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version 1.1, 2002/06/18 09:37:11

version 1.2, 2002/06/24 18:03:48

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<!ELEMENT s

<!ATTLIST s

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<info>

<author>Girolamo Cardano</author>

<author>Cardano, Girolamo</author>

<title>De proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum</title>

<place>Lyons</place>

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<s>Proportionem in proportionem duci e&longs;t, quoties recto ordine <lb/>tres quantitates in ei&longs;dem collo <expan abbr="can&ttilde;">cantur</expan>: ut &longs;int tres quan <lb/>

<arrow.to.target n="fig1"></arrow.to.target><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s>

<figure id="fig1"></figure><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s>

<s>Tertiadecima diffinitio.</s>

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<s>Velut &longs;i comparentur a b & b c ad d, inde tota <lb/>

<arrow.to.target n="fig2"></arrow.to.target><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con <lb/><expan abbr="iunctã">iunctam</expan> ex duabus proportionibus a b ad d & b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. Hoc & duo &longs;equentes &longs;icut & du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon<lb/>&longs;trabitur e&longs;&longs;e. nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>

<figure id="fig2"></figure><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con <lb/><expan abbr="iunctã">iunctam</expan> ex duabus proportionibus a b ad d & b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. Hoc & duo &longs;equentes &longs;icut & du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon<lb/>&longs;trabitur e&longs;&longs;e. nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>

<s>Quintadecima diffinitio.</s>

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<s>Inter quantitatem, & defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. Sit a b linea, & detra<lb/>cta quantitas b c, non maior a b & d &longs;it alia quæuis quantitas eiu&longs;<lb/>

<arrow.to.target n="fig3"></arrow.to.target><lb/><expan abbr="d&etilde;">dem</expan> generis, dico quòd inter d & b c e&longs;t propor<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. Sed ut b c e&longs;t defectus, nulla e&longs;t propor<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, & <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s>

<figure id="fig3"></figure><lb/><expan abbr="d&etilde;">dem</expan> generis, dico quòd inter d & b c e&longs;t propor<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. Sed ut b c e&longs;t defectus, nulla e&longs;t propor<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, & <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s>

<s>Quinta animi communis &longs;ententia.</s>

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<s>Sit proportio lineæ a ad lineam b, ut anguli cad angulum d, &longs;ta<lb/>

<arrow.to.target n="marg6"></arrow.to.target><lb/>tuatur e monas in genere a <lb/>

<arrow.to.target n="fig4"></arrow.to.target><lb/>b, & fiat fad e, ut cad d, & du <lb/>

<arrow.to.target n="marg7"></arrow.to.target><lb/>catur<gap/>a in f & b in e, & pro<lb/>ducantur g & h. Quia ergo <lb/>

<arrow.to.target n="marg8"></arrow.to.target><lb/>fe&longs;t proportio ip&longs;a, erit g ad <lb/>

<arrow.to.target n="marg9"></arrow.to.target><lb/>a ut c ad d, &longs;ed h e&longs;t æqualis <lb/>b, igitur a ad h ut ad b. Du<lb/>cta ergo dicetur proportio a <lb/>

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<s>Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s>

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<s>Sint a b c quantitates dico proportio<lb/>

<arrow.to.target n="fig5"></arrow.to.target><lb/>nem a ad c, produci ex proportione a ad b </s>

<figure id="fig5"></figure><lb/>nem a ad c, produci ex proportione a ad b </s>

<s>

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<s>Sit proportio a ad c, & b ad d, & &longs;int c & d <lb/>

<arrow.to.target n="fig6"></arrow.to.target><lb/>æquales, dico quòd proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, & b ad <lb/>d diui&longs;o compo&longs;ito per duplam. Quia enim </s>

<figure id="fig6"></figure><lb/>æquales, dico quòd proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, & b ad <lb/>d diui&longs;o compo&longs;ito per duplam. Quia enim </s>

<s>

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<s>Sint propo&longs;itæ proportiones a ad c & <lb/>

<arrow.to.target n="fig7"></arrow.to.target><lb/>b ad d, & a&longs;&longs;umo e ad c, iuxta ea quæ Eu<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s>

<figure id="fig7"></figure><lb/>b ad d, & a&longs;&longs;umo e ad c, iuxta ea quæ Eu<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s>

<s>

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<s>Omnium quatuor quantitatum propo&longs;ita <lb/>prima, quæ non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, quàm alia ad aliam <lb/>

<arrow.to.target n="fig8"></arrow.to.target><lb/>erit proportio confu&longs;a illarum, ut pro<lb/>ducti ex aggregato primæ & tertiæ in

<figure id="fig8"></figure><lb/>erit proportio confu&longs;a illarum, ut pro<lb/>ducti ex aggregato primæ & tertiæ in

<pb pagenum="15"/>tertiam, ad productum ex aggregato tertiæ & omiotatæ ad &longs;ecun<lb/>dam in ip&longs;am quartam.</s>

<s>Hæc magis reducit confu&longs;am proportionem ad notitiam, quàm, <lb/>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu&ecedil; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, duæ <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. Summatur ergo a b, c, d & e, & non &longs;it maior propor<lb/>tio d ad e, quàm a b ad c, & &longs;tatuatur tunc prima a b, &longs;ecunda c, ter<lb/>tia d, quarta e, & po&longs;tquam non e&longs;t minor ratio a b ad c, quàm d ad <lb/>e, &longs;umatur a f ad c, ut d ad e. licet enim hoc facere. Dico quod pro<lb/>portio confufa a b & d ad c & e e&longs;t uelut producti ex aggregato a b <lb/>& d in d ad productum ex aggregato a f & d in e. Statuatur aggre<lb/>

<arrow.to.target n="marg49"></arrow.to.target><lb/>gatum a b & d linea a d prima quantitas, & aggregatum a f & d, <lb/>

<arrow.to.target n="fig9"></arrow.to.target><lb/>a d &longs;ecunda quantitas, & d tertia, <lb/>& c quarta, & ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri<lb/>mam propo&longs;itionem g ad h pro<lb/>

<figure id="fig9"></figure><lb/>a d &longs;ecunda quantitas, & d tertia, <lb/>& c quarta, & ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri<lb/>mam propo&longs;itionem g ad h pro<lb/>

<arrow.to.target n="marg50"></arrow.to.target><lb/>ducta ex proportionibus a b d ad <lb/>a f d, & d ad c. Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. Proportio uerò a b d ad a f d, & <lb/>a f d ad e c producunt proportio<lb/>nem a b d ad c & e per &longs;ecundam propo&longs;itionem, harum igitur con<lb/>&longs;u&longs;a a b ad c, & d ad e, & e&longs;t proportio a b d ad c & e, producuntur <lb/>ex proportionibus a b d ad a f d, & d ad e. Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, & d ad e, quod erat demon&longs;trandum.</s>

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<s>Propo&longs;itio decima&longs;eptima.</s>

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<s>

<arrow.to.target n="marg54"></arrow.to.target><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua<lb/>

<arrow.to.target n="fig10"></arrow.to.target><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. Rur&longs;us, quia b quadrati ad c quadratum, <lb/>

<figure id="fig10"></figure><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. Rur&longs;us, quia b quadrati ad c quadratum, <lb/>

<arrow.to.target n="marg55"></arrow.to.target><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, & ita d ad e, igitur <lb/>

<arrow.to.target n="marg56"></arrow.to.target><lb/>gnomonum b c cum quadrato d ad aggrega<lb/>tum c d e quadratorum, ut d ad e, &longs;ed c gno<lb/>mo cum d quadrato perficit c quadratum, <lb/>& c quadratum cum gnomone b perficit <lb/>quadratum b, igitur proportio quadrati b <lb/>ad quadrata c d e, ut d quadrati a d e. Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. Hæc proponitur ab Archimede in libro de quadrato <lb/>æquali parabolæ, & minus generaliter & pluribus demon&longs;tratur. <lb/>Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad<lb/>damque aliud quod ex hoc &longs;equitur.<lb/>

<arrow.to.target n="marg57"></arrow.to.target></s>

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<s>Si fuerint quotlibet <expan abbr="quãtitates">quantitates</expan> omnes analogæ præter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui primæ & &longs;ecundæ <lb/>ad &longs;ecundam, erit proportio primæ ad aggregatum omnium alia<lb/>rum ueluti penultimæ ad ultimam.</s>

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<s>Hæc enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/>exemplo autem in numeris à latere <lb/>po&longs;ito uides declarationem. nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui primæ & &longs;ecundæ ad ip&longs;am &longs;e<lb/>cundam &longs;cilicet ad 54.</s>

<figure id="fig11"></figure><lb/>exemplo autem in numeris à latere <lb/>po&longs;ito uides declarationem. nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui primæ & &longs;ecundæ ad ip&longs;am &longs;e<lb/>cundam &longs;cilicet ad 54.</s>

<s>

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<s>Si fu erint aliquot quantitates arithmeticæ omiologæ, quarum <lb/>exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upple<lb/>menta ad &ecedil;qualitatem maximè adiungantur, erunt quadrata omni<lb/>um quantitatum æqualium adiecto rur&longs;us quadrato primæ cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan<lb/>titatum eiu&longs;dem tripla aggregato quadra<lb/>

<arrow.to.target n="fig12"></arrow.to.target><lb/>torum omnium quantitatum primi ordinis <lb/>

<figure id="fig12"></figure><lb/>torum omnium quantitatum primi ordinis <lb/>

<arrow.to.target n="marg66"></arrow.to.target><lb/>pariter acceptis.</s>

<s>Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. Arithmetica di&longs;po&longs;it&ecedil; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&ecedil; &longs;it h, &longs;it &ecedil;qualis diffe<lb/>renti&ecedil; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oens</expan> <lb/>fiant &ecedil;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&ecedil; <lb/>à maiori. E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti

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<s>Cùm enim quantitates hæ non fuerint &ecedil;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun

<pb pagenum="22"/>dam harum, quod proportio primæ ad <expan abbr="quartã">quartam</expan> producitur ex pro<lb/>portione primæ ad &longs;ecundam, &longs;ecund&ecedil; ad tertiam, & terti&ecedil; ad quar <lb/>tam: ergo non ex &longs;olis proportionibus primæ ad &longs;ecundam, & ter<lb/>tiæ ad quartam, & &longs;imiliter ex prima harum proportio prim&ecedil; ad &longs;e<lb/>cundam, & tertiæ ad quartam producunt proportionem producti <lb/>primæ in &longs;ecundam ad productum tertiæ in quartam. Et in multi<lb/>plicatione proportio, quæ &longs;olet e&longs;&longs;e inter producta illa, & e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. Sint igitur quantitates a b c d, <lb/>& &longs;it b æqualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/>

<arrow.to.target n="fig13"></arrow.to.target><lb/>tio a ad d producta ex proportioni<lb/>bus a ad b, b ad c, & c ad d, producan<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. proportio c ad f, erit igitur pro<lb/>portio e ad f, &longs;i multiplicetur per pro<lb/>portionem b ad c eadem quæ prius, & </s>

<figure id="fig13"></figure><lb/>tio a ad d producta ex proportioni<lb/>bus a ad b, b ad c, & c ad d, producan<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. proportio c ad f, erit igitur pro<lb/>portio e ad f, &longs;i multiplicetur per pro<lb/>portionem b ad c eadem quæ prius, & </s>

<s>

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<s>Sit motus in circulo &longs;eu per circulum in orbe cuius &longs;it centrum, <lb/>&longs;it c mundi centrum: igitur ex diffinitione circuli tantum di&longs;tabit a, <lb/>quantum b ab ip&longs;o c: &longs;ed in motu naturali per pr&ecedil;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius,

<arrow.to.target n="fig14"></arrow.to.target><lb/>non naturalis. nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. Sit modo rota e f g, di <lb/>co enon moueri motu circulari nam linea <lb/>e <expan abbr="clõgior">clongior</expan> e&longs;t g c, ergo recta mouetur ad cen <lb/>trum non circa centrum. Indicio etiamid <lb/>e&longs;t: quòd &longs;i in e ponatur fru&longs;tum aliquod <lb/>in&longs;igne plumbi in motu ad g per f de&longs;cen<lb/>det raptim: at dum ex g in e magna cum dif<lb/>ficultate, igitur motus hic non e&longs;t naturalis, <lb/>nec circularis. nihil etiam hoc modo &longs;ponte mouetur. Sed cum non <lb/>moueatur per rectam naturaliter, nec æquidi&longs;tans à centro per cir<lb/>culum relinquitur, ut moueatur motu uiolento, aut mi&longs;to, &longs;ed non <lb/>ex uoluntario, cum nullo modo moueatur æquidi&longs;tans à centro, <lb/>&longs;ed &longs;emper ab e lineæ ad centrum fiant breuiores, liquet e&longs;&longs;e mo<lb/>tum uiolentum: aut mi&longs;tum ex naturali, & uiolento.</s>

<figure id="fig14"></figure><lb/>non naturalis. nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. Sit modo rota e f g, di <lb/>co enon moueri motu circulari nam linea <lb/>e <expan abbr="clõgior">clongior</expan> e&longs;t g c, ergo recta mouetur ad cen <lb/>trum non circa centrum. Indicio etiamid <lb/>e&longs;t: quòd &longs;i in e ponatur fru&longs;tum aliquod <lb/>in&longs;igne plumbi in motu ad g per f de&longs;cen<lb/>det raptim: at dum ex g in e magna cum dif<lb/>ficultate, igitur motus hic non e&longs;t naturalis, <lb/>nec circularis. nihil etiam hoc modo &longs;ponte mouetur. Sed cum non <lb/>moueatur per rectam naturaliter, nec æquidi&longs;tans à centro per cir<lb/>culum relinquitur, ut moueatur motu uiolento, aut mi&longs;to, &longs;ed non <lb/>ex uoluntario, cum nullo modo moueatur æquidi&longs;tans à centro, <lb/>&longs;ed &longs;emper ab e lineæ ad centrum fiant breuiores, liquet e&longs;&longs;e mo<lb/>tum uiolentum: aut mi&longs;tum ex naturali, & uiolento.</s>

<s>Propo&longs;itio uige&longs;imaquinta.</s>

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<s>Sit mobile a cui partes &longs;ubiaceant directæ b, & &longs;it graue. Et pa<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen<lb/>dunt ante a, & trahunt partes c & d adh&ecedil;rentes &longs;ecum, atque ita e c d f <lb/>

<arrow.to.target n="fig15"></arrow.to.target><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g & h cum a tran&longs;it in b, ne detur uacuum, tran<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio <lb/>reimpetu inferius.</s>

<figure id="fig15"></figure><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g & h cum a tran&longs;it in b, ne detur uacuum, tran<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio <lb/>reimpetu inferius.</s>

<s>

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<s>Ex quo patet, quòd motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe<lb/>

<arrow.to.target n="marg86"></arrow.to.target><lb/>cie, ut cùm quis iacit lapidem è turri: uel ex occulto naturali, & uio<lb/>lento manife&longs;to: uelut cùm quis iacit lapidem, & de&longs;cendit po&longs;tmo <lb/>

<arrow.to.target n="fig16"></arrow.to.target><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, & naturali oc<lb/>culto: uel ratione medij, & hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis,

<figure id="fig16"></figure><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, & naturali oc<lb/>culto: uel ratione medij, & hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis,

<pb pagenum="27"/>& medij ad mobile, ac medium, quod re&longs;i&longs;tit. Quarto ex motibus <lb/>imperfectis natura &longs;ua, & non e&longs;t uera mi&longs;tio, & hoc apparet in mo<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque æquales, neque perfe <lb/>ctè circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. Etideo de<lb/>mon&longs;trationes illæ Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s>

<s>Propo&longs;itio trige&longs;ima&longs;ecunda.</s>

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<s>Sint duo mobilia a & b magnitudine, & forma omnino paria, <lb/>& &longs;int media c & d, exempli gratia: & pertran&longs;eant æquale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei quæ e&longs;t raritatis c ad raritatem d. Quia <lb/>enim feruntur æqualiter, nam in æquali tem<lb/>

<arrow.to.target n="fig17"></arrow.to.target><lb/>pore, &longs;eu eodem æqualia &longs;patia pertran&longs;e<lb/>unt, erit proportio potentiæ a cum &longs;uo auxi<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten<lb/>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t æqua<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>quæ e&longs;t raritatis c ad raritatem d.</s>

<figure id="fig17"></figure><lb/>pore, &longs;eu eodem æqualia &longs;patia pertran&longs;e<lb/>unt, erit proportio potentiæ a cum &longs;uo auxi<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten<lb/>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t æqua<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>quæ e&longs;t raritatis c ad raritatem d.</s>

<s>SCHOLIVM PRIMVM.</s>

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<s>Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra<lb/>

<arrow.to.target n="fig18"></arrow.to.target><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, & effi<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua<lb/>draginta octo, tota igitur proportio, qua mo<lb/>uebitur a in c, erit centum nonaginta duorum, & hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. Pro<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</s>

<figure id="fig18"></figure><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, & effi<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua<lb/>draginta octo, tota igitur proportio, qua mo<lb/>uebitur a in c, erit centum nonaginta duorum, & hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. Pro<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</s>

<s>Quòd &longs;i quis neget tantundem augere c actionem a, quanto mi<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, & &longs;it proportio b ad a dupli<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam&longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicatæ. <lb/>Sed ex duabus proportionibus a ad c, & actionis ex c ad a produ<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptimã">decimam&longs;eptimam</expan> &longs;exti Elemento<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, & actio<lb/>nis a in c, quare æqualis f, igitur proportio b ad a duplicata ei, quæ <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s>

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<s>Sit cubus a b c eius quadrata, &longs;uperficies a <lb/>

<arrow.to.target n="fig19"></arrow.to.target><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, & toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>

<figure id="fig19"></figure><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, & toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>

<s>

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<s>Quoniam facta uariatione in hypate, quæ e&longs;t <lb/>in Diapa&longs;on, uel bis Díapa&longs;on maiore interual<lb/>

<arrow.to.target n="fig20"></arrow.to.target><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in<lb/>teruallum ex c in d, igitur maior e&longs;t b d, quàm a c <lb/>ergo &longs;ingulæ uoces inter b & d magis di&longs;tant, <lb/>

<figure id="fig20"></figure><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in<lb/>teruallum ex c in d, igitur maior e&longs;t b d, quàm a c <lb/>ergo &longs;ingulæ uoces inter b & d magis di&longs;tant, <lb/>

<arrow.to.target n="fig21"></arrow.to.target><lb/>quàm inter a & c, & quanto magis appropin<lb/>quant ad d, igitur d maius e&longs;t quàm b. Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b & <lb/>cip&longs;o a. O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. Motus autem e&longs;t res, quies, <lb/>priuatio.</s>

<figure id="fig21"></figure><lb/>quàm inter a & c, & quanto magis appropin<lb/>quant ad d, igitur d maius e&longs;t quàm b. Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b & <lb/>cip&longs;o a. O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. Motus autem e&longs;t res, quies, <lb/>priuatio.</s>

<s>Secundum &longs;ic: nam remi&longs;sio mota non feriet aurem, ideò &longs;onum <lb/>non pariet ob nimiam tarditatem. At in uelo ci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, & non contrahitur ni&longs;i per mu&longs;cu<lb/>los, igitur contentio illa finem habet. Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in aëre <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut fermè ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s>

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<s>Tertium &longs;ic &longs;it a b humi<lb/>lior uox, quæ excre&longs;cat &longs;e<lb/>mitonio minore &longs;olum in <lb/>c, & &longs;it d e dupla ad ab &longs;e<lb/>

<arrow.to.target n="fig22"></arrow.to.target><lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, & in acutis ubi ex<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui&longs;u<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cen dunt ad bis diapa <lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. Etideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga

<figure id="fig22"></figure><lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, & in acutis ubi ex<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui&longs;u<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cen dunt ad bis diapa <lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. Etideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga

<pb pagenum="30"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera<lb/>tur, à quibu&longs;dam non omnino feratur.</s>

<s>SCHOLIVM.</s>

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<s>

<arrow.to.target n="marg96"></arrow.to.target><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra<lb/>

<arrow.to.target n="fig23"></arrow.to.target><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. Manife&longs;tum e&longs;t autem, quod hic <lb/>

<figure id="fig23"></figure><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. Manife&longs;tum e&longs;t autem, quod hic <lb/>

<arrow.to.target n="marg97"></arrow.to.target><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/>

<arrow.to.target n="marg98"></arrow.to.target></s>

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<s>Ex hoc patet cur naues & currus ab initio tardè & difficulter mo <lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s>

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<s>Sint duæ magnitudines a & b, & &longs;it a maior <lb/>

<arrow.to.target n="fig24"></arrow.to.target><lb/>b, & &longs;umatur exempli gratia a quater cum b &longs;e<lb/>mel, & b quater cum a &longs;emel, dico, quod propor <lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s>

<figure id="fig24"></figure><lb/>b, & &longs;umatur exempli gratia a quater cum b &longs;e<lb/>mel, & b quater cum a &longs;emel, dico, quod propor <lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s>

<s>

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<s>Sæpe contingit, ut quinque homines moueant nauim, & &longs;patium <lb/>ad tempus notum, & etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. Sit ergo a numerus hominum, b na<lb/>

<arrow.to.target n="fig25"></arrow.to.target><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, & g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, & h &longs;patium, dico, quod erit f motor, &longs;eu numerus

<figure id="fig25"></figure><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, & g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, & h &longs;patium, dico, quod erit f motor, &longs;eu numerus

<pb pagenum="34"/>hominum notus. Quoniam ergo notum e&longs;t a & c, quia e&longs;t æquale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per præcedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. Sed per eadem ut temporis d ad &longs;patium illud, <lb/>ita g ad h, ergo cum nota &longs;int d e f g erit etiam h, & ita conuertendo.</s>

<s>Propo&longs;itio quadrage&longs;imaquinta.</s>

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<s>Archimedes nititur huic fundamento, quod pondera, quæ pro<lb/>portionem mutuam habent, ut di&longs;tantiæ à libella a, quæ &longs;u&longs;pen<lb/>duntur, æqualiter ponderant, &longs;it ergo libella a b, & &longs;u&longs;pen&longs;a in a cen <lb/>trum mundi c, ad quod dirigitur pondus, & liquet, quod ip&longs;um <lb/>non &longs;e inclin abit ex uige&longs;imatertia propo&longs;itione. Si ergo ponantur <lb/>lo co lineæ b d in e & f, & &longs;it proportio e b <lb/>

<arrow.to.target n="fig26"></arrow.to.target><lb/>ad b f, ut g ad h, dico, quòd erit æquili<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et |&longs;u&longs;pen&longs;um h, moueretur in re<lb/>cta e h per eandem, quia ergo retinetur, mo<lb/>uetur per obliquam h k, & &longs;umatur in pro<lb/>pin quum punctum in b e, & n in æquali di<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon<lb/>dere h, & in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, & ita in b erit penè infinita: quia quanta ui ex<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o æqualis m p, & n b, & b m æquales, ut uerò g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s>

<figure id="fig26"></figure><lb/>ad b f, ut g ad h, dico, quòd erit æquili<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et |&longs;u&longs;pen&longs;um h, moueretur in re<lb/>cta e h per eandem, quia ergo retinetur, mo<lb/>uetur per obliquam h k, & &longs;umatur in pro<lb/>pin quum punctum in b e, & n in æquali di<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon<lb/>dere h, & in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, & ita in b erit penè infinita: quia quanta ui ex<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o æqualis m p, & n b, & b m æquales, ut uerò g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s>

<s>

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<s>Etrota, quanto uelocius mouetur in ambitu, tanto mi <lb/>norem habet uim: &longs;ed propter aërem, qui &longs;ecum circum<lb/>

<arrow.to.target n="fig27"></arrow.to.target><lb/>fertur, mouetur magno impetu, & magnas facit læ&longs;iones. <lb/>Ideò hoc in cono non accidit.</s>

<figure id="fig27"></figure><lb/>fertur, mouetur magno impetu, & magnas facit læ&longs;iones. <lb/>Ideò hoc in cono non accidit.</s>

<s>

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<s>Sint duo mobilia a & b in eodem pun<lb/>

<arrow.to.target n="fig28"></arrow.to.target><lb/>cto, quæ æqualiter uer&longs;us candem partem <lb/>moueantur æqualibus in temporibus, inui <lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho <lb/>rum differentia &longs;it h. Dum ita que a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu<lb/>li ad c d b, ut g ad h, & b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem & cum circulo & b c & c d b diffe<lb/>rentia, & &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a & b &longs;int in d ultra perfectas circulationes uterque eorum pertran <lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un

<figure id="fig28"></figure><lb/>cto, quæ æqualiter uer&longs;us candem partem <lb/>moueantur æqualibus in temporibus, inui <lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho <lb/>rum differentia &longs;it h. Dum ita que a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu<lb/>li ad c d b, ut g ad h, & b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem & cum circulo & b c & c d b diffe<lb/>rentia, & &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a & b &longs;int in d ultra perfectas circulationes uterque eorum pertran <lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un

<pb pagenum="37"/>guem. Hoc declarato ponatur m &longs;patium compofitum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, & erit ex de<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, & &longs;it p &longs;pa<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, & <lb/>&longs;it q differentia o, & p qu&ecedil; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m & b, n, erit m ad n, ut a ad b, & eadem ratio<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m & n, & q, differentia o & p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana <lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. Reuertor igitur ad propo&longs;i<lb/>tum, cum &longs;it m ad o, ut o ad q, & m ad o, ut n ad p, ex &longs;extadecima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;extadecima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, & diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon<lb/>&longs;trandum.</s>

<s>

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<s>Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. Dico quod primum redibunt in numero producto ex <lb/>&longs;eptem quinque & duobus, qui &longs;unt numeri primi, & erit ille nume<lb/>rus &longs;eptuaginta annorum. Nam in &longs;eptuaginta annis a perficiet tri<lb/>gintaquinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in trigintaquinque annis igitur b & c per<lb/>ficient perfectos circuitus, ergo <expan abbr="redib&utilde;t">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat trigintaquinque<lb/>aliter non fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum

<pb pagenum="38"/>ergo iam &longs;upponatur numerari a b & c non numerabitur a b a, er<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b & c. Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/>

<arrow.to.target n="fig29"></arrow.to.target><lb/>quo illorum temporum, auferantur perfe<lb/>ctæ circulationes, & <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi<lb/>tur oportebit ut hæ portiones &longs;int æqua<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. Rur&longs;us dicantur conuenire in annis qua</s>

<figure id="fig29"></figure><lb/>quo illorum temporum, auferantur perfe<lb/>ctæ circulationes, & <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi<lb/>tur oportebit ut hæ portiones &longs;int æqua<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. Rur&longs;us dicantur conuenire in annis qua</s>

<s>

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<s>Sint in circulo a b c d e f g: a & b iuncta, & in primo congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cunque<lb/>tempora</expan> &longs;int æqualia, erunt & circuitus totidem numero, & exce&longs;<lb/>&longs;us æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb/>

<arrow.to.target n="fig30"></arrow.to.target><lb/>k l. Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>& portionis fuerit commen&longs;um circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commen&longs;a</expan> ad circulum, </s>

<figure id="fig30"></figure><lb/>k l. Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>& portionis fuerit commen&longs;um circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commen&longs;a</expan> ad circulum, </s>

<s>

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<s>Quod uerò non contingat coniungi in alio loco, neque tem<lb/>pore &longs;it, ut a b iungantur in c, & &longs;it reuolutio a triplex integra, & b

