longs;umus, &longs;emper ad <expan abbr="&longs;ept&etilde;trionem">&longs;eptentrionem</expan> proijcimus. </s> <s id="id.001970">Quod &longs;i ad meridiem <lb/>perrexerimus, occurret inhabitabilis (vt falsò putat) terra, prius quam. <lb/></s> <s id="id.001971">vmbra meridiana in Boream vergens deficiat. </s> <s id="id.001972">quæ &longs;igna &longs;unt no&longs;tram habi­ <lb/>tationem e&longs;&longs;e citra Zonam torridam, in Boreali parte. </s> <s id="id.001973">Quæ autem &longs;ub Vr­ <lb/>&longs;a, ide&longs;t &longs;ub polo arctico, ob nimium frigus inho&longs;pita omninò habetur, nam</s></p><p type="main"> <s id="id.001974"><emph type="italics"/>Quod latus mundi nebulæ, <expan abbr="malus&qacute;">malusque</expan>; <lb/>Iupiter vrget.<emph.end type="italics"/></s></p><p type="main"> <s id="id.001975">Verumtamen, quæ &longs;ub <expan abbr="vtroq;">vtroque</expan> polo partes &longs;unt adhuc incognitæ manent.</s></p><p type="main"> <s id="id.001976"><arrow.to.target n="marg158"/></s></p><p type="margin"> <s id="id.001977"><margin.target id="marg158"/>158</s></p><p type="main"> <s id="id.001978">Eodem cap: <emph type="italics"/>(Fertur autem, & corona &longs;ecundam hunc locum, videtur enim &longs;u­ <lb/>per caput e&longs;&longs;e nohis, cum fuerit &longs;ecundum meridianum)<emph.end type="italics"/> con&longs;tellatio videlicet, <lb/>quæ corona Ariadnæ dicitur, hæc cum in cœlo manife&longs;tè &longs;it Borealis, <expan abbr="no-&longs;tro&qacute;">no­ <lb/>&longs;troque</expan>; vertici noctu, quando meridianum pertran&longs;it, incumbat: clarè indi­ <lb/>cat nos <expan abbr="quoq;">quoque</expan> e&longs;&longs;e Boreales.<!-- KEEP S--></s></p><p type="main"> <s id="id.001979"><arrow.to.target n="marg159"/></s></p><p type="margin"> <s id="id.001980"><margin.target id="marg159"/>159</s></p><p type="main"> <s id="id.001981">Eodem cap. <emph type="italics"/>(Et quidem ad latitudinem <expan abbr="v&longs;q;">v&longs;que</expan> ad inhabitabilia &longs;cimus hahita-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.111.jpg" pagenum="111"/><emph type="italics"/>tam, hic enim propter frigus non amplius habitant, illic autem propter æ&longs;tum)<emph.end type="italics"/> <lb/>illic autem, ide&longs;t &longs;ub Zona torrida, compertum autem e&longs;t nunc totam ferè <lb/>torridam Zonam, & quidem alicubi percommodè habitari, cuius cau&longs;æ &longs;unt <lb/>quatuor, quæ ip&longs;um latuerunt. </s> <s id="id.001982">prima <expan abbr="ea&qacute;">eaque</expan>; toti Zonæ torridæ communis, <lb/>e&longs;t perpetuum æquinoctium, quo Sol tantum &longs;upra, quantum infra terram <lb/>immoratur. </s> <s id="id.001983">accedit, quòd Sol nocturno tempore maximè ad imum cœli fe­ <lb/>ratur, <expan abbr="plurimum&qacute;">plurimumque</expan>; ab horizonte, <expan abbr="&longs;upero&qacute;">&longs;uperoque</expan>; hemi&longs;pherio recedat. </s> <s id="id.001984">atque ob <lb/>hanc &longs;olam rationem Campanus in &longs;ua &longs;phæra Zonam hanc putat maximè <lb/>e&longs;&longs;e habitabilem: quamuis hæc &longs;ola cau&longs;a, vt quotidiana docet experientia, <lb/>non &longs;ufficiat. </s> <s id="id.001985">&longs;ecunda &longs;unt pluuiæ, quæ alicubi quotidie &longs;tata hora decidunt. <lb/></s> <s id="id.001986">tertia venti, qui veluti flabella quædam aerem agitant. </s> <s id="id.001987">quarta præalti mon­ <lb/>tes perpetuis niuibus ob&longs;iti. </s> <s id="id.001988">quæ quatuor torridam hanc pa&longs;&longs;im refrigerant, <lb/>atque habitabilem reddunt.</s></p><p type="main"> <s id="id.001989"><arrow.to.target n="marg160"/></s></p><p type="margin"> <s id="id.001990"><margin.target id="marg160"/>160.a</s></p><p type="main"> <s id="id.001991">Summæ 2. cap. 3. de <expan abbr="v&etilde;tis">ventis</expan> <emph type="italics"/>(Oportet autem de &longs;itu &longs;imul rationes ex de&longs;criptio <lb/>ne con&longs;iderare)<emph.end type="italics"/> ide&longs;t rationes ventorum ex de&longs;criptione, ide&longs;t in figura ali­ <lb/>qua, vt in &longs;equenti con&longs;iderare; &longs;olet enim Ari&longs;t. figuras, imò demon&longs;tratio­ <lb/>nes ip&longs;as Mathematicorum, de&longs;criptiones appellare, vt &longs;æpius in Logicis <lb/>monuimus.</s></p><p type="main"> <s id="id.001992"><emph type="italics"/>De&longs;criptus &longs;it igitur, vt clarior res euadat horizontis circulus quapropter, & <lb/>rotundus)<emph.end type="italics"/> vt in &longs;equenti figura circulus A G B H, de&longs;criptus horizontem <lb/>referret,</s></p><figure id="id.009.01.111.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.111.1.jpg" place="text"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.112.jpg" pagenum="112"/><p type="main"> <s id="id.001993"><emph type="italics"/>Oportet autem ip&longs;ius alteram portionem intelligere, quæ nobis habitatur; quæ <lb/>eodem modo diuidi poterit)<emph.end type="italics"/> ide&longs;t oportet intelligere ip&longs;ius horizontis, vel ter­ <lb/>ræ habitatæ partem, quæ quamuis rotunda non &longs;it, poterit tamen, ac &longs;i ro­ <lb/>tunda e&longs;&longs;et in figura circulari repre&longs;entari, <expan abbr="atq;">atque</expan> in plures partes eo modo, <lb/>quo circulus &longs;ecatur, &longs;ecari.</s></p><p type="main"> <s id="id.001994"><emph type="italics"/>Supponatur autem primò contraria &longs;ecundum locum, e&longs;&longs;e plurimum di&longs;tantia <lb/>&longs;ecundum locum; &longs;icut &longs;ecundum &longs;peciem contraria, plurimum di&longs;tant &longs;ecundum <lb/>&longs;peciem. </s> <s id="id.001995">plurimum autem di&longs;tant &longs;ecundum locum, quæ per diametrum opponuntur, <lb/>&longs;it igitur vbi A, occidens æquinoctionalis, contrarius autem huic locus vltimus B, <lb/>ortus æquinoctionalis)<emph.end type="italics"/> ide&longs;t in &longs;equenti figura ducta diametro B A. in altera <lb/>ip&longs;ius extremitate vbi A. &longs;it occa&longs;us æquinoctialis, qui fit Sole exi&longs;tente in <lb/>alterutro æquinoctio; huic igitur per diametrum opponatur ortus æquino­ <lb/>ctialis in B. qui pariter contingit tempore æquinoctiorum: linea autem B A, <lb/>refert ip&longs;um æquatorem.</s></p><p type="main"> <s id="id.001996"><emph type="italics"/>Alia autem diameter hanc perpendiculariter &longs;ecet, cuius punctum illud, in quo <lb/>G, &longs;it Vr&longs;a: huic autem contrarium ex oppo&longs;ito illud, in quo H, meridies)<emph.end type="italics"/> hæc dia­ <lb/>meter erit ip&longs;a linea meridiana. </s> <s id="id.001997">pro Vr&longs;a verò intelligit &longs;eptentrionem, <lb/>quod ibi &longs;it Vr&longs;æ con&longs;tellatio.</s></p><p type="main"> <s id="id.001998"><emph type="italics"/>Id autem, in quo F, ortus æ&longs;tiualis; in quo verò E, occidens æ&longs;tiualis)<emph.end type="italics"/> quæ duo <lb/>puncta iunguntur linea F E, quæ refert &longs;ectionem tropici, Cancri cum ho­ <lb/>rizonte: ortus enim, & occa&longs;us æ&longs;tiualis contingunt Sole Cancri tropicum <lb/>percurrente.</s></p><p type="main"> <s id="id.001999"><emph type="italics"/>Id autem, in quo D, oriens byemalis; vbi verò C, occidens hyemalis)<emph.end type="italics"/> linea au­ <lb/>tem D C, erit &longs;ectio tropici Capricorni, & horizontis; Sole enim hunc tro­ <lb/>picum attingente ortus, & occa&longs;us hybernus fiunt.</s></p><p type="main"> <s id="id.002000"><emph type="italics"/>Ab F, autem ducatur diameter ad C, & à D, ad E. quoniam igitur plurimum <lb/>di&longs;tantia &longs;ecundum locum, contraria &longs;unt &longs;ecundum locum: plurimum autem di­ <lb/>stantia, quæ &longs;ecundum diametrum; nece&longs;&longs;arium e&longs;t, & flatuum hos inuicem con­ <lb/>trarios e&longs;&longs;e, <expan abbr="quicunq;">quicunque</expan> &longs;ecundum diametrum exi&longs;tunt. </s> <s id="id.002001">vocantur autem &longs;ecundum po­ <lb/>&longs;itionem locorum venti &longs;ic; Zephyrus quidem ab A, hoc enim e&longs;t occidens æquino­ <lb/>ctialis. </s> <s id="id.002002">Boreas autem, & Aparetias à G. hic enim Vr&longs;a, contrarius autem huic <lb/>Notus ab H. <!-- KEEP S--></s> <s id="id.002003">Meridies enim e&longs;t hic, à quo flat, & H, ip&longs;i G, contrarium e&longs;t; &longs;ecun­ <lb/>dum enim diametrum &longs;unt. </s> <s id="id.002004">Ab F, autem Cæcias; hic enim oriens æ&longs;tiuus e&longs;t; cui <lb/>contrarius est, non qui flat ab E, &longs;ed qui à C. Libs, i&longs;te enim ab occidente hyemali <lb/>flat; <expan abbr="est&qacute;">estque</expan>, illi contrarius, quia &longs;ecundum diametrum illi opponitur. </s> <s id="id.002005">Qui verò à D, <lb/>Eurus, i&longs;te enim ab horiente hyberno flat, vicinus existens Noto, vnde & &longs;æpè Eu­ <lb/>ronoti flare dicuntur: <expan abbr="cõtrarius">contrarius</expan> autem huic, non qui à C. Libs, &longs;ed qui ab E, quem <lb/>vocant, hi quidem Arge&longs;ten, hi autem Olympium, alij verò Scironem; iste enim ab <lb/>occidente æ&longs;tiuo flat, & &longs;ecundum diametrum ip&longs;i &longs;olus opponitur. </s> <s id="id.002006">Venti igitur, qui <lb/>&longs;ecundum diametrum po&longs;iti &longs;unt, & quibus alij aduer&longs;antur, ij &longs;unt. </s> <s id="id.002007">Alij autem <lb/>&longs;unt, &longs;ecundum quos non &longs;unt contrarij venti, ab I, quem vocant Tra&longs;ciam, qui me­ <lb/>dius e&longs;t inter Argesten, & Apparitiam, à K, autem, quem vocant Me&longs;en, Mtdius <lb/>enim e&longs;t Cæciæ, & Aparetiæ. </s> <s id="id.002008">Diameter autem K I, iuxta circulum &longs;emper con&longs;pi­ <lb/>cuum e&longs;&longs;e &longs;olet, non tamen exactè)<emph.end type="italics"/> ide&longs;t linea K I, &longs;olet in horizonte referre <lb/>diametrum circuli omnium &longs;emper apparentium maximi, eo quod &longs;it ferè <lb/>&longs;ub diametro illius, in qualibet enim &longs;phæra obliqua, ide&longs;t, in qua polus cle­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.113.jpg" pagenum="113"/>natur, intelligunt A&longs;tronomi circulum quendam &longs;emper apparentium ma­ <lb/>ximum, quem de&longs;cribunt ex ip&longs;o polo, tanquam centro, & interuailo v&longs;que <lb/>ad horizontem, circa ip&longs;um polum: hunc appellant &longs;emper apparentium, <lb/>maximum, quia intra hunc alios quamplurimos concipiunt circa eundem <lb/>polum, quorum minores &longs;emper &longs;unt polo propinquiores. </s> <s id="id.002009">huius igitur dia­ <lb/>metrum vult Ari&longs;t. per lineam, quæ à K, in I, duceretur (quamuis non exa­ <lb/>ctè) repre&longs;entari.</s></p><p type="main"> <s id="id.002010"><emph type="italics"/>Contrarij autem non &longs;unt his &longs;latibus, <expan abbr="neq;">neque</expan> ip&longs;i Me&longs;e, &longs;piraret enim <expan abbr="vtiq;">vtique</expan> aliquis <lb/>ab eo, in quo M. hoc enim illi e&longs;t &longs;ecundum diametrum; <expan abbr="neq;">neque</expan> Tra&longs;ciæ ab N, enim, <lb/>quod punctum per diametrum aduer&longs;um illi e&longs;t, &longs;piraret. </s> <s id="id.002011">Ni&longs;i ab eo veniat, qui ta­ <lb/>men non longè progreditur ventus quidam, quem accolæ Phæniciam vocant. </s> <s id="id.002012">maxi­ <lb/>mè igitur præcipui, & definiti venti hi &longs;unt: <expan abbr="hoc&qacute;">hocque</expan>, modo di&longs;po&longs;iti)<emph.end type="italics"/> &longs;upradicta por­ <lb/>rò omnia ex &longs;equenti figura optimè poterunt intelligi, quam diligenti ope­ <lb/>ra ad mentem Ari&longs;t. ex græcis codicibus re&longs;tituere conatus &longs;um, cum ani­ <lb/>maduerterem figuras val dè deprauatas pa&longs;&longs;im apud <expan abbr="cõmentatores">commentatores</expan> reperiri. <lb/></s> <s id="id.002013">Porrò ad literam M, in figura &longs;crip&longs;i ventum Libonotum, quem Ari&longs;t. qui­ <lb/>dem non ponit propter ip&longs;ius paruitatem; imò apertè dicit Hele&longs;pontum <lb/>non habere contrarium: &longs;ed feci, vt completum ventorum numerum, quem <lb/>alij tradunt, haberemus.</s></p></chap><chap><p type="head"> <s id="id.002014"><emph type="italics"/>Ex Tertio Meteororum.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002015"><arrow.to.target n="marg161"/></s></p><p type="margin"> <s id="id.002016"><margin.target id="marg161"/>160.b</s></p><p type="main"> <s id="id.002017">Antequam textuum explicationem aggrediar, illud animaduerten <lb/>dum e&longs;t, <expan abbr="vbicunq;">vbicunque</expan> interpretatio antiqua vtitur verbis, refractio, <lb/>& refrangere; ibi Vicomercatum in &longs;ua interpretatione meritò, <lb/>& propriè v&longs;um e&longs;&longs;e verbis; reflexio, & reflecti: differunt enim <lb/>valdè apud Opticos refractio, & reflexio, vt etiam refrangere, & reflectere. <lb/></s> <s id="id.002018">propterea optimè hoc loco Olympiodorus di&longs;tinguit inter <foreign lang="greek">anaxlasin, xai <lb/>diaxlasin,</foreign> reflexionem, & refractionem. </s> <s id="id.002019">Reflexio enim fit ex repercu&longs;&longs;o, vt <lb/>quando lumen Solis incidens in aliquod &longs;peculum, inde re&longs;ilit in oppo&longs;itum <lb/>parietem, illud re&longs;ilire e&longs;t propriè per&longs;pectiuis reflecti, vnde reflexio. </s> <s id="id.002020">Re­ <lb/>fractio autem fit ex tran&longs;pectu: vt quando lapis, qui e&longs;t in aqua, emittit <lb/>fuam &longs;peciem ad oculum, qui e&longs;t in aere, tunc enim, quia &longs;pecies lapidis re­ <lb/>pre&longs;entatiua non tendit recta ad oculum, &longs;ed in confinio aquæ, & aeris fran­ <lb/>gitur, dicitur fieri refractio, & refrangi, in refractione igitur requiruntur <lb/>duo media, per quæ &longs;iat vi&longs;io, quæ &longs;int diuer&longs;æ den&longs;itatis, vt &longs;unt aqua, & <lb/>aer: vapor, exhalatio, & aer: vitrum, & aer, &c. </s> <s id="id.002021">quando igitur videmus <lb/>Solem, aut Lunam per vapores, aut exhalationes fit refractio, quia den&longs;ior <lb/>e&longs;t vapor, & exhalatio, quam aer.</s></p><p type="main"> <s id="id.002022">Notandum etiam Aream, de qua mox dicam explicari po&longs;&longs;e tam per re­ <lb/>flexionem, quàm per refractionem: per reflexionem, quia &longs;upponunt Philo­ <lb/>&longs;ophi e&longs;&longs;e in acre rorido innumcra &longs;pecula parua inuicem valdè proxima, <lb/>ide&longs;t guttulas, per quas re&longs;lectatur ad oculum no&longs;trum &longs;pecies &longs;yderis. </s> <s id="id.002023">per <lb/>re&longs;ractionem verò, vt vult Vitellio, quia &longs;umit totum illum aerem humi­ <lb/>dum magis den&longs;um e&longs;&longs;e aere paro, qui e&longs;t circa oculos no&longs;tros, & hoc modo <lb/>con&longs;tituit diucr&longs;a media in den&longs;itate, per quam fiat vi&longs;io; corpus inquam <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.114.jpg" pagenum="114"/>illud humidum den&longs;ius, & aerem deinde circa oculum rarius. </s> <s id="id.002024">Vicomerca­ <lb/>tus igitur quamuis vtatur voce reflexionis in Halone, non tamen ex prædi­ <lb/>ctis videtur reprchenden dus.</s></p><p type="main"> <s id="id.002025"><arrow.to.target n="marg162"/></s></p><p type="margin"> <s id="id.002026"><margin.target id="marg162"/>161</s></p><p type="main"> <s id="id.002027">Summæ 2. cap. 2. De Areæ figura <emph type="italics"/>(Refrangitur autem à con&longs;i&longs;tente caligine <lb/>circa Solem, aut Lunam vi&longs;us; quapropter non ex oppo&longs;ito &longs;icut iris, apparet. </s> <s id="id.002028"><expan abbr="Vn-diq;">Vn­ <lb/>dique</expan> autem &longs;imiliter refracto, nece&longs;&longs;e e&longs;t circulum e&longs;&longs;e, aut circuli partem. </s> <s id="id.002029">ab co­ <lb/>dem enim &longs;igno ad idem &longs;ignum æquales frangentur &longs;uper circuli lineam &longs;emper. </s> <s id="id.002030">&longs;it<emph.end type="italics"/> <lb/><figure id="id.009.01.114.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.114.1.jpg" place="text"/> <lb/><emph type="italics"/>enim à puncto, in quo A, ad B, fracta, & ea, quæ est <lb/>A C B, & quæ A F B, & quæ A D B, æquales autem <lb/>& hæ A C, A F, A D, inuiccm. </s> <s id="id.002031">& quæ ad B, inui­ <lb/>cem &longs;cilicet C B, E B, D B. & protrahatur A E B, <lb/>quare trianguli æquales, etenim &longs;uper æqualem, quæ <lb/>e&longs;t A E B, ducantur autem <expan abbr="perp&etilde;diculares">perpendiculares</expan> ad A E B, <lb/>ex angulis; à C, quidem, quæ e&longs;t C E; ab F, autem, <lb/>quæ e&longs;t F E; à D, autem, quæ e&longs;t D E, æquales itaque <lb/>hæ, in æqualibus enim triăgulis, & in vno plano om­ <lb/>nes, ad rectum emm omnes ei, quæ e&longs;t A E B. & ad <lb/>vnum punctum E, copulantur, circulus igitur erit <lb/>de&longs;cripta, centrum autem E. &longs;it autem B, quidem Sol, <lb/>A, autem vi&longs;us, quæ autem e&longs;t circa C D F, circun­ <lb/>ferentia nubes, à qua refrangitur vi&longs;us ad Solem)<emph.end type="italics"/> <lb/>quia &longs;uppono Aream, &longs;iue Halonem fieri per re­ <lb/>fractionem, vt vult etiam Vitellio, propterea <lb/><expan abbr="præmitt&etilde;dum">præmittendum</expan> e&longs;t principium quoddam, quo tra­ <lb/>ctatio de refractione innititur; e&longs;t autem huiu&longs;­ <lb/>modi; ea, quæ <expan abbr="vid&etilde;tur">videntur</expan> per refractionem, &longs;iue &longs;ub <lb/>aliquo refractionis angulo, manentibus nobis & <lb/>a&longs;tro, & medio ij&longs;dem in locis, non po&longs;&longs;unt vide­ <lb/>ri &longs;ub diuer&longs;o angulo à priori, nec per con&longs;e<expan abbr="qu&etilde;s">quens</expan> <lb/>alibi apparere. </s> <s id="id.002032">v. <!-- REMOVE S-->g. <!-- REMOVE S-->Sol (vt in præ&longs;enti figura) <lb/>videatur ab oculo A, media nube C D F, &longs;ub an­ <lb/>gulo refractionis B C A, vel B F A, & alijs &longs;imilibus angulis in eadem nube; <lb/>manente igitur oculo A, & a&longs;tro B, necnon nube C D E. eodem in loco, im­ <lb/>po&longs;&longs;ibile e&longs;t Solem videri ab eodem oculo &longs;ub diuer&longs;o angulo à priori, nec <lb/>con&longs;equenter alibi apparere, quam in B. <!-- KEEP S--></s> <s id="id.002033">Nunc ad textus declarationem, in <lb/>quo continetur Geometrica demon&longs;tratio rotunditatis Areæ, quam &longs;ic bre­ <lb/>uiter prius veteres excogitarunt: Viderunt primò Solem in Area apparere <lb/>in orbem, & con&longs;imiliter: hinc intulerunt nece&longs;&longs;e e&longs;&longs;e apparere etiam per <lb/>con&longs;imiles, &longs;iue æquales refractionis angulos; quia diuer&longs;i anguli, diuer&longs;am <lb/>etiam <expan abbr="appar&etilde;tiam">apparentiam</expan> efficiunt: atqui con&longs;imiles, &longs;iue æquales refractionis an­ <lb/>gulos nece&longs;&longs;e e&longs;t in circulum <expan abbr="cõ&longs;titui">con&longs;titui</expan>, vt mox con&longs;tabit; cau&longs;a igitur rotun­ <lb/>ditatis huius, e&longs;t angulorum refractionis æqualitas. </s> <s id="id.002034">Sed iam textum Ari&longs;t. <lb/>qui geometricam huius rci continet demon&longs;trationem, explicemus. </s> <s id="id.002035">Suppo­ <lb/>nit igitur primò Ari&longs;t. lineas vi&longs;uales à &longs;ydere B, ad oculos no&longs;tros A, per <lb/>nubem roridam C D F, procedentes, in nube con&longs;imiliter refrangi, ide&longs;t <expan abbr="vn-diq;">vn­ <lb/>dique</expan> circa Solem, Lunamuè facere angulos refractionis æquales. </s> <s id="id.002036">quod etiam <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.115.jpg" pagenum="115"/>patet ex 48. 10. Vitellionis; vt in figura, in qua &longs;ydus B, oculus A, nubes <lb/>C D F, radij vi&longs;uales tres refracti in nube &longs;int B C A, B D A, B E A, facien­ <lb/>tes con&longs;imilem refractionem, ide&longs;t angulos refractos B C A, B D A, B E A, <lb/>æquales in punctis C, D, F: <expan abbr="atq;">atque</expan> hoc e&longs;t con&longs;imilem facere refractionem. <lb/></s> <s id="id.002037">Supponit &longs;ecundò lineas à &longs;ydere ad nubem, v&longs;que exten&longs;as e&longs;&longs;e æquales, vt <lb/>&longs;unt B C, B D, B F: &longs;imiliter reliquas tres à nube ad vi&longs;um A. pares e&longs;&longs;e C A, <lb/>D A, F A. his &longs;uppo&longs;itis, &longs;i deinde protrahatur recta A B, ab oculo ad &longs;ydus, <lb/>exurgunt tria triangula omninò æqualia, & &longs;imilia, cuni duo latera vnius <lb/>&longs;int æqualia duobus alterius <expan abbr="vtrunq;">vtrunque</expan> vtrique, & angulus angulo, & præterea <lb/>ba&longs;is &longs;it communis; ideò per quartam primi &longs;unt omninò æqualia. </s> <s id="id.002038">ducan­ <lb/>tur nunc ex angulis C, D, F, tres perpendiculares ad rectam A B, quæ &longs;int <lb/>C E, D E, F E, in figura; quæ tres nece&longs;&longs;ariò erunt æquales, cum &longs;int ductæ <lb/>ab angulis æqualibus æqualium triangulorum ad communem ba&longs;im, & di­ <lb/>uident nece&longs;&longs;ariò ba&longs;im in eodem puncto E, cum diuidant triangula æqua­ <lb/>lia proportionaliter; <expan abbr="erunt&qacute;">eruntque</expan>; propterea hæ tres rectæ in eodem plano, quod <lb/>in nube concipitur ex 5. 11. Quare &longs;i concipiamus &longs;uperficiem, &longs;iue planum <lb/>delineari circa E, ad interuallum linearum æqualium C E, D E, F E, de­ <lb/>&longs;criptus erit circulus per 9. tertij, cuius circumferentia C D F. <!-- KEEP S--></s> <s id="id.002039">Ex quibus <lb/>patet tria illa puncta C, D, E, per quæ Sol tran&longs;paret e&longs;&longs;e in orbem di&longs;po&longs;i­ <lb/>ta. </s> <s id="id.002040">cau&longs;a igitur rotunditatis Areæ, e&longs;t &longs;imilitudo angulorum refractionis, <lb/>quibus Sol tran&longs;paret: vel ideo rotunda e&longs;t, quia &longs;imiles anguli nece&longs;&longs;ariò <lb/>in orbem con&longs;tituuntur, vt o&longs;ten&longs;um e&longs;t. </s> <s id="id.002041">Eadem ratione omnia alia puncta <lb/>eiu&longs;dem <expan abbr="circũferentiæ">circunferentiæ</expan> &longs;unt puncta, per quæ Sol videtur refractè; & hoc mo­ <lb/>do ad &longs;imilitudinem trium linearum A C B, A D B, A F B, refractarum, in­ <lb/>finitæ <expan abbr="vndiq;">vndique</expan> intelligendæ &longs;unt, quarum aliæ refrangantur in circunferentia <lb/>prædicta, aliæ verò in alia periphæria maiori, aliæ etiam in minori, ita vt <lb/>ex tota nube fiant refractiones circulares plurimæ, ex quibus in nube area <lb/>con&longs;tituatur. </s> <s id="id.002042"><expan abbr="Atq;">Atque</expan> hæc cur Halonis figura orbicularis videatur, rationem <lb/>reddunt, <expan abbr="vna&qacute;">vnaque</expan>; textui lucem afferunt.