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Colored diff for /texts/archimedes/xml/Attic/bianc_locam_01_la_1615.xml between version 1.72 and 1.73

version 1.72, 2002/09/09 22:05:53

version 1.73, 2002/09/13 18:22:04

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<s>BONONIÆ M. D C. X V.</s>

Line 44

<s>Superiorum permi&longs;&longs;u.</s>

<s>Sumptibus Hieronymi Tamburini.</s><pb pagenum="3"/><figure/>

<s>Sumptibus Hieronymi Tamburini.</s><pb/><pb pagenum="3"/>

<s>ILLVSTRISSIMO <lb/>AC NOBILISSIMO</s>

<s>PETROFRANCISCO <lb/>MALASPINAE</s>

<s>ÆDIFICIORVM MARCHIONI.</s><figure/>

<s>ÆDIFICIORVM MARCHIONI.</s>

<s><emph type="italics"/>En tandem Illustriß. Marchio opus no<lb/>strum de Locis Mathematicis apud Ari<lb/>stotelem, vnà cum Tractatione de natura <lb/>&longs;cientiarum Mathematicarum, necnon <lb/>Clarorum Mathematicor&ubreve; Chronologia; <lb/>quod tibi Mec&oelig;nati meo munificenti&longs;simo iure meritò <lb/>dicare, ac &longs;ub clarißimi tui nominis patrocinio in lucem <lb/>dare con&longs;titui. </s>

Line 84

<s><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/><figure id="fig1"/></s><pb pagenum="5"/><figure/>

<s><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/></s><pb pagenum="5"/>

<s>Liber de &longs;e ip&longs;o.</s>

<s><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"/></s><figure/>

<s><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"/></s>

<s>Ego Iordanus Ca&longs;&longs;ini Præpo&longs;itus Prouincialis Prouinciæ Venetæ Societatis <lb/>Ie&longs;u, ex auctoritate Adm. Reuer. P. nc&longs;tro Præpo&longs;iti Generalis P. Claudij <lb/>Aquæuiuæ, facultatem concedo, vt hoc opus P. 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>

Line 246

<s>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>Pr&ecedil;cipua qu&ecedil;dam, aut noua, autre&longs;taurata, <lb/>quæ obiter pertractantur.<lb/><arrow.to.target n="table1"/></s><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"/>S<gap/>ytala 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><figure/><pb pagenum="12"/>

<s><emph type="italics"/>PRIMVS INDEX LOCORVM ARIST.<emph.end type="italics"/></s>

Line 1442

<s>7. <emph type="italics"/>Clarorum Mathematicorum Chronicon.<emph.end type="italics"/></s>

<s><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s><figure/><pb pagenum="22"/>

<s><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s><pb pagenum="22"/>

<s><emph type="italics"/>ALTER INDEX<emph.end type="italics"/></s>

Line 1880

<s><emph type="bold"/>Finis Secundi Indicis.<emph.end type="bold"/></s>

<s>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><figure/><pb pagenum="25"/>

<s><emph type="italics"/>TERTIVS INDEX ALPHABETICVS,<emph.end type="italics"/></s>

<s><emph type="italics"/>cuius numeri re&longs;pondent numeris marginalibus Operis.<emph.end type="italics"/><lb/><arrow.to.target n="table2"/></s><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 C&oelig;lorum.<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"/>C&oelig;lorum 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 C&oelig;lo 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 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 denari&ubreve; 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 C&oelig;lorum.<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="quotc&ubreve;q;">quotc&ubreve;que</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"/>Fu<gap/>ium l<gap/>ctorum 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 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, ei&queacute; 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"/>Li<gap/>ea, 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"/>Mater<gap/>a intelligibilis. fusè verò. explicatur in tract. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mathema<gap/>icæ 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"/>Mathematic<gap/>s 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 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"/>Monochordiux<gap/>.<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 c&oelig;lestium 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 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 dignita<gap/>em, 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 bominibus. probl.<emph.end type="italics"/> 5.</cell><cell/></row><row><cell><emph type="italics"/>Cur in Sole euane&longs;<gap/>at. 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, eas&queacute; ceteris &longs;cientijs præponeb&abreve;t.<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"/>Qv adra<gap/>ura 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<gap/>antiquæ &longs;<gap/>uris 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 C&oelig;li 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 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"/>Tunic<gap/> oculi. 408. in tractatu de Pupi<gap/>la.<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>

<s>Finis Tertij Indicis.</s><figure/><pb pagenum="31"/>

<s>Finis Tertij Indicis.</s><pb pagenum="31"/>

<s>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><pb pagenum="33"/><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 c&oelig;le&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><figure/>

<s>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><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 c&oelig;le&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>

<s>MATHEMATICA</s>

Line 1898

<s>PRÆDICAMENTORVM</s>

<s>Per ordinem declarata.</s><figure/>

<s>Per ordinem declarata.</s>

Line 1926

<s>& 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 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>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><figure/>

<s>Sit, v.g. 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>

Line 1984

<s>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="fig2"/><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>

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 place="text" xlink:href="figures-la/009.01.035.1.tif"/><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>& 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.</s>

Line 2078

<s>&longs;ic; commen&longs;. </s>

<s>magnitudines dicuntur, quas <lb/><figure id="fig3"/><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. 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. 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="fig4"/><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. g. &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>magnitudines dicuntur, quas <lb/><figure place="text" xlink:href="figures-la/009.01.037.1.tif"/><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. 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. 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 place="text" xlink:href="figures-la/009.01.037.2.tif"/><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. g. &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>

Line 2100

<s>inter lineas incommen&longs;. </s>

<s>&longs;unt diameter, & latus eiu&longs;<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti <lb/><figure id="fig5"/><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. g. <lb/>latus B C, præcisè omnino metiatur. </s>

<s>&longs;unt diameter, & latus eiu&longs;<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti <lb/><figure place="text" xlink:href="figures-la/009.01.037.3.tif"/><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. g. <lb/>latus B C, præcisè omnino metiatur. </s>

Line 2148

<s>e&longs;t au<lb/>tem figura in omnibus textibus deprauata, quam &longs;ic puto <expan abbr="rè&longs;t&imacr;tuendam">rè&longs;tintuendam</expan> e&longs;&longs;e <lb/>ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. </s>

<s>&longs;it I&longs;o&longs;ce<lb/><figure id="fig6"/><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>&longs;it I&longs;o&longs;ce<lb/><figure place="text" xlink:href="figures-la/009.01.038.1.tif"/><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>facto centro <lb/>in A, de&longs;cribatur circulus A B C, tran&longs;iens per puncta <lb/>C B, iam &longs;ic. </s>

Line 2208

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="ill&utilde;">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="fig7"/><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s>

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="ill&utilde;">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure place="text" xlink:href="figures-la/009.01.039.1.tif"/><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s>

<s>&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 pagenum="40"/><expan abbr="linear&utilde;">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. </s>

Line 2216

<s>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="n&utilde;quam">nunquam</expan> tamen licet dicere angulum A C B, <lb/>vel C B A. </s>

<s>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/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu<lb/><figure id="fig8"/><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>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/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu<lb/><figure place="text" xlink:href="figures-la/009.01.040.1.tif"/><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>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. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure id="fig9"/><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="fig10"/><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. &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-qu&utilde;">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>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. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure place="text" xlink:href="figures-la/009.01.040.2.tif"/><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 place="text" xlink:href="figures-la/009.01.040.3.tif"/><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. &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-qu&utilde;">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>

Line 2290

<s>vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s>

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"/><lb/>probat Euclides in 28. primi Elem.

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure place="text" xlink:href="figures-la/009.01.042.1.tif"/><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="nimir&utilde;">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>

Line 2344

<s><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>

<s>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="fig12"/><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>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 place="text" xlink:href="figures-la/009.01.043.1.tif"/><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>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>

Line 2426

<s>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. 11. primi Phy&longs;ic. 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="diametr&utilde;">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 pagenum="45"/><figure id="fig13"/><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>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. 11. primi Phy&longs;ic. 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="diametr&utilde;">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 pagenum="45"/><figure place="text" xlink:href="figures-la/009.01.045.1.tif"/><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>

Line 2434

<s><expan abbr="hucu&longs;q;">hucu&longs;que</expan> be<lb/>nè procedit Hippocrates. </s>

<s>&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="fig14"/><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>&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 place="text" xlink:href="figures-la/009.01.045.2.tif"/><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>& <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="diametror&utilde;">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>

Line 2486

<s>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="fig15"/><lb/>micirculo. </s>

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 place="text" xlink:href="figures-la/009.01.046.1.tif"/><lb/>micirculo. </s>

<s>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>

Line 2560

<s>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>

<s>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="fig16"/><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>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 place="text" xlink:href="figures-la/009.01.048.1.tif"/><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>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>

Line 2568

<s><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>

<s>Per altera parte longius, intelligit numerum, qui producitur à duobus <lb/><figure id="fig17"/><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>Per altera parte longius, intelligit numerum, qui producitur à duobus <lb/><figure place="text" xlink:href="figures-la/009.01.048.2.tif"/><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>atque hæc <lb/>&longs;unt, quæ ex Mathematicis petenda erant, ad huius <lb/>loci intelligentiam.</s>

Line 2602

<s>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="fig18"/><lb/>cen&longs;ui. </s>

de&longs;umitur, quam propterea primo loco exponendam <lb/><figure place="text" xlink:href="figures-la/009.01.049.1.tif"/><lb/>cen&longs;ui. </s>

<s>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>Verum, <lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tant&utilde;">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="fig19"/><lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan<lb/>tia. </s>

<s>Verum, <lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tant&utilde;">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 place="text" xlink:href="figures-la/009.01.049.2.tif"/><lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan<lb/>tia. </s>

<s>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 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>

Line 2650

<s>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 pagenum="51"/>dum quid &longs;it alterna proportio. </s>

<s>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="fig20"/><lb/>& con&longs;equentis ad con&longs;equentem. </s>

<s>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 place="text" xlink:href="figures-la/009.01.051.1.tif"/><lb/>& con&longs;equentis ad con&longs;equentem. </s>

<s>Explico, exponantur qua<lb/>tuor quantitates proportionales, v.g. 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>

Line 2722

<s>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>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="fig21"/><lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></s>

<s>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 place="text" xlink:href="figures-la/009.01.052.1.tif"/><lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></s>

<s>4. qui e&longs;t numerus quadratus huius figuræ, <figure id="fig22"/>, deinde <lb/>tertius binarius ducitur in prædictum quadratum 4. & pro<lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua<lb/>zerna<gap/>ij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe<lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb 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="fig23"/><lb/>qui e&longs;t quadratus. </s>

<s>4. qui e&longs;t numerus quadratus huius figuræ, <figure place="text" xlink:href="figures-la/009.01.052.2.tif"/>, deinde <lb/>tertius binarius ducitur in prædictum quadratum 4. & pro<lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua<lb/>zerna<gap/>ij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe<lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb 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 place="text" xlink:href="figures-la/009.01.053.1.tif"/><lb/>qui e&longs;t quadratus. </s>

<s>quo deinde ducto in tertium ter<lb/>narium, producitur 27. qui e&longs;t cubus, & refert &longs;igu<lb/>ram cubicam hanc. </s>

Line 2740

<s>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>v.g. 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="fig24"/><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>

Line 2748

<s>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>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 pagenum="54"/><figure id="fig25"/><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-n&utilde;">&longs;o<lb/>num</expan> chordæ dimidiæ A C, ha<lb/>bebit eandem rationem, <expan abbr="nimir&utilde;">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s>

<s>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 pagenum="54"/><figure place="text" xlink:href="figures-la/009.01.054.1.tif"/><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-n&utilde;">&longs;o<lb/>num</expan> chordæ dimidiæ A C, ha<lb/>bebit eandem rationem, <expan abbr="nimir&utilde;">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s>

<s>&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>

Line 2766

<s>Bry&longs;o itaque, <lb/>vt tradit Alexander, in hunc modum conabatur quadrare <expan abbr="circul&utilde;">circulum</expan>. </s>

<s>&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="fig26"/><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="quadrat&utilde;">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>&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 place="text" xlink:href="figures-la/009.01.054.2.tif"/><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="quadrat&utilde;">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>Iam &longs;ic o&longs;tendebat i&longs;tud <lb/>medium quadratum e&longs;&longs;e æquale circu<lb/>lo propo&longs;ito. </s>

Line 2784

<s>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="fig27"/><lb/>mus. </s>

mentem probè penetrare poteri<lb/><figure place="text" xlink:href="figures-la/009.01.055.1.tif"/><lb/>mus. </s>

<s>&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>

Line 2880

<s>Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli. </s>

<s>Quoniam enim in 13. <lb/>Aequep. 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="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure id="fig28"/><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;ol&utilde;">&longs;olum</expan> e&longs;t in vtra<lb/>que, quod erat demon&longs;trandum. </s>

<s>Quoniam enim in 13. <lb/>Aequep. 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="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure place="text" xlink:href="figures-la/009.01.057.1.tif"/><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;ol&utilde;">&longs;olum</expan> e&longs;t in vtra<lb/>que, quod erat demon&longs;trandum. </s>

Line 3114

<s>&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><figure/>

Line 3196

<s>Et paulo po&longs;t <emph type="italics"/>(Qutmadmod&ubreve; &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>

Line 3274

<s>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 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><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="fig29"/><lb/>cipiamus, accipe exemplum. </s>

<s>Sint duæ chordæ <lb/>A, & B, æqualis cra&longs;&longs;itici, & æquè ten&longs;æ. </s>

Line 3350

<s>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>per angulum in &longs;emicirculo intelligas eum, qui <lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum <lb/><figure id="fig30"/><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>per angulum in &longs;emicirculo intelligas eum, qui <lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum <lb/><figure place="text" xlink:href="figures-la/009.01.065.1.tif"/><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>quod Euclides eodem pror&longs;us medio, quod <lb/>Ari&longs;t.

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<s>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>vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb pagenum="67"/><figure id="fig31"/><lb/>circa angulos æquales, v. g. 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>

<s>vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb pagenum="67"/><figure place="text" xlink:href="figures-la/009.01.067.1.tif"/><lb/>circa angulos æquales, v. g. 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>

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exemplo mathematico <lb/>o&longs;tendere, difficile e&longs;&longs;e di&longs;putare, aut <gap/> <expan abbr="rgum&etilde;tari">rgumentari</expan>, ni&longs;i prius rectè a&longs;&longs;ignetur <pb pagenum="69"/>definitio illius rei, de qua di&longs;&longs;eritur. </s>

<s>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="fig32"/><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. </s>

<s>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 place="text" xlink:href="figures-la/009.01.069.1.tif"/><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. </s>

<s>Inquit ergo Ari&longs;t. </s>

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<s>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.

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<s>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>

<s><emph type="italics"/>Log<gap/>corum locorum finis.<emph.end type="italics"/></s><figure/><pb pagenum="72"/>

<s><emph type="italics"/>Log<gap/>corum locorum finis.<emph.end type="italics"/></s><pb pagenum="72"/>

<s>EX PRIMO LIBRO</s>

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1. quam nece&longs;&longs;e e&longs;t, con&longs;ulas. </s>

<s>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="fig33"/><lb/>A C, &longs;it perpendiculare <expan abbr="c&utilde;">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="c&utilde;">cum</expan> latus A C, <lb/>&longs;it perpendiculare (quod Ari&longs;t.

<s>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 place="text" xlink:href="figures-la/009.01.073.1.tif"/><lb/>A C, &longs;it perpendiculare <expan abbr="c&utilde;">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="c&utilde;">cum</expan> latus A C, <lb/>&longs;it perpendiculare (quod Ari&longs;t.

dicit, cum <expan abbr="re-ct&utilde;">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>

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<s>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="fig34"/><lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de <lb/>reliquis imparibus. </s>

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 place="text" xlink:href="figures-la/009.01.074.1.tif"/><lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de <lb/>reliquis imparibus. </s>

<s>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="fig35"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="fig36"/> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s>

<s>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 place="text" xlink:href="figures-la/009.01.074.2.tif"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure place="text" xlink:href="figures-la/009.01.074.3.tif"/> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s>

<s>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.

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<s>7. Arith<lb/>metices lordani, vbi i&longs;tud idem demon&longs;trat, quæ e&longs;t hæc. </s>

<s>&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="fig37"/><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. 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;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

<s>&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 place="text" xlink:href="figures-la/009.01.074.4.tif"/><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. 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;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

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<s>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>&longs;i igitur vnitati binarius coaceruetur, fit numerus <lb/><figure id="fig38"/><lb/>triangularis, vt in prima figura. </s>

<s>&longs;i igitur vnitati binarius coaceruetur, fit numerus <lb/><figure place="text" xlink:href="figures-la/009.01.075.1.tif"/><lb/>triangularis, vt in prima figura. </s>

<s>&longs;i huic ternario <lb/>coaceruetur &longs;equens par, fiet altera &longs;pecies, ni<lb/>mirum <gap/>exagonus numerus, vt in &longs;ecunda figu<lb/>ra. </s>

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<s>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><figure/><pb pagenum="76"/>

23.</s><pb pagenum="76"/>

<s><emph type="italics"/>Ex Quinto Phy&longs;icorum.<emph.end type="italics"/></s>

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<s>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="fig39"/><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 C&oelig;lum 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>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 place="text" xlink:href="figures-la/009.01.076.1.tif"/><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 C&oelig;lum 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>&longs;ed impo&longs;&longs;ibile est; non <lb/>est igitur circulariter verti infinit&<gap/>, 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>

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<s>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 con<gap/>luere aqua ad magis concauum: ma<lb/>gis autem concauum e&longs;t, quod centro propinquius est. </s>

<s>ducantur ergo ex centro A,<emph.end type="italics"/><lb/><figure id="fig40"/><lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C, <lb/>duct<gap/> igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis, <lb/>quæ ex centro. </s>

<s>ducantur ergo ex centro A,<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.078.1.tif"/><lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C, <lb/>duct<gap/> igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis, <lb/>quæ ex centro. </s>

<s>magis igitur concauus locus e&longs;t, quare <lb/>influet aqua, donec <expan abbr="vtiq;">vtique</expan> æquetur. </s>

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<s>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>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 <gap/>b 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="fig41"/><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;egment&utilde;">&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>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 <gap/>b 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 place="text" xlink:href="figures-la/009.01.079.1.tif"/><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;egment&utilde;">&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>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 c&oelig;le&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 c&oelig;lum di&longs;&longs;ecare, quod accideret, &longs;i pro<lb/>prio motu veluti pri&longs;ces per aquam progrederentur.</s>

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<s>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>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="fig42"/><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>

<s>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 place="text" xlink:href="figures-la/009.01.080.1.tif"/><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>

Line 4104

<s>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>Terram e&longs;&longs;e rotundam, <expan abbr="atq;">atque</expan> <lb/>re&longs;pectu c&oelig;le&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="fig43"/><lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s>

<s>Terram e&longs;&longs;e rotundam, <expan abbr="atq;">atque</expan> <lb/>re&longs;pectu c&oelig;le&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 place="text" xlink:href="figures-la/009.01.082.1.tif"/><lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s>

<s>&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;tror&utilde;">a&longs;trorum</expan> ver&longs;icalium B, in C, quæ mul<lb/>tum ab inuicem di&longs;tant. </s>

Line 4184

<s>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>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="fig44"/><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>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 place="text" xlink:href="figures-la/009.01.084.1.tif"/><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>g. A, in hac <lb/>figura) in plano, quotquot fuerint con&longs;tituti, &longs;unt æqua<lb/>les quatuor rectis, ex coroll. </s>

Line 4192

<s>&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>Quadratum etiam replere lo<lb/><figure id="fig45"/><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>

<s>Quadratum etiam replere lo<lb/><figure place="text" xlink:href="figures-la/009.01.084.2.tif"/><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>

<s>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>præter has tres <lb/><figure id="fig46"/><lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol<lb/>&longs;it. </s>

<s>præter has tres <lb/><figure place="text" xlink:href="figures-la/009.01.084.3.tif"/><lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol<lb/>&longs;it. </s>

<s>cuius demon&longs;trationem perfectam videre pote<lb/>ris in fine commentarij P. Clauij &longs;uper 4. Elem.

Line 4208

<s>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>quare quartum <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> figura relinquetur; quod e&longs;t ab&longs;urdum.</s><figure/><pb pagenum="85"/>

<s>quare quartum <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> figura relinquetur; quod e&longs;t ab&longs;urdum.</s><pb pagenum="85"/>

<s><emph type="italics"/>Admirabilis quædam A&pgrave;um industria.<emph.end type="italics"/></s>

Line 4348

<s>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>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="fig47"/><lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in m<gap/>dio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;c<gap/>ndat; vtitur, præter alias, etism <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;<lb/>modi. </s>

<s>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 place="text" xlink:href="figures-la/009.01.087.1.tif"/><lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in m<gap/>dio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;c<gap/>ndat; vtitur, præter alias, etism <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;<lb/>modi. </s>

<s>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. &longs;it terra in &longs;igu<lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c<gap/><pb 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>

Line 4416

<s>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>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="fig48"/><lb/>magis elongantur, tan<lb/>to maius corpus <expan abbr="illu-minãs">illu<lb/>minans</expan> intercipiunt. </s>

<s>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 place="text" xlink:href="figures-la/009.01.089.1.tif"/><lb/>magis elongantur, tan<lb/>to maius corpus <expan abbr="illu-minãs">illu<lb/>minans</expan> intercipiunt. </s>

<s>ha<lb/>ctenus de magnitudine <lb/>terræ ad Solem. </s>

Line 4552

<s>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>&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><figure/>

<s>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>

Line 4866

<s>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>de&longs;cribatur enim figura <lb/><figure id="fig49"/><pb 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>de&longs;cribatur enim figura <lb/><figure place="text" xlink:href="figures-la/009.01.100.1.tif"/><pb 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>Cauca&longs;o F V. </s>

Line 5254

<s>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>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="fig50"/><lb/>Maior circulus &longs;it c&oelig;lum, 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>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 place="text" xlink:href="figures-la/009.01.109.1.tif"/><lb/>Maior circulus &longs;it c&oelig;lum, 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>

Line 5338

figuras, imò demon&longs;tratio<lb/>nes ip&longs;as Mathematicorum, de&longs;criptiones appellare, vt &longs;æpius in Logicis <lb/>monuimus.</s>

<s><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><figure/><pb pagenum="112"/>

<s><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>

Line 5422

<s>ab co<lb/>dem enim &longs;igno ad idem &longs;ignum æquales frangentur &longs;uper circuli lineam &longs;emper. </s>

<s>&longs;it<emph.end type="italics"/><lb/><figure id="fig51"/><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>&longs;it<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.114.1.tif"/><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>& 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 tri&abreve;gulis, & 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, ncc per con&longs;<gap/><expan abbr="qu&etilde;s">quens</expan> <lb/>alibi apparere. </s>

Line 5472

<s>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>&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb pagenum="116"/><figure id="fig52"/><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>&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb pagenum="116"/><figure place="text" xlink:href="figures-la/009.01.116.1.tif"/><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>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 c&oelig;lum 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>

Line 5496

<s>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;phæ<gap/>æ, qui <lb/>maximus &longs;it in quo A, differet enim <lb/>mbil &longs;i quod<gap/><expan abbr="&ubreve;q;">&ubreve;que</expan> eorum, quæ &longs;uper <lb/>G K, &longs;ecundum triangul&ubreve; K M G, <lb/>erectum fuerit planum. </s>

Line 5514

<s>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="fig53"/><lb/>pote&longs;t) vtitur. </s>

<s>v. g. 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>

Line 5560

<s>&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/></s>

<s>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. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure id="fig54"/><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 <gap/>iametrum <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 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.</s>

<s>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. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure place="text" xlink:href="figures-la/009.01.118.1.tif"/><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 <gap/>iametrum <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 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.</s>

Line 5580

<s>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>

<s>Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta, <lb/>nam &longs;ecundum veritatem po&longs;&longs;<gap/>nt in eadem nube con&longs;titui plures anguli <lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alter&utilde;">alterum</expan>; vt in figura præ<lb/><figure id="fig55"/><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="illor&utilde;">illorum</expan> vtrun<lb/>que Sol Iridem efficere. </s>

<s>Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta, <lb/>nam &longs;ecundum veritatem po&longs;&longs;<gap/>nt in eadem nube con&longs;titui plures anguli <lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alter&utilde;">alterum</expan>; vt in figura præ<lb/><figure place="text" xlink:href="figures-la/009.01.119.1.tif"/><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="illor&utilde;">illorum</expan> vtrun<lb/>que Sol Iridem efficere. </s>

Line 5606

<s>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="fig56"/><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. </s>

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 place="text" xlink:href="figures-la/009.01.120.1.tif"/><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. </s>

<s>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. </s>

Line 5658

<s>Atque hoc accidit Sole, vel Luna in horizonte exi&longs;tentibus; quod <lb/>erat primo loco demon&longs;trandum.</s>

<s>Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum <lb/><figure id="fig57"/><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>Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum <lb/><figure place="text" xlink:href="figures-la/009.01.122.1.tif"/><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><expan abbr="ea&longs;d&etilde;">ea&longs;dem</expan> autem proprietates habcbant etiam <lb/>triangula Ari&longs;t.