<pb pagenum="42"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b funt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/>

<arrow.to.target n="fig31"></arrow.to.target><lb/>puncto. Si enim primùm in eodem pun<lb/>cto, &, gratia exempli, in quatuor annis <lb/>congruit enim, & b dicamus quod per<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. Cum enim tempora di<lb/>uer&longs;a diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/>

<figure id="fig31"></figure><lb/>puncto. Si enim primùm in eodem pun<lb/>cto, &, gratia exempli, in quatuor annis <lb/>congruit enim, & b dicamus quod per<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. Cum enim tempora di<lb/>uer&longs;a diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/>

<arrow.to.target n="table13"></arrow.to.target><lb/>numeri in eadem ratione. Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, & diui&longs;o 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non pote&longs;t e&longs;&longs;e æquale 1/3. Si enim per <lb/>præcedentem repetuntur, ergo non po&longs;<lb/>&longs;unt redire, doneciterum coniung antur in ip&longs;o a. Si enim aliter &longs;it <lb/>ut ex e, igitur e c e&longs;t æqualis a c pars toti, quod contingere non po<lb/>te&longs;t. Sin uerò coniunctio fiat in d, igitur per præcedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um<lb/>pti primi numeri. Veluti in exemplo con&longs;tituimus, quod a, & b <lb/>conueniunt in c in decem annis, & a c e&longs;t tertia pars circuitus: er<lb/>go in triginta annis conueniunt in a, & in quadraginta rur&longs;us in c. <lb/>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con<lb/>gre&longs;&longs;u, & diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, & &longs;i per 3 exiret 13 1/3, & mani<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu<lb/>merum, utpote 4 & exit numerus cum eadem parte &longs;cilicet 6 1/3 & <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro<lb/>portione. Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo<lb/>dem modo.</s>

<table>

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<s>Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & &longs;int a d, a e inconimen&longs;æ, dico quòd a b c nunquam con<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à prim o: &longs;i non con

<arrow.to.target n="fig32"></arrow.to.target><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/>Et quia hæ con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, & &longs;unt &longs;patia a d, & b cum c, & <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge<lb/>nere quantitatis a d, & a e per quinqua<lb/>ge&longs;imam, harum ergo erunt commen&longs;æ: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>à Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s>

<figure id="fig32"></figure><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/>Et quia hæ con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, & &longs;unt &longs;patia a d, & b cum c, & <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge<lb/>nere quantitatis a d, & a e per quinqua<lb/>ge&longs;imam, harum ergo erunt commen&longs;æ: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>à Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s>

<s>Propo&longs;itio quinquage&longs;imatertia.</s>

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<s>Sit orbis a b cuius cen<lb/>

<arrow.to.target n="fig33"></arrow.to.target><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu <lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or<lb/>biculus &longs;olidus aut &longs;emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

<figure id="fig33"></figure><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu <lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or<lb/>biculus &longs;olidus aut &longs;emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

<s>Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo</s>

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<s>Quoniam enim &longs;uperficies circuli, ut ab <lb/>

<arrow.to.target n="fig34"></arrow.to.target><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi</s>

<figure id="fig34"></figure><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi</s>

<s>

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<s>Secunda difficultas e&longs;t maior, & magis pertinet ad nos, & e&longs;t, <lb/>quòd non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliquã">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperaturæ &longs;implicium <lb/>medicamentorum cogno &longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. Sed &longs;i ordines &longs;er<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>&longs;it Arithmetica, uel Geometrica, uel Mu&longs;ica, & nihil mirum e&longs;&longs;et, <lb/>quod e&longs;&longs;et Mu&longs;ica, ut aliâs docuimus, ubitractauimus de differen<lb/>tia inter &longs;en&longs;um auditus, et ui&longs;us. Sed quia de hac nullus medicus ui <lb/>detur intellexi&longs;&longs;e, omittam hanc tractationem. Et quanquàm Gale<lb/>nus po&longs;sit uideri non exi&longs;tima&longs;&longs;e, quòd hi ordines non &longs;eruent <lb/>proportionem ullam, quia non au&longs;us e&longs;t tractare de temperamen<lb/>to medicamentorum compo&longs;itorum per rationem temperamen<lb/>ti &longs;implicium, nihilominus &longs;uppo&longs;ito quod ita e&longs;&longs;et, quod &longs;eruetur <lb/>altera proportionum, uolo o&longs;tendere rationem componendi in <lb/>utraque proportione & Arithmetica, & Geometrica. Ex quo &longs;e<lb/>quitur, quod Aueroes quàm o&longs;citanter tractauerit in quinto &longs;uo<lb/>rum collectaneorum de hoc, & non di&longs;tinguit, neque docet pri<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, & <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, quàm malè agant qui ei in arduis tan<lb/>tum tribuunt fidei, & authoritatis, &longs;ed hæc e&longs;t infelicitas no&longs;tra, & <lb/>ira Deorum. Suppo&longs;ito ergo quod primò ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/>

<arrow.to.target n="fig35"></arrow.to.target><lb/>& a &longs;it calida in primo gradu, & b in ter<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u<lb/>perficie rectangula b, hoc igitur erit to<lb/>tum mi&longs;tum, & quia quantitas medicamenti non mutatur quæ e&longs;t <lb/>a, b, ergo talia corpora æquantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo

<figure id="fig35"></figure><lb/>& a &longs;it calida in primo gradu, & b in ter<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u<lb/>perficie rectangula b, hoc igitur erit to<lb/>tum mi&longs;tum, & quia quantitas medicamenti non mutatur quæ e&longs;t <lb/>a, b, ergo talia corpora æquantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo

<pb pagenum="47"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paruæ huiu&longs;modi, & reliquæ, traduxi autem illas ad hunc <lb/>locuin, “quia pendent ex demon&longs;tratione hac: “duc numerum ordi<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti<lb/>tatum, exibit numerus ordinis compo&longs;iti. Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. Secunda &longs;i ex pluribus diuer&longs;arum, qualitatum, & ordi<lb/>num temperatum efficere uelis, duc quæ &longs;unt eiu&longs;dem qualitatis in <lb/>&longs;uas quantitates, & iunge, quod fit, diuide per numerum or dinis <lb/>medicamenti contrarij, exibit quantitas illius, &longs;ub qua &longs;i iungatur, <lb/>fiet medicamentum temperatum. Tertia cum nolueris ex tempera<lb/>to, & alio cuiu&longs;cunque ordinis medicamen conficere ordinis re<lb/>mi&longs;sionis, detrahe numerum ordinis eius, quod conficere uis ex nu <lb/>mero ordinis eius, quod habes, & cum re&longs;iduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit e&longs;t numerus quantita<lb/>tis medicamenti non temperati in comparatione ad temperatum.” <lb/>Ex his potes propo&longs;itis quibu&longs;cunque medicamentis conficere <lb/>antidotum &longs;ub quo cunque ordine remi&longs;siore potenti&longs;simo ex il<lb/>lis. Quarta in compo&longs;itione, quæ non fermente&longs;cit calida, calidis <lb/>iuncta &longs;emper opus augent, ut mel cum pipere. Quæ autem &longs;ub mi <lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s>

<s>Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, & quo ad numerum etiam in primo, & &longs;ecun <lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat à primo, quantum primus ab æqualitate, quia <lb/>unum & duo &longs;eruant proportionem, & æqualem di&longs;tantiam, &longs;ed in <lb/>cæteris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, & quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/>

<arrow.to.target n="fig36"></arrow.to.target><lb/>&longs;cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia<lb/>rum, & medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag

<figure id="fig36"></figure><lb/>&longs;cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia<lb/>rum, & medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag

<pb pagenum="48"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio &longs;ecundi ordinis. Secundum exemplum &longs;int eædem medicinæ, <lb/>& &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, & <lb/>fiunt &longs;exdecim, & duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex <lb/>decim, & remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecund i gradus uides ergo di&longs;crimen. rur&longs;us <lb/>&longs;int ambæ medicinæ calidæ, & ducemus, ut prius in tertio exem<lb/>plo, ubi proportio &longs;it Arithmetica iungendo duodecim cum qua<lb/>tuor, & fient &longs;exdecim, diuide per &longs;ex, exeunt duo, & duæ tertiæ, er<lb/>go erit calida in medio tertij gradus, rur&longs;us in quarto exemplo iun <lb/>gemus &longs;edecim cum quatuor, & fient uiginti, diuide per &longs;ex exi<lb/>bunt tria & tertia, & ita erit in medio tertij gradus, ut prius, &longs;ed &longs;i <lb/>ille quatuor unciæ e&longs;&longs;ent calidæ in quarto gradu, & illæ duæ unciæ <lb/>in &longs;ecundo gradu, ut prius ducendo quatuor in quatuor fiunt &longs;ex<lb/>decim, & duo in duo fiunt quatuor, iunge, & fient uiginti, diuide <lb/>per &longs;ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus &longs;ecundum proportionem Arithmeticam, &longs;ed &longs;ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua<lb/>tuor ut prius, &longs;cilicet productum duorum in duo fiunt triginta &longs;ex, <lb/>diuide per &longs;ex, exeunt &longs;ex, & quia &longs;ex ad quatuor maiorem habent <lb/>proportionem, quàm octo ad &longs;ex ideo hæc medicina erit calida ul<lb/>tra medium quarti gradus, iam ergo uides rationem, & differen<lb/>tiam horum.</s>

<s>Quod &longs;i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, re&longs;pondeo, quòd ueri&longs;imilius e&longs;t de <lb/>Arithmetica, quia illa proportio etiam quod &longs;it minor quatuor ad <lb/>trium, quàm trium ad duo, & multò minor quàm duo ad unum ni<lb/>hilominus longè plus operatur, quia tertius ordo iam incipit e&longs;&longs;e <lb/>præter naturam, & uidemus, quod læ&longs;io facta in uulnerato, etiam <lb/>quòd &longs;it quadruplo minor, plus nocet longè, quàm in &longs;ano qua<lb/>druplo maior: quia termini præter naturam &longs;unt ualdè angu&longs;ti in <lb/>comparatione ad latitudinem naturalem, &longs;icut etiam uidemus in<lb/>tendendis chordis &longs;corpionum, quod ultima pars e&longs;t breuis & ta<lb/>men homini tantam difficultatem adijcit. Notandum e&longs;t etiam, <lb/>quòd ob hoc diui&longs;erunt ordines in tres partes, uelut gingiber e&longs;t <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, & ita euphorbium e&longs;t calidum in principio quarti gra<lb/>dus, &longs;ed in fine principij piper, in prin cipio principij aqua &longs;epara<lb/>tionis in medio quarti ordinis, &longs;ed oleum chalcanthi factum ea ar<lb/>te, ut exurat paleas, &longs;icut ignis e&longs;t calidum in fine quarti ordinis, & <lb/>ita &longs;ufficiet diuidere propter eandem cau&longs;am primum, & &longs;ecun

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<s>O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob aëris motum, ubi uerò hæret e&longs;t ac &longs;i quie&longs;cat. <lb/>Eadem autem e&longs;t ratio in motis uiolenter, & naturaliter dum &ecedil;qua<lb/>li impetu feruntur. Sed &longs;ubitò po&longs;t etiam, quod motus æqualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im<lb/>

<arrow.to.target n="fig37"></arrow.to.target><lb/>pedimentum naturale. Sed &longs;i uis mouens fuerit <lb/>adeò ualida ut proportio incrementi ex aëre &longs;it <lb/>maior, quàm impedimentum, & in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo<lb/>cior fiet naturali, ut in &longs;phæris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad præ&longs;entem <lb/>motum ratio e&longs;t eadem. Quicun que ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue æquiponderans. Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad &longs;e inuicem, & ad a qualis mo<lb/>tuum ob di&longs;tantiam intentorum. Experimentum ergo docet, quòd <lb/>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, & ex palmo quarta pars ponderis, ergo &longs;e ha<lb/>bent prope portionem.</s>

<figure id="fig37"></figure><lb/>pedimentum naturale. Sed &longs;i uis mouens fuerit <lb/>adeò ualida ut proportio incrementi ex aëre &longs;it <lb/>maior, quàm impedimentum, & in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo<lb/>cior fiet naturali, ut in &longs;phæris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad præ&longs;entem <lb/>motum ratio e&longs;t eadem. Quicun que ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue æquiponderans. Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad &longs;e inuicem, & ad a qualis mo<lb/>tuum ob di&longs;tantiam intentorum. Experimentum ergo docet, quòd <lb/>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, & ex palmo quarta pars ponderis, ergo &longs;e ha<lb/>bent prope portionem.</s>

<s>Propo&longs;itio quinquage&longs;imaoctaua.</s>

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<s>Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. At ca<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean<lb/>dem partem, &longs;eu aliam. Qui uerò naturalis e&longs;t, debilis <lb/>

<arrow.to.target n="fig38"></arrow.to.target><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="cõtinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aquæ maris fluit ac refluit: aliter ne<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo <lb/>tus, qui &longs;unt multo tardiores non latentnos. Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="cõ&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis a&eumle;ris <lb/>e&longs;&longs;et continuus, in hora aër procederet ob ambitum terræ millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, imò ne po&longs;sit quidem, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i adeò uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. Præterea tantus impetus nun<lb/>quam à minore motu, aut cau&longs;a &longs;uperaretur, adeò ut &longs;emper flatum <lb/>aëris orientalem &longs;entiremus. Quotidie etiam aduenire ad nos aë<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Bythínia, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terra que flo <lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. A'&longs;en&longs;u quidem, quoniam nebul&ecedil;, <lb/>quæ in aëre mouentur, primùm non in eandem partem &longs;emper mo <lb/>uentur: nun quam autem adeò celeriter: at &longs;i aër &longs;ic circumuoluere<lb/>tur, mouerentur & illa, qu&ecedil; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. Nechis, quæ <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti<lb/>aliqua in regione no&longs;tra directa &longs;æuiret, ut aër &longs;ingulis diebus la<lb/>be ea infectus ad nos deferretur. Moueri uerò aërem &longs;emper mani<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua & c&oelig;lum naturaliter perpetuò mouentur, quare etiam <lb/>aër. Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rectà, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc &longs;equitur.</s>

<figure id="fig38"></figure><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="cõtinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aquæ maris fluit ac refluit: aliter ne<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo <lb/>tus, qui &longs;unt multo tardiores non latentnos. Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="cõ&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis a&eumle;ris <lb/>e&longs;&longs;et continuus, in hora aër procederet ob ambitum terræ millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, imò ne po&longs;sit quidem, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i adeò uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. Præterea tantus impetus nun<lb/>quam à minore motu, aut cau&longs;a &longs;uperaretur, adeò ut &longs;emper flatum <lb/>aëris orientalem &longs;entiremus. Quotidie etiam aduenire ad nos aë<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Bythínia, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terra que flo <lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. A'&longs;en&longs;u quidem, quoniam nebul&ecedil;, <lb/>quæ in aëre mouentur, primùm non in eandem partem &longs;emper mo <lb/>uentur: nun quam autem adeò celeriter: at &longs;i aër &longs;ic circumuoluere<lb/>tur, mouerentur & illa, qu&ecedil; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. Nechis, quæ <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti<lb/>aliqua in regione no&longs;tra directa &longs;æuiret, ut aër &longs;ingulis diebus la<lb/>be ea infectus ad nos deferretur. Moueri uerò aërem &longs;emper mani<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua & c&oelig;lum naturaliter perpetuò mouentur, quare etiam <lb/>aër. Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rectà, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc &longs;equitur.</s>

<s>Propo&longs;itio quin quage&longs;imanona.</s>

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<s>

<arrow.to.target n="marg157"></arrow.to.target><lb/>

<arrow.to.target n="fig39"></arrow.to.target><lb/>to cum naturali coniuncto: & &longs;it terminus naturalis e, <lb/>

<figure id="fig39"></figure><lb/>to cum naturali coniuncto: & &longs;it terminus naturalis e, <lb/>

<arrow.to.target n="marg158"></arrow.to.target><lb/>& uiolenti d: uter que in directo c, dico, quod tardius per<lb/>ueniet ad c quam d, uel e. De e manife&longs;tum e&longs;t, quoniam <lb/>motus aëris, qui intendit motum a, diuíditur in partem, <lb/>quæ iuuat motum ad d, & partem, quæ mouetur ad e, <lb/>igitur fit minor adiectio. Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rectæ: quare tardius perueniet ad c quàm ad e du <lb/>plici ratione. Dico etiam, quod tardius ad c quàm d. Quia enim <lb/>uis, quæ fert ad d repugnat ei, quæ fert ad e, & uis, quæ fert ad e, re<lb/>pugnat ei quæ fert ad d, igitur tardius perueniet ad c, quàm d. Nec <lb/>potes dicere, quòd uis, quæ fert ad c adiuuet ad motum è regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tardí<lb/>or ab&longs;olutè motu ad d. Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen<lb/>rum ob cau&longs;am dictam. Vtrùm uerò motus ad c ab&longs;olutè &longs;it tardi<lb/>or, quàm ad d, non &longs;uppo&longs;ito, quod c e &longs;it æqualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s>

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<s>Ex hoc patet, quod motus æquidi&longs;tantis mobilis, finis e&longs;t mini<lb/>

<arrow.to.target n="marg159"></arrow.to.target><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. Velut &longs;i a mo <lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/>

<arrow.to.target n="fig40"></arrow.to.target><lb/>e&longs;t motus actu: uiolentus autem æqualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/>

<figure id="fig40"></figure><lb/>e&longs;t motus actu: uiolentus autem æqualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/>

<arrow.to.target n="marg160"></arrow.to.target><lb/>e&longs;&longs;et minor dimidio palmi. Et etiam quòd e&longs;&longs;et minor, quia ut di<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t æqualis uni eorum non impedito <lb/>uel minor.</s>

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<s>Propo&longs;itio &longs;exage&longs;ima.</s>

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<s>Sit a mobile, grauitatis centrum b, cuius pars ei pro<lb/>

<arrow.to.target n="marg161"></arrow.to.target><lb/>

<arrow.to.target n="fig41"></arrow.to.target><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na<lb/>tura partis, & totius: totius autem terræ natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/>

<figure id="fig41"></figure><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na<lb/>tura partis, & totius: totius autem terræ natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/>

<arrow.to.target n="marg162"></arrow.to.target><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. Sed pars pro<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita

<pb pagenum="52"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio<lb/>rem partem.<lb/>

<arrow.to.target n="marg163"></arrow.to.target></s>

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<s>Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub<lb/>&longs;tantia, &longs;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</s>

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<s>Con&longs;titutum e&longs;t inuenire proportionem uirium, quæ eleuant <lb/>

<arrow.to.target n="marg175"></arrow.to.target><lb/>pondus ad uires, quæ ip&longs;um in plano leui trahere po&longs;<lb/>

<arrow.to.target n="fig42"></arrow.to.target><lb/>&longs;unt. Vires enim, quæ eleuant pondus a &longs;unt eædem <lb/>puta b, quæ uero trahunt c, &longs;ed hæ po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>& maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/>Duæ uerò po&longs;tremæ cau&longs;æ etiam per &longs;e per&longs;picuæ &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla<lb/>num tangit. Nos præterea &longs;upponimus planum æquale undique <lb/>leue durum, & corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe<lb/>rens, & uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>quòd in hoc ca&longs;u b e&longs;t duplum ad c. Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, & &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci<lb/>tur circa medium, nam plana &longs;uperficies parum differt à rotunda <lb/>terræ ob terræ magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s>

<figure id="fig42"></figure><lb/>&longs;unt. Vires enim, quæ eleuant pondus a &longs;unt eædem <lb/>puta b, quæ uero trahunt c, &longs;ed hæ po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>& maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/>Duæ uerò po&longs;tremæ cau&longs;æ etiam per &longs;e per&longs;picuæ &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla<lb/>num tangit. Nos præterea &longs;upponimus planum æquale undique <lb/>leue durum, & corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe<lb/>rens, & uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>quòd in hoc ca&longs;u b e&longs;t duplum ad c. Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, & &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci<lb/>tur circa medium, nam plana &longs;uperficies parum differt à rotunda <lb/>terræ ob terræ magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s>

<s>Propo&longs;itio &longs;exage&longs;imatertia.</s>

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<s>Sit graue a b c alligatum funibus in d ef, dico, <lb/>

<arrow.to.target n="fig43"></arrow.to.target><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua<lb/>igitur attractio c per d e&longs;t debilior, quàm per f. Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s>

<figure id="fig43"></figure><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua<lb/>igitur attractio c per d e&longs;t debilior, quàm per f. Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s>

<s>

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<s>

<arrow.to.target n="marg179"></arrow.to.target><lb/>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb/>diuidere medium. Quia ergo &longs;i tangat in puncto facillime moue<lb/>tur, &longs;i in linea paulò difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici<lb/>lius mouetur. Sit ergo mobile a b, quod moueatur uer&longs;us c, & quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me<lb/>

<arrow.to.target n="fig44"></arrow.to.target><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, & longius a b, tanto difficilius <lb/>

<figure id="fig44"></figure><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, & longius a b, tanto difficilius <lb/>

<arrow.to.target n="marg180"></arrow.to.target><lb/>mouetur. Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s>

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<s><margin.target id="marg180"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s>

<s>Propo&longs;itio &longs;exage&longs;imaquinta.</s>

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<s>Sit eptagonus a b d f g e c, & &longs;ubten&longs;æ b <lb/>

<arrow.to.target n="fig45"></arrow.to.target><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono æ<lb/>quilatero, & æquiangulo) b c & e finuicem <lb/>æquales: & item d c, & d e æquales: & &longs;i du<lb/>cerentur b e & c f inuicem æquales: & ad a c <lb/>& d g: quare cum angulus cb d con&longs;i&longs;tatin </s>

<figure id="fig45"></figure><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono æ<lb/>quilatero, & æquiangulo) b c & e finuicem <lb/>æquales: & item d c, & d e æquales: & &longs;i du<lb/>cerentur b e & c f inuicem æquales: & ad a c <lb/>& d g: quare cum angulus cb d con&longs;i&longs;tatin </s>

<s>

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<s>Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, & du <lb/>cantur perpendiculares a f, d g & c d, & &longs;it c e i ca 1 pos, & quia ut <lb/>

<arrow.to.target n="marg192"></arrow.to.target><lb/>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) & <lb/>quia d h e&longs;t dimidium d e, erit d h, & g f <lb/>

<arrow.to.target n="fig46"></arrow.to.target><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di<lb/>ui&longs;a 1/2 pos per 1 pos, & exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>& eius quadratum 1/4/(1 quad). igitur cum qua<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, quæ <lb/>e&longs;t 1/2p: 1/2/(1 pos) & f g quæ e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/>

<figure id="fig46"></figure><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di<lb/>ui&longs;a 1/2 pos per 1 pos, & exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>& eius quadratum 1/4/(1 quad). igitur cum qua<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, quæ <lb/>e&longs;t 1/2p: 1/2/(1 pos) & f g quæ e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/>

<arrow.to.target n="marg193"></arrow.to.target><lb/>compo&longs;itum ex quadratis c g & g d erit 2 p: 1/pos c a uerò e&longs;t æqua<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, & angulus b c e ei duplus, quare cum c f a &longs;it re<lb/>ctus erit ex trige&longs;ima&longs;ecunda primi Elementorum f a c tres &longs;epti<lb/>mæ unius recti, ergo d a c 6/7 unius recti, d c a uerò 2/7 unius recti, quia <lb/>

<arrow.to.target n="marg194"></arrow.to.target><lb/>e&longs;t &longs;eptima pars duorum rectorum, ígitur a d c e&longs;t 6/7 unius recti: igi<lb/>tur c d e&longs;t æqualis c a, ergo quadratum quadrato: igitur 1 quad. p: 2 <lb/>pos p: 1, æquatur 2 p: 1/(1 pos) igitur 1 quad. p: 2 pos, æquantur 1 p: 1/(1 pos). <lb/>Quare 1 cub. p: 2 quad. æquatur 1 pos p: 1. <lb/>

<arrow.to.target n="fig47"></arrow.to.target><lb/>Sit etiam angulus a duplus b, & b c dupla <lb/>b a: & erit per eadem proportio a c, & a b <lb/>ad c b, ut c b ad c a. Ponamus ergo ab 1, erit <lb/>b c 2, & a c 1 pos, & a c, a b 1 pos p: 1, & du<lb/>cta in a c fit 1 quad. p: 1 pos, & hoc e&longs;t æquale 4 quadrato b c per re<lb/>flexæ proportionis diffinitionem. Igitur a c e&longs;t <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s>

<figure id="fig47"></figure><lb/>Sit etiam angulus a duplus b, & b c dupla <lb/>b a: & erit per eadem proportio a c, & a b <lb/>ad c b, ut c b ad c a. Ponamus ergo ab 1, erit <lb/>b c 2, & a c 1 pos, & a c, a b 1 pos p: 1, & du<lb/>cta in a c fit 1 quad. p: 1 pos, & hoc e&longs;t æquale 4 quadrato b c per re<lb/>flexæ proportionis diffinitionem. Igitur a c e&longs;t <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s>

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<s><margin.target id="marg194"></margin.target>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s>

<s>Propo&longs;itio &longs;exage&longs;ima&longs;eptima.</s>

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<s>Si fuerit linea bifariam diui&longs;a, eique in longum alia addita, & rur<lb/>

<arrow.to.target n="marg231"></arrow.to.target><lb/>&longs;us alia detracta, fueritque totius cum addita ad eam, quæ addita e&longs;t <lb/>ueluti re&longs;idui ad detractam erit lineæ com<lb/>

<arrow.to.target n="fig48"></arrow.to.target><lb/>po&longs;itæ ex addita, & dimidia ad dimidiam

<figure id="fig48"></figure><lb/>po&longs;itæ ex addita, & dimidia ad dimidiam

<pb pagenum="60"/>ip&longs;am uelut dimidiæ ad differentiam eius, & detractæ. Rur&longs;usque li<lb/>neæ compo&longs;itæ ex dimidio & re&longs;iduo dimidiæ ac detractæ ad li<lb/>neam compo&longs;itam ex addita & detracta ut re&longs;idui dimidiæ, & de<lb/>tractæ ad partem detractam. Et rur&longs;us totius compo&longs;itæ ad com<lb/>po&longs;itam ex dimidia & addita, uelut compo&longs;itæ ex addita, & diffe<lb/>rentia ad ip&longs;am additam. Velut &longs;it propo&longs;ita a b per æqualia diui&longs;a <lb/>in c, addita b d, & detracta b e, &longs;it proportio a d ad d b, ut a e ad e b, <lb/>dico e&longs;&longs;e, ut c d ad cb, ita ab ad c e. Et ut a e ad e d ut c e ad e b. Etite<lb/>

<arrow.to.target n="marg232"></arrow.to.target><lb/>rum ut a d ad c d uelut e d ad d b. In parabole proportio partium <lb/>diametri ad uerticem terminantium duplicata e&longs;t proportioni li<lb/>nearum ab ei&longs;dem punctis ordinatim ductarum ad ip&longs;am &longs;ectio<lb/>

<arrow.to.target n="marg233"></arrow.to.target><lb/>nem. In hyperbole autem & ellip&longs;i & circuli circumferentia erit <lb/>quadratorum linearum ordinatim ductarum inter &longs;e uelut rectan<lb/>