</s></p><p type="head"> <s id="id.002043"><emph type="italics"/>Summæ 2. cap. 4. De Iridis figura.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002044"><arrow.to.target n="marg163"/></s></p><p type="margin"> <s id="id.002045"><margin.target id="marg163"/>162</s></p><p type="main"> <s id="id.002046"><emph type="italics"/>Qvod autem <expan abbr="neq;">neque</expan> circulum po&longs;&longs;ibile &longs;it fieri Iridis, <expan abbr="neq;">neque</expan> maiorem &longs;emicir­ <lb/>culo portionem, & de alijs accidentibus circa ip&longs;am, ex de&longs;criptione <lb/>erit con&longs;iderantibus manife&longs;tum)<emph.end type="italics"/> In Logicis &longs;æpius monui Ari&longs;t. per <lb/>de&longs;criptiones intelligere geometricas demon&longs;trationes, quod <lb/>etiam hoc loco confirmatur, vbi Geometrica demon&longs;tratione quam de&longs;cri­ <lb/>ptionem appellat, Iridis figuræ accidentia o&longs;tendit; nimirum cur &longs;it quidem <lb/>circularis, nunquam tamen circulus integer, imò <expan abbr="neq;">neque</expan> &longs;emicirculo vnquam <lb/>maior, &longs;ed tamen &longs;emicirculo minor.</s></p><p type="main"> <s id="id.002047"><arrow.to.target n="marg164"/></s></p><p type="margin"> <s id="id.002048"><margin.target id="marg164"/>163</s></p><p type="main"> <s id="id.002049">Ibidem <emph type="italics"/>(Hemi&longs;pberio enim exi&longs;t exte &longs;uper horizontis circulum in quo A. cen­ <lb/>tro autem K, alio autem quodam oriente puncto, in quo G, &longs;i lineæ, quæ à K, &longs;ecun­ <lb/>dum conum excidentes faciant velut axem lineam in qua G K, & à K. ad M, co­ <lb/>pulatæ refrangantur ab hemi&longs;phærio ad G, &longs;uper maiorem angulum, circuli circun­ <lb/>ferentiam incident lineæ, quæ à K, & &longs;i quidem in ortu, aut in occa&longs;u a&longs;tri reflexio <lb/>fiat, &longs;emicirculus ab <expan abbr="horizõte">horizonte</expan> a&longs;&longs;umetur &longs;uper terram factus. </s> <s id="id.002050">&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.116.jpg" pagenum="116"/><figure id="id.009.01.116.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.116.1.jpg" place="text"/> <lb/><emph type="italics"/>&longs;emper &longs;emicirculo, minus autem, <lb/>cum in meridie fuerit a&longs;trum)<emph.end type="italics"/> quod <lb/>&longs;upra monui, iterum moneo, <expan abbr="re-tin&etilde;dam">re­ <lb/>tinendam</expan> vocem reflexionis, <expan abbr="quã-uis">quan­ <lb/>uis</expan> in antiqua tran&longs;latione lega­ <lb/>tur refractio, e&longs;t enim apud om­ <lb/>nes in confe&longs;&longs;o Iridem fieri per <lb/>reflexionem. </s> <s id="id.002051">E&longs;t igitur in &longs;upe­ <lb/>riori figura, quam textui, vt par <lb/>erat re&longs;titui, horizon G K O. cuius centrum K. in quo e&longs;t vi&longs;us no&longs;ter, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; <lb/>hemi&longs;phærium no&longs;trum in arcu G A M O, repræ&longs;entatum, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; nubes rori­ <lb/>da, in qua Iris appareat, vbi M, quod punctum M, nubem referens, in figu­ <lb/>ra ponitur in hemi&longs;phærij ambitu, quod cœlum repræ&longs;entat, cum tamen <lb/>nubes parum à terra &longs;ubuchatur; id enim ad demon&longs;trationem ferè perinde <lb/>e&longs;t. </s> <s id="id.002052">in oriente G, &longs;it a&longs;trum. </s> <s id="id.002053">&longs;i ergò lineæ vi&longs;uales à K, ad M, nubem tenden­ <lb/>tes reflectantur &longs;uper maiorem angulum M K G, ad G, erit reflexarum vna <lb/>veluti M G. <!-- KEEP S--></s> <s id="id.002054">Porro omnes lineæ viluales, quæ adnubem M, incidunt, nece&longs;­ <lb/>&longs;ariò, vt probabo, cadent in ambitum circularem. </s> <s id="id.002055">debemus enim innume­ <lb/>ras lineas im aginari à K, in coni figuram excidentes, cuius vertex &longs;it in K, <lb/>& axis G K O, quas omnes repræ&longs;entat vna K M, <expan abbr="melius&qacute;">meliusque</expan>; repræ&longs;entabit, fi <lb/>cogitemus axem G K O, circa polos G, O, manentes circumuolui, <expan abbr="&longs;ecum&qacute;">&longs;ecumque</expan>; <lb/>lineam K M, circumducere. </s> <s id="id.002056">in hac etiam giratione linea K M, tran&longs;ibit per <lb/>omnes illas lineas, quas imaginabamur; <expan abbr="de&longs;cribet&qacute;">de&longs;cribetque</expan>; conum, quem illæ con­ <lb/>formare debebant. </s> <s id="id.002057">In prædicta autem axis volutatione, extremum M, li­ <lb/>neæ K M, nece&longs;&longs;ariò de&longs;cribit circulum, qui e&longs;t circulus Iridis, & e&longs;t ba&longs;is <lb/>memorati coni.</s></p><p type="main"> <s id="id.002058">Si igitur oriente, vel occidente a&longs;tro fiat iris, Iris erit &longs;emicirculus, ide&longs;t <lb/>illa &longs;emi&longs;&longs;is circuli pr&ecedil;dicti (quem horizon bifariam diuidit) quæ &longs;upra ter­ <lb/>ram extabit. </s> <s id="id.002059">&longs;i autem a&longs;trum eleuatum &longs;upra horizontem fuerit, quando fit <lb/>iris, erit &longs;emper arcus Iridis &longs;emicirculo minor; <expan abbr="tunc&qacute;">tuncque</expan>; minimus <expan abbr="cũ">cum</expan> a&longs;trum <lb/><expan abbr="meridianũ">meridianum</expan> <expan abbr="circulũ">circulum</expan> occupauerit. </s> <s id="id.002060">h&ecedil;c tria &longs;unt, quæ deinceps <expan abbr="probãda">probanda</expan> recipit.</s></p><p type="main"> <s id="id.002061"><arrow.to.target n="marg165"/></s></p><p type="margin"> <s id="id.002062"><margin.target id="marg165"/>264</s></p><figure id="id.009.01.116.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.116.2.jpg" place="text"/><p type="main"> <s id="id.002063">Ibidem <emph type="italics"/>(Sit enim in <expan abbr="ori&etilde;te">oriente</expan> pri­ <lb/>mum vbi G, & refracta &longs;it K M, <lb/>ad G, & planum erectum &longs;it in quo <lb/>A, à triangulo in quo G K M, cir­ <lb/>culus igitur erit &longs;ectio &longs;phæræ, qui <lb/>maximus &longs;it in quo A, differet enim <lb/>mbil &longs;i quodc<expan abbr="ŭq;">ŭque</expan> eorum, quæ &longs;uper <lb/>G K, &longs;ecundum triangulŭ K M G, <lb/>erectum fuerit planum. </s> <s id="id.002064">lineæ igitur <lb/>ab ijs, quæ G, K, ductæ in bac ratio­ <lb/>ne non cen&longs;tituentur ad aliud, & <lb/>aliud punctum, quàm &longs;emicirculi <lb/>in quo A. <!-- KEEP S--></s> <s id="id.002065">Quoniam enim puncta <lb/>G, K, data &longs;unt, & quæ K M, vtique data erit; & quæ M G, ad M K; datam igi­ <lb/>tur circunferentiam tanget M, fit <expan abbr="itaq;">itaque</expan> bæc in qua M N, quare &longs;ectio circunferen-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.117.jpg" pagenum="117"/><emph type="italics"/>tiarum data e&longs;t. </s> <s id="id.002066">apud autem aliud punctum, quam ip&longs;ius M N, circunferentiæ, ab <lb/>ij&longs;dem punctis, eadem ratio in eodem plano non con&longs;i&longs;tit)<emph.end type="italics"/> eorum omnium, quæ <lb/>demon&longs;tranda &longs;unt, præmittenda &longs;unt duo nece&longs;&longs;aria fundamenta. </s> <s id="id.002067">Primum <lb/>e&longs;t; ea, quæ videmus per reflexionem &longs;ub quopiam angulo, manentibus no­ <lb/>bis &longs;peculo, & obiecto ij&longs;dem in locis, non po&longs;&longs;unt videri &longs;ub alio diuer&longs;o <lb/>angulo, nec alibi con&longs;equenter apparere. </s> <s id="id.002068">v. <!-- REMOVE S-->g. <!-- REMOVE S-->in &longs;uperiori figura, quam <lb/>textui re&longs;tituimus exi&longs;tente Sole in G, oculo in K, & nube in M. ex qua ra­ <lb/>dius Solis G M, re&longs;lectatur ad vi&longs;um in K, per <expan abbr="lineã">lineam</expan> M K, &longs;ub angulo G M K, <lb/>impo&longs;&longs;ibile e&longs;t manentibus illis, vt dixi, videri Solem in nube M, &longs;ub diuer­ <lb/>&longs;o angulo à priori, nec alibi apparere. </s> <s id="id.002069">Alterum e&longs;t apud Opticos vulga­ <lb/>tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden­ <lb/>tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo <lb/>reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri <lb/><figure id="id.009.01.117.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.117.1.jpg" place="text"/> <lb/>pote&longs;t) vtitur. </s> <s id="id.002070">v. <!-- REMOVE S-->g. <!-- REMOVE S-->in figura præ&longs;enti &longs;it &longs;pe­ <lb/>culum C D E, obiectum A, oculus B, linea in­ <lb/>cidentiæ e&longs;t A D, & angulus pariter inciden­ <lb/>tiæ e&longs;t A D C. linea verò D B, e&longs;t linea refle­ <lb/>xionis, & angulus pariter reflexionis e&longs;t B D­ <lb/>E, qui duo anguli ni&longs;i fuerint æquales, nun­ <lb/>quam videbitur obiectum A, ab oculo B, hinc <lb/>e&longs;t, quod aliquando po&longs;ito &longs;peculo, obiectum <lb/>quamuis illi aduer&longs;um, à nobis pariter ante <lb/>&longs;peculum con&longs;titutis, videri nequit, quia &longs;ci­ <lb/>licet in tali po&longs;itione &longs;peculi, obiecti, & no&longs;tri, nulla linea incidentiæ, ide&longs;t, <lb/>quæ ab obiecto in &longs;peculum tendit, facere pote&longs;t angulum cum &longs;peculo, qui <lb/>dicitur angulus incidentiæ, æqualem angulo illi, quem facit linea eadem re­ <lb/>flexa à &longs;peculo ad oculum, quem dicunt angulum re&longs;lexionis. </s> <s id="id.002071">Cum ergo in <lb/>Iride videamus colorem Solis per reflexionem, tunc &longs;olum apparebit Iris, <lb/>quando Sol, nubes, & oculus fuerint in ea con&longs;titutione, qua radius <expan abbr="incid&etilde;s">incidens</expan> <lb/>nubi, & radius à nube repercu&longs;&longs;us faciant pares angulos. </s> <s id="id.002072">Et quia quando <lb/>nubes ro&longs;cida perpendiculariter opponitur Soli, & nobis, po&longs;&longs;unt &longs;ieri præ­ <lb/>dicti anguli æquales non in vno loco nubis, &longs;ed in pluribus, con&longs;titutis ta­ <lb/>men in circuli periphæria, hinc fit, quod Solis color reflectatur ex pluribus <lb/>locis in orbem con&longs;titutis, quæ reflexio e&longs;t ip&longs;ius Iridis arcus. </s> <s id="id.002073">ex Vitellion <lb/>63. 10. Totam autem figuræ Iridis demon&longs;trationem &longs;ic breuiter puto ad­ <lb/>inuentam e&longs;&longs;e. </s> <s id="id.002074">cum Sol in Iride videatur in orbem, <expan abbr="atq;">atque</expan> con&longs;imiliter, ne ce&longs;­ <lb/>&longs;e e&longs;t id prouenire ex angulis reflexionum con&longs;imilibus, &longs;iue æqualibus: di&longs;­ <lb/>&longs;imiles enim anguli, di&longs;&longs;imilem <expan abbr="vtiq;">vtique</expan> efficiunt Solis <expan abbr="appar&etilde;tiam">apparentiam</expan>. </s> <s id="id.002075">atqui con­ <lb/>&longs;imiles anguli, &longs;iue æquales, non ni&longs;i in orbem po&longs;&longs;unt con&longs;titui; igitur an­ <lb/>gulorum æqualitas cau&longs;a erit rotundationis arcus. </s> <s id="id.002076">h&ecedil;c e&longs;t &longs;umma totius di­ <lb/>&longs;cur&longs;us, quem pluribus, & nimis ob&longs;curè Ari&longs;t. explicat.</s></p><p type="main"> <s id="id.002077">Inquit igitur Ari&longs;t. &longs;it enim in oriente, &c. </s> <s id="id.002078">vbi aggreditur probare vnum <lb/>ex tribus illis, quæ &longs;upra propo&longs;uit, nimirum tunc Iridem e&longs;&longs;e &longs;emicircu­ <lb/>lum, quando a&longs;trum fuerit in oriente, &longs;iue in horizonte, vbi G. &longs;i igitur per <lb/>triangulum G M K, intelligamus <expan abbr="planũ">planum</expan> exten&longs;um, in quo A, in figura, adeo <lb/>magnum, vt totum &longs;ecet hemi&longs;phærium, faciet in &longs;uperficie hemi&longs;phærij &longs;e­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.118.jpg" pagenum="118"/>ctionem, quæ erit portio maximi circuli, per 6. Theodo&longs;ij, cum planum &longs;e­ <lb/>cans hemi&longs;phærium, tran&longs;eat per <expan abbr="centrũ">centrum</expan> ip&longs;ius, quæ &longs;ectio, &longs;iue circuli por­ <lb/>tio repræ&longs;entatur in figura, per &longs;emicirculum in quo A, &longs;iue in quo G A M­ <lb/>R O. nihil autem refert quodcunque intelligas planum &longs;uper axem G K O, <lb/>tran&longs;iens &longs;iue per triangulum G K M, &longs;iue per aliud illi &longs;imile. </s> <s id="id.002079">Præmitten­ <lb/>dum præterea non po&longs;&longs;e in &longs;emicirculo &longs;uperiori, quod e&longs;t planum, & &longs;ectio <lb/>trianguli G K M, poni alias duas lineas. </s> <s id="id.002080">v. <!-- REMOVE S-->g. <!-- REMOVE S-->G R, K R, ad aliud punctum, <lb/>vti e&longs;t R, quæ habeant eandem inuicem proportionem, quam habent prio­ <lb/>res duæ G M, K M, quod probatur, quia &longs;i &longs;int vt G M, ad K M, ita G R, ad <lb/>K R, cum G R, &longs;it centro K, propinquior quam G M, erit etiam eadem G R, <lb/>longior ip&longs;a G M, per 15. 3. & tamen deberet e&longs;&longs;e æqualis illi; quemadmo­ <lb/>dum K M, e&longs;t æqualis alteri K R; nequeunt autem duæ lineæ inæquales inui­ <lb/>cem, habere eandem rationem ad duas inuicem æquales: ergo non habent <lb/>eandem rationem G M, & K M, quam habent G R, & K R. quod &longs;i punctum <lb/>R, &longs;umatur &longs;upra M, erit &longs;imilis <expan abbr="demõ&longs;tratio">demon&longs;tratio</expan>, &longs;i literæ M, & R, loca permu­ <lb/>tent. </s> <s id="id.002081">his po&longs;itis, ait <emph type="italics"/>(Quoniam enim G, K, puncta data &longs;unt, & c.)<emph.end type="italics"/> ide&longs;t data <lb/>&longs;unt po&longs;itione, cum notum &longs;it vbi &longs;int. </s> <s id="id.002082">G, enim e&longs;t in ortu. </s> <s id="id.002083">K, verò in centro <lb/>horizontis, &longs;equitur, quod etiam linea G K, cuins ip&longs;a &longs;unt extrema, data <lb/>&longs;it, & po&longs;itione, & magnitudine, per 26. Datorum Euclidis. <!-- KEEP S--></s> <s id="id.002084">eadem quoque <lb/>ratione data erit K M, linea; &longs;iue quia e&longs;t æqualis ip&longs;i G K, &longs;iue quia per <lb/>a&longs;trolabium po&longs;&longs;umus ip&longs;ius longitudinem, & po&longs;itionem inue&longs;tigare; qua­ <lb/>re & punctum M, datum erit per 27. Datorum, quare & linea G M, data <lb/>erit quoad &longs;itum, & magnitudinem per 26. Datorum. <!-- KEEP S--></s> <s id="id.002085">Quare per primam <lb/>Datorum erit data proportio linearum G M, M K, punctum <expan abbr="itaq;">itaque</expan> M, tange t <lb/>ambitum datum, qui ba&longs;is e&longs;t coni, quem linea K M, de&longs;cribit in reuolutio­ <lb/>ne axis G K O, &longs;uper polis G, O. cum enim data &longs;it K M, po&longs;itu, & magni­ <lb/>tudine, <expan abbr="ea&qacute;">eaque</expan>; &longs;it latus prædicti coni, &longs;equitur periphæriam, vel ambitum ba­ <lb/>&longs;is coni e&longs;&longs;e datum per &longs;imilem definitionem 5. definitioni Datorum. <!-- KEEP S--></s> <s id="id.002086">&longs;it <expan abbr="au-t&etilde;">au­ <lb/>tem</expan> ambitus ille in figura &longs;equenti notatus literis L M N. qui ambitus L M N, <lb/>non e&longs;t <expan abbr="concipi&etilde;dus">concipiendus</expan> in eodem plano &longs;emicirculi G A N O, quemadmodum <lb/>falsò pingitur in figura; &longs;ed debemus ip&longs;um concipere tanquam erectum ad <lb/>angulos rectos cum prædicto &longs;emicirculo, necnon cum horizonte G K O. <lb/><!-- KEEP S--></s> <s id="id.002087">Iam &longs;i <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O, <lb/>punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum <lb/>inquit Ariltot. <!-- REMOVE S-->linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure id="id.009.01.118.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.118.1.jpg" place="text"/> <lb/>Erit præterea &longs;ectio circunferentiarum ho­ <lb/>rizontis, & huius amb tus data, cuius extre­ <lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan> <lb/>in figura non &longs;olum horizontis diametrum <lb/>G K O, &longs;ed etiam circunferentiam (in qua <lb/>circunferentia e&longs;&longs;ent duo illa puncta L, & N, <lb/>vt in præ&longs;enti de&longs;criptione melius intelli ge­ <lb/>tur, in qua horizon G N O L, & ambitus <lb/>prædictus e&longs;t L M N, qui debet intelligi ele­ <lb/>uatus &longs;upra horizontem perpendiculariter) <lb/>tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.119.jpg" pagenum="119"/>linea N P L, cuius extrema puncta &longs;unt L, N, quæ data erunt, cum &longs;int ex­ <lb/>trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum <lb/>M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio <lb/>N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.<!-- KEEP S--></s> </p><p type="main"> <s id="id.002088">Illud nunc in memoriam <expan abbr="reuocãdum">reuocandum</expan>, quod paulò ante probaui, nimirum <lb/>proportionem linearum G M, K M, non po&longs;&longs;e &longs;eruari in alijs lineis, quæ &longs;int <lb/>in eodem plano trianguli G M K, &longs;i ducantur ab ij&longs;dem punctis G, K. pote&longs;t <lb/>tamen &longs;eruari in alijs duabus, quæ cadant in prædictum ambitum, &longs;iue <expan abbr="cir-cunfer&etilde;tiam">cir­ <lb/>cunferentiam</expan> L M N, <expan abbr="quæ&qacute;">quæque</expan>; &longs;int in alio plano, <expan abbr="quã">quam</expan> in plano trianguli G M K, <lb/>quod tamen tran&longs;eat per axem G K O, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vnum ex planis illis, de quibus <lb/>&longs;upra dictum e&longs;t. </s> <s id="id.002089">Verumenimuerò ad quid probatio hæc? </s> <s id="id.002090">non po&longs;&longs;e duas <lb/>alias lineas in eodem plano, &c.? exi&longs;timo Ari&longs;t. idcircò hoc proba&longs;&longs;e, quia <lb/>&longs;i aliæ duæ lineæ habentes eandem rationem, po&longs;&longs;ent collocari in eodem <lb/>plano; e&longs;&longs;ent <expan abbr="permutãdo">permutando</expan> illæ duæ (in priori figura) G R, R K. <expan abbr="vtraq;">vtraque</expan> <expan abbr="vtriq;">vtrique</expan> <lb/>æquales prioribus G M, M K, per quas videtur Iris, cum enim K R, &longs;it æqua­ <lb/>lis ip&longs;i K M, erit, & G M, æqualis ip&longs;i G R, per 7. 5. & in eius &longs;cholio. </s> <s id="id.002091">qua­ <lb/>re natura ageret tam per lineas breui&longs;&longs;imas <expan abbr="ag&etilde;do">agendo</expan> per has, quam per illas, <lb/><expan abbr="hoc&qacute;">hocque</expan>; pacto perhas etiam Iris videri po&longs;&longs;et. </s> <s id="id.002092">cum ergò con&longs;tet non po&longs;&longs;e has <lb/>e&longs;&longs;e prioribus proportionales, &longs;ed maiorem, vel minorem, alteram illarum, <lb/>quàm &longs;it G M, &longs;equitur, quod non faciunt angulum æqualem angulo G M K, <lb/>&longs;ub quo videtur Iris, <expan abbr="nimirũ">nimirum</expan> angulum G R K, qui &longs;it æqualis angulo G M K; <lb/>habet enim Iris hunc angulum determinatum, ita vt &longs;ub maiori, vel mino­ <lb/>ri videri nequeat; ex 10. Bapti&longs;ta Porta. </s> <s id="id.002093">&longs;i autem punctum R, e&longs;&longs;et infra M, <lb/>angulus G R K, e&longs;&longs;et minor angulo Iridis G M K, &longs;i verò &longs;upra e&longs;&longs;et maior <lb/>eodem, quod vel ad &longs;en&longs;um patere pote&longs;t in quouis circulo, <expan abbr="id&qacute;">idque</expan>; &longs;ufficiat, ne <lb/>longior euadat hæc tractatio. </s> <s id="id.002094">Fortè etiam addi pote&longs;t, quod alibi exi&longs;ten­ <lb/>te puncto R, quàm in M, non po&longs;&longs;ent anguli incidentiæ, & reflexionis e&longs;&longs;e <lb/>æquales, quæ cau&longs;a e&longs;&longs;et cur &longs;ub alio angulo, quam prædicto G M K, Iris <lb/>non appareret.</s></p><p type="main"> <s id="id.002095">Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta, <lb/>nam &longs;ecundum veritatem po&longs;&longs;unt in eadem nube con&longs;titui plures anguli <lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alterũ">alterum</expan>; vt in figura præ­ <lb/><figure id="id.009.01.119.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.119.1.jpg" place="text"/> <lb/>&longs;enti, &longs;i nubes e&longs;&longs;et vbi B D. <lb/>oculus in C, Sol in A. e&longs;&longs;ent <lb/>duo anguli A B C, A D C, æ­ <lb/>quales per 33. 3. qui tamen <lb/>non &longs;unt in gyrum con&longs;tituti, <lb/>po&longs;&longs;et igitur, per <expan abbr="illorũ">illorum</expan> vtrun­ <lb/>que Sol Iridem efficere. </s> <s id="id.002096">atque <lb/>animaduer&longs;io h&ecedil;c videtur ma­ <lb/>gni <expan abbr="mom&etilde;ti">momenti</expan> e&longs;&longs;e, ad Iridis <expan abbr="de-mon&longs;tration&etilde;">de­ <lb/>mon&longs;trationem</expan> con&longs;tituendam: <lb/>cum hinc v&longs;itatæ demon&longs;tra­ <lb/>tiones infringatur. </s> <s id="id.002097">Fortè confu giendum e&longs;t ad illud, quod Maurolycus, & <lb/>10. Bapti&longs;ta Porta ob&longs;eruarunt; debere <expan abbr="nimirũ">nimirum</expan> di&longs;tantiam ab oculo ad cen­ <lb/>trum Iridis e&longs;&longs;e æqualem altitudini, &longs;iue &longs;emidiametro Iridis. </s> <s id="id.002098">Ita vt non &longs;o­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.120.jpg" pagenum="120"/>lum requiratur idem angulus, &longs;ed etiam tanta Iridis altitudo, <expan abbr="quãta">quanta</expan> requi­ <lb/>ritur vt angulus in orbem con&longs;tituatur, ex quo Iris po&longs;&longs;it apparere. </s> <s id="id.002099">hæc à <lb/>nemine hactenus animaduer&longs;a placuit addere, vt ex ijs demon&longs;tratio Iridis <lb/>omnibus numeris aliquando ab&longs;olui po&longs;&longs;it, quod infra (ni fallor, fauente <lb/>Deo) præ&longs;tabimus. <lb/><arrow.to.target n="marg166"/></s></p><p type="margin"> <s id="id.002100"><margin.target id="marg166"/>165</s></p><p type="main"> <s id="id.002101">Ibidem <emph type="italics"/>(Extraponatur igitur quædam linea, quæ D B, & &longs;eindatur vt M G, ad­ <lb/>M K, &longs;ic quæ D, ad B, maior autem quæ M G, ea quàm M K, quoniam &longs;uper ma­ <lb/>iorem angulum reflexio coni, maiori enim angulo &longs;ubtenditur trianguli M K G. <lb/><!-- KEEP S--></s> <s id="id.002102">Maior igitur e&longs;t & ip&longs;a D, ip&longs;a B. addatur igitur ad eam, quæ B, ea in qua F, vt <lb/>&longs;it quod D, ad B, quæ B F, ad D. <!-- KEEP S--></s> <s id="id.002103">Deinde quod F, ad K G, quæ B, ad aliam fiat, <lb/>quæ K P. & à P, ad M, copuletur quæ P M, erit igitur P. polus circuli, ad quem <lb/>lineæ, quæ à K, incidunt)<emph.end type="italics"/> <expan abbr="hucu&longs;q;">hucu&longs;que</expan> o&longs;tendit lineas vi&longs;uales cadere ad M, pun­ <lb/>ctum in Iridis periphæriam, pergit deinceps inue&longs;tigare polum, & po&longs;tea <lb/>centrum eiu&longs;dem ambitus, vtraque autem exi&longs;tere in horizonte reperit, vt <lb/>hinc inferat Iridis portionem illam, quæ oriente Sole &longs;upra horizontem ap­ <lb/>paret, e&longs;&longs;e &longs;emicirculum, vt propo&longs;uerat. </s> <s id="id.002104">Differt autem polus circuli à cen­ <lb/>tro eiu&longs;dem circuli. </s> <s id="id.002105">polus e&longs;t punctum extra planum circuli, ex quo tamen <lb/>vt <expan abbr="c&etilde;tro">centro</expan> adhibito circino circuli periphæria de&longs;cribi pote&longs;t; &longs;ic polus æqua­ <lb/>toris e&longs;t idem, qui polus mundi: <expan abbr="centrũ">centrum</expan> verò e&longs;t in plano &longs;ui cir culi, &longs;ic cen­ <lb/>trum æquatoris e&longs;t idem cum centro mundi, cum æquator per illud incedat.