Line 5676

<s>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. </s>

<s>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 pagenum="123"/><figure id="fig58"/><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>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 pagenum="123"/><figure place="text" xlink:href="figures-la/009.01.123.1.tif"/><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>&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>

Line 5726

<s>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><figure/><pb pagenum="125"/>

<s>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><pb pagenum="125"/>

<s><emph type="italics"/>Additio de Iride.<emph.end type="italics"/></s>

Line 5744

<s>non in hoc &longs;itu, & ouali figura, &longs;ed <lb/><figure id="fig59"/><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 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>non in hoc &longs;itu, & ouali figura, &longs;ed <lb/><figure place="text" xlink:href="figures-la/009.01.125.1.tif"/><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 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>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>

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<s>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>

<s>Octauo loco Problemata nonnulla re&longs;oluemus.</s><figure/><pb pagenum="128"/>

<s>Octauo loco Problemata nonnulla re&longs;oluemus.</s><pb pagenum="128"/>

<s><emph type="italics"/>Problema Primum.<emph.end type="italics"/></s>

Line 5822

<s><emph type="italics"/>Problema Secundum.<emph.end type="italics"/></s>

<s>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="fig60"/><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><pb pagenum="129"/>

<s>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 place="text" xlink:href="figures-la/009.01.128.1.tif"/><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><pb pagenum="129"/>

<s>Cur quanto Sol altior e&longs;t, tanto inferior, <expan abbr="tanto&qacute;">tantoque</expan>; &longs;emicir<lb/>culo minor appareat Iris?</s>

<s><emph type="italics"/>Problema Tertium.<emph.end type="italics"/></s>

<s>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="fig61"/><lb/>Sol E, oculus C, <expan abbr="centr&utilde;&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>

<s>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 place="text" xlink:href="figures-la/009.01.129.1.tif"/><lb/>Sol E, oculus C, <expan abbr="centr&utilde;&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>

<s>Cur Iris in&longs;equentes fugit, fugientes verò in&longs;equitur?</s>

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<s>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>

<s>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><figure/><pb pagenum="130"/>

<s>Curlunares Irides fiunt rariores?</s>

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<s>&longs;ed <lb/>&longs;icut recto in quantum rectum multa accidunt, vt tangere æ<gap/>eam &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="fig62"/><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>

&pacute;robat li<lb/><figure place="text" xlink:href="figures-la/009.01.131.1.tif"/><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>v. g. 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>

Line 5926

<s>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. </s>

<s>probat autem hac ferè ra<lb/><figure id="fig63"/><lb/>tione. </s>

<s>probat autem hac ferè ra<lb/><figure place="text" xlink:href="figures-la/009.01.132.1.tif"/><lb/>tione. </s>

<s>&longs;it &longs;phæra A B C, quæ tangat quodpiam planum <lb/>in duobus punctis A, B, &longs;i fieri pote&longs;t. </s>

Line 5964

<s>loquitur igitur hic Philo &longs;ophus non de quadratione circuli, <pb 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>per 13. 6. inue<lb/><figure id="fig64"/><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>per 13. 6. inue<lb/><figure place="text" xlink:href="figures-la/009.01.133.1.tif"/><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>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>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="fig65"/><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. </s>

<s>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 place="text" xlink:href="figures-la/009.01.133.2.tif"/><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. </s>

<s>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="fig66"/><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. </s>

<s>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 place="text" xlink:href="figures-la/009.01.133.3.tif"/><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. </s>

<s>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>

Line 6070

<s>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. 1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure id="fig67"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

<s>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. 1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure place="text" xlink:href="figures-la/009.01.135.1.tif"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

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3. propo&longs;it. </s>

<s>11. & 12. & infra <lb/>Problem. 7. &longs;ectionis 31.</s><figure/><pb pagenum="137"/>

<s>11. & 12. & infra <lb/>Problem. 7. &longs;ectionis 31.</s><pb pagenum="137"/>

<s>EX PRIMO</s>

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<s>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><figure/><pb pagenum="140"/>

23. de hac commen&longs;urabilitate, & incommen<lb/>&longs;urabilitate tractata &longs;unt.</s><pb pagenum="140"/>

<s><emph type="italics"/>Ex Quinto Metaphy&longs;icæ.<emph.end type="italics"/></s>

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<s>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><figure/>

23.</s><figure place="text" xlink:href="figures-la/009.01.141.1.tif"/>

Line 6440

<s>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="fig68"/><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. g. 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.

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 place="text" xlink:href="figures-la/009.01.142.1.tif"/><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. g. 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. g. 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>

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<s>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="fig69"/><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>tales &longs;unt quatuor anguli <lb/>ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet, <lb/><figure place="text" xlink:href="figures-la/009.01.142.2.tif"/><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>hoc <lb/>&longs;en&longs;i&longs;&longs;e Ari&longs;t.

Line 6474

<s>nunc &longs;olùm <pb pagenum="143"/><figure id="fig70"/><lb/>hæc addenda &longs;unt. </s>

<s>nunc &longs;olùm <pb pagenum="143"/><figure place="text" xlink:href="figures-la/009.01.143.1.tif"/><lb/>hæc addenda &longs;unt. </s>

<s>Re&longs;pondet Ari&longs;t.

Line 6822

<s><expan abbr="Atq;">Atque</expan> ex hac <lb/>mira circuli proprietate, <expan abbr="pleraq;">pleraque</expan> miraculorum accidunt circuli motioni<lb/>bus, vt in &longs;equentibus quæ&longs;tionibus manife&longs;tum erit.</s>

<s>Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus, <lb/>& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um <lb/><figure id="fig71"/><lb/>mouetur; alterum verò vbi B, retror<lb/>&longs;um, efficiunt nonnulli, vt ab vnica mo<lb/>tione multi contrariò &longs;imul mouean<lb/>tur denticulati circuli: vt &longs;unt ij, quos <lb/>in locis proponunt &longs;acris, quorum alij <lb/>&longs;unt ænei, alij ferrei. </s>

<s>Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus, <lb/>& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um <lb/><figure place="text" xlink:href="figures-la/009.01.150.1.tif"/><lb/>mouetur; alterum verò vbi B, retror<lb/>&longs;um, efficiunt nonnulli, vt ab vnica mo<lb/>tione multi contrariò &longs;imul mouean<lb/>tur denticulati circuli: vt &longs;unt ij, quos <lb/>in locis proponunt &longs;acris, quorum alij <lb/>&longs;unt ænei, alij ferrei. </s>

<s>&longs;i enim circulus <lb/>A B, alterum circulum C D, contige<lb/>rit, mota diametro A B, ita vt A, an<lb/>tror&longs;um eat, commouebit alteram dia<lb/>metrum C D, ita vt C, retror&longs;um, hoc e&longs;t in contrarium ip&longs;i A, veniat, in <lb/>contrarium igitur mouebitur &longs;ecundus circulus C D, ad circulum A B, & <lb/>rur&longs;us circulus E F. in contrarium ip&longs;i C D, commouebitur ab ip&longs;o C D, ob <lb/>eandem rationem. </s>

Line 6838

<s>horum autem cau<lb/>&longs;a, quoniam quæ circulum de&longs;cribit linea, duabus fertur lationibus, quæ nul<lb/>lam inuicem obtinent analogiam: quod antequam probemus, &longs;ciendum <lb/>e&longs;t, quod, quidquid duobus motibus inuicem proportionatis, mouetur, ne<lb/>ce&longs;&longs;e e&longs;t, quod motu exillis mixto progrediatur per lineam rectam, quæ dia<lb/>meter e&longs;t quadrilateri, cuius latera habeant illam proportionem, quam <lb/>duo illi motus. </s>

<s>&longs;it enim in figura proportio lateris A B, ad latus A C, quam <lb/>ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; la<lb/>tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo<lb/>tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B, <expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> ver&longs;us M C, <pb pagenum="151"/><figure id="fig72"/><lb/>ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr <lb/>ad B. his duabus lationibus A, latum. </s>

<s>&longs;it enim in figura proportio lateris A B, ad latus A C, quam <lb/>ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; la<lb/>tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo<lb/>tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B, <expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> ver&longs;us M C, <pb pagenum="151"/><figure place="text" xlink:href="figures-la/009.01.151.1.tif"/><lb/>ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr <lb/>ad B. his duabus lationibus A, latum. </s>

<s>nece&longs;&longs;a<lb/>niò motu mixto progre dietur per diametrum <lb/>A M, quod &longs;ic probari pote&longs;t; &longs;it iam A, mo<lb/>tum primo motu <expan abbr="v&longs;q;">v&longs;que</expan> ad D, linea verò ex &longs;e<lb/>cundo motu &longs;it in G F E, quo motu punctum <lb/>A, quod erat in D, <expan abbr="tract&utilde;">tractum</expan> erit in F. quod pun<lb/>ctum e&longs;t in diametro A M, quoniam enim mo<lb/>uetur duobus motibus, cum lineis A B, A C, proportionalibus, motus au<lb/>tem <expan abbr="hucu&longs;q;">hucu&longs;que</expan> &longs;unt A D, A E, quæ debent e&longs;&longs;e proportionales, cum A B, A C <lb/>compleatur rectangulum A D F E, erunt &longs;imiliter proportionalia F E, D E, <lb/>cum &longs;int æqualia duobus D A, A E, quare per 26. 6. cum quadrilaterum <lb/>paruum A D F E, &longs;it &longs;imile toti A B M C, erit A M, <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> diameter, ergò <lb/>punctum F, in quo e&longs;t A, e&longs;t in diametro A M. eodem modo, de quouis pun<lb/>cto in linea A B, ad quod A, perueniat, probabitur ab altero motu de&longs;cen<lb/>di&longs;&longs;e v&longs;que ad diametrum. </s>

Line 6850

<s>&longs;i enim dixeris illud po&longs;&longs;e de&longs;cribere rectam lineam, tunc circa rectam il<lb/>lam tanquam diametrum de&longs;cribam quadrilaterum, & po&longs;tea o&longs;tendam, vt <lb/>proximè o&longs;ten&longs;um e&longs;t, illud latum e&longs;&longs;e &longs;ecundum laterum illius proportio<lb/>nem, quare impo&longs;&longs;ibile e&longs;t id, quod mouetur duabus lationibus nullam in<lb/>uicem rationem habentibus, ferri per lineam rectam: quapropter <expan abbr="dic&etilde;dum">dicendum</expan> <lb/>e&longs;t hoc modo <expan abbr="lat&utilde;">latum</expan>, nece&longs;&longs;ariò ferri circulariter, &longs;iue per lineam circularem. <lb/></s>

<s>Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente <lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus <lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, & <lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir<lb/><figure id="fig73"/><lb/>culum, nunquam ad diametri perpendiculum <lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a <lb/>à centro perpendicularis diametro. </s>

<s>Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente <lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus <lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, & <lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir<lb/><figure place="text" xlink:href="figures-la/009.01.151.2.tif"/><lb/>culum, nunquam ad diametri perpendiculum <lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a <lb/>à centro perpendicularis diametro. </s>

<s>&longs;it circuli <lb/>figura A B C D, in qua extremum diametri <lb/>B, feratur ad alterum extremum vbi D, per <lb/>ip&longs;ius diametri B D, circumuolutionem circa <lb/>centrum F, nece&longs;&longs;e e&longs;t aliquando B, perueniat <lb/>ad C. &longs;i igitur B, feretur duabus lationibus <lb/>aliquo modo proportionatis, v. g. vt e&longs;t pro<lb/>portio lateris B E, ad E C, latus, &longs;equeretur <lb/>ex demon&longs;tratis ip&longs;um B, ferri per <expan abbr="rectã">rectam</expan> B C, <lb/>quæ diameter e&longs;&longs;et quadrilateri B E C G. &longs;ed <pb pagenum="152"/>quia in nu&longs;la proportione fertur, propterea per circularem lineam B E C, <lb/>progreditur ad C, ita vt ip&longs;a diameter B D, in po&longs;itione A C, fiat perpendi<lb/>cularis priori diametro B D. ex quibus &longs;equitur eam moueri duobus moti<lb/>bus nullam rationem habentibus; quod erat intentum.</s>

Line 6874

<s>quoniam <lb/>enim propius e&longs;t manenti eius, quæ minor e&longs;t extremum, quàm extremum <lb/>maioris, propterea plus à centro, cui propius e&longs;t, retrahitur à priori mo<lb/>tu, <expan abbr="hinc&qacute;">hincque</expan>; motus eius tardior redditur, ide&longs;t, quia centro propius e&longs;t; hinc <lb/>fit, vt extremum illud de&longs;cribat lineam circularem quidem, &longs;ed tamen <lb/>curuiorem quam de&longs;cribat extremum longioris lineæ, quæ circulum minus <lb/>curuum, &longs;eu magis ad rectam lineam accedentem delineat. </s>

<s>omni quidem <lb/>igitur lineæ circulum de&longs;cribenti i&longs;tud accidit, vt duobus feratur motioni<lb/>bus; vna quidem, quæ illi naturalis, ac &longs;ecundum circunferentiam, qua re<lb/>ctà tenderet ni&longs;i impediretur: altera verò, quæ illi innaturalis, qua in tran&longs;<lb/>uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor <lb/>autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma<lb/>ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro <lb/><figure id="fig74"/><lb/>vicinior, quod quodammodo retra<lb/>hit à motu naturali, propterea ma<lb/>gis vincitur, quàm remotior. </s>

<s>omni quidem <lb/>igitur lineæ circulum de&longs;cribenti i&longs;tud accidit, vt duobus feratur motioni<lb/>bus; vna quidem, quæ illi naturalis, ac &longs;ecundum circunferentiam, qua re<lb/>ctà tenderet ni&longs;i impediretur: altera verò, quæ illi innaturalis, qua in tran&longs;<lb/>uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor <lb/>autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma<lb/>ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro <lb/><figure place="text" xlink:href="figures-la/009.01.152.1.tif"/><lb/>vicinior, quod quodammodo retra<lb/>hit à motu naturali, propterea ma<lb/>gis vincitur, quàm remotior. </s>

<s>Quod <lb/>ex his erit <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan>. </s>

Line 6920

<s>Cvr autem maiores libræ minoribus &longs;int exactiores, palàm e&longs;t ex <lb/>præmi&longs;&longs;is principijs. </s>

<s>con&longs;iderare enim oportet, quod in motu li<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ <lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri <lb/><figure id="fig75"/><lb/>à centro exeuntes, vt in figura cerne<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum <lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

<s>con&longs;iderare enim oportet, quod in motu li<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ <lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri <lb/><figure place="text" xlink:href="figures-la/009.01.154.1.tif"/><lb/>à centro exeuntes, vt in figura cerne<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum <lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

<s>In <lb/>eadem porrò figura libra maior &longs;it <lb/>A B. minor verò circa idem &longs;partum <lb/>C, &longs;it F G. </s>

Line 6948

<s>Si autem deor&longs;um constitutum <lb/>fuerit, non a&longs;cendit, &longs;ed manet? </s>

<s>An quia &longs;ur&longs;um &longs;parto quidem exi&longs;ten<lb/>te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum, <lb/>quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li<lb/>bram diuidit ad ip&longs;um perpendicul&ubreve;, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"/><lb/><figure id="fig76"/><lb/><emph type="italics"/>&longs;it libra vbi recta B C, &longs;partum autem <lb/>A D: hoc igitur &longs;ur&longs;um erecto, perpendi<lb/>culum erit vbi A D M. &longs;i igitur in ip&longs;o B, <lb/>ponatur onus, B, quidem de&longs;cendet vbi E; <lb/>C, autem a&longs;cendat vbi H, quamobrem ea, <lb/>quæ bifariam libram &longs;ecat, primò quidem <lb/>erit D M, ip&longs;ius perpendiculi; incumben<lb/>te autem onere erit D G, quare libræ ip<lb/>&longs;ius vbi E H, quod extra A M, perpendi<lb/>culum e&longs;t, vbi e&longs;t D D H, maius e&longs;t dimidio. </s>

<s>An quia &longs;ur&longs;um &longs;parto quidem exi&longs;ten<lb/>te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum, <lb/>quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li<lb/>bram diuidit ad ip&longs;um perpendicul&ubreve;, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.155.1.tif"/><lb/><emph type="italics"/>&longs;it libra vbi recta B C, &longs;partum autem <lb/>A D: hoc igitur &longs;ur&longs;um erecto, perpendi<lb/>culum erit vbi A D M. &longs;i igitur in ip&longs;o B, <lb/>ponatur onus, B, quidem de&longs;cendet vbi E; <lb/>C, autem a&longs;cendat vbi H, quamobrem ea, <lb/>quæ bifariam libram &longs;ecat, primò quidem <lb/>erit D M, ip&longs;ius perpendiculi; incumben<lb/>te autem onere erit D G, quare libræ ip<lb/>&longs;ius vbi E H, quod extra A M, perpendi<lb/>culum e&longs;t, vbi e&longs;t D D H, maius e&longs;t dimidio. </s>

<s>&longs;i igitur amoueatur onus ab ip&longs;o E, ne<lb/>ce&longs;&longs;e e&longs;t H, deor&longs;um ferri, minus enim e&longs;t ip&longs;um E D. &longs;i quidem igitur &longs;ur&longs;um ha<lb/>buerit &longs;partum, propter hoc a&longs;cendit libra. </s>

<s>&longs;i autem deor&longs;um fuerit, id quod &longs;ub<lb/>stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm <lb/>quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.<emph.end type="italics"/><lb/><figure id="fig77"/><lb/><emph type="italics"/>&longs;it libra vbi N G, perpendiculum autem <lb/>K L M, bifariam igitur &longs;ecatur N G. im<lb/>po&longs;ito autem onere in ip&longs;o N, erit quidem <lb/>N, vbi O, ip&longs;um autem G, vbi R; K L, au<lb/>tem vbi K P, quare maius e&longs;t L P O, quàm <lb/>L R, ip&longs;o P L. </s>

<s>&longs;i autem deor&longs;um fuerit, id quod &longs;ub<lb/>stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm <lb/>quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.155.2.tif"/><lb/><emph type="italics"/>&longs;it libra vbi N G, perpendiculum autem <lb/>K L M, bifariam igitur &longs;ecatur N G. im<lb/>po&longs;ito autem onere in ip&longs;o N, erit quidem <lb/>N, vbi O, ip&longs;um autem G, vbi R; K L, au<lb/>tem vbi K P, quare maius e&longs;t L P O, quàm <lb/>L R, ip&longs;o P L. </s>

<s>Ablato igitur onere, ne<lb/>ce&longs;&longs;e e&longs;t manere; incumbit enim, ceu onus <lb/>exce&longs;&longs;us medietatis in quo P L.)<emph.end type="italics"/> Aduer<lb/>te textum græcum e&longs;&longs;e mendo&longs;um, la<lb/>tinum vero mendo&longs;i&longs;&longs;imum. </s>

Line 6962

<s>Porrò quoniam Piccolomineus, <lb/>& &longs;i plurimum, vt ip&longs;e fatetur, in&longs;udauerit, non tamen &longs;olutionem huius <lb/>quæ&longs;tionis e&longs;t a&longs;&longs;ecutus, eam tibi ex Mechanicis Guidibaldi tradam. </s>

<s>Ari&longs;t<lb/>igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones, <lb/>vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra. <lb/><figure id="fig78"/><lb/>vt in præ&longs;enti figura, &longs;it libra B C, cuius <lb/>&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur<lb/>&longs;um, ita vt in puncto A, &longs;it affixum perpen<lb/>diculum, & circa idem punctum A, tan<lb/>quam circa centrum tota libra circum<lb/>uertatur. </s>

<s>Ari&longs;t<lb/>igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones, <lb/>vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra. <lb/><figure place="text" xlink:href="figures-la/009.01.155.3.tif"/><lb/>vt in præ&longs;enti figura, &longs;it libra B C, cuius <lb/>&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur<lb/>&longs;um, ita vt in puncto A, &longs;it affixum perpen<lb/>diculum, & circa idem punctum A, tan<lb/>quam circa centrum tota libra circum<lb/>uertatur. </s>

<s>hæc e&longs;t prima libræ collocatio. </s>

<s>&longs;it deinde libra B C, cuius &longs;partum, <lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb pagenum="156"/><figure id="fig79"/><lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem <lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

<s>&longs;it deinde libra B C, cuius &longs;partum, <lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb pagenum="156"/><figure place="text" xlink:href="figures-la/009.01.156.1.tif"/><lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem <lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

<s>Quærit <lb/>igitur, cur &longs;i in libra &longs;ur&longs;um <expan abbr="hab&etilde;te">habente</expan> per<lb/>pendiculum, & centrum, ponatur ex vna <lb/>parte onus quodpiam, v. g. in parte B, vt in prima textus figura factum e&longs;t, <lb/>libra de primo &longs;itu B C, mouetur ad &longs;itum E H, &longs;ed tamen ablato pondere <lb/>reuertitur &longs;ua &longs;pontè ad pri&longs;tinum &longs;itum B C. &longs;i autem in libra, cuius per<lb/>pendiculum, ac centrum deor&longs;um &longs;it, vt in &longs;ecunda figura textus, pondus <lb/>imponatur, ip&longs;a quidem à &longs;itu B C, ad &longs;itum O R, transferretur; verumta<lb/>men ablato onere, <expan abbr="nõ">non</expan> amplius ad priorem po&longs;itionem, vti prior, reucrtitur.</s>

Line 7032

<s>vt in &longs;ubiectis figuris apparet. </s>

<s>In prima, vectis e&longs;t A B, fultu<lb/><figure id="fig80"/><lb/>ra E, onus C. potentia autem &longs;eu vis, <lb/>&longs;eu aliud pondus <expan abbr="mou&etilde;s">mouens</expan> &longs;it vbi D. quæ <lb/>deor&longs;um in D, præmens eleuabit &longs;ur<lb/>&longs;um ex altera parte onus C. & vectis <lb/>circa fulturam E, tanquam centrum <lb/>conuertetur. </s>

<s>In prima, vectis e&longs;t A B, fultu<lb/><figure place="text" xlink:href="figures-la/009.01.158.1.tif"/><lb/>ra E, onus C. potentia autem &longs;eu vis, <lb/>&longs;eu aliud pondus <expan abbr="mou&etilde;s">mouens</expan> &longs;it vbi D. quæ <lb/>deor&longs;um in D, præmens eleuabit &longs;ur<lb/>&longs;um ex altera parte onus C. & vectis <lb/>circa fulturam E, tanquam centrum <lb/>conuertetur. </s>

<s>In altera figura pondus <lb/>e&longs;t inter fulturam, & potentiam, ful<lb/>tura autem in altera extremitate, vt <lb/>patet in figura, hic autem potentia <lb/>non præmit deor&longs;um in D: &longs;ed &longs;ur&longs;um <lb/>vectem eleuando pondus C, attollitur. <lb/></s>

Line 7092

vectem &longs;ecundi ge<lb/>neris ad &longs;olutionem non adhibuit, vt par erat, & propterea ob&longs;cu<lb/>rior eua&longs;it, ita etiam in præ&longs;enti, qu&ecedil;&longs;tionem ad ve ctem primi ge<lb/>neris reducit, quæ ad alterum reducendà erat: <expan abbr="atq;">atque</expan> hinc ob&longs;curi<lb/>tas, atque prolixitas &longs;olutionis manauit. </s>

<s>E&longs;t enim propriè Temo, &longs;iue gu<lb/>bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo <lb/><figure id="fig81"/><lb/>in&longs;trumentum in extrema nanis par<lb/>te, &longs;eu puppi affixum, vt in figura prç<lb/>&longs;enti vides tabellam, in qua B C D, <lb/>cuius manubrium A B, intra nauim <lb/>recipitur, quæ tabella, &longs;eu temo in <lb/>duobus cardinibus, vbi C, & D, cir<lb/>cumuertitur à Nauis gubernatore, <lb/>manubrium vbi A, tractante; ex qua <lb/>conuer&longs;ione nauigium, quò vult ip&longs;e <lb/>gubernator facilè dirigit, ip&longs;umque <lb/>nauigium huc illuc quamuis adeò magnum ip&longs;e &longs;olus impellit, & agitat. </s>

<s>E&longs;t enim propriè Temo, &longs;iue gu<lb/>bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo <lb/><figure place="text" xlink:href="figures-la/009.01.159.1.tif"/><lb/>in&longs;trumentum in extrema nanis par<lb/>te, &longs;eu puppi affixum, vt in figura prç<lb/>&longs;enti vides tabellam, in qua B C D, <lb/>cuius manubrium A B, intra nauim <lb/>recipitur, quæ tabella, &longs;eu temo in <lb/>duobus cardinibus, vbi C, & D, cir<lb/>cumuertitur à Nauis gubernatore, <lb/>manubrium vbi A, tractante; ex qua <lb/>conuer&longs;ione nauigium, quò vult ip&longs;e <lb/>gubernator facilè dirigit, ip&longs;umque <lb/>nauigium huc illuc quamuis adeò magnum ip&longs;e &longs;olus impellit, & agitat. </s>

<s>e&longs;t <lb/>enim temo vectis, cuius auxilio vires mirum in modum augentur, nam to<lb/>ta A B, e&longs;t ip&longs;a Vectis longitudo, cuius hypomoclion e&longs;t mare, cui contra<pb pagenum="160"/>nititur tabella B E; onus autem e&longs;t puppis, quod onus præ&longs;ertim in cardini<lb/>bus C D, mouenti re&longs;i&longs;tit, & quod præcipuè mouere gubernator intendit. <lb/></s>