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<s>Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei æquidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectione<gap/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

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<pb pagenum="61"/>dit: linea uerò tangens uerticem hyperbolis ad quam ordinatæ <lb/>

<arrow.to.target n="marg240"></arrow.to.target><lb/>po&longs;&longs;unt, Recta appellabitur. Datarecta linea po&longs;itione, aliaque ma <lb/>gnitudine data & angülo parabolen, & hyperbolen, & ellip&longs;im, <lb/>& contrapo&longs;itas circa datam po&longs;itione tanquàm diametrum de<lb/>&longs;cribere tanquàm cono erecto, ut angulus ad uerticem &longs;ectionis <lb/>comprehen&longs;us &longs;it, & per rectam rectangulum æquale comprehen<lb/>datur quadrato datæ lineæ magnitudine. Si linea in duas partes <lb/>

<arrow.to.target n="marg241"></arrow.to.target><lb/>diuidatur, eique utrinque æquales lineæ adiun<lb/>

<arrow.to.target n="fig49"></arrow.to.target><lb/>gantur erit rectangulum ex partibus totius æ<lb/>quale rectangulis partium prioris lineæ, & ex <lb/>priore linea cum una adiecta in eam, quæ adiecta e&longs;t. Si hyperbo <lb/>

<figure id="fig49"></figure><lb/>gantur erit rectangulum ex partibus totius æ<lb/>quale rectangulis partium prioris lineæ, & ex <lb/>priore linea cum una adiecta in eam, quæ adiecta e&longs;t. Si hyperbo <lb/>

<arrow.to.target n="marg242"></arrow.to.target><lb/>len recta linea in uertice contingat, & utrinque ab&longs;cindatur, quan<lb/>tum e&longs;t, quod pote&longs;t in quartam partem rectanguli ex diametro <lb/>tran&longs;uer&longs;a hyperbolis, quæ exterius adiacetin eam, quæ recta dici<lb/>tur, ad quam, quæ ordinatim ducuntur, &longs;unt æquidi&longs;tantes lineæ, <lb/>quæ à &longs;ectionis centro ad terminos contingentis ducuntur &longs;emper <lb/>ip&longs;i &longs;ectioni magis appropinquabunt, nec unquam conuenient: & <lb/>ob id a&longs;ymptoton appellantur. Nec ullæ aliæ intra <expan abbr="angul&utilde;">angulum</expan> illum <lb/>

<arrow.to.target n="marg243"></arrow.to.target><lb/>inueniri poterunt. Vnde etiam intra <expan abbr="dat&utilde;">datum</expan> angulum de&longs;cribere do<lb/>cemur hyperbolen cuius anguli latera &longs;int a&longs;ymptota. A&longs;ymptotis <lb/>

<arrow.to.target n="marg244"></arrow.to.target><lb/>duabus propo&longs;itis uni hyperboli, in finitas alías eidem a&longs;ymptotas <lb/>inuenire. Duabus rectis a&longs;ymptotis infinitas &longs;ubijci po&longs;&longs;e hyperbo <lb/>les illis rectis, & inter &longs;e a&longs;ymptotas. Cum in duabus &longs;uperficie<lb/>

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<s>Propo&longs;itio &longs;eptuage&longs;ima.</s>

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<s>Velut &longs;int a b c primi ordi<lb/>

<arrow.to.target n="fig50"></arrow.to.target><lb/>nis, & d ef &longs;ecundi, & &longs;it 28, </s>

<figure id="fig50"></figure><lb/>nis, & d ef &longs;ecundi, & &longs;it 28, </s>

<s>

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<s>Sint duo pondera æqualia in plano a & b, & &longs;it <lb/>

<arrow.to.target n="fig51"></arrow.to.target><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt æqualia: &longs;u&longs;pendantur, & erunt <lb/>æqualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, & b quie&longs;cens in plano e&longs;t di <lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi<lb/>tur per communem animi &longs;ententiam a & b in pla<lb/>no &longs;unt æqualia.</s>

<figure id="fig51"></figure><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt æqualia: &longs;u&longs;pendantur, & erunt <lb/>æqualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, & b quie&longs;cens in plano e&longs;t di <lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi<lb/>tur per communem animi &longs;ententiam a & b in pla<lb/>no &longs;unt æqualia.</s>

<s>

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<s>Sit pondus a in terra æquale b eiu&longs;dem naturæ magnitudinis fi<lb/>

<arrow.to.target n="marg266"></arrow.to.target><lb/>guræ, & eodem in &longs;itu, quod &longs;it in aqua porrò a, &longs;i e&longs;&longs;et affixum ter<lb/>ræ oportet, ut conuellatur, aut di&longs;&longs;oluatur aut frangatur. Et clarum <lb/>

<arrow.to.target n="fig52"></arrow.to.target><lb/>e&longs;t, quod totum ictum excipit. Si uerò <lb/>affixum non &longs;it, euertitur, & tanto mino<lb/>rem partem excipit ictus, quanto faci<lb/>lior e&longs;t ad euer&longs;ionem. Vnde nata fabu<lb/>la de quercu, quæ cum immobilis e&longs;&longs;et, <lb/>& &longs;taret uento euer&longs;a e&longs;t, arundo flecten<lb/>do &longs;e, cecidit quidem, &longs;ed non e&longs;t eradi<lb/>cata. Sermo igitur e&longs;t de b in&longs;identi aqu&ecedil; <lb/>in comparatione ad a, quando excipit <lb/>plenum ictum. Cum ergo b tangitur, ex<lb/>cipit plenum ictum illo in&longs;tanti, &longs;ed quia <lb/>non excipitur ictus cedente materia, & <lb/>antequam materia cedat b mouetur loco, quia in&longs;idet aquæ, ergo <lb/>non excipit ictum. Proponatur ergo, quod moueatur b per c&longs;pa<lb/>tium in d tempore, & &longs;it, ut idem b ab e ui trahatur per idem &longs;pa<lb/>tium in eodem tempore ex loco directo ad eandem partem: qua<lb/>lis ergo proportio e ad b, & aërem, qui cum eo re&longs;i&longs;tit, talis propor<lb/>tio ictus f grauis puta in a ad ictum Y in b. Quia per demon&longs;tra<lb/>

<figure id="fig52"></figure><lb/>e&longs;t, quod totum ictum excipit. Si uerò <lb/>affixum non &longs;it, euertitur, & tanto mino<lb/>rem partem excipit ictus, quanto faci<lb/>lior e&longs;t ad euer&longs;ionem. Vnde nata fabu<lb/>la de quercu, quæ cum immobilis e&longs;&longs;et, <lb/>& &longs;taret uento euer&longs;a e&longs;t, arundo flecten<lb/>do &longs;e, cecidit quidem, &longs;ed non e&longs;t eradi<lb/>cata. Sermo igitur e&longs;t de b in&longs;identi aqu&ecedil; <lb/>in comparatione ad a, quando excipit <lb/>plenum ictum. Cum ergo b tangitur, ex<lb/>cipit plenum ictum illo in&longs;tanti, &longs;ed quia <lb/>non excipitur ictus cedente materia, & <lb/>antequam materia cedat b mouetur loco, quia in&longs;idet aquæ, ergo <lb/>non excipit ictum. Proponatur ergo, quod moueatur b per c&longs;pa<lb/>tium in d tempore, & &longs;it, ut idem b ab e ui trahatur per idem &longs;pa<lb/>tium in eodem tempore ex loco directo ad eandem partem: qua<lb/>lis ergo proportio e ad b, & aërem, qui cum eo re&longs;i&longs;tit, talis propor<lb/>tio ictus f grauis puta in a ad ictum Y in b. Quia per demon&longs;tra<lb/>

<arrow.to.target n="marg267"></arrow.to.target><lb/>ta &longs;uperius proportio f ad a producitur ex proportionibus e ad b, <lb/>

<arrow.to.target n="marg268"></arrow.to.target><lb/>& a ad e, ergo diui&longs;a proportione f ad a per proportionem c ad b <lb/>exibit proportio ictus Y in a ad ictum Y in b quod erat demon<lb/>&longs;trandum.</s>

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<s><margin.target id="marg268"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 42. & 43. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s>Ex hoc patet, quod b quanto mollius, leuius, & &longs;trictius in imo, <lb/>

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<s>Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di

<pb pagenum="67"/>rigitur, tendunturqúe uela ac expanduntur &longs;umma in parte mali, <lb/>tunc motus e&longs;t ueloci&longs;simus: fingamus autem, quod omnia ad <lb/>idem tendant præter uentum, qui non directus &longs;it ad puppim, &longs;ed <lb/>à latere, ut uides, & temo &longs;itin contrarium tantundem directus, & <lb/>&longs;upponamus pro nune, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/>

<arrow.to.target n="fig53"></arrow.to.target><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quæritur igitur proportio motus b c ad mo<lb/>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb/>rectus &longs;it, & manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, i&nacute;de noto angulo h c f <lb/>per ea, quæ tradita &longs;unt ab A&longs;trologis de &longs;inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k æqualis c h, igitur c k erit maior e g, &longs;i ergo perambula<lb/>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim æqualis c h, erit temporis ad <lb/>tempus proportio nota. Quod autem in æquali tempore mouebi<lb/>tur nauis per c k & h c patet ex a&longs;&longs;umpto inferius declarando.</s>

<figure id="fig53"></figure><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quæritur igitur proportio motus b c ad mo<lb/>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb/>rectus &longs;it, & manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, i&nacute;de noto angulo h c f <lb/>per ea, quæ tradita &longs;unt ab A&longs;trologis de &longs;inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k æqualis c h, igitur c k erit maior e g, &longs;i ergo perambula<lb/>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim æqualis c h, erit temporis ad <lb/>tempus proportio nota. Quod autem in æquali tempore mouebi<lb/>tur nauis per c k & h c patet ex a&longs;&longs;umpto inferius declarando.</s>

<s>

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<s>Hæc uídetur &longs;imilis &longs;uperiori cuidam propo&longs;itioni, &longs;ed tamen in <lb/>

<arrow.to.target n="marg280"></arrow.to.target><lb/>hoc differt, quoniam in c a &longs;upponimus nauim moueri, ut concu<lb/>tiat, hic autem iuxta motum &longs;olum: ut proponamus b nauim ferri <lb/>

<arrow.to.target n="fig54"></arrow.to.target><lb/>uer&longs;us a uento recto ex b in a: &longs;it autem uentus ex <lb/>cin a mouens nauim ex b in a: nòn enim moue<lb/>bit ut quidam putant in ratione c a ad b a: ut &longs;i ca <lb/>&longs;it &longs;exquiquarta ad b a, ut æquali impetu ex b & <lb/>c flante uento moueretur tardius per c a, quam <lb/>per b a, quia æqualiter ex &longs;uppo&longs;ito: ergo tanto <lb/>tardius c fertur in a, quam b in idem quanto lon<lb/>gior e&longs;t c a, b a igitur &longs;i b perueniet in a in qua<lb/>tuor diebus c perueniet in idem a in quinque <lb/>diebus. Hoc enim e&longs;t per &longs;e manife&longs;tum: &longs;ed non quærimus id, &longs;ed <lb/>ut uento c a æquali per c a ei, qui e&longs;t b a per b a, ubi b moueatur uen <lb/>to c a per b a, quanto tardius mouebitur. Mouebitur. n. tardius ad <lb/>a per b a, quam per c a, at per c a tardius, quam ex b in a per æqua<lb/>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen <lb/>tum ex b in a. Quærimus ergo compo&longs;itionem horum, ut &longs;it c <lb/>nauis, quæ debeat transferri ad a per uentum ex b, & &longs;equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, & tardius ex b per <lb/>uentum ex cin a. Ergo malus, qui in prora e&longs;t conuoluto eo, qui <lb/>e&longs;t in puppi, ut etiam Ari&longs;toteles docet tantundem nititur ad re<lb/>

<figure id="fig54"></figure><lb/>uer&longs;us a uento recto ex b in a: &longs;it autem uentus ex <lb/>cin a mouens nauim ex b in a: nòn enim moue<lb/>bit ut quidam putant in ratione c a ad b a: ut &longs;i ca <lb/>&longs;it &longs;exquiquarta ad b a, ut æquali impetu ex b & <lb/>c flante uento moueretur tardius per c a, quam <lb/>per b a, quia æqualiter ex &longs;uppo&longs;ito: ergo tanto <lb/>tardius c fertur in a, quam b in idem quanto lon<lb/>gior e&longs;t c a, b a igitur &longs;i b perueniet in a in qua<lb/>tuor diebus c perueniet in idem a in quinque <lb/>diebus. Hoc enim e&longs;t per &longs;e manife&longs;tum: &longs;ed non quærimus id, &longs;ed <lb/>ut uento c a æquali per c a ei, qui e&longs;t b a per b a, ubi b moueatur uen <lb/>to c a per b a, quanto tardius mouebitur. Mouebitur. n. tardius ad <lb/>a per b a, quam per c a, at per c a tardius, quam ex b in a per æqua<lb/>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen <lb/>tum ex b in a. Quærimus ergo compo&longs;itionem horum, ut &longs;it c <lb/>nauis, quæ debeat transferri ad a per uentum ex b, & &longs;equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, & tardius ex b per <lb/>uentum ex cin a. Ergo malus, qui in prora e&longs;t conuoluto eo, qui <lb/>e&longs;t in puppi, ut etiam Ari&longs;toteles docet tantundem nititur ad re<lb/>

<arrow.to.target n="marg281"></arrow.to.target><lb/>ctum ex cin æquidi&longs;tantem locum ab a quantum c di&longs;tat ab con<lb/>tra temo, qui in puppi e&longs;t dirigitur ad h, & &longs;i ualidius &longs;it uentus e<lb/>tiam adiuuante temonem, &longs;eu contra nitente, quantum licet mo<lb/>bili pondere nauis ad id latus, premitur enim nauis, qua&longs;i &longs;ubmer<lb/>gi debeat, uento in aduer&longs;um premente, ut &longs;i uentus repente huic <lb/>contrarius exoriatur, <expan abbr="pericul&utilde;">periculum</expan> &longs;ubeat, ne obruatur. Cum ergo uen<lb/>tus ex b feratur, æquidi&longs;tans c h, & c feratur per temonem in k, & ab <lb/>oppo&longs;itis æqualis actio &longs;equatur, imò tota impeditur, ex c in h fere<lb/>tur iuxta proportionem anguli, quem con&longs;tituit h c cum a c ad to<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob

<pb pagenum="70"/>uim uenti, & uiæ longitudinem, angulus uerò h c a &longs;it &longs;exta re<lb/>cti pars, feretur ex c uer&longs;us a ad quantitatem b a in quatuorde<lb/>cim horis: igitur rur&longs;us quanta e&longs;t proportio c a ad b a tan<lb/>tum e&longs;t temporis, in quo fertur ex c ad a ad quatuordecim horas <lb/>per uentum b a.</s>

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<s><margin.target id="marg281"></margin.target>Q<emph type="italics"/>uæ&longs;t.<emph.end type="italics"/> 7. M<emph type="italics"/>echanica.<emph.end type="italics"/></s>

<s>Propo&longs;itio octuage&longs;imaprima.</s>

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<s>Sit nauis in a itura in b (uentus rectus ad c, medius ad e) per <expan abbr="ob-liqu&utilde;">ob<lb/>liquum</expan>, cum ergo tardius moueatur per a e quàm a c & per a b, quam <lb/>per a d, & &longs;int ad perpendiculum b e, b d quas con&longs;tat e&longs;&longs;e breui&longs;si<lb/>mas earum, quæ ad a c & ad a d. Queritur igitur quando uelocius <lb/>

<arrow.to.target n="fig55"></arrow.to.target><lb/>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb/>an cum per a b &longs;impliciter. Et con&longs;tat quod a d & d b <lb/>longiores &longs;unt a b, i&longs;tud enim demon&longs;tratum e&longs;t ab <lb/>Euclide in primo Elementorum, dico modo a c, & </s>

<figure id="fig55"></figure><lb/>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb/>an cum per a b &longs;impliciter. Et con&longs;tat quod a d & d b <lb/>longiores &longs;unt a b, i&longs;tud enim demon&longs;tratum e&longs;t ab <lb/>Euclide in primo Elementorum, dico modo a c, & </s>

<s>

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<s>Non me later Ari&longs;totelem exi&longs;timare centrum mundi e&longs;&longs;e cen<lb/>trum terræ illudque proba&longs;&longs;e, quod tamen ex demon&longs;tratione no&longs;tra <lb/>mathematica apparet nunc&longs;ubijciam, & quid ad illius rationes di<lb/>cendum &longs;it, aliâs etiam dicendum erit: nam liber hic, ut mathemati<lb/>ca decet, e&longs;&longs;e debet ab omnibus contentionibus ab&longs;olutus. Con<lb/>&longs;tat &longs;anè non e&longs;&longs;e propriam uim lapidis illius, ut qui non &longs;it circum<lb/>&longs;criptus &longs;ed fru&longs;tulum quoduis id pote&longs;t, neque per &longs;e, &longs;ed in ferro & <lb/>pendulo, nec fieri pote&longs;t, ut &longs;it illius <expan abbr="tãquam">tanquam</expan> &longs;peciei unius lapidum, <lb/>&longs;ed qua&longs;i perfectæ portionis cuiu&longs;dam generis terræ, quæ ab&longs;olu<lb/>ta &longs;it, cuius indicium e&longs;t illius copia, neque enim ullibi non inuenitur, <lb/>& ubi ferrum effoditur, ut in Ilua In&longs;ula Tyrrheno mari, e&longs;t ergo fer <lb/>

<arrow.to.target n="fig56"></arrow.to.target><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge<lb/>nere, ubi uim f&oelig;cundam acceperit à ma&longs;cu<lb/>lo &longs;cilicet Herculeo lapide, quærit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> quæ<lb/>rit, ut quie&longs;cere po&longs;sit. Vt ergo quie&longs;cat à <lb/>motu c&oelig;li qui e&longs;t ab Oriente in Occiden<lb/>tem iuxta axis c&oelig;li &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;simè moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre<lb/>mit uchementer, adeò ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;erè & <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa c&oelig;li &longs;entiatur circa centrum mundi. Cuius indi<lb/>cio e&longs;t Oceani maris, aquarum fluxus & refluxus. Duos ergo ha<lb/>bet motus terra perfecta, &longs;eu ferrum lapide Herculeo <expan abbr="imbut&utilde;">imbutum</expan> &longs;ub<lb/>ordinatos imperfectum perfecto: perfectus e&longs;t, ut de&longs;cendat ad cen <lb/>trum terræ, ut ibi quie&longs;cat: imperfectum, cum à perfecto prohibe<lb/>tur, ut quie&longs;cat &longs;altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet &longs;i &longs;ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum &longs;itu tamen de&longs;cen&longs;ui ad terræ centrum proximiore, ut &longs;æpi<lb/>us &longs;uperius declarauimus, dum de motu grauium & præcipuè li<lb/>bræ, & centro grauitatis loqueremur. Quibus demon&longs;tratis tum <lb/>experimento tum ratione à Fortunio Affaytato Cremonen&longs;i Me<lb/>dico, cum per hæc po&longs;tmodum cogeretur fateri acum ad polum

<figure id="fig56"></figure><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge<lb/>nere, ubi uim f&oelig;cundam acceperit à ma&longs;cu<lb/>lo &longs;cilicet Herculeo lapide, quærit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> quæ<lb/>rit, ut quie&longs;cere po&longs;sit. Vt ergo quie&longs;cat à <lb/>motu c&oelig;li qui e&longs;t ab Oriente in Occiden<lb/>tem iuxta axis c&oelig;li &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;simè moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre<lb/>mit uchementer, adeò ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;erè & <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa c&oelig;li &longs;entiatur circa centrum mundi. Cuius indi<lb/>cio e&longs;t Oceani maris, aquarum fluxus & refluxus. Duos ergo ha<lb/>bet motus terra perfecta, &longs;eu ferrum lapide Herculeo <expan abbr="imbut&utilde;">imbutum</expan> &longs;ub<lb/>ordinatos imperfectum perfecto: perfectus e&longs;t, ut de&longs;cendat ad cen <lb/>trum terræ, ut ibi quie&longs;cat: imperfectum, cum à perfecto prohibe<lb/>tur, ut quie&longs;cat &longs;altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet &longs;i &longs;ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum &longs;itu tamen de&longs;cen&longs;ui ad terræ centrum proximiore, ut &longs;æpi<lb/>us &longs;uperius declarauimus, dum de motu grauium & præcipuè li<lb/>bræ, & centro grauitatis loqueremur. Quibus demon&longs;tratis tum <lb/>experimento tum ratione à Fortunio Affaytato Cremonen&longs;i Me<lb/>dico, cum per hæc po&longs;tmodum cogeretur fateri acum ad polum

<pb pagenum="74"/>tendere, cum tamen tendat à dextro latere &longs;cilicet ab Oriente no<lb/>uem partibus, &longs;eu decima parte unius recti in centro terræ, quæ e&longs;t <lb/>quadrage&longs;ima totius ambitus c&oelig;li. Statuatur centrum mundia, & <lb/>b a c axis, &longs;ecundum quam mouetur motu diurno, ital a dextra exit <lb/>oriens, k a &longs;ini&longs;tra occidens, & &longs;tatuatur d centrum terræ, &longs;eu &longs;uprà <lb/>&longs;eu infrà, non tamen in linea b c, &longs;ed uel &longs;uprà in dextra parte, uel in<lb/>frà in &longs;ini&longs;tra, ita ut ducta linea per illud punctum arcus b g &longs;it no<lb/>uem partium. Con&longs;tituta ergo acu in e puncto, ubilinea h ad g &longs;ecat <lb/>peripheriam terr&ecedil; dico, quod acus dirigetur per h g, & non per b c, <lb/>nam acus mouetur ad centrum per eam, & in eo &longs;itu tota dirigitur, <lb/>quia omnes partes grauis con&longs;entiunt in motu principij grauitatis <lb/>ad centrum, hoc enim demon&longs;tratum: nixus ergo e&longs;t ut moueatur <lb/>per c d, & in eo nixu qui e&longs;t quies cu&longs;to dit lineam axis, quæ e&longs;t a b, <lb/>ut quie&longs;cat, ergo non quie&longs;cet, ni&longs;i in linea d g, quod erat demon<lb/>&longs;trandum. Quæ autem &longs;equuntur ex his corrolaria omnia concor<lb/>dant cum experimentis. Ergo hic &longs;ermo e&longs;t demon&longs;tratiuus, ut e<lb/>nim bene dixit Auerroes: Sermo demon&longs;tratiuus &longs;atisfacit omni<lb/>bus problematibus quæ <expan abbr="cõtingunt">contingunt</expan> circa principale quæ&longs;itum. Ex <lb/>hoc ergo patet, quod angulus di&longs;tantia d ab a in latitudine e&longs;t de ci<lb/>ma pars recti, et quod quanto magis di&longs;tatin longitudine centrum <lb/>terræ à centro mundi, tanto etiam minus di&longs;tatin latitudine. Hæc <lb/>enim &longs;unt demon&longs;trata clarè in mathematicis. Vnde fieri po&longs;&longs;et <lb/>quod hæc quantitas di&longs;tantiæ e&longs;&longs;et res, per quam exigua etiam &longs;i <lb/>non e&longs;&longs;et maior quatuor digitis &longs;ufficeret, modo etiam per ualde <lb/>paruum &longs;patium di&longs;taret ab eodem in longitudine. De cau&longs;a au<lb/>tem huius differentiæ aliâs dicendum erit, hiclo cus non e&longs;t, &longs;ed &longs;uf<lb/>ficit &longs;cire quod ita &longs;it, quod &longs;i mobilis &longs;it punctus d, clarum e&longs;t ali<lb/>quando futurum ut minus di&longs;tet g à b, aliquando ut &longs;it idem. Et <lb/>quali&longs;cunque motus &longs;it, nece&longs;&longs;e e&longs;t eam di&longs;tantiam uariari.</s>

<s>Propo&longs;itio octuage&longs;imaquinta.</s>

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<s>Sit aurum a, & liquor b, quæ repleant uas c, & <lb/>pondus amborum &longs;it librarum quadraginta, & <lb/>

<arrow.to.target n="fig57"></arrow.to.target><lb/>uas repletum liquore &longs;olo &longs;it librarum xxix, au<lb/>rum autem &longs;it ponderis librarum xij, igitur reli<lb/>quum erit ponderis xxviij, differentia ergo ua<lb/>&longs;is pleni, & non pleni liquore e&longs;t libra una, pon<lb/>dus auri e&longs;t librarum duodecim: dico quod au<lb/>ri pondus e&longs;t duode cuplum ponderi liquoris, &

<figure id="fig57"></figure><lb/>uas repletum liquore &longs;olo &longs;it librarum xxix, au<lb/>rum autem &longs;it ponderis librarum xij, igitur reli<lb/>quum erit ponderis xxviij, differentia ergo ua<lb/>&longs;is pleni, & non pleni liquore e&longs;t libra una, pon<lb/>dus auri e&longs;t librarum duodecim: dico quod au<lb/>ri pondus e&longs;t duode cuplum ponderi liquoris, &

<pb pagenum="75"/>&longs;i fui&longs;&longs;et pondus amborum libræ xxxix, manentibus reliquis, &longs;eque <lb/>retur quod pondus liquoris e&longs;&longs;et xxvij, & quia plenum uas &longs;uppo<lb/>nitur e&longs;&longs;e librarum xxix, e&longs;&longs;et differentia libræij, at auri pondus e&longs;t <lb/>libræ xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu<lb/>pla. Nam &longs;i uas plenum liquore ex &longs;uppo&longs;ito e&longs;t librarum xxix, & <lb/>cum auro xl, gratia exempli, & auri pondus e&longs;t xij, igitur liquoris <lb/>pondus e&longs;t xxviij librarum: &longs;ed cum liquor &longs;it corpus &longs;imilium par<lb/>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum ade&longs;t <lb/>aurum, liquor occupat xxviij partes cxxxix, totius ua&longs;is igitur au<lb/>rum continet unam partem tantum, & cum aurum pondus habeat <lb/>librarum xij, & liquor unius: quia totum uas cxxxix librarum dum <lb/>e&longs;t plenum, & e&longs;t diui&longs;um in xxix partes, igitur pondus unius par<lb/>tis liquoris e&longs;t una libra, igitur pondus auri e&longs;t duode cuplum ad <lb/>pondus liquoris quod fuit propo&longs;itum.</s>

<s>

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<s>Ex hoc patet &longs;olutio problematis cuiu&longs;dam propo&longs;iti aliasque mi <lb/>nus bene &longs;oluti cùm cau&longs;am habeat manife&longs;ti&longs;simam, &longs;cilicet quod

<pb pagenum="76"/>ua&longs;e aqua pleno impo&longs;itis &longs;en&longs;im centum aureis coronatis nihil ef<lb/>funditur, non quod quicquam ab&longs;umatur in metallo, &longs;ed cau&longs;a e&longs;t <lb/>quod cum aurum &longs;it duplum pondere ferro, erit ex demon&longs;tratis <lb/>&longs;ex decuplum ad pondus aquæ. Igitur cum &longs;it proportio ponderis <lb/>auri ad differentiam &longs;patij eadem, &longs;i &longs;it uas aquæ ponderis libræ <lb/>unius & mediæ, erit pondus totum xxiij unciarum, igitur aqua de<lb/>ficiet &longs;olum ex decimaoctaua parte &longs;eu cre&longs;cet ex impo&longs;itione auri, <lb/>&longs;ed illa pars in tumore aquæ ab&longs;umitur, <expan abbr="nõ">non</expan> &longs;olum, quia <lb/>