</s></p><p type="main"> <s id="id.002106">Dicit <expan abbr="itaq;">itaque</expan> Ari&longs;t. cum data &longs;it proportio linearum K M, & M G, in &longs;upe­ <lb/>riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex­ <lb/><figure id="id.009.01.120.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.120.1.jpg" place="text"/> <lb/>ponatur alia linea recta B D. quæ diui­ <lb/>datur in partes B, & D. proportionales <lb/>cum lineis K M, G M, per 10. 6. cum <lb/>ergo K M, &longs;it minor quàm G M, per 19. <lb/>primi, quia in triangulo G M K, oppo­ <lb/>nitur minori angulo, erit <expan abbr="quoq;">quoque</expan> B, minor quàm D, addatur iam ip&longs;i B. linea <lb/>nea F, ita vt &longs;it tota F B, tertia proportionalis ad duas B, & D, per 11. 6. <lb/>hoc ordine, vt F B, ad D. ita D, ad B. <!-- KEEP S--></s> <s id="id.002107">Deinde vt &longs;e habet F, ad K G. ita &longs;it <lb/>B, ad aliam, quæ &longs;it K P, in eadem figura per 12. 6. & à puncto P, ad M, iun­ <lb/>gatur recta P M. <!-- KEEP S--></s> <s id="id.002108">Dico P, e&longs;&longs;e polum circuli, quem dixi Iridis, & in quem li­ <lb/>neæ à K, procedentes turbinis formam effingunt, probatur autem ab Ari&longs;t. <lb/>in &longs;equentibus.</s></p><p type="main"> <s id="id.002109"><arrow.to.target n="marg167"/></s></p><p type="margin"> <s id="id.002110"><margin.target id="marg167"/>166</s></p><p type="main"> <s id="id.002111">Ibidem <emph type="italics"/>(Erit etiam, quod quæ F, ad K G. & quæ B, ad K P. & quæ D, ad P M. <lb/>non enim &longs;it, &longs;ed aut ad minorem, aut ad maiorem ea, quæ P M, nibil emm differet. <lb/></s> <s id="id.002112">&longs;it enim ad P R. eandem ergo rationem G K, & K P, & P R, inuicem habebunt, <lb/>quam quæ F, B, D: quæ autem F, B, D, proportionales crant, quod quidem D, ad <lb/>B. quæ F B, ad D: quare quod quæ P G, ad P R, quæ P R, ad eam, quæ P K. &longs;i igi­ <lb/>tur ab ijs, quæ K G, quæ G R, & K R, ad R, coniungantur, coniunctæ hæ eandem <lb/>habebunt rationem, quam quæ G P, ad eam, quæ P R, circa eundem enim angulum <lb/>P, proportion aliter, & quæ trianguli G P R, & eius, qui K R P. quare & quæ G R, <lb/>ad eam quæ K R, eandem rationem habebit, quam & quæ G P, ad eam quæ P R, <lb/>habet autem & quæ M G, ad M K, eam rationem, quam quæ D, ad eam quæ B, <lb/>quare ambæ à punctis G K, non &longs;olum ad circun&longs;erentiam M N, con&longs;tituentur ean­ <lb/>aem habentes rationem, &longs;ed & alibi, quod quidem impo&longs;&longs;ibile)<emph.end type="italics"/> incipit, vt dixi, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.121.jpg" pagenum="121"/>probare P, e&longs;&longs;e polum prædicti ambitus, &longs;ic. </s> <s id="id.002113">Primò enim &longs;ciendum in præ­ <lb/>mi&longs;&longs;a con&longs;tructione e&longs;&longs;e, vt F, ad G K, & B, ad K P, ita D, ad P M. nam &longs;i non <lb/>&longs;it eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiu&longs;dem D, <lb/>ad aliam maiorem, vel minorem ipfa P M. &longs;it ad minorem P R. nihil enim <lb/>refert &longs;iue dixeris habere eandem rationem ad minorem, &longs;iue ad maiorem, <lb/>ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. &longs;ed li­ <lb/>neæ F, B, D, erant proportionales <expan abbr="compon&etilde;do">componendo</expan> hoc modo, vt F B, ad D, ita <lb/>D, ad B: quare &longs;imiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. &longs;i igi­ <lb/>tur à punctis G, & K, figuræ nu. </s> <s id="id.002114">164. <expan abbr="iungãtur">iungantur</expan> lineæ ad R, quæ &longs;int G R, K R, <lb/>erit vt G R, ad K R, ita G P, ad P R. quia orta <expan abbr="sũt">sunt</expan> duo <expan abbr="triãgula">triangula</expan> G P R, K P R, <lb/>quæ habent eundem angulum ad P. & latera proportionalia circa dictum <lb/>angulum. </s> <s id="id.002115">e&longs;t etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in <lb/>minori, ex con&longs;tructione, quare per 6. 6. erunt illa duo triangula æquian­ <lb/>gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia; <lb/>quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita <lb/>B, ad D. & ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R, <lb/>ad R G, intra eandem circunferentiam, & in eodem plano: quod e&longs;&longs;e im­ <lb/>po&longs;&longs;ibile &longs;upra o&longs;tendimus, hoc autem impo&longs;&longs;ibile, &longs;equitur &longs;i neges e&longs;&longs;e vt <lb/>F, ad G K; & B, ad K P, ita D, ad P M.<!-- KEEP S--></s></p><p type="main"> <s id="id.002116"><arrow.to.target n="marg168"/></s></p><p type="margin"> <s id="id.002117"><margin.target id="marg168"/>167</s></p><p type="main"> <s id="id.002118">Ibidem <emph type="italics"/>(Quoniă igitur quæ D, <expan abbr="neq;">neque</expan> ad minorem ea, quæ P M, <expan abbr="neq;">neque</expan> ad maiorem <lb/>(&longs;imiliter enim demon&longs;ir abimus) palam e&longs;t, quod ad ip&longs;am <expan abbr="vtiq;">vtique</expan> erit, in qua P M, <lb/>quare erit, quod quæ M P, ad P K, quæ P G, ad M P. <!-- KEEP S--></s> <s id="id.002119">Si igitur eo in quo P, polo <lb/>vtens, di&longs;tantia autem ea, in qua P M, circulns de&longs;cribatur, omnes angulos attin­ <lb/>get, quos reflexæ faciunt, quæ à K, G. &longs;i autem non, &longs;imiliter o&longs;tendentur eandem <lb/>babere rationem, quæ alibi, quam in &longs;emicirculo con&longs;tituuntur; quod quidem erat <lb/>impo&longs;&longs;ibile)<emph.end type="italics"/> quoniam igitur, inquit, linea D, <expan abbr="neq;">neque</expan> ad minorem, <expan abbr="neq;">neque</expan> ad ma­ <lb/>iorem quam P M, habet eam rationem, quæ e&longs;t ip&longs;ius F, ad G K, aut ip&longs;ius <lb/>B, ad K P. &longs;imiliter enim demon&longs;tratur ab&longs;urdum &longs;equi. </s> <s id="id.002120">palàm e&longs;t, quoniam <lb/>erit D, ad P M, vt prædictæ ad prædictas: quare componendo, & permu­ <lb/>tando, erunt tandem vt G P, ad P M, ita P M, ad P K, & ita G M, ad M K, <lb/>a&longs;&longs;ump&longs;imus enim in con&longs;tructione e&longs;&longs;e G M, ad M K, ita F B, ad D, & D, ad <lb/>B. quare cum &longs;it vt G M, ad M K, ita F B, ad D. & G P, ad P M. & P M, ad <lb/>K P; erunt per 11. 5. vt G M, ad M K. ita G P, ad P M. & P M, ad P K. &longs;i quis <lb/>igitur vtens puncto P, tanquam polo, & interuallo P M, circulum de&longs;cribat, <lb/>omnes angulos reflexionis attinget, quos faciunt lineæ productæ à K, & re­ <lb/>flexæ ab M, ad G. harum enim infinitam multitudinem debemus imaginari <lb/>à K, ad infinita puncta M, produci in ambitu illo con&longs;tituta, <expan abbr="re&longs;lecti&qacute;">re&longs;lectique</expan>; ad G. <lb/>&longs;i enim non attingat omnes illos angulos, &longs;equitur, vt &longs;upra, in eodem &longs;emi­ <lb/>circulo <expan abbr="cõ&longs;titui">con&longs;titui</expan> po&longs;&longs;e duas alias rectas proportionales prioribus G M, M K, <lb/>quod e&longs;t impo&longs;&longs;ibile. </s> <s id="id.002121">Porrò &longs;ub angulo G M K, linearum G M, M K, Iris <lb/>apparet: quare apparebit etiam &longs;ub alijs omnibus, quæ à punctis G K, duci <lb/>po&longs;&longs;unt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum <lb/>non de&longs;inant in eundem <expan abbr="&longs;emicirculũ">&longs;emicirculum</expan>, &longs;ed in ambitum Iridis M N, in quo M, <lb/>punctum imaginamur circumduci. </s> <s id="id.002122">Ex quibus pater P, e&longs;&longs;e polum Iridis, ex <lb/>quo per puncta M, vbi &longs;it reflexio, de&longs;cribitur arcus attingens omnes Iridis <lb/>reflexiones.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.122.jpg" pagenum="122"/><p type="main"> <s id="id.002123"><arrow.to.target n="marg169"/></s></p><p type="margin"> <s id="id.002124"><margin.target id="marg169"/>168</s></p><p type="main"> <s id="id.002125">Ibidem <emph type="italics"/>(Si igitur circumducas &longs;emicirculŭ, in quo A, circa diametrum in qua <lb/>G K P, que à G, K, reflexæ ad id in quo M; in omnibus planis &longs;imiliter &longs;e habebunt, <lb/>& æqualem facient angulam, qui K M G, & quem etiam facient angulum, quæ <lb/>K P, & P M, &longs;uper eam, quæ G P, &longs;emper æqualis erit. </s> <s id="id.002126">Trianguli igitur &longs;uper eam, <lb/>quæ G P, æquales ei, qui G M P. con&longs;i&longs;tunt. </s> <s id="id.002127">horum autem perpendiculares ad idem <lb/>&longs;ignum cadent eius, quæ G P, & æquales erunt, cadunt ad <foreign lang="greek">w,</foreign> centrum ergò circuli <lb/><foreign lang="greek">w</foreign> &longs;emicirculus autem, qui circa M N, ab&longs;ectus e&longs;t ab horizonte)<emph.end type="italics"/> hac vitima <lb/>textus parte concludit Iridis portionem &longs;upra horizontem a&longs;tro <expan abbr="ori&etilde;te">oriente</expan> exi­ <lb/>&longs;tentem e&longs;&longs;e &longs;emicirculum, hoc modo; &longs;i igitur imaginatione circumducas <lb/>&longs;emicirculum, in quo A, circa diametrum horizontis G K P, in hac circum­ <lb/>uolutione duæ lineæ G M, M K, in omnibus planis con&longs;titui po&longs;&longs;ibilibus cir­ <lb/>ca prædictam diametrum, quæ &longs;upra etiam fieri à triangulis infinitis dixi­ <lb/>mus, &longs;ucce&longs;&longs;iuè erunt; &longs;iue percurrent &longs;imiliter omnia illa plana, & facient <lb/>vbique angulum Iridis K M G, eundem: pariter duæ lineæ K P, P M, facient <lb/>vndique eundem angulum K P M. quare omnia triangula in predictis planis <lb/>imaginata, & <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> &longs;uper linea G P, &longs;imilia ip&longs;i G M P, & æqualia erunt; <lb/>&longs;i igitur ab angulis ip&longs;orum, in quibus M, ductæ &longs;int perpendiculares ad la­ <lb/>tus G P, omnes cadent in idem punctum <foreign lang="greek">w,</foreign> vt in figura; <expan abbr="quarũ">quarum</expan> vna erit M <foreign lang="greek">w,</foreign> <lb/>quæ tamen cæteras omnes repre&longs;entabit, <expan abbr="eis&qacute;">eisque</expan>; omnibus in volutatione axis <lb/>G K <foreign lang="greek">w,</foreign> coincidit; erunt autem omnes æquales, quandoquidem &longs;unt trian­ <lb/>gulorum æqualium. </s> <s id="id.002128"><expan abbr="erunt&qacute;">eruntque</expan>; in eodem eiu&longs;dem circuli plano, & punctum <foreign lang="greek">w,</foreign> <lb/>erit centrum ip&longs;ius. </s> <s id="id.002129">&longs;imilia dicta &longs;unt in Halone. </s> <s id="id.002130">Cum ergò ip&longs;ius centrum <lb/><foreign lang="greek">a</foreign>, &longs;it in diametro horizontis G K <foreign lang="greek">w</foreign> P O, manife&longs;tum fit portionem eius, quæ <lb/>&longs;upra horizontem eminet, e&longs;&longs;e &longs;emicirculum, qui in figura notatur lineis <lb/>L M N. <!-- KEEP S--></s> <s id="id.002131">Atque hoc accidit Sole, vel Luna in horizonte exi&longs;tentibus; quod <lb/>erat primo loco demon&longs;trandum.</s></p><p type="main"> <s id="id.002132">Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum <lb/><figure id="id.009.01.122.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.122.1.jpg" place="text"/> <lb/>ad M, ducatur M P, faciens angulum K P M, æqua­ <lb/>lem angulo G M K, per 23. primi, erunt enim duo <lb/>triangula <expan abbr="æquiãgula">æquiangula</expan> G P M, K P M, angulus enim <lb/>P, e&longs;t communis, angulus verò M K P, e&longs;t æqualis <lb/>duobus G, & G M K, per 32. primi, ergo etiam <lb/>duobus ad M, &longs;iue toti G M P, & reliquus K M P, <lb/>reliquo, quare per 4.6. latera circa angulos æqua­ <lb/>les proportionalia erunt, & omologa G M, ad M K, ita G P, ad P M, quæ <lb/>æqualibus augulis &longs;ubtenduntur. </s> <s id="id.002133"><expan abbr="ea&longs;d&etilde;">ea&longs;dem</expan> autem proprietates habcbant etiam <lb/>triangula Ari&longs;t. in figura, de qua paulò ante dicebam. </s> <s id="id.002134">Verba illa <emph type="italics"/>(Quæ ali­ <lb/>b quam in &longs;emicirculo constituuntur)<emph.end type="italics"/> &longs;unt perperam in antiqua tran&longs;latione <lb/>tran&longs;lata, nam Græcè &longs;ic, <foreign lang="greek">ai alloqi tou_ hmixoxlnou/ sunisamenai,</foreign> transferenda <lb/>e&longs;&longs;ent, quæ in alio circuli loco concurrunt.</s></p><p type="main"> <s id="id.002135"><arrow.to.target n="marg170"/></s></p><p type="margin"> <s id="id.002136"><margin.target id="marg170"/>169</s></p><p type="main"> <s id="id.002137">Ibidem <emph type="italics"/>(Iterum &longs;it horizon quidem in quo A C. oriatur autem &longs;upra hunc G, <lb/>axis autem &longs;it nunc in quo G P. <!-- KEEP S--></s> <s id="id.002138">Alia igitur omnia &longs;imiliter o&longs;tendentur vt & prius. <lb/></s> <s id="id.002139">Polus autem circuli, in quo P, erit &longs;ub horizonte eo, in quo A C, eleuato puncto, <lb/>in quo G. in eadem autem & polus, & centrum circuli, & terminantis nunc ortum, <lb/>e&longs;t enim i&longs;te, in quo G P. <!-- KEEP S--></s> <s id="id.002140">Quoniam autem &longs;upra diametrum, quæ A C, quod K G, <lb/>centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="greek">w,</foreign><emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.123.jpg" pagenum="123"/><figure id="id.009.01.123.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.123.1.jpg" place="text"/> <lb/><emph type="italics"/>Quare minor erit &longs;uperior &longs;ectio &longs;emicir­ <lb/>culo, in qua S T, (nam Q S T, &longs;emicir­ <lb/>culus est, nunc autem inter&longs;ectus e&longs;t ab <lb/>horizonte A C; <expan abbr="itaq;">itaque</expan> Q S, di&longs;parens erit) <lb/>eleuato ip&longs;o Sole)<emph.end type="italics"/> demon&longs;trat propo&longs;i­ <lb/>tionem &longs;ecundam nimirum Sole &longs;upra <lb/>horizontem elcuato, ambitum Iridis <lb/>e&longs;&longs;e minorem circuli portionem, &longs;iue <lb/>&longs;emicirculo minorem. </s> <s id="id.002141">&longs;it igitur in fi­ <lb/>gura &longs;uperiori, quam textui <expan abbr="cõgruen-tem">congruen­ <lb/>tem</expan> re&longs;tituimus, linea A C, horizon­ <lb/>talis, &longs;upra quam Sol &longs;it eleuatus in <lb/>circulo altitudinis in loco G, axis au­ <lb/>rem coni, quem reflexè faciunt &longs;it <lb/>G K <foreign lang="greek">w</foreign> P. alia igitur omnia, quæ &longs;upra exi&longs;tente in ortu a&longs;tro o&longs;ten&longs;a &longs;unt, hic <lb/>pariter o&longs;tendi po&longs;&longs;unt, &longs;cilicet Iridem fieri tantum per lineas proportiona­ <lb/>les, & æquales lineis G M, M K, quia Iris videri nequit, ni&longs;i in tali, ac deter­ <lb/>minata reflexione, & angulo, vt initio &longs;uppo&longs;ui; & quia lineæ illis propor­ <lb/>tionales non po&longs;&longs;unt alibi con&longs;titui, quam in ambitu circulari, & in diuer&longs;is <lb/>planis, &longs;equitur, vt &longs;upra Iridem e&longs;&longs;e circularem M N L; <expan abbr="eius&qacute;">eiusque</expan>; polum P, & <lb/>centrum <foreign lang="greek">w,</foreign> inueniemus &longs;imiliter in axe G K <foreign lang="greek">w</foreign> P, & quia axis hic &longs;ecat hori­ <lb/>zontem in K, in hac vltima figura propter eleuationem Solis &longs;upra A C, in <lb/>G, &longs;equitur partem axis, in qua <foreign lang="greek">w,</foreign> & P, exi&longs;tunt, infra horizontem deprimi. <lb/></s> <s id="id.002142">& quia (vt pater ex 64. 10. Vitell.) & P, polus, & centrum <foreign lang="greek">w,</foreign> Iridis, & cen­ <lb/>crum K, circuli horizontis, cuius &longs;cilicet diameter e&longs;&longs;et A K S, & Sol, &longs;unt <lb/>in eadem linea G K <foreign lang="greek">w</foreign> P, &longs;i centrum Iridis <foreign lang="greek">w,</foreign> &longs;it infra horizontem, patet mi­ <lb/>norem circuli portionem, quam &longs;it &longs;emicirculus &longs;upra horizontem eminere, <lb/>in qua po&longs;ui literas S L T, nam Q S L T R, e&longs;t &longs;emicirculus, cuius pars con­ <lb/>tenta inter duos arcus Q S, & T R, e&longs;t infra horizontem. </s> <s id="id.002143">debemus autem <lb/>hunc &longs;emicirculum, & hanc portionem ip&longs;ius S L T, extantem &longs;upra hori­ <lb/>zontem imaginari erectam e&longs;&longs;e, vt planum ip&longs;ius circuli faciat angulos re­ <lb/>ctos &longs;iue &longs;it perpendiculare cum axe G K P; & <expan abbr="circulũ">circulum</expan> altitudinis A G M N, <lb/>modo fungi vice horizontis. </s> <s id="id.002144">&longs;ic enim &longs;ola portio S L T, appareret nobis, <expan abbr="e&longs;-&longs;et&qacute;">e&longs;­ <lb/>&longs;etque</expan>; rationabiliter con&longs;tituta. </s> <s id="id.002145">Ex quibus 2. Ari&longs;t. propo&longs;itio manife&longs;ta e&longs;t.</s></p><p type="main"> <s id="id.002146"><arrow.to.target n="marg171"/></s></p><p type="margin"> <s id="id.002147"><margin.target id="marg171"/>180</s></p><p type="main"> <s id="id.002148">Ibidem <emph type="italics"/>(Minima autem cum in meridie, quanto enim &longs;uperius G, tanto in&longs;e­ <lb/>rius & polus, & centrum circuli erit)<emph.end type="italics"/> probat tertiam propo&longs;itionem, nimi­ <lb/>rum Sole exi&longs;tente in meridie minimam <expan abbr="omniũ">omnium</expan> e&longs;&longs;e Iridis arcus portionem: <lb/>ratio autem e&longs;t, quia tunc G, &longs;iue Sol, e&longs;t alti&longs;&longs;imus &longs;upra horizontem, & <lb/>con&longs;equenter <foreign lang="greek">w;</foreign> centrum Iridis e&longs;t depre&longs;si&longs;&longs;imum, quare tunc maxima cir­ <lb/>culi Iridis portio ab&longs;condetur, & proinde minima apparebit, quod erat vl­ <lb/>timo <expan abbr="demõ&longs;trandum">demon&longs;trandum</expan>. </s> <s id="id.002149">Non me latet has Ari&longs;t. figurationes e&longs;&longs;e apud Olym­ <lb/>piodorum nonnullis obiectionibus obnoxias, &longs;ed cum facilè dilui po&longs;&longs;int, & <lb/>etiam &longs;i non diluantur, &longs;aluetur tamen veritas Ari&longs;totelicæ demon&longs;tratio­ <lb/>nis, breuitati &longs;tudens, con&longs;ultò eas prætermitto.</s></p><p type="main"> <s id="id.002150">Aduertendum præterea Vicomercatum inordinatè citare librum Dato­ <lb/>rum Euclidis, & <expan abbr="quandoq;">quandoque</expan> etiam malè citare Euclidem ip&longs;um. </s> <s id="id.002151">peius verò <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.124.jpg" pagenum="124"/>faciunt ij, qui has demon&longs;trrationes <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla libri Datorum mentione ex­ <lb/>plicare conantur, cum manife&longs;tè illo innitantur.</s></p><p type="main"> <s id="id.002152">Cæterum &longs;i quis breues, ac dilucidas harum rerum demon&longs;trationes re­ <lb/>quirat, is legat 74. 75. 76. propo&longs;itiones 10. Vitell. <!-- REMOVE S-->vel &longs;equentem no&longs;tram <lb/>de Iride additionem. </s> <s id="id.002153">ego enim longiorem hanc, <expan abbr="atq;">atque</expan> impeditam Ari&longs;t. tra­ <lb/>ctationem in gratiam textus illius, vt in&longs;tituti mei ratio po&longs;tulabat, per&longs;e­ <lb/>quutus &longs;um.<!-- KEEP S--></s></p><p type="main"> <s id="id.002154"><arrow.to.target n="marg172"/></s></p><p type="margin"> <s id="id.002155"><margin.target id="marg172"/>181</s></p><p type="main"> <s id="id.002156">Ibidem <emph type="italics"/>(Quod autem in minoribus quidem diebus ijs, qui po&longs;t æquinoctium au­ <lb/>tumnale <expan abbr="cõtingit">contingit</expan> &longs;emper fieri Iridem: in longioribus aurem diebus ijs qui ab æqui­ <lb/>noctio altero, ad æquinoctium alterum circa meridiem non fit Iris, can&longs;a est, quia <lb/>quæ ad Vr&longs;am &longs;ectiones omnes maiores &longs;unt &longs;emicirculo, & &longs;emper ad maiores quod <lb/>autem e&longs;t occultum, paruum: quæ autem ad æquatoris meridiem &longs;ectiones, quæ qui­ <lb/>dem &longs;upra &longs;ectio, parua; quæ autem &longs;ub terra magna, & &longs;emper maiores, quæ lon­ <lb/>gius. </s> <s id="id.002157">quare in ijs, qui ad æ&longs;tiuas ver&longs;iones diebus propter magnitudinem &longs;ectionis, <lb/>antequam veniat G, ad medium &longs;ectionis, infra iam pœnitus fit P; propterea quod <lb/>longè di&longs;tat à terra meridies propter magnitudinem &longs;ectionis. </s> <s id="id.002158">In ijs autem diebus, <lb/>qui ad hyemates ver&longs;iones, quia non multŭ &longs;unt &longs;upra terram &longs;ectiones cir culorum, <lb/>contrarium nece&longs;&longs;arium fieri, modicum enim eleuato in quo G, in meridie fit Sol)<emph.end type="italics"/> <lb/>quærit cur po&longs;t æquinoctium autumnale v&longs;que ad vernum, hoc e&longs;t hyemali <lb/>tempore, Iris appareat etiam Sole meridiem occupante: reliquo autem <lb/>tempore æ&longs;tiuo, quod e&longs;t ab æquinoctio verno ad autumnale appareat tan­ <lb/>tum Sole vel in ortu, aut occa&longs;u exi&longs;tente, vel parum &longs;upra terram &longs;ublato. <lb/></s> <s id="id.002159">cau&longs;a autem huius refert in &longs;ectiones parallelorum circulorum, quos Sol <lb/>diurno motu inter <expan abbr="vtrunq;">vtrunque</expan> <expan abbr="tropicũ">tropicum</expan> de&longs;cribit: nam &longs;ectiones parallelorum, <lb/>qui &longs;unt ad Vr&longs;am, ide&longs;t in parte &longs;phæræ Boreali, qui omnes &longs;unt inter æqua­ <lb/>torem, & tropicum Cancri; &longs;ectiones inquam horum circulorum, quæ &longs;unt <lb/>&longs;upra horizontem, maiores &longs;unt &longs;ectionibus infra horizontem depre&longs;&longs;is, & <lb/>&longs;emper eò maiores, quò propiores &longs;unt Cancro, ita vt magna yaldè &longs;it ea <lb/>portio, quæ e&longs;t &longs;upra terram, exigua verò admodum, quæ infra (intelligan­ <lb/>tur hæc in &longs;phæra obliqua, cuius polus eleuetur grad. <!