Line 7140

<s><expan abbr="atq;">atque</expan> hinc oritur nece&longs;&longs;itas <expan abbr="e&utilde;">eum</expan> po&longs;teriori parti affigendi. </s>

<s>&longs;ubdit po&longs;tea aliam <lb/>eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma<lb/>ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ <lb/>&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in <lb/><figure id="fig82"/><lb/>ad&longs;cripta figura intueri licet; in qua duæ <lb/>lineæ A B, A C, continent angulum A, cui <lb/>angulo &longs;ubtenduntur tres lineæ parallelæ <lb/>F G, D E, B C, quarum B C, maxima e&longs;t, <lb/>quia ibi maiores, &longs;iue remotiores &longs;unt ab <lb/>angulo A, duæ rectæ A B, A C, ip&longs;um con<lb/>tinentes, quod Geometricè per 4. 6. pro<lb/>bari pote&longs;t. </s>

<s>&longs;ubdit po&longs;tea aliam <lb/>eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma<lb/>ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ <lb/>&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in <lb/><figure place="text" xlink:href="figures-la/009.01.161.1.tif"/><lb/>ad&longs;cripta figura intueri licet; in qua duæ <lb/>lineæ A B, A C, continent angulum A, cui <lb/>angulo &longs;ubtenduntur tres lineæ parallelæ <lb/>F G, D E, B C, quarum B C, maxima e&longs;t, <lb/>quia ibi maiores, &longs;iue remotiores &longs;unt ab <lb/>angulo A, duæ rectæ A B, A C, ip&longs;um con<lb/>tinentes, quod Geometricè per 4. 6. pro<lb/>bari pote&longs;t. </s>

<s>&longs;ic etiam facta motione, vel <lb/>parua in puppi, tota nauis transfertur ad <lb/>alium &longs;itum, ita vt prora multum aliò transferatur, quod non accideret, &longs;i <lb/>eadem motio fieret ad medium nauigij. </s>

Line 7162

<s>B E, enim ponitur <lb/>æqualis ip&longs;i A D, &longs;ed minus interuallum propter re&longs;i&longs;tentiam aquæ ex &longs;up<lb/>po&longs;itione percurret, quale e&longs;t B F, quod minus e&longs;t quàm A D, quare etiam li<lb/>nea B G, abbreuiabitur, <expan abbr="erit&qacute;">eritque</expan>; veluti F Y, quæ etiam erit minor ip&longs;a D G, <lb/>quæ facta e&longs;t D Y, propter duo &longs;indlia triangula D Y A, B Y F, &longs;imilia au<lb/>tem triangula &longs;unt ea, quorum anguli vnius &longs;unt æquales angulis alterius, <lb/>quo po&longs;ito &longs;unt etiam latera vnius proportionalia lateribus alterius, vt pa<lb/>tet ex prima definitione 6. necnon ex quarta eiu&longs;dem demon&longs;tratione. </s>

<s>hæc <pb pagenum="162"/><figure id="fig83"/><lb/>quidem duo triangula &longs;unt &longs;imi<lb/>lia, & rectè concluditur F Y, mi<lb/>nus e&longs;&longs;e quàm D Y, &longs;ed tamen <lb/>non videntur i&longs;ta propo&longs;itum <lb/>o&longs;tendere, quod erat, plus nauim <lb/>procedere, quàm palmulam re<lb/>trocedere. </s>

<s>hæc <pb pagenum="162"/><figure place="text" xlink:href="figures-la/009.01.162.1.tif"/><lb/>quidem duo triangula &longs;unt &longs;imi<lb/>lia, & rectè concluditur F Y, mi<lb/>nus e&longs;&longs;e quàm D Y, &longs;ed tamen <lb/>non videntur i&longs;ta propo&longs;itum <lb/>o&longs;tendere, quod erat, plus nauim <lb/>procedere, quàm palmulam re<lb/>trocedere. </s>

<s>Fateor quidem tex<lb/>tum hunc e&longs;&longs;e ob&longs;curi&longs;&longs;imum, <lb/><expan abbr="id&qacute;">idque</expan>; propterea fortè quia e&longs;t admodum corruptus, præ&longs;ertim circa chara<lb/>cteres, qui corrigendi &longs;unt vti nos facimus. </s>

Line 7200

<s>Duæ <expan abbr="itaq;">itaque</expan> rectæ lineæ ponantur æquales A B, & D E, quæ quidem <lb/>in C, puncto medio &longs;e inuicem &longs;ecent, & connectantur A B, & D E: remus <lb/>autem in initio vnius remigationis po&longs;itionem habeat rectam lineam A B, <lb/><expan abbr="&longs;it&qacute;">&longs;itque</expan>; A, manubrium; B, palmula; C, verò &longs;ealmus. </s>

<s>Cum igitur A, remi ca<lb/>put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i <lb/><figure id="fig84"/><lb/>enim ibi fuerit; remus igitur po&longs;itionem <lb/>habebit rectam lineam D E; & quoniam <lb/>contrapo&longs;iti anguli, qui ad C, æquales &longs;unt, <lb/>& duo latera A C, & D C, trianguli A D C, <lb/>duobus lateribus B C, & C E, trianguli B<lb/>E C, æqualia etiam &longs;unt: reliqui igitur an<lb/>guli, <expan abbr="atq;">atque</expan> ba&longs;es ip&longs;orum <expan abbr="triãgulorum">triangulorum</expan> æqua<lb/>les erunt per 4. propo&longs;itionem primi libri <lb/>Euclidis, & propterea tantum &longs;patium per<lb/>curret B, quantum A: &longs;calmus verò C, im<lb/>motus omninò erit: & nauigium idcircò, in <lb/>quo ip&longs;e &longs;calmus, immotum etiam erit con<lb/>tra hypothe&longs;im. </s>

<s>Cum igitur A, remi ca<lb/>put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i <lb/><figure place="text" xlink:href="figures-la/009.01.163.1.tif"/><lb/>enim ibi fuerit; remus igitur po&longs;itionem <lb/>habebit rectam lineam D E; & quoniam <lb/>contrapo&longs;iti anguli, qui ad C, æquales &longs;unt, <lb/>& duo latera A C, & D C, trianguli A D C, <lb/>duobus lateribus B C, & C E, trianguli B<lb/>E C, æqualia etiam &longs;unt: reliqui igitur an<lb/>guli, <expan abbr="atq;">atque</expan> ba&longs;es ip&longs;orum <expan abbr="triãgulorum">triangulorum</expan> æqua<lb/>les erunt per 4. propo&longs;itionem primi libri <lb/>Euclidis, & propterea tantum &longs;patium per<lb/>curret B, quantum A: &longs;calmus verò C, im<lb/>motus omninò erit: & nauigium idcircò, in <lb/>quo ip&longs;e &longs;calmus, immotum etiam erit con<lb/>tra hypothe&longs;im. </s>

<s>&longs;upponitur enim in que&longs;tio<lb/>ne, quod nauigium illa remigatione in anteriora moueatur, remi verò pal<lb/>mula retrocedat. </s>

Line 7246

<s>Si Remiges nauigium mouere po&longs;&longs;unt, maius &longs;emper &longs;pa<lb/>tium remi manubrium percurrit, quàm nauigium.</s>

<s>Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij <lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i<lb/><figure id="fig85"/><lb/>tum rectitudinis habeat E F. </s>

<s>Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij <lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i<lb/><figure place="text" xlink:href="figures-la/009.01.164.1.tif"/><lb/>tum rectitudinis habeat E F. </s>

<s>Spatium <lb/>itaque, quod A, conficit, curna linea <lb/>&longs;it A E, cui recta linea re&longs;pondeat A Z, in re<lb/>ctam E F, perpendieularis. </s>

Line 7266

<s>Nam &longs;i a loco &longs;uo dimota fuerit: <lb/>&longs;patium igitur permeet C D, ad po&longs;teriora: quo quidem decur&longs;o, <lb/>remus A C, po&longs;itionem rectitudmis habeat F D, &longs;calmus <expan abbr="itaq;">itaque</expan> B, tran&longs;latus <lb/>erit in G. </s>

<s>Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb/><figure id="fig86"/><lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H, <lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper <lb/>E R; ip&longs;arum verò rectarum linearum E R, & <lb/>A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo<lb/>niam A K, id &longs;patium e&longs;t, quod motu proprio re<lb/>mi manubrium permeauit, curuilineo enim re<lb/>&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t, <lb/>quod nauigium confecit: ip&longs;æ igitur rectæ lineæ <lb/>H K, & B G, æquales erunt. </s>

<s>Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb/><figure place="text" xlink:href="figures-la/009.01.165.1.tif"/><lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H, <lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper <lb/>E R; ip&longs;arum verò rectarum linearum E R, & <lb/>A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo<lb/>niam A K, id &longs;patium e&longs;t, quod motu proprio re<lb/>mi manubrium permeauit, curuilineo enim re<lb/>&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t, <lb/>quod nauigium confecit: ip&longs;æ igitur rectæ lineæ <lb/>H K, & B G, æquales erunt. </s>

<s>Atqui in duobus æqui<lb/>angulis triangulis E B C, & B A K, vel per 26. <lb/>propo&longs;itionem primi Euclidis, vel 4. 6. æquales <lb/>e&longs;&longs;e concludes A K, & E C, rectas lineas: quapro<lb/>pter æqualis erit E C, rectæ B G, per communem <lb/>&longs;ententiam: eidem autem B G, æqualis e&longs;t E Z, <lb/>in parallelogrammo, per 34. propo&longs;itionem ip<lb/>&longs;ius primi libri: æqualis igitur erit recta E Z, re<lb/>ctæ E C, pars toti, quod e&longs;t impo&longs;&longs;ibile. </s>

Line 7278

<s>Remus enim incipiente motu po&longs;itionem habeat A C, de&longs;inente <lb/>verò rectitudinis &longs;itum F G. &longs;calmus igitur B, propter nauigij <lb/>motum, &longs;patium con&longs;iciet B D. </s>

<s>Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C, <lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb pagenum="166"/><figure id="fig87"/><lb/>decur&longs;um motu proprio &longs;patij B D, duplum <lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat <lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

<s>Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C, <lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb pagenum="166"/><figure place="text" xlink:href="figures-la/009.01.166.1.tif"/><lb/>decur&longs;um motu proprio &longs;patij B D, duplum <lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat <lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

<s>Di<lb/>co ip&longs;as rectas lineas B D, & C H, æquales <lb/>e&longs;&longs;e. </s>

Line 7322

<s>Si celerius feratur nauigium, quàm remi manubrium, mo<lb/>uebitur palmula in vlteriora, <expan abbr="nil&qacute;">nilque</expan>; vnquam retroce<lb/>det, <expan abbr="id&qacute;">idque</expan>; &longs;patium decurret, quo nauigij motus <lb/>motum manubrij &longs;uperat.</s>

<s>Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente <lb/><figure id="fig88"/><lb/>verò <expan abbr="&longs;it&utilde;">&longs;itum</expan> rectitudinis F G. &longs;cal<lb/>mus igitur B, propter nauigij <lb/>motum tran&longs;latus, erit in D, &longs;it <lb/><expan abbr="itaq;">itaque</expan> &longs;patium B D, maius quàm A H, à re<lb/>mi manubrio motu proprio decur&longs;um: &longs;ic <lb/>enim celerius dicetur ferri <expan abbr="nauigi&utilde;">nauigium</expan>, quàm <lb/>manubrium. </s>

<s>Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente <lb/><figure place="text" xlink:href="figures-la/009.01.167.1.tif"/><lb/>verò <expan abbr="&longs;it&utilde;">&longs;itum</expan> rectitudinis F G. &longs;cal<lb/>mus igitur B, propter nauigij <lb/>motum tran&longs;latus, erit in D, &longs;it <lb/><expan abbr="itaq;">itaque</expan> &longs;patium B D, maius quàm A H, à re<lb/>mi manubrio motu proprio decur&longs;um: &longs;ic <lb/>enim celerius dicetur ferri <expan abbr="nauigi&utilde;">nauigium</expan>, quàm <lb/>manubrium. </s>

<s>Dico, quòd palmula C, in <lb/>vlteriora mouebitur. </s>

Line 7376

<s>mouens <lb/>potentia e&longs;t ventus, qui mouet in antenna F C G. quanto igitur &longs;ublimior <lb/>e&longs;t antenna, tanto longior euadit vectis E C, <expan abbr="tanto&qacute;">tantoque</expan>; maiores fiunt venti <lb/>vires. </s>

<s>dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb pagenum="169"/><figure id="fig89"/><lb/>veluti centro parte E. quia ve<lb/>rò ob maris liquiditatem na<lb/>uis minimè obfi&longs;tit, &longs;ed facilè <lb/>cedens à ventis vrgetur, hinc <lb/>fit, vt meritò dixerim pondus <lb/>nauis e&longs;&longs;e ad D, fulcimentum <lb/>verò ad E.</s>

<s>dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb pagenum="169"/><figure place="text" xlink:href="figures-la/009.01.169.1.tif"/><lb/>veluti centro parte E. quia ve<lb/>rò ob maris liquiditatem na<lb/>uis minimè obfi&longs;tit, &longs;ed facilè <lb/>cedens à ventis vrgetur, hinc <lb/>fit, vt meritò dixerim pondus <lb/>nauis e&longs;&longs;e ad D, fulcimentum <lb/>verò ad E.</s>

<s>Quæ&longs;tio &longs;eptima, & &longs;atis pec <lb/>&longs;e clara e&longs;t; <expan abbr="neq;">neque</expan> Mathemati<lb/>ci e&longs;t eam exponere.</s><figure/>

<s>Quæ&longs;tio &longs;eptima, & &longs;atis pec <lb/>&longs;e clara e&longs;t; <expan abbr="neq;">neque</expan> Mathemati<lb/>ci e&longs;t eam exponere.</s>

<s><emph type="italics"/>QVÆSTIO OCTAVA<emph.end type="italics"/></s>

Line 7416

<s>& quia &longs;icut diameter ad diametrum, ita maio<lb/>ris circuli periphæria ad minoris periphæriam (vt po&longs;tea o&longs;tendam) & quia <lb/>quo <expan abbr="lõgior">longior</expan> diameter e&longs;t, eò facilius, vt initio probaui, mouetur, fit vt etiam <lb/>periphæria maioris facilius, quàm minoris moueatur, &longs;iue dixeris, quod an<lb/>gulus maioris circuli ad angulum minoris nutum quendam habet; & quia <lb/>facilius mouetur angulus maioris, quàm minoris, fit, vt maior rota adhi<lb/>beatur ad minorem mouendam: & quia intra maiorem infinitæ circa idem <lb/>centrum concipi po&longs;&longs;unt, hinc fit, vt rotæ maiores facilius moueantur, & <lb/>motæ moueant cæteras intra &longs;e contentas. </s>

<s>quod dictum e&longs;t de nutu anguli <lb/>maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi <lb/><figure id="fig90"/><lb/>pro minore angulo intelligendus e&longs;t arcus C B, <lb/>pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo<lb/>catur angulus, quoniam angulo A, qui e&longs;t in cen<lb/>tro opponuntur. </s>

<s>quod dictum e&longs;t de nutu anguli <lb/>maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi <lb/><figure place="text" xlink:href="figures-la/009.01.170.1.tif"/><lb/>pro minore angulo intelligendus e&longs;t arcus C B, <lb/>pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo<lb/>catur angulus, quoniam angulo A, qui e&longs;t in cen<lb/>tro opponuntur. </s>

<s>Atque hæc &longs;ufficiant deijs, quæ <lb/>primo modo moueutur.</s>

Line 7462

<s>3. probatum e&longs;t ex Archimede, & Gui<lb/>do Vbaldo, di&longs;tantiæ autem &longs;int æquales, er<gap/>nt etiam pondus, & potentia <lb/>æqualia, ide&longs;t, &longs;i pondus e&longs;&longs;et vnius libræ, &longs;u&longs;tineretur à tanta vi, <expan abbr="quãta">quanta</expan> opus <lb/>e&longs;t ad libram vnam &longs;u&longs;tinendam, & non amplius. </s>

<s>vt autem clarè appareat <lb/>vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua <lb/><figure id="fig91"/><lb/>pondus D, ductario funi D C B E, alligatum. </s>

<s>vt autem clarè appareat <lb/>vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua <lb/><figure place="text" xlink:href="figures-la/009.01.171.1.tif"/><lb/>pondus D, ductario funi D C B E, alligatum. </s>

<s>poten<lb/>tia <expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan> E. axis autem erit diameter rotulæ B A C, <lb/>nam potentia premit rotulam in B, & pondus in C, & <lb/>cum rotula &longs;u&longs;tineatur in A, à &longs;u&longs;pen&longs;orio F A. erit <lb/>punctum A, hypomoclion, quia in motu vectis eua<lb/>dit centrum, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; punctum manens. </s>

Line 7518

<s>hoc illi vtebantur &longs;ecreto, cum <lb/>literas ad Imperatores &longs;uos mi&longs;&longs;as, ho&longs;tibus occultas e&longs;&longs;e volebant.</s>

<s>Ex quibus conijcere licet &longs;cytalam fui&longs;&longs;e lignum oblongum, & teres, &longs;iue <lb/>vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus e&longs;&longs;ent <lb/>margines duo aliquantulum prominentes, ceu binæ rotæ, cum ip&longs;o tamen <lb/>continuæ, & connexæ, vt cum ip&longs;o &longs;imul conuoluerentur; non tamen tan<lb/><figure id="fig92"/><lb/>quam circa axem. </s>

<s>cuius hanc accipe fi<lb/>guram. </s>

Line 7636

<s>quærit igitur, <lb/>cur quanto gracilius fucrit corpus &longs;ucculæ A B, tanto facilius vertitur. <lb/></s>

<s>Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem, <pb pagenum="176"/><figure id="fig93"/><lb/>cuius hypomoclion e&longs;t in medio <lb/>&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ; <lb/>potentia verò e&longs;t in &longs;ummitatibus <lb/>collopum, vt in C, E, F, D, pon<lb/>dus verò e&longs;t vbi funis ductarius <lb/>cum onere pendet è &longs;uccula in &longs;u<lb/>perficie nimirum, vt vbi L, quare <lb/>pars vectis inter axim, & &longs;uperfi<lb/>ciem &longs;ucculæ eadem e&longs;t, quæ inter <lb/>hypomoclium, & pondus. </s>

<s>Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem, <pb pagenum="176"/><figure place="text" xlink:href="figures-la/009.01.176.1.tif"/><lb/>cuius hypomoclion e&longs;t in medio <lb/>&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ; <lb/>potentia verò e&longs;t in &longs;ummitatibus <lb/>collopum, vt in C, E, F, D, pon<lb/>dus verò e&longs;t vbi funis ductarius <lb/>cum onere pendet è &longs;uccula in &longs;u<lb/>perficie nimirum, vt vbi L, quare <lb/>pars vectis inter axim, & &longs;uperfi<lb/>ciem &longs;ucculæ eadem e&longs;t, quæ inter <lb/>hypomoclium, & pondus. </s>

<s>quanto <lb/>igitur &longs;ucculæ corpus gracilius fuerit, tanto hæc pars minuetur; & con&longs;e<lb/>quenter altera inter hypomoclium, & potentiam productior euadet: eaque <lb/>propter facilius à motore ver&longs;abitur.</s>

Line 7682

<s>in A, verò onus ip&longs;ius ligni, deor&longs;um tendens <lb/>premit, quod nunc e&longs;t in&longs;tar potentiæ motricis, quare A, & C, &longs;unt &longs;ibi in<lb/>uicem, & potentiæ, & pondera. </s>

<s>In minori autem ligno, onus ligni in D, <pb pagenum="177"/><figure id="fig94"/><lb/>fultura manus in E, potentia alterius ma<lb/>nus in F. iam inquir Ari&longs;t.

<s>In minori autem ligno, onus ligni in D, <pb pagenum="177"/><figure place="text" xlink:href="figures-la/009.01.177.1.tif"/><lb/>fultura manus in E, potentia alterius ma<lb/>nus in F. iam inquir Ari&longs;t.

maius lignum <lb/>A B C, magis flectitur, quamuis cra&longs;&longs;ius <lb/>&longs;it, quàm lignum D E F, quod e&longs;t tenuuius, <lb/>&longs;ed multò breuius; quia in maiori onus <lb/>ip&longs;ius ligni, quod circa A, deor&longs;um pre<lb/>mit <expan abbr="lõgius">longius</expan> di&longs;tat ab hypomoclio B, quàm <lb/>in minori ligno. </s>

Line 7708

<s>Verum &longs;atius e&longs;t cum Guido Vbaldo reducere cuneum ad duos &longs;ecundi <lb/>generis vectes, quorum fultura &longs;it in cunei apice extremo, pondus verò in<lb/>tra vectem, ea nimirum pars ligni, que à cuneo vrgetur, ac diuellitur. </s>

<s>cuneo <lb/>præterea vires adduntur ex valida mallei percu&longs;&longs;ione; malleus autem ip&longs;e <lb/>magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet <lb/>ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus <lb/>fit valentior: <expan abbr="hoc&qacute;">hocque</expan>; modo paruos cunci vectes maiores con&longs;equuntur vires, <lb/><figure id="fig95"/><lb/>quàm ip&longs;a vectium magnitudo po&longs;tulet. <lb/></s>

<s>cuneo <lb/>præterea vires adduntur ex valida mallei percu&longs;&longs;ione; malleus autem ip&longs;e <lb/>magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet <lb/>ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus <lb/>fit valentior: <expan abbr="hoc&qacute;">hocque</expan>; modo paruos cunci vectes maiores con&longs;equuntur vires, <lb/><figure place="text" xlink:href="figures-la/009.01.177.2.tif"/><lb/>quàm ip&longs;a vectium magnitudo po&longs;tulet. <lb/></s>

<s>&longs;it cuneus A B C. lignum autem &longs;cinden<lb/>dum D E F G, <expan abbr="vectes&qacute;">vectesque</expan> duo &longs;int A C, & <lb/>B C, quorum commune hypomoclion e&longs;t <lb/>in C, onus autem vectis B C, e&longs;t pars li<lb/>gni G, hæc enim ip&longs;i contranititur, <expan abbr="atq;">atque</expan> <lb/>ab eo expellitur. </s>

Line 7738

<s>Auxilio huius in&longs;trumenti machinato<lb/>res parua vi attollunt ingentia pondera. </s>

<s>communiter autem con&longs;tat ex plu<lb/><figure id="fig96"/><lb/>ribus orbiculis, qui partim &longs;uperiori loculamento, <lb/>partim infetiori inditi &longs;unt, per quos orbiculos cer<lb/>ta lege circumductus e&longs;t ductarius funis, qui deinde <lb/>in &longs;ui po&longs;trema parte à potentia tractus omnes illos <lb/>orbiculos, per quos tran&longs;it circumuoluens inferius <lb/>loculamentum, cui appen&longs;um e&longs;t pondus, vnà cum <lb/>ponderc attollit. </s>

<s>communiter autem con&longs;tat ex plu<lb/><figure place="text" xlink:href="figures-la/009.01.178.1.tif"/><lb/>ribus orbiculis, qui partim &longs;uperiori loculamento, <lb/>partim infetiori inditi &longs;unt, per quos orbiculos cer<lb/>ta lege circumductus e&longs;t ductarius funis, qui deinde <lb/>in &longs;ui po&longs;trema parte à potentia tractus omnes illos <lb/>orbiculos, per quos tran&longs;it circumuoluens inferius <lb/>loculamentum, cui appen&longs;um e&longs;t pondus, vnà cum <lb/>ponderc attollit. </s>

<s>figuram &longs;implicis trochleæ, con<lb/>&longs;tantis &longs;cilicet ex duobus tantum orbiculis, facilita<lb/>tis cau&longs;a exhibebo, in hac enim melius apparebit, <lb/>qua ratione trochlea ad vectem reducatur. </s>

Line 7832

<s>Ex his autem po&longs;tremis verbis de&longs;criptio<lb/>nem antiquæ &longs;ecuris, &longs;ic puto eruendam. </s>

<s>Ducatur primo recta A C, quæ <lb/><figure id="fig97"/><lb/>erit in&longs;tar manubrij &longs;ecuris. </s>

<s>Ducatur primo recta A C, quæ <lb/><figure place="text" xlink:href="figures-la/009.01.180.1.tif"/><lb/>erit in&longs;tar manubrij &longs;ecuris. </s>

<s>de<lb/>inde ex centro C, interuallo. </s>

Line 7842

<s>Quod autem tam angulus E, quàm angulus F, &longs;int æquales duabus tertijs <lb/>vnius angulirecti, &longs;iue quod idem e&longs;t angulo trianguli æquilateri, manife<lb/>ftum erithoc modo. </s>

<s>De&longs;cribatur iterum &longs;ecuralis figura prædicto mode, <lb/><gap/><expan abbr="itq;">itque</expan> ea A B C. ducantur præterea ad &longs;ingulos angulos tres rectæ A B, B C, <lb/>C A, quæ con&longs;tituunt trianguium æquilaterum A B C, tria enim ip&longs;ius late<lb/><figure id="fig98"/><lb/>ra &longs;ubtendunt tres arcus æquales A B, B C, C A, <lb/>&longs;unt enim tres &longs;extantes æqualium circulorum, <lb/><gap/>t facilè colligi pote&longs;t ex 15. 4. ex quo etiam &longs;e<lb/>quitur tres ilias circulorum portiones, quas re<lb/>ctè cum &longs;uis arcubus con&longs;tituunt e&longs;&longs;e inuicem <lb/>æquales, & limiles portiones nimirum A B E, <lb/>B C D, C A F. hinc pr&ecedil;terea &longs;equitur angulos ip<lb/>&longs;arum e&longs;&longs;e inuicem æquales, angulos, v.g. A B E, <lb/>C B D, mixtos e&longs;&longs;e æquales, quod facilè e&longs;t per imaginatiam &longs;uperpo&longs;itio<lb/>nem demon&longs;trare. </s>