<arrow.to.target n="fig58"></arrow.to.target><lb/>dum aureos imponimus plana &longs;olum &longs;it, &longs;ed quia non ex <lb/>quauis rotunditate defluit, aliter in urceo tam exiguo <lb/>non po&longs;&longs;et apparere rotunda: quod enim rotunditas to<lb/>tius terræ, quæ etiam planam o&longs;tendit totam unam re<lb/>gionem ad rotun ditatem quæ apparet in exiguo urceo <lb/>aquæ. E&longs;t igitur rotunditas illa potius ob lentorem aqu&ecedil; qui auge<lb/>tur à lentore argenti, & etiam magis auri, cum &longs;en&longs;u digitorum per<lb/>cipiatur.</s>

<figure id="fig58"></figure><lb/>dum aureos imponimus plana &longs;olum &longs;it, &longs;ed quia non ex <lb/>quauis rotunditate defluit, aliter in urceo tam exiguo <lb/>non po&longs;&longs;et apparere rotunda: quod enim rotunditas to<lb/>tius terræ, quæ etiam planam o&longs;tendit totam unam re<lb/>gionem ad rotun ditatem quæ apparet in exiguo urceo <lb/>aquæ. E&longs;t igitur rotunditas illa potius ob lentorem aqu&ecedil; qui auge<lb/>tur à lentore argenti, & etiam magis auri, cum &longs;en&longs;u digitorum per<lb/>cipiatur.</s>

<s>

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<s>

<arrow.to.target n="marg298"></arrow.to.target><lb/>& operaberis per regulam de con&longs;olatione monetarum, quas po<lb/>nemus infrà, & fient auri partes octo & argen <lb/>

<arrow.to.target n="fig59"></arrow.to.target><lb/>ti partes iij, nam cum duxeris iij in octo pon<lb/>dus argenti fiet xxiiij, & cum duxeris viij in <lb/>xij, pondus auri fiet xcvi, igitur totum pon<lb/>dus erit cxx, diuidendum per xi, aggregatum <lb/>partium auri & argenti, ita uero uncia ad unciam, ut tota corona mi <lb/>&longs;ta ad coronam puram auri & argenti.</s>

<figure id="fig59"></figure><lb/>ti partes iij, nam cum duxeris iij in octo pon<lb/>dus argenti fiet xxiiij, & cum duxeris viij in <lb/>xij, pondus auri fiet xcvi, igitur totum pon<lb/>dus erit cxx, diuidendum per xi, aggregatum <lb/>partium auri & argenti, ita uero uncia ad unciam, ut tota corona mi <lb/>&longs;ta ad coronam puram auri & argenti.</s>

<s><margin.target id="marg298"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 178.</s>

<s>Ex hoc etiam patet modus <expan abbr="cogno&longs;c&etilde;di">cogno&longs;cendi</expan> proportionem grauium <lb/>

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<s>Et &longs;imiliter &longs;ciemus per hoc accipere partes diuer&longs;orum, qu&ecedil; iun <lb/>

<arrow.to.target n="marg300"></arrow.to.target><lb/>ctæ faciant con&longs;titutum pondus. Velut uolo facere ma&longs;&longs;am ex mel<lb/>

<arrow.to.target n="fig60"></arrow.to.target><lb/>le & aqua, quæ impleat uas, quod mellis contineat <lb/>quindecim, aquæ duodecim, uolo ut contentum &longs;it <lb/>ponderis quatuorde cim, operabor, ut in <expan abbr="cõ&longs;olatio-nibus">con&longs;olatio<lb/>nibus</expan>, ponam duas partes mellis & unam aquæ, ut <lb/>uides in operatione à latere.</s>

<figure id="fig60"></figure><lb/>le & aqua, quæ impleat uas, quod mellis contineat <lb/>quindecim, aquæ duodecim, uolo ut contentum &longs;it <lb/>ponderis quatuorde cim, operabor, ut in <expan abbr="cõ&longs;olatio-nibus">con&longs;olatio<lb/>nibus</expan>, ponam duas partes mellis & unam aquæ, ut <lb/>uides in operatione à latere.</s>

<s>Propo&longs;itio octuage&longs;ima&longs;exta.</s>

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<s>Capiantur tres quartæ cir culorum magnorum a b, a c, b c, & alia <lb/>

<arrow.to.target n="marg301"></arrow.to.target><lb/>b d ad rectos angulos <expan abbr="er&utilde;tque">eruntque</expan> uici&longs;sim poli, & ducatur per medium <lb/>parallelus, erit ergo e f æqualis e g, & f e æqualis f g, &longs;ed ba&longs;is c g e&longs;t <lb/>

<arrow.to.target n="fig61"></arrow.to.target><lb/>quarta circuli, & ba&longs;is c b dimidium quartæ <lb/>circuli eo quod tota b a e&longs;t quarta circuli, igi<lb/>tur per modum 25 primi Elementorum quæ <lb/>tenet, erit angulus c f g maior oppo&longs;ito c f b. <lb/>Hoc autem tenet in eiu&longs;dem rationis &longs;uperfi<lb/>ciebus, quales &longs;unt hæ, quæ &longs;unt &longs;uperficies eiu&longs;dem &longs;ph&ecedil;ræ. po&longs;&longs;et <lb/>etiam demon&longs;trari per modum quartæ primi Elementorum. Et eti<lb/>am con&longs;tituta &longs;phæra e f g, cuius hic circulus e&longs;&longs;et maior circulus, & <lb/>non tangeret ni&longs;i in illa linea &longs;phæra maiorem, & utrin que &longs;ecaret eo<lb/>dem circulo. Et etiam per cordas & trigonos rectilineos, auxilio <lb/><expan abbr="tam&etilde;">tamen</expan> regulæ dialecticæ. Ex hoc &longs;equitur auxilio regulæ dialecticæ, <lb/>

<figure id="fig61"></figure><lb/>quarta circuli, & ba&longs;is c b dimidium quartæ <lb/>circuli eo quod tota b a e&longs;t quarta circuli, igi<lb/>tur per modum 25 primi Elementorum quæ <lb/>tenet, erit angulus c f g maior oppo&longs;ito c f b. <lb/>Hoc autem tenet in eiu&longs;dem rationis &longs;uperfi<lb/>ciebus, quales &longs;unt hæ, quæ &longs;unt &longs;uperficies eiu&longs;dem &longs;ph&ecedil;ræ. po&longs;&longs;et <lb/>etiam demon&longs;trari per modum quartæ primi Elementorum. Et eti<lb/>am con&longs;tituta &longs;phæra e f g, cuius hic circulus e&longs;&longs;et maior circulus, & <lb/>non tangeret ni&longs;i in illa linea &longs;phæra maiorem, & utrin que &longs;ecaret eo<lb/>dem circulo. Et etiam per cordas & trigonos rectilineos, auxilio <lb/><expan abbr="tam&etilde;">tamen</expan> regulæ dialecticæ. Ex hoc &longs;equitur auxilio regulæ dialecticæ, <lb/>

<arrow.to.target n="fig62"></arrow.to.target><lb/>quod in omnibus parallelis a c d & e f g cum b c circulo <lb/>maiore, & per aliam regulam dialecticam in omnibus cira <lb/>culis inæqualibus inter &longs;e ad æquales angulos &longs;ecanti<lb/>bus & ex tertia demum regula dialectica, &longs;equitur in o<lb/>mnibus circulis in æqualibus &longs;e &longs;ecantibus ad quemuis <lb/>angulum in &longs;phæræ &longs;uperficie. Sunt autem hæ regulæ mediæ inter <lb/>axiomata & demon&longs;trata. Et ex logica propria illi arti. In plano au<lb/>

<figure id="fig62"></figure><lb/>quod in omnibus parallelis a c d & e f g cum b c circulo <lb/>maiore, & per aliam regulam dialecticam in omnibus cira <lb/>culis inæqualibus inter &longs;e ad æquales angulos &longs;ecanti<lb/>bus & ex tertia demum regula dialectica, &longs;equitur in o<lb/>mnibus circulis in æqualibus &longs;e &longs;ecantibus ad quemuis <lb/>angulum in &longs;phæræ &longs;uperficie. Sunt autem hæ regulæ mediæ inter <lb/>axiomata & demon&longs;trata. Et ex logica propria illi arti. In plano au<lb/>

<arrow.to.target n="marg302"></arrow.to.target><lb/>tem &longs;patium d b c minus e&longs;t a b c, &longs;ed &longs;patium c b d e&longs;t unum, ergo <lb/>per communem animi &longs;ententiam &longs;patium a b d, maius e&longs;t &longs;patio <lb/>c b c, quod fuit probandum.</s>

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<s><margin.target id="marg302"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>terd <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s>Propo&longs;itio octuage&longs;ima&longs;eptima.</s>

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<s>Sit in aheno a b c d in imo e dena <lb/>

<arrow.to.target n="fig63"></arrow.to.target><lb/>rius argenteus cera affixus uel cla<lb/>uo, quem uideat ex h impo&longs;ita aqua <lb/>clara u&longs;que ad f, uideat ex k, igitur per <lb/>aquam deflectitur à perpendiculo <lb/>per angulum k f n, & in l, per angu<lb/>lum l g o cre&longs;cente aqua demum in <lb/>labro m a p, & &longs;it e annexus, & tabu <lb/>la h k l m &longs;it affixa &longs;olo uel pondere <lb/>firma foraminibus obliquis infrà <lb/>&longs;pectantibus, & per a a&longs;picientibus extremitatem e. Po&longs;&longs;umus ergo <lb/>imaginari primum, quòd omnes inclinationes &longs;int à perpendicu<lb/>lari, dum exit aqua, & ita denarius uideretur, uel in &longs;uperficie aquæ <lb/>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb/>e&longs;t. Secundus modus e&longs;t, ut radius delatus e a flectatur ad k uell, & <lb/>hoc non quia in a non e&longs;t mutatio medij. Tertius e&longs;t, ut linea ex ocu <lb/>lo ducta perueniat per punctum a ad &longs;uperficiem aquæ, & ex ea <lb/>per directum ad denarium, & tunc quia oculus iudicat &longs;e uidere <lb/>per rectam, ideo iudicabit &longs;e uidere per l a g in q, eo quod &longs;emper in <lb/>directo loci in quo e&longs;t e. At quoniam non ex qua cunque di&longs;tantia ui<lb/>detur e, &longs;ed ex longinquiore loco, ubi uas fuerit humilius quod li<lb/>neæ ad a ex oculo, quanto a fuerit humilius, tanto propius ip&longs;i e <lb/>procedunt. Et uer&longs;a uice lineæ ex e ad a, quanto e e&longs;t humilius ad <lb/>quencunque locum inflectuntur, tanto inferius <expan abbr="cad&utilde;t">cadunt</expan>. Ergo cum fue <lb/>rint ad æquilibrium h, magis di&longs;tabunt ab e, & ita e magis procul <lb/>uidebitur. Cau&longs;a ergo triplex e&longs;t humilitas, uel altitudo ua&longs;is: humi <lb/>litas uel altitudo aquæ: & labri ua&longs;is altitudo. Sed han crelinquere <lb/>po&longs;&longs;umus. Difficultas ergo experimenti etiam rectè facti e&longs;t, quo<lb/>niam po&longs;ito ua&longs;e n c d &longs;olum, ut altitudo &longs;it tantum n e, procul ma<lb/>gis uidebitur e, quàm &longs;i uas &longs;it a b c d, & totum plenum. Vbi autem <lb/>uas fit a b c d, magis procul uidebitur e cum &longs;uerit totum plenum, <lb/>quam cum fuerit plena &longs;ola pars n c d. Sic difficile e&longs;t con&longs;iderare <lb/>an altitudo aquæ faciat ad ui&longs;ionem procul, cum in humiliore, &longs;ed <lb/>di&longs;sipari ua&longs;e longius uideatur in pauca, quia labrum non ob&longs;tat: <lb/>in eodem autem longius in pluri aqua, quia labrum etiam non ob<lb/>&longs;tat, &longs;ed alia ratione. Vt ergo uideamus hoc experimentum, capie

<figure id="fig63"></figure><lb/>rius argenteus cera affixus uel cla<lb/>uo, quem uideat ex h impo&longs;ita aqua <lb/>clara u&longs;que ad f, uideat ex k, igitur per <lb/>aquam deflectitur à perpendiculo <lb/>per angulum k f n, & in l, per angu<lb/>lum l g o cre&longs;cente aqua demum in <lb/>labro m a p, & &longs;it e annexus, & tabu <lb/>la h k l m &longs;it affixa &longs;olo uel pondere <lb/>firma foraminibus obliquis infrà <lb/>&longs;pectantibus, & per a a&longs;picientibus extremitatem e. Po&longs;&longs;umus ergo <lb/>imaginari primum, quòd omnes inclinationes &longs;int à perpendicu<lb/>lari, dum exit aqua, & ita denarius uideretur, uel in &longs;uperficie aquæ <lb/>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb/>e&longs;t. Secundus modus e&longs;t, ut radius delatus e a flectatur ad k uell, & <lb/>hoc non quia in a non e&longs;t mutatio medij. Tertius e&longs;t, ut linea ex ocu <lb/>lo ducta perueniat per punctum a ad &longs;uperficiem aquæ, & ex ea <lb/>per directum ad denarium, & tunc quia oculus iudicat &longs;e uidere <lb/>per rectam, ideo iudicabit &longs;e uidere per l a g in q, eo quod &longs;emper in <lb/>directo loci in quo e&longs;t e. At quoniam non ex qua cunque di&longs;tantia ui<lb/>detur e, &longs;ed ex longinquiore loco, ubi uas fuerit humilius quod li<lb/>neæ ad a ex oculo, quanto a fuerit humilius, tanto propius ip&longs;i e <lb/>procedunt. Et uer&longs;a uice lineæ ex e ad a, quanto e e&longs;t humilius ad <lb/>quencunque locum inflectuntur, tanto inferius <expan abbr="cad&utilde;t">cadunt</expan>. Ergo cum fue <lb/>rint ad æquilibrium h, magis di&longs;tabunt ab e, & ita e magis procul <lb/>uidebitur. Cau&longs;a ergo triplex e&longs;t humilitas, uel altitudo ua&longs;is: humi <lb/>litas uel altitudo aquæ: & labri ua&longs;is altitudo. Sed han crelinquere <lb/>po&longs;&longs;umus. Difficultas ergo experimenti etiam rectè facti e&longs;t, quo<lb/>niam po&longs;ito ua&longs;e n c d &longs;olum, ut altitudo &longs;it tantum n e, procul ma<lb/>gis uidebitur e, quàm &longs;i uas &longs;it a b c d, & totum plenum. Vbi autem <lb/>uas fit a b c d, magis procul uidebitur e cum &longs;uerit totum plenum, <lb/>quam cum fuerit plena &longs;ola pars n c d. Sic difficile e&longs;t con&longs;iderare <lb/>an altitudo aquæ faciat ad ui&longs;ionem procul, cum in humiliore, &longs;ed <lb/>di&longs;sipari ua&longs;e longius uideatur in pauca, quia labrum non ob&longs;tat: <lb/>in eodem autem longius in pluri aqua, quia labrum etiam non ob<lb/>&longs;tat, &longs;ed alia ratione. Vt ergo uideamus hoc experimentum, capie

<pb pagenum="79"/>mus duo ua&longs;a a b c d duplum h k l m &longs;ub eadem proportione alti<lb/>tudinis & latitudinis, & collo cabimus ita ut p n radius æquidi&longs;tet <lb/>f e, & collo cabimus tabulas cum foraminibus, ut prius, & g f p q <lb/>

<arrow.to.target n="fig64"></arrow.to.target><lb/>in æquilibrio, in de uidebimus, an q p &longs;it æqualis aut breuior, nam <lb/>longior e&longs;&longs;e non pote&longs;t, quoniam inflectitur a minore aqua, ideo <lb/>angulus p h q non pote&longs;t e&longs;&longs;e maior f a g, &longs;uppo&longs;ita p h æquali a f: <lb/>quod &longs;i non e&longs;&longs;et, &longs;ufficeret, ut q & p e&longs;&longs;ent in æquilibrio uno, & f g <lb/>alio. Sed ueritas e&longs;t quod à maiore aqua maior fit reflexio: tum <lb/>quia in his, quæ &longs;unt &longs;ecundum naturam corpoream, & &longs;ub&longs;tan<lb/>tiam den&longs;am, aut tenuem uarietas quantitatis uariat uires: tum <lb/>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb/>fundo elatus. Igitur his cognitis experimentum fiat cum ua&longs;e ple<lb/>no. Et (ut dixi) con&longs;iderabimus proportionem anguli f a g ad far, <lb/>&longs;eu f e c quæ &longs;anè e&longs;t no tabilis: adeò ut &longs;it maior proportio aquæ ad <lb/>aërem comparatione grauium quàm lucis.</s>

<figure id="fig64"></figure><lb/>in æquilibrio, in de uidebimus, an q p &longs;it æqualis aut breuior, nam <lb/>longior e&longs;&longs;e non pote&longs;t, quoniam inflectitur a minore aqua, ideo <lb/>angulus p h q non pote&longs;t e&longs;&longs;e maior f a g, &longs;uppo&longs;ita p h æquali a f: <lb/>quod &longs;i non e&longs;&longs;et, &longs;ufficeret, ut q & p e&longs;&longs;ent in æquilibrio uno, & f g <lb/>alio. Sed ueritas e&longs;t quod à maiore aqua maior fit reflexio: tum <lb/>quia in his, quæ &longs;unt &longs;ecundum naturam corpoream, & &longs;ub&longs;tan<lb/>tiam den&longs;am, aut tenuem uarietas quantitatis uariat uires: tum <lb/>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb/>fundo elatus. Igitur his cognitis experimentum fiat cum ua&longs;e ple<lb/>no. Et (ut dixi) con&longs;iderabimus proportionem anguli f a g ad far, <lb/>&longs;eu f e c quæ &longs;anè e&longs;t no tabilis: adeò ut &longs;it maior proportio aquæ ad <lb/>aërem comparatione grauium quàm lucis.</s>

<s>

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<s>Sit uiolentum a quod moueatur per b c d e, e &longs;patium, & quia <lb/>uiolentum contrà nititur naturali, cadat ergo in planum in e: &longs;unt <lb/>ergo tria con&longs;ideran da, primum quod, ut dixi aliâs, motus uiolen<lb/>tus pro certa di&longs;tantia augetur, & cau&longs;am ibireddidi, ut potè u&longs;que <lb/>ad c, &longs;ed hoc e&longs;&longs;et difficile cognitu. Secundum, quod ubi in cipit de<lb/>cre&longs;cere, &longs;emper magis ac magis decre&longs;cit propter naturalem ni<lb/>xum contra operantem. Tertium quod ubi de&longs;cendere in cipit, ibi <lb/>e&longs;t æqualis uis uiolentum motum agens cum naturali. Certum e&longs;t <lb/>etiam quod motus æqualis intelligitur erecta ad perpendiculum <lb/>e f, donec occurrat a d: & diui&longs;a tota b f per tempus, locus ergo, in <lb/>quo mouetur per tantum &longs;patium, dicitur locus motus æqualis:

<pb pagenum="83"/>qui &longs;it gratia exempli g h, cuius medium proportione &longs;it k, di<lb/>co k con&longs;i&longs;tere propiorem f, quàm b, etiam&longs;i æqualiter mouere<lb/>tur. Primum quòd in tota g f declinat, & totus motus e&longs;t lentior, <lb/>quàm in tota b g, & tamen tardatur tantundem, ergo per commu<lb/>nem animi &longs;ententiam, k e&longs;t propior f, quàm b. Secundò, quia per <lb/>&longs;ecundum &longs;uppo &longs;itum motus a uer&longs;us f, continuè fit lentior, igitur <lb/>per communem animi &longs;ententiam multò longius e&longs;t tempus mo<lb/>tus a k, quam f, & tanto maius &longs;patium. Tertiò, quia motus ex b uer <lb/>&longs;us caugetur, & &longs;i e&longs;&longs;et æqualis adhuc multò e&longs;&longs;et breuior k f quam <lb/>a k, igitur multò magis hoc modo, & triplicata ratione. Si ergo b k <lb/>

<arrow.to.target n="fig65"></arrow.to.target><lb/>e&longs;&longs;et &longs;exquiquarta &longs;olum ip&longs;i k f, <lb/>erit b k dupla: fermè ex triplicata <lb/>ratione ip&longs;i k f, & iuxta eundem <lb/>modum ponemus mediam uim <lb/>xlvi pa&longs;sibus à &longs;corpione a quam <lb/>& hoc modo erit propèid quod e&longs;t.</s>

<figure id="fig65"></figure><lb/>e&longs;&longs;et &longs;exquiquarta &longs;olum ip&longs;i k f, <lb/>erit b k dupla: fermè ex triplicata <lb/>ratione ip&longs;i k f, & iuxta eundem <lb/>modum ponemus mediam uim <lb/>xlvi pa&longs;sibus à &longs;corpione a quam <lb/>& hoc modo erit propèid quod e&longs;t.</s>

<s>SCHOLIVM.</s>

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<s>Hic non pauca &longs;unt <expan abbr="cõ&longs;ideranda">con&longs;ideranda</expan>: Primum <lb/>

<arrow.to.target n="fig66"></arrow.to.target><lb/>quòd hoc intelligi pote&longs;t, uel de motibus at<lb/>tractionis, uel impul&longs;ionis, uel inuer&longs;ionis. <lb/>Secundum quod omne, quod impellitur &longs;uperiùs, tantundem gra<lb/>uat attractum, quantum ad de&longs;cen&longs;um, &longs;i &longs;it rotundum, nam qua<lb/>drata, <expan abbr="etiã">etiam</expan> alia non de&longs;cendunt &longs;ponte in decliui, & &longs;i &longs;it locus ualdè

<figure id="fig66"></figure><lb/>quòd hoc intelligi pote&longs;t, uel de motibus at<lb/>tractionis, uel impul&longs;ionis, uel inuer&longs;ionis. <lb/>Secundum quod omne, quod impellitur &longs;uperiùs, tantundem gra<lb/>uat attractum, quantum ad de&longs;cen&longs;um, &longs;i &longs;it rotundum, nam qua<lb/>drata, <expan abbr="etiã">etiam</expan> alia non de&longs;cendunt &longs;ponte in decliui, & &longs;i &longs;it locus ualdè

<pb pagenum="84"/>decliuis, tanto minus de&longs;cendunt, quanto &longs;unt latiora. Quia tamen <lb/>omnia difficiliùs de&longs;cendunt &longs;phæricis, & facilius quàm in plano, <lb/>ubi ponderant ni&longs;i per dimidium grauitatis, ideò proportio hæc <lb/>con&longs;tat ex proportione anguli de&longs;cen&longs;us ad totum rectum, & ma<lb/>gnitudine &longs;uperficiei, qua incumbit ad pondus comparata. Omne <lb/>enim graue, quanto grauius tam ad quietem, quàm ad motum na<lb/>turalem potentius e&longs;t: hoc enim per&longs;picuum e&longs;t, quia quieti natu<lb/>rali motus uiolentus, & motui naturali quies uiolenta opponitur: <lb/>quia ergo maiore ui opus e&longs;t ad motum præter naturam, ergo &longs;e<lb/>cundum naturam etiam maiore ui quie&longs;cit. A&longs;&longs;ump&longs;imus ergo cu<lb/>bum, ut magis notum. Sphæra igitur in omni decliui de&longs;cendit, <lb/>quia ut dictum e&longs;t, nil habet quod re&longs;i&longs;tat ad motum: & ip&longs;a gra<lb/>uior e&longs;t in decliui, quàm in plano, quia c pun<lb/>ctus cadit ultra e, ergo punctus contactus, & <lb/>

<arrow.to.target n="fig67"></arrow.to.target><lb/>centrum grauitatis, & centrum mundi, non &longs;unt <lb/>in una linea. Si enim b c contangeretur, e&longs;&longs;et b c <lb/>plana. Si uerò tangit, angulus e&longs;t maior angulo <lb/>contactus, ergo cum nece&longs;&longs;arium &longs;it, æquidi&longs;ta<lb/>re aliter non e&longs;&longs;et &longs;phæricum, oportet, ut eleue<lb/>tur ex parte c, & de&longs;cendat uer&longs;us b, & ideò ut <lb/>continuetur motus. Si uerò &longs;it in linea conta<lb/>ctus b c f, & æquidi&longs;tet non erit, ut dixi punctus <lb/>contactus in linea centrorum, &longs;ed in a c, cum &longs;uppo&longs;itum &longs;it lineam <lb/>a d e&longs;&longs;e lineam centrorum: maior e&longs;t ergo portio g c e, quàm re&longs;i<lb/>duum, ergo de&longs;cendet in b. Cubus uerò non de&longs;cendet, ni&longs;i cum di<lb/>midium d addito, quod inter cipitur inter lineam mediam, & quæ à <lb/>centro mundi ad punctum medium contactus u&longs;que quò perueniat <lb/>ad oppo&longs;itam partem, eam habuerit proportionem ad idem me<lb/>dium eadem portione detracta, quem iuncta proportioni an guli <lb/>declinationis ad re&longs;iduum recti dimidiam proportionem efficiat. <lb/>Eademque ratio aliorum planorum. Dico præterea quòd motus <lb/>&longs;phæræ, & etiam corporum rectarum &longs;uperficierum in de&longs;cen&longs;u <lb/>alius e&longs;t æqualis, & alius inæqualis, & qua&longs;i à latere, uelut &longs;i angu<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, & maximè &longs;i non retineatur æqualiter, & difficile &longs;it in medio <lb/>retinere, propterea prolap&longs;us hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>quàm in medio, non &longs;olum ob hanc æqualitatem, & complexum <lb/>meliorem, &longs;ed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa <lb/>ciliùs cohibentur, & <expan abbr="deducun&ttilde;">deducuntur</expan> diui&longs;i in partes, <08> &longs;i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trahãtur">trahantur</expan>. Et ideo uin cula in rami cibus duplicia dextra, & &longs;ini <lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamë longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s>