-- REMOVE S-->45. circiter) quare <lb/>quando a&longs;trum G, con&longs;cenderit meridiem, adeò P, polus Iridis, & etiam <foreign lang="greek">w,</foreign> <lb/>centrum eius infra terram deprimitur, vt aut nihil, aut in&longs;en&longs;ibile quid de <lb/>Iridis ambitu &longs;upra terram eleuari po&longs;&longs;it, contrarium accidit in parallelis <lb/>meridionalibus, quia eorum &longs;ectiones &longs;uperiores &longs;unt &longs;emper inferioribus <lb/>minores, quapropter etiam &longs;i a&longs;trum ad meridiem eleuetur, parum tamen <lb/>attollitur, & con&longs;equenter centrum <foreign lang="greek">w,</foreign> Iridis parum infra horizontem <lb/>de&longs;cendit, ac propterea etiam in meridie pars ip&longs;ius &longs;atis ma­ <lb/>gna con&longs;picitur. </s> <s id="id.002160">quæ omnia adhibita &longs;phæra materia­ <lb/>li, eaque a&longs;tronomicè ad &longs;uam eleuationem <lb/>accommodata, nullo negotio li­ <lb/>cebit intueri.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.125.jpg" pagenum="125"/><p type="head"> <s id="id.002161"><emph type="italics"/>Additio de Iride.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002162">Cvm &longs;uperior Ari&longs;tot. de Iride tractatio ob&longs;cura, ac tricis pluribus <lb/>impedita eua&longs;erit, <expan abbr="cum&qacute;">cumque</expan>; aliorum etiam demon&longs;trationes aliqua <lb/>ex parte vacilient, vi&longs;um e&longs;t breuiter expeditam, <expan abbr="atq;">atque</expan> ab&longs;olutam <lb/>ip&longs;ius apponere demon&longs;trationem. </s> <s id="id.002163">Cum igitur in cœle&longs;ti arcu <lb/>duo poti&longs;&longs;imum &longs;int, quæ &longs;ui admiratione <expan abbr="Philo&longs;ophorũ">Philo&longs;ophorum</expan> animos in &longs;ui con­ <lb/>templationem alliciant, colores, &longs;cilicet, & figura: nos mirabilem illam co­ <lb/>lorum triadem, tanquam alienam, phy&longs;icis relinquentes, de figura ip&longs;ius iu­ <lb/>re mathematico di&longs;&longs;eremus: rotunditatis &longs;cilicet Iridis cau&longs;am opticis ra­ <lb/>tionibus venabimur, cur aliquando &longs;emicirculus, aliquando &longs;emicirculo mi­ <lb/>nor appareat. </s> <s id="id.002164">vt igitur ordine procedamus.</s></p><p type="main"> <s id="id.002165">Primo loco aduertendum e&longs;t tria ad Iridis vi&longs;ionem e&longs;&longs;e nece&longs;&longs;aria, So­ <lb/>lem, oculum, & nubem tenuem, ac ro&longs;cidam, quæ &longs;cilicet minutis guttulis <lb/>iam &longs;cateat; hac enim ratione guttulæ illæ innumera erunt veluti parua <lb/>&longs;pecula, quæ lumen Solis ob paruitatem imperfecto quodam modo repre­ <lb/>&longs;entare po&longs;&longs;int, ex tali enim repre&longs;entatione Iris apparet. </s> <s id="id.002166">quæ tria debent <lb/>e&longs;&longs;e ita di&longs;po&longs;ita, vt Sol, oculus, & centrum Iridis &longs;int in eadem recta linea <lb/>con&longs;tituta, <expan abbr="oculus&qacute;">oculusque</expan>; medium locum, inter Solem, & Iridis <expan abbr="c&etilde;trum">centrum</expan> obtineat, <lb/>vt in prima figura videre e&longs;t, in qua Sol vbi A, oculus in C. nubes verò <lb/>G H L E, in qua apparet Iris in arcu E B F, quem debemus concipere e&longs;&longs;e <lb/>in rece&longs;&longs;u, vt pictores aiunt, depictum. </s> <s id="id.002167">i. </s> <s id="id.002168">non in hoc &longs;itu, & ouali figura, &longs;ed <lb/><figure id="id.009.01.125.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.125.1.jpg" place="text"/> <lb/>e&longs;&longs;e perfectè &longs;emicircularem, <expan abbr="habere&qacute;">habereque</expan>; talem po&longs;itionem, vt pars ip&longs;ius B F, <lb/>&longs;it citra chartam eleuata, <expan abbr="ip&longs;i&qacute;">ip&longs;ique</expan>; perpendicularis, pars verò E B, vltra pagi­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.126.jpg" pagenum="126"/>nam rectà recedat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; diameter Iridis E F, faciat angulos rectos cum <lb/>linea horizontali A C L, in quo &longs;itu oculo C, totus ex oppo&longs;ito directè &longs;pe­ <lb/>ctaretur, non aliter ac Iridem ip&longs;am con&longs;picere &longs;olemus. </s> <s id="id.002169">Quod autem ne­ <lb/>ce&longs;&longs;aria &longs;it nubes ro&longs;cida, pulcherrima hac experientia <expan abbr="cõprobatur">comprobatur</expan>: &longs;i enim <lb/>in Sole po&longs;iti ore aquam efflantes leui a&longs;pergine aerem Soli, ac nobis ad­ <lb/>uer&longs;um irroremus, actutum Iridis arcum guttulis illis, quamuis volitanti­ <lb/>bus inhærcntem &longs;umma voluptate &longs;pectabimus. </s> <s id="id.002170">Quod præterea oculus no­ <lb/>&longs;ter, cum Iridem videmus, medius &longs;it inter Solem, & Iridis centrum, expe­ <lb/>rimento diuturno, manife&longs;tum e&longs;t.</s></p><p type="main"> <s id="id.002171">Secundò, notandum e&longs;t, arcum per reflexionem fieri: quod quidem pri­ <lb/>mo eadem experientia, qua præcedens conclu&longs;io confirmatur: deinde, quia <lb/>Iridem &longs;emper in oppo&longs;ita Soli, ac nobis parte <expan abbr="cõ&longs;picimus">con&longs;picimus</expan>; quemadmodum <lb/>in eadem figura o&longs;tenditur, quod aliter quàm per reflexionem fieri nequit.</s></p><p type="main"> <s id="id.002172">Tertiò, &longs;ciendum e&longs;t ex Maurolyco, & 10. Bapti&longs;ta Porta, tantam e&longs;&longs;e di­ <lb/>&longs;tantiam C D, ab oculo ad centrum arcus, quanta e&longs;t altitudo, &longs;eu &longs;emidia­ <lb/>meter D B, ob&longs;eruarunt enim ip&longs;i angulos D C B, & C B D, e&longs;&longs;e &longs;emirectos, <lb/>& proinde æquales, & con&longs;equenter duo latera C D, D B, trianguli C D B, <lb/>per 6. 1. æqualia &longs;unt.</s></p><p type="main"> <s id="id.002173">Quartò, con&longs;iderandum e&longs;t lineas A B, A D, ob maximam Solis ab Iride <lb/>di&longs;tantiam in&longs;en&longs;ibiliter differre; & ideò &longs;upponi po&longs;&longs;unt æquidi&longs;tantes, <lb/>quare angulus A B C, qui æqualis e&longs;t alterno B C D, &longs;umi pote&longs;t ab&longs;que vllo <lb/>errore pro &longs;emirecto. </s> <s id="id.002174">hic autem angulus A B C, dicitur angulus reflexionis <lb/>Iridis, &longs;ub tali enim reflexione lumen Solis occurrens nubi in B, reflectitur <lb/>ad oculum C.<!-- KEEP S--></s></p><p type="main"> <s id="id.002175">Quintò, &longs;equitur ex prædictis arcum videri &longs;emper &longs;ub &longs;tato, ac determi­ <lb/>nato reflexionis angulo, &longs;cilicet &longs;ub &longs;emirecto, <expan abbr="neq;">neque</expan> po&longs;&longs;e per alium videri. <lb/></s> <s id="id.002176">quod etiam probari pote&longs;t ex Ari&longs;t. quia nimirum videmus arcum apparere <lb/>con&longs;imiliter in ambitu circulari, ergò nece&longs;&longs;ariò apparebit <expan abbr="vbiq;">vbique</expan> in toto il­ <lb/>lo ambitu per con&longs;imilem reflexionem, &longs;iue per æquales reflexionis angulos, <lb/>pro quibus omnibus vnus cernitur in figura angulus A B C.<!-- KEEP S--></s></p><p type="main"> <s id="id.002177">Sextò, ad Iridis vi&longs;ionem, præter ea, requiri aeris rorantis multiplica­ <lb/>tionem; &longs;icuti enim nebulam videre nequimus, ni&longs;i aer exhalatione illa in­ <lb/>fectus multus &longs;it ante oculum no&longs;trum: &longs;ic etiam exi&longs;timo ad Iridis appari­ <lb/>tionem, opus e&longs;&longs;e plurima nube rore&longs;cente, vt ex multiplicatione guttula­ <lb/>rum, quarum aliæ po&longs;t alias &longs;int, totus tandem Iris appareat. </s> <s id="id.002178">quia paucæ <lb/>guttulæ, etiam &longs;i quælibet illarum aliquid Iridis efficeret, ob paruitatem <lb/>tamen illarum, nulla arcus figura &longs;pectaretur. </s> <s id="id.002179">Quod &longs;i ante oculum pluri­ <lb/>mæ &longs;int in toto aere aliæ po&longs;t alias, tunc &longs;e mutuò iuuantes, obiectum &longs;atis <lb/>&longs;en&longs;ibile, quoc Iris e&longs;t, efficere po&longs;&longs;unt. </s> <s id="id.002180">Adde, quod etiam ex tali guttula­ <lb/>rum multiplicatione, aer opacatur, quæ opacatio plurimum iuuat ad Iri­ <lb/>dem &longs;pectandam.</s></p><p type="main"> <s id="id.002181">Septimò, Iridis rotundationis cau&longs;am ex præmi&longs;&longs;is con&longs;tare poti&longs;&longs;imum <lb/>ex duabus. </s> <s id="id.002182">primò, ex angulo reflexionis determinato, qui videlicet &longs;it ferè <lb/>&longs;emirectus. </s> <s id="id.002183">&longs;ecundò, ex paribus di&longs;tantijs C D, D B, huiu&longs;modi enim plures <lb/>anguli, qui ad Iridem &longs;unt nece&longs;&longs;arij (debent enim &longs;ingulæ Iridis partes &longs;ub <lb/>huiu&longs;modi angulo repre&longs;entari) non po&longs;&longs;unt aliter quàm in gyrum <expan abbr="cõ&longs;titui">con&longs;titui</expan>, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.127.jpg" pagenum="127"/>quem gyrum optimè concipiemus, &longs;i imaginemur triangulum A B C, cir­ <lb/>cumuerti circa lineam horizontalem A C L, fixam, tanquam circa axem. </s> <s id="id.002184">in <lb/>hac enim conuer&longs;ione angulus Iridis B, de&longs;cribet circulum, qui erit Iris, & <lb/>pertr an&longs;ibit omnes angulos, qui in tali Solis, oculi, ac nubis &longs;itu, arcum ef­ <lb/>ficere &longs;unt idonei.</s></p><p type="main"> <s id="id.002185">Sed contra prædicta de angulo Iridis determinato eadem nobis obijcies, <lb/>quæ nos &longs;upra ad finem numeri 164. Ari&longs;t. & alijs obiecimus, plures <expan abbr="nimi-rũ">nimi­ <lb/>rum</expan> po&longs;&longs;e con&longs;titui angulos æquales angulo Iridis B, in plano trianguli A B C, <lb/>qui non &longs;int in eodem orbe con&longs;tituti, in quo &longs;unt omnes anguli B. <!-- KEEP S--></s> <s id="id.002186">Iridem <lb/>reflectentes, <expan abbr="quiq;">quique</expan> reflexionem faciant ad eundem oculum C, vnde &longs;equitur <lb/>prædictam Iridis altitudinem non e&longs;&longs;e, vti diximus, determinatam, cum <lb/>po&longs;&longs;it angulus B, alios &longs;ibi æquales tam &longs;upra, quàm infra habere, qua ra­ <lb/>tione deberet etiam Iris, & altius, & inferius apparere.</s></p><p type="main"> <s id="id.002187">Huic dubitationi re&longs;pondeo, quod quamuis huiu&longs;modi plures anguli <lb/>æquales fiant, non tamen Iridis generationi ob&longs;tant, quinimò ad eam valdè <lb/>nece&longs;&longs;arij &longs;unt; <expan abbr="cũ">cum</expan> enim omnes &longs;int in <expan abbr="circunfer&etilde;tia">circunferentia</expan> circuli A C D B, quar­ <lb/>tæ figuræ num. </s> <s id="id.002188">164. quæ modo in&longs;picienda e&longs;t, vt &longs;unt in ea anguli A D C, <lb/>A B C; quæ circunferentia ob &longs;ui circuli immen&longs;itatem ad &longs;en&longs;um e&longs;t in&longs;tar <lb/>lineæ rectæ, fit vt omnes illi anguli tàm qui &longs;upra B, quàm qui infra &longs;unt, <lb/>&longs;int quoad &longs;en&longs;um in eadem recta C D B, ante vi&longs;um proten&longs;a, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; Iris, qui <lb/>apparet in D, & in B, &c. </s> <s id="id.002189">ob medij rorantis multiplicationem vnam <expan abbr="tãtùm">tantùm</expan> <lb/>oculo Iridem repre&longs;entet. </s> <s id="id.002190">locus tamen, in quo apparet, & vbi e&longs;t angulus <lb/>B, qui propriè Iridis appellatur, e&longs;t in tanta di&longs;tantia à centro arcus, quan­ <lb/>ta e&longs;t ab eodem centro ad oculum, vt &longs;upra dictum e&longs;t.</s></p><p type="main"> <s id="id.002191">Quod verò alibi extra circunferentiam illius circuli, poni nequeat angu­ <lb/>lus æqualis angulo B, præ&longs;entis figuræ, qui re&longs;lectat ad C. patet &longs;ic, &longs;it enim <lb/>angulus A N O, &longs;emirectas, & ideò æqualis angulo B, erunt ergo B C, N O, <lb/>parallelæ, quare non concurrent ambæ ad C, &longs;ed altera ad E, altera verò ad <lb/>O, quæ propterea oculo in O, po&longs;ito Iridem efficeret, non autem oculo C: <lb/><expan abbr="&longs;ic&qacute;">&longs;icque</expan>; oculus C, & oculus O, viderent diuer&longs;os arcus. </s> <s id="id.002192">eodem modo o&longs;tendi <lb/>pote&longs;t, <expan abbr="neq;">neque</expan> in &longs;uperiori parte nubis vbi P, con&longs;titui po&longs;&longs;e angulum æqualem <lb/>angulo B, qui oculo C, Iridem valeat o&longs;tendere. </s> <s id="id.002193">Ex quibus &longs;atis patefacta <lb/>e&longs;t cau&longs;a rotunditatis arcus, angulus &longs;cilicet determinatus cum di&longs;tantia­ <lb/>rum C D, D B, paritate, necnon cum medij rorantis &longs;ufficienti multiplica­ <lb/>tione. </s> <s id="id.002194">Ex his etiam Iridis definitio in hunc modum concinnari pote&longs;t, Iris <lb/>e&longs;t arcus multicolor in nube rorida, ex radiorum Solis, aut Lunæ reflexio­ <lb/>ne &longs;ab &longs;tatuto angulo effulgens.</s></p><p type="main"> <s id="id.002195">Octauo loco Problemata nonnulla re&longs;oluemus.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.128.jpg" pagenum="128"/><p type="head"> <s id="id.002196"><emph type="italics"/>Problema Primum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.002197">Cur oriente, aut occumbente Sole, Iris &longs;emicirculus e&longs;t?</s></p><p type="main"> <s id="id.002198">Cau&longs;a huius hæc e&longs;t; &longs;upra enim dictum e&longs;t, in omni Iridis appari­ <lb/>tione tria hæc, Solem, oculum, & Iridis centrum e&longs;&longs;e in eadem re­ <lb/>cta linea, v. <!-- REMOVE S-->g. <!-- REMOVE S-->in linea A C D, præcedentis figuræ, cum igitur Sol <lb/>tam oriens, quam occidens &longs;it in horizonte, v. <!-- REMOVE S-->g. <!-- REMOVE S-->in A, horizontis <lb/>puncto, &longs;imiliter oculus &longs;it in C, horizontis centro, con&longs;ectarium e&longs;t, cen­ <lb/>trum etiam Iridis D, e&longs;&longs;e pariter in horizontis &longs;uperficie, quare &longs;ecabitur <lb/>ab horizonte per centrum, vnde etiam &longs;equitur ip&longs;ius Iridis portionem <lb/>E B F, quæ &longs;upra horizontem extat e&longs;&longs;e &longs;emicirculum. </s> <s id="id.002199">Quod &longs;i horizon non <lb/>ob&longs;taret, <expan abbr="integrũ">integrum</expan> Iris compleret orbem, <expan abbr="cerneretur&qacute;">cernereturque</expan>; toto ambitu B F M E.<!-- KEEP S--></s></p><p type="head"> <s id="id.002200">An <expan abbr="quando&qacute;">quandoque</expan>; maior &longs;emicirculo appareat?</s></p><p type="head"> <s id="id.002201"><emph type="italics"/>Problema Secundum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002202">Maior quidem, imò etiam integer circulus, &longs;ed ab oculo in &longs;ummitate <lb/>montis con&longs;tituto, <expan abbr="Sole&qacute;">Soleque</expan>; iam multum eleuato videri pote&longs;t, vt in <lb/>hac &longs;ecunda figura cernitur, vbi euecto Sole ad locum E, &longs;upra horizontem <lb/><figure id="id.009.01.128.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.128.1.jpg" place="text"/> <lb/>A B, poterit oculus in vertice montis C, po&longs;itus Iridem F G H I, comple­ <lb/>tam videre, quia infra lineam E C D, in qua exi&longs;tunt Sol, oculus, & Iridis <lb/>centrum, nihil e&longs;t ad partes D, vbi nubes irrorat, quod Iridis apparitioni <lb/>&longs;it impedimento.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.129.jpg" pagenum="129"/><p type="head"> <s id="id.002203">Cur quanto Sol altior e&longs;t, tanto inferior, <expan abbr="tanto&qacute;">tantoque</expan>; &longs;emicir­ <lb/>culo minor appareat Iris?<!-- KEEP S--></s></p><p type="head"> <s id="id.002204"><emph type="italics"/>Problema Tertium.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002205">Qvia eleuato Sole ad E, vt in hac tertia figura, nece&longs;&longs;ario centrum Iri­ <lb/>dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D. <lb/><figure id="id.009.01.129.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.129.1.jpg" place="text"/> <lb/>Sol E, oculus C, <expan abbr="centrũ&qacute;">centrunque</expan>; Iridis D, exi&longs;tant: vnde nece&longs;&longs;ariò &longs;equitur Iridis <lb/>portionem F G H, &longs;upra horizontem extantem, &longs;emicirculo minorem e&longs;&longs;e.</s></p><p type="head"> <s id="id.002206">Cur Iris in&longs;equentes fugit, fugientes verò in&longs;equitur?</s></p><p type="head"> <s id="id.002207"><emph type="italics"/>Problema Quartum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002208">Pvlcherrimum i&longs;tud phænomenon primus omnium Philippus Mendæus <lb/>Platonis di&longs;cipulus, ob&longs;eruauit; Cuius ratio e&longs;t, quia arcus non ni&longs;i &longs;ub <lb/>determinato angulo, di&longs;tantijs etiam illis paribus, ac tandem idone a a&longs;per­ <lb/>gino&longs;æ nubis multiplicatione &longs;pectatur; quapropter &longs;i quis per aerem to­ <lb/>tum <expan abbr="vndiq;">vndique</expan> ro&longs;cidum inambulet, <expan abbr="vbicunq;">vbicunque</expan> illi anguli, <expan abbr="illæ&qacute;">illæque</expan>; conditiones af­ <lb/>fuerint Iris apparebit: quod &longs;i in aperta planitie obequitans arcu con&longs;pe­ <lb/>cto, additis equo calcaribus citatum cur&longs;um ad eum direxerit, fugientem <lb/>ante &longs;e Iridem &longs;umma cum iucunditate mirabitur.</s></p><p type="main"> <s id="id.002209">Ex dictis pr&ecedil;tere a patet, &longs;impliciter nimis eos hallucinari, qui exi&longs;timant <lb/>in plana, aut concaua nubis &longs;uperficie Iridem tantummodo apparere po&longs;&longs;e.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.130.jpg" pagenum="130"/><p type="head"> <s id="id.002210">Curlunares Irides fiunt rariores?</s></p><p type="head"> <s id="id.002211"><emph type="italics"/>Problema Quintum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002212">Qvoniam iuxta plenilunia tantum, cum &longs;cilicet Luna plurimo lumine <lb/>abundat, quod Iridem efficere debet, contingunt: præterea quia cum <lb/>lunare lumen debile &longs;it, ni&longs;i aliæ cau&longs;æ perfectæ admodum concur­ <lb/>rant, quod rarò accidit, Iris nullo modo effulgere valet. </s> <s id="id.002213">Hactenus de Iri­ <lb/>dis figura &longs;it &longs;atis.</s></p><p type="head"> <s id="id.002214"><emph type="italics"/>Summa 2. cap. 5. De Parelio.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002215"><arrow.to.target n="marg173"/></s></p><p type="margin"> <s id="id.002216"><margin.target id="marg173"/>182</s></p><p type="main"> <s id="id.002217">Textus <emph type="italics"/>(Fiunt autem vt diximus, & Virgæ, & Parelia in ortu, & oc­ <lb/>ca&longs;u, & nec &longs;upra Solem, nec infra, &longs;ed ex lateribus, nec propè admo­ <lb/>dum, nec procul omninò. </s> <s id="id.002218">propinquam enim concretionem Sol di&longs;&longs;oluit: <lb/>&longs;i autem procul ab&longs;it, a&longs;pectus non reflectetur, &longs;i enim à paruo &longs;peculo <lb/>procul protenditur imbecillus fit. </s> <s id="id.002219">quare, & Coronæ è regione Solis non fiunt. </s> <s id="id.002220">&longs;i igi­ <lb/>tur &longs;upra fuerit, & proxima; eam Sol di&longs;&longs;oluet: &longs;i verò procul a&longs;pectus minor <lb/>quam vt reflecti po&longs;&longs;it in Solem non-incidet; à latere autem fieri pote&longs;t, vt &longs;pecu­ <lb/>lum ita distet à Sole, vt non &longs;oluatur, & a&longs;pectus totus ad eum perueniat, eo quod <lb/>ad terram dum fertur, qua&longs;i per immen&longs;um feratur, peruenire nequeat. </s> <s id="id.002221">&longs;ub Sole <lb/>verò non fit, quia cum ad terram propius acce&longs;&longs;erit à Sole di&longs;&longs;oluitur, cum medium <lb/>cœli tenuerit a&longs;pectus di&longs;trahitur. </s> <s id="id.002222">omninò ne à latere quidem, Sole medium cœli <lb/>tenente, efficitur, quia a&longs; pectus &longs;ub terram non fertur, quare exiguus ad &longs;peculum <lb/>producitur, & qui reftectitur pror&longs;us imbecillis redditur)<emph.end type="italics"/> ibi <emph type="italics"/>(propinquam enim <lb/>concretionem Sol di&longs;&longs;oluit)<emph.end type="italics"/> rationes, quas affert circa Parelia videntur (auda­ <lb/>cter loquar) admodum debiles. </s> <s id="id.002223">præ&longs;ens ea e&longs;t, vt Parelium non fiat propè <lb/>Solem, quia illa nubis concretio, quæ Parelio nece&longs;&longs;aria e&longs;t, nequit adeo So­ <lb/>li propinqua e&longs;&longs;e, quia nimirum Sol ob propinquitatem eam di&longs;&longs;olueret; &longs;ed <lb/>quis non videt eam nubem, quam vulgò exi&longs;timamus e&longs;&longs;e Soli propinquam, <lb/>&longs;eu qua&longs;i inter nos, & Solem tantum, imò etiam minus aliquando à Sole ve­ <lb/>rè di&longs;tare, quàm alia, quàm vulgò remotiorem à Sole putabimus? </s> <s id="id.002224">præte­ <lb/>rea omnes nubes no&longs;tri horizontis re vera æquidi&longs;tare à Sole certum e&longs;t, ob <lb/>maximam enim Solis di&longs;tantiam totus no&longs;ter horizon phy&longs;icus e&longs;t in&longs;en&longs;i­ <lb/>bilis quantitatis ad Solem, & vnius puncti vicem gerit.</s></p><p type="main"> <s id="id.002225">Ibi verò <emph type="italics"/>(Si autem procul ab&longs;it, &c.)<emph.end type="italics"/> reddit rationem, cur parelium non <lb/>appareat in nube à Sole valde remota &longs;ecundum vulgarem æ&longs;timationem, <lb/>vnde vulgarem etiam rationem affert, ait enim, nubem illam e&longs;&longs;e veluti &longs;pe­ <lb/>culum Solis repre&longs;entatiuum, &longs;peculum autem tàm longè à Sole po&longs;itum, <lb/>reddi debile, & proptereá non po&longs;&longs;e Solis imaginem referre: Verùm ratio <lb/>hæc nulla e&longs;&longs;e videtur, quis enim ignorat non propterea e&longs;&longs;e remotius à So­ <lb/>le, quamuis maiorem habere videatur à Sole lateralem di&longs;tantiam, vt pau­ <lb/>lò ante dixi? </s> <s id="id.002226">Eandem rationem illi dubitationi accommodat, cur <expan abbr="neq;">neque</expan> vi­ <lb/>deatur &longs;upra Solem, quamuis non ei quadret, pote&longs;t enim aliqua nubes vi­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.131.jpg" pagenum="131"/>deri &longs;upra Solem, quæ tamen remotior &longs;it à Sole, quam illa, in qua Parclium <lb/>gignitur. </s> <s id="id.002227">Ait po&longs;tea <emph type="italics"/>(A latere autem, &c.)<emph.end type="italics"/> cur appareat in nube fatis Soli <lb/>à latere vicina, in di&longs;tantiam à Sole refert: &longs;ed quæ dudum dicta &longs;unt, i&longs;tud <lb/><expan abbr="quoq;">quoque</expan> refellunt. </s> <s id="id.002228">Verba illa <emph type="italics"/>(Eo quod ad terram dum fertur qua&longs;i per immen&longs;um <lb/>feratur, peruenire nequeat)<emph.end type="italics"/> videntur alieno loco dicta; &longs;imilia præcedentibus <lb/>&longs;unt reliqua, præ&longs;ertim quæ ibi <emph type="italics"/>(Sub Sole verò non fit, quia cum ad terram pro­ <lb/>pius acce&longs;&longs;erit)<emph.end type="italics"/> cur non videatur infra Solem, rationem quandam, quæ fortè <lb/>inanis e&longs;t reddit; nunquid enim non po&longs;&longs;umus tam infra Solem, quàm &longs;upra <lb/>ita &longs;peculum accommodare, vt Solem no&longs;tris vi&longs;ibus remittat? </s> <s id="id.002229">huic certè <lb/>Optice tota repugnat. </s> <s id="id.002230">Cum igitur Mathematica ratione hæ rationes non <lb/>con&longs;i&longs;tant, alias alij excogitent. </s> <s id="id.002231">Mirum tamen e&longs;t, omnes, quos viderim <lb/>commentatores, eas tanquam optimas admittere.