<s>De&longs;cribatur iterum &longs;ecuralis figura prædicto mode, <lb/><gap/><expan abbr="itq;">itque</expan> ea A B C. ducantur præterea ad &longs;ingulos angulos tres rectæ A B, B C, <lb/>C A, quæ con&longs;tituunt trianguium æquilaterum A B C, tria enim ip&longs;ius late<lb/><figure place="text" xlink:href="figures-la/009.01.180.2.tif"/><lb/>ra &longs;ubtendunt tres arcus æquales A B, B C, C A, <lb/>&longs;unt enim tres &longs;extantes æqualium circulorum, <lb/><gap/>t facilè colligi pote&longs;t ex 15. 4. ex quo etiam &longs;e<lb/>quitur tres ilias circulorum portiones, quas re<lb/>ctè cum &longs;uis arcubus con&longs;tituunt e&longs;&longs;e inuicem <lb/>æquales, & limiles portiones nimirum A B E, <lb/>B C D, C A F. hinc pr&ecedil;terea &longs;equitur angulos ip<lb/>&longs;arum e&longs;&longs;e inuicem æquales, angulos, v.g. A B E, <lb/>C B D, mixtos e&longs;&longs;e æquales, quod facilè e&longs;t per imaginatiam &longs;uperpo&longs;itio<lb/>nem demon&longs;trare. </s>

Line 7856

<s>quod etiam ma<lb/>nife &longs;tum &longs;ignum e&longs;t &longs;ecuris figuram a me re&longs;titutam e&longs;&longs;e illam veterem, de <lb/>qua idem Proclus loquitur.</s>

<s>Re&longs;tat, vt de antiquæ bipennis etiam figura di&longs;&longs;eramus; quæ nihil aliud <lb/>erat, quàm duplex &longs;ecuris, &longs;iue &longs;ecuris anceps, qualis e&longs;t præ&longs;ens figura, vt <lb/><figure id="fig99"/><lb/>propterea etiam &longs;æpius <expan abbr="bip&etilde;nis">bipennis</expan> ip<lb/>&longs;a &longs;ecuris appelletur. </s>

<s>Re&longs;tat, vt de antiquæ bipennis etiam figura di&longs;&longs;eramus; quæ nihil aliud <lb/>erat, quàm duplex &longs;ecuris, &longs;iue &longs;ecuris anceps, qualis e&longs;t præ&longs;ens figura, vt <lb/><figure place="text" xlink:href="figures-la/009.01.181.1.tif"/><lb/>propterea etiam &longs;æpius <expan abbr="bip&etilde;nis">bipennis</expan> ip<lb/>&longs;a &longs;ecuris appelletur. </s>

<s>dicitur enim <lb/>bipennis, qua&longs;i binis pinnis, quæ &longs;e<lb/>cures erant, con&longs;tet, vt & Græcis <lb/><foreign lang="greek">dixteros</foreign> dicebatur. </s>

Line 7888

<s>hoc autem ordine legenda e&longs;t: lectio à manubrio <lb/>incipiat, deinde legatur carmen; forti&longs;&longs;imæ Deæ, quod &longs;ub&longs;e<lb/>quatur; dedit Epeus, & &longs;ic in orbem lectio, <expan abbr="v&longs;q;">v&longs;que</expan> ad me<lb/>dium circumducatur. </s>

<s>hæc &longs;unt, quæ præ&longs;ertim <lb/>in gratiam eorum, qui &longs;uaui&longs;&longs;imo an<lb/>tiquitatis &longs;tudio tenentur, la<lb/>tere nolui.</s><figure/><pb pagenum="182"/>

<s>hæc &longs;unt, quæ præ&longs;ertim <lb/>in gratiam eorum, qui &longs;uaui&longs;&longs;imo an<lb/>tiquitatis &longs;tudio tenentur, la<lb/>tere nolui.</s><pb pagenum="182"/>

<s><emph type="italics"/>Simmiæ Rhodij <lb/>Bipennis.<emph.end type="italics"/></s><figure/><pb pagenum="183"/>

<s><emph type="italics"/>Simmiæ Rhodij <lb/>Bipennis.<emph.end type="italics"/></s><figure place="text" xlink:href="figures-la/009.01.182.1.tif"/><pb pagenum="183"/>

<s><emph type="italics"/>QVÆSTIO XX.<emph.end type="italics"/></s>

Line 7904

loco, magna mihi licuit cum delectatione col<lb/>ligere: quod etiam antiquitatis &longs;tudio&longs;is pergratum fore non du<lb/>bito: <expan abbr="atq;">atque</expan> hinc etiam ineptas, <expan abbr="atq;">atque</expan> ad&longs;cititias textus huius figuras tanquam <lb/>adulterinas reijcere; <expan abbr="in&qacute;">inque</expan>; earum locum veras re&longs;tituere licebit. </s>

<s>erat igitur <lb/><figure id="fig100"/><lb/>&longs;tatera, quantum ex Ari&longs;t.

<s>erat igitur <lb/><figure place="text" xlink:href="figures-la/009.01.183.1.tif"/><lb/>&longs;tatera, quantum ex Ari&longs;t.

conijcio <lb/>primum ha&longs;ta oblonga, qualis e&longs;t in <lb/>præ&longs;enti figura A B, ex cuius altero <lb/>extremo B, pendebat appendicu<lb/>lum, quod propriè æquipondium <lb/>dicitur: ex altera verò extremitate <lb/>A, lanx vna pendebat; in qua carnes, aliæuè merces ponderabantur: in me<lb/>dia <expan abbr="deniq;">denique</expan> ha&longs;ta paribus interuallis plures trutinæ, ex quibus &longs;ingulis modo <lb/>hac, modo illa, prout pondus emptoris po&longs;tulabat &longs;u&longs;pendebatur, <expan abbr="atq;">atque</expan> in<lb/>terim tantum mercis lanci imponebatur, donec æquipondio præpondera<lb/>ret in æquilibrio. </s>

Line 7976

<s>Vel etiam, <lb/>quia digiti propter carnis mollitiem cedentem nequeunt dentem <lb/>&longs;<gap/>miter circumplecti; &longs;errum verò, cum vndique durum æque &longs;it, nec ce<lb/>dens, melius dentem comprçhendit. </s>

<s>Aut tandem, quia forceps hæc duos <lb/>in &longs;e continet contrarios vectes; quorum, vnum tantum e&longs;t hypomoclion, <lb/><figure id="fig101"/><lb/>corum &longs;cilicet connexio; Virtute igitur <lb/>vectis arctius dentem per&longs;tringunt, <expan abbr="atq;">atque</expan> <lb/>adeò obtinent, <expan abbr="atq;">atque</expan> hinc etiam facilius <lb/>commouent. </s>

<s>Aut tandem, quia forceps hæc duos <lb/>in &longs;e continet contrarios vectes; quorum, vnum tantum e&longs;t hypomoclion, <lb/><figure place="text" xlink:href="figures-la/009.01.184.1.tif"/><lb/>corum &longs;cilicet connexio; Virtute igitur <lb/>vectis arctius dentem per&longs;tringunt, <expan abbr="atq;">atque</expan> <lb/>adeò obtinent, <expan abbr="atq;">atque</expan> hinc etiam facilius <lb/>commouent. </s>

<s>&longs;it dentiforcip is figura, ex<lb/>po&longs;ita, cuius alterum extremum, vbi &longs;unt <lb/>A, B, e&longs;t illud, quod binis &longs;emicirculis <lb/>concurrentibus dentem arctè <expan abbr="cõ&longs;tringit">con&longs;tringit</expan>, <pb pagenum="185"/>& commouet. </s>

Line 8014

<s>vecte autem facilè onera quælibet <expan abbr="obuiãtia">obuiantia</expan> diuelluntur. <lb/></s>

<s>qui duo vectes vnicum habent hypomoclion ip&longs;am &longs;cilicet connexionem <lb/><figure id="fig102"/><lb/>A. vectes &longs;unt binæ in&longs;trumenti ha&longs;tæ, F A D, <lb/>E A C. <expan abbr="dilatãdo">dilatando</expan> igitur extrema C D, deducun<lb/>tur etiam alia extrema F, E, & impo&longs;ita nuce in <lb/>hiatu K, quæuis potentia con&longs;tringendo C, D, <lb/>con&longs;tringet &longs;imul F, E, <expan abbr="ip&longs;am&qacute;">ip&longs;amque</expan>; nucem confrin<lb/>get. </s>

<s>qui duo vectes vnicum habent hypomoclion ip&longs;am &longs;cilicet connexionem <lb/><figure place="text" xlink:href="figures-la/009.01.185.1.tif"/><lb/>A. vectes &longs;unt binæ in&longs;trumenti ha&longs;tæ, F A D, <lb/>E A C. <expan abbr="dilatãdo">dilatando</expan> igitur extrema C D, deducun<lb/>tur etiam alia extrema F, E, & impo&longs;ita nuce in <lb/>hiatu K, quæuis potentia con&longs;tringendo C, D, <lb/>con&longs;tringet &longs;imul F, E, <expan abbr="ip&longs;am&qacute;">ip&longs;amque</expan>; nucem confrin<lb/>get. </s>

<s>quod igitur cum percu&longs;&longs;ione feci&longs;&longs;et pon<lb/>dus mallei, id valentiori vectium virtute efficiunt F A D, E A C. quanto au<lb/>tem locus nucis K, propinquior fuerit hypomoclio A, tanto celerius <lb/>confringitur, quia partes vectium A C, A D, tunc à centro <lb/>A, productiores fiunt, ide&longs;t multò maiores fiunt, <lb/>quàm &longs;int di&longs;tantiæ inter nucem, & cen<lb/>trum A, quod maximè poten<lb/>tiam iuuat.</s>

Line 8030

<s>Rhombus ex definitione 23. primi Elem.

e&longs;t figura æquilatera qui<lb/><figure id="fig103"/><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ<lb/>&longs;ens figura A B D C. </s>

e&longs;t figura æquilatera qui<lb/><figure place="text" xlink:href="figures-la/009.01.186.1.tif"/><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ<lb/>&longs;ens figura A B D C. </s>

<s>In præ&longs;enti porrò quæ&longs;tione <lb/>&longs;upponitur punctum A, quod e&longs;t vnum extremum <lb/>in rhombo moueri &longs;uper latus A B, ver&longs;us B, & &longs;i<lb/>militer interim æqua velocitate moueri alterum <lb/>extremum B, &longs;uper idem latus A B_{2} ver&longs;us A, & in<lb/>terim dum hæc duo puncta hoc modo &longs;ibi obuiam <lb/>procedunt, moueri latus totum A B, eadem ve<lb/>locitate, ver&longs;us latus C D, ita vt &longs;emper ip&longs;i C D, <lb/>æquidi&longs;ter, <expan abbr="de&longs;cendat&qacute;">de&longs;cendatque</expan>; per latera A C, B D, quo<lb/>u&longs;que ip&longs;i C D, congruat.</s>

Line 8074

<s>& propterea punctum B, minus interuallum, quale e&longs;t B C, percurret, quan<lb/>do A, maius A D, percurrit.</s>

<s>Ad &longs;ecundam verò quæ&longs;tionis partem, re&longs;pondeo con&longs;iderandum e&longs;&longs;e <lb/>latus B A, moueri vnico motu ad D C, quare à nullo impedi<lb/>tur, vnde nihil mirum videri debet, quòd ip&longs;um vnica <lb/>latione maius conficiat &longs;pacium quàm B, quod <lb/>quamuis duplici pellatur motu, vnus <lb/>tamen ab altero inhibetur.</s><figure/><pb pagenum="188"/>

<s><emph type="italics"/>QVÆSTIO XXIIII.<emph.end type="italics"/></s>

Line 8092

<s><expan abbr="dict&utilde;">dictum</expan> e&longs;t: <lb/>eandem igitur, vt proximè dixi habebunt etiam proportionem illæ lineæ, <lb/>quæ à &longs;ingulis &longs;eor&longs;um orbibus reuolutis de&longs;ignabuntur. </s>

<s>Quod præterea &longs;e<lb/>cundum æqualem conuoluuntur, quando circa idem po&longs;iti fuerint centrum, <lb/>manife&longs;tum e&longs;t, ita tamen, vt aliquando ambæ æquales &longs;int ei, &longs;ecundum <lb/>quam &longs;olus maior conuolueretur; aliquando verò &longs;ecundum quam minor. <lb/><figure id="fig104"/><lb/>&longs;it enim circulus maior quidem vbi <lb/>D F C, minor verò vbi E G B, <expan abbr="vtriq;">vtrique</expan> <lb/>autem centrum A, linea, &longs;ecundum <lb/>quam quadrans F C, maioris per &longs;e <lb/>rotaretur, &longs;it F L. linea verò, &longs;ecun<lb/>dum quam <expan abbr="quadrãs">quadrans</expan> G B, minoris &longs;e<lb/>iuncti à maiori, volutaretur &longs;it G K, <lb/>quæ æqualis e&longs;t dicto quadranti G B, <lb/>&longs;icut etiam F I, æqualis e&longs;t quadran<lb/>ti F C. &longs;i quis igitur impellat mino<lb/>rem orbem mouens &longs;imul commune <lb/>centrum A, cui maior e&longs;t circumpo<lb/>&longs;itus, donec diameter A B, perpendicularis &longs;it lineæ G K, in puncto K. tunc <lb/>pariter diameter maioris A C, erit perpendicularis lineæ F L, in puncto L. <lb/></s>

<s>Quod præterea &longs;e<lb/>cundum æqualem conuoluuntur, quando circa idem po&longs;iti fuerint centrum, <lb/>manife&longs;tum e&longs;t, ita tamen, vt aliquando ambæ æquales &longs;int ei, &longs;ecundum <lb/>quam &longs;olus maior conuolueretur; aliquando verò &longs;ecundum quam minor. <lb/><figure place="text" xlink:href="figures-la/009.01.188.1.tif"/><lb/>&longs;it enim circulus maior quidem vbi <lb/>D F C, minor verò vbi E G B, <expan abbr="vtriq;">vtrique</expan> <lb/>autem centrum A, linea, &longs;ecundum <lb/>quam quadrans F C, maioris per &longs;e <lb/>rotaretur, &longs;it F L. linea verò, &longs;ecun<lb/>dum quam <expan abbr="quadrãs">quadrans</expan> G B, minoris &longs;e<lb/>iuncti à maiori, volutaretur &longs;it G K, <lb/>quæ æqualis e&longs;t dicto quadranti G B, <lb/>&longs;icut etiam F I, æqualis e&longs;t quadran<lb/>ti F C. &longs;i quis igitur impellat mino<lb/>rem orbem mouens &longs;imul commune <lb/>centrum A, cui maior e&longs;t circumpo<lb/>&longs;itus, donec diameter A B, perpendicularis &longs;it lineæ G K, in puncto K. tunc <lb/>pariter diameter maioris A C, erit perpendicularis lineæ F L, in puncto L. <lb/></s>

<s>G K, autem, & F L, nece&longs;&longs;ariò erunt æquales per 34. primi, æquales igitur <lb/>lineas hoc modo peragrarunt inæquales circunferentiæ, &longs;iue quadrantes <lb/>G B, F C. &longs;i autem quadrantes hoc præ&longs;tant, manife&longs;tum e&longs;t, quod & toti <lb/>ambitus idem efficiunt, quare quando tota periphæria G B E G, fuerit re<lb/>uoluta etiam tota F C D F, &longs;uum orbem <expan abbr="complet&utilde;">completum</expan> habebit. </s>

Line 8148

<s>Ad &longs;ecundum verò dicendum extendi illos funes non per diametrum, &longs;ed <lb/>ex oppo&longs;ito, quia hoc modo ligna ip&longs;ius lecti minus di&longs;trahuntur: facilè <lb/>enim ex natura &longs;ua ligna hæc ab inuicem &longs;ecundum longum &longs;eparantur; ar<lb/>ctius autem ductis funibus per tran&longs;uer&longs;um, quàm per diametrum inuicem <lb/>con&longs;tringuntur: præterea, quia &longs;ic etiam funes minus laborant, cum &longs;int eo<lb/>rum ductus breuiores; & quia debent &longs;u&longs;tinere onus ftragulorum, <expan abbr="atq;">atque</expan> cul<lb/>cìtrarum, &longs;ic certè ex hoc onere minus laborabunt &longs;i tran&longs;uer&longs;im, quàm &longs;i <lb/>diametraliter &longs;ubtendantur.</s>

<s>Tertia demum ratio e&longs;t, quia hac ratione minus re&longs;tium ab&longs;umitur, quæ <lb/><figure id="fig105"/><lb/>vt benè intelligatur, de&longs;eriba<lb/>tur lectuli figura A F G K, & <lb/>bifariam diuidatur latus F G, <lb/>in B. & quia tota F G, dupla <lb/>e&longs;t ip&longs;ius A F, erit dimidium <lb/>F B, æquale ip&longs;i A F. & propte<lb/>rea tot erunt foramina, quibus <lb/>funes immittuntur in F B, quot <lb/>in A F. <expan abbr="ext&etilde;dunt">extendunt</expan> autem funem <lb/>hoc modo incipiunt ab A, & <lb/>ducunt ad B, po&longs;tea per C, re<pb pagenum="191"/>nertuntur ad D; hinc flectunt per H, v&longs;que ad E, & per G, angulum iterum <lb/>de&longs;cendunt ad M, à quo recta tendunt in F, hinc per 2. deducunt ad 3. à quo <lb/>foramine, per foramen 4. reflexum faciunt ad 5. à quo iterum per B, de&longs;cen<lb/>dunt ad angulum K, <expan abbr="ibi&qacute;">ibique</expan>; alterum funis extremum de&longs;init: <expan abbr="hoc&qacute;">hocque</expan>; modo duo <lb/>anguli A, & K, re&longs;tis habent capita, & re&longs;tes exten&longs;æ &longs;unt non diametrali<lb/>ter, &longs;ed tran&longs;uer&longs;im.</s>

<s>Tertia demum ratio e&longs;t, quia hac ratione minus re&longs;tium ab&longs;umitur, quæ <lb/><figure place="text" xlink:href="figures-la/009.01.190.1.tif"/><lb/>vt benè intelligatur, de&longs;eriba<lb/>tur lectuli figura A F G K, & <lb/>bifariam diuidatur latus F G, <lb/>in B. & quia tota F G, dupla <lb/>e&longs;t ip&longs;ius A F, erit dimidium <lb/>F B, æquale ip&longs;i A F. & propte<lb/>rea tot erunt foramina, quibus <lb/>funes immittuntur in F B, quot <lb/>in A F. <expan abbr="ext&etilde;dunt">extendunt</expan> autem funem <lb/>hoc modo incipiunt ab A, & <lb/>ducunt ad B, po&longs;tea per C, re<pb pagenum="191"/>nertuntur ad D; hinc flectunt per H, v&longs;que ad E, & per G, angulum iterum <lb/>de&longs;cendunt ad M, à quo recta tendunt in F, hinc per 2. deducunt ad 3. à quo <lb/>foramine, per foramen 4. reflexum faciunt ad 5. à quo iterum per B, de&longs;cen<lb/>dunt ad angulum K, <expan abbr="ibi&qacute;">ibique</expan>; alterum funis extremum de&longs;init: <expan abbr="hoc&qacute;">hocque</expan>; modo duo <lb/>anguli A, & K, re&longs;tis habent capita, & re&longs;tes exten&longs;æ &longs;unt non diametrali<lb/>ter, &longs;ed tran&longs;uer&longs;im.</s>

<s>Notandum autem, quod re&longs;tes æquales &longs;unt cum &longs;uis curuaturis. </s>

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<s>cum igitur in toto lecto &longs;int huiu&longs;modi <lb/>octo re&longs;tes, erit omnium &longs;umma pedum 34. ferè. </s>

<s>&longs;i autem &longs;eeundum diame<lb/>trum extendantur re&longs;tes, vti factum e&longs;t in lectulo A B C D, neutiquam re<lb/>&longs;tes omnes &longs;imul &longs;uperiori quantitati adæquabuntur, &longs;ed illam longè &longs;upe<lb/><figure id="fig106"/><lb/>rabunt. </s>

<s>Sit igitur lectus A B<lb/>C D, in quo diametraliter du<lb/>ctæ &longs;int re&longs;tes B D, E H, & re<lb/>liquæ, vt in figura. </s>

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quamuis tex. ip&longs;ius propter nimiam tam in græ<lb/>cis, quàm in latinis codicibus corruptionem, totus re&longs;titui nequiuerit.</s>

<s><emph type="italics"/>QVÆSTIO XXVI.<emph.end type="italics"/></s>

Line 8198

<s>pondus enim totius ligni vergit ferè ad alteram ligni <lb/>medietatem, quæ ab hypomoclio productior cuadit, quapropter ad onus <lb/>i&longs;tud æquilibrandum, opus e&longs;t alia potentia in altero extremo. </s>

<s>&longs;it lignum <lb/><figure id="fig107"/><lb/>A B, &longs;u&longs;pen&longs;um ex medio C. <lb/>hoc modo lignum ponderi<lb/>bus libratum &longs;uis manet in <lb/>æquilibrio, pote&longs;tque à &longs;ola <lb/>potentia illud eleuante etiam deferri: quia A, & B, extrema &longs;e mutuò &longs;u&longs;ti<lb/><figure id="fig108"/><lb/>nent. </s>

<s>&longs;it lignum <lb/><figure place="text" xlink:href="figures-la/009.01.192.1.tif"/><lb/>A B, &longs;u&longs;pen&longs;um ex medio C. <lb/>hoc modo lignum ponderi<lb/>bus libratum &longs;uis manet in <lb/>æquilibrio, pote&longs;tque à &longs;ola <lb/>potentia illud eleuante etiam deferri: quia A, & B, extrema &longs;e mutuò &longs;u&longs;ti<lb/><figure place="text" xlink:href="figures-la/009.01.192.2.tif"/><lb/>nent. </s>

<s>quod &longs;i non ex medio eleuaretur, <lb/>&longs;ed ab extremo, vt in &longs;ecunda figura, <lb/>eleuans potentia ex C, æqualis oportet, <lb/>vt &longs;it præcedenti; &longs;ed præterea opus e&longs;t <lb/>alia vi, quæ in B, æquiponderet alteri <lb/>extremo A, quod magis grauitat, quo ab C, longius fuerit; & hoc modo in <lb/>æquilibrio con&longs;titutum, & con&longs;eruatum poterit non &longs;olum eleuari, &longs;ed <lb/>etiam circumferri.</s>

Line 8232

<s>E&longs;t autem tolleno <expan abbr="in&longs;trument&utilde;">in&longs;trumentum</expan> hauriendæ è puteo aquæ <lb/>idoneum, quo ru&longs;tici pa&longs;&longs;im vtuntur: <expan abbr="id&qacute;">idque</expan>; iuxta puteos &longs;tabile, ac firmum <lb/>con&longs;truunt, quale à figura &longs;equenti refertur. </s>

<s>vbi puteus F, tolleno con&longs;tat <lb/><figure id="fig109"/><lb/>erecto tigno D C, & tran&longs;<lb/>uer&longs;a ha&longs;ta A C B, vnà cum <lb/>fune B E, & hydria E. ap<lb/>ponitur præterea onus &longs;a<lb/>tis graue ad <expan abbr="part&etilde;">partem</expan> A, quale <lb/>e&longs;t G. ha&longs;ta porrò A B, ve<lb/>luti vectis circa <expan abbr="p&utilde;ctum">punctum</expan> C, <lb/>tanquam hypomoclion, <lb/><expan abbr="&longs;us&qacute;">&longs;usque</expan>; <expan abbr="de&qacute;">deque</expan>; agitur, à poten<lb/>tia funem B E, trahente. <lb/></s>

<s>vbi puteus F, tolleno con&longs;tat <lb/><figure place="text" xlink:href="figures-la/009.01.193.1.tif"/><lb/>erecto tigno D C, & tran&longs;<lb/>uer&longs;a ha&longs;ta A C B, vnà cum <lb/>fune B E, & hydria E. ap<lb/>ponitur præterea onus &longs;a<lb/>tis graue ad <expan abbr="part&etilde;">partem</expan> A, quale <lb/>e&longs;t G. ha&longs;ta porrò A B, ve<lb/>luti vectis circa <expan abbr="p&utilde;ctum">punctum</expan> C, <lb/>tanquam hypomoclion, <lb/><expan abbr="&longs;us&qacute;">&longs;usque</expan>; <expan abbr="de&qacute;">deque</expan>; agitur, à poten<lb/>tia funem B E, trahente. <lb/></s>

<s>&longs;ed iam textus exponatur.</s>

Line 8268

<s>Cvm &longs;edemus, præcipuè &longs;i commodè &longs;edeamus, &longs;olemus duos angu<lb/>los rectos facere, vnum quidem, quem facit thorax cum femore; <lb/>alterum quem facit femur cum crure, vt in figura thorax &longs;it A B, <lb/><figure id="fig110"/><lb/>femur B C, crus C D, anguli duo recti &longs;unt B, <lb/>& C. </s>