<figure id="fig67"></figure><lb/>centrum grauitatis, & centrum mundi, non &longs;unt <lb/>in una linea. Si enim b c contangeretur, e&longs;&longs;et b c <lb/>plana. Si uerò tangit, angulus e&longs;t maior angulo <lb/>contactus, ergo cum nece&longs;&longs;arium &longs;it, æquidi&longs;ta<lb/>re aliter non e&longs;&longs;et &longs;phæricum, oportet, ut eleue<lb/>tur ex parte c, & de&longs;cendat uer&longs;us b, & ideò ut <lb/>continuetur motus. Si uerò &longs;it in linea conta<lb/>ctus b c f, & æquidi&longs;tet non erit, ut dixi punctus <lb/>contactus in linea centrorum, &longs;ed in a c, cum &longs;uppo&longs;itum &longs;it lineam <lb/>a d e&longs;&longs;e lineam centrorum: maior e&longs;t ergo portio g c e, quàm re&longs;i<lb/>duum, ergo de&longs;cendet in b. Cubus uerò non de&longs;cendet, ni&longs;i cum di<lb/>midium d addito, quod inter cipitur inter lineam mediam, & quæ à <lb/>centro mundi ad punctum medium contactus u&longs;que quò perueniat <lb/>ad oppo&longs;itam partem, eam habuerit proportionem ad idem me<lb/>dium eadem portione detracta, quem iuncta proportioni an guli <lb/>declinationis ad re&longs;iduum recti dimidiam proportionem efficiat. <lb/>Eademque ratio aliorum planorum. Dico præterea quòd motus <lb/>&longs;phæræ, & etiam corporum rectarum &longs;uperficierum in de&longs;cen&longs;u <lb/>alius e&longs;t æqualis, & alius inæqualis, & qua&longs;i à latere, uelut &longs;i angu<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, & maximè &longs;i non retineatur æqualiter, & difficile &longs;it in medio <lb/>retinere, propterea prolap&longs;us hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>quàm in medio, non &longs;olum ob hanc æqualitatem, & complexum <lb/>meliorem, &longs;ed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa <lb/>ciliùs cohibentur, & <expan abbr="deducun&ttilde;">deducuntur</expan> diui&longs;i in partes, <08> &longs;i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trahãtur">trahantur</expan>. Et ideo uin cula in rami cibus duplicia dextra, & &longs;ini <lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamë longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s>

<s>

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<s>Ex hoc etiam &longs;equitur, <lb/>

<arrow.to.target n="fig68"></arrow.to.target><lb/>quod cùm omne graue <lb/>&longs;pontè &longs;emper appropin<lb/>quet centro mundi, & a &longs;i <lb/>moueretur per planum e, <lb/>magis remoueretur à cen<lb/>tro mundi, ut per e c per ea <lb/>quæ diximus, & quoniam <lb/>linea ex centro mundi ad <lb/>c longior e&longs;t, quàm ad e, <lb/>multò pote&longs;t enim e&longs;&longs;e, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, & <lb/>ctiam maior. ergo poterit <lb/>e&longs;&longs;e adeò parum decliuis <lb/>linea c d, ut c punctus ma<lb/>gis di&longs;ter à centro mundi, <lb/>quàm d, & tamen feretur <lb/>ex d in c motu naturali, ut demon&longs;tratum e&longs;t, ergo per purum mo<lb/>tum naturalem poterit a remoueri à centro mundi. Hoc uolui pro<lb/>ponere, ut intelligeres in plano uero c e non moueri a &longs;ponte, quia <lb/>c nece&longs;&longs;ariò altior e&longs;t d: &longs;i ergo mouebitur, non erit c e recta, &longs;ed <lb/>pars proportionis circuli &longs;uperficiei terræ, quæ &longs;en&longs;u à recta di&longs;tin<lb/>gui non poterit. Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s>

<figure id="fig68"></figure><lb/>quod cùm omne graue <lb/>&longs;pontè &longs;emper appropin<lb/>quet centro mundi, & a &longs;i <lb/>moueretur per planum e, <lb/>magis remoueretur à cen<lb/>tro mundi, ut per e c per ea <lb/>quæ diximus, & quoniam <lb/>linea ex centro mundi ad <lb/>c longior e&longs;t, quàm ad e, <lb/>multò pote&longs;t enim e&longs;&longs;e, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, & <lb/>ctiam maior. ergo poterit <lb/>e&longs;&longs;e adeò parum decliuis <lb/>linea c d, ut c punctus ma<lb/>gis di&longs;ter à centro mundi, <lb/>quàm d, & tamen feretur <lb/>ex d in c motu naturali, ut demon&longs;tratum e&longs;t, ergo per purum mo<lb/>tum naturalem poterit a remoueri à centro mundi. Hoc uolui pro<lb/>ponere, ut intelligeres in plano uero c e non moueri a &longs;ponte, quia <lb/>c nece&longs;&longs;ariò altior e&longs;t d: &longs;i ergo mouebitur, non erit c e recta, &longs;ed <lb/>pars proportionis circuli &longs;uperficiei terræ, quæ &longs;en&longs;u à recta di&longs;tin<lb/>gui non poterit. Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s>

<s>

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<pb pagenum="88"/>& d cognitæ &longs;unt erit & b c, quod e&longs;t primum. Per hæc eadem pro<lb/>bantur quatuor &longs;equentes partes eodem modo. Sexta &longs;ic: &longs;it pro<lb/>portio a c ad c b, nota igitur in comparatione ad monadem, &longs;ed pro <lb/>portio a c ad c b b a e&longs;t monas, igitur proportio a c ad a b nota e&longs;t, <lb/>quoniam aliter non po&longs;&longs;et dici proportio a c ad b c nota. Aliter, &longs;it <lb/>proportio a c ad c b e nota, ex &longs;uppo&longs;ito igitur conuer&longs;a nota quæ <lb/>&longs;it f ex f, igitur in a c fit b c ex g in a c, fiat a b ergo ex a c in f g fit a, cigi <lb/>tur f g e&longs;t monas, f autem nota e&longs;t, igitur in comparatione ad mona<lb/>

<arrow.to.target n="marg321"></arrow.to.target><lb/>dem, ergo re&longs;iduum g notum. Cum uerò proportio a c ad c b com<lb/>ponatur ex proportione a b b c ad b c, & proportio b c ad b c &longs;it <lb/>monas, & proportio a c ad b c nota, erit proportio a b ad b c cogni <lb/>

<arrow.to.target n="marg322"></arrow.to.target><lb/>ta, & monade minor proportione a c ad b c. Per idem octaua pars <lb/>

<arrow.to.target n="fig69"></arrow.to.target><lb/>demon&longs;trabitur. Inde &longs;it proportio a ad b, & ad c no<lb/>ta, erit ergo b, & c ad a nota, quare b c ad a nota, &longs;ed <lb/>

<figure id="fig69"></figure><lb/>demon&longs;trabitur. Inde &longs;it proportio a ad b, & ad c no<lb/>ta, erit ergo b, & c ad a nota, quare b c ad a nota, &longs;ed <lb/>

<arrow.to.target n="marg323"></arrow.to.target><lb/>hæc e&longs;t conuer&longs;a ad b c confu&longs;a, igitur proportio a <lb/>ad b confu&longs;a nota e&longs;t. Vltimum &longs;it, &longs;int a b c omiologæ, & &longs;int a & b <lb/>

<arrow.to.target n="marg324"></arrow.to.target><lb/>notæ duo, quod c nota e&longs;t, nam a b, &longs;i notæ &longs;unt, nota e&longs;t proportio <lb/>earum. Ergo & proportio b ad c ergo per primam partem huius <lb/>

<arrow.to.target n="marg325"></arrow.to.target><lb/>cum &longs;it b nota, exit & c. Et &longs;i ponantur a c notæ, dico, quòd b nota <lb/>erit: nam proportio a c ad c nota e&longs;t, quæ &longs;it d, igitur d ad monadem <lb/>ut a ad c, ergo latus notum erit, quod ductum in c producit b, b igi<lb/>

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<s><margin.target id="marg326"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 2. A<emph type="italics"/>nimi <lb/>&longs;ententia.<emph.end type="italics"/></s>

<s>Propo&longs;itio nonage&longs;imaquinta.</s>

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<s>

<arrow.to.target n="marg328"></arrow.to.target><lb/>e&longs;t: &longs;it igitur primum a b c trigonus orthogonius: & &longs;it a rectus, & <lb/>proportio <expan abbr="duor&utilde;">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb/>

<arrow.to.target n="marg329"></arrow.to.target><lb/>

<arrow.to.target n="fig70"></arrow.to.target><lb/>erunt: nam &longs;it proportio, gratia exempli, <lb/>a b ad b c, erit ergo quadrati a b ad qua<lb/>dratum b c cognita, quia duplicata: at <lb/>quadrata a b, & a c perficiunt quadratum <lb/>b c, igitur proportio quadrati a b ad a c et <lb/>e&longs;t 1 p: cognita erit, quare & a b ad a c, & <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>e&longs;t primum. Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a cigi

<figure id="fig70"></figure><lb/>erunt: nam &longs;it proportio, gratia exempli, <lb/>a b ad b c, erit ergo quadrati a b ad qua<lb/>dratum b c cognita, quia duplicata: at <lb/>quadrata a b, & a c perficiunt quadratum <lb/>b c, igitur proportio quadrati a b ad a c et <lb/>e&longs;t 1 p: cognita erit, quare & a b ad a c, & <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>e&longs;t primum. Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a cigi

<pb pagenum="89"/>tur &longs;i a b ponatur 1 b c erit 2, & a c <02> 3. Rur&longs;us ponatur angulus b <lb/>duplus angulo c quali&longs;cunque &longs;it, erit per demon&longs;trata &longs;uperius pro<lb/>portio a b b c ad a c, ut a c ad a b, &longs;i igitur nota &longs;it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per præcedentem. Ergo per <lb/>eandem omnia nota &longs;cilicet b c ad b a, & b c ad c a. Et &longs;i e&longs;&longs;et nota <lb/>proportio a b ad b c, dico, quod e&longs;&longs;ent nota omnia, nam nota e&longs;&longs;et <lb/>a b, & b c, & quod fit ex a b in ip&longs;um aggregatum. Sed hoc e&longs;t æ<lb/>

<arrow.to.target n="marg330"></arrow.to.target><lb/>quale quadrato a c, igitur notum e&longs;t quadratum a c ergo a c: igitur <lb/>proportio a b b c ad a c, & a c ad a b. Vt &longs;i a b e&longs;&longs;et 4 b c 5, e&longs;&longs;et a b b c <lb/>9 ducta in a b, quæ e&longs;t, fit 36, cuius latus e&longs;t b a c &longs;cilicet. Et &longs;i e&longs;&longs;et <lb/>trigonus aliquis in cir culo, cuius proportio duorum laterum &longs;it co <lb/>gnita ad dimetientem relata, &longs;equitur per demon&longs;trata &longs;upe<lb/>rius, quod etiam tertium latus erit cognitum in comparatione ad <lb/>eadem, & ideo etiam proportio illorum laterum ad unguem co<lb/>gnita erit.</s>

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<s><margin.target id="marg330"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 17.</s>

<s>Multa præterea cognita e&longs;&longs;ent in hoc genere, quæ nunc præter<lb/>

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<s>Sit &longs;ol a, & per&longs;picuum den&longs;um, exempli gratia, ut ampula <lb/>magna aqua plena b c d, & &longs;i &longs;it rotunda accendit ignem ex ad<lb/>uer&longs;o ut in e. Dico ergo in b c d e&longs;&longs;e quatuor genera luminis. Pri<lb/>mum quod e&longs;t ualidius, & rectà tran&longs;it, ualidius enim e&longs;t, quod <lb/>tran&longs;it quàm quod tran&longs;ire non pote&longs;t, & etiam quia, ut dixi, <lb/>ignem accen dit. Secundum e&longs;t quod colligitur in ampula, & dein<lb/>de &longs;pargitur <expan abbr="circ&utilde;circà">circuncircà</expan>, nam id ualidius e&longs;t, quia penetrat, & re&longs;ilit <lb/>quàm quod non penetrat, aut &longs;i penetrat, non &longs;pargitur, & hoc dif<lb/>funditur circa uas, necreflectitur rectè, &longs;ed qua&longs;i intro colligitur, & <lb/>diuer&longs;a ratione diffunditur, e&longs;t tamen imbecillius primo, ut dictum <lb/>e&longs;t. Tertium genus e&longs;t, quod illuminat intus ingrediendo, &longs;ed non <lb/>&longs;pargitur, & hoc e&longs;t debilius &longs;ecundo, quia <expan abbr="nõ">non</expan> pote&longs;t &longs;pargi. Quar<lb/>

<arrow.to.target n="fig71"></arrow.to.target><lb/>tum e&longs;t, quod non ingreditur omnino, &longs;ed refle<lb/>ctitur, i&longs;tud e&longs;t ab&longs;que dubio imbecillimum, quo<lb/>niam penetrare non pote&longs;t. Et licet in &longs;peculis <lb/>concauis radius reflexus uideatur e&longs;&longs;e ualidior, <lb/>&longs;tatim enim accendit ignem, hoc non contin<lb/>git, ni&longs;i quia in &longs;peculo cauo radij omnes col

<figure id="fig71"></figure><lb/>tum e&longs;t, quod non ingreditur omnino, &longs;ed refle<lb/>ctitur, i&longs;tud e&longs;t ab&longs;que dubio imbecillimum, quo<lb/>niam penetrare non pote&longs;t. Et licet in &longs;peculis <lb/>concauis radius reflexus uideatur e&longs;&longs;e ualidior, <lb/>&longs;tatim enim accendit ignem, hoc non contin<lb/>git, ni&longs;i quia in &longs;peculo cauo radij omnes col

<pb pagenum="90"/><expan abbr="ligun&ttilde;">liguntur</expan> ob <expan abbr="opac&utilde;">opacum</expan>, quod à tergo e&longs;t, neque <expan abbr="&longs;pargun&ttilde;">&longs;parguntur</expan>, neque <expan abbr="tran&longs;e&utilde;t">tran&longs;eunt</expan>, neque<lb/>combibuntur, ut ita dicam &longs;ed omnes <expan abbr="reflect&utilde;tur">reflectuntur</expan>. Ex quo colligitur <lb/>quin cuplex ordo radiorum iuxta rationem uirium, primus e&longs;t refle <lb/><expan abbr="xor&utilde;">xorum</expan> à &longs;peculo <expan abbr="cõcauo">concauo</expan>, & hi &longs;unt <expan abbr="pot&etilde;ti&longs;simi">potenti&longs;simi</expan> ob <expan abbr="ration&etilde;">rationem</expan> <expan abbr="dictã">dictam</expan>, po&longs;t <lb/>quos &longs;unt radij, qui tran&longs;eunt per per&longs;picuum maximè rotundum, <lb/>qui & ip&longs;i generant ignem, & debiliorem primo, deinde reliqui <lb/>tres &longs;equentes &longs;upradicti. Sextus e&longs;t radiorum, qui reflectuntur à <lb/>rebus non nitidis, ut à muris, & tabulis, nam omnia dura reflectunt <lb/>& etiam mollium pleraque, & hæc reflexio e&longs;t fermè infinita, & ob id <lb/>cubicula etiam in angulis illuminantur.</s>

<s>

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<s>

<arrow.to.target n="marg339"></arrow.to.target><lb/>lut ouata ip&longs;um mouetur à quauis ui, &longs;ed &longs;i in&longs;ideat per &longs;uperfi<lb/>ciem, quanto maior e&longs;t, & humilior, tanto difficilius mouetur, <lb/>ideò in corpore uiginti ba&longs;ium, quòd inter regularia uocata, plu<lb/>res habet, &longs;uperficies pro ratione æqualis ponderis, motus erit <lb/>longe facilior. Alia cau&longs;a e&longs;t inæqualitas partium, unde quæ ro<lb/>tunda &longs;unt, quia prominent, facile mouentur, & cum partes me<lb/>diæ in&longs;i&longs;tant plano, quanto minores erunt tanto facilius moue<lb/>buntur ratione ponderis. Vnde patet, quòd corpora ouata faci<lb/>lius mouentur, etiam quàm &longs;phærica, habent enim partem me<lb/>diam minorem, & paria &longs;unt ratione ince&longs;&longs;us plani, &longs;ed aëris mul<lb/>titudine tardius, quoniam enim &longs;phæra &longs;ub æquali ambitu plus <lb/>continet corporis, ergo ouatum æquale &longs;phæræ habet maio<lb/>rem ambitum ip&longs;a &longs;phæra. Hæc autem à Theone partim de<lb/>mon&longs;trata &longs;unt, partim ab Archimede, & partim à nobis, ergo <lb/>motus ouati e&longs;t fermè æqualis motui &longs;phæræ, & tardior e&longs;t con<lb/>

<arrow.to.target n="fig72"></arrow.to.target><lb/>citatus, quàm &longs;phæræ, quia à ma<lb/>iore excipitur aëre, & partes exte<lb/>riores non ita incumbunt in me<lb/>dium &longs;ecundum longitudinem. Cu<lb/>bus uero tardior e&longs;t propter æqua<lb/>litatem, & latitudinem &longs;uperficiei in<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro<lb/>pter has cau&longs;as conus ambligonius, <lb/>& quanto magis fuerit, ratio uero <lb/>eleuationis e&longs;t, ut &longs;it cubus b c, cuius <lb/>medium grauitatis &longs;it b &longs;uper pla

<figure id="fig72"></figure><lb/>citatus, quàm &longs;phæræ, quia à ma<lb/>iore excipitur aëre, & partes exte<lb/>riores non ita incumbunt in me<lb/>dium &longs;ecundum longitudinem. Cu<lb/>bus uero tardior e&longs;t propter æqua<lb/>litatem, & latitudinem &longs;uperficiei in<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro<lb/>pter has cau&longs;as conus ambligonius, <lb/>& quanto magis fuerit, ratio uero <lb/>eleuationis e&longs;t, ut &longs;it cubus b c, cuius <lb/>medium grauitatis &longs;it b &longs;uper pla

<pb pagenum="92"/>no de, & eleuetur ex a, & manife&longs;tum e&longs;t, quod in&longs;idebit per totam <lb/>lineam c f ip&longs;i plano, & proportio grauitatis totius &longs;u&longs;pen&longs;i in com <lb/>paratione ad grauitatem eius, qui inuertit, e&longs;t, uelut proportio par<lb/>tis terminatæ ad lineam c f uer&longs;us eum, qui eleuat ad partem, quæ <lb/>ultra e&longs;t, cum uerò hæ partes notæ &longs;int iuxta perpendiculum ex <lb/>centro grauitatis, manife&longs;tum e&longs;t, quod &longs;ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque &longs;itu ad pondus eius, dum &longs;u<lb/>&longs;penditur, & clarum e&longs;t, quòd cùm centrum, & medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s>

<s>Propo&longs;itio nonage&longs;imaoctaua.</s>

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<s>Sit a b, quæ &longs;i appen&longs;a e&longs;&longs;et ad æquidi<lb/>

<arrow.to.target n="fig73"></arrow.to.target><lb/>&longs;tantem terræ &longs;uperficiei, nulla ui po&longs;&longs;et ele </s>

<figure id="fig73"></figure><lb/>&longs;tantem terræ &longs;uperficiei, nulla ui po&longs;&longs;et ele </s>

<s>

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<s>Solent gemmarij uendere adamantem ponderis unius grani <lb/>uno coronato, duorum autem granorum tribus coronatis, qua<lb/>tuor autem, gratia exempli, quadraginta coronatis, qu&ecedil;ritur quan<lb/>tum ualebit adamas octo granorum, quoniam ergo proportio <lb/>non &longs;eruatur. E&longs;t enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>quàm tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. gratia exempli, in prima ad ditione fuit <lb/>unum granum, & acqui&longs;iuit proportionem triplam, in &longs;ecunda fue <lb/>runt duo grana, &longs;i ergo acqui&longs;i&longs;&longs;et &longs;olum &longs;excuplam proportio<lb/>nem, haberemus intentum. Propterea in i&longs;to ca&longs;u oportet demon<lb/>&longs;trare forma Geometrica, &longs;uppo&longs;ito, quòd &longs;it figura recta ex uno la <lb/>

<arrow.to.target n="fig74"></arrow.to.target><lb/>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, & ex <lb/>æquali b a c addito fiat b d tripla b c, & ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, & iuxta rationem erit in infinitum. <lb/>Siue &longs;it parabole, &longs;iue hiperbole, &longs;eu &longs;it alia coincidentium.</s>

<figure id="fig74"></figure><lb/>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, & ex <lb/>æquali b a c addito fiat b d tripla b c, & ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, & iuxta rationem erit in infinitum. <lb/>Siue &longs;it parabole, &longs;iue hiperbole, &longs;eu &longs;it alia coincidentium.</s>

<s>SCHOLIVM.</s>

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<s>Si ab accliue, &longs;eu decliue in quo d ad attra<lb/>

<arrow.to.target n="marg358"></arrow.to.target><lb/>

<arrow.to.target n="marg359"></arrow.to.target><lb/>

<arrow.to.target n="fig75"></arrow.to.target><lb/>hendum, cuius nota e&longs;t ex &longs;uperioribus dif<lb/>ficultas in plano ratione figuræ con&longs;tante, er<lb/>go ea quæritur proportio a&longs;cen&longs;us, & quo<lb/>niam terminus ad perpendiculum e&longs;t dupla <lb/>proportio, & iam grauitas in plano e&longs;t dimidium, ideò quicquid <lb/>acquiritur in eleuatione e&longs;t in comparatione ad illud dimidium, <lb/>cum ergo attractio &longs;ecundum eandem proportionem augeatur, er<lb/>go &longs;emper maior difficultas augebitur, ergo ab initio minimum

<figure id="fig75"></figure><lb/>hendum, cuius nota e&longs;t ex &longs;uperioribus dif<lb/>ficultas in plano ratione figuræ con&longs;tante, er<lb/>go ea quæritur proportio a&longs;cen&longs;us, & quo<lb/>niam terminus ad perpendiculum e&longs;t dupla <lb/>proportio, & iam grauitas in plano e&longs;t dimidium, ideò quicquid <lb/>acquiritur in eleuatione e&longs;t in comparatione ad illud dimidium, <lb/>cum ergo attractio &longs;ecundum eandem proportionem augeatur, er<lb/>go &longs;emper maior difficultas augebitur, ergo ab initio minimum

<pb pagenum="96"/>erit di&longs;crimen ab attractione in plano. Exempli gratia &longs;it, ut graue d <lb/>in plano &longs;it, ut quin que, & &longs;u&longs;pen&longs;um decem, ergo in medio angulo <lb/>erit penè &longs;eptem, &longs;ed &longs;eptem minus longe <expan abbr="di&longs;tãt">di&longs;tant</expan> à quin que, quàm de<lb/>cem ad &longs;eptem, ergo in &longs;ecunda parte plus longè augebitur difficul <lb/>tas attractionis &longs;upra difficultatem in medio angulo accliui, quam <lb/>in prima parte à plano ad medium accliue, & quoniam planum in <lb/>plano de&longs;cendit, tanto uehementius, quanto difficilius attrahitur, <lb/>ergo planum in decliui &longs;ublimi longe maiore impetu feretur infrà <lb/>quam &longs;it proportio anguli ad angulum. Exempli gratia, planum in <lb/>medio angulo, &longs;i incipiat de&longs;cendere in dodrante multo lentius, <lb/>quàm pro dimidio uirium de&longs;cen&longs;us totius anguli, imò initium de<lb/>&longs;cen&longs;us e&longs;t à medio recti ad unguem, ubi omnia plana &longs;int, & duri&longs;<lb/>&longs;ima, & cau&longs;a huius e&longs;t, quia omne graue tendit ad centrum, quòd <lb/>maior pars ip&longs;ius grauis e&longs;t ultra medium grauitatis in decliui <lb/>humiliore.</s>

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<s><margin.target id="marg359"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 62. & <lb/>64. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s>Propo&longs;itio cente&longs;imaquinta.</s>

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<s>Sit circulus a b c, cuius dimetiens, nota b d &longs;it b, erit ergo latus <lb/>

<arrow.to.target n="marg364"></arrow.to.target><lb/>

<arrow.to.target n="fig76"></arrow.to.target><lb/>exagoni a b dimidium b d, id e&longs;t 3. igitur <lb/>cum angulus a &longs;it rectus, erit a d <02> 27 latus <lb/>trianguli. Et latus quadrati per eandem <02><lb/>18. Vt latus exagoni &longs;it <02> 9. Quadrati <02> 18 <lb/>Trianguli <02> 27, & ita pote&longs;tate &longs;e habent <lb/>hæc ut 1. 2. 3. Et &longs;unt nota. Et quia latus d e c <lb/>agoni e&longs;t <02> 11 1/4 m, 1 1/2. & ip&longs;um erit notum. <lb/>Quare latus pentagoni e&longs;t <02> v 22 1/2 m: <02><lb/>101 1/4 notum. Et iam notum fuit latus epta<lb/>goni. Habebimus igitur latera Trianguli

<figure id="fig76"></figure><lb/>exagoni a b dimidium b d, id e&longs;t 3. igitur <lb/>cum angulus a &longs;it rectus, erit a d <02> 27 latus <lb/>trianguli. Et latus quadrati per eandem <02><lb/>18. Vt latus exagoni &longs;it <02> 9. Quadrati <02> 18 <lb/>Trianguli <02> 27, & ita pote&longs;tate &longs;e habent <lb/>hæc ut 1. 2. 3. Et &longs;unt nota. Et quia latus d e c <lb/>agoni e&longs;t <02> 11 1/4 m, 1 1/2. & ip&longs;um erit notum. <lb/>Quare latus pentagoni e&longs;t <02> v 22 1/2 m: <02><lb/>101 1/4 notum. Et iam notum fuit latus epta<lb/>goni. Habebimus igitur latera Trianguli

<pb pagenum="98"/>quadrati pentagoni, & eptagoni æquilaterorum nota: & etiam <lb/>&longs;ubten&longs;orum duobus ex his. Sit, gratia exempli, a b 3 & b c <02> 11 1/4m: <lb/>1 1/2, ut prius, & ponatur b d diameter, erit ad <02> 27 & c d <02> v 22 1/2 m: <lb/><02> 101 1/4, quam ducemus in a b, & fiet <02> v 202 1/2 m: <02> 8201 1/4. Duce<lb/>mus itidem <02> 27 a d in b c <02> 11 1/4 m: 1 1/2 fiet <02> 303 3/4m: <02> 60 3/4, hoc to<lb/>tum diuide per 66, quæ e&longs;t b: fiet a c <02> 8 7/16 m: <02> 1 11/16 p: <02> v: 5 45/72 m: <02><lb/>6 1701/5184. Nec credas te errare, quoniam latus pentagoni e&longs;&longs;et, ac &longs;i an<lb/>gulus b rectus e&longs;&longs;et: &longs;ed quia e&longs;t obtu&longs;us, ideo a c e&longs;t alia linea, & <lb/>maior latere pentagoni. Et &longs;imiliter &longs;i a b, & a c notæ e&longs;&longs;ent, utpo<lb/>

<arrow.to.target n="marg365"></arrow.to.target><lb/>te a b 3, ut prius a c 5 dico, quòd b c nota e&longs;t: nam a d erit <02> 27, & <lb/>quia ex b d in a c fit 30, fiet ex b c in a d pos <02> 27, et ex a b in c d <02> 324 <lb/>m: 9 quad. igitur 30 m: pos <02> 27 æquantur <02> 324 m: 9 quad. quare <lb/>900 p: 27 quad. m: pos <02> 97200 <expan abbr="æquãtur">æquantur</expan> 324 m: 9 quad. igitur 576 <lb/>p: 16 quad. &ecedil;quantur pos <02> 97200. Quadratum igitur p: 36 &ecedil;quan<lb/>tur pos <02> 379 11/16, erit ergo b c <02> v: <02> 94 59/64 p: <02> 58 59/64 & &longs;imiliter &longs;i a c <lb/>&longs;it nota, puta 4 erit a b &longs;ubten&longs;a dimidio arcus a c nota. Erit enim a e <lb/>2 ergo d e 3 p: <02> 5 et b e 3 m: <02> 5, <expan abbr="igi&ttilde;">igitur</expan> a b <02> v: 18 m, <02> 180. Igitur hoc <lb/>modo diuidendo, iungendo, & detrahendo habebimus ex quatu<lb/>or illis &longs;implicibus trianguli quadrati. Pentagoni, & eptagoni in <lb/>numeras linearum magnitudines in circulo. Et &longs;imiliter quouis mo <lb/>do, ut dictum e&longs;t, in quauis figura æquilatera, utpote &longs;uppo&longs;ito <lb/>