</s></p><p type="head"> <s id="id.002232"><emph type="italics"/>In quarto Meteororum nihil Mathematicum occurrit.<emph.end type="italics"/></s></p></chap><chap><p type="head"> <s id="id.002233"><emph type="italics"/>EX LIB. PRIMO DE ANIMA.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002234"><arrow.to.target n="marg174"/></s></p><p type="margin"> <s id="id.002235"><margin.target id="marg174"/>183</s></p><p type="main"> <s id="id.002236">Tex. 11. <emph type="italics"/>(Videtur autem non &longs;olum ip&longs;um quid e&longs;t cogno&longs;cere vtile e&longs;&longs;e <lb/>ad cogno&longs;cendas cau&longs;as accidentium &longs;ub&longs;tantijs: &longs;icut in Mathemati­ <lb/>cis quid rectum, & quid obliqaum, aut quid linea, & planum, ad co­ <lb/>gno&longs; cendum quot rectis, trianguli anguli &longs;unt æquales)<emph.end type="italics"/> quid &longs;it <expan abbr="vnum-quodq;">vnum­ <lb/>quodque</expan> ex prædictis patet tum ex definitionibus primi Elem. tum ex com­ <lb/>mentarijs ip&longs;arum; quamuis autem ibi non definiatur <expan abbr="rectũ">rectum</expan>, nec obliquum <lb/>in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue <lb/>planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi­ <lb/>nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int <lb/>tres anguli cuiufuis trianguli. </s> <s id="id.002237">vide quæ de hac æqualitate &longs;crip&longs;i lib, primo <lb/>Priorum, &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.002238"><arrow.to.target n="marg175"/></s></p><p type="margin"> <s id="id.002239"><margin.target id="marg175"/>184</s></p><p type="main"> <s id="id.002240">Tex. 13. <emph type="italics"/>(Si igitur e&longs;t aliqua animæ operatio, aut pa&longs;&longs;io propria, continget vti­ <lb/>que ip&longs;am &longs;eparari: &longs;i verò nulla e&longs;t propria ip&longs;ius non vtique erit &longs;eparabilis. </s> <s id="id.002241">&longs;ed <lb/>&longs;icut recto in quantum rectum multa accidunt, vt tangere æneam &longs;phæram &longs;ecun­ <lb/>dum punctum, non tamen tanget hoc, rectum ip&longs;um &longs;eparatum: in&longs;eparabile enim, <lb/>&longs;i quidem cum corpore quodam &longs;emper e&longs;t)<emph.end type="italics"/> Propo&longs;itio 2. tertij Elem. &pacute;robat li­ <lb/><figure id="id.009.01.131.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.131.1.jpg" place="text"/> <lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro­ <lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s> <s id="id.002242">v. <!-- REMOVE S-->g. <!-- REMOVE S-->puncta A B, quantum­ <lb/>uis &longs;ibi inuicem propinqua fnerint, attamen &longs;i line a <lb/>A B, ea coniungat, ip&longs;a cadet intra circulum, & <lb/>veluti chorda &longs;ubtendet arcum A B, quantulum­ <lb/>cunque. </s> <s id="id.002243">ex qua demon&longs;tratione colligitur in corol­ <lb/>lario eius lineam rectam tangentem circulum ip­ <lb/>&longs;um in vnico puncto tangere. </s> <s id="id.002244">v. <!-- REMOVE S-->g. <!-- REMOVE S-->rectam C D, tan­ <lb/>gere circulum in puncto E. &longs;i enim dixeris tangere <lb/>in duobus admodum propinquis, vt in E F, tunc non erit amplius tangens, <lb/>&longs;ed &longs;ecans, quia vt modo dixi, pars lineæ rectæ, quæ <expan abbr="cõiungeret">coniungeret</expan> puncta E F, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.132.jpg" pagenum="132"/>intra circulum per &longs;ecundam præallegatam caderet, quod e&longs;t ab&longs;urdum, <lb/>quia contra hypothe&longs;im, cum &longs;upponamus illam &longs;olùm tangere, non autem <lb/>&longs;ecare circulum. </s> <s id="id.002245">Ex hac Euclidis doctrina Theodo&longs;ius primo &longs;phæricorum, <lb/>propo&longs;itione 3. probat planum, &longs;iue &longs;uperficiem planam tangere &longs;phæram <lb/>in vnico puncto, vt hoc loco innuit Philo&longs;ophus. <!-- KEEP S--></s> <s id="id.002246">probat autem hac ferè ra­ <lb/><figure id="id.009.01.132.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.132.1.jpg" place="text"/> <lb/>tione. </s> <s id="id.002247">&longs;it &longs;phæra A B C, quæ tangat quodpiam planum <lb/>in duobus punctis A, B, &longs;i fieri pote&longs;t. </s> <s id="id.002248">per quæ duo pun­ <lb/>cta intelligatur ducta recta linea A B, intelligatur <expan abbr="etiã">etiam</expan> <lb/>circulus A B C, qui &longs;ecet &longs;phæram per centrum C. & <lb/>per puncta A, B, ergo ex demon&longs;tratis ab Euclide li­ <lb/>nea A B, quæ coniungit puncta A B, cadet intra prædi­ <lb/>ctum circulum; &longs;ed linea hæc e&longs;t in plano tangente ex <lb/>&longs;uppo&longs;itione, circulus verò in &longs;phæra; ergò cum linea <lb/>cadat intra circulum, cadet etiam nece&longs;&longs;ariò planum <lb/>in quo e&longs;t linea, & cum linea cadat intra circulum, cadet etiam nece&longs;&longs;ariò <lb/>intra &longs;phæram; <expan abbr="idem&qacute;">idemque</expan>; faciet planum, quod eam nece&longs;&longs;ariò &longs;equatur, ergò <lb/>planum &longs;ecat &longs;phæram, non autem tangit, quod e&longs;t ab&longs;urdum, quia contra <lb/>hypothe&longs;im, &longs;upponunt autem Mathematici, entia hæc mathematica e&longs;&longs;e <lb/>perfecta, qualia in &longs;ublunaribus fortè non reperiuntur; ænea enim &longs;phæra <lb/>nulla erit perfectè rotunda, vel planum aliquod perfectè complanatum, vt <lb/>ip&longs;i &longs;upponunt, eò quod materiæ imperfectio, ac ruditas id nequaquam pa­ <lb/>tiatur. </s> <s id="id.002249">quare cum huiu&longs;inodi entia non reperiantur ab&longs;tracta ab impura hac <lb/>materia, nullum erit inquit Ari&longs;t. ab&longs;tractum planum, quod po&longs;&longs;it mathe­ <lb/>maticè, <expan abbr="atq;">atque</expan> adeò in vnico puncto mathematico &longs;phæram tangere. </s> <s id="id.002250"><expan abbr="hucu&longs;q;">hucu&longs;que</expan> <lb/>nece&longs;&longs;aria &longs;unt mathematica ad huius loci <expan abbr="intelligentiã">intelligentiam</expan>. </s> <s id="id.002251">ex quibus ea etiam, <lb/>quæ ad phy&longs;icum &longs;pectant manife&longs;ta fiunt, nimirum &longs;icut entia mathemati­ <lb/>ca à materia non exi&longs;tunt &longs;eparata, quia &longs;ic nullam haberent operationem; <lb/>ita etiam anima, &longs;i nullam habet propriam operationem non exi&longs;tet à cor­ <lb/>pore &longs;eparata.</s></p></chap><chap><p type="head"> <s id="id.002252"><emph type="italics"/>Ex Secundo de Anima.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002253"><arrow.to.target n="marg176"/></s></p><p type="margin"> <s id="id.002254"><margin.target id="marg176"/>185</s></p><p type="main"> <s id="id.002255">Tex. 12. <emph type="italics"/>(Non enim &longs;olum ip&longs;um, quod&longs;it, oportet definitiuam rationem <lb/>oftendere, &longs;icut plures definitionum dicunt, &longs;ed & cau&longs;am ine&longs;&longs;e, & ap­ <lb/>parere. </s> <s id="id.002256">nunc autem, vt conclu&longs;iones rationes definitionum &longs;unt, vt quid <lb/>tetragoni&longs;mus? </s> <s id="id.002257">æquale altera parte longiori rectangulum æquilaterum <lb/>e&longs;&longs;e, talis autem definitio ratio conclu&longs;ionis. </s> <s id="id.002258">dicens autem, quod tetragoni&longs;mus e&longs;t <lb/>medij inuentio rei cau&longs;am dicit)<emph.end type="italics"/> aggre&longs;&longs;urus Ari&longs;t. animæ definitionem præ­ <lb/>mittit duplicem e&longs;&longs;e definitionem, alteram &longs;cilicet, quæ explicat &longs;olum rei <lb/>e&longs;&longs;entiam, quam dicunt formalem definitionem; alteram verò, quæ præte­ <lb/>rea explicat etiam rei cau&longs;am, quam dicunt cau&longs;alem definitionem: vtram­ <lb/>que autem exemplo Geometrico explicat.</s></p><p type="main"> <s id="id.002259">In cap. igitur de relatione plura &longs;crip&longs;i de tetragoni&longs;mo, &longs;eu qua dratio­ <lb/>ne circuli, quæ huc &longs;pectant. </s> <s id="id.002260">propterea nunc tantum propria huius loci <expan abbr="de-clarãda">de­ <lb/>claranda</expan> re&longs;tant. </s> <s id="id.002261">loquitur igitur hic Philo &longs;ophus non de quadratione circuli, <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.133.jpg" pagenum="133"/>&longs;ed figuræ rectilineæ illius, quæ dicitur Altera parte longior, qualis e&longs;t præ­ <lb/>&longs;ens figura A B C D, cuius quadrandæ ratio e&longs;t huiu&longs;modi. </s> <s id="id.002262">per 13. 6. inue­ <lb/><figure id="id.009.01.133.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.133.1.jpg" place="text"/> <lb/>niatur recta linea media proportionalis inter <lb/>duo latera figuræ A B, B C, <expan abbr="ea&qacute;">eaque</expan>; &longs;it B D, in &longs;e­ <lb/>quenti figura. </s> <s id="id.002263">e&longs;&longs;e autem mediam proportio­ <lb/>nalem nihil aliud e&longs;t quam ita e&longs;&longs;e A B, ad B D, <lb/>&longs;icut B D, ad B C. <expan abbr="dicitur&qacute;">diciturque</expan>; media proportio­ <lb/>nalis, quia in hac habitudine medium locum obtinet. </s> <s id="id.002264">quadratum autem li­ <lb/>neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu­ <lb/>ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto. <lb/><figure id="id.009.01.133.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.133.2.jpg" place="text"/> <lb/>accipe duo latera A B, & B C, <expan abbr="quadrãdi">quadrandi</expan> rectan­ <lb/>guli, <expan abbr="ea&qacute;">eaque</expan>; in directum con&longs;titue, vt vnicam re­ <lb/>ctam con&longs;tituant A C, vt apparet in figura; de­ <lb/>inde diui&longs;a tota A C, bifariam in E, facto cen­ <lb/>tro in E, de&longs;cribe &longs;emicirculum &longs;uper lineam <lb/>A C, demum à puncto B, in quo duo latera con­ <lb/>iunguntur, erigatur linea perpendicularis <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad periphæriam, quæ &longs;it B D. hæc enim B D, e&longs;t media proportionalis inter <lb/>latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo­ <lb/>que obtinet B D, ad B C. <!-- KEEP S--></s> <s id="id.002265">Quadratum igitur huius B D, hoc e&longs;t quadratum, <lb/>cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit <lb/><figure id="id.009.01.133.3.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.133.3.jpg" place="text"/> <lb/>dato &longs;uperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per­ <lb/>acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri <lb/>A B C D. <!-- KEEP S--></s> <s id="id.002266">Vides igitur, qua ratione quadratum con&longs;ti­ <lb/>tuatur æquale dato quadrilatero; & qua rationem inuen­ <lb/>tio illius mediæ proportionalis &longs;it cau&longs;a quadraturæ re­ <lb/>ctanguli, & proinde &longs;i quis dicat quadrationem hanc e&longs;&longs;e <lb/>effectionem rectanguli æquilateri, ide&longs;t quadrati, æqualis dato quadrilate­ <lb/>ro, hic definitionem formalem &longs;olum afferet: quæ definitio, vt dixit in Lo­ <lb/>gicis, e&longs;t in&longs;tar conclu&longs;ionis. </s> <s id="id.002267">&longs;i quis verò dicat tetragoni&longs;mum hunc quadri­ <lb/>lateri dati e&longs;&longs;e mediæ prædictæ inuentionem cau&longs;alem afferet definitionem, <lb/>cum rei cau&longs;am dicat. </s> <s id="id.002268">Aduerte 10. Grammaticum immeritò accu&longs;are Ale­ <lb/>xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor­ <lb/>tionalis tradi in 2. Elem. nam verè in 14. 2. traditur talis inuentio, quam­ <lb/>uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen ea <lb/>reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb/>14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in­ <lb/>uentionem.</s></p><p type="main"> <s id="id.002269"><arrow.to.target n="marg177"/></s></p><p type="margin"> <s id="id.002270"><margin.target id="marg177"/>186</s></p><p type="main"> <s id="id.002271">Tex. 86. <emph type="italics"/>(Acutum mouet &longs;en&longs;um in tempore pauco multùm: graue autem in <lb/>multo parùm; non igitur velox e&longs;t acutum, graue autem tardum, fed fit illius qui­ <lb/>dem propter velocitatem motus huiu&longs;modi, huius autem propter tarditatem)<emph.end type="italics"/> vide <lb/>quæ de hac re primo topic. </s> <s id="id.002272">cap. 13. dicta &longs;unt, illa enim omnia in hunc lo­ <lb/>cum quadrant. </s> <s id="id.002273">Verum occurrit illa dubitatio; quod cum Ari&longs;t. ibi dicat <lb/><emph type="italics"/>(Vox acuta quidem velox)<emph.end type="italics"/> hic autem <emph type="italics"/>(Non igitur velox e&longs;t acutum<emph.end type="italics"/>) repugnan­ <lb/>tia dicere videtur. </s> <s id="id.002274">cui dubitationi &longs;ic occurrendum; vt dicamus ibi Philo­ <lb/>&longs;ophum dicere vocem acutam e&longs;&longs;e velocem, quatenus acumen vocis oritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.134.jpg" pagenum="134"/>ex velocitate motus aerem impellentis. </s> <s id="id.002275">hic verò di&longs;tinguere acutum à ve­ <lb/>loci, tanquam effectum à cau&longs;a.</s></p><p type="main"> <s id="id.002276"><arrow.to.target n="marg178"/></s></p><p type="margin"> <s id="id.002277"><margin.target id="marg178"/>187</s></p><p type="main"> <s id="id.002278">Tex. 159. <emph type="italics"/>(Apparent autem, & fal&longs;a, de quibus &longs;imul exi&longs;timationem veram <lb/>habet, vt apparet &longs;ol vnius pedis, per&longs;ua&longs;um autem e&longs;t, eum maiorem e&longs;&longs;e babitata)<emph.end type="italics"/> <lb/>habitata, ide&longs;t terra habitata. </s> <s id="id.002279">Vide, quæ cap. 3. &longs;ummæ 1. primi Meteor. <lb/><!-- KEEP S--></s> <s id="id.002280">Item capite 5. &longs;ummæ 2. de Solis magnitudine &longs;crip&longs;i, ea enim huic loco <lb/>abundè &longs;atisfaciunt.</s></p></chap><chap><p type="head"> <s id="id.002281"><emph type="italics"/>Ex Tertio de Anima.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002282"><arrow.to.target n="marg179"/></s></p><p type="margin"> <s id="id.002283"><margin.target id="marg179"/>188</s></p><p type="main"> <s id="id.002284">Tex. 21. <emph type="italics"/>(Vt incommen&longs;urabile, & diameter)<emph.end type="italics"/> vide, quæ de incom­ <lb/>men&longs;uratione diametri, & co&longs;tæ &longs;cripta &longs;unt lib. 1. Priorum, cap. <lb/>23. vnde &longs;atis huic loco fieri pote&longs;t.</s></p><p type="main"> <s id="id.002285"><arrow.to.target n="marg180"/></s></p><p type="margin"> <s id="id.002286"><margin.target id="marg180"/>189</s></p><p type="main"> <s id="id.002287">Tex 25. <emph type="italics"/>(Punctum autem, & omnis diui&longs;io, & &longs;ic indiui&longs;ibile mon­ <lb/>&longs;tratur &longs;i cut priuatio)<emph.end type="italics"/> punctum enim cum &longs;it terminus lineæ, e&longs;t negatio vl­ <lb/>terioris lineæ <emph type="italics"/>(Et omnis diui&longs;io)<emph.end type="italics"/> innuit his verbis præter punctum, lineam <lb/>etiam, & &longs;uperficiem, nam quemadmodum punctus oritur ex diui&longs;ione li­ <lb/>neæ, ita linea ex diui&longs;ione &longs;uperficiei, & &longs;uperficies ex diui&longs;ione corporis. <lb/></s> <s id="id.002288">& quamuis punctum, linea, &longs;uperficies, &longs;int indiui&longs;ibilia, mon&longs;trantur ta­ <lb/>men quatenus &longs;unt priuationes, &longs;eu negationes, illud vlterioris lineæ, i&longs;ta <lb/>vlterioris &longs;uperficiei, hæc tandem vlterioris corporis.</s></p><p type="main"> <s id="id.002289"><arrow.to.target n="marg181"/></s></p><p type="margin"> <s id="id.002290"><margin.target id="marg181"/>190</s></p><p type="main"> <s id="id.002291">Tex. 32. <emph type="italics"/>(Sit igitur vt A, quidem album, ad B, quod nigrum; &longs;ic C, ad D; qua­ <lb/>re & permutatim)<emph.end type="italics"/> ide&longs;t, quare & permutando (vt aiunt Geometræ) erit vt <lb/>A, ad C, ita B, ad D, hunc argumentandi modum à permutata proportio­ <lb/>ne explicaui in primo Po&longs;ter. cap. 5. tex. <!-- REMOVE S-->13. dicitur etiam alterna ratio; <lb/>& definitur ab Euclide definitione 12, 5.<!-- KEEP S--></s> </p></chap><chap><p type="head"> <s id="id.002292"><emph type="italics"/>Ex Libro de Sen&longs;u.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002293"><arrow.to.target n="marg182"/></s></p><p type="margin"> <s id="id.002294"><margin.target id="marg182"/>191</s></p><p type="main"> <s id="id.002295">Cap, 6. <emph type="italics"/>(Et qui in Die&longs;i &longs;onus latet, quamuis continuum exi&longs;tentem audit <lb/>omnem cantum, di&longs;t antia enim eius ad extremos &longs;onos latet)<emph.end type="italics"/> quid &longs;it <lb/>Die&longs;is apud Mu&longs;icos explicatum e&longs;t primo Po&longs;ter. tex. <!-- REMOVE S-->38. cum <lb/>autem Die&longs;is &longs;it minima di&longs;tantia, &longs;eu vt loquuntur Mu&longs;ici, mini­ <lb/>mum <expan abbr="interuallũ">interuallum</expan> inter duas voces, hinc fit vt hæc minima di&longs;tantia inter ex­ <lb/>tremos &longs;onos non exaudiatur, quemadmodum nec minima particula alicu­ <lb/>ius magni corporis à longè vi&longs;i <expan abbr="nõ">non</expan> percipitur, &longs;ed latetinter extrema illius.</s> </p><p type="main"> <s id="id.002296"><arrow.to.target n="marg183"/></s></p><p type="margin"> <s id="id.002297"><margin.target id="marg183"/>192</s></p><p type="main"> <s id="id.002298">Cap. 8. <emph type="italics"/>(<expan abbr="Vnumquodq;">Vnumquodque</expan> magis e&longs;t &longs;entire &longs;implex exi&longs;tens, quàm mixtum, velut <lb/>vinum non temperatum, quàm temperatum; & mel, & colorem, & neten &longs;olam. <lb/></s> <s id="id.002299">quàm in diapa&longs;on, quia ob&longs;curant &longs;e inuicem)<emph.end type="italics"/> nete apud veteres mu&longs;icos erat <lb/>in mu&longs;icis in&longs;trumentis omnium chordarum acuti&longs;&longs;ima, cuiu&longs;modi apud <lb/>nos e&longs;t, quam vulgò canto appellant. </s> <s id="id.002300">Hypate verò erat chorda omnium <lb/>graui&longs;&longs;ima, qualis e&longs;t ea, quam modo Ba&longs;&longs;o vocant. </s> <s id="id.002301">hæ duæ &longs;imul pul&longs;atæ <lb/>edebant conionantiam, quæ Diapa&longs;on dicitur, & vulgò octaua. </s> <s id="id.002302">ex quibus <lb/>&longs;en&longs;us verberum Ari&longs;t. manife&longs;tus e&longs;t.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.135.jpg" pagenum="135"/><p type="main"> <s id="id.002303"><arrow.to.target n="marg184"/></s></p><p type="margin"> <s id="id.002304"><margin.target id="marg184"/>193</s></p><p type="main"> <s id="id.002305">Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa­ <lb/>&longs;on, explicaui in primo Po&longs;ter. tex. <!-- REMOVE S-->1. Diapente verò e&longs;t con&longs;onantia ex duo­ <lb/><figure id="id.009.01.135.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.135.1.jpg" place="text"/> <lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s> <s id="id.002306">v. <!-- REMOVE S-->g. <!-- REMOVE S-->&longs;int duæ <lb/>chordæ æqualis cra&longs;fitiei, <expan abbr="atq;">atque</expan> æquè ten&longs;æ: vna <lb/>tamen habeat ad alteram proportionem &longs;e&longs;­ <lb/>quialteram, vt in figura apparet; &longs;i &longs;imul pul­ <lb/>&longs;entur, edent con&longs;onantiam Diapente. <!-- KEEP S--></s> <s id="id.002307">vulgò autem quinta.</s></p></chap><chap><p type="head"> <s id="id.002308"><emph type="italics"/>Ex Libro de Memoria, & remini&longs;centia.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002309"><arrow.to.target n="marg185"/></s></p><p type="margin"> <s id="id.002310"><margin.target id="marg185"/>194</s></p><p type="main"> <s id="id.002311">Cap. 1. <emph type="italics"/>(Sic meminit eos, qui trianguli, quod duobus rectis æquales)<emph.end type="italics"/> ide&longs;t <lb/>&longs;ic meminit tres angulos cuiu&longs;uis trianguli &longs;imul &longs;umptos æqua­ <lb/>les e&longs;&longs;e duobus angulis rectis &longs;imul &longs;umptis. </s> <s id="id.002312">lege annotata primo <lb/>Po&longs;ter. &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.002313"><arrow.to.target n="marg186"/></s></p><p type="margin"> <s id="id.002314"><margin.target id="marg186"/>195</s></p><p type="main"> <s id="id.002315">Cap. 3. <emph type="italics"/>(Sunt facilè remini&longs;cibilia, <expan abbr="quæcunq;">quæcunque</expan> habent ordinationem aliquam, <lb/>vt mathemata)<emph.end type="italics"/> h&ecedil;c Philo&longs;ophus dicensfp ectabat ad mirabilem illam, ac per­ <lb/>petuam de mon&longs;trationum connexionem, qua Geometræ omnes, & præci­ <lb/>puè Euclides opera &longs;ua ab initio ad finem v&longs;que, diuino planè ingenij acu­ <lb/>mine deduxerunt.</s> </p></chap><chap><p type="head"> <s id="id.002316"><emph type="italics"/>Ex Libro de Somnijs.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002317"><arrow.to.target n="marg187"/></s></p><p type="margin"> <s id="id.002318"><margin.target id="marg187"/>196</s></p><p type="main"> <s id="id.002319">Cap. 2. <emph type="italics"/>(Cur autem fallimur, cau&longs;a e&longs;t, quoniam non &longs;olum cum &longs;en&longs;ibile <lb/>mouetur apparent quælibet, &longs;ed etiam cum &longs;en&longs;us ip&longs;e mouetur, &longs;i eodem <lb/>modo moueatur, quemadmodum à &longs;en&longs;ibili. </s> <s id="id.002320">dico autem velut terra vi­ <lb/>detur nauigantibus moueri, dummodo vi&longs;us ab alio)<emph.end type="italics"/> reddit rationem, <lb/>cur nauigantibus videatur terra ip&longs;a moueri, ac retrocedere, non autem <lb/>ip&longs;i nauigantes, quin potius ip&longs;i fibi &longs;tare videantur. </s> <s id="id.002321">cau&longs;am igitur eam e&longs;­ <lb/>&longs;e ait, quia ex motu nauis, terra ip&longs;a manente, accidit, vt eodem modo im­ <lb/>mutetur &longs;en&longs;us vi&longs;us, ac &longs;i terra ip&longs;a moueretur, vi&longs;us verò quie&longs;ceret. <lb/></s> <s id="id.002322">At cur eodem modo afficitur &longs;en&longs;us? </s> <s id="id.002323">Per&longs;pectiuirationem e&longs;&longs;e dicunt, quia <lb/>ea, quæ circa oculum &longs;unt, vt nauis, & ea, quæ in naui &longs;unt, non mutant &longs;i­ <lb/>tum re&longs;pectu oculi, quemadmodum facerent, &longs;i nos ip&longs;i &longs;ine naui progrede­ <lb/>remur. </s> <s id="id.002324">arbores autem, & reliqua, quæ in terra &longs;unt, variant &longs;itum re&longs;pectu <lb/>oculi, non &longs;ecus, ac &longs;i ip&longs;æ arbores retro deferrentur. </s> <s id="id.002325">propterea igitur vi&longs;us <lb/>tunc arbores remeare iudicat, quia quæ circa oculum &longs;unt re&longs;pectu ip&longs;ius <lb/>oculi non mouentur, &longs;iue non variant &longs;itum ad ip&longs;um; ex variatione enim <lb/>&longs;itus rei re&longs;pectu oculi, percipimus cuiu&longs;uis rei localem motum.