<s>Cvm &longs;edemus, præcipuè &longs;i commodè &longs;edeamus, &longs;olemus duos angu<lb/>los rectos facere, vnum quidem, quem facit thorax cum femore; <lb/>alterum quem facit femur cum crure, vt in figura thorax &longs;it A B, <lb/><figure place="text" xlink:href="figures-la/009.01.194.1.tif"/><lb/>femur B C, crus C D, anguli duo recti &longs;unt B, <lb/>& C. </s>

<s>Quærit igitur, cur quando &longs;urgere volumus angu<lb/>los ho&longs;ce rectos in acutos commutamus, nam crus re<lb/>trahimus &longs;ub femur ad acutum angulum, v. g. ad po&longs;itio<lb/>nem C F. <expan abbr="fit&qacute;">fitque</expan>; acutus angulus B C F. &longs;imiliter thoracem <lb/>femori aptamus ad acutum angulum E B C, alioquin &longs;ur<lb/>gere non valemus? </s>

Line 8304

<s>&longs;ed cau&longs;a e&longs;t, quia in <lb/>vortice aqua ip &longs;a &longs;piratim circumcurrens tandem in centrum, vbi demer<lb/>gitur de&longs;cendit; nece&longs;&longs;e igitur e&longs;t, vt etiam ea, quæ in ip&longs;a &longs;unt, &longs;imul cum <lb/>illa ad centrum per plures conuolutiones deducantur. </s>

<s>Cæterum &longs;i quis ve<lb/>lit Mechanicam facultatem &longs;eriò aggredi, nequaquam paucis his ab Ari&longs;t. <lb/>traditis, <expan abbr="eis&qacute;">eisque</expan>; leui brachio pertractatis, contentus &longs;it: verùm Archimedem <lb/>de Aquæponderantibus, Commandinum, ac Lucam Valerium de centro <lb/>grauitatis &longs;olidorum, ac tandem Guidi Vbaldi Mechanica adeat, vbi hu<lb/>ius &longs;cientiæ admiranda plurima, <expan abbr="ea&qacute;">eaque</expan>; firmi&longs;&longs;imè demon&longs;trata reperiet.</s><figure/>

<s><emph type="italics"/>IN LIBELLVM DE MVNDO<emph.end type="italics"/></s>

Line 8348

hoc loco, deinde Strabo, Pomponius Mela, Plinius, Solinus, & alij <lb/>plures idem &longs;en&longs;erunt. </s>

<s>Lunam &longs;cilicet eam habere vim in mare, vt pars il<lb/>la, quæ Lunæ &longs;irbiacet, &longs;iue quam Luna radijs ferit, æftuet, & turgeat; non <lb/>aliter pars maris huic antipoda, & auer&longs;a, quamuis tota terræ moles inter <lb/><figure id="fig111"/><lb/>ip&longs;am, & Lunam interpona<lb/>tur, æ&longs;tuat, fluxumque, ac re<lb/>fluxum quamuis priori mi<lb/>norem, efficit. </s>

<s>Lunam &longs;cilicet eam habere vim in mare, vt pars il<lb/>la, quæ Lunæ &longs;irbiacet, &longs;iue quam Luna radijs ferit, æftuet, & turgeat; non <lb/>aliter pars maris huic antipoda, & auer&longs;a, quamuis tota terræ moles inter <lb/><figure place="text" xlink:href="figures-la/009.01.196.1.tif"/><lb/>ip&longs;am, & Lunam interpona<lb/>tur, æ&longs;tuat, fluxumque, ac re<lb/>fluxum quamuis priori mi<lb/>norem, efficit. </s>

<s>quæ omnia <lb/>melius in figura cernentur; <lb/>vbi infra Lunam vides tumo<lb/>rem A, ex quo fluxus deriua<lb/>tur. </s>

Line 8434

<s>qui quam<lb/>uis &longs;int minus quàm perpendiculares efficaces, obtinent tamen non modi<lb/>cam vim. </s>

<s>Ex &longs;tellis igitur circa A, & C, reflecti pote&longs;t ex quarto fundamen<lb/>to lumen <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> luminaris ad D, quod &longs;atis e&longs;t efficax, cùm ferè perpendi<pb pagenum="199"/><figure id="fig112"/><lb/>culariter terræ D, incidat. </s>

<s>Ex &longs;tellis igitur circa A, & C, reflecti pote&longs;t ex quarto fundamen<lb/>to lumen <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> luminaris ad D, quod &longs;atis e&longs;t efficax, cùm ferè perpendi<pb pagenum="199"/><figure place="text" xlink:href="figures-la/009.01.199.1.tif"/><lb/>culariter terræ D, incidat. </s>

<s>quamuis autem ex &longs;tellis F, E, lumen aliquod ad <lb/>D, tran&longs;mittatur, tamen cum obliquè admodum illi accidat, nihil penè ef<lb/>ficere valet. </s>

Line 8584

<s>Secundus locus <emph type="italics"/>(Idem etiam contingit in figuris planis, quæ à lineis rationa<lb/>libus procreantur: nam omnes huiu&longs;modi figuræ erunt etiam inuicem commen&longs;ura<lb/>biles, quare <expan abbr="&emacr;adem">enadem</expan> ratione, qua in lineis proximè v&longs;i &longs;umus, &longs;equetur earum com<lb/>munem men&longs;uram e&longs;&longs;e pariter indiuiduam.<emph.end type="italics"/></s>

<s>Sciendum e&longs;t omnes lineas <expan abbr="cõmen&longs;urabiles">commen&longs;urabiles</expan> longitudine, e&longs;&longs;e etiam com<lb/>men&longs;urabiles (vt aiunt Geometræ) potentia, ide&longs;t &longs;ecundum quadrata ea<pb pagenum="203"/>rum, &longs;iue dicas quadrata <expan abbr="quoq;">quoque</expan> earum e&longs;&longs;e commen&longs;urabilia, v. g. linea dua<lb/><figure id="fig113"/><lb/>rum vnciarum, & linea trium vnciarum &longs;unt <lb/>commen&longs;urabiles longitudine, & potentia, <lb/>quia potentia lineæ duarum vnciarum, &longs;iue <lb/><expan abbr="quadrat&utilde;">quadratum</expan>, e&longs;t quatuor vnciarum &longs;uperficia<lb/>lium: & quadratum lineæ trium vnciarum, <lb/>e&longs;t nouem vnciarum quadratarum, vt patet <lb/>in figuris, quorum quadratorum communis <lb/>men&longs;ura e&longs;t vncia vna quadrata. </s>

<s>Sciendum e&longs;t omnes lineas <expan abbr="cõmen&longs;urabiles">commen&longs;urabiles</expan> longitudine, e&longs;&longs;e etiam com<lb/>men&longs;urabiles (vt aiunt Geometræ) potentia, ide&longs;t &longs;ecundum quadrata ea<pb pagenum="203"/>rum, &longs;iue dicas quadrata <expan abbr="quoq;">quoque</expan> earum e&longs;&longs;e commen&longs;urabilia, v. g. linea dua<lb/><figure place="text" xlink:href="figures-la/009.01.203.1.tif"/><lb/>rum vnciarum, & linea trium vnciarum &longs;unt <lb/>commen&longs;urabiles longitudine, & potentia, <lb/>quia potentia lineæ duarum vnciarum, &longs;iue <lb/><expan abbr="quadrat&utilde;">quadratum</expan>, e&longs;t quatuor vnciarum &longs;uperficia<lb/>lium: & quadratum lineæ trium vnciarum, <lb/>e&longs;t nouem vnciarum quadratarum, vt patet <lb/>in figuris, quorum quadratorum communis <lb/>men&longs;ura e&longs;t vncia vna quadrata. </s>

Line 8604

&longs;unt explicanda: <lb/>vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini<lb/>bus, quæ Apotomæ.</s>

<s>Propo&longs;ita igitur linea quapiam, v. g. trium palmorum qualis e&longs;t linea A, <lb/>po&longs;&longs;unt inueniri quamplurimæ lineæ, quarum aliæ &longs;int illi longitudine com<lb/><figure id="fig114"/><lb/>men&longs;urabiles, &longs;iue quæ cum expo&longs;ita A, ha<lb/>beant communem men&longs;uram. </s>

<s>Propo&longs;ita igitur linea quapiam, v. g. trium palmorum qualis e&longs;t linea A, <lb/>po&longs;&longs;unt inueniri quamplurimæ lineæ, quarum aliæ &longs;int illi longitudine com<lb/><figure place="text" xlink:href="figures-la/009.01.203.2.tif"/><lb/>men&longs;urabiles, &longs;iue quæ cum expo&longs;ita A, ha<lb/>beant communem men&longs;uram. </s>

Line 8614

<s>Cæterum lineam primò expo&longs;itam, vt e&longs;t in præ<lb/>&longs;entia A, quod e&longs;&longs;et notæ quantitatis, Græci appella<lb/>runt <foreign lang="greek">Ph/thg,</foreign> ide&longs;t rationalem, quemadmodum Latini <lb/>eam appellant.</s>

Line 8624

<s>lineæ dicuntur <expan abbr="quadrat&utilde;">quadratum</expan> illius.</s>

<s>Quæ igitur rationali propo&longs;itæ &longs;unt commen&longs;urabiles aliquo modo, &longs;iue <lb/>longitudine, & potentia (<expan abbr="quæcunq;">quæcunque</expan> enim commen&longs;urabilis e&longs;t longitudine, <lb/>e&longs;t etiam potentia) &longs;iue potentia &longs;olùm, rationales ip&longs;æ quoque dicuntur. <pb pagenum="204"/><figure id="fig115"/><lb/>Aliæ verò (quarum permultæ in decimo reperiun<lb/>tur) quæ nec longitudine, nec potentia illi &longs;unt <lb/>commen&longs;urabiles, irrationales appellantur, qua<lb/>lis e&longs;&longs;et media proportionalis E F, inter duas A, <lb/>& C D, in præ&longs;enti figura ex 11. 10.</s>

<s>Quæ igitur rationali propo&longs;itæ &longs;unt commen&longs;urabiles aliquo modo, &longs;iue <lb/>longitudine, & potentia (<expan abbr="quæcunq;">quæcunque</expan> enim commen&longs;urabilis e&longs;t longitudine, <lb/>e&longs;t etiam potentia) &longs;iue potentia &longs;olùm, rationales ip&longs;æ quoque dicuntur. <pb pagenum="204"/><figure place="text" xlink:href="figures-la/009.01.204.1.tif"/><lb/>Aliæ verò (quarum permultæ in decimo reperiun<lb/>tur) quæ nec longitudine, nec potentia illi &longs;unt <lb/>commen&longs;urabiles, irrationales appellantur, qua<lb/>lis e&longs;&longs;et media proportionalis E F, inter duas A, <lb/>& C D, in præ&longs;enti figura ex 11. 10.</s>

<s>Sciendum præterea ex 37. 10. & &longs;equentibus, <lb/>quod ex duabus lineis rationalibus re&longs;pectu rationalis expo&longs;itæ. </s>

<s>v. g. A, com<lb/>men&longs;urabilibus inuicem tantum potentia, componitur linea, quæ cum ea<lb/><figure id="fig116"/><lb/>dem expo&longs;ita e&longs;t irrationalis, <expan abbr="vocatur&qacute;">vocaturque</expan>; ex <lb/>duobus nominibus, &longs;iue Binomium, vt &longs;i ex <lb/>latere A, & diametro C D, componatur li<lb/>nea A C D, erit irrationalis cum rationali <lb/>A, <expan abbr="dicetur&qacute;">diceturque</expan>; binomium. </s>

<s>v. g. A, com<lb/>men&longs;urabilibus inuicem tantum potentia, componitur linea, quæ cum ea<lb/><figure place="text" xlink:href="figures-la/009.01.204.2.tif"/><lb/>dem expo&longs;ita e&longs;t irrationalis, <expan abbr="vocatur&qacute;">vocaturque</expan>; ex <lb/>duobus nominibus, &longs;iue Binomium, vt &longs;i ex <lb/>latere A, & diametro C D, componatur li<lb/>nea A C D, erit irrationalis cum rationali <lb/>A, <expan abbr="dicetur&qacute;">diceturque</expan>; binomium. </s>

Line 8672

<s>Refellit hoc loco &longs;uperiores rationes in tribus locis præmi&longs;&longs;is allatas, <lb/>quibus nonnulli probabant quantitatem ex indiuiduis con&longs;tare, & proinde <lb/>concedenda e&longs;&longs;e quædam Quanta, omninò atoma; &longs;ic igitur inquit. </s>

<s>Quod <lb/>verò de commen&longs;urabilibus lineis dicunt, omues videlicet vnica quadam, <lb/><expan abbr="eadem&qacute;">eademque</expan>; determinata men&longs;ura men&longs;urari oportere, fal&longs;um omninò e&longs;t, & <lb/>contra mathematicorum dogmata, non enim Geometræ hoc a&longs;&longs;erunt, cùm <lb/>ip&longs;orum demon&longs;trationibus a duer&longs;etur; &longs;ed <expan abbr="tant&utilde;">tantum</expan> dicunt omnes lineas, quæ <lb/>ad inuicem &longs;unt commen&longs;urabiles, commen&longs;urari, vna <expan abbr="eadem&qacute;">eademque</expan>; men&longs;ura, <lb/><figure id="fig117"/><lb/>&longs;ed non tamen vnica, ide&longs;t non vnica, ac determi<lb/>nata. </s>

<s>Quod <lb/>verò de commen&longs;urabilibus lineis dicunt, omues videlicet vnica quadam, <lb/><expan abbr="eadem&qacute;">eademque</expan>; determinata men&longs;ura men&longs;urari oportere, fal&longs;um omninò e&longs;t, & <lb/>contra mathematicorum dogmata, non enim Geometræ hoc a&longs;&longs;erunt, cùm <lb/>ip&longs;orum demon&longs;trationibus a duer&longs;etur; &longs;ed <expan abbr="tant&utilde;">tantum</expan> dicunt omnes lineas, quæ <lb/>ad inuicem &longs;unt commen&longs;urabiles, commen&longs;urari, vna <expan abbr="eadem&qacute;">eademque</expan>; men&longs;ura, <lb/><figure place="text" xlink:href="figures-la/009.01.205.1.tif"/><lb/>&longs;ed non tamen vnica, ide&longs;t non vnica, ac determi<lb/>nata. </s>

<s>po&longs;&longs;unt enim e&longs;&longs;e plures <expan abbr="eædem&qacute;">eædemque</expan>; men&longs;uræ <lb/>communes plurium quantitatum commen&longs;ura<lb/>bilium, vt præ&longs;entium trium linearum 4. 6. 8. <lb/>communis <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> e&longs;t linea 2. binarius enim tres <lb/>numeros 4.6. & 8. men&longs;urat. </s>

Line 8682

<s>Quintus locus <emph type="italics"/>(Ob rectæ verò lineæ motum in &longs;emicirculum, quam nece&longs;&longs;e e&longs;t <lb/>in rectum ita diuidere, vt infinitæ circunferentiæ, & interualla totidem inuenian<lb/><gap/>ur)<emph.end type="italics"/> Interpres latinus &longs;ic vertit <emph type="italics"/>(Ob rectæ verò lineæ motum in &longs;emicirculum <lb/>diuiduas non credere, &c.)<emph.end type="italics"/> vbi verba illa <emph type="italics"/>(Diuiduas non credere)<emph.end type="italics"/> pro arbittio, <lb/>ac &longs;ine ratione, imò contra rationem addidit: tum quia in Græco textu non <lb/>extant, tum quia &longs;en&longs;us totius &longs;ententiæ is e&longs;t, vt potius debui&longs;&longs;et affirmati<lb/>uè dicere <emph type="italics"/>(Diuiduas credere)<emph.end type="italics"/> nam Ari&longs;toteles videtur &longs;ic <expan abbr="argum&etilde;tari">argumentari</expan>, quan<lb/><figure id="fig118"/><lb/>do recta linea A B, vt in appo&longs;ita figura mo<lb/>uetur intrando in &longs;emicirculum C A D B, ita <lb/>vt primò &longs;it in &longs;itu A B, &longs;ecundò in E F, tertiò <lb/>in G H, & &longs;imiliter in alijs omnibus &longs;emicir<lb/>culi locis, nece&longs;&longs;ariò accidit, vt inf<gap/>itæ p<gap/>ri<lb/>ph&ecedil;riæ, quales <expan abbr="s&utilde;t">sunt</expan> A B, E A B F, G E A B F H, <lb/>cadant inter infinitas partes lineæ ingredien<lb/>tis, vt &longs;unt A B, E F, G H, <expan abbr="atq;">atque</expan> tom tota recta <lb/>ingrediens, quàm totus &longs;emicirculus, diuidatur in partes infinitas, ita vt <lb/>nulla pars lineæ rectæ, <expan abbr="neq;">neque</expan> vlla &longs;emicirculi &longs;uper&longs;it, quæ &longs;e &longs;e mutuò non <lb/>diuidantur, ergò nihil tam in linea, quàm in &longs;emicirculo remanet, quod non <lb/>&longs;ecetur: tota igitur linea recta, & periphæria illa diuidua e&longs;t, quam ob rem <lb/>nullo modo con&longs;tare pote&longs;t ex indiuiduis, ex quibus manife&longs;tum e&longs;t perpe<lb/>ram additamentum illud factum e&longs;&longs;e, & &longs;imul ratio, & textus Ari&longs;t. <gap/>adem <lb/>opera patefacta &longs;unt.</s><pb pagenum="206"/>

<s>Quintus locus <emph type="italics"/>(Ob rectæ verò lineæ motum in &longs;emicirculum, quam nece&longs;&longs;e e&longs;t <lb/>in rectum ita diuidere, vt infinitæ circunferentiæ, & interualla totidem inuenian<lb/><gap/>ur)<emph.end type="italics"/> Interpres latinus &longs;ic vertit <emph type="italics"/>(Ob rectæ verò lineæ motum in &longs;emicirculum <lb/>diuiduas non credere, &c.)<emph.end type="italics"/> vbi verba illa <emph type="italics"/>(Diuiduas non credere)<emph.end type="italics"/> pro arbittio, <lb/>ac &longs;ine ratione, imò contra rationem addidit: tum quia in Græco textu non <lb/>extant, tum quia &longs;en&longs;us totius &longs;ententiæ is e&longs;t, vt potius debui&longs;&longs;et affirmati<lb/>uè dicere <emph type="italics"/>(Diuiduas credere)<emph.end type="italics"/> nam Ari&longs;toteles videtur &longs;ic <expan abbr="argum&etilde;tari">argumentari</expan>, quan<lb/><figure place="text" xlink:href="figures-la/009.01.205.2.tif"/><lb/>do recta linea A B, vt in appo&longs;ita figura mo<lb/>uetur intrando in &longs;emicirculum C A D B, ita <lb/>vt primò &longs;it in &longs;itu A B, &longs;ecundò in E F, tertiò <lb/>in G H, & &longs;imiliter in alijs omnibus &longs;emicir<lb/>culi locis, nece&longs;&longs;ariò accidit, vt inf<gap/>itæ p<gap/>ri<lb/>ph&ecedil;riæ, quales <expan abbr="s&utilde;t">sunt</expan> A B, E A B F, G E A B F H, <lb/>cadant inter infinitas partes lineæ ingredien<lb/>tis, vt &longs;unt A B, E F, G H, <expan abbr="atq;">atque</expan> tom tota recta <lb/>ingrediens, quàm totus &longs;emicirculus, diuidatur in partes infinitas, ita vt <lb/>nulla pars lineæ rectæ, <expan abbr="neq;">neque</expan> vlla &longs;emicirculi &longs;uper&longs;it, quæ &longs;e &longs;e mutuò non <lb/>diuidantur, ergò nihil tam in linea, quàm in &longs;emicirculo remanet, quod non <lb/>&longs;ecetur: tota igitur linea recta, & periphæria illa diuidua e&longs;t, quam ob rem <lb/>nullo modo con&longs;tare pote&longs;t ex indiuiduis, ex quibus manife&longs;tum e&longs;t perpe<lb/>ram additamentum illud factum e&longs;&longs;e, & &longs;imul ratio, & textus Ari&longs;t. <gap/>adem <lb/>opera patefacta &longs;unt.</s><pb pagenum="206"/>

Line 8692

<s>Atque bæc Mathematicorum &longs;cita, multò magis ab omnibus conce&longs;&longs;a &longs;unt, quàm <lb/>illorum dicta.<emph.end type="italics"/></s>

<s>Hæc e&longs;t alia ratio, qua probat totam circuli periphæriam e&longs;&longs;e diuiduam. <lb/><figure id="fig119"/><lb/>&longs;int enim duo circuli æquales primum in eo<lb/>dem loco, <expan abbr="vocentur&qacute;">vocenturque</expan>; A, & B, deinde circu<lb/>lus B, moueatur, & di&longs;cedat à circulo A, ma<lb/>nente; &longs;tatim <expan abbr="namq;">namque</expan> pars egre&longs;&longs;a E F G, erit <lb/>maior &longs;emicirculo, & &longs;emper fiet maior, ac <lb/>maior. </s>

<s>Hæc e&longs;t alia ratio, qua probat totam circuli periphæriam e&longs;&longs;e diuiduam. <lb/><figure place="text" xlink:href="figures-la/009.01.206.1.tif"/><lb/>&longs;int enim duo circuli æquales primum in eo<lb/>dem loco, <expan abbr="vocentur&qacute;">vocenturque</expan>; A, & B, deinde circu<lb/>lus B, moueatur, & di&longs;cedat à circulo A, ma<lb/>nente; &longs;tatim <expan abbr="namq;">namque</expan> pars egre&longs;&longs;a E F G, erit <lb/>maior &longs;emicirculo, & &longs;emper fiet maior, ac <lb/>maior. </s>

<s><expan abbr="atq;">atque</expan> in tali motu omnes partes egre<lb/>dientis circuli &longs;ecantur ab omnibus partibus <lb/>circuli manentis. </s>

Line 8746

<s>Nonus Iocus, cuius latinam interpretationem, cum admodum e&longs;&longs;et de<lb/>prauata ex græco textu, in hunc modum correxi <emph type="italics"/>(Præterea cùm circa maio<lb/>rem lstitudinem facit applicata, æquale ei, quod ab indiuidua, & pedali copulatis <lb/>circa bipedalem, minorem faciet latitudinem, quàm &longs;it indiuidua: erit minus, quod <lb/>circa ind<gap/>uiduam)<emph.end type="italics"/> ide&longs;t cùm minor linea applicata cum maiore, latitudinem <lb/><figure id="fig120"/><lb/>faciat. </s>

<s>v. g. linea minor A B, applicata cum ma<lb/>iori B C, vt in figura, ita vt contineant figuram <lb/>A B C D. </s>

Line 8756

<s>Minor A B, facit latitudinem figuræ, <lb/>maior verò B C, facit longitudinem. </s>

<s>Iam cum <lb/>aduer&longs;arij velint extare huiu&longs;modi lineas ato<lb/>mas, con&longs;tituatur figura &longs;ub vna ex illis, quæ &longs;it v. g. A B, & altera maiori, <lb/>quæ &longs;it pedalis, v. g. B C, vt in præcedenti figura, &longs;umatur deinde linea bi<lb/><figure id="fig121"/><lb/>pedalis E F, cui per 45. primi ap<lb/>plicetur &longs;patium E F G H, æquale <lb/>&longs;patio &longs;uperiori A B C D, nece&longs;&longs;a<lb/>riò latitudo E H, huius &longs;ecundæ fi<lb/>guræ minor erit quàm latitudo il<lb/>lius, hoc e&longs;t minor, quàm &longs;it indinidua A B, quod e&longs;t ab&longs;urdum. </s>

<s>Iam cum <lb/>aduer&longs;arij velint extare huiu&longs;modi lineas ato<lb/>mas, con&longs;tituatur figura &longs;ub vna ex illis, quæ &longs;it v. g. A B, & altera maiori, <lb/>quæ &longs;it pedalis, v. g. B C, vt in præcedenti figura, &longs;umatur deinde linea bi<lb/><figure place="text" xlink:href="figures-la/009.01.207.2.tif"/><lb/>pedalis E F, cui per 45. primi ap<lb/>plicetur &longs;patium E F G H, æquale <lb/>&longs;patio &longs;uperiori A B C D, nece&longs;&longs;a<lb/>riò latitudo E H, huius &longs;ecundæ fi<lb/>guræ minor erit quàm latitudo il<lb/>lius, hoc e&longs;t minor, quàm &longs;it indinidua A B, quod e&longs;t ab&longs;urdum. </s>

Line 8778

<s>Ex 22. primi Elem.

ex tribus datis lineis, quarum quælibet duæ &longs;int, re<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui<lb/><figure id="fig122"/><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in<lb/>diuiduæ lineæ &longs;int æquales. </s>

ex tribus datis lineis, quarum quælibet duæ &longs;int, re<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui<lb/><figure place="text" xlink:href="figures-la/009.01.207.3.tif"/><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in<lb/>diuiduæ lineæ &longs;int æquales. </s>

<s>&longs;it igitur ex eis triangulum A B C, <lb/>&longs;i igitur ab angulo A, ducatur perpendicularis A D, ad ba&longs;im <lb/>B C, eam bif<gap/>riam &longs;ecabit ex &longs;cholio 26. primi, erit igitur li<lb/>nea B C, &longs;ecabilis, contra quam aduer&longs;arij opinantur.</s>