<arrow.to.target n="fig77"></arrow.to.target><lb/>quod de&longs;criptum &longs;it nonangulum in <lb/>circulo æquilaterum, quod etiam erit <lb/>æquiangulum, & &longs;it arcus a b duplus <lb/>arcui a c, erit angulus a c b duplus an<lb/>gulo a b c, & angulus b a c in portione <lb/>b d e c &longs;excuplus a b c, & triplus a c b. <lb/>Erit ergo per demon&longs;trata proportio <lb/>

<figure id="fig77"></figure><lb/>quod de&longs;criptum &longs;it nonangulum in <lb/>circulo æquilaterum, quod etiam erit <lb/>æquiangulum, & &longs;it arcus a b duplus <lb/>arcui a c, erit angulus a c b duplus an<lb/>gulo a b c, & angulus b a c in portione <lb/>b d e c &longs;excuplus a b c, & triplus a c b. <lb/>Erit ergo per demon&longs;trata proportio <lb/>

<arrow.to.target n="marg366"></arrow.to.target><lb/>b a ad a c, uelut a c, & c b, ad a b: pro<lb/>portio autem a b arcus ad a c, ex &longs;up<lb/>po&longs;ito maior e&longs;t proportione rectæ a b ad a c, igitur etiam propor<lb/>tione a c & c b ad a b, ergo duo latera trianguli ad tertium minorem <lb/>habent proportionem, quam arcus ad arcum, quanto rectæ ad re<lb/>ctam minor e&longs;t. Sit rur&longs;us in triangulo b e d quomodolibet modo <lb/>&longs;it angulus b d e quadruplus angulo b e d, & diuidatur d per &ecedil;qua<lb/>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, &longs;ed e f ad <lb/>

<arrow.to.target n="marg367"></arrow.to.target><lb/>f b ut d e ad d b. igitur proportio b d, d e ad f b <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex propor<lb/>tionibus e f ad f d, & e d ad d b. Proportio igitur b d, d e ad f b, ut <lb/>producti ex e f in e d ad productum ex d fin d b. Rur&longs;us ponamus, <lb/>

<arrow.to.target n="marg368"></arrow.to.target><lb/>quod in quadrangulo a b c d primæ figuræ &longs;it a b 4 b c 3 c d 5 ad 6 <lb/>dico, quòd &longs;pacium contentum erit notum. Ductis rectis a c & b d

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<s>Sit modo obtu&longs;i angulus a b c, & nota latera &longs;ingula, & angu<lb/>lus a b c, & producantur latera ad perpendicu<lb/>

<arrow.to.target n="fig78"></arrow.to.target><lb/>lum, ut &longs;int d & e recti, & quia anguli ad a &longs;unt <lb/>æquales, erunt anguli e b a, & d e a &longs;emper æ<lb/>

<figure id="fig78"></figure><lb/>lum, ut &longs;int d & e recti, & quia anguli ad a &longs;unt <lb/>æquales, erunt anguli e b a, & d e a &longs;emper æ<lb/>

<arrow.to.target n="marg372"></arrow.to.target><lb/>quales. Et hoc idem contingit in acuti angulis <lb/>triangulis intus, & e&longs;t utile mechanicum: & <lb/>quia a b c notus e&longs;t, & d notus, erunt anguli tri <lb/>goni d b c noti: & &longs;i fuerit angulus a notus, <expan abbr="er&utilde;t">erunt</expan> anguli d a c & e a b <lb/>noti, & ideo anguli e b a, & d c a: & &longs;emper notum, quod fit ex b a <lb/>in a d, uel c a in a e, &longs;unt enim &ecedil;qualia inter &longs;e: etiam notæ ad & a c, <lb/>quoniam duplum horum e&longs;t exce&longs;&longs;us quadrati b c &longs;uper quadrata <lb/>a b, & a c. Quod uerò proponiturà Monteregio de cognitione an<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/>

<arrow.to.target n="marg373"></arrow.to.target><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>

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<s><margin.target id="marg373"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>&longs;e<lb/>cundi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s>Et iterum ponamus, quòd proportio a c c b ad a b &longs;it qualis a b <lb/>ad a c, dico quòd angulus c duplus e&longs;t angulo b. Si non ducatur c d <lb/>

<arrow.to.target n="fig79"></arrow.to.target><lb/>faciens angulum d c b duplum b, erit igitur pro<lb/>portio d c c b ad d b, ut d b ad d c. Maior e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>d c, quàm a c, aut æqualis, aut minor, &longs;i æqualis, <lb/>igitur maior proportio d c c b ad b d quàm b a, <lb/>igitur maior proportio b d ad d c quam b a ad a c <lb/>ad a c & æquales &longs;unt igitur b d maior d a pars toto, quod e&longs;&longs;e non <lb/>pote&longs;t. Si uerò d c ponatur maior a c, magis ex hoc &longs;equitur b d ma<lb/>iorem e&longs;&longs;e b a. Quod &longs;i minor &longs;it d c quàm a c. Ex demon&longs;tratio<lb/>ne ip&longs;ius reflexæ proportionis patet hoc contingere non po&longs;&longs;e. <lb/>Et &longs;imiliter patet conuer&longs;as in reliquis etiam ueras e&longs;&longs;e, non &longs;olum

<figure id="fig79"></figure><lb/>faciens angulum d c b duplum b, erit igitur pro<lb/>portio d c c b ad d b, ut d b ad d c. Maior e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>d c, quàm a c, aut æqualis, aut minor, &longs;i æqualis, <lb/>igitur maior proportio d c c b ad b d quàm b a, <lb/>igitur maior proportio b d ad d c quam b a ad a c <lb/>ad a c & æquales &longs;unt igitur b d maior d a pars toto, quod e&longs;&longs;e non <lb/>pote&longs;t. Si uerò d c ponatur maior a c, magis ex hoc &longs;equitur b d ma<lb/>iorem e&longs;&longs;e b a. Quod &longs;i minor &longs;it d c quàm a c. Ex demon&longs;tratio<lb/>ne ip&longs;ius reflexæ proportionis patet hoc contingere non po&longs;&longs;e. <lb/>Et &longs;imiliter patet conuer&longs;as in reliquis etiam ueras e&longs;&longs;e, non &longs;olum

<pb pagenum="100"/>in proportionibus noti&longs;simis angulorum &longs;ed etiam in coniuncti<lb/>one & detractione. Et e&longs;t ex &longs;ubtili&longs;simis operationibus, quæ ho<lb/>mini in hoc genere eueniant.</s>

<s>Propo&longs;itio cente&longs;ima&longs;eptima.</s>

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<s>Per præcedentem moto puncto a uer&longs;us c &longs;emper u&longs; que ad e, c ma <lb/>

<arrow.to.target n="marg380"></arrow.to.target><lb/>gis di&longs;tat <expan abbr="p&utilde;ctum">punctum</expan> a linea a e, quàm à puncto a uer&longs;us, quia linea n h <lb/>maior e&longs;t n a, & per eandem dum appropinquat ad c cum e c fiat <lb/>&ecedil;qualis ea, maius fit in crementum in a e, quàm re&longs;pectu lineæ tran&longs;<lb/>uer&longs;alis. Volo ergo inuenire punctum hoc in quo fit mutatio: & <lb/>diuido arcum ac per æqualia in f, & dico illum e&longs;&longs;e punctum quæ<lb/>&longs;itum: accepto quouis puncto in e f, puta k, duco g o h p &ecedil;quidi&longs;tan

<arrow.to.target n="fig80"></arrow.to.target><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/>

<figure id="fig80"></figure><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/>

<arrow.to.target n="marg381"></arrow.to.target><lb/>& anguli f e a, & f e c &ecedil;quales, igitur uter <lb/>

<arrow.to.target n="marg382"></arrow.to.target><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &ecedil;qua <lb/>

<arrow.to.target n="marg383"></arrow.to.target><lb/>lis n k, igitur c q æqualis e n, quare h p <lb/>æqualis g o, &longs;ed quod fit ex o k in k g e&longs;t <lb/>

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<s>Propo&longs;itio cente&longs;imanona.</s>

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<arrow.to.target n="marg388"></arrow.to.target><lb/>a r, quæ e&longs;t minor dimidio e r, & ideò minor e r, quæ e&longs;t maior di<lb/>midio, ut demon&longs;tratum e&longs;t, & etiam minor r f, quæ æqualis e&longs;t r e <lb/>

<arrow.to.target n="marg389"></arrow.to.target><lb/>per demon&longs;trata rur&longs;us: & hic e&longs;t naturalis ut palam e&longs;t: alter præ<lb/>ter <expan abbr="naturã">naturam</expan>, & e&longs;t ferri ad latus, quoniam hoc e&longs;t <expan abbr="propri&utilde;">proprium</expan> immortali<lb/>bus: cun que hic &longs;it ad latus e&longs;t etiam <expan abbr="cõtra">contra</expan> naturam, quia magis di&longs;tat <lb/>a centro, nam e f e&longs;t longior c r, &longs;i ergo r ferretur in f, moueretur à <lb/>centro, & contra naturam. Dum ergo fertur ex a in f, multo lentius

<pb pagenum="102"/>fertur, quàm ex f in c: uelo cius autem ex c u&longs;que ad medium: nam <lb/>plurimum de&longs;cendit. Ex h ad b autem celerrimè, quoniam de&longs;cen<lb/>dit, & appropinquat lineæ a b, ut uter que motus &longs;it naturalis. Non <lb/>ergo mouetur pr&ecedil;ter naturam ni&longs;i quatenus longius recedit à linea <lb/>a b, unde in inferiore parte mouetur ad eandem, ideò de parte c b <lb/>tota per&longs;picua e&longs;t ratio, cur facillimè de&longs;cendat, &longs;imiliter & tota, <lb/>hoc enim e&longs;t demon&longs;tratum. Similiter & quare difficillimè feratur <lb/>ex b u&longs; que ad p, & ultra p u&longs; que ad directum r f: at de motu ex a in f, <lb/>quod debeat ferri, quia plus remouetur, quam de&longs;cendat, nulla e&longs;t <lb/>ratio: ut nec cur ex oppo&longs;ito f ad a difficilem &longs;e præ&longs;tet: & hoc e&longs;t, <lb/>quia tertiam rationem etiam ip&longs;e Ari&longs;toteles, & qui eum &longs;equuti <lb/>&longs;unt, prætermi&longs;it. Ea autem e&longs;t, quod dum fertur ad g, uel f etiam li<lb/>cet non de&longs;cendat magis, quàm remoueatur, ex a <lb/>

<arrow.to.target n="fig81"></arrow.to.target><lb/>ad centrum terræ tamen magis appropinquat. <lb/>Quia enim e a e&longs;t &ecedil;qualis e c, quoniam prodeunt <lb/>à centro circuli eiu&longs;dem, & b e, & e c &longs;unt maio<lb/>res b c, ideò b a erit maior b c, e&longs;t autem b cen<lb/>

<figure id="fig81"></figure><lb/>ad centrum terræ tamen magis appropinquat. <lb/>Quia enim e a e&longs;t &ecedil;qualis e c, quoniam prodeunt <lb/>à centro circuli eiu&longs;dem, & b e, & e c &longs;unt maio<lb/>res b c, ideò b a erit maior b c, e&longs;t autem b cen<lb/>

<arrow.to.target n="marg390"></arrow.to.target><lb/>trum mundi, ergo a motum ad c, appropin qua<lb/>uit ip&longs;i b</s>

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<s>Dico etiam quod libra ex chalybe tenui&longs;simo, <lb/>& quanto <expan abbr="leuior&utilde;">leuiorum</expan> concharum, & longioris iugi <lb/>10 exactior, quoniam lances illæ minori exce&longs;&longs;u <lb/>mouentur, quia plus di&longs;tant ab hypomochlio. <lb/>Sit ergo libra, cuius iugum a b trutin a c: lances d & e, alia libra, <lb/>cuius lances h, & k, & l m longiores, iugum f g. Con&longs;tat, quod <lb/>qualis proportio f g ad a b, talis ambitus, ad ambitum: motus er<lb/>go &longs;i &longs;it æqualis utrarumque, igitur a tanto minore proportione <lb/>

<arrow.to.target n="fig82"></arrow.to.target>

<pb pagenum="103"/>mouebitur in h, quam in d, uelut &longs;it proportio f g ad a b dupla, ut <lb/>ergo æqualiter moueantur, &longs;i &longs;it dupla &longs;exquiquarta in d cum lan<lb/>ce ad e uacuam, erit in h &longs;exquialtera, & mouebit æquali tempore. <lb/>Ergo iuxta hoc fient libræ, quæ examinabunt decimam, & uige&longs;i<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in precio&longs;is rebus, & me<lb/>dicamentis potentibus, & longè magis in mechanicis experimen<lb/>tis, & maximè quæ ad demon&longs;trationem pertinent magnitudinis <lb/>&longs;uperficierum, & con&longs;tat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materiæ illius, & lancium, nam tanto maior redditur propor <lb/>tio ponderis exigui, & in firmitate iugi ac rectitudine. ideò debet <lb/>fieri ex chalybe purgato, durato ac tenui&longs;simo, natura que leui, & ut c <lb/>&longs;it in medio, & mobilis f g.</s>

<s>Con&longs;iderandum e&longs;t demum an f l & g m &longs;int grauiores f h, & <lb/>g k. Vt enim grauiores extiterint minus facilè mouentur. Viden<lb/>tur autem mihi, qui de his con&longs;crip&longs;erunt perperam contemp&longs;i&longs;&longs;e <lb/>hoc, con&longs;tat enim, quòd dum l de&longs;cendit, remouetur a b n c tru<lb/>tina, & m, quæ a&longs;cendit contra appropinquat. Videtur autem hoc <lb/>bifariam contra naturam: nam ut diximus pondus applicat &longs;e ad <lb/>rectam n c, quia uer&longs;us centrum, & etiam quia facit angulum ob<lb/>tu&longs;um, cum deberet, ut ab initio &longs;altem con&longs;tituere cum iugo re<lb/>ctum. Et de m nihil mirum e&longs;t, cum acutum, ut &longs;e ad lineam, quæ ad <lb/>centrum retrahat. Huiu&longs;modi præterij&longs;&longs;e Ari&longs;totelem, demiror, <lb/>quæ nimis fuerunt in con&longs;picuo, ut dubitem ne non &longs;uus &longs;it ille li<lb/>ber, qui eius penè nihil &longs;apiat præter ob&longs;curitatem. Tentan<lb/>dum e&longs;t igitur horum cau&longs;as a&longs;signare. nam quæ huiu&longs;modi po<lb/>te&longs;t e&longs;&longs;e doctrina ni&longs;i perfecta fuerit, in omnibus etenim nece&longs;&longs;e e&longs;t <lb/>aut omnia &longs;cire, aut ignorare. In hoc igitur dico, quod h f, &longs;eu l f, <lb/>&longs;emper æquidi&longs;tant n c trutinæ, ergo cum angulus f c n in clina<lb/>to iugo fiat obtu&longs;us de&longs;cendente pondere, & n c g a&longs;cendente pon<lb/>dere fiat acutus, ergo angulus l f c tantundem fiet obtu&longs;ior, & m g c <lb/>acutior, quanto anguli ad c tales &longs;unt. Et cau&longs;a e&longs;t quia n c ratio<lb/>ne ponderis e&longs;t directa ad centrum, ergo oportet, ut pondera l, uel <lb/>h, & m, uel k, &longs;i debent tendere ad centrum, ut f l, & g m æquidi<lb/>&longs;tent n c, ni&longs;i quantum e&longs;t pro di&longs;tantia f, à puncto c, & g a b eodem, <lb/>quæ comparata ad <expan abbr="centr&utilde;">centrum</expan> terr&ecedil;, &longs;eu mundi, e&longs;t in&longs;en&longs;ibilis omnino. <lb/>Circa hæc <expan abbr="notand&utilde;">notandum</expan> i&longs;tud mirabile fcilicet, quod ratio motus, quan<lb/>tumuis exigua &longs;ufficit ad motus <expan abbr="mod&utilde;">modum</expan>, licet uelo citas <expan abbr="p&etilde;deat">pendeat</expan> ex gra <lb/>uitate, & alijs. Et quae graue, quod expers e&longs;t &longs;en&longs;us, debeat &longs;equi ratio <lb/>nem Geometricam uix &longs;apientibus <expan abbr="cognitã">cognitam</expan>, cau&longs;a tamen una e&longs;t, & <lb/>per&longs;picua: <expan abbr="nã">nam</expan> omne graue e&longs;t in linea à centro <expan abbr="m&utilde;di">mundi</expan>: &longs;i <expan abbr="a&utilde;t">aunt</expan> medium <lb/>grauis &longs;it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu&ecedil; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em

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<s>Si duæ &longs;phæræ ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">aem</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/>

<arrow.to.target n="fig83"></arrow.to.target><lb/>

<arrow.to.target n="marg391"></arrow.to.target></s>

<s>Supponitur quod ex eodem loco. Sermo enim <lb/>ab&longs;urda &longs;ub interpretatione nunquam ni&longs;i ab inui<lb/>dio&longs;o, uel imperito intelligi debet. Sit ergo a tripla <lb/>ad b, &longs;phærula ad &longs;phærulam ex plumbo ambæ fer<lb/>ro uel lapide eiu&longs;dem generis, dico, quòd inæquali <lb/>tempore peruenient ad planum c d. Nam a propor<lb/>tionem habet ad b, ut uiginti&longs;eptem ad unum. pro<lb/>portio autem &longs;patij a ad &longs;patium b nonupla e&longs;t, & <lb/>proportio den&longs;itatis aëris ad aërem e&longs;t tripla, propterea quod den<lb/>&longs;itas illa multiplicatur propter impetus magnitudinem. nam &longs;i ro<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du<lb/>plo, quàm &longs;it robur, ut quinque percutiat baculo, ut duo: propter <lb/>den&longs;itatem ergo maiorem aëris in a, quam in b: & quoniam &longs;i &longs;ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aerr</expan> &longs;ub a, quam &longs;ub b, igitur proportio <lb/>erit comparanda longitudini à centro a ad longitudinem a centro <lb/>b, quæ e&longs;t tripla. Si ergo &longs;ubtripla e&longs;t ratio motus b ad a, quod <lb/>ad medium attinet, tripla autem propter uelo citatem di&longs;ce&longs;&longs;us aë<lb/>ris à medio grauitatis, quod e&longs;t in &longs;uperficie e regione centri graui<lb/>tatis in linea ad centrum mundi, ut dictum e&longs;t in præcedenti: mani<lb/>fe&longs;tum e&longs;t, quod a, & b inæquali tempore peruenient ad &longs;ubie<lb/>ctum planum, & æquidi&longs;tans centris eorum. Similiter & in aqua:

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<s>

<arrow.to.target n="marg392"></arrow.to.target><lb/>dentur rem nauticam quòd ad remos attinet, referre in longitu<lb/>dinem partis, quæ &longs;calmum tanquàm hypomochlium interiacet <lb/>& manum: ea enim circa medium nauis cum illa ibi &longs;it latior ma<lb/>ior e&longs;t. Sed & qui lembos ducunt, & in puppe magis di&longs;tant à <lb/>&longs;calmo & in prora, quàm in medio nauis, nec tamen uelo cius il<lb/>lam agunt: non quòd ratio illa fal&longs;a &longs;it, &longs;ed quia uelo cius ferun<lb/>tur etiam ob aliam cau&longs;am, quàm &longs;it hæc, & magis uniuer&longs;alem. <lb/>Primum igitur &longs;umamus, quod &longs;uperiùs demon&longs;tratum e&longs;t &longs;cili<lb/>

<arrow.to.target n="marg393"></arrow.to.target><lb/>cet, quòd ubi pondus aliquod æquale undique tanquam in li<lb/>bra &longs;u&longs;pen&longs;um fuerit, proportio ponderis partium inæqualium <lb/>ad duas partes æquales, e&longs;t confu&longs;a ex proportione longitudi<lb/>nis earundem, & quadrato eiu&longs;dem proportionis. Sit ergo diui<lb/>&longs;a a b in c, & fiat c e æqualis c a: proportio igitur ponderis b e ad <lb/>pondus e a e&longs;t compo&longs;ita ex proportione b e ad e a, & quadrato <lb/>

<arrow.to.target n="fig84"></arrow.to.target><lb/>eius <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> longitudinem. at po&longs;ita agi <lb/>na d g in medio a b, proportio ponderis b e <lb/>ad pondus ea e&longs;t, ueluti longitudinis b e <lb/>ad e a, igitur proportio <expan abbr="põderis">ponderis</expan> b e ad e a, <lb/>cum agina e&longs;t extra medium in c, e&longs;t tanto <lb/>maior proportione b c ad ea, quantum e&longs;t quadratum illius pro<lb/>

<figure id="fig84"></figure><lb/>eius <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> longitudinem. at po&longs;ita agi <lb/>na d g in medio a b, proportio ponderis b e <lb/>ad pondus ea e&longs;t, ueluti longitudinis b e <lb/>ad e a, igitur proportio <expan abbr="põderis">ponderis</expan> b e ad e a, <lb/>cum agina e&longs;t extra medium in c, e&longs;t tanto <lb/>maior proportione b c ad ea, quantum e&longs;t quadratum illius pro<lb/>

<arrow.to.target n="marg394"></arrow.to.target><lb/>portionis, ergo b e pondus maius e&longs;t, cum agina e&longs;t in c, quàm in d. <lb/>igitur per <expan abbr="commun&etilde;">communem</expan> animi <expan abbr="&longs;ententiã">&longs;ententiam</expan> addito communi pondere a e, <lb/>erit pondus a b minus &longs;emper cum agina e&longs;t in d, <08> in ullo alio lo<lb/>co a b. Ergo pondus a b apprehen&longs;um in d <expan abbr="mouebi&ttilde;">mouebitur</expan> a b æquali ui <lb/>

<arrow.to.target n="marg395"></arrow.to.target><lb/>maiore proportione, <08> in ullo alio loco. Ha&longs;tile ergo in medio ap<lb/>prehen&longs;um maiore ui mouebitur, quàm in ulla alia parte. Et &longs;i gra

<pb pagenum="106"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. Semper igitur ob hanc cau<lb/>&longs;am mota ex medio grauitatis, &longs;eu uelo, &longs;eu ramo, &longs;eu manu uelo<lb/>cius mouentur, quàm ex alijs partibus. In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;tcteles. Propter hoc igi <lb/>tur, qui malum in naui collo carunt tantùm unum, in medio fermè <lb/>eum collocarunt, ut antiqui: & qui duos aut tres, maiorem cra&longs;sio<lb/>

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<s><margin.target id="marg396"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 82.</s>

<s>Propo&longs;itio cente&longs;imaduodecima.</s>

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<s>Iam uerò <expan abbr="cõ&longs;ideremus">con&longs;ideremus</expan>, quòd propo&longs;itum e&longs;t, non &longs;olum in com<lb/>

<arrow.to.target n="marg397"></arrow.to.target><lb/>paratione ad medium, &longs;ed extremorum inuicem, mi&longs;&longs;a enim ab imo <lb/>uelo cius feruntur, quàm à medio non &longs;olum manu, &longs;ed &longs;corpioni<lb/>bus, & arcubus. Videmus & hoc ob&longs;eruare pueros uirgam lon<lb/>gius iacentes non ex medio, &longs;ed imo apprehen&longs;am, quoniam pars <lb/>ip&longs;a anterior, & quæ manu apprehen&longs;a e&longs;t, uehementi impetu emit<lb/>titur: & ut recipit impetum magis æqualem, longius fertur, nam <lb/>quod emittitur proportionem habet ad &longs;patium. Cum ergo appre <lb/>hen&longs;a in medio uirga &longs;olum medietate anteriore impetum recipiat <lb/>per &longs;e, ob id minus fertur: at impetus &longs;equitur proportionem, ut ui<lb/>&longs;um e&longs;t, quæ e&longs;t circa medium ob leuitatem ponderis. In leuibus <lb/>ergo maius &longs;patium &longs;uperabunt emi&longs;&longs;a ex imo, quoniam propor<lb/>tio &longs;patij eadem e&longs;t ad duplum, & ad dimidium. igitur ex imo fer<lb/>me duplum etiam &longs;patij &longs;uperabit: non tamen omnino quia maio<lb/>rem, ut dixi proportionem habet ad id, quod ex medio comprehen <lb/>&longs;um e&longs;t. At in leuibus non e&longs;t nece&longs;&longs;arium, ut ex medio apprehen<lb/>dantur, quoniam etiam cum incremento illo ponderis iam leuia <lb/>&longs;unt: plus ergo facit longitudo eius, quod eiaculatur, quàm impe<lb/>

<arrow.to.target n="fig85"></arrow.to.target><lb/>tus, cuius demon&longs;tratio e&longs;t hæc. Sit uirga <lb/>a b apprehen&longs;a in medio ponderis unciæ <lb/>mediæ, & in a d, ut &longs;it d a palmus, & uige&longs;i<lb/>ma pars totius a b, erit ergo re&longs;iduum ad duplum, a d nonuplum, <lb/>

<figure id="fig85"></figure><lb/>tus, cuius demon&longs;tratio e&longs;t hæc. Sit uirga <lb/>a b apprehen&longs;a in medio ponderis unciæ <lb/>mediæ, & in a d, ut &longs;it d a palmus, & uige&longs;i<lb/>ma pars totius a b, erit ergo re&longs;iduum ad duplum, a d nonuplum, <lb/>

<arrow.to.target n="marg398"></arrow.to.target><lb/>& a b tota unciarum quin que cum dimidia, &longs;i igitur grauetur, quia in <lb/>&longs;itu recto e&longs;t mediæ unciæ, in æquidi&longs;tanti terræ, quin que unciarum <lb/>cum dimidio, erit in &longs;itu dimidij recti unciarum trium. E&longs;t igitur <lb/>proportio &longs;excupla, &longs;i apprehendatur in medio, & ad æquidi&longs;tan<lb/>tem, ad apprehen&longs;am in imo, & ad angulum medium: at emi&longs;&longs;a ex <lb/>

<arrow.to.target n="marg399"></arrow.to.target><lb/>a d habet totum aërem a b circumdantem impul&longs;um ex c b &longs;olum <lb/>dimidium reliqua pars ui trahitur, ergo proportio &longs;patij a b, erit <lb/>&longs;exdecupla fermè &longs;patio b c, quoniam e&longs;t triplicata corporis ad cor <lb/>pus eius, quæ e&longs;t longitudinis ad longitudinem, & quadruplicata

<pb pagenum="107"/>re&longs;pectu aëris a c, qui re&longs;i&longs;tit apprehen&longs;a a b in c. Et iam minus fere<lb/>batur quinta parte, ideo longius eiaculabitur triplo ex a, quàm ex <lb/>c. Nec tamen maiore impetu, quia obliquè fertur, & quæ obliquè <lb/><expan abbr="feri&utilde;t">feriunt</expan>, minore cum impetu feriunt: at que eo magis &longs;i leuia fuerint: ab <lb/>aëre enim circumambiente perturbantur, & in incertum trudun<lb/>tur. Quæ ergo grauia &longs;unt ex medio emi&longs;&longs;a, & ad æquidi&longs;tantem <lb/>longius feruntur, & maiore cum impetu, quia magis directè: leuia <lb/>autem longius ex imo, &longs;ed minore cum impetu, &longs;i aliqua cau&longs;a à re<lb/>cto, & æquidi&longs;tante declinauerint. At &longs;i à &longs;uprema parte, & iuxta <lb/>cu&longs;pidem, neque procul feruntur, neque cum impetu ob cau&longs;as di<lb/>ctas. Eadem quoque ratio e&longs;t omnium machinarum: ideò oblon<lb/>g&ecedil;longius eiaculantur, quoniam proportionem &longs;eruant ad cana<lb/>