</s></p><p type="main"> <s id="id.002326"><arrow.to.target n="marg188"/></s></p><p type="margin"> <s id="id.002327"><margin.target id="marg188"/>197</s></p><p type="main"> <s id="id.002328">Cap. 3. <emph type="italics"/>(Quemadmodum igitur, &longs;i quem lateat &longs;uppo&longs;itus oculo digitus, non <lb/>&longs;olum app trebit, &longs;ed etiam putabitur duo, quod e&longs;t vnum. </s> <s id="id.002329">Si verò non lateat appa­ <lb/>rebit quidem, non putabitur tamen)<emph.end type="italics"/> e&longs;t hæc optica deceptio, quæ tunc accidit, <lb/>cum aliquod obiectum intuentes, interim digito alterum oculum &longs;ur&longs;um <lb/>pellimus, ita vt oculi propterea varient &longs;itum re&longs;pectu obiecti, &longs;iue non eo­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.136.jpg" pagenum="136"/>dem &longs;itu vterque obiectum intueatur, hoc e&longs;t, vt optici aiunt, axes vi&longs;uales <lb/>non amplius concurrunt &longs;imul in rem vi&longs;am. </s> <s id="id.002330">Vnde &longs;equitur &longs;peciem rei in­ <lb/>tentionalem oculis vario &longs;itu affectis imprimi, ac proinde eam eundem &longs;i­ <lb/>tum in vtroque oculo minimè obtinere, &longs;ed ea, quæ oculo à &longs;uo naturali <lb/>&longs;latu dimoto accidit ab altera alterius oculi differt; quapropter vario <lb/>ctiam modo, duplici nimirum, obiectum repre&longs;entant. </s> <s id="id.002331">atque hæc <lb/>ip&longs;a cau&longs;a e&longs;t, cur illud, quod vnum tantum e&longs;t, duo tamen <lb/>emoto oculorum altero, videatur. </s> <s id="id.002332">Vide Alhaze­ <lb/>num lib. 3. propo&longs;it. </s> <s id="id.002333">11. & 12. & infra <lb/>Problem. 7. &longs;ectionis 31.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.137.jpg" pagenum="137"/><chap><p type="head"> <s id="id.002334">EX PRIMO <lb/>METAPHYSICAE.</s></p><p type="main"> <s id="id.002335"><arrow.to.target n="marg189"/></s></p><p type="margin"> <s id="id.002336"><margin.target id="marg189"/>198</s></p><p type="main"> <s id="id.002337">Capite 1. <emph type="italics"/>(Cirea Aegyptum Mathematicæ artes constitutæ &longs;unt; illic <lb/>enim gens Sacerdotum vacare permittitur)<emph.end type="italics"/> Notanda maximè no­ <lb/>bilis Mathematicarum origo, cum ab Aegyptiorum Sacerdoti­ <lb/>bus te&longs;te Philo&longs;opho fuerint adinuentæ, quibus occa&longs;ionem præ­ <lb/>buit anniuer&longs;aria agrorum ob Nili innundationem, diui&longs;io: cum enim iam <lb/>perplures dimetiendorum agrorum rationes repertæ fui&longs;&longs;ent, Sacerdotes <lb/>ip&longs;i, quibus per otium licebat, illarum praxium demon&longs;tr ationes cœperunt <lb/>perue&longs;tigare, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; paulatim po&longs;tea Geometria amplius exculta adoleuit; <lb/>quæ deinde ij&longs;dem ad res a&longs;tronomicas per&longs;crutandas <expan abbr="adium&etilde;to">adiumento</expan> fuit, <expan abbr="hac&qacute;">hacque</expan>; <lb/>ratione reliquas etiam in mathematicas inciderunt.</s></p><p type="main"> <s id="id.002338"><arrow.to.target n="marg190"/></s></p><p type="margin"> <s id="id.002339"><margin.target id="marg190"/>199</s></p><p type="main"> <s id="id.002340">Cap. 2. <emph type="italics"/>(Sicut de præ&longs;tigio&longs;is, quæ per &longs;e mouentur, illi qui nondum &longs;peculati <lb/>&longs;unt cau&longs;am<emph.end type="italics"/>) verbis illis (<emph type="italics"/>Quæ per &longs;emouentur<emph.end type="italics"/>) vnica dictio <expan abbr="Græcare&longs;põdet">Græcare&longs;pondet</expan>, <lb/>Automata. <!-- KEEP S--></s> <s id="id.002341">erant autem Automata apud veteres Gr&ecedil;cos machinæ qu&ecedil;dam, <lb/>quæ à Mathematicis Mechanicæ artis occultis quibu&longs;dam ingenijs, ea arte <lb/>con&longs;truebantur, vt à &longs;eip&longs;is de loco ad locum, ac &longs;i viuæ e&longs;&longs;ent &longs;pontè pro­ <lb/>grederentur; vnde, & automata, qua&longs;i &longs;pontanca dicebantur. </s> <s id="id.002342">Extat adhuc <lb/>de huiu&longs;modi machinis liber Heronis Alexandrini, quem nuper ex græco <lb/>latinum reddidit docti&longs;&longs;imus Abbas Gua&longs;tallenfis. </s> <s id="id.002343">de huiu&longs;modi artificio&longs;is <lb/>operibus, quibus &longs;æpè pri&longs;ci ita admirationi fuere, vt præ&longs;tigia quædam ar­ <lb/>tificium ignorantibus, viderentur, intelligit hoc loco Ari&longs;t.<!-- KEEP S--></s></p><p type="main"> <s id="id.002344"><arrow.to.target n="marg191"/></s></p><p type="margin"> <s id="id.002345"><margin.target id="marg191"/>200</s></p><p type="main"> <s id="id.002346">Cap. 3. (<emph type="italics"/>Aut de &longs;ol&longs;titijs<emph.end type="italics"/>) quid &longs;ol&longs;titium, cur dicatur &longs;ol&longs;titium, & cur Sol <lb/>in <expan abbr="vtroq;">vtroque</expan> topico, quoad dierum incrementum, ac decrementum, & quoad <lb/>eleuationem eius, aut depre&longs;&longs;ionem meridianam, videatur moras trahere, <lb/>quamuis no&longs;trum &longs;it explicare, ob rei tamen facilitatem omittantur. </s> <s id="id.002347">Hoc <lb/>tantum &longs;cias velim &longs;ol&longs;titiorum cau&longs;am e&longs;&longs;e Zodiaci ad Tropicos longio­ <lb/>rem adhæ&longs;ionem, ide&longs;t, quòd Zodiacus propè contactum tropicorum ab ijs <lb/>parum recedat, cum ergo Sol motu proprio &longs;emper per Zodiacum inam­ <lb/>bulet, fit vt ip&longs;e <expan abbr="quoq;">quoque</expan> pariter modicum à tropicis remoueatur, imò pluri­ <lb/>mum &longs;ecus illos incedat, ita vt eo tempore, quo ad eos paulatim accedit, <lb/>aut ab eis paulatim recedit, qua&longs;i &longs;tare, &longs;iue quie&longs;cere apud eo&longs;dem videa­ <lb/>tur: <expan abbr="atq;">atque</expan> hinc etiam quantitas <expan abbr="dierũ">dierum</expan>, ac noctium videatur ferè nihil variari; <lb/>& noua elcuatio, aut depre&longs;&longs;io Solis &longs;upra horizontem nuila ferè appareat.</s></p><p type="main"> <s id="id.002348"><arrow.to.target n="marg192"/></s></p><p type="margin"> <s id="id.002349"><margin.target id="marg192"/>201</s></p><p type="main"> <s id="id.002350">Ibidem (<emph type="italics"/>Aut de diametri incommen&longs;ur abilitate, admirabile enim omnibus vi­ <lb/>detur, &longs;i quid, cum non &longs;it minimum non men&longs;uretur, decet autem in contrarium, <lb/>& in melius &longs;ecundum prouerbium con&longs;umare, quemadmodŭ in his fit, cum di&longs;cant, <lb/>nihil enim magis vir Geometricus admiraretur, quàm &longs;i diamcter commen&longs;urabi­ <lb/>lis &longs;ieret<emph.end type="italics"/>) vide quæ de hac commen&longs;urabilitate &longs;crip&longs;i lib. 1. Priorum, &longs;ect. </s> <s id="id.002351">1. <lb/>cap. 1. Videtur inquit mirum à principio Geometriam aggredienti diame­ <lb/>trum, & latus eiu&longs;dem quadrati non commen&longs;urari, cum in neutro eorum <lb/>detur minimum, &longs;eu indiui&longs;ibile, videtur enim omne diui&longs;ibile po&longs;&longs;e men&longs;u­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.138.jpg" pagenum="138"/>rari. </s> <s id="id.002352">po&longs;tea tamen cum in Geometria ver&longs;atus fuerit, maximè admirare­ <lb/>tur, &longs;i audiret diametrum e&longs;&longs;e lateri commen&longs;urabilem.</s></p><p type="main"> <s id="id.002353"><arrow.to.target n="marg193"/></s></p><p type="margin"> <s id="id.002354"><margin.target id="marg193"/>202</s></p><p type="main"> <s id="id.002355">Summa 2. cap. 3. (<emph type="italics"/>Pythagorici primi Mathematicis operam dedere, hæc præpo­ <lb/>nebant, & in cis enutriti, eorum principia, entium <expan abbr="quoq;">quoque</expan> cunctorum putant e&longs;&longs;e <lb/>principia<emph.end type="italics"/>) vtinam no&longs;trates Philo&longs;ophi Pythagoricos imitarentur; enimue­ <lb/>rò multò melius & &longs;ibi, & Philo&longs;ophiæ con&longs;ulerent. </s> <s id="id.002356">At verò non &longs;ine ma­ <lb/>gno artium, <expan abbr="atq;">atque</expan> di&longs;ciplinarum omnium di&longs;pendio à plurimis hac tempe­ <lb/>&longs;tate de&longs;pectui habentur; &longs;ed quid mirum cum quas &longs;cientiarum omnium <lb/>alumni Pythagorei omnibus &longs;cientijs anteferebant; eas no&longs;tri &longs;eculi quam­ <lb/>plures omnibus alijs facultatibus po&longs;thabeant.</s></p><p type="main"> <s id="id.002357"><arrow.to.target n="marg194"/></s></p><p type="margin"> <s id="id.002358"><margin.target id="marg194"/>203</s></p><p type="main"> <s id="id.002359">Tex. 47. (<emph type="italics"/>Qui Geometriam di&longs;cit aliqua præ&longs;cire contingit<emph.end type="italics"/>) ide&longs;t definitio­ <lb/>nes, po&longs;tulata, axiomata, quæ &longs;unt tria principiorum genera, ex quibus to­ <lb/>ta Geometria deducitur.</s></p></chap><chap><p type="head"> <s id="id.002360"><emph type="italics"/>Ex Secundo Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002361"><arrow.to.target n="marg195"/></s></p><p type="margin"> <s id="id.002362"><margin.target id="marg195"/>204</s></p><p type="main"> <s id="id.002363">Tex. 14. (<emph type="italics"/>Quantam verò vim con&longs;uetudo habeat, leges declar ant, in qui­ <lb/>bus fabulo&longs;a, ac puerilia plus po&longs;&longs;unt propter con&longs;uetudinem, quàm &longs;i <lb/>ea cogno&longs;ceremus<emph.end type="italics"/>) per leges intelligit cantilenas illas, quas vete­ <lb/>res Mu&longs;ici leges appellabant, eò quòd eas &longs;olas, cæteris abroga­ <lb/>tis liceret lata lege decantari. </s> <s id="id.002364">Vide declarationem problematis 15. & 28. <lb/>&longs;ect. </s> <s id="id.002365">19. <expan abbr="problematũ">problematum</expan> vbi <expan abbr="tanquã">tanquam</expan> in proprio loco i&longs;ta fu&longs;ius pertractabuntur.</s></p></chap><chap><p type="head"> <s id="id.002366"><emph type="italics"/>Ex Tertio Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002367"><arrow.to.target n="marg196"/></s></p><p type="margin"> <s id="id.002368"><margin.target id="marg196"/>205</s></p><p type="main"> <s id="id.002369">Tex. 3. Verba huius textus, cum &longs;atis per&longs;picua &longs;int, ac parum ma­ <lb/>thematicis indigeant, omittenda duxi. </s> <s id="id.002370">Quod ad mathematicas <lb/>attinet, ait, eas non demon&longs;trare, nec per cau&longs;am finalem, nec <lb/>per efficientem (quod intelligendum e&longs;t de Mathematicis puris, <lb/>& &longs;peculatiuis nam mathematicæ practicæ reliquas etiam cau&longs;as, efficien­ <lb/>tem, & finalem nece&longs;&longs;ariò habere debent, quapropter &longs;ophi&longs;ta quidam no­ <lb/>mine Ari&longs;tippus, eas irridebat, <expan abbr="atq;">atque</expan> adeo illiberalibus, ac &longs;edentarijs arti­ <lb/>bus po&longs;thabebat, quæ cau&longs;am efficientem, quia &longs;cilicet operantur, & fina­ <lb/>lem &longs;cilicet quæ&longs;tum &longs;ibi proponunt. </s> <s id="id.002371">fuit autem i&longs;te ex Plutarcho, & Laer­ <lb/>tio primus, qui pacto pretio doceret, <expan abbr="philo&longs;ophiam&qacute;">philo&longs;ophiamque</expan>; faceret quæ&longs;tuo&longs;am: <lb/><expan abbr="ideo&qacute;">ideoque</expan>; mathematicas paruipendebat, quòd neglecta cau&longs;a efficiente, nihil <lb/>efficerent; & finali, nihil lucrarentur. </s> <s id="id.002372">videas igitur quales &longs;int pulcherrima­ <lb/>rum facultatum contemptores, ij nimirum, qui philo&longs;ophiæ, aut lucri, aut <lb/>ambitionis cau&longs;a dant operam. </s> <s id="id.002373">Quod autem Mathematicæ nihil efficiant, <lb/><expan abbr="nihil&qacute;">nihilque</expan>; lucrentur, ne videamur vtile paruifacere, e&longs;t omninò fal&longs;um: &longs;unt <lb/>enim plures mathematicæ practicæ, quæ innumera, <expan abbr="atq;">atque</expan> <expan abbr="admirãda">admiranda</expan> efficiunt <lb/>opera, <expan abbr="quæ&qacute;">quæque</expan>; magnos quæ&longs;tus quotidie faciunt. </s> <s id="id.002374">huiu&longs;modi &longs;unt Geometria <lb/>practica, qua men&longs;urationes omnes vel &longs;olo vi&longs;u perficiuntur. </s> <s id="id.002375">Arithmeti­ <lb/>ca, cuius v&longs;us quàm latè patet? </s> <s id="id.002376">Mu&longs;ica practica, qua quotidie ip&longs;i oblecta­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.139.jpg" pagenum="139"/>mur; <expan abbr="Deo&qacute;">Deoque</expan>; Optimo Maximo laudes debitas concinimus. </s> <s id="id.002377">Mechanica pra­ <lb/>ctica, cuius ope ingentia pondera, vel exigua vi, <expan abbr="inuita&qacute;">inuitaque</expan>; natura &longs;u&longs;g; <expan abbr="de&qacute;">deque</expan>; <lb/>commouentur. </s> <s id="id.002378">Per&longs;pectiua, quæ Pictoribus, & Architectoribus adeo in&longs;er­ <lb/>uit, vt <expan abbr="ab&longs;q;">ab&longs;que</expan> ea nihil fermè audeant. </s> <s id="id.002379">A&longs;tronomia tandem, &longs;i in praxim de­ <lb/>ducatur, ex vna &longs;olum eclyp&longs;ium prædictione, quantam vniuer&longs;o orbi ad­ <lb/>mirationem parit? </s> <s id="id.002380">mitto hanc &longs;olam dierum, men&longs;ium, & annorum di&longs;tri­ <lb/>butionem, ac temporum emendationem exhibere, rem adeò Reipublicæ <lb/>Chri&longs;tianæ nece&longs;&longs;ariam.</s></p><p type="main"> <s id="id.002381"><arrow.to.target n="marg197"/></s></p><p type="margin"> <s id="id.002382"><margin.target id="marg197"/>206</s></p><p type="main"> <s id="id.002383">Eodem tex. <!-- REMOVE S-->3. (<emph type="italics"/>Item & in cæteris tunc &longs;cire vnumquodque arbitramur torum, <lb/>quorum &longs;unt demonstrationes, cum quid e&longs;t &longs;ciamus, vt puta quid tetragoni&longs;mus, <lb/>quòd inuentio mediæ<emph.end type="italics"/>) eadem reperies &longs;uperius in &longs;ecundo de Anima, tex. <!-- REMOVE S-->12. <lb/>fu&longs;ius explicata.</s> </p><p type="main"> <s id="id.002384"><arrow.to.target n="marg198"/></s></p><p type="margin"> <s id="id.002385"><margin.target id="marg198"/>207</s></p><p type="main"> <s id="id.002386">Tex. 8. (<emph type="italics"/>Si enim in hoc differret &longs;olum Geometria à Geodæ&longs;ia, quod hæc quidem <lb/>eorum e&longs;t, quæ &longs;entimus, illa verò non &longs;en&longs;ibilium e&longs;t<emph.end type="italics"/>) Geodæ&longs;ia e&longs;t pars Geo­ <lb/>metriæ practicæ, ea &longs;cilicet, quæ circa diui&longs;ionem &longs;uperficierum ver&longs;atur. <lb/></s> <s id="id.002387">audi Pedia&longs;mum de men&longs;uratione: Terræ inquit men&longs;uratio in duas partes <lb/>diuiditur, Geometriam &longs;cilicet, & Geodæ&longs;iam: Areæ <expan abbr="namq;">namque</expan> &longs;ecundum ar­ <lb/>tem men&longs;uratio, & terræ men&longs;uratio e&longs;t, & meritò Geometria vocatur. <lb/></s> <s id="id.002388">Vnius verò, & eiu&longs;dem areæ, &longs;eu loci diui&longs;io inter diuer&longs;as per&longs;onas, parti­ <lb/>tio quædam e&longs;t terræ, & iure optimo Geodæ&longs;ia appellatur. </s> <s id="id.002389">hæcille. </s> <s id="id.002390">dicitur <lb/>autem Geodæ&longs;ia à <foreign lang="greek">gea</foreign>, terra, & <foreign lang="greek">da/iw</foreign>, diuido. </s> <s id="id.002391">Vocabulum tamen i&longs;tud Geo­ <lb/>dæ&longs;iæ fuit po&longs;tea ad latiorem tran&longs;latum &longs;ignificationem: extat enim Geo­ <lb/>dæ&longs;ia Heronis Mechanici antiqui &longs;criptoris, quampridem Baroccius lati­ <lb/>nitate donauit, quæ quidem ars e&longs;t eadem cum Geometria practica, cum <lb/>non &longs;olum diui&longs;iones, &longs;ed men&longs;urationes omnes etiam per dioptricam f­ <lb/>cultatem, &longs;eu per lineas vi&longs;uales doceat inue&longs;tigare.</s></p></chap><chap><p type="head"> <s id="id.002392"><emph type="italics"/>Ex Quarto Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002393"><arrow.to.target n="marg199"/></s></p><p type="margin"> <s id="id.002394"><margin.target id="marg199"/>208</s></p><p type="main"> <s id="id.002395">Tex. 4. <emph type="italics"/>(Philo&longs;ophus <expan abbr="namq;">namque</expan> e&longs;t, vt ille, qui Mathematicus dicitur, & <lb/>bæc enim habet partes: ac prima quædam, & &longs;ecunda &longs;cientia e&longs;t: cæ <lb/>teræ <expan abbr="quoq;">quoque</expan> con&longs;equenter in mathematibus<emph.end type="italics"/>) inter mathematicas pri­ <lb/>mæ &longs;cientiæ &longs;unt Geometria, & Arithmetica, quia ip&longs;æ à cæteris <lb/>nulla ratione dependent; imò cæteræ ip&longs;is innituntur, quæ &longs;ecundæ hoc lo­ <lb/>co appellantur, hæ &longs;unt Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia. <!-- KEEP S--></s> <s id="id.002396">illas <lb/>duas recentiores &longs;ubalternantes, has verò &longs;ecundas &longs;ubalternatas vocant. <lb/></s> <s id="id.002397">Exempla &longs;ubalternationum varia attuli in Logicis tex. <!-- REMOVE S-->20. & 23. primi Po­ <lb/>&longs;ter. <!-- REMOVE S-->vbi clarè licet intueri quid &longs;it &longs;ubalternatio, vnde etiam præ&longs;ens lo­ <lb/>cus illu&longs;tratur.</s> </p><p type="main"> <s id="id.002398"><arrow.to.target n="marg200"/></s></p><p type="margin"> <s id="id.002399"><margin.target id="marg200"/>209</s></p><p type="main"> <s id="id.002400">Tex. 28. (<emph type="italics"/>Vti diametrum commen&longs;urabilem e&longs;&longs;e<emph.end type="italics"/>) legenda &longs;unt ea, quæ libro <lb/>primo Priorum, &longs;ecto 1. cap. 23. de hac commen&longs;urabilitate, & incommen­ <lb/>&longs;urabilitate tractata &longs;unt.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.140.jpg" pagenum="140"/><chap><p type="head"> <s id="id.002401"><emph type="italics"/>Ex Quinto Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002402"><arrow.to.target n="marg201"/></s></p><p type="margin"> <s id="id.002403"><margin.target id="marg201"/>210</s></p><p type="main"> <s id="id.002404">Tex. 2. (<emph type="italics"/>Alia verò cau&longs;a e&longs;t forma, & exemplar: hæc autem e&longs;t ratio ip­ <lb/>&longs;ius quid erat e&longs;&longs;e, & borum genera, vt ip&longs;ius Diapa&longs;on duo ad vnum, <lb/>& &longs;impliciter numerus, & partes, quæ in ra ione &longs;unt<emph.end type="italics"/>) affert exem­ <lb/>plum cau&longs;æ formalis ex Mu&longs;ica petitum; <expan abbr="ait&qacute;">aitque</expan>; cau&longs;am formalem <lb/>illius con&longs;onantiæ, quæ Diapa&longs;on dicitur, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; omnium perfecti&longs;&longs;ima, e&longs;&longs;e <lb/>duplam proportionem, ide&longs;t, quæ e&longs;t inter duo, & vnum, id, quod omnes <lb/>Mu&longs;ici <expan abbr="fat&etilde;tur">fatentur</expan>. </s> <s id="id.002405">quod vtinelius intelligas, repete, quæ in 2. Po&longs;ter. ad tex. <!-- REMOVE S-->1. <lb/>&longs;cripta &longs;unt: necnon quæ in libro de Sen&longs;u in cap. 8. Amplius inquit can&longs;am <lb/>formalem genericam eiu&longs;dem Diapa&longs;on e&longs;&longs;e numerum, & partes numeri, <lb/>&longs;ub numero enim continentur & duo, & vnum. </s> <s id="id.002406">Occurrit hoc loco vnum <lb/>magnopere notandum, videlicet tam con&longs;onantias, quam di&longs;&longs;onantias ha­ <lb/>bere proportiones numerorum, hoc tamen di&longs;crimine, quod con&longs;onantiæ <lb/>habent &longs;olùm proportiones numerorum eorum, qui quaternario continen­ <lb/>tur, ex veterum præ&longs;ertim Pythagoreorum &longs;ententia, qui propterea vltra <lb/>quaternarium progredi vetabant. </s> <s id="id.002407">Recentiores tamen y&longs;que ad &longs;enarium <lb/>procedunt, quippe, qui omnes vocum con&longs;onantias admittunt, quæ pro­ <lb/>portionibus numerorum &longs;enario contentorum præditæ &longs;int. </s> <s id="id.002408">Di&longs;&longs;onantiæ <lb/>verò &longs;eoundum pri&longs;cos habent proportiones numerorum extra quaterna­ <lb/>rium progredientium, iuxta no&longs;tros autem extra &longs;enarium. </s> <s id="id.002409">qua de re pluri­ <lb/>bus Zarlinus colloquio 2. definit. </s> <s id="id.002410">3.<!-- KEEP S--></s></p><p type="main"> <s id="id.002411"><arrow.to.target n="marg202"/></s></p><p type="margin"> <s id="id.002412"><margin.target id="marg202"/>211</s></p><p type="main"> <s id="id.002413">Tex. 3. <emph type="italics"/>(Partes <expan abbr="quoq;">quoque</expan> totius<emph.end type="italics"/>) ide&longs;t &longs;unt inateria; loquitur enim de cau&longs;a <lb/>materiali. </s> <s id="id.002414">libuit locum hunc annotare in gratiam Geometricarum demon­ <lb/>&longs;trationum, quorum media &longs;æpè &longs;unt ex cau&longs;a materiali &longs;umpta, quod ta­ <lb/>men non ita ab omnibus ob&longs;eruatur, <expan abbr="quotie&longs;cunq;">quotie&longs;cunque</expan> enim probant affe ctio­ <lb/>nem quampiam de aliquo &longs;ubiecto, ex eo, quod &longs;ubiectum illud &longs;it, vel di­ <lb/>midium alicuius, vel duplum, vel reliquum, vel tertia pars, & his &longs;imilia, <lb/>erit talis ratio in genere cau&longs;æ materialis. </s> <s id="id.002415"><expan abbr="neq;">neque</expan> e&longs;t cur recentiores quidam, <lb/>naturalibus &longs;cientijs a&longs;&longs;ueti, negent huiu&longs;modi materiam veram e&longs;&longs;emate­ <lb/>riam, ac proinde neq, Geometricas demon&longs;trationes veras e&longs;&longs;e demon&longs;tra­ <lb/>tiones; dicendum enim talem quidem materiam non e&longs;&longs;e veram materiam <lb/>phy&longs;icam, & proinde illas demon&longs;trationes <expan abbr="nõ">non</expan> e&longs;&longs;e veras naturales demon­ <lb/>&longs;irationes, e&longs;&longs;e tamen veram materiam intelligibilem, quæ Geometriæ &longs;u­ <lb/>bijcitur, & proinde demon&longs;trationes illas veras e&longs;&longs;e demon&longs;trationes Geo­ <lb/>metricas; id quod Ari&longs;t. &longs;æpius in libris Po&longs;ter, apertè &longs;ignificat, tum a&longs;&longs;er­ <lb/>tionibus, tum exemplis quamplurimis. </s> <s id="id.002416">Quapropter cauendum e&longs;t illis, ne <lb/>ingrati animi notam incurrant, dum pulcherrimam artem re&longs;olutoriam, <lb/>quam Ari&longs;t. à Mathematicis acceptam omnibus &longs;cientijs accommodauit <lb/>(vt initio Priorum o&longs;ten&longs;um e&longs;t) eam ip&longs;i ita alijs facultatibus adaptent, vt <lb/>Mathematicis ip&longs;is, ex quibus orta, & &longs;ub quibus adoleuit, pulla ratione <lb/>conuenire poi&longs;it. </s> <s id="id.002417">De hac materia fu&longs;ius infra in additamento de natura Ma­ <lb/>thematicarum.