Line 8788

<s>Vndecimus locus <emph type="italics"/>(Si quadratum ex quatuor indiuiduis con&longs;t<gap/>tuatur diame<gap/><expan abbr="õ">om</expan> <lb/>protracta, & perpendiculari ducta, quadrati ce&longs;ta potentia <expan abbr="perp&etilde;dicularem">perpendicularem</expan>, diame-<emph.end type="italics"/><pb pagenum="208"/><emph type="italics"/>trum&queacute; mediam æquat: quare non erit minima. </s>

<s>neque duplum erit &longs;patium à diame<lb/>tro con&longs;urgens illius, quod ab indiuidua procreatur: nans æquali ablato, reliquum <lb/>erit minus indiutdua, nam &longs;i æqualis, diameter quadruplum de&longs;criberet, &c.)<emph.end type="italics"/><lb/><figure id="fig123"/><lb/>ide&longs;t &longs;i per 46 primi quadratum. </s>

<s>neque duplum erit &longs;patium à diame<lb/>tro con&longs;urgens illius, quod ab indiuidua procreatur: nans æquali ablato, reliquum <lb/>erit minus indiutdua, nam &longs;i æqualis, diameter quadruplum de&longs;criberet, &c.)<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.208.1.tif"/><lb/>ide&longs;t &longs;i per 46 primi quadratum. </s>

<s>v.g. A B C D, ex qua<lb/>tuor in&longs;ecabilibus componatur, cuius diametro B C, <lb/>perpendicularis A E, in&longs;i&longs;tat, erit per 47. primi qua<lb/>dratum lineæ A B, æquale quadratis <expan abbr="linear&utilde;">linearum</expan> A E, E B, <lb/>quare tam E B, quàm A E, minores erunt ip&longs;a A B; <lb/>quare ip&longs;a non erit minima cum &longs;it indiuidua, quod e&longs;t <lb/>ab&longs;urdum. </s>

Line 8864

<s>In 2. 3. & corollario eius demon&longs;tratur circuli peripheriam tangere re<lb/>ctam lineam in vnico puncto. </s>

<s>iam &longs;i linea con&longs;taret ex punctis indiuiduis <lb/>tanquam partibus, po&longs;&longs;et circulus <expan abbr="tãgere">tangere</expan> rectam lineam in duobus punctis. <lb/><figure id="fig124"/><lb/>Sit circulus, cuius centrum A, tangens lineam <lb/>rectam B C, con&longs;tantem ex punctis, quorum vnus <lb/>&longs;it in extremo D, lineæ B D, alterum verò in E, <lb/>principio lineæ E C, circulus A, tangere poterit <lb/>in F, termino communi vtriu&longs;que lineæ, hocque <lb/>modo tanget <expan abbr="vtrunq;">vtrunque</expan> punctum D, & E, quod e&longs;t <lb/>impo&longs;&longs;ibile per 2. 3. &longs;equitur igitur <expan abbr="neq;">neque</expan> illa duo puncta D, E, &longs;e mutuò tan<lb/>gere, & eadem ratione nulla alia <expan abbr="p&utilde;cta">puncta</expan> eiu&longs;dem lineæ, ex quibus manife&longs;tum <lb/>e&longs;t, impo&longs;&longs;ibile e&longs;&longs;e, lineam ex huiu&longs;modi punctis con&longs;tare po&longs;&longs;e.</s>

<s>iam &longs;i linea con&longs;taret ex punctis indiuiduis <lb/>tanquam partibus, po&longs;&longs;et circulus <expan abbr="tãgere">tangere</expan> rectam lineam in duobus punctis. <lb/><figure place="text" xlink:href="figures-la/009.01.209.1.tif"/><lb/>Sit circulus, cuius centrum A, tangens lineam <lb/>rectam B C, con&longs;tantem ex punctis, quorum vnus <lb/>&longs;it in extremo D, lineæ B D, alterum verò in E, <lb/>principio lineæ E C, circulus A, tangere poterit <lb/>in F, termino communi vtriu&longs;que lineæ, hocque <lb/>modo tanget <expan abbr="vtrunq;">vtrunque</expan> punctum D, & E, quod e&longs;t <lb/>impo&longs;&longs;ibile per 2. 3. &longs;equitur igitur <expan abbr="neq;">neque</expan> illa duo puncta D, E, &longs;e mutuò tan<lb/>gere, & eadem ratione nulla alia <expan abbr="p&utilde;cta">puncta</expan> eiu&longs;dem lineæ, ex quibus manife&longs;tum <lb/>e&longs;t, impo&longs;&longs;ibile e&longs;&longs;e, lineam ex huiu&longs;modi punctis con&longs;tare po&longs;&longs;e.</s>

<s>Reliqua huius opu&longs;culi, quamuis Mathematica alicui videri po&longs;&longs;int, <lb/>non tamen &longs;unt, non enim linearibus indigent demon&longs;trationi<lb/>bus, <expan abbr="neq;">neque</expan> ex Geometriæ principijs procedunt. </s>

<s>ad Phy&longs;i<lb/>cum igitur pertinebunt, cuius e&longs;t di&longs;putare, num <lb/>indiuidua exi&longs;tant, & quomodo in quanti<lb/>tate, <expan abbr="id&qacute;">idque</expan>; rationibus aliunde, quàm <lb/>ex Geometria deductis.</s><figure/><pb pagenum="210"/>

<s><emph type="italics"/>In Librum de Propriet. Elementorum.<emph.end type="italics"/></s>

Line 8878

<s><emph type="italics"/>In Librum de Cau&longs;is.<emph.end type="italics"/></s>

<s>Alterum de cau&longs;is libellum pariter prætermi&longs;i, cum is vocibus Arabi<lb/>cam barbariem redolentibus &longs;cateat: phra&longs;is præterca, & qu&ecedil;dam <lb/>de Deo dicta, planè indicant authorem non e&longs;&longs;e Ari&longs;tctelem; &longs;ed potius <lb/>Arabem quempiam.</s><figure/><pb pagenum="211"/>

<s>EX LIBRO NONO</s>

Line 8939

<s>factum e&longs;t autem, vt eo<lb/>rum vnum quendam arboris cuiu&longs;dam ramum attingeret, <expan abbr="ei&qacute;">eique</expan>; &longs;tatim adhæ<lb/>reret; quod illicò Araneus optimè per&longs;en&longs;it, quippe quod filum illud vi&longs;ce<lb/>ribus eius ex altero capite affigeretur, atque per filum illud, alijs ommi&longs;&longs;is, <lb/>&longs;ubitò, vti egregius funambulus accurrit, &longs;ed tamen pedibus &longs;ur&longs;um, dor &longs;o <lb/>autem deor&longs;um, non &longs;upra filum, &longs;ed infra ad ramum illum &longs;e contulit, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; <lb/>me ho&longs;tem &longs;uum fuga &longs;æpius elu&longs;it. </s>

<s>Ex qna repetita &longs;æpius ob&longs;eruatione lu<lb/>ce clarius comperi Araneum non &longs;implex filum, &longs;ed ramo&longs;um, ac multiplex <lb/>emittere, <expan abbr="atq;">atque</expan> aliquando ex &longs;ece&longs;&longs;u etiam ip&longs;o duo &longs;imul eijcere, <expan abbr="alter&utilde;">alterum</expan> quo <lb/><figure id="fig125"/><lb/>&longs;u&longs;pendatur, alterum <lb/>verò, quod &longs;orte hac, <lb/><expan abbr="atq;">atque</expan> illac volitans, ali<lb/>cui rei occurrat, <expan abbr="atq;">atque</expan> <lb/>hæreat, per quod po<lb/>&longs;tea<gap/>ip&longs;e incedens, ad <lb/>locum &longs;ibi prius inac<lb/>ce&longs;&longs;um, aditum parat. <lb/></s>

<s>Ex qna repetita &longs;æpius ob&longs;eruatione lu<lb/>ce clarius comperi Araneum non &longs;implex filum, &longs;ed ramo&longs;um, ac multiplex <lb/>emittere, <expan abbr="atq;">atque</expan> aliquando ex &longs;ece&longs;&longs;u etiam ip&longs;o duo &longs;imul eijcere, <expan abbr="alter&utilde;">alterum</expan> quo <lb/><figure place="text" xlink:href="figures-la/009.01.212.1.tif"/><lb/>&longs;u&longs;pendatur, alterum <lb/>verò, quod &longs;orte hac, <lb/><expan abbr="atq;">atque</expan> illac volitans, ali<lb/>cui rei occurrat, <expan abbr="atq;">atque</expan> <lb/>hæreat, per quod po<lb/>&longs;tea<gap/>ip&longs;e incedens, ad <lb/>locum &longs;ibi prius inac<lb/>ce&longs;&longs;um, aditum parat. <lb/></s>

<s>qua inre fures eos per<lb/>bellè imitatur, qui <lb/>&longs;chalas ex funibus con<lb/>textas, ac hamis fer<lb/>reis munitas, ad fene<lb/>&longs;tras proijciunt, vt per <lb/>cas ibi affixas con&longs;cen<lb/>dere queant. </s>

Line 8971

<s>quoniam autem æqualia &longs;unt membra, ne<lb/>ce&longs;&longs;e e&longs;t inflecti id, quod manet, aut in poplite, aut in conflexione)<emph.end type="italics"/> Vult probare <lb/>in gre&longs;&longs;u nece&longs;&longs;ariam e&longs;&longs;e aliquam flexionem membrorum. </s>

<s>verum prius <lb/>&longs;ciendum, quod lineam hypotenu&longs;am, quemadmodum etiam Athenæus lib. <lb/>10. te&longs;tatur, eam appellant geometræ, quæ in triangulo rectangulo recto <lb/>angulo &longs;ubtenditur, vnde & denominata e&longs;t hypotenu&longs;a, ide&longs;t &longs;ubten&longs;a, vt <lb/><figure id="fig126"/><lb/>in triangulo A B C, cuius angulus B, rectus &longs;it, recta <lb/>A C, angulo recto B, &longs;ubten&longs;a, hypotenu&longs;a dicitur. <lb/></s>

<s>verum prius <lb/>&longs;ciendum, quod lineam hypotenu&longs;am, quemadmodum etiam Athenæus lib. <lb/>10. te&longs;tatur, eam appellant geometræ, quæ in triangulo rectangulo recto <lb/>angulo &longs;ubtenditur, vnde & denominata e&longs;t hypotenu&longs;a, ide&longs;t &longs;ubten&longs;a, vt <lb/><figure place="text" xlink:href="figures-la/009.01.213.1.tif"/><lb/>in triangulo A B C, cuius angulus B, rectus &longs;it, recta <lb/>A C, angulo recto B, &longs;ubten&longs;a, hypotenu&longs;a dicitur. <lb/></s>

Line 8979

igitur ait, quod antequam animal ambulare in<lb/>cipiat, dum &longs;cilicet manet, habet crura, quæ manent <lb/>recta, &longs;iue perpendicularia horizonti, cum autem in<lb/>cipit progredi nece&longs;&longs;e e&longs;t <expan abbr="vtr&utilde;q;">vtrunque</expan> crus inclinari ad ho<lb/>rizontem. </s>

<s>nam primum crus in ingre&longs;lu prolatum fit <lb/>hypotenu&longs;a, quia &longs;cilicet &longs;ubtendit angulum rectum, <lb/>quem facit alterum crus adhuc quie&longs;cens, cum hori<pb pagenum="214"/>zonte; vt in &longs;uperiori triangulo, &longs;i concipiamus crura fui&longs;&longs;e duo latera A B, <lb/>A D, quæ manente animali, fui&longs;&longs;ent ambo &longs;imul in &longs;itu A B, perpendicula<lb/>ria horizonti; incipiens autem animal ambulare, proferat primo crus A D, <lb/>A D, fiet hypotenu&longs;a trianguli A B C, & quia crus hoc A D, factum hypo<lb/>tenu&longs;a æquale e&longs;t alteri manenti A B, nequit totius veræ hypotenufæ A C, <lb/>officio fungi, quæ æquiualet toti A D, & præterea interiacenti D C, vrea au<lb/><figure id="fig127"/><lb/>tem hypotenu&longs;a debet e&longs;&longs;e maior, quia opponitun <lb/>maiori angulo nimirum recto B, quam latus A B, <lb/>quod angulo acuto C, opponitur per 19. primi, & <lb/>propterea ni&longs;i alterum &longs;ub&longs;equens crus A B, incli<lb/>netur, vt in &longs;ecunda figura, non pote&longs;t hypotenu&longs;a <lb/>A D, terram attingere, <expan abbr="atq;">atque</expan> hac de cau&longs;a nece&longs;&longs;e <lb/>e&longs;t, vt initio gre&longs;&longs;us <expan abbr="vtrumq;">vtrumque</expan> crus, quod prius per<lb/>pendiculare erat, inclinetur; inclinato igitur crure <lb/>A B, antror&longs;um tunc prolatum crus A C, terram <lb/>contingit, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; factus e&longs;t primus gre&longs;&longs;us B C.</s>

<s>nam primum crus in ingre&longs;lu prolatum fit <lb/>hypotenu&longs;a, quia &longs;cilicet &longs;ubtendit angulum rectum, <lb/>quem facit alterum crus adhuc quie&longs;cens, cum hori<pb pagenum="214"/>zonte; vt in &longs;uperiori triangulo, &longs;i concipiamus crura fui&longs;&longs;e duo latera A B, <lb/>A D, quæ manente animali, fui&longs;&longs;ent ambo &longs;imul in &longs;itu A B, perpendicula<lb/>ria horizonti; incipiens autem animal ambulare, proferat primo crus A D, <lb/>A D, fiet hypotenu&longs;a trianguli A B C, & quia crus hoc A D, factum hypo<lb/>tenu&longs;a æquale e&longs;t alteri manenti A B, nequit totius veræ hypotenufæ A C, <lb/>officio fungi, quæ æquiualet toti A D, & præterea interiacenti D C, vrea au<lb/><figure place="text" xlink:href="figures-la/009.01.214.1.tif"/><lb/>tem hypotenu&longs;a debet e&longs;&longs;e maior, quia opponitun <lb/>maiori angulo nimirum recto B, quam latus A B, <lb/>quod angulo acuto C, opponitur per 19. primi, & <lb/>propterea ni&longs;i alterum &longs;ub&longs;equens crus A B, incli<lb/>netur, vt in &longs;ecunda figura, non pote&longs;t hypotenu&longs;a <lb/>A D, terram attingere, <expan abbr="atq;">atque</expan> hac de cau&longs;a nece&longs;&longs;e <lb/>e&longs;t, vt initio gre&longs;&longs;us <expan abbr="vtrumq;">vtrumque</expan> crus, quod prius per<lb/>pendiculare erat, inclinetur; inclinato igitur crure <lb/>A B, antror&longs;um tunc prolatum crus A C, terram <lb/>contingit, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; factus e&longs;t primus gre&longs;&longs;us B C.</s>

Line 9009

<s>crura enim in gre&longs;&longs;u fiunt <lb/>latera &longs;uperiora trianguli i&longs;o&longs;celis, vt in &longs;ecunda figura patuit, cuius ba&longs;is <lb/>e&longs;t pa&longs;&longs;us. </s>

<s>& tunc caput ambulantis fit inferius, quàm antequam gradere<lb/><figure id="fig128"/><lb/>tur; quia tunc ambo crura erant horizonti perpen<lb/>dicularia. </s>

<s>& tunc caput ambulantis fit inferius, quàm antequam gradere<lb/><figure place="text" xlink:href="figures-la/009.01.215.1.tif"/><lb/>tur; quia tunc ambo crura erant horizonti perpen<lb/>dicularia. </s>

<s>quando autem caput fuerit in linea <expan abbr="per-p&etilde;diculari">per<lb/>pendiculari</expan> trianguli i&longs;o&longs;celis, tunc erit inferius quàm <lb/>alibi, vt in pr&ecedil;&longs;enti figura, linea <expan abbr="perp&etilde;dicularis">perpendicularis</expan> trian<lb/>guli huius i&longs;o&longs;celis e&longs;t linea A E, quia ba&longs;i B C, per<lb/>pendicularis incidit; quando igitur caput ambulan<lb/>tis. </s>

Line 9029

<s>Cap. 1. <emph type="italics"/>(Primum quidem in animalibus: oportet enim &longs;i moueatur aliqua <lb/>particularum quie&longs;cere ahquam, & propter hoc, & flexus animalibus <lb/>in&longs;unt: tanquam enim centro vtuntur flex: bus & fit tota pars, in qua <lb/>e&longs;t flexus & vna, & duæ; & recta, & flexa, quæ permutatur potentia, <lb/>& actu, propter flexum. </s>

<s>cum autem flectitur, & mouetur, hoc quidem &longs;ignum mo-<emph.end type="italics"/><lb/><figure id="fig129"/><lb/><emph type="italics"/>etur, illud autem manet in flexibus, quemadmodum <expan abbr="vtiq;">vtique</expan> &longs;i dia<lb/>metri, quæ quidem A D, maneat, quæ cutem B, moueatur, & <lb/>fiat A C, &longs;ed hic quidem videtur, &longs;ecundum omnem modum in<lb/>diui&longs;ibile e&longs;&longs;e centrum. </s>

<s>cum autem flectitur, & mouetur, hoc quidem &longs;ignum mo-<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.215.2.tif"/><lb/><emph type="italics"/>etur, illud autem manet in flexibus, quemadmodum <expan abbr="vtiq;">vtique</expan> &longs;i dia<lb/>metri, quæ quidem A D, maneat, quæ cutem B, moueatur, & <lb/>fiat A C, &longs;ed hic quidem videtur, &longs;ecundum omnem modum in<lb/>diui&longs;ibile e&longs;&longs;e centrum. </s>

<s>etenim moueri, vt aiunt, fingunt in ip&longs;is, <lb/>non enim mouetur mathematicorum aliquid.<emph.end type="italics"/></s>

Line 9113

<s>Lib. 1. cap.

7. <emph type="italics"/>(Faber enim, & Geometra diuer&longs;o modo rectum angulum <lb/><expan abbr="vtriq;">vtrique</expan> con&longs;iderant: ille quatenus <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad opus vtile e&longs;t, hic verò cum ve<lb/>ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)<emph.end type="italics"/> Id quod dicit Ari&longs;t. <lb/>confirmatur ex eo, quod Fabri omnes vtuntur amu&longs;&longs;i, &longs;eu norma, <lb/><figure id="fig130"/><lb/>quæ nihil aliud e&longs;t quàm angulus rectus, quæ vulgò <lb/>&longs;quadra dicitur, vt eius auxilio angulum ip&longs;um re<lb/>ctum in opus conferant, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; normæ, aut amu&longs;&longs;is du<lb/>ctu &longs;ua ip&longs;i opera ad angulos rectos, ide&longs;t quadrata, <lb/>conficiunt. </s>

7. <emph type="italics"/>(Faber enim, & Geometra diuer&longs;o modo rectum angulum <lb/><expan abbr="vtriq;">vtrique</expan> con&longs;iderant: ille quatenus <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad opus vtile e&longs;t, hic verò cum ve<lb/>ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)<emph.end type="italics"/> Id quod dicit Ari&longs;t. <lb/>confirmatur ex eo, quod Fabri omnes vtuntur amu&longs;&longs;i, &longs;eu norma, <lb/><figure place="text" xlink:href="figures-la/009.01.216.1.tif"/><lb/>quæ nihil aliud e&longs;t quàm angulus rectus, quæ vulgò <lb/>&longs;quadra dicitur, vt eius auxilio angulum ip&longs;um re<lb/>ctum in opus conferant, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; normæ, aut amu&longs;&longs;is du<lb/>ctu &longs;ua ip&longs;i opera ad angulos rectos, ide&longs;t quadrata, <lb/>conficiunt. </s>

<s>Geometra verò con&longs;iderat eundem an<lb/>gulum, quatenus fit à linea &longs;uper lineam aliam per<lb/>pendiculariter in&longs;i&longs;tente, vt e&longs;t in definit. </s>

Line 9133

<s>&longs;imiliter inter hos terminos 3. 6. 9. 12. e&longs;t arith<lb/>metica analogia, cùm omnes ternario numero &longs;uperent præcedentes, & à <lb/>&longs;equentibus &longs;uperentur. </s>

<s>Porrò apud Mathematicos tria &longs;unt genera pro<lb/>portionum, &longs;iue medietatum, Arithmetica quam modo &longs;uppo&longs;ui; Geome<lb/>trica, & Harmonica, quas inferius oblata occa&longs;ione opportunius explicabo.</s><figure/>

Line 9765

<s><emph type="italics"/>In Oeconomicis nihil Mathematicum reperi.<emph.end type="italics"/></s><figure/><pb pagenum="227"/>

<s><emph type="italics"/>In Oeconomicis nihil Mathematicum reperi.<emph.end type="italics"/></s><pb pagenum="227"/>

<s><emph type="italics"/>EX PROBLEMATIBVS ARIST,<emph.end type="italics"/></s>

Line 9851

<s>quemadmodum faciunt; qui <gap/>a&longs;a militaria <lb/>partiuntur. </s>

<s>At cæteræ lineæ, quæ per lineas compo&longs;itam figuram &longs;ecant, eam cor<lb/>rumpunt: committitur enim rectilmea figura in argulis, vel &longs;ecundum angulos)<emph.end type="italics"/><lb/>Vt rectè problema hoc percipiamus, proponenda e&longs;t figura rectilinea, & <lb/>vna ex ijs, quæ paralle logramma dicuntur, vt &longs;unt Quadratum, Quadrila<lb/><figure id="fig131"/><lb/>terum, Rhombus, Rhomboides, cuiu&longs;mo<lb/>di e&longs;t præ&longs;ens, aliter verba Ari&longs;t.