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<s>Propo&longs;itio cente&longs;imatertia decima.</s>

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<s>Id e&longs;t, ue&longs;tigia per cu&longs;sit pedibus, ante que illa puluis pedibus ex<lb/>cu&longs;&longs;us (ue&longs;tigia &longs;cilicet relinquentibus) ingrederetur. Principalis <lb/>autem cau&longs;a uelo citatis e&longs;t agens, uelut equi. Sed inter <expan abbr="h&utilde;c">hunc</expan> motum <lb/>& priorem medius e&longs;t Scitalæ uocatæ, nam ut in primo axis proci<lb/>dit & rotundum à &longs;uperficie circumagitur, licet axis etiam circum<lb/>ducatur, ut axis, & rota, aut &longs;phæra duplici motu moueantur, fci<lb/>licet antror&longs;um, & circumcirca, in rota currus duo ijdem motus <lb/>&longs;int, axis quo que antror&longs;um moueatur, &longs;ed non circumagatur: unde <lb/>impeditior e&longs;t hic motus: ita in Scytala utrun que utro que motu mo<lb/>uetur, & circumcirca, & antror&longs;um, at que id commune e&longs;t, cum pri<lb/>mo ita axis mouet rotas, non rotæ axem, quòd &longs;ecundo motui ro<lb/>tarum in curru proprium e&longs;t, ut tantum degenerent à primo motu, <lb/>quanto leuius uertuntur, quàm in &longs;ecundo motu. Trahitur ergo <lb/>

<arrow.to.target n="fig86"></arrow.to.target><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri<lb/>bus. Propterea in primo motu trahi<lb/>tur, uel impellitur à &longs;uperficie: in &longs;e<lb/>cundo a b axe, &longs;ed non affixo rotis, unde ægrè trahuntur in &longs;cyta<lb/>la ab axe affixo rot&ecedil;. Quare leuius quàm in curru, difficilius quàm <lb/>in rota uel &longs;phæra à &longs;uperficie extima circumacta. Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. E&longs;t enim hic &longs;imilior <lb/>primo, quia contrarius, in primo enim procedit rota, & uertitur à <lb/>circumferentia, hic quie&longs;cit rota, & mouetur ab axe. Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rotæ. Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir<lb/>cunferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta<lb/>tem. Sed & trochlea caret colloppibus. Ergo uerum e&longs;t, quod o<lb/>mnia rotunda facilius circumaguntur, &longs;ed uaria ratione: nam plus <lb/>mota &longs;uper aliquo plano, ut in plau&longs;tris & &longs;cytalis: minus in &longs;uccu<lb/>lis, & rotis acuentibus ferrum, & molis: nam & &longs;i rotun ditatem iu<lb/>uet ob æqualitatem ad conuer&longs;ionem, non tamen in his e&longs;t ad eò

<figure id="fig86"></figure><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri<lb/>bus. Propterea in primo motu trahi<lb/>tur, uel impellitur à &longs;uperficie: in &longs;e<lb/>cundo a b axe, &longs;ed non affixo rotis, unde ægrè trahuntur in &longs;cyta<lb/>la ab axe affixo rot&ecedil;. Quare leuius quàm in curru, difficilius quàm <lb/>in rota uel &longs;phæra à &longs;uperficie extima circumacta. Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. E&longs;t enim hic &longs;imilior <lb/>primo, quia contrarius, in primo enim procedit rota, & uertitur à <lb/>circumferentia, hic quie&longs;cit rota, & mouetur ab axe. Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rotæ. Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir<lb/>cunferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta<lb/>tem. Sed & trochlea caret colloppibus. Ergo uerum e&longs;t, quod o<lb/>mnia rotunda facilius circumaguntur, &longs;ed uaria ratione: nam plus <lb/>mota &longs;uper aliquo plano, ut in plau&longs;tris & &longs;cytalis: minus in &longs;uccu<lb/>lis, & rotis acuentibus ferrum, & molis: nam & &longs;i rotun ditatem iu<lb/>uet ob æqualitatem ad conuer&longs;ionem, non tamen in his e&longs;t ad eò

<pb pagenum="110"/>utilis. Vtilitas ergo prima e&longs;t, cum circumuertitur in plano, uelut <lb/>in rotis &longs;cytalis, & &longs;phæris. Secunda quæ minor e&longs;t, cum à &longs;uperfi<lb/>cie circumuertitur, ut in trochleis. Tertia cum à coloppis, quæ mi<lb/>nima e&longs;t omnium, ut in &longs;ucculis. Motus autem c&oelig;li non e&longs;t ex tri<lb/>plici primo genere, cum &longs;it in loco, & non ad locum, neque ut rotæ <lb/>molaris: nam ille e&longs;t ex axe: necut in tro chlea: nam in ea axis quie&longs;<lb/>citip&longs;um autem c&oelig;lum circa axem non uertitur, &longs;ed cum axe, &longs;i ta<lb/>men in&longs;ecabilis linea circumagi pote&longs;t dici. Relinquitur ergo, ut <lb/>C&oelig;li motus propior &longs;it motui &longs;ucculæ, quàm alij motui. Differt <lb/>ab eo in hoc, quod in &longs;uccula mouetur axis ab orbe: at in c&oelig;lo <lb/>ut non mouetur ab axe, ita nec axis ab orbe: cun que &longs;it motus &longs;im<lb/>plici&longs;simus, in alio genere collocandus e&longs;t: quando quidem in illo <lb/>nulla pars po&longs;sit dici primo, quod <expan abbr="nece&longs;&longs;ari&utilde;">nece&longs;&longs;arium</expan> e&longs;t in uno quo que <expan abbr="hor&utilde;">horum</expan>.</s>

<s>Propo&longs;itio cente&longs;imaquinta decima.</s>

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<arrow.to.target n="marg408"></arrow.to.target><lb/>&longs;is (uocant cerbatanas) non pote&longs;t &longs;atisfacere, cum tamen idem &longs;e<lb/>quatur in his, ut in illis uidetur, qua&longs;i uis e&longs;&longs;e in &longs;phærula &longs;ic emi&longs;<lb/>&longs;a, & non in aëre, quemadmodum dicebamus, coniuncto e&longs;&longs;e. Ex <lb/>quo nece&longs;&longs;e e&longs;&longs;et, ut quod longius ferretur, etiam ualidiores ictus <lb/>

<arrow.to.target n="fig87"></arrow.to.target><lb/>inferret, hoc autem <lb/>non ita &longs;e habet, &longs;ed <lb/>ictus magnitud o <lb/>ex robore machi<lb/>narum tam ignea<lb/>rum, quam &longs;corpio <lb/>num pendet, nam <lb/>&longs;it a &longs;corpio ma<lb/>gnus, &longs;ed tenuis, ex <lb/>hòc palam e&longs;t lon<lb/>gius mittere &longs;agit<lb/>tam, quòd à parua, <lb/>& breui, quantun<lb/>uis cra&longs;&longs;a non lon<lb/>ge mittitur: at uerò <lb/>quod b cra&longs;&longs;us & paruus maiore cum impetu mittat o&longs;tenditur <lb/>nam ea pondera &longs;agittæ mouet, quæ non pote&longs;t mouere a, igitur b <lb/>ualidiore robore mouet, quam a. Prætera illud o&longs;ten dit iugum fu<lb/>nis arcus cra&longs;siora duriora, quæ maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui à puero tendi poterit. Non e&longs;t ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. Eadem ergo cum ratio &longs;it in <lb/>machinis igneis, cra&longs;siores enim, & latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris &longs;phæræ capaces: <lb/>non &longs;olum ob mag nitudinem &longs;phæræ magis illæ concutiunt, &longs;ed, <lb/>ut dixi, ob maiorem impetus uim: cau&longs;a ergo e&longs;t manife&longs;ta in his, <lb/>&longs;ed non cau&longs;a, qua longius ferantur in longiore canali. Sed uide

<figure id="fig87"></figure><lb/>inferret, hoc autem <lb/>non ita &longs;e habet, &longs;ed <lb/>ictus magnitud o <lb/>ex robore machi<lb/>narum tam ignea<lb/>rum, quam &longs;corpio <lb/>num pendet, nam <lb/>&longs;it a &longs;corpio ma<lb/>gnus, &longs;ed tenuis, ex <lb/>hòc palam e&longs;t lon<lb/>gius mittere &longs;agit<lb/>tam, quòd à parua, <lb/>& breui, quantun<lb/>uis cra&longs;&longs;a non lon<lb/>ge mittitur: at uerò <lb/>quod b cra&longs;&longs;us & paruus maiore cum impetu mittat o&longs;tenditur <lb/>nam ea pondera &longs;agittæ mouet, quæ non pote&longs;t mouere a, igitur b <lb/>ualidiore robore mouet, quam a. Prætera illud o&longs;ten dit iugum fu<lb/>nis arcus cra&longs;siora duriora, quæ maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui à puero tendi poterit. Non e&longs;t ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. Eadem ergo cum ratio &longs;it in <lb/>machinis igneis, cra&longs;siores enim, & latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris &longs;phæræ capaces: <lb/>non &longs;olum ob mag nitudinem &longs;phæræ magis illæ concutiunt, &longs;ed, <lb/>ut dixi, ob maiorem impetus uim: cau&longs;a ergo e&longs;t manife&longs;ta in his, <lb/>&longs;ed non cau&longs;a, qua longius ferantur in longiore canali. Sed uide

<pb pagenum="112"/>tur una, eadem que e&longs;&longs;e ratio in utri&longs;que. Con&longs;tituatur can alis a b <lb/>lo&nacute;gior, & c d breuior, ut &longs;it &longs;exqui alter a b ad c d, & &longs;it rur&longs;us <lb/>

<arrow.to.target n="fig88"></arrow.to.target><lb/>&longs;phærulæ locus e in longiore, <lb/>&longs;exqui alter in di&longs;tantia a b, qua <lb/>lis e&longs;t in f a d, & erit per dicta <lb/>ab Euclide in quinto, ac &longs;exqui <lb/>altera c f. Po&longs;&longs;emus igitur di<lb/>cere, quod uelut ab hypomo<lb/>chlio longiore &longs;patio circuma<lb/>gitur pondus: ita & a b c, & f. <lb/>Sed rur&longs;us incidimus in id, ut <lb/>maiore impetu feratur e quàm f. Ideo &longs;i concedatur maiore ferri ex <lb/>e, quam ex f non &longs;equitur, ut celerius, aut maiore impetu. Percutit <lb/>puer pugno quanta ui pote&longs;t ac celerrimè, uir robu&longs;tus lentè, & mi<lb/>nore impetu, &longs;ed tamen ictus longè maior e&longs;t. E&longs;t enim ictus robur <lb/>non à uelo citate &longs;olum, &longs;ed maiore ex ponderis grauitate, quæ &longs;ola <lb/>premit, urget, & frangit etiam &longs;ine motu. Solum ergo id re&longs;tat du<lb/>bium, cur &longs;i grauius e&longs;t, moueatur eodem ferm é impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con <lb/>cutit, aut qua&longs;&longs;at, &longs;ed grauitas ad hoc plus facit impetu. Palea maxi<lb/>mo impetu demi&longs;&longs;a non ferit, non ledit, & celerius de&longs;cendit, fer<lb/>rum &longs;ola grauitate actum, imò etiam temperato ictu lædit graui<lb/>ter, qua&longs;&longs;at, & frangit: ita que f maiore indiget quantitate pyrij pulue<lb/>ris, quàm e: &longs;iquidem tertia parte ponderis &longs;uæ &longs;phæræ: at maius <lb/>e&longs;t pondus f quam e, ergo maius pondus pulueris f quàm e, ergo <lb/>maior uehementia ictus, &longs;iquidem ea &longs;equitur, robur cau&longs;æ mouen <lb/>tis &longs;im pliciter: ut concludamus longitudinem ictus &longs;equi propor<lb/>tionem motoris ad motum, &longs;ed uehementia robur motoris: nam &longs;i <lb/>ex portione mouet æquale pondus maiore cum impetu mouet, <lb/>quoniam maior e&longs;t proportio: &longs;i minore igitur pondus maius e&longs;t, <lb/>&, ut dixi plus facit magnitudo ponderis cum leui ictu, quàm ma<lb/>gnitudo ictus cum leui pondere. Quæ ergo feruntur per longio<lb/>res canales maiore impetu feruntur, & &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> aëris moti <lb/>per longius <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="u&itilde;s">uins</expan> <lb/>motus confirmata e&longs;t, & proportio eius, quòd mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uerò f <expan abbr="mot&utilde;">motum</expan> minore propor<lb/>tione <expan abbr="ict&utilde;">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, proa, ut dixi, <expan abbr="tãto">tanto</expan> grauius, e&longs;t quod ferit. Quod <lb/><expan abbr="aut&etilde;">autem</expan> minus <expan abbr="ext&etilde;datur">extendatur</expan> machina a b quam c d, <expan abbr="n&utilde;c">nunc</expan> <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan> oporter.</s>

<figure id="fig88"></figure><lb/>&longs;phærulæ locus e in longiore, <lb/>&longs;exqui alter in di&longs;tantia a b, qua <lb/>lis e&longs;t in f a d, & erit per dicta <lb/>ab Euclide in quinto, ac &longs;exqui <lb/>altera c f. Po&longs;&longs;emus igitur di<lb/>cere, quod uelut ab hypomo<lb/>chlio longiore &longs;patio circuma<lb/>gitur pondus: ita & a b c, & f. <lb/>Sed rur&longs;us incidimus in id, ut <lb/>maiore impetu feratur e quàm f. Ideo &longs;i concedatur maiore ferri ex <lb/>e, quam ex f non &longs;equitur, ut celerius, aut maiore impetu. Percutit <lb/>puer pugno quanta ui pote&longs;t ac celerrimè, uir robu&longs;tus lentè, & mi<lb/>nore impetu, &longs;ed tamen ictus longè maior e&longs;t. E&longs;t enim ictus robur <lb/>non à uelo citate &longs;olum, &longs;ed maiore ex ponderis grauitate, quæ &longs;ola <lb/>premit, urget, & frangit etiam &longs;ine motu. Solum ergo id re&longs;tat du<lb/>bium, cur &longs;i grauius e&longs;t, moueatur eodem ferm é impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con <lb/>cutit, aut qua&longs;&longs;at, &longs;ed grauitas ad hoc plus facit impetu. Palea maxi<lb/>mo impetu demi&longs;&longs;a non ferit, non ledit, & celerius de&longs;cendit, fer<lb/>rum &longs;ola grauitate actum, imò etiam temperato ictu lædit graui<lb/>ter, qua&longs;&longs;at, & frangit: ita que f maiore indiget quantitate pyrij pulue<lb/>ris, quàm e: &longs;iquidem tertia parte ponderis &longs;uæ &longs;phæræ: at maius <lb/>e&longs;t pondus f quam e, ergo maius pondus pulueris f quàm e, ergo <lb/>maior uehementia ictus, &longs;iquidem ea &longs;equitur, robur cau&longs;æ mouen <lb/>tis &longs;im pliciter: ut concludamus longitudinem ictus &longs;equi propor<lb/>tionem motoris ad motum, &longs;ed uehementia robur motoris: nam &longs;i <lb/>ex portione mouet æquale pondus maiore cum impetu mouet, <lb/>quoniam maior e&longs;t proportio: &longs;i minore igitur pondus maius e&longs;t, <lb/>&, ut dixi plus facit magnitudo ponderis cum leui ictu, quàm ma<lb/>gnitudo ictus cum leui pondere. Quæ ergo feruntur per longio<lb/>res canales maiore impetu feruntur, & &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> aëris moti <lb/>per longius <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="u&itilde;s">uins</expan> <lb/>motus confirmata e&longs;t, & proportio eius, quòd mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uerò f <expan abbr="mot&utilde;">motum</expan> minore propor<lb/>tione <expan abbr="ict&utilde;">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, proa, ut dixi, <expan abbr="tãto">tanto</expan> grauius, e&longs;t quod ferit. Quod <lb/><expan abbr="aut&etilde;">autem</expan> minus <expan abbr="ext&etilde;datur">extendatur</expan> machina a b quam c d, <expan abbr="n&utilde;c">nunc</expan> <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan> oporter.</s>

<s>Propo&longs;itio cente&longs;imadecima&longs;eptima.</s>

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<s>Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui &longs;i <lb/>

<arrow.to.target n="marg410"></arrow.to.target><lb/>e&longs;&longs;et in c d <expan abbr="pariet&etilde;">parietem</expan> e&longs;&longs;e ad perpendiculum, & <lb/>ualidi&longs;simus, &longs;in uero in f g abraderet, & <expan abbr="nõ">non</expan> <lb/><expan abbr="cõqua&longs;&longs;aret">conqua&longs;&longs;aret</expan>. Quæritur ergo ex b d e muro <lb/>qualis excipietur? erit ergo proportio anguli c d a ad <expan abbr="angul&utilde;">angulum</expan> b d a, <lb/>ueluti ictus a d in d c ad <expan abbr="ict&utilde;">ictum</expan> in b d, <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <expan abbr="a&utilde;t">aunt</expan> &longs;equi proportio<lb/>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõ&longs;tat">con&longs;tat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoniã">quoniam</expan> &longs;i b d c &longs;it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit re&longs;iduus ad <expan abbr="dimidi&utilde;">dimidium</expan> b <lb/>d a nonuplus ip&longs;i dimidio, & ad <expan abbr="quartã">quartam</expan> <expan abbr="part&etilde;">partem</expan> habebit proportionem <lb/><expan abbr="decemnou&etilde;">decemnouem</expan> ad <expan abbr="un&utilde;">unum</expan>. Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb/>ictus &qring;d tres, cuius demon&longs;tratio h&ecedil;c e&longs;t. Supponamus <expan abbr="proportion&etilde;">proportionem</expan> <lb/>b d c ad <expan abbr="&qtilde;rtam">quartam</expan> <expan abbr="part&etilde;">partem</expan> a d b ad dito re&longs;iduo ad b d c e&longs;&longs;e <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="decuplã">decuplam</expan>: <lb/><expan abbr="t&utilde;c">tunc</expan> ex duob. ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu&ecedil; in b e, <expan abbr="etiã">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="cõqua&longs;&longs;ata">conqua&longs;&longs;ata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum <08> in b d e <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> decima pars in ambitu d, & illa <lb/>erit ergo <expan abbr="tã">tam</expan> di&longs;&longs;oluta, & infirma ex &longs;uppo&longs;ito, <08> e&longs;t tota b e: &longs;ed ex &longs;e <lb/>cundo ictu decuplo magis <expan abbr="cõqua&longs;&longs;abi&ttilde;">conqua&longs;&longs;abitur</expan> illa pars, <08> b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">qua&longs;&longs;abitur</expan> ex duob. ictibus c d turris, <08> b e, & ita in <lb/>tribus: ex <expan abbr="dec&etilde;">decem</expan> millibus ergo ictibus <expan abbr="etiã">etiam</expan> ad amu&longs;sim directis, <expan abbr="c&utilde;">cum</expan> ta <lb/><expan abbr="m&etilde;id">menid</expan> uix fieri po&longs;sit in <expan abbr="tãta">tanta</expan> multitudine <expan abbr="nõ">non</expan> plus <expan abbr="cõminue&ttilde;">comminuetur</expan> b d e, <08><lb/>ex decë c d <expan abbr="&ptilde;ter">pnter</expan> <expan abbr="quã">quam</expan> <expan abbr="exigu&utilde;">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in &longs;uperficie. Imo ut <expan abbr="declarat&utilde;">declaratum</expan> <lb/>e&longs;t multo minus repetita ratione multiplicis. Ob id in arce <expan abbr="Medio-lan&etilde;&longs;i">Medio<lb/>lanen&longs;i</expan> exterius lapidibus uiuis in <expan abbr="rotund&utilde;">rotundum</expan> diducta &longs;uperficie inter<lb/>

<arrow.to.target n="fig89"></arrow.to.target><lb/>uallo que <expan abbr="&qtilde;">quae</expan> drato hunc in <expan abbr="mod&utilde;">modum</expan> munit&ecedil; &longs;unt altiores tur <lb/>res. Fiat ergo murus cuius proportio a d c ad b d a &longs;it &longs;ex <lb/>quitertia, erit que angulus b d c <expan abbr="dodrãs">dodrans</expan> recti, & <expan abbr="par&utilde;">parum</expan> incli <lb/>natis, <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> b d c erit quarta pars recti, & &longs;it tant&ecedil; ma<lb/>gnitudinis, at que duritiei, ac adeò benè coniunctus fer<lb/>

<figure id="fig89"></figure><lb/>uallo que <expan abbr="&qtilde;">quae</expan> drato hunc in <expan abbr="mod&utilde;">modum</expan> munit&ecedil; &longs;unt altiores tur <lb/>res. Fiat ergo murus cuius proportio a d c ad b d a &longs;it &longs;ex <lb/>quitertia, erit que angulus b d c <expan abbr="dodrãs">dodrans</expan> recti, & <expan abbr="par&utilde;">parum</expan> incli <lb/>natis, <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> b d c erit quarta pars recti, & &longs;it tant&ecedil; ma<lb/>gnitudinis, at que duritiei, ac adeò benè coniunctus fer<lb/>

<arrow.to.target n="table16"></arrow.to.target><lb/>reis cathenis, ac &longs;tolonibus, ut po&longs;sit re&longs;i&longs;tere <expan abbr="machinar&utilde;">machinarum</expan> <expan abbr="fe-renti&utilde;">fe<lb/>rentium</expan> <expan abbr="&longs;ph&ecedil;rã">&longs;ph&ecedil;ram</expan> <expan abbr="librar&utilde;">librarum</expan> ducentarum (quæ &longs;anè maximæ &longs;unt) <lb/>quin quaginta: <expan abbr="t&utilde;c">tunc</expan> cum proportio &longs;exquitertia nouies repeti<lb/>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb/>ictibus, fiet proportio decupla quinquies producta, qu&ecedil; e&longs;t cen <lb/><expan abbr="t&utilde;">tum</expan> millium ad <expan abbr="un&utilde;">unum</expan> in quadraginta quin que ictibus. <expan abbr="Antequã">Antequam</expan> <lb/>ergo peruenit ad quinquaginta ictus rectos nece&longs;&longs;e erit, ut

<pb pagenum="115"/>multo plures centum millibus ictus excipiat ante <08> euertatur, quæ <lb/>recta &longs;i e&longs;&longs;et quin quaginta &longs;olùm potui&longs;&longs;et &longs;u&longs;tinere. Quæ ergo hu <lb/>mana potentia &longs;ufficeret. In arce Medio <expan abbr="lan&etilde;&longs;i">lanen&longs;i</expan> uidimus uix attactas <lb/>in illis extuberationibus lapideis. Sed quoniam hic occurritur per <lb/>inclinationem machinarum, ideò de hoc <expan abbr="&longs;ermon&etilde;">&longs;ermonem</expan> &longs;um habiturus.</s>

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<table>

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<s>Huiu&longs;ce cau&longs;a <expan abbr="excogitar&utilde;t">excogitarunt</expan>, ut ictus ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <expan abbr="dirigere&ttilde;">dirigeretur</expan>, & <lb/>

<arrow.to.target n="marg411"></arrow.to.target><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &ecedil;quali angulo a b c, longè <expan abbr="tñ">tnm</expan> maior e&longs;t uis <lb/>a b <08> d e duplici cau&longs;a, & <expan abbr="quoniã">quoniam</expan> a b e&longs;t <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> nat uram impetus <lb/>

<arrow.to.target n="fig90"></arrow.to.target><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&ecedil; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ict&utilde;">ictum</expan>, in e non retineat. Sed caui <lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere pote&longs;t <expan abbr="c&utilde;">cum</expan> ha&longs;ta. Huic ergo &longs;olertiæ, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="prim&utilde;">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, & ma <lb/>ximè &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t <08> exone <lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, & ad <expan abbr="collimand&utilde;">collimandum</expan> decedit parte <expan abbr="ui-ri&utilde;">ui<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paru&utilde;">paruum</expan> &longs;it in ductu <expan abbr="tñ">tnm</expan>, & <expan abbr="ictu&utilde;">ictuum</expan> mul <lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. Habet & <expan abbr="cõmo">commo</expan> <lb/>dum &longs;itus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="&longs;uppo&longs;itã">&longs;uppo&longs;itam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>&longs;u&longs;tinet: adeò ut omnib. <expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen<lb/>diculo, et &ecedil;quidi&longs;tanti ad <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. Venetus. S. aliter Patauij cauit, <lb/>uidetur que, quae &longs;apienti&longs;simus &longs;it, & eandem &longs;equatur ubi que normam, <lb/>po&longs;t <08> in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="tot&utilde;">totum</expan> urbis ambitum formauit, & fo&longs;&longs;a la <lb/>ta, ac pro fundi&longs;sima aqua que perenni muniuit, & <expan abbr="&longs;ummã">&longs;ummam</expan> muri partem <lb/><expan abbr="rotundã">rotundam</expan> in hunc <expan abbr="mod&utilde;">modum</expan> effecit <expan abbr="cauã">cauam</expan> que interius undi que, ne cuniculis <lb/>

<figure id="fig90"></figure><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&ecedil; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ict&utilde;">ictum</expan>, in e non retineat. Sed caui <lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere pote&longs;t <expan abbr="c&utilde;">cum</expan> ha&longs;ta. Huic ergo &longs;olertiæ, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="prim&utilde;">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, & ma <lb/>ximè &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t <08> exone <lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, & ad <expan abbr="collimand&utilde;">collimandum</expan> decedit parte <expan abbr="ui-ri&utilde;">ui<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paru&utilde;">paruum</expan> &longs;it in ductu <expan abbr="tñ">tnm</expan>, & <expan abbr="ictu&utilde;">ictuum</expan> mul <lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. Habet & <expan abbr="cõmo">commo</expan> <lb/>dum &longs;itus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="&longs;uppo&longs;itã">&longs;uppo&longs;itam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>&longs;u&longs;tinet: adeò ut omnib. <expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen<lb/>diculo, et &ecedil;quidi&longs;tanti ad <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. Venetus. S. aliter Patauij cauit, <lb/>uidetur que, quae &longs;apienti&longs;simus &longs;it, & eandem &longs;equatur ubi que normam, <lb/>po&longs;t <08> in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="tot&utilde;">totum</expan> urbis ambitum formauit, & fo&longs;&longs;a la <lb/>ta, ac pro fundi&longs;sima aqua que perenni muniuit, & <expan abbr="&longs;ummã">&longs;ummam</expan> muri partem <lb/><expan abbr="rotundã">rotundam</expan> in hunc <expan abbr="mod&utilde;">modum</expan> effecit <expan abbr="cauã">cauam</expan> que interius undi que, ne cuniculis <lb/>