</s></p><p type="main"> <s id="id.002418"><arrow.to.target n="marg203"/></s></p><p type="margin"> <s id="id.002419"><margin.target id="marg203"/>212</s></p><p type="main"> <s id="id.002420">Tex. 3. (<emph type="italics"/>Et ip&longs;ius Diapa&longs;on duplum, & numerus<emph.end type="italics"/>) &longs;cilicet cau&longs;æ formales <lb/>&longs;unt, quemadmodum &longs;upra tex. <!-- REMOVE S-->2. huius cap. explicatum e&longs;t.</s> </p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.141.jpg" pagenum="141"/><p type="main"> <s id="id.002421"><arrow.to.target n="marg204"/></s></p><p type="margin"> <s id="id.002422"><margin.target id="marg204"/>213</s></p><p type="main"> <s id="id.002423">Tex. 4. (<emph type="italics"/>Similiter autem figurationum <expan abbr="quoq;">quoque</expan> elementa dicuntur, ac &longs;impliciter <lb/>demon&longs;trationum primæ enim demon&longs;trationes, quæ in pluribus demonstr ationbus <lb/>in&longs;unt, hæc elementa demon&longs;trationum dicuntur<emph.end type="italics"/>) verbo (<emph type="italics"/>Figurationum<emph.end type="italics"/>) &longs;iue <expan abbr="de-&longs;criptionũ">de­ <lb/>&longs;criptionum</expan>, Ari&longs;t, intelligere demon&longs;trationes Geometricas, &longs;æpius dictum <lb/>e&longs;t, præ&longs;ertim in Logicis, & ex hoc loco pariter confirmatur. </s> <s id="id.002424">Ex hoc por­ <lb/>rò loco illud innote&longs;cit dignum, quod præcipuè à Mathematico non igno­ <lb/>retur, quæ nam &longs;int demon&longs;trationes illæ, quæ nomine <expan abbr="elementorũ">elementorum</expan> debeant <lb/>appelllari, necnon cau&longs;a cur Euclides &longs;uum opus elementa nuncupauerit, <lb/>&longs;unt enim illæ, quæ in pluribus demon&longs;trationibus in&longs;unt, ide&longs;t, quæ &longs;æpius <lb/>in alijs demon&longs;trationibus citantur, vti &longs;unt præcipuè &longs;ex priores libri Eu­ <lb/>clidis: <expan abbr="atq;">atque</expan> hac ratione elementa appellantur.</s></p><p type="main"> <s id="id.002425"><arrow.to.target n="marg205"/></s></p><p type="margin"> <s id="id.002426"><margin.target id="marg205"/>214</s></p><p type="main"> <s id="id.002427">Tex. 12. <emph type="italics"/>(Principium <expan abbr="itaq;">itaque</expan> &longs;cibilis, circa <expan abbr="vnumquodq;">vnumquodque</expan> ip&longs;um vnum, non e&longs;t au­ <lb/>tem idem in cunct is generibus vnum, &longs;ed hic quidem die&longs;is, hic verò vocalis, aut <lb/>muta)<emph.end type="italics"/> ide&longs;t, in Mu&longs;ica quidem principium omnium, & elementum e&longs;t die­ <lb/>&longs;is, quæ e&longs;t minima vox, aut &longs;onus, qui &longs;ub Mu&longs;ici con&longs;iderationem cadat. <lb/></s> <s id="id.002428">Porrò ad tex. <!-- REMOVE S-->38. primi Po&longs;ter. de die&longs;i plura &longs;unt dicta.</s> </p><p type="main"> <s id="id.002429"><arrow.to.target n="marg206"/></s></p><p type="margin"> <s id="id.002430"><margin.target id="marg206"/>215</s></p><p type="main"> <s id="id.002431">Tex. 17. <emph type="italics"/>(Veluti diametrum commen&longs;urabilem e&longs;&longs;e impo&longs;&longs;ibile est)<emph.end type="italics"/> huius expo­ <lb/>&longs;itionem inuenies 1. Priorum, &longs;ecto 1. cap. 23.</s></p><figure id="id.009.01.141.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.141.1.jpg" place="text"/><p type="main"> <s id="id.002432"><arrow.to.target n="marg207"/></s></p><p type="margin"> <s id="id.002433"><margin.target id="marg207"/>216</s></p><p type="main"> <s id="id.002434">Tex. eodem <emph type="italics"/>(Metaphoricè autem, quæ in Geometria po­ <lb/>tentia dicitur)<emph.end type="italics"/> potentiam vnius lineæ appellant Geometræ <lb/>quadratum illius, ide&longs;t quadratum &longs;uper ip&longs;am con&longs;tru­ <lb/>ctum. </s> <s id="id.002435">v. <!-- REMOVE S-->g. <!-- REMOVE S-->quadratum in quo C, dicitur potentia lineæ <lb/>D B, quia &longs;uper illam con&longs;tructum e&longs;t.</s> </p><p type="main"> <s id="id.002436"><arrow.to.target n="marg208"/></s></p><p type="margin"> <s id="id.002437"><margin.target id="marg208"/>217</s></p><p type="main"> <s id="id.002438">Tex. 34. (<emph type="italics"/>Quemadmodum dicitur diametrum e&longs;&longs;e commen&longs;urabilem<emph.end type="italics"/>) vide an­ <lb/>notata 1. Priorum, fecto 1. cap. 23.</s></p><p type="main"> <s id="id.002439"><arrow.to.target n="marg209"/></s></p><p type="margin"> <s id="id.002440"><margin.target id="marg209"/>218</s></p><p type="main"> <s id="id.002441">Tex. 35. (<emph type="italics"/>Vt triangulo duos rectos habere<emph.end type="italics"/>) ide&longs;t affectio trianguli e&longs;t habe­ <lb/>re tres angulos æquales duobus rectis angulis. </s> <s id="id.002442">Vide declarationem huius <lb/>lib. primo Priornm, &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p></chap><chap><p type="head"> <s id="id.002443"><emph type="italics"/>Ex Sexto Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002444"><arrow.to.target n="marg210"/></s></p><p type="margin"> <s id="id.002445"><margin.target id="marg210"/>219</s></p><p type="main"> <s id="id.002446">Tex. 1. (<emph type="italics"/>Mathematicorum quoque principia, elementa, & cau&longs;æ &longs;unt<emph.end type="italics"/>) <lb/>notanda &longs;unt hæc aduer&longs;us quo&longs;dam, qui negant in Mathemati­ <lb/>cis cau&longs;as reperiri, vt hinc <expan abbr="quoq;">quoque</expan> illis &longs;cientiam auferant. </s> <s id="id.002447">enim­ <lb/>uerò apertè patet eos falli ex toto hoc Ari&longs;t. di&longs;cur&longs;u.</s></p></chap><chap><p type="head"> <s id="id.002448"><emph type="italics"/>Ex Nono Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002449"><arrow.to.target n="marg211"/></s></p><p type="margin"> <s id="id.002450"><margin.target id="marg211"/>220</s></p><p type="main"> <s id="id.002451"><emph type="italics"/>Vt &longs;i quis dicat diametrum po&longs;&longs;e commen&longs;arari, non tamen commen&longs;u­ <lb/>rabitur<emph.end type="italics"/>) & paulò infra (<emph type="italics"/>Commen&longs;urari enim impo&longs;&longs;ibile e&longs;t<emph.end type="italics"/>) expo&longs;i­ <lb/>tionem horum reperies 1. Priorum, &longs;ecto 1. cap. 23.</s></p><p type="main"> <s id="id.002452"><arrow.to.target n="marg212"/></s></p><p type="margin"> <s id="id.002453"><margin.target id="marg212"/>221</s></p><p type="main"> <s id="id.002454">Tex. 20. (<emph type="italics"/>De&longs;eriptiones <expan abbr="quoq;">quoque</expan> actu inueniuntur, diuidentes nanque <lb/>inuenirent, quod &longs;i diui&longs;æ e&longs;&longs;ent, manife&longs;i è e&longs;&longs;ent, nunc autem in&longs;unt potentia, cur <lb/>triangulus duo recti? </s> <s id="id.002455">quia qui circa vnum punctum anguli duobus rectis æquales<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.142.jpg" pagenum="142"/><emph type="italics"/>&longs;unt, &longs;i igitur quæ ad latus educeretur, videnti mox e&longs;&longs;et manife&longs;tum<emph.end type="italics"/>) per de&longs;cri­ <lb/>ptiones, vel figurationes, vel de&longs;ignationes intelligendas e&longs;&longs;e demon&longs;tra­ <lb/>tiones Geometricas &longs;æpius &longs;upra dictum e&longs;t, & pariter ex hoc loco com­ <lb/>probatur. </s> <s id="id.002456">Dicit igitur, quod demon&longs;trationes &longs;uas Geometræ inueniunt, <lb/>reducendo ad actum ea, quæ erant in potentia, diuidentes enim educunt in <lb/>actum, figuras, angulos, lineas, & cætera huiu&longs;inodi, quæ prius &longs;olùm erat <lb/>in potentia, ex quibus po&longs;tea &longs;uas demon&longs;trationes perficiunt (<emph type="italics"/>Cur triangu­ <lb/>lus duo recti<emph.end type="italics"/>) affert exemplum eius, quod proximè dixerat, &longs;cilicet Geome­ <lb/>tras demon&longs;trare producendo ad actum entia quædam Mathematica, quod <lb/>exemplum, vt intelligas ijs opus habes, quæ primo Priorum, &longs;ecto 3. cap. 1. <lb/>con&longs;cripta &longs;unt (<emph type="italics"/>Cur triangulus duo recti?<emph.end type="italics"/>) ide&longs;t, cur triangulus habet tres <lb/>angulos æquales duobus rectis angulis (<emph type="italics"/>Quia qui circa vnum punctum anguli <lb/>duobus rectis angulis æquales &longs;unt<emph.end type="italics"/>) ni&longs;i hoc dictum ad bonum trahatur &longs;en&longs;um, <lb/><figure id="id.009.01.142.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.142.1.jpg" place="text"/> <lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. <!-- REMOVE S-->g. <!-- REMOVE S-->A, &longs;unt con&longs;tituti, æquales &longs;unt <lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis, <lb/>vt patet ex corollario 2. 15. primi Elem. quot­ <lb/>quot enim anguli con&longs;tituantur ad punctum A, <lb/>omnes &longs;imul erunt æquales quatuor rectis, quos <lb/>faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim <lb/>illi congruent his quatuor rectis: &longs;ed Ari&longs;t. &longs;en­ <lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti­ <lb/>tutos, v. <!-- REMOVE S-->g. <!-- REMOVE S-->ad partes &longs;uperiores lineæ B C, e&longs;&longs;e <lb/>æquales duobus rectis B A D, D A C, vt o&longs;tenditur in 13. primi, necnon <lb/>etiam patere pote&longs;t ex corollario 2. 15. eiu&longs;dem. </s> <s id="id.002457">tales &longs;unt quatuor anguli <lb/>ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet, <lb/><figure id="id.009.01.142.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.142.2.jpg" place="text"/> <lb/>&longs;unt æquales duobus rectis B A D, D A C, <lb/>tales etiam &longs;unt in hac &longs;ecunda figura tres <lb/>anguli B C A, A C D, D C E, qui quidem <lb/>æquales &longs;unt duobus rectis angulis. </s> <s id="id.002458">hoc <lb/>&longs;en&longs;i&longs;&longs;e Ari&longs;t. patet ex demon&longs;tratione 32. <lb/>primi, quæ demon&longs;trat <expan abbr="memoratã">memoratam</expan> ab Ari­ <lb/>&longs;tot. trianguli affectionem, & ad quam <lb/>propterea ip&longs;e &longs;pectabat, cuius figura e&longs;t <lb/>eadem cum hac &longs;ecunda, in qua Euclides o&longs;tendit prædictos tres angulos <lb/>æquari duobus rectis. </s> <s id="id.002459">&longs;ubdit po&longs;tea, &longs;i igitur linea C D, quæ ad latus A B, <lb/>parallela e&longs;t in potentia, educeretur in actum, videnti mox e&longs;&longs;et manife&longs;tum <lb/>tres angulos trianguli A B C, e&longs;&longs;e pares duobus rectis. </s> <s id="id.002460">ducta enim C D, pa­ <lb/>rallela lateri B A, apparet &longs;tatim angulus A, æqualis angulo A C D, & an­ <lb/>gulus B, angulo D C E; cum reliquus verò <expan abbr="triãguli">trianguli</expan> angulus B C A, &longs;it apud <lb/>prædictos duos ad idem punctum C, con&longs;tit utus; <expan abbr="atq;">atque</expan> omnes hi tres duobus <lb/>rectis æquentur, mox in&longs;picienti talem figur ationem manife&longs;tum fit tres an­ <lb/>gulos illius trianguli e&longs;&longs;e duobus rectis æqu ales.</s></p><p type="main"> <s id="id.002461"><arrow.to.target n="marg213"/></s></p><p type="margin"> <s id="id.002462"><margin.target id="marg213"/>222</s></p><p type="main"> <s id="id.002463">Ibidem (<emph type="italics"/>Cur in &longs;emicirculo vniuer&longs;aliter rectus? </s> <s id="id.002464">quia &longs;i tres æquales, & quæ <lb/>ba&longs;is e&longs;t duo, & quæ ex medio &longs;upra stat recta, videnti manifestum erit ei, qui illud <lb/>&longs;ciat<emph.end type="italics"/>) In 2. Po&longs;ter. tex. <!-- REMOVE S-->11. inuenies hu ius loci expo&longs;itionem. </s> <s id="id.002465">nunc &longs;olùm <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.143.jpg" pagenum="143"/><figure id="id.009.01.143.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/009-01-figures/009.01.143.1.jpg" place="text"/> <lb/>hæc addenda &longs;unt. </s> <s id="id.002466">Re&longs;pondet Ari&longs;t. quæ­ <lb/>&longs;ito pr&ecedil;cedenti, cur &longs;cilicet angulus in &longs;e­ <lb/>micirculo &longs;it rectus, qualis e&longs;t in figura <lb/>angulus A C B, <expan abbr="dicit&qacute;">dicitque</expan>; cau&longs;am e&longs;&longs;e, quia <lb/>in figura tres lineæ &longs;unt æquales, duæ ni­ <lb/>mirum, in quas ba&longs;is B A, diuiditur, quæ <lb/>&longs;unt B D, D A, & tertia, quæ ex medio <lb/>ba&longs;is erigitur, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; D C, cum omnes &longs;int <lb/>&longs;emidia metri ciu&longs;dem circuli. </s> <s id="id.002467">educta <expan abbr="itaq;">itaque</expan> linea D C, de potentia in actum, <lb/>&longs;i cuipiam trium harum linearum æqualitas innote&longs;cat, continuò ei etiam <lb/>manife&longs;tum erit angulum A C B, in &longs;emicirculo, e&longs;&longs;e rectum. </s> <s id="id.002468">quia &longs;tatim ap­ <lb/>parent duo i&longs;o&longs;celia B D C, A D C, quorum anguli ad ba&longs;es B C, A C, &longs;unt <lb/>æquales inuicem; & anguli duo ad D, &longs;unt dupli duorum <expan abbr="angulorũ">angulorum</expan> A C D, <lb/>D C B, ex quibus conflatur totus angulus A C B, ergo duo anguli ad D, &longs;unt <lb/>dupli anguli B C A, &longs;ed duo anguli ad D, &longs;unt æquales duobus rectis, ergo <lb/>duo recti &longs;unt dupli anguli A C B, ergo angulus B C A, e&longs;t dimidium duo­ <lb/>rum rectorum. </s> <s id="id.002469">cum autem omnes recti &longs;int æquales, con&longs;ectarium e&longs;t dimi­ <lb/>dium duorum rectorum e&longs;&longs;e angulum rectum. </s> <s id="id.002470">patet igitur, qua ratione ex <lb/>ductu linearum prædictarum actu, manife&longs;tum fiat angulum in &longs;emicirculo <lb/>A C B, e&longs;&longs;e rectum. </s> <s id="id.002471">ne mireris &longs;i vulgatam tran&longs;lationem antiquam non <lb/>&longs;um &longs;equutus, indigebat enim correctione, quam iuxta græcum exem­ <lb/>plar adhibui.</s></p><p type="main"> <s id="id.002472"><arrow.to.target n="marg214"/></s></p><p type="margin"> <s id="id.002473"><margin.target id="marg214"/>223</s></p><p type="main"> <s id="id.002474">Tex. 22. (<emph type="italics"/>Vt puta &longs;i triangulum non putet mutari, non opinabitur modo duos <lb/>rectos habere, modo non, mutaretur enim<emph.end type="italics"/>) quia nimirum huius habemus &longs;cien­ <lb/>tiam per demon&longs;trationem 32. primi Elementorum. <!-- KEEP S--></s> <s id="id.002475">quomodo autem tri­ <lb/>angulus habeat duos rectos, ide&longs;t tres angulos æquales duobus rectis angu­ <lb/>lis, explicatum e&longs;t primo Priorum, &longs;ecto 3. cap. 1.<!-- KEEP S--></s></p><p type="main"> <s id="id.002476"><arrow.to.target n="marg215"/></s></p><p type="margin"> <s id="id.002477"><margin.target id="marg215"/>224</s></p><p type="main"> <s id="id.002478">Ibidem (<emph type="italics"/>Verum aliquid quidem, aliquid verò non, vt puta parem numerum <lb/>primum nullum e&longs;&longs;e; aut quo&longs;dam quidem, quo&longs;dam verò non<emph.end type="italics"/>) definitione 11. <lb/>7. Elem. &longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri­ <lb/>mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa­ <lb/>res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men­ <lb/>&longs;uraret. </s> <s id="id.002479">quaternarium autem, &longs;enarium, &c. </s> <s id="id.002480">pares, non e&longs;&longs;e primos, cum <lb/>eos non &longs;ola vnitas, &longs;ed alius numerus metiatur: quaternarium enim bina­ <lb/>rius bis replicatus men&longs;urat: &longs;enarium men&longs;urat & binarius, & ternarius: <lb/>quare verum erit exi&longs;timare inter pares numeros aliquos e&longs;&longs;e primos, ide&longs;t <lb/>binarium, aliquos verò non, ide&longs;t cæteros pares vltra binarium.</s></p></chap><chap><p type="head"> <s id="id.002481"><emph type="italics"/>Ex Decimo Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002482"><arrow.to.target n="marg216"/></s></p><p type="margin"> <s id="id.002483"><margin.target id="marg216"/>225</s></p><p type="main"> <s id="id.002484">Tex. 4. (<emph type="italics"/>Ac etiam motum &longs;implici, & veloti&longs;&longs;imo motu men&longs;urant, mi­ <lb/>nimum enim tempus hic habet. </s> <s id="id.002485">quapropter in A&longs;trologia tale vnŭ prin­ <lb/>cipium, & men&longs;ura e&longs;t. </s> <s id="id.002486">motum enim æqualem, & veloci&longs;&longs;iniŭ œli &longs;up­ <lb/>ponunt, ad quem cæteros tudicant<emph.end type="italics"/>) intelligit motum diureum, quam <lb/>primo cœlo, &longs;eu mobili a&longs;eribunt, hic enim veloci&longs;&longs;imus e&longs;t omnium reli­<pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.144.jpg" pagenum="144"/>quorum cœle&longs;tium motuum, ac &longs;impliciffimus, & valdè vniformis, ac regu­ <lb/>laris, & propterea minimum habet tempus, ide&longs;t tempus vnius diei natura­ <lb/>lis, quo tempore totum primum mobile circulationem integram perficit. <lb/></s> <s id="id.002487">per minimum tempus, po&longs;&longs;uut etiam intelligi partes diei, quæ &longs;unt horæ, & <lb/>horarum partes. </s> <s id="id.002488">con&longs;iderant hunc motum in circulo æquàtoris, quia æqua­ <lb/>tor motu primi mobilis, &longs;eu diurno vniformiter, ae maximè regulatiter <lb/>mouetur: hac de cau&longs;a hunc motum tanquam reliquorum men&longs;uram, ac <lb/>normam meritò a&longs;&longs;ump&longs;erunt.</s></p><p type="main"> <s id="id.002489"><arrow.to.target n="marg217"/></s></p><p type="margin"> <s id="id.002490"><margin.target id="marg217"/>226</s></p><p type="main"> <s id="id.002491">Ibidem <emph type="italics"/>(Et in Mu&longs;ica Die&longs;is primus &longs;en&longs;ibilis &longs;onus, quia minimum)<emph.end type="italics"/> ide&longs;t mi­ <lb/>nimum interuallum, quod à Mu&longs;icis con&longs;ideretur, e&longs;t men&longs;ura maiorum in­ <lb/>teruallorum. </s> <s id="id.002492">ad tex. <!-- REMOVE S-->38. primi Po&longs;ter. &longs;atis dictum e&longs;t de Die&longs;i, quæ videas.</s> </p><p type="main"> <s id="id.002493"><arrow.to.target n="marg218"/></s></p><p type="margin"> <s id="id.002494"><margin.target id="marg218"/>227</s></p><p type="main"> <s id="id.002495">Eodem tex. <!-- REMOVE S-->&longs;ed cap. 3. <emph type="italics"/>(Nox &longs;emper autem men&longs;ura numero vnum e&longs;t, verum <lb/>aliquando plura, vt puta die&longs;es duæ, non quidem &longs;ecundum cuditum, &longs;ed in ratio­ <lb/>nibus, & voces plures, quibus men&longs;uramus, & diameter duobus men&longs;uratur, & la­ <lb/>tus, & omnes magnitudines)<emph.end type="italics"/> ita corrigenda e&longs;t antiqua tran&longs;latio. </s> <s id="id.002496">quid die&longs;is <lb/>dictum &longs;it ad tex. <!-- REMOVE S-->38. primi Po&longs;ter. quando autem ait <emph type="italics"/>(Vt puta duæ die&longs;es)<emph.end type="italics"/> <lb/>ide&longs;t duæ die&longs;es &longs;unt men&longs;ura vnius interualli mu&longs;ici, qui tonus appellatur: <lb/>quæ quidem duæ die&longs;es non &longs;unt men&longs;ura &longs;en&longs;ibilis, quæ &longs;cilicet auribus per­ <lb/>cipiatur, &longs;ed tantummodò exi&longs;tunt in numerorum proportionibus, ibi per <lb/>intellectum excogitatis, quando ait <emph type="italics"/>(Et voces plures quibus men&longs;uramus)<emph.end type="italics"/> <lb/>quando vtimur eodem interuallo, &longs;iue eadem voce ad cantus men&longs;uram, <lb/>tunc &longs;unt plures men&longs;uræ numero, quamuis vna tantum &longs;pecie. Ait <emph type="italics"/>(Et dia­ <lb/>meter duobus men&longs;uratur)<emph.end type="italics"/> v. <!-- REMOVE S-->g. <!-- REMOVE S-->duobus &longs;emidiametris: vel duobus pedibus. <lb/></s> <s id="id.002497">& latus pariter quadrati, duobus. </s> <s id="id.002498">v. <!-- REMOVE S-->g. <!-- REMOVE S-->pedibus mensuratur; eode<expan abbr="m&qacute;">mque</expan>; mo­ <lb/>do reliquæ omnes magnitudines po&longs;&longs;unt ab eadem men&longs;ura &longs;æipius replica­ <lb/>ta men&longs;urari.</s> </p><p type="main"> <s id="id.002499"><arrow.to.target n="marg219"/></s></p><p type="margin"> <s id="id.002500"><margin.target id="marg219"/>228</s></p><p type="main"> <s id="id.002501">Eodem tex. <emph type="italics"/>(Semper autem men&longs;ura eiu&longs;dem generis e&longs;t, magnitudinum nam­ <lb/>que magnitudo, & &longs;ecundum vnumquedque, longitudinis longitudo<emph.end type="italics"/>) ex his ratio <lb/>manife&longs;ta apparet, cur Geometræ practici men&longs;urent longitudines per ali­ <lb/>quam longitudinem, vt puta per vlnam, digitum, vnciam, &c. </s> <s id="id.002502">&longs;uperficies <lb/>etiam per aliquam &longs;uperficiem, &longs;ed quæ &longs;it quadrata, vt puta per vlnam qua­ <lb/>dratam, palmum quadratum, &c. </s> <s id="id.002503">corpora <expan abbr="quoq;">quoque</expan> per corpus, quod tamen <lb/>&longs;it cubus, vt per vlnam cubicam, palmum cubicum, vnciam cubicam, &c.</s></p><p type="main"> <s id="id.002504"><arrow.to.target n="marg220"/></s></p><p type="margin"> <s id="id.002505"><margin.target id="marg220"/>229</s></p><p type="main"> <s id="id.002506">Tex. 11. <emph type="italics"/>(Similia verò &longs;i cum non &longs;int eadem &longs;impliciter, nec &longs;ecundum &longs;ab&longs;t an­ <lb/>tiam &longs;ubiectam in d. <!-- KEEP S--></s> <s id="id.002507">fferentia &longs;ecundum formam eadem &longs;it: quemadmodum quadra­ <lb/>tum maius minori &longs;imile e&longs;t, & lineæ inæquales, hæ enim &longs;imiles quidem, verŭ non <lb/>cædem &longs;impliciter &longs;unt)<emph.end type="italics"/> Prima definitio &longs;exti definit &longs;imiles figuras eas e&longs;&longs;e, <lb/>quæ &longs;unt æquiangulæ inuicem, & quæ habent latera proportionalia circa <lb/>æquales angulos. </s> <s id="id.002508">cum ergò quadratum maius, & minus &longs;int æquiangula, <lb/>quia habent omnes angulos rectos; & præterea habeant latera circa æqua­ <lb/>les angulos proportionalia, &longs;icut enim latera maioris quadrati circa vnum <lb/>angulum rectum &longs;unt in proportione æqualitatis; ita <expan abbr="quoq;">quoque</expan> latera minoris <lb/>circa vnum angulum rectum &longs;unt illis proportionalia, cum &longs;int inuicem pa­ <lb/>riter in proportione æqualitatis, erunt nece&longs;&longs;ariò &longs;imilia hæc duo quadrata. <lb/></s> <s id="id.002509">duæ ctiam, exempli gratia, lineæ rectæ &longs;unt inuicem &longs;imiles, quamuis vna <lb/>&longs;it maior altera.