<s>At cæteræ lineæ, quæ per lineas compo&longs;itam figuram &longs;ecant, eam cor<lb/>rumpunt: committitur enim rectilmea figura in argulis, vel &longs;ecundum angulos)<emph.end type="italics"/><lb/>Vt rectè problema hoc percipiamus, proponenda e&longs;t figura rectilinea, & <lb/>vna ex ijs, quæ paralle logramma dicuntur, vt &longs;unt Quadratum, Quadrila<lb/><figure place="text" xlink:href="figures-la/009.01.228.1.tif"/><lb/>terum, Rhombus, Rhomboides, cuiu&longs;mo<lb/>di e&longs;t præ&longs;ens, aliter verba Ari&longs;t.

illi non <lb/>&longs;emper quadrarent, quia illarum diameter <lb/>illas &longs;emper bifariam non &longs;e caret. </s>

Line 10013

cur cum Sol eodem <lb/>vigore feratur, non idem tamen incrementum, decrementumuè vmbrarum <lb/>exultet? </s>

<s>pro cuius intelligentia, ac &longs;olutione in&longs;picienda e&longs;t figura &longs;equens: <lb/>in qua &longs;emicirculus, &longs;iue arcus A B C, &longs;it is, per quem Sol incedit, dum ele<lb/>uatur &longs;upra horizontem H I; & quia Sol vniformiter &longs;candit hunc arcum, <pb pagenum="231"/><figure id="fig132"/><lb/>&longs;int duo arcus æquales A B, B C. <lb/>in plano autem horizontis ere<lb/>ctum &longs;it corpus G D, quod à So<lb/>le con&longs;piciatur, &longs;iue illuminetur. <lb/></s>

<s>pro cuius intelligentia, ac &longs;olutione in&longs;picienda e&longs;t figura &longs;equens: <lb/>in qua &longs;emicirculus, &longs;iue arcus A B C, &longs;it is, per quem Sol incedit, dum ele<lb/>uatur &longs;upra horizontem H I; & quia Sol vniformiter &longs;candit hunc arcum, <pb pagenum="231"/><figure place="text" xlink:href="figures-la/009.01.231.1.tif"/><lb/>&longs;int duo arcus æquales A B, B C. <lb/>in plano autem horizontis ere<lb/>ctum &longs;it corpus G D, quod à So<lb/>le con&longs;piciatur, &longs;iue illuminetur. <lb/></s>

<s><expan abbr="&longs;it&qacute;">&longs;itque</expan>; primum Sol in A. radius ip<lb/>&longs;ius per verticem D, tran&longs;iliens <lb/>erit A D I; vmbra verò erit G I. <lb/></s>

Line 10113

<s>Primò igitur illud Per&longs;pectiuus principium &longs;tatuendum e&longs;t, ex <lb/>quolibet corporis lucidi puncto, ad quodlibet medij punctum, lumen rectis <lb/>lineis quoquouer&longs;us emicare, ita vt lumen à quouis puncto lucidi, tanquam <lb/>à centro <expan abbr="circumquaq;">circumquaque</expan> effu&longs;um in modum &longs;phæræ diffundatur.</s>

<s>Secundò, quò magis duorum vicinorum circulorum peripheriæ augen<lb/>tur, eò magis ad vnius circuli &longs;imilitudinem accedere; vt in hac figura cer<lb/><figure id="fig133"/><lb/>nere licet, vbi &longs;unt primò duo parui <lb/>circuli circa centra A, & B, de&longs;cripti, <lb/>quorum circunferentiæ cre&longs;cant <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad circunferentias C D, & E F, quo in<lb/>cremento po&longs;ito, &longs;tatim vel ad &longs;en&longs;um <lb/>manife&longs;tum e&longs;t, has duas maiores pe<lb/>ripherias, magis rcferre vnius circuli <lb/>&longs;imilitudinem, quàm referant duo pa<lb/>rui circelli. </s>

<s>Secundò, quò magis duorum vicinorum circulorum peripheriæ augen<lb/>tur, eò magis ad vnius circuli &longs;imilitudinem accedere; vt in hac figura cer<lb/><figure place="text" xlink:href="figures-la/009.01.233.1.tif"/><lb/>nere licet, vbi &longs;unt primò duo parui <lb/>circuli circa centra A, & B, de&longs;cripti, <lb/>quorum circunferentiæ cre&longs;cant <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad circunferentias C D, & E F, quo in<lb/>cremento po&longs;ito, &longs;tatim vel ad &longs;en&longs;um <lb/>manife&longs;tum e&longs;t, has duas maiores pe<lb/>ripherias, magis rcferre vnius circuli <lb/>&longs;imilitudinem, quàm referant duo pa<lb/>rui circelli. </s>

<s>quod &longs;i e&longs;&longs;ent tres circelli, <lb/>qui augerentur, magis adhuc vnicum <lb/>circulum imitarentur; <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; deinceps, <lb/>quò plures eò perfectius: & quò magis <pb pagenum="234"/>etiam augentur eò perfectius. </s>

Line 10121

<s>quod de circulis dictum e&longs;t, intelligi etiam de<lb/>bet de omnibus alijs figuris eiu&longs;dem &longs;peciei, vt de duabus ellip&longs;ibus, aut <lb/>de duobus triangulis, &c.</s>

<s>Tertiò, lumen Solis per foramen tam <expan abbr="exigu&utilde;">exiguum</expan>, quod &longs;it in&longs;tar puncti tran&longs;<lb/>mi&longs;&longs;um, <expan abbr="figurã">figuram</expan> Solis rotundam videlicet, quamuis conuer&longs;am referre; quod <lb/><figure id="fig134"/><lb/>hac de&longs;criptione patefiet. </s>

<s>Tertiò, lumen Solis per foramen tam <expan abbr="exigu&utilde;">exiguum</expan>, quod &longs;it in&longs;tar puncti tran&longs;<lb/>mi&longs;&longs;um, <expan abbr="figurã">figuram</expan> Solis rotundam videlicet, quamuis conuer&longs;am referre; quod <lb/><figure place="text" xlink:href="figures-la/009.01.234.1.tif"/><lb/>hac de&longs;criptione patefiet. </s>

<s>&longs;it Sol vbi A B, fora<lb/>men in&longs;tar puncti vbi E. illuminatio, in planum <lb/>radio Solis perpendiculariter obiectum, &longs;it C I D. <lb/>aio eam e&longs;&longs;e in&longs;tar Solis rotundam, inuer&longs;am ta<lb/>men. </s>

Line 10149

<s>per punctum igitur D, ducantur radij A D E, B D K. per pun<lb/>ctum verò C, ducantur alij A C H, B C F, qui cùm ab extremitatibus Solis <lb/>profluant, reliquos omnes radios intra &longs;e continebunt. </s>

<s>ex tertia igitur pr&ecedil;<lb/>notatione per punctum C, procedit rotunda illuminatio, cuius diameter <lb/><figure id="fig135"/><lb/>F H, &longs;imiliter per punctum D, illuminatio <lb/>rotunda emanat, cuius diameter K E, & pa<lb/>riter ex omnibus alijs rimulæ punctis ro<lb/>tundi &longs;plendores in &longs;uperficiem, vbi F E, <lb/>tran&longs;mittuntur. </s>

<s>ex tertia igitur pr&ecedil;<lb/>notatione per punctum C, procedit rotunda illuminatio, cuius diameter <lb/><figure place="text" xlink:href="figures-la/009.01.235.1.tif"/><lb/>F H, &longs;imiliter per punctum D, illuminatio <lb/>rotunda emanat, cuius diameter K E, & pa<lb/>riter ex omnibus alijs rimulæ punctis ro<lb/>tundi &longs;plendores in &longs;uperficiem, vbi F E, <lb/>tran&longs;mittuntur. </s>

<s>Iam dicimus has duas illu<lb/>minationes ex prænotatis &longs;ecundo loco, <lb/>quàm longius planum F E, à foramine de<lb/>&longs;titerit, vt &longs;i e&longs;&longs;et in L M, ad vnius circuli <lb/>rotunditatem magis accedere, vt apparet <lb/>in plano L M, vbi maiores factæ &longs;unt illumi<lb/>nationes, & ideò magis ad vnam circula<lb/>rem accedunt. </s>

Line 10253

<s>Interim tamen huius loci explicationem hanc accipe. </s>

<s>Cur <lb/>cùm Luna &longs;emiplena e&longs;t, linea illa, quæ terminus e&longs;t partis illuminatæ, & <lb/>partis ob&longs;curæ, <expan abbr="quæ&qacute;">quæque</expan>; Lunam bifariam diuidit, videtur linea recta, cùm ta<lb/>men non &longs;it; cùm enim fit in globo&longs;a &longs;uperficie Lunæ, nece&longs;&longs;ariò circularis <lb/><figure id="fig136"/><lb/>e&longs;t? </s>

<s>vt autem rem hanc melius intelli<lb/>gamus, præ&longs;ens figura illuminationis <lb/>Lunæ in&longs;picienda e&longs;t: vbi oculus no&longs;ter <lb/>e&longs;t in centro mundi A; vnde varias Lu<lb/>næ illuminationes a&longs;picit: è quibus <lb/>octo tantum, in figura &longs;unt depictæ: in <lb/>quibus videre e&longs;t Lunæ &longs;emper dimi<lb/>dium illud, &longs;iue hemi&longs;phærium, quod <lb/>Solem a&longs;picit, e&longs;&longs;e illuminatum, cuius <lb/>terminus, &longs;iue ba&longs;is e&longs;t linea K L, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>con&longs;inium illu&longs;tratæ partis, & opacæ. <lb/></s>

Line 10311

<s>An quod oriens primo vmbram ter<lb/>ræ æquidi&longs;tantem reddit, ac infinitam p&ecedil;nè protrahit, deinde longam, & po&longs;tea mi<lb/>norem &longs;ubinde? </s>

<s>quia linea recta, quæ de &longs;uperiori puncto elicitur, interius cadit.<emph.end type="italics"/><lb/><figure id="fig137"/><lb/><emph type="italics"/>&longs;it Gnomon A B. Sol, vbi C, & vbi D. <lb/>radius igitur ex C, prefici&longs;cens, e&longs;t C F, <lb/>& exterius proced t, quàm radius D E. <lb/>e&longs;t autem vmbra B E, Sole &longs;ublimiori <lb/>exi&longs;tente: vmbra verò B F, Sole humi<lb/>liori. </s>

<s>quia linea recta, quæ de &longs;uperiori puncto elicitur, interius cadit.<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.239.1.tif"/><lb/><emph type="italics"/>&longs;it Gnomon A B. Sol, vbi C, & vbi D. <lb/>radius igitur ex C, prefici&longs;cens, e&longs;t C F, <lb/>& exterius proced t, quàm radius D E. <lb/>e&longs;t autem vmbra B E, Sole &longs;ublimiori <lb/>exi&longs;tente: vmbra verò B F, Sole humi<lb/>liori. </s>

<s>ergò quò Sol altior fuerit, eò mi<lb/>nor vmbra erit, minima&queacute; tunc erit, cum <lb/>Sol &longs;uper caput no&longs;trum ver&longs;abitur)<emph.end type="italics"/><lb/>Problema præ&longs;ens e&longs;t idem <expan abbr="c&utilde;">cum</expan> quar<lb/>to huius &longs;ectionis: eadem igitur ex<lb/>po&longs;itio <expan abbr="vtriq;">vtrique</expan> in&longs;eruiat. </s>

Line 10329

<s><expan abbr="itaq;">itaque</expan> nece&longs;&longs;e e&longs;t radium à &longs;upe<lb/>riore procedentem intus cadere. </s>

<s>&longs;it Gnomon A D, Luna B Sol C, Lunæ radius B F.<emph.end type="italics"/><pb pagenum="240"/><figure id="fig138"/><lb/><emph type="italics"/>ergò vmbra Lunæ D F, Solis radius <lb/>C E; vmbra igitur nece&longs;&longs;ariò minor <lb/>e&longs;t, e&longs;t enim D E.)<emph.end type="italics"/> de hac re vide <lb/>Spheram P. Clauij, cap.

<s>&longs;it Gnomon A D, Luna B Sol C, Lunæ radius B F.<emph.end type="italics"/><pb pagenum="240"/><figure place="text" xlink:href="figures-la/009.01.240.1.tif"/><lb/><emph type="italics"/>ergò vmbra Lunæ D F, Solis radius <lb/>C E; vmbra igitur nece&longs;&longs;ariò minor <lb/>e&longs;t, e&longs;t enim D E.)<emph.end type="italics"/> de hac re vide <lb/>Spheram P. Clauij, cap.

de Ordine <lb/>c&oelig;lorum, vnde huius textus expo<lb/>&longs;itionem in hunc modum licebit af<lb/>ferre. </s>

Line 10375

<s>tempore igitur &longs;olaris defectus, <lb/>intra parietes v&longs;piam, clau&longs;is omnibus fene&longs;tris, admittatur Solis radius, <lb/>per angu&longs;tum foramen rotundum; <expan abbr="excipiatur&qacute;">excipiaturque</expan>; radius hic in plana tabella, <lb/>vbi certo quantum Sol defecerit ad vnguem, licet videre, <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla intui<lb/>tus mole&longs;tia, ac tam perfectè, atque fi in c&oelig;lo coram ade&longs;&longs;es) hæcille, licet <lb/>autem videre, quia illuminatio in tabella excepta, quæ alias &longs;olet e&longs;&longs;e cir<lb/>cularis, erit tempore eclyp&longs;is ip&longs;a pariter cum Sole defectiua, <expan abbr="atq;">atque</expan> in&longs;tar fal<lb/>catæ lunulæ. </s>

<s>deinde &longs;ubdit; verum hoc omninò &longs;cire nece&longs;&longs;arium e&longs;t, con<lb/>trario modo apparere defectum illum in tabula per radios Solis, quàm in <lb/>c&oelig;lo contingit: hoc e&longs;t, &longs;i in c&oelig;lo &longs;uperior pars deliquium patiatur, in ra<lb/>dijs apparebit inferior deficere, vt ratio exigit optica) <expan abbr="hucu&longs;q;">hucu&longs;que</expan> Gemma Fri<lb/>&longs;ius; ex quo etiam placuit accipere totius huius <expan abbr="experi&etilde;tiæ">experientiæ</expan> figuram, quam <lb/><figure id="fig139"/><lb/>ip&longs;e <expan abbr="cuiu&longs;dã">cuiu&longs;dam</expan> eclyp&longs;is an<lb/>ni 1544. apponit. </s>

<s>deinde &longs;ubdit; verum hoc omninò &longs;cire nece&longs;&longs;arium e&longs;t, con<lb/>trario modo apparere defectum illum in tabula per radios Solis, quàm in <lb/>c&oelig;lo contingit: hoc e&longs;t, &longs;i in c&oelig;lo &longs;uperior pars deliquium patiatur, in ra<lb/>dijs apparebit inferior deficere, vt ratio exigit optica) <expan abbr="hucu&longs;q;">hucu&longs;que</expan> Gemma Fri<lb/>&longs;ius; ex quo etiam placuit accipere totius huius <expan abbr="experi&etilde;tiæ">experientiæ</expan> figuram, quam <lb/><figure place="text" xlink:href="figures-la/009.01.241.1.tif"/><lb/>ip&longs;e <expan abbr="cuiu&longs;dã">cuiu&longs;dam</expan> eclyp&longs;is an<lb/>ni 1544. apponit. </s>

<s>e&longs;t au<lb/>tem hæc. </s>

Line 10425

<s>deinde ait: &longs;i igitur nihil aliud e&longs;t <lb/>vmbra, quam id, quod non a&longs;picitur à Sole, & à Sole tamen videamus illu<lb/>minari totam bullæ ba&longs;im circulariter, nece&longs;&longs;e e&longs;t totam etiam bullam <expan abbr="vn-diq;">vn<lb/>dique</expan> à Sole illuminari, & con&longs;pici, quod tamen impo&longs;&longs;ibile e&longs;&longs;e demon&longs;tra<lb/>tur ab opticis: ip&longs;i enim demon&longs;trant, nullum corpus, quantumuis mini<lb/>mum, totum po&longs;&longs;e circum&longs;pici à quamuis maximo illuminante. </s>

<s>quod qui<lb/>dem antiquitus demon&longs;trauit Ari&longs;tarchus Samius in libello de di&longs;tantijs So<lb/><figure id="fig140"/><lb/>lis, & Lunæ: & <lb/>po&longs;tea Vitellio <lb/>lib.

<s>quod qui<lb/>dem antiquitus demon&longs;trauit Ari&longs;tarchus Samius in libello de di&longs;tantijs So<lb/><figure place="text" xlink:href="figures-la/009.01.242.1.tif"/><lb/>lis, & Lunæ: & <lb/>po&longs;tea Vitellio <lb/>lib.

2. propo&longs;. </s>

Line 10465

<s>&longs;atis e&longs;&longs;e exi&longs;timo per paraphrafim præ&longs;ens problema ex<lb/>ponere, ex qua tamen, vbi opus fuerit, textus corrigatur. </s>

<s>Cur ea, quæ in <lb/>terram cadunt, <expan abbr="atq;">atque</expan> re&longs;iliunt angulos ad planitiem, faciunt &longs;imiles vtraque <lb/><figure id="fig141"/><lb/>ex parte, qua planum tetigerint? </s>

<s>Cur ea, quæ in <lb/>terram cadunt, <expan abbr="atq;">atque</expan> re&longs;iliunt angulos ad planitiem, faciunt &longs;imiles vtraque <lb/><figure place="text" xlink:href="figures-la/009.01.243.1.tif"/><lb/>ex parte, qua planum tetigerint? </s>

<s>v. g. &longs;i <lb/>corpus quodpiam cadat ex puncto D, per <lb/>lineam D C, &longs;uper planum A B, ex puncto <lb/>C, vbi cæciderat, re&longs;ilit per lineam C E, <lb/>ita vt faciat duos angulos æquales vtrin<lb/>que ad punctum. </s>

Line 10485

<s>Duobus porrò modis grauia &longs;uper terræ planitiem cadunt; aut enim per<lb/>pendiculariter, & fecundum mundi diametrum decidunt; aut obliquè, &longs;eu <lb/>in latera. </s>

<s>quæ igitur primo modo de&longs;cendunt, ide&longs;t perpendiculariter, &longs;eu <lb/>quæ angulos rectos cum plano faciunt, ea etiam re&longs;iliunt perpendiculariter, <lb/>&longs;eu ad angulos rectos, & ideo nece&longs;&longs;ariò per candein lineam, qua decide<lb/>rant, repercutiuntur; cuius cau&longs;a e&longs;&longs;e pote&longs;t, quia diameter &longs;cilicet mundi, <lb/>per quam delap&longs;a &longs;unt, ea bifariam diuidit, vt in figura, graue E, per D C, <lb/><figure id="fig142"/><lb/>plano A B, perpendicularem de&longs;cendat; <lb/>quæ <expan abbr="perp&etilde;dicularis">perpendicularis</expan> coincidit cum mun<lb/>di diametro, perpendicula enim omnia <lb/>ad mundi centrum tendunt; graue igi<lb/>tur E, dum puncto C, alliditur, diuidi<lb/>tur bifariam, à diametro mundi D C: <lb/>vnde & in æquilibrio con&longs;tituitur, ita vt <lb/>nulla &longs;it maior ratio, cur ad partem <lb/>vnam, quàm ad alteram re&longs;ultet, & ideo <lb/><expan abbr="conueni&etilde;s">conueniens</expan> e&longs;t, ip&longs;um per eandem lineam <lb/>D C, reuerti; &longs;ic enim faciet etiam an<lb/>gulos incidentiæ, & reflexionis æquales. <lb/></s>

<s>quæ igitur primo modo de&longs;cendunt, ide&longs;t perpendiculariter, &longs;eu <lb/>quæ angulos rectos cum plano faciunt, ea etiam re&longs;iliunt perpendiculariter, <lb/>&longs;eu ad angulos rectos, & ideo nece&longs;&longs;ariò per candein lineam, qua decide<lb/>rant, repercutiuntur; cuius cau&longs;a e&longs;&longs;e pote&longs;t, quia diameter &longs;cilicet mundi, <lb/>per quam delap&longs;a &longs;unt, ea bifariam diuidit, vt in figura, graue E, per D C, <lb/><figure place="text" xlink:href="figures-la/009.01.244.1.tif"/><lb/>plano A B, perpendicularem de&longs;cendat; <lb/>quæ <expan abbr="perp&etilde;dicularis">perpendicularis</expan> coincidit cum mun<lb/>di diametro, perpendicula enim omnia <lb/>ad mundi centrum tendunt; graue igi<lb/>tur E, dum puncto C, alliditur, diuidi<lb/>tur bifariam, à diametro mundi D C: <lb/>vnde & in æquilibrio con&longs;tituitur, ita vt <lb/>nulla &longs;it maior ratio, cur ad partem <lb/>vnam, quàm ad alteram re&longs;ultet, & ideo <lb/><expan abbr="conueni&etilde;s">conueniens</expan> e&longs;t, ip&longs;um per eandem lineam <lb/>D C, reuerti; &longs;ic enim faciet etiam an<lb/>gulos incidentiæ, & reflexionis æquales. <lb/></s>

<s>quæ verò &longs;ecundo modo decidunt, ide&longs;t <lb/>obliquè, & &longs;ecundum latera: quoniam non &longs;ecundum perpendiculum, &longs;ed <lb/>ex puncto extra perpendiculum po&longs;ito, planum feriunt, accidit vt à puncto <lb/>incidentiæ C, vt in priori figura, in contrariam partem repul&longs;a re&longs;ulcent; <lb/>de&longs;cenderant enim ex D, & in contrariam partem, &longs;cilicet ad E, priotis fi<lb/>guræ reflectuntur. </s>

Line 10587

<s>In 2. problema. </s>

<s>In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/><lb/><figure id="fig143"/><lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit, <lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non <lb/>tamen de&longs;cribit quadratum duplum quadrati il<lb/>lius, &longs;ed quadruplum: vt probatur in &longs;cholio 4. <lb/>2. Elem.

<s>In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.246.1.tif"/><lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit, <lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non <lb/>tamen de&longs;cribit quadratum duplum quadrati il<lb/>lius, &longs;ed quadruplum: vt probatur in &longs;cholio 4. <lb/>2. Elem.

& videre e&longs;t in hac figura, vbi linea A B, <lb/>e&longs;t dupla lineæ A C. quadratum verò lineæ A B, <lb/>&longs;cilicet quadratum A B D E, e&longs;t <expan abbr="quadrupl&utilde;">quadruplum</expan> qua<lb/>drati lineæ A C, quadrati nimirum A C F G. re<lb/>liqua huius textus manife&longs;ta &longs;unt</s>

Line 10613

<s>Cape duas chordas æreas, ex ijs &longs;cilicet, quas in cytharis adhibere <expan abbr="&longs;ol&etilde;t">&longs;olent</expan>; <lb/>nam quæ ex inte&longs;tinis ouium fiunt, vt plurimum aut falfæ &longs;unt, aut aeris mu<lb/>tationi obnoxiæ. </s>

<s>&longs;int hæ duæ chordæ æquales omninò, atque æquè inten&longs;æ, <lb/><figure id="fig144"/><lb/>ita vt &longs;int vni&longs;onæ, hoc e&longs;t, vna <expan abbr="tan-t&utilde;">tan<lb/>tum</expan> vox &longs;it. </s>

<s>&longs;int hæ duæ chordæ æquales omninò, atque æquè inten&longs;æ, <lb/><figure place="text" xlink:href="figures-la/009.01.247.1.tif"/><lb/>ita vt &longs;int vni&longs;onæ, hoc e&longs;t, vna <expan abbr="tan-t&utilde;">tan<lb/>tum</expan> vox &longs;it. </s>

<s>quamuis duæ fides. </s>

Line 10637

<s>ex quibus <pb pagenum="248"/>prima, Vt, & vltima, Fa, quæ octaua e&longs;t, con&longs;onantiam diapa&longs;on, aut du<lb/>plam, aut octauam reddunt.</s>

<s>Rur&longs;us eadem chorda C D, diuidatur in tres partes æquales in punctis <lb/><figure id="fig145"/><lb/>F, G. </s>

<s>Rur&longs;us eadem chorda C D, diuidatur in tres partes æquales in punctis <lb/><figure place="text" xlink:href="figures-la/009.01.248.1.tif"/><lb/>F, G. </s>

<s>F D igitur erit duæ ter<lb/>tiæ tam totius C D, quàm to<lb/>tius A B. ponatur iam tactus <lb/>in F, <expan abbr="percutiãtur&qacute;">percutianturque</expan>; &longs;imul A B, <lb/>& F D; audietur con&longs;onantia <lb/>&longs;uauis admodum, & perfecta <lb/>quidem, &longs;ed <expan abbr="nõ">non</expan> tamen, vt Dia<lb/>pa&longs;on. </s>

Line 10649

<s>vulgò quinta, quia ex prima voce, Vt, <lb/>& quinta, Sol, con&longs;tat.</s>

<s>Eadem iterum chorda in quatuor æquas partes &longs;ecetur in punctis H, E, I, <lb/><figure id="fig146"/><lb/>ita vt chorda H D, &longs;it tres quartæ <lb/>totius A D. facto deinde tactu in <lb/>H, pul&longs;entur &longs;imul A B, H D, & au<lb/>dietur con&longs;onantia quidem, &longs;ed <lb/>duabus præcedentibus imperfe<lb/>ctior. </s>

<s>Eadem iterum chorda in quatuor æquas partes &longs;ecetur in punctis H, E, I, <lb/><figure place="text" xlink:href="figures-la/009.01.248.2.tif"/><lb/>ita vt chorda H D, &longs;it tres quartæ <lb/>totius A D. facto deinde tactu in <lb/>H, pul&longs;entur &longs;imul A B, H D, & au<lb/>dietur con&longs;onantia quidem, &longs;ed <lb/>duabus præcedentibus imperfe<lb/>ctior. </s>

<s>hæc antiquitus Diate&longs;&longs;aron, <lb/>ide&longs;t per quatuor, &longs;ubaudi chor<lb/>das, aut voces, &longs;imili ratione, qua &longs;uperiores dicta fuit. </s>

Line 10691

<s>Illud po&longs;tremò loco non ignorandum, aliter has voces Vt, Re, &c. </s>

<s>vete<lb/>res illos Græcos denomina&longs;&longs;e, nam primam, ide&longs;t graui&longs;&longs;imam vocem, &longs;iue <lb/>chordam, quam modò Vt, dicunt, eam ip&longs;i Hypaten vocarunt, & reliquas <lb/>ordine &longs;equenti.</s>

<s>Vt, Hypate — ide&longs;t Principalis.</s>

Line 10731

<s>re&longs;pondet inde prouenire, quia hæc vox <lb/>cantandi principium e&longs;t. </s>

<s>vbi per cantandi principium puto ip&longs;um intellige<lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vtip&longs;e ait primo <lb/><figure id="fig147"/><lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vt ip&longs;e ait primo <lb/>Po&longs;ter. cap.

<s>vbi per cantandi principium puto ip&longs;um intellige<lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vtip&longs;e ait primo <lb/><figure place="text" xlink:href="figures-la/009.01.250.2.tif"/><lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vt ip&longs;e ait primo <lb/>Po&longs;ter. cap.

38. die&longs;is h&ecedil;c, &longs;iue &longs;emitonium dicitur <lb/>principium cantus, quia minimum e&longs;t omnium in<lb/>teruallorum, quæ voce po&longs;&longs;int exprimi: <expan abbr="atq;">atque</expan> ex eo <lb/>alia interualla con&longs;tant, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; veluti illorum ele<lb/>mentum. </s>

Line 10813

<s>ergò hypa<lb/>te magis antiphonum, quam nete reddebat; nam vt acutum vim de&longs;iderat plenio<lb/>rem, &longs;ic graue exprimi facilius pote&longs;t)<emph.end type="italics"/> Propter quid, inquit, antiqui&longs;&longs;imi Mu<lb/>&longs;icorum cùm ex &longs;emptem tantum chordis mu&longs;ica in&longs;frumenta <expan abbr="componer&etilde;t">componerent</expan>, <lb/>neten non hypatèn omittebant? </s>

<s>&longs;upra octo voces, &longs;eu chordas recen&longs;ui, <expan abbr="ra-tionem&qacute;">ra<lb/>tionemque</expan>; ip&longs;arum, vnà cum antiquis appellationibus explicani, quarum <pb pagenum="252"/><figure id="fig148"/><lb/>prima e&longs;t hypate, vltima verò note, quibus re<lb/>petitis facilè e&longs;t intelligere, quod re&longs;pondet <lb/>Ari&longs;t. </s>

<s>&longs;upra octo voces, &longs;eu chordas recen&longs;ui, <expan abbr="ra-tionem&qacute;">ra<lb/>tionemque</expan>; ip&longs;arum, vnà cum antiquis appellationibus explicani, quarum <pb pagenum="252"/><figure place="text" xlink:href="figures-la/009.01.252.1.tif"/><lb/>prima e&longs;t hypate, vltima verò note, quibus re<lb/>petitis facilè e&longs;t intelligere, quod re&longs;pondet <lb/>Ari&longs;t. </s>

<s>Re&longs;pondet enim id non omninò verum <lb/>cen&longs;eri debere, nam vtramque quidem hypa<lb/>tem, &longs;cilicet & neten a&longs;&longs;umebant; triten verò <lb/><expan abbr="omittebãt">omittebant</expan>. </s>

Line 10974

<s>In dupla enim propor<lb/>tione altera quantitas diuiditur tantum bifariam, vt &longs;uperius patuit: diui&longs;io <lb/>porrò bifariam, &longs;iuc in partes æquales e&longs;t prima <expan abbr="omni&utilde;">omnium</expan>, quia magis ad vni

<pb pagenum="255"/><expan abbr="tate&etilde;">tatem</expan>, siue ad indiui&longs;um, & ad æquale accedit, cùm in partes æquas diuidat: <lb/>& cùm ad eam opus &longs;it unica tantum diuisione. </s>

<s>In alijs proportionalibus, vt <lb/>in tripla, opus e&longs;t duabus diui&longs;ionibus, vt supra patuit: &longs;imiliter in alijs, &longs;e&longs;<lb/>quialtera, &longs;e&longs;quitertia, opus e&longs;t pluribus diui&longs;ionibus: cùm igitur ip&longs;a præ<lb/>cæteris magis ad æqualitatem, &amp; vnitatem accedat, facit,vt voces ip&longs;sius <lb/>videantur ferè æquales, hoc e&longs;t ferè eædem, & vni&longs;onæ, &amp; eiu&longs;dem naturæ: <lb/>id, quod etiam in fi&longs;stolis adeò verum e&longs;t, vt aliquando aures decipiant, cùm <lb/>nimirium aures iudicent duas fi&longs;stulas e&longs;&longs;e aut vnam tantum, aut duas vni<lb/>&longs;onas, quæ tamen re vera &longs;unt in con&longs;onantia Diapa&longs;on.<s><arrow.to.target n="marg363a"/></s>

<s>In alijs proportionalibus, vt <lb/>in tripla, opus e&longs;t duabus diui&longs;ionibus, vt supra patuit: &longs;imiliter in alijs, &longs;e&longs;<lb/>quialtera, &longs;e&longs;quitertia, opus e&longs;t pluribus diui&longs;ionibus: cùm igitur ip&longs;a præ<lb/>cæteris magis ad æqualitatem, &amp; vnitatem accedat, facit,vt voces ip&longs;sius <lb/>videantur ferè æquales, hoc e&longs;t ferè eædem, & vni&longs;onæ, &amp; eiu&longs;dem naturæ: <lb/>id, quod etiam in fi&longs;stolis adeò verum e&longs;t, vt aliquando aures decipiant, cùm <lb/>nimirium aures iudicent duas fi&longs;stulas e&longs;&longs;e aut vnam tantum, aut duas vni<lb/>&longs;onas, quæ tamen re vera &longs;unt in con&longs;onantia Diapa&longs;on.<arrow.to.target n="marg363a"/></s>

Line 11002

<s>quæ genera ab invicem di&longs;tinguebantur, &longs;ecundum variam Te<lb/>trachordorum <expan abbr="coñ">con&longs;titutionem</expan>&longs;titutionem; ex tetrachordis enim totam &longs;eriem, &longs;eu Mo<lb/>nochordium, &longs;eu regulam harmonicam componebant. </s>

<s>Erat autem tetra<lb/>chordum intervallum Diate&longs;&longs;aron, con&longs;tans ex quatuor chordis, &longs;eu voci<lb/>bus, vt, re, mi, fa; <expan abbr="quaru&utilde;">quarum</expan> vocum interualla vnius tetrachordi generis, vnius, <lb/>differebant ab interuallis alterius tetrachordi alterius generis, v.g. in ge<lb/>nere Diatonico erat huiu&longs;modi tetrachordium, cuius primum interualllum <lb/>erat &longs;omitonium, reliqua verò duo erant to<lb/>ni. </s>

<s>Erat autem tetra<lb/>chordum intervallum Diate&longs;&longs;aron, con&longs;tans ex quatuor chordis, &longs;eu voci<lb/>bus, vt, re, mi, fa; <expan abbr="quaru&utilde;">quarum</expan> vocum interualla vnius tetrachordi generis, vnius, <lb/>differebant ab interuallis alterius tetrachordi alterius generis, v.g. in ge<lb/>nere Diatonico erat huiu&longs;modi tetrachordium, cuius primum interualllum <lb/><figure place="text" xlink:href="figures-la/009.01.255.1.tif"/>erat &longs;omitonium, reliqua verò duo erant to<lb/>ni. </s>

<s>& à prima voce Hypate, ad vltimam Me<lb/>&longs;en, erat con&longs;onantia Diate&longs;&longs;eron huic tetra<lb/>chordo addebant aliud &longs;imile, mediante tono <lb/>vno inter vtrunque: ita vt ex duobus confla<lb/>retur tota diapa&longs;on à gravi hypate, ad &longs;upre<lb/>mam Neten. </s>

Line 11011

<pb pagenum="256"/><figure id="fig149"/><lb/><expan abbr="chromatic&utilde;">chromaticum</expan> inter chordas &longs;ui tetrachor<lb/>di <expan abbr="&longs;equ&etilde;tia">&longs;equentia</expan> interualla &longs;eruabat. </s>

<pb pagenum="256"/><figure place="text" xlink:href="figures-la/009.01.256.1.tif"/><lb/><expan abbr="chromatic&utilde;">chromaticum</expan> inter chordas &longs;ui tetrachor<lb/>di <expan abbr="&longs;equ&etilde;tia">&longs;equentia</expan> interualla &longs;eruabat. </s>

<s><expan abbr="trihemi-toni&utilde;">trihemi<lb/>tonium</expan> autem interuallum ex tribus &longs;emi<lb/>tonijs con&longs;tabat, &longs;eu ex vno toto, & vno <lb/>&longs;emitonio ex duobus huiu&longs;modi tetra<lb/>chordis &longs;uum monochordium, &longs;eu &longs;uas <lb/>octo voces, &longs;eu &longs;uam Diapa&longs;on genus <lb/>chromaticum componebat. </s>

<s>Enharmo<lb/>nicum tandem genus tetrachordo vtebatur, cuius interualla erant ea, quæ <lb/>&longs;equuntur.</s><figure/>

<s>Enharmo<lb/>nicum tandem genus tetrachordo vtebatur, cuius interualla erant ea, quæ <lb/>&longs;equuntur.</s><figure place="text" xlink:href="figures-la/009.01.256.2.tif"/>

<s>Erat hic etiam inter hypate, & Me&longs;e Diate&longs;&longs;aron; huic aliud tetrachor<lb/>dum pariter addebatur, vt in alijs generibus, ex quibus tota Diapa&longs;on <expan abbr="cõ-flabatur">con<lb/>flabatur</expan>. </s>

Line 11331

4. tonus, &longs;en mo<lb/>dus erat quædam cantus con&longs;titutio, ab hypate <expan abbr="v&longs;q;">v&longs;que</expan> ad netem, proprio rith<lb/>mo modificata: ita vt modos Dorius alium rithmum, à Phrygio, & reliquis <lb/>di&longs;crepantem haberet. </s>

<s>quilibet preterea modus &longs;uam certam &longs;edem in Mo<lb/>nochordio obtinebat, vnde &longs;equebatur vnum e&longs;&longs;e reliquis omnibus <lb/>grauiorem, alium e&longs;&longs;e omnium acuti&longs;&longs;imum, reliquos verò in<lb/>termedios, alijs grauiores fui&longs;&longs;e, vt in &longs;equenti figura, <lb/>in qua, tanquam in tabella, omnia, quæ de hi&longs;ce <lb/>modis dici &longs;olent, per&longs;picuè licet <lb/>intucri.</s><pb pagenum="262"/>

<s><emph type="italics"/>ORDO ANTIQVORVM MODORVM.<emph.end type="italics"/><lb/><arrow.to.target n="table6"/></s><pb pagenum="263"/><table><table.target id="table6"/><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>T.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell>S.</cell><cell>S.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell></row><row><cell/><cell/><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell></row><row><cell/><cell/><cell/><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell></row><row><cell/><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos</cell><cell>Parhypate</cell></row><row><cell/><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell/><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell></row><row><cell/><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell></row><row><cell>Nete.</cell><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypatc.</cell><cell>Pro&longs;lamb.</cell></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell></row><row><cell>Paranete.</cell><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Hypermix</cell></row><row><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>— —</cell><cell>tolydius.</cell></row><row><cell>Trite.</cell><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Mixtolyd.</cell><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/></row><row><cell>Me&longs;e.</cell><cell>Lychanos.</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Lydius.</cell><cell/><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/></row><row><cell>Lychanos</cell><cell>Parhypat.</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Phrygius.</cell><cell/><cell/><cell/></row><row><cell>S.</cell><cell>S.</cell><cell>S.</cell><cell>— —</cell><cell/><cell/><cell/><cell/></row><row><cell>Parhypate</cell><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Dorius.</cell><cell/><cell/><cell/><cell/></row><row><cell>T.</cell><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>Hypate.</cell><cell>Pro&longs;lamb.</cell><cell>Hypolyd.</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>T.</cell><cell>— —</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>Pro&longs;lamb.</cell><cell>Hypophry</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>— —</cell><cell>gius.</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>Hypodor.</cell><cell/><cell/><cell/><cell/><cell/><cell/><cell/></row></table>

Line 11811

<s>quibus repetitis melius &longs;equentem pa<lb/>raphra&longs;im percipies. </s>

<s>Verum ante omnia antiquæ lyræ ex antiquis monu<lb/>mentis figuram oculis &longs;ubijciam.</s><figure/>

<s>Verum ante omnia antiquæ lyræ ex antiquis monu<lb/>mentis figuram oculis &longs;ubijciam.</s><figure place="text" xlink:href="figures-la/009.01.270.1.tif"/>

<s>Porrò, vt tradit Vincentius Galilæus in &longs;uis Dialogis, erat eius figura, <lb/>qua&longs;i ex caprino capite con&longs;tructa, cuius duo brachia erant capræ cornua; <lb/>inferior pars cranium, quæ tota ba&longs;i complanatæ ita &longs;uperponebatur, vt in <lb/>quouis po&longs;ita plano recta con&longs;i&longs;teret, <expan abbr="neq;">neque</expan> vt ge&longs;taretur, opus erat. </s>

Line 12163

<s>vt autem &longs;pecies <expan abbr="vtrumq;">vtrumque</expan> oculum con&longs;imiliter <expan abbr="inform&etilde;t">informent</expan>, ne<lb/><figure id="fig150"/><lb/>ce&longs;&longs;e e&longs;t, vt axes vi&longs;uales, quales <lb/>&longs;unt in appo&longs;ita figura, C B, D B, <lb/>ab oculis C, D, ad obiectum B, <lb/>ducti, in ip &longs;o obiecto B. imò in <lb/>eodem ip&longs;ius puncto vniantur: <lb/>quoties. </s>

<s>vt autem &longs;pecies <expan abbr="vtrumq;">vtrumque</expan> oculum con&longs;imiliter <expan abbr="inform&etilde;t">informent</expan>, ne<lb/><figure place="text" xlink:href="figures-la/009.01.276.1.tif"/><lb/>ce&longs;&longs;e e&longs;t, vt axes vi&longs;uales, quales <lb/>&longs;unt in appo&longs;ita figura, C B, D B, <lb/>ab oculis C, D, ad obiectum B, <lb/>ducti, in ip &longs;o obiecto B. imò in <lb/>eodem ip&longs;ius puncto vniantur: <lb/>quoties. </s>

Line 12231

<s><emph type="italics"/>De humani Oculi Pupilla.<emph.end type="italics"/></s>

<s>Vt ea, quæ dicenda &longs;unt, facilius percipi po&longs;&longs;int, nece&longs;&longs;e e&longs;t breuiter <lb/>oculi fabricam, in qua mirum totius naturæ Opificis artificium elu<lb/>cet, auxilio &longs;equentium figurarum præmittere.</s><figure/>

<s>A B C, cornea. </s>

Line 12449

<s><expan abbr="Atq;">Atque</expan> hæc &longs;unt, quæ nuper circa oculi pupillam ob&longs;eruata, huic loco ad<lb/>denda exi&longs;timaui.</s>

<s>DE <lb/>MATHEMATICARVM <lb/>NATVRA DISSERTATIO.</s>

Line 12457

<s>AD ILLVSTRISSIMVM AC NOBILISSIMVM <lb/>PETRVMFRANCISCVM MALASPINAM <lb/>ÆDIFICIORVM MARCHIONEM.</s>

<s><emph type="italics"/>Authore eodem Io&longs;epho Blancano è Societate IESV, <lb/>Mathematicarum in Parmen&longs;i Academia profe&longs;&longs;ore.<emph.end type="italics"/></s><figure/>

<s>BONONIÆ M. DC. XV.</s>

Line 12465

<s>Superiorum permi&longs;&longs;u.</s>

<s>Sumptibus Hieronymi Tamburini.</s><pb pagenum="3"/><figure/>

<s>Sumptibus Hieronymi Tamburini.</s><pb/><pb pagenum="3"/>

<s>ILLVSTRISSIMO <lb/>AC NOBILISSIMO <lb/>PETROFRANCISCO <lb/>MALASPINAE</s>

<s><emph type="italics"/>ÆDIFICIORVM MARCHIONI.<emph.end type="italics"/></s><figure/>

<s><emph type="italics"/>ÆDIFICIORVM MARCHIONI.<emph.end type="italics"/></s>

<s>Io&longs;eph Blancanus è Societate Ie&longs;u. S.P.D.</s>

Line 12501

<s><emph type="italics"/>Parmæ Idibus Nouembris M. DC. XIIII.<emph.end type="italics"/><lb/><figure id="fig151"/></s><pb pagenum="5"/><figure/>

<s><emph type="italics"/>Parmæ Idibus Nouembris M. DC. XIIII.<emph.end type="italics"/><lb/></s><pb pagenum="5"/>

<s><emph type="bold"/>ADDITAMENTVM.<emph.end type="bold"/></s>

<s><emph type="italics"/>DE NATVRA SCIENTIARVM<emph.end type="italics"/></s>

<s><emph type="italics"/>MATHEMATICARVM.<emph.end type="italics"/></s><figure/>

<s><emph type="italics"/>MATHEMATICARVM.<emph.end type="italics"/></s>

<s>Qvoniam in hoc Opere multa ad Mathematicarum natu<lb/>r am &longs;pectantia &longs;par&longs;im dicta &longs;unt, non ab re, <expan abbr="neq;">neque</expan> ingratum <lb/>Lectori fore duxi, ea quodammodo huc in vnum congerere, <lb/>quæ ad earum naturam ritè percipiendam nece&longs;&longs;aria e&longs;&longs;e vi<lb/>derentur. </s>

Line 13837

<s>Vnde &longs;it, vt to<lb/>ta hæc adeò digna, <expan abbr="atq;">atque</expan> admiranda cognitio &longs;it <lb/>mera veritas, quæ omnium &longs;cientiarum finis <lb/>atque animæ no&longs;træ cibus e&longs;t.</s>

<s>APPENDIX.</s>

Line 14047

<s>Curandum igitur e&longs;t, <lb/>à recentioribus Platonicis, vt tandem aliquan<lb/>do, ne à quopiam actum agatur latina <lb/>voce, ac luce pariter donetur.</s>

<s>CLARORVM <lb/>MATHEMATICORVM <lb/>CHRONOLOGIA</s>

Line 14055

<s><emph type="italics"/>Omißis tum fabulo&longs;is, tum ob nimiam antiquitatem incertis, <lb/>veluti &longs;unt ea, quæ de Athlante, Zoroa&longs;tro, Endimione, <lb/>Orpheo, Lino, alijs&queacute; traduntur.<emph.end type="italics"/></s>

<s>Iubal verò pater canentium cithara, & organo, hoc e&longs;t Mu&longs;icæ <lb/>auctor, omi&longs;&longs;us e&longs;t eò, quòd nimio interuallo <lb/>cæteros antecedat.</s><figure/><pb pagenum="39"/>

<s><emph type="italics"/>PRIMVM SECVLVM INCIPICNS<emph.end type="italics"/><lb/><arrow.to.target n="table7"/></s><table><table.target id="table7"/><row><cell>Ab Orbe cond. anno</cell><cell>3237</cell></row><row><cell>Ante primam Olymp. ann.</cell><cell>76</cell></row><row><cell>Ante Vrbem cond. ann.</cell><cell>100</cell></row><row><cell>Ante Chri&longs;ti natiuitatem ann.</cell><cell>852</cell></row><row><cell>Io&longs;aphat Iudæorum Rege.</cell><cell/></row><row><cell>Aremulo Latinorum Rege.</cell><cell/></row></table>

Line 14109

<s><emph type="italics"/>SECVNDVM SECVLVM INCIPIENS<emph.end type="italics"/><lb/><arrow.to.target n="table8"/></s><table><table.target id="table8"/><row><cell>Ab Vrbecondita.</cell><cell/></row><row><cell>Ab Orbe cond. ann.</cell><cell>3337</cell></row><row><cell>Olympiadis 6. ann.</cell><cell>4</cell></row><row><cell>Ante Chri&longs;ti natiuitatem ann.</cell><cell>752</cell></row><row><cell>Ozia ludæorum Rege.</cell><cell/></row><row><cell>Romulo Latinorum Rege.</cell><cell/></row></table>

Line 14385

<s>CRATISTVS, qui &longs;olo naturæ acumine, quoduis geometricum Pro<lb/>blema, quamuis difficile re&longs;oluebat. </s>

<s>Proclus.</s><figure/><pb pagenum="43"/>

<s>Proclus.</s><pb pagenum="43"/>

<s><emph type="italics"/>QVINTVM SECVLVM INCIPIENS<emph.end type="italics"/><lb/><arrow.to.target n="table11"/></s><table><table.target id="table11"/><row><cell>Ab Vrbe cond. ann.</cell><cell>301</cell></row><row><cell>Ante Chri&longs;tum ann.</cell><cell>452</cell></row><row><cell>P. Sextio, T. Memnio Con&longs;&longs;.</cell><cell/></row></table>

Line 14811

<s>ob&longs;eruauit primam &longs;tellam Arietis po&longs;t &longs;ectionem vernalem gr. 2.</s>

<s><emph type="italics"/>SEPTIMVM SECVLVM INCIPIENS<emph.end type="italics"/><lb/><arrow.to.target n="table13"/></s><table><table.target id="table13"/><row><cell>Ab Vrbe cond. anno</cell><cell>501</cell></row><row><cell>Ante Chri&longs;ti natiuitatem ann.</cell><cell>252</cell></row></table>

Line 15447

<s>canticum Eccle&longs;ia&longs;ticum ordi<lb/>nauit, qui ab eo denominatur. </s>

<s>choro etiam modum con&longs;tituit.</s><figure/><pb pagenum="55"/>

<s>choro etiam modum con&longs;tituit.</s><pb pagenum="55"/>

<s><emph type="italics"/>DECIMUMSEXTVM SECVLVM<emph.end type="italics"/><lb/><arrow.to.target n="table21"/></s><table><table.target id="table21"/><row><cell>Septimum verò Chri. ab ann. Chri&longs;ti</cell><cell>601</cell></row><row><cell>Gregorio magno &longs;um. Pont.</cell><cell/></row><row><cell>Mauritio Imperatore orientis.</cell><cell/></row><row><cell>Longobardis in Italia regnantibus.</cell><cell/></row></table>

Line 15549

<s>BEN MVSA Arabs. </s>

<s>de figuris planis, & &longs;phæricis.</s><figure/><pb pagenum="57"/>

<s>de figuris planis, & &longs;phæricis.</s><pb pagenum="57"/>

<s><emph type="italics"/>VIGESIMVM SECVLVM<emph.end type="italics"/><lb/><arrow.to.target n="table25"/></s><table><table.target id="table25"/><row><cell>Vndecimum <expan abbr="verõ">verom</expan> Chr. ab ann. Chri&longs;ti</cell><cell>1001</cell></row><row><cell>Silue&longs;tro &longs;um. Pont.</cell><cell/></row><row><cell>Impp. Ottone 3. occid.</cell><cell/></row><row><cell>Ba&longs;ilio, & Con&longs;t. orient.</cell><cell/></row></table>

Line 16342

<pb/>

<s><emph type="italics"/>Errata, quæ Lectorem &longs;isterent, &longs;ic corrigantur.<emph.end type="italics"/><lb/><arrow.to.target n="table32"/></s><pb/><table><table.target id="table32"/><row><cell><emph type="italics"/>pag. linea. erratum. correctum.<emph.end type="italics"/></cell><cell/><cell/><cell/></row><row><cell>4</cell><cell>2</cell><cell>per&longs;ua&longs;i&longs;ti</cell><cell>perua&longs;i&longs;ti</cell></row><row><cell>18</cell><cell>22</cell><cell>quadratum</cell><cell>quadrangulum</cell></row><row><cell>36</cell><cell>34</cell><cell>inue&longs;tare</cell><cell>inueftigare</cell></row><row><cell>38</cell><cell>15</cell><cell>A G,</cell><cell>A C G,</cell></row><row><cell>39</cell><cell>22</cell><cell>æquatiorem</cell><cell>æquatorem</cell></row><row><cell>51</cell><cell>46</cell><cell><expan abbr="vnoq;">vnoque</expan></cell><cell><expan abbr="vnoquoq;">vnoquoque</expan></cell></row><row><cell>70</cell><cell>in figura defiderantur duæ lineolæ rectæ A I, & I E,</cell><cell/><cell/></row><row><cell>80</cell><cell>13</cell><cell>Tex.</cell><cell>Textus 59.</cell></row><row><cell>ibid.</cell><cell>in figura pro O, ponatur C,</cell><cell/><cell/></row><row><cell>82</cell><cell>13</cell><cell>Tex.</cell><cell>Tex. 110.</cell></row><row><cell>ibid.</cell><cell>45</cell><cell>Vr&longs;am</cell><cell>ad Vr&longs;am</cell></row><row><cell>83</cell><cell>42</cell><cell>Tex.</cell><cell>Tex. 65.</cell></row><row><cell>107</cell><cell>10</cell><cell>fides</cell><cell>fidem</cell></row><row><cell>ibid.</cell><cell>11</cell><cell>&longs;axiatritu</cell><cell>&longs;ci&longs;&longs;a, & attrita</cell></row><row><cell>166</cell><cell>in 2 figura de&longs;ideratur linea P R.</cell><cell/><cell/></row><row><cell>127</cell><cell>28</cell><cell>L,</cell><cell>C,</cell></row><row><cell>155</cell><cell>in 2. figura dee&longs;t linea K P:</cell><cell/><cell/></row><row><cell/><cell>altera verò, quæ dextror&longs;um vergit, &longs;uperfluaeft.</cell><cell/><cell/></row><row><cell>220</cell><cell>45</cell><cell>quadratum</cell><cell>quadrangulum</cell></row><row><cell>221</cell><cell>1</cell><cell>quadratum</cell><cell>quadrangulum</cell></row><row><cell>231</cell><cell>30</cell><cell>incrementa</cell><cell>decrementa</cell></row><row><cell>ibid.</cell><cell>31</cell><cell>decrementis</cell><cell>incrementis</cell></row><row><cell>235</cell><cell>19</cell><cell>L O,</cell><cell>M N,</cell></row><row><cell>ibid.</cell><cell>21</cell><cell>N M,</cell><cell>O L,</cell></row><row><cell>ibid.</cell><cell>34</cell><cell>H L,</cell><cell>N L,</cell></row><row><cell>237</cell><cell>in figura linea K D L, deberet directè ad punctum A, tendere</cell><cell/><cell/></row><row><cell>256</cell><cell>4</cell><cell>toto</cell><cell>tono</cell></row><row><cell>264</cell><cell>14</cell><cell>numeris</cell><cell>neruis</cell></row><row><cell>266</cell><cell>44</cell><cell>cæteris</cell><cell>cæteri</cell></row><row><cell>267</cell><cell>45</cell><cell>indicium</cell><cell>indicium</cell></row><row><cell>274</cell><cell>27</cell><cell>contrà verum</cell><cell>contrà, verum</cell></row><row><cell>279</cell><cell>30</cell><cell>MKINQ,</cell><cell>MHINQ,</cell></row><row><cell/><cell/><cell><emph type="italics"/>In Additamento.<emph.end type="italics"/></cell><cell/></row><row><cell>9</cell><cell>5</cell><cell>vncìa</cell><cell>vnica</cell></row><row><cell>9</cell><cell>28</cell><cell>proportionali</cell><cell>proportionalis</cell></row><row><cell>14</cell><cell>28</cell><cell>diuionem</cell><cell>diui&longs;ionem</cell></row><row><cell>18</cell><cell>22</cell><cell>78.</cell><cell>7. 8.</cell></row></table><figure/>

<s>BONONIÆ</s>

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<s>Superiorum permi&longs;&longs;u.<lb/><figure id="fig152"/></s> </chap> </body> <back/> </text></archimedes>

<s>Superiorum permi&longs;&longs;u.<lb/></s> </chap> </body> <back/> </text></archimedes>

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