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.145.jpg" pagenum="145"/><p type="main"> <s id="id.002510"><arrow.to.target n="marg221"/></s></p><p type="margin"> <s id="id.002511"><margin.target id="marg221"/>230</s></p><p type="main"> <s id="id.002512">Eodem tex. <emph type="italics"/>(Tertium &longs;icut illa, quæ in Mathematicis)<emph.end type="italics"/> tertium &longs;cilicet mo­ <lb/>dum diuer&longs;i, ponit in entibus Mathematicis, &longs;icut enim po&longs;uit idem e&longs;&longs;e in <lb/>Mathematicis, quando duæ figuræ &longs;unt &longs;imiles, & æquales: ita ex oppo&longs;ito <lb/>diuer&longs;um erit in Mathematicis, quando duæ figuræ fuerint di&longs;&longs;imiles, & in­ <lb/>æquales, <expan abbr="dicentur&qacute;">dicenturque</expan>; diuer&longs;æ, in quo con&longs;i&longs;tat &longs;imilitudo figurarum dictum <lb/>e&longs;t in præcedenti expo&longs;itione.</s></p></chap><chap><p type="head"> <s id="id.002513"><emph type="italics"/>Ex Vndecimo Metaphy&longs;iæ.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002514"><arrow.to.target n="marg222"/></s></p><p type="margin"> <s id="id.002515"><margin.target id="marg222"/>231</s></p><p type="main"> <s id="id.002516">Svmma r. </s> <s id="id.002517">cap. 2. <emph type="italics"/>(Si quis verò lineas, aut quæ has &longs;equuntur, dico autem <lb/>primas &longs;uperficies principia e&longs;&longs;e ponat. </s> <s id="id.002518">bæc non &longs;unt &longs;ub&longs;tantiæ &longs;eparabiles, <lb/>verùm &longs;ectiones, & diui&longs;iones, illæ quidem in &longs;uperficierum, hæc verò cor­ <lb/>porum, puncta verò linearum &longs;unt, & etiam ip&longs;arum earumdem termini; <lb/>hæc autem omnia in alijs &longs;unt, & nihil &longs;eparabile e&longs;t)<emph.end type="italics"/> ait puncta oriri ex &longs;ectio­ <lb/>ne lineæ, quamuis &longs;int etiam termini illius; lineas verò oriri ex diui&longs;ione <lb/>&longs;uperficierum, quamuis &longs;int etiam termini illarum. </s> <s id="id.002519">&longs;uperficies <expan abbr="quoq;">quoque</expan> oriri <lb/>ex diui&longs;ione corporum, quamuis &longs;int etiam termini, illorum. </s> <s id="id.002520">Hæc placuit <lb/>annotare propter <expan abbr="ip&longs;orũ">ip&longs;orum</expan> conuenientiam <expan abbr="cũ">cum</expan> ijs, quæ à Geometris traduntur.</s></p><p type="main"> <s id="id.002521"><arrow.to.target n="marg223"/></s></p><p type="margin"> <s id="id.002522"><margin.target id="marg223"/>232</s></p><p type="main"> <s id="id.002523">Summa 3. cap. 2. <emph type="italics"/>(Vt puta &longs;ub Cane fiat frigus)<emph.end type="italics"/> ideft &longs;ub ortum Canis cœ­ <lb/>læ&longs;tis, &longs;eu Caniculæ. <!-- KEEP S--></s> <s id="id.002524">Vide quæ libro &longs;ecundo Meteororum, &longs;umma 2. cap. 2. <lb/>de hac &longs;tella &longs;crip&longs;imus.</s></p></chap><chap><p type="head"> <s id="id.002525"><emph type="italics"/>Ex Duodecimo Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002526"><arrow.to.target n="marg224"/></s></p><p type="margin"> <s id="id.002527"><margin.target id="marg224"/>233</s></p><p type="main"> <s id="id.002528">Tex. 44. <emph type="italics"/>(Pluralitatem verò lationum ex peculiari&longs;&longs;ima Philo&longs;ophie <lb/>Mathematicarum &longs;cientiarum, videlicet ex A&longs;tronomia con&longs;iderandum <lb/>est: hæc enim de &longs;ub&longs;tantia &longs;en&longs;ibili quidem, ac &longs;empiterna &longs;peculatur)<emph.end type="italics"/> <lb/>pluralitatem nimirum cœle&longs;tium motuum petendam e&longs;&longs;e a&longs;&longs;erit <lb/>ex præcipua totius Philo&longs;ophiæ parte, quam ait e&longs;&longs;e A&longs;tronomiam. </s> <s id="id.002529">dignum <lb/>porrò con&longs;ideratione e&longs;t, quanti faciat Ari&longs;t. Mathematicas di&longs;ciplinas, ac <lb/>præcipuè &longs;yderalem &longs;cientiam.</s></p><p type="main"> <s id="id.002530"><arrow.to.target n="marg225"/></s></p><p type="margin"> <s id="id.002531"><margin.target id="marg225"/>234</s></p><p type="main"> <s id="id.002532">Tex. 45. <emph type="italics"/>(Eudoxus igitur Solis, & Lunæ lationem po&longs;uit fieri à tribus orbibus, <lb/>quorum primus quidem e&longs;&longs;et, qui inerrantium &longs;iellarum; &longs;ecundus verò &longs;ecunduns <lb/>id, quod per medium Zodiacum; tertius tandem, &longs;ecundum quem qui in latitudine <lb/>Zodiaci obliquatur. </s> <s id="id.002533">in maiori autem latitudine obliquari eum &longs;ecundum quem Lu­ <lb/>na, quàm eum &longs;ecundum quem Sol &longs;ertur)<emph.end type="italics"/> Eudoxi tempore nondum &longs;atis ex­ <lb/>culta fuerat A&longs;tronomia, vt propterea minimè mirandum &longs;it, eum hoc lo­ <lb/>co imperfecta admodum circa c&ecedil;le&longs;tia tradere. </s> <s id="id.002534">omittit enim in Sole orbem <lb/>motum augis conficientem; necnon duos eccentricos, qui &longs;olis anomaliam, <lb/><expan abbr="atq;">atque</expan> eccentricitatis variationem excu&longs;ant. </s> <s id="id.002535">attribuit præterea Soli motum <lb/>quendam in latitudinem, quod fal&longs;um e&longs;t omninò, cum Sol perpetuò directè <lb/>&longs;ub eclyptica incedat. </s> <s id="id.002536">In Luna pariter plures nece&longs;&longs;arios illi orbes ad motus <lb/>ip&longs;ius &longs;aluandos prætermittit. </s> <s id="id.002537">Ex &longs;ententia tamen Tychonis Brahe hos or­ <lb/>bes, ac circulos tanquam ab inuicem di&longs;tinctos abrogare debemus.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.146.jpg" pagenum="146"/><p type="main"> <s id="id.002538"><arrow.to.target n="marg226"/></s></p><p type="margin"> <s id="id.002539"><margin.target id="marg226"/>235</s></p><p type="main"> <s id="id.002540">Tex. 46. <emph type="italics"/>(Errantium verò &longs;tellarum <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> in quatuor &longs;phæris, quarura <lb/>primam quidem, & &longs;ecundam eandem illis e&longs;&longs;e: etenim, quæ fix arum eft eam illam <lb/>e&longs;&longs;e, quæomnes fert: at cam, quæ &longs;ub ip&longs;a ordinata e&longs;t, ac quæ &longs;ecuxdum Zodiacum <lb/>lationem habet, communem omnibus e&longs;&longs;e. </s> <s id="id.002541">Tertiæ verò omnium polos in eo, quod <lb/>per medium Zodiacum e&longs;t. </s> <s id="id.002542">Quartæ autem lationem &longs;ecundum eum, qui obliquatus <lb/>ad medusm eius e&longs;t; e&longs;&longs;e verò tertiæ &longs;phæræ polos aliarum quidem proprios, Veneris <lb/>autem, & Mercurij eo&longs;dem)<emph.end type="italics"/> pergit tradere theoriam reliquorum errantium <lb/><expan abbr="quinq;">quinque</expan> &longs;yderum, &longs;ecundum mentem Eudoxi, qui propriè Planetæ dicuntur: <lb/>Sol autem, & Luna hoc nomine non e&longs;t complexus, eo quod ip&longs;a mereantur <lb/>potius duo mundi luminaria appellari, quàm cum c&ecedil;teris &longs;tellis in ordinem <lb/>redigi. </s> <s id="id.002543">Reliquis igitur <expan abbr="quinq;">quinque</expan> erronibus &longs;ingulis quatuor &longs;phæris attribue­ <lb/>bat, quarum prima, & &longs;ecunda eodem modo &longs;e habebant, ac in Sole, & Lu­ <lb/>na, etenim octaua &longs;phæra, &longs;eu firmamentum, quod affixa &longs;ibi &longs;ydera differt <lb/>communicabat, &longs;ecundum ip&longs;um reliquis inferioribus &longs;phæris motum &longs;uum <lb/>peculiarem, videlicet diurnum, quo ab oriente in occidentem tota c&ecedil;li ma­ <lb/>china conuertebatur. </s> <s id="id.002544">fecundam eam facit, quæ Planetas omnes &longs;ecundum <lb/>Zodiaci longitudinem ab occidente in <expan abbr="ori&etilde;tem">orientem</expan> vehebat, quæ pariter eodem <lb/>modo &longs;e habet in &longs;ingulis. </s> <s id="id.002545">Tertiam verò eam confinxit, cuius poli e&longs;&longs;ent in <lb/>eclyptica, in quibus cita, ab eclyptica vltrò, <expan abbr="citro&qacute;">citroque</expan>; dilataretur. </s> <s id="id.002546">Quartam <lb/>demum po&longs;uit, quæ tertiam bifariam &longs;ecaret, <expan abbr="eam&qacute;">eamque</expan>; tali motu cieret, ne ab <lb/>eclyptica plus iu&longs;to ver&longs;us mundi polos exorbitaret. </s> <s id="id.002547">porrò in reliquis vo­ <lb/>luit polos tertij orbis e&longs;&longs;e peculiares, Veneri autem, & Mercurio eo&longs;dem <lb/>e&longs;&longs;e, ide&longs;t e&longs;&longs;e in eadem linea. </s> <s id="id.002548">Ex mente igitur Eudoxi cœle&longs;tes orbes in <lb/>vniuer&longs;um 27. numerantur, in Sole &longs;imul, ac Luna 6. in reliquis quinque er­ <lb/>rantibus 20. <expan abbr="atq;">atque</expan> octauæ &longs;phæræ 1. Non me later, has Eudoxi po&longs;itiones, <lb/>ob ratas po&longs;teriorum a&longs;tronomorum ob&longs;eruationes non &longs;ub&longs;i&longs;tere. </s> <s id="id.002549">at verò <lb/>hic non ip&longs;ius placita, &longs;ed præcipuè textus intelligentiam per&longs;equor.</s></p><p type="main"> <s id="id.002550"><arrow.to.target n="marg227"/></s></p><p type="margin"> <s id="id.002551"><margin.target id="marg227"/>236</s></p><p type="main"> <s id="id.002552">Tex. 47. <emph type="italics"/>(At Calippus &longs;itum quidem &longs;phærarum eundem Eudoxo ponebat, hoe <lb/>e&longs;t di&longs;tantiarum ordinem. </s> <s id="id.002553">pluralitatem autem &longs;tellæ quidem Iouis, ac Saturni ean­ <lb/>dem illi attribuebat. </s> <s id="id.002554">Solis verò, & Lunæ duas adbuc putabat &longs;phæras addendas <lb/>e&longs;&longs;e, &longs;i quis eorum, quæ &longs;en&longs;ibilitcr apparent, can&longs;as a&longs;&longs;ignare debeat. </s> <s id="id.002555">Cæteris ve­ <lb/>rò errantium vnicuique vnam. </s> <s id="id.002556">nece&longs;&longs;e verò e&longs;&longs;e, &longs;i debent omnes &longs;imul po&longs;itæ, quæ <lb/>apparent reddere, &longs;ecundam <expan abbr="vnamquamq;">vnamquamque</expan> errantium alteras &longs;phæras vna paucie­ <lb/>res e&longs;&longs;e, quæ reuoluant, & ad idem po&longs;itione &longs;emper primam eius astri &longs;phæram, <lb/>quod inferius ordinatum e&longs;t, con&longs;tituant. </s> <s id="id.002557">boc enim modo &longs;olùm contingit errantium <lb/>lationem omnia facere. </s> <s id="id.002558">Cùmigitur, in quibus ip&longs;a quidem feruntur &longs;phæris, hæ <lb/>quidem octo, bæverò <expan abbr="vigintiquinq;">vigintiquinque</expan> &longs;int. </s> <s id="id.002559">horum &longs;ane non oportet illas &longs;olas reuo­ <lb/>lai, in quiòus fertur, quod infimè ordinatum e&longs;t. </s> <s id="id.002560">quæ quidem duarum &longs;phærarum <lb/>primas reuoluant, &longs;ex erunt. </s> <s id="id.002561">quæ verò pe&longs;teriorum quatuor, &longs;exdecim. </s> <s id="id.002562">cunctarum <lb/>verò numerus, tùm earum quæ ferunt, tùm quæ reuoluunt eas, quinquaginta quin­ <lb/>que. </s> <s id="id.002563">quòd &longs;i Lunæ, & Soli, non addat aliquis quos diximus motus, omnes &longs;phæræ <lb/>erunt &longs;eptem, & quadraginta. </s> <s id="id.002564">pluralitas <expan abbr="itaq;">itaque</expan> &longs;phærarum tanta &longs;it)<emph.end type="italics"/> textum hunc <lb/>per paraphra&longs;im &longs;ic explico; Calippus igitur eundem quidem ordinem, at­ <lb/>que di&longs;tantiam &longs;phærarum cum Eudoxo ponebat: <expan abbr="eandem&qacute;">eandemque</expan>; pluralitatem <lb/>orbium mouentium Saturnum, ac Jouem; quatuor <expan abbr="nimirũ">nimirum</expan> <expan abbr="vnicuiq;">vnicuique</expan> eorum. <lb/></s> <s id="id.002565">&longs;ed putabat &longs;oli duas addendas, ac Lunæ &longs;imiliter, &longs;i quis eorum <expan abbr="appar&etilde;tias">apparentias</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.147.jpg" pagenum="147"/>&longs;aluare vellet. </s> <s id="id.002566">cæteris verò errantium, Marti, Veneri, & Mercurio <expan abbr="vnicuiq;">vnicuique</expan> <lb/>vnam. </s> <s id="id.002567">nece&longs;&longs;e præterea exi&longs;timabat e&longs;&longs;e, vt prædictæ omnes &longs;phæræ &longs;imul <lb/>apparentias omnes excu&longs;arent, addendas e&longs;&longs;e alias &longs;ingulis planetis toti­ <lb/>dem &longs;phæras vna minus, quas Reuoluentes appellabat; ita vt qui quatuor <lb/>Mouentes &longs;phæras habuif&longs;et, tribus præterea reuoluentibus opus haberet: <lb/>quæ &longs;phæræ reuoluentes id præ&longs;tabant, vt qua&longs;i priores Mouentes ita in of­ <lb/>ficio continerent, vt priori po&longs;itioni a&longs;trum, quod interiori orbi affigebur <lb/>&longs;uo tempore re&longs;tituerent, vt Alexander exponit. </s> <s id="id.002568">hoc enim &longs;olummodo po&longs;­ <lb/>&longs;ibile putabat omnes errantium lationes nos imitari po&longs;&longs;e. </s> <s id="id.002569">Cum igitur mo­ <lb/>uentes &longs;phæræ illæ quidem Saturni, ac Iouis &longs;int octo; reliquorum verò vi­ <lb/>gintiquinque, nam reliqui Planetæ <expan abbr="quinq;">quinque</expan> &longs;inguli &longs;phæras <expan abbr="quinq;">quinque</expan> mouentes <lb/>habent, quæ omnes &longs;imul numerum <expan abbr="vigintiquinq;">vigintiquinque</expan> explent: quarum omnium <lb/>&longs;olæ inferiores, quibus a&longs;trum affixum volebat, non indigebant reuoluente, <lb/>&longs;equitur duorum &longs;uperiorum Saturni, & Iouis, quorum octo erant mouen­ <lb/>tes, &longs;ex debere e&longs;&longs;e reuoluentes. </s> <s id="id.002570">Inferiorum verò quatuor planetarum re­ <lb/>uoluentes erunt &longs;exdecim: &longs;ed hoc loco Ari&longs;t. memoria fallit, deberet enim <lb/>dicere, reliquorum <expan abbr="quinq;">quinque</expan> planetarum reuoluentes erunt vigintì, &longs;unt enim <lb/>planetæ &longs;eptem, quorum Saturno, ac Ioui &longs;upremis &longs;ex reuoluentes attri­ <lb/>buit habita ratione &longs;phæratum mouentium; reliquis igitur <expan abbr="quinq;">quinque</expan> planetis <lb/>habita ratione &longs;uorum orbium mouentium, 25. cum &longs;inguli habeant <expan abbr="quinq;">quinque</expan> <lb/>mouentes, habebunt ex prælcripto Calippi &longs;inguli 4. reuoluentes; ac pro­ <lb/>inde 20. in vniuer&longs;um erunt reuoluentes. </s> <s id="id.002571">Omnium igitur &longs;phærarum tam <lb/>mouentium, quàm reuoluentium &longs;ummam ait, &longs;ed perperam, e&longs;&longs;e quinqua­ <lb/>gintaquinque; cum enim mouentes Saturni, & Iouis &longs;int 8. reliquorum au­ <lb/>tem 25. reuoluentes verò Saturni, & Iouis &longs;int 6. reliquorum autem, vt ip­ <lb/>&longs;e memoria fal&longs;us ponit, &longs;exdecim, conflant quidem &longs;ummam prædictam, <lb/>&longs;ed illi in memoria reuocandus e&longs;t, planeta ille, quem oblitus e&longs;t, cuius &longs;unt <lb/>quatuor reuoluentes, qui prioribus additi &longs;phærarum errantium numerum <lb/>quinquaginta nouem con&longs;tituent: quibus etiam addenda e&longs;t octaua &longs;phæra, <lb/>&longs;eu firmamentum, quod inerrantium &longs;edes e&longs;t, non enim &longs;olum errantium, <lb/>&longs;ed omnium cœle&longs;tium orbium numerum inue&longs;tigare volebat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; e&longs;&longs;ent <lb/>omnes &longs;ecundum Calippum &longs;ph&ecedil;ræ &longs;exaginta. </s> <s id="id.002572">Quod &longs;i Lunæ, & Soli non ad­ <lb/>dantur &longs;ingulis duo mouentes, vt facit Calippus, <expan abbr="neq;">neque</expan> con&longs;equenter quatuor <lb/>illis debiti reuoluentes non erunt omnes, 55. verùm, detractis octo prædi­ <lb/>ctis, erunt tantum 47. &longs;eu vt melius loquatur non erunt in vniuer&longs;um, 60. &longs;ed <lb/>52. tantum. </s> <s id="id.002573">Hactenus de numero cœlorum.</s></p></chap><chap><p type="head"> <s id="id.002574"><emph type="italics"/>Ex Decimotertio Metaphy&longs;icæ.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s id="id.002575"><arrow.to.target n="marg228"/></s></p><p type="margin"> <s id="id.002576"><margin.target id="marg228"/>237</s></p><p type="main"> <s id="id.002577">Svmma 1. cap. 3. <emph type="italics"/>(Qui dicunt Mathematicas &longs;cientias nihil de bono, vel <lb/>pulchro dicere, fal&longs;um dicunt. </s> <s id="id.002578">dicunt. </s> <s id="id.002579">n. </s> <s id="id.002580">& maximè <expan abbr="o&longs;t&etilde;dunt">o&longs;tendunt</expan>. </s> <s id="id.002581">nam & &longs;i non <lb/>nominant, quia tamen opera, & rationes ostendunt, non ne dicunt de eis? <lb/></s> <s id="id.002582">pulchra <expan abbr="namq;">namque</expan> maximè &longs;pecies &longs;unt, ordo, commen&longs;uratio, & definuŭ, quæ <lb/>maximè à Mathematicis &longs;cientijs o&longs;tenduntur, &c.)<emph.end type="italics"/> placuit hæc in Mathemati­ <lb/>carum commendationem, ac defen&longs;ionem apponere, cum non de&longs;int hac <lb/>no&longs;tra tempe&longs;tate ageometrcti complures, qui cas libenter &longs;ugillare &longs;olent.</s></p></chap><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.148.jpg" pagenum="148"/><chap><p type="head"> <s id="id.002583"><emph type="italics"/>IN MECHANICAS QVÆSTIONES.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002584">Qvidquid Mathematicum in his quæ&longs;tionibus occurret, illud, vt <lb/>plurimum per paraphra&longs;im exponemus, ita tamen, vt tex. <!-- REMOVE S-->Ari&longs;t. <lb/>& figuræ textui re&longs;pondentes per eam, quantum fieri poterit re­ <lb/>&longs;tituantur, & &longs;i quæ &longs;e offerent difficilia, pro viribus &longs;oluantur. <lb/></s> <s id="id.002585">E&longs;t autem nonnullis in locis textus tam græcus, quàm latinus adeò corrup­ <lb/>tus, ac deprauatus, vt nullo modo emendari queat.</s></p><p type="head"> <s id="id.002586"><emph type="italics"/>Caput Primum.<emph.end type="italics"/></s></p><p type="head"> <s id="id.002587">Quæ &longs;it artis Mechanicæ facultas.</s></p><p type="main"> <s id="id.002588"><arrow.to.target n="marg229"/></s></p><p type="margin"> <s id="id.002589"><margin.target id="marg229"/>238</s></p><p type="main"> <s id="id.002590">Eorum, quæ miraculo &longs;unt, alia quidem natura contingunt, <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; ea, <lb/>quorum ignorantur cau&longs;æ: alia verò &longs;unt, quæ præter naturam per <lb/>artificium aliquod ad hominum vtilitatem perficiuntur, in multis <lb/><expan abbr="namq;">namque</expan> natura ei, quod nobis v&longs;ui e&longs;&longs;e pote&longs;t, contrarium facit, quod <lb/>inde oritur, quia natura eundem &longs;emper, ac &longs;implicem &longs;eruat modum: quod <lb/>autem nobis vtile e&longs;t, plurimas &longs;ubit varietates. </s> <s id="id.002591">quando igitur quippiam <lb/>præter naturam facere opportuerit, illud, quod faciendum e&longs;t, difficultate <lb/>&longs;ua nos remoratur, <expan abbr="arte&qacute;">arteque</expan>; propterea indigemus. </s> <s id="id.002592">quamobrem eam artis <lb/>partem, quæ huiu&longs;modi &longs;uccurrit difficultatibus, Mechanicam appellamus. <lb/></s> <s id="id.002593">Cæterùm optimè Antiphon Poeta in hunc modum cecinit;</s></p><p type="head"> <s id="id.002594"><emph type="italics"/>Arte &longs;uperamus ea, in quibus à natura vincimur.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002595">Quemadmodum accidit, cum minora &longs;uperant maiora, & quæcunque exi­ <lb/>guam vim habentia, magna tamen mouent pondera, & omnia ferè illa, quæ <lb/>&longs;ub ea cadunt problemata, quæ mechanica nuncupari <expan abbr="&longs;ol&etilde;t">&longs;olent</expan>. </s> <s id="id.002596">&longs;unt autem hæc <lb/><expan abbr="neq;">neque</expan> naturalibus omninò quæ&longs;tionibus eadem, <expan abbr="neq;">neque</expan> &longs;eiugata valde: verùm <lb/>mathematicarum contemplationum, <expan abbr="naturaliumq;">naturaliumque</expan> communia. </s> <s id="id.002597">Po&longs;tea in <lb/>græcis codicibus hæc &longs;equuntur (<foreign lang="greek">to\ men ga\r w_c di\a twn maqhmatixw_n dh/log: to\ <lb/>de pevi\o\, di\a tw_n fuszxw_n</foreign>) ide&longs;t, &longs;i quidem quomodo &longs;int, &longs;eu qua ratione <lb/>exi&longs;tant, manife&longs;tum e&longs;t per Mathematica: illud verò circa quod ver&longs;antur, <lb/>hoc e&longs;t obiectum, de quo pertractant Mechanicæ quæ&longs;tiones per &longs;cientias <lb/>phy&longs;icas habetur, ide&longs;t res naturalis e&longs;t; e&longs;t enim pondus, & vis, aut poten­ <lb/>tia pondus ip&longs;um mouens, quatenus quanta &longs;unt; &longs;iue dixeris e&longs;t quantitas <lb/>ponderum, <expan abbr="atq;">atque</expan> potentiarum. </s> <s id="id.002598">Mathematicæ enim mediæ, de quorum nu­ <lb/>mero e&longs;t facultas Mechanica, con&longs;iderant quantitatem rei alicuius <lb/>determinatæ, &longs;ic A&longs;tronomia circa cœle&longs;tium corporum, <expan abbr="mo-tuum&qacute;">mo­ <lb/>tuumque</expan>; quantitates, Per&longs;pectiua circa linearum vi&longs;ua­ <lb/>lium; Mu&longs;ica circa &longs;onorum quantitates ver­ <lb/>&longs;antur. </s> <s id="id.002599">quæ placuit annotare, vt &longs;cien­ <lb/>tiæ huius naturam per&longs;pectam <lb/>haberemus.</s></p><pb xlink:href="http://archimedes.fas.harvard.edu/images/009-01-pageimg/009.01.149.jpg" pagenum="149"/><p type="head"> <s id="id.002600">De dignitatibus, <expan abbr="admirandis&qacute;">admirandisque</expan>; Circuli proprietatibus.</s></p><p type="head"> <s id="id.002601"><emph type="italics"/>Cap. Secundum.<emph.end type="italics"/></s></p><p type="main"> <s id="id.002602"><arrow.to.target n="marg230"/></s></p><p type="margin"> <s id="id.002603"><margin.target id="marg230"/>239</s></p><p type="main"> <s id="id.002604">Cvm vellet Ari&longs;t. mirabilium effectuum, quos in Mechanicis admi­ <lb/>ramur, cau&longs;am referre in circulum: meritò ante omnia de admi­ <lb/>randa ip&longs;ius circuli natura di&longs;&longs;erit, quo minus mirum deinde vi­ <lb/>deatur prædictas mirabiles-operationes exip&longs;o procedere. </s> <s id="id.002605">quan­ <lb/>doquidem exadmiranda cau&longs;a admirabiles effectus prodire debeant. </s> <s id="id.002606">qua­ <lb/>lia &longs;unt ea, quæ circa yectem, cum magna <expan abbr="omniũ">omnium</expan> admiratione contingunt. <lb/></s> <s id="id.002607">videmus enim exiguam pror&longs;us vim ingens pondus, quod <expan abbr="ab&longs;q;">ab&longs;que</expan> vecte mini­ <lb/>mè mouere po&longs;&longs;et, addito etiam ip&longs;ius vectis pondere, facilè <expan abbr="quocunq;">quocunque</expan> vo­ <lb/>luerit propellere. </s> <s id="id.002608">quod quidem auditu ab&longs;urdum foret, ni&longs;i vi&longs;u con&longs;taret. <lb/></s> <s id="id.002609">omnium autem huiu&longs;modi cau&longs;æ principium circulus obtinet: & hoc qui­ <lb/>dem meritò, ex admirabili enim, quippiam mirandum accidere rationi <lb/>omninò con&longs;entaneum eft.</s></p><p type="main"> <s id="id.002610">Primò igitur maxim&e