<arrow.to.target n="fig91"></arrow.to.target><lb/>po&longs;&longs;et euerti, à lateribus uerò humiles, ac cra&longs;si&longs;simas turres, ut nul <lb/>la ui po&longs;&longs;ent dirui, eas que tormentis bellicis, undi que latera lu&longs;trantib. <lb/>reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libellã">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&ecedil;rulæ non tangerent <expan abbr="mur&utilde;">murum</expan>: nam <expan abbr="c&utilde;">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;<lb/>&longs;uum, excedat <expan abbr="a&utilde;t">aunt</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, & aggeris angulus ma <lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="t&utilde;">tum</expan> magis adiecta cra&longs;sitie machin&ecedil; fieri non pote&longs;t, <lb/>utictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. Eam ob cau&longs;am <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&ecedil;difici&utilde;">&ecedil;dificium</expan> ul<lb/>

<figure id="fig91"></figure><lb/>po&longs;&longs;et euerti, à lateribus uerò humiles, ac cra&longs;si&longs;simas turres, ut nul <lb/>la ui po&longs;&longs;ent dirui, eas que tormentis bellicis, undi que latera lu&longs;trantib. <lb/>reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libellã">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&ecedil;rulæ non tangerent <expan abbr="mur&utilde;">murum</expan>: nam <expan abbr="c&utilde;">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;<lb/>&longs;uum, excedat <expan abbr="a&utilde;t">aunt</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, & aggeris angulus ma <lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="t&utilde;">tum</expan> magis adiecta cra&longs;sitie machin&ecedil; fieri non pote&longs;t, <lb/>utictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. Eam ob cau&longs;am <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&ecedil;difici&utilde;">&ecedil;dificium</expan> ul<lb/>

<arrow.to.target n="fig92"></arrow.to.target><lb/>lum, aut planta, uel colliculus e&longs;&longs;et cir<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo h&ecedil;c urbs, ne tota &ecedil;dificijs euer<lb/>&longs;is concidat. <expan abbr="Turcar&utilde;">Turcarum</expan> enim Princeps di<lb/>dicit, ut in Nouo ca&longs;tro in Melit&ecedil; In&longs;ul&ecedil; <lb/>arce S. Elmi appellata plu&longs; <08> mille icti<lb/>bus in &longs;ingulos dies imo M D obtundere

<figure id="fig92"></figure><lb/>lum, aut planta, uel colliculus e&longs;&longs;et cir<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo h&ecedil;c urbs, ne tota &ecedil;dificijs euer<lb/>&longs;is concidat. <expan abbr="Turcar&utilde;">Turcarum</expan> enim Princeps di<lb/>dicit, ut in Nouo ca&longs;tro in Melit&ecedil; In&longs;ul&ecedil; <lb/>arce S. Elmi appellata plu&longs; <08> mille icti<lb/>bus in &longs;ingulos dies imo M D obtundere

<pb pagenum="116"/>munitiones. Eum que impetum producere ad quindecim dies, & ui<lb/>ginti tum etiam longius, ut facilè domos omnes euertat, homines <lb/>occidat: &longs;i qui &longs;uper&longs;unt tot in commodis obruuntur uigilijs, fame, <lb/>&longs;iti, puluere, ut inutiles red dantur. Ideò huic <expan abbr="incõmodo">incommodo</expan> occurrunt <lb/>aggeribus intra m&oelig;nia erectis, in quos uis <expan abbr="torm&etilde;torum">tormentorum</expan> igneorum <lb/>emoritur. Sed dices, cur ergo non pro muris erigere eos præ&longs;tat, & <lb/>minore &longs;umptu &longs;atis? quoniam &longs;ubruuntur à fo&longs;&longs;oribus facillimè, &longs;<gap/><lb/>ad illos peruenire po&longs;sit ho&longs;tis. Ideò intra m &oelig;nia utili&longs;simi &longs;unt, pro<lb/>m&oelig;nijs parum pro&longs;unt. Quod uerò ad te&longs;tudines attinet, &longs;ub qui<lb/>bus <expan abbr="lat&etilde;t">latent</expan> fo&longs;&longs;ores machinæ laterales, & à fronte & ignes, & aqua al<lb/>tior prohibent omnino iniuriam, qu&ecedil; ab his imminet. Cæterum hu<lb/>iu&longs;modi cum in longum <expan abbr="differun&ttilde;">differuntur</expan> morbis, illuuie, <expan abbr="incõmodis">incommodis</expan>, plu<lb/>uijs, frigoribus omnino <expan abbr="di&longs;&longs;olu&utilde;tur">di&longs;&longs;oluuntur</expan>, ut nulla multitudo huic operi <lb/>&longs;ufficere po&longs;sit. Rhodus, Alba regia, Melita, Ca&longs;trum <expan abbr="nou&utilde;">nouum</expan>, Byzan <lb/>tium, &longs;i diferri potui&longs;&longs;ent tempora, non ce&longs;si&longs;&longs;ent uictori quantum<lb/>uis &longs;uperbo. Vicit pertinacia, audacia que &longs;umma, <expan abbr="Corcyrã">Corcyram</expan>, Viennam <lb/>capere <expan abbr="nõ">non</expan> potuit, quoniam in <expan abbr="long&utilde;">longum</expan> trahebatur oppugnatio. Mul <lb/>tæ machinæ, & pauci homines prædæ ob&longs;e&longs;&longs;orum expo&longs;itæ &longs;unt: <lb/>pauc&ecedil;, & pauci homines ob&longs;idebuntur potius, quam ob&longs;idebunt. <lb/>Exercitus magnus di&longs;&longs;oluitur, & &longs;emetip&longs;um con&longs;umit, &longs;i nulla fiat <lb/>acce&longs;sio aut exigua quomodo &longs;tabit: &longs;i magna auxilia omnia cor<lb/>rumpuntur. Contrà ob&longs;e&longs;sis auxilia &longs;i ueniant lu&longs;trata, & munita, et <lb/>omnibus nece&longs;&longs;arijs ornata uiri integri <expan abbr="cõtra">contra</expan> fatigatos, & fe&longs;&longs;os cor <lb/>pore, armati contra inermes, alacres contra torpidos &longs;uperueniunt. <lb/>Ob id præcipuum e&longs;t auxilium pr&ecedil;ter h&ecedil;c his, qui oppugnantur co <lb/>pia militum, qui per initia nun <08> quie&longs;cant diu noctu que, <expan abbr="uer&utilde;">uerum</expan> noctu <lb/>duo tubicines per&longs;æpe <expan abbr="exercit&utilde;">exercitum</expan> <expan abbr="in&longs;omn&etilde;">in&longs;omnem</expan> in armis tota nocte <expan abbr="cõtine">contine</expan> <lb/><expan abbr="b&utilde;t">bunt</expan>. Serio <expan abbr="a&utilde;t">aunt</expan> die pugnare, & noctu <expan abbr="c&utilde;">cum</expan> minimè id <expan abbr="&longs;perãt">&longs;perant</expan>, & fatigati <lb/>&longs;unt: mira euenire &longs;olent in his in&longs;peratis, ac audacibus eruptionib. <lb/>per&longs;&ecedil;pe <expan abbr="etiã">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à cepto de&longs;inat ho&longs;tis, aut <expan abbr="loc&utilde;">locum</expan> occupet &longs;ibi <expan abbr="relict&utilde;">relictum</expan> po<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. nam <expan abbr="experiment&utilde;">experimentum</expan> frequens do cuit, ubi illæ ma <lb/>gn&ecedil; uires &longs;uo arbitrio <expan abbr="loc&utilde;">locum</expan>, <expan abbr="qu&etilde;">quem</expan> <expan abbr="eleger&utilde;t">elegerunt</expan> obtinere potuerint, <expan abbr="tand&etilde;">tandem</expan> <lb/>potiri locis <expan abbr="quãtumuis">quantumuis</expan> munitis in hoc &qring;d diximus <expan abbr="cõtra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/>Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="c&utilde;">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quor&utilde;">quorum</expan> duo &longs;unt ex <lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pn&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tol-li&ttilde;">tol<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to <expan abbr="d&utilde;">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&ecedil;dificior&utilde;">&ecedil;dificiorum</expan>: & <expan abbr="&qtilde;uo">quauo</expan> cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, & crates <lb/><expan abbr="c&utilde;">cum</expan> &longs;agittarijs: his omnib. <expan abbr="&longs;atisfact&utilde;">&longs;atisfactum</expan> puto, pr&ecedil;ter <08> oppugnationi pro<lb/>pter <expan abbr="humilitat&etilde;">humilitatem</expan> <expan abbr="muror&utilde;">murorum</expan>: <expan abbr="nã">nam</expan> lignis <expan abbr="opplen&ttilde;">opplentur</expan>, at que fa&longs;ciculis, terra que fo&longs; <lb/>&longs;&ecedil;: nihil. n. re&longs;i&longs;tit immen&longs;&ecedil; illi pote&longs;tati, & crudelitati <expan abbr="&longs;&ecedil;ui&longs;simor&utilde;">&longs;&ecedil;ui&longs;simorum</expan> ty <lb/><expan abbr="rãnor&utilde;">rannorum</expan>. <expan abbr="Ver&utilde;">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificio&longs;is ignib. eru

<pb pagenum="117"/>untur. Et longum e&longs;t opus &longs;iue per paucos, &longs;iue per multos quis ef<lb/>ficere conetur: ut non minus exigat temporis, quàm ob&longs;idio: nam <lb/>multitudine unus alterum impedit, & mortui uiuos, ut omnino res <lb/>&longs;it non &longs;peranda ni&longs;i aduer&longs;us inerti&longs;simos. Pontes euertunt machi <lb/>næ, ignes que. Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis con&longs;i&longs;tere non po&longs;&longs;unt. Inde à defen&longs;oribus propul&longs;antur &longs;ari&longs;<lb/>&longs;is, telis, ignibus, tran&longs;uer&longs;is trabibus, machinis: illudque accedit com <lb/>modi, ut quanto plures eo facilius excutiantur. Dixi non debere <lb/>uereri maxima etiam præterid, quoniam & i&longs;t&ecedil; ip&longs;&ecedil; tanto &longs;anguine <lb/>acqui&longs;it&ecedil; tanto deorum & hominum iniuria modica &longs;cintilla ignis <lb/>&longs;ine munitionibus, exercitibus, &longs;iue machinis, ab&longs;que terræ <expan abbr="cõcu&longs;sio-ne">concu&longs;sio<lb/>ne</expan>, aut inundatione, uel pe&longs;te euertuntur. In illam mi&longs;eram lachry<lb/>mam patris &longs;cintilla ignis inferni, cùm Deo placuerit, <expan abbr="mitti&ttilde;">mittitur</expan>, ex qua, <lb/>quod <expan abbr="coalit&utilde;">coalitum</expan> e&longs;t, multis &longs;eculis imperium luxu, crudelitate, &longs;tultitia <lb/>unius filij, uix uno lu&longs;tro toto di&longs;&longs;oluitur. Hanc <expan abbr="&longs;cintillã">&longs;cintillam</expan> cum felici <lb/>etiam genio &longs;ecum ex utero detulit Alexander Magnus. In alijs alij <lb/>genium &longs;ortiti &longs;unt, alij <expan abbr="&longs;cintillã">&longs;cintillam</expan> detulere ab Orco. Ex imperio A&longs;&longs;y <lb/>riorum per luxum Sardanapalus: ex Medorum per <expan abbr="&longs;cintillã">&longs;cintillam</expan> A&longs;tya<lb/>ges: ex <expan abbr="Per&longs;ar&utilde;">Per&longs;arum</expan> per &longs;tultitiam Darius: ex <expan abbr="Romanor&utilde;">Romanorum</expan> Honorius. Di <lb/>ces, h&ecedil;c quid ad proportionem? Imò uelut machina ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita &longs;cintilla illa infer <lb/>ni ignis &longs;emini magni tyranni indita euertit at que di&longs;&longs;oluit totum re<lb/>gnum &longs;ine machinis, ut dixi, uel exercitibus ullis, & quod maius e&longs;t <lb/>remedio nullo. Sed puerulo indito luxus, ignauiæ, crudelitatis at que<lb/>&longs;tultiti&ecedil; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu <lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, &qring;d m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&ecedil; & in Cremonen&longs;i arce. <lb/>Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="præ&longs;idi&utilde;">præ&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag <lb/>geri confidant, nam & pauci homines tanto labori non &longs;ufficerent, <lb/>& agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;t&utilde;">angu&longs;tum</expan> <lb/>cogeret. At in urbibus contra eueniet: muris enim erectis altius ma <lb/>chinæ lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="tot&utilde;">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&ecedil;, quo <expan abbr="cad&etilde;te">cadente</expan>, & fo&longs;&longs;a impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&ecedil;be&ttilde;">pr&ecedil;betur</expan>: imò percul&longs;i defen

<pb pagenum="118"/>&longs;ores &longs;æpe muneris &longs;ui obliui&longs;cuntur, de&longs;ertaque ea parte liberum <lb/>ingre&longs;&longs;um ho&longs;tibus exhibent. Tum uerò magis, quod non confi<lb/>dunt animo <expan abbr="nõ">non</expan> ad id parato, po&longs;&longs;e aggerem &longs;ufficientem, & in tam <lb/>breui tempore ex&longs;truere, & etiam intelligunt, antequam erigatur, <lb/>patere à lateribus introitum ho&longs;tibus.</s>

Line 6200

Line 6198

<s>Propo&longs;itio cente&longs;imauige&longs;ima.</s>

Line 6219

Line 6217

<s>Sint mali in naui a b c, ad b e, c fuentus è regione g h k etiam ad <lb/>perpendiculum feratur, ut anguli g d a, h e b, k f c &longs;int æquales, dico <lb/>tamen diuer&longs;o modo affici: nam cum premitur a uer&longs;us l, c premi<lb/>tur uer&longs;us f: at &longs;i prematur cuer&longs;us n a, premitur uer&longs;us d, at &longs;i pre<lb/>

<arrow.to.target n="fig93"></arrow.to.target><lb/>matur b uer&longs;us m, & a uer<lb/>&longs;us l, &longs;ed non quantum ex <lb/>g d, & cuer&longs;us n, &longs;ed non <lb/>quantum ex k f, ab eodem <lb/>ergo uento contrarij mo<lb/>tus efficiuntur ex uelorum <lb/>diuer&longs;itate, etenim per uen <lb/>tum d feretur ad meridiem <lb/>nauis, & per uelum f ad Se <lb/>ptentrionem etiam didu<lb/>cto auxilio e l a ui, quanto <lb/>magis cum illo: & &longs;i uen<lb/>tus excipiatur in f uelo, <lb/>non iuuabit clauus, & &longs;i in <lb/>d dirigetur, & temperabitur motus, & &longs;i in e medio modo. Ergo &longs;i <lb/>uentus feratur rectè iuuabit, ut dici &longs;olet omnibus, & plenis uelis <lb/>excipere, &longs;i ex obliquo demittere antennam puppis, &longs;in autem ual<lb/>de obliqu us &longs;it, &longs;olo proræ uelo utemur. Si ualidior quàm oportet <lb/>humiliore. Atque hæc po&longs;tmodum &longs;unt diligenter numeranda, ac <lb/>metienda: nunc &longs;ufficiat cau&longs;am reddidi&longs;&longs;e, & admonui&longs;&longs;e diuer&longs;i<lb/>tatis motuum, quæ ex uelis contingit: nam eò fertur nauis, quò <lb/>prora dirigitur. Ergo cum puppis tanto feratur uer&longs;us meridiem <lb/>a b, quanto prora uer&longs;us meridiem a d, & quanto puppis fertur uer <lb/>&longs;us <expan abbr="meridi&etilde;">meridiem</expan>, tanto prora fertur uer&longs;us boream, igitur quanto prora <lb/>fertur uer&longs;us meridiem a d, tanto uer&longs;us boream a b f, &longs;ed &longs;itus claui <lb/>pote&longs;t multo plus in comparatione ueli d, quam f &longs;cilicet, quia di<lb/>&longs;tantia a b a e&longs;t o a, & di&longs;tantia e c e&longs;t o c, tanto plus ergo pote&longs;t cla<lb/>ui &longs;itus in comparatione ad uelum d, quam f, quanta e&longs;t proportio

<figure id="fig93"></figure><lb/>matur b uer&longs;us m, & a uer<lb/>&longs;us l, &longs;ed non quantum ex <lb/>g d, & cuer&longs;us n, &longs;ed non <lb/>quantum ex k f, ab eodem <lb/>ergo uento contrarij mo<lb/>tus efficiuntur ex uelorum <lb/>diuer&longs;itate, etenim per uen <lb/>tum d feretur ad meridiem <lb/>nauis, & per uelum f ad Se <lb/>ptentrionem etiam didu<lb/>cto auxilio e l a ui, quanto <lb/>magis cum illo: & &longs;i uen<lb/>tus excipiatur in f uelo, <lb/>non iuuabit clauus, & &longs;i in <lb/>d dirigetur, & temperabitur motus, & &longs;i in e medio modo. Ergo &longs;i <lb/>uentus feratur rectè iuuabit, ut dici &longs;olet omnibus, & plenis uelis <lb/>excipere, &longs;i ex obliquo demittere antennam puppis, &longs;in autem ual<lb/>de obliqu us &longs;it, &longs;olo proræ uelo utemur. Si ualidior quàm oportet <lb/>humiliore. Atque hæc po&longs;tmodum &longs;unt diligenter numeranda, ac <lb/>metienda: nunc &longs;ufficiat cau&longs;am reddidi&longs;&longs;e, & admonui&longs;&longs;e diuer&longs;i<lb/>tatis motuum, quæ ex uelis contingit: nam eò fertur nauis, quò <lb/>prora dirigitur. Ergo cum puppis tanto feratur uer&longs;us meridiem <lb/>a b, quanto prora uer&longs;us meridiem a d, & quanto puppis fertur uer <lb/>&longs;us <expan abbr="meridi&etilde;">meridiem</expan>, tanto prora fertur uer&longs;us boream, igitur quanto prora <lb/>fertur uer&longs;us meridiem a d, tanto uer&longs;us boream a b f, &longs;ed &longs;itus claui <lb/>pote&longs;t multo plus in comparatione ueli d, quam f &longs;cilicet, quia di<lb/>&longs;tantia a b a e&longs;t o a, & di&longs;tantia e c e&longs;t o c, tanto plus ergo pote&longs;t cla<lb/>ui &longs;itus in comparatione ad uelum d, quam f, quanta e&longs;t proportio

<pb pagenum="119"/>o a, ad o c, igitur clauus e&longs;t longè potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. Sed ut extrema <lb/>&longs;e habent, ita medium eorum comparatione, igitur malus b e uali<lb/>dior e&longs;t, multo d a, & infirmior c f. Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>era&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca<lb/>tum acatius. At ut etiam docui c b non e&longs;t in medio, nec æquidi&longs;tat <lb/>ab a d & c f, &longs;ed in clinatur ad proram ideoque imbecillior: cum ergo <lb/>&longs;it æqualium, & paulo maiorum uirium, quàm c f, & tutior, & me<lb/>lius agatur per <expan abbr="clau&utilde;">clauum</expan> quàm c f, & &longs;it a d nimis iu&longs;to imbecillis, pro<lb/>pterea b e mali, & ueli maximus e&longs;t u&longs;us: adeò mali nomen per an<lb/>tonoma&longs;iam de ip&longs;o &longs;impliciter intelligatur.</s>

<s>Propo&longs;itio cente&longs;imauige&longs;imaprima.</s>

Line 6258

Line 6256

<s>Quibus con&longs;titutis, qui &longs;tabunt iuxta l, & m longitudines aëris <lb/>moti, & loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex<lb/>tremis, &longs;tantes, & alti tangentur à uento agitato. Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per<lb/>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigeræ <lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi

<pb pagenum="120"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/>

<arrow.to.target n="fig94"></arrow.to.target><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. Sed &longs;i humili loco &longs;e<lb/>deant, quoniam aër de&longs;cen <lb/>dit, ex l & m uer&longs;us x, & <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refri-gerabun&ttilde;">refri<lb/>gerabuntur</expan> ualde. Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="er&utilde;t">erunt</expan> iuxta h & k minus <expan abbr="re-frigerabun&ttilde;">re<lb/>frigerabuntur</expan> utri&longs;que, &longs;ed pau <lb/>lulum in reditibus propin <lb/>quis, & neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque<lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla<lb/>tur h k. Rur&longs;us &longs;i b h k fue<lb/>rit grauior eodem, ut de<lb/>&longs;cendat tanto impetu, <expan abbr="quã-to">quan<lb/>to</expan> a&longs;cendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige<lb/>rabit, & procul, <expan abbr="nõ">non</expan> ob uim <lb/>ualidiorem, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="ue-hem&etilde;tibus">ue<lb/>hementibus</expan> fiet colli&longs;io par <lb/>tium aëris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius longè flabellum lento motu <lb/>agitatum, aut ex materia leui. Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, quæ ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: & &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum aëris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s>

<figure id="fig94"></figure><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. Sed &longs;i humili loco &longs;e<lb/>deant, quoniam aër de&longs;cen <lb/>dit, ex l & m uer&longs;us x, & <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refri-gerabun&ttilde;">refri<lb/>gerabuntur</expan> ualde. Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="er&utilde;t">erunt</expan> iuxta h & k minus <expan abbr="re-frigerabun&ttilde;">re<lb/>frigerabuntur</expan> utri&longs;que, &longs;ed pau <lb/>lulum in reditibus propin <lb/>quis, & neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque<lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla<lb/>tur h k. Rur&longs;us &longs;i b h k fue<lb/>rit grauior eodem, ut de<lb/>&longs;cendat tanto impetu, <expan abbr="quã-to">quan<lb/>to</expan> a&longs;cendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige<lb/>rabit, & procul, <expan abbr="nõ">non</expan> ob uim <lb/>ualidiorem, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="ue-hem&etilde;tibus">ue<lb/>hementibus</expan> fiet colli&longs;io par <lb/>tium aëris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius longè flabellum lento motu <lb/>agitatum, aut ex materia leui. Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, quæ ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: & &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum aëris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s>

<s>Propo&longs;itio cente&longs;imauige&longs;ima&longs;ecunda.</s>

Line 6513

Line 6511

<s>Sit horizon a d b æquinoctij circulus <lb/>

<arrow.to.target n="fig95"></arrow.to.target><lb/>a k f eclyptica c g, & punctus ortus in ea g. <lb/>& c initium arietis, & g b amplitudo ortiua <lb/>& c e, c f quartæ circulorum, ut &longs;it e f maxi<lb/>ma &longs;olis declinatio, & polus mundi borea<lb/>lis l, quia igitur l d nota e&longs;t ex &longs;uppo&longs;ito, & <lb/>l k quadrans erit k h <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> ad dimidium <lb/>circuli notum. Quia uerò æquinoctium, & <lb/>Meridianus &longs;ecant &longs;e ad angulos rectos, & <lb/>b a æquidi&longs;tat ab utro que polo, erit b polus <lb/>h d, quare b k, quarta circuli, & angulus k <lb/>rectus. Igitur &longs;umus in di&longs;po&longs;itione tabula<lb/>rum primi mobilis, ergo etiam oppo&longs;itus <lb/>triangulus, qui ei e&longs;t æqualis, & &ecedil;quiangu<lb/>lus in eadem di&longs;po&longs;itione b m d, quare cum <lb/>data &longs;it g n declinatio <expan abbr="p&utilde;cti">puncti</expan> g dati, datus erit, & arcus g b quæ&longs;itus.</s>

<figure id="fig95"></figure><lb/>a k f eclyptica c g, & punctus ortus in ea g. <lb/>& c initium arietis, & g b amplitudo ortiua <lb/>& c e, c f quartæ circulorum, ut &longs;it e f maxi<lb/>ma &longs;olis declinatio, & polus mundi borea<lb/>lis l, quia igitur l d nota e&longs;t ex &longs;uppo&longs;ito, & <lb/>l k quadrans erit k h <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> ad dimidium <lb/>circuli notum. Quia uerò æquinoctium, & <lb/>Meridianus &longs;ecant &longs;e ad angulos rectos, & <lb/>b a æquidi&longs;tat ab utro que polo, erit b polus <lb/>h d, quare b k, quarta circuli, & angulus k <lb/>rectus. Igitur &longs;umus in di&longs;po&longs;itione tabula<lb/>rum primi mobilis, ergo etiam oppo&longs;itus <lb/>triangulus, qui ei e&longs;t æqualis, & &ecedil;quiangu<lb/>lus in eadem di&longs;po&longs;itione b m d, quare cum <lb/>data &longs;it g n declinatio <expan abbr="p&utilde;cti">puncti</expan> g dati, datus erit, & arcus g b quæ&longs;itus.</s>

<s>Propo&longs;itio cente&longs;imauige&longs;imaoctaua.</s>

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<s>

<arrow.to.target n="marg442"></arrow.to.target><lb/>erit re&longs;iduum quart&ecedil; circuli, & &longs;imiliter h k <lb/>

<arrow.to.target n="fig96"></arrow.to.target><lb/>nota, quia declinatio puncti dati in eclypti <lb/>ca e&longs;t n nota dies, & locus &longs;olis ex &longs;uppo&longs;i<lb/>to ergo nota fh <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> quart&ecedil; circuli no<lb/>ta e&longs;t <expan abbr="etiã">etiam</expan> g e, quæ e&longs;t &ecedil;qualis altitudini po<lb/>li ex &longs;uppo&longs;ito, ergo re&longs;iduum quadrantis <lb/>f g, ergo triangulus f g h notorum laterum <lb/>ergo notus angulus f, ergo arcus k e di&longs;tan <lb/>

<figure id="fig96"></figure><lb/>nota, quia declinatio puncti dati in eclypti <lb/>ca e&longs;t n nota dies, & locus &longs;olis ex &longs;uppo&longs;i<lb/>to ergo nota fh <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> quart&ecedil; circuli no<lb/>ta e&longs;t <expan abbr="etiã">etiam</expan> g e, quæ e&longs;t &ecedil;qualis altitudini po<lb/>li ex &longs;uppo&longs;ito, ergo re&longs;iduum quadrantis <lb/>f g, ergo triangulus f g h notorum laterum <lb/>ergo notus angulus f, ergo arcus k e di&longs;tan <lb/>

<arrow.to.target n="marg443"></arrow.to.target><lb/>tia &longs;umpta in æquinoctij circulo puncti h, <lb/>cui &longs;imilis e&longs;t arcus h m ex parallelo h m, nam quando k perueniet <lb/>

<arrow.to.target n="marg444"></arrow.to.target><lb/>in e h perueniet in m, & in æquali tempore, qua diui&longs;a per quinde<lb/>cim gradus, habebimus horas <expan abbr="di&longs;tãti&ecedil;">di&longs;tanti&ecedil;</expan> &longs;olis à Meridie ante, uel po&longs;t, <lb/>& minuta horarum dando quibuslibet gradibus quatuor minuta <lb/>horæ, & quibuslibet minutis graduum quatuor &longs;ecunda horæ, & <lb/>ita habebimus tempus exacti&longs;simum à Meridie in quacunque regi<lb/>one, & in quacunque hora diei.</s>

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<s><margin.target id="marg444"></margin.target>D<emph type="italics"/>e<emph.end type="italics"/> T<emph type="italics"/>riang.<emph.end type="italics"/><lb/>M<emph type="italics"/>onteregij.<emph.end type="italics"/></s>

<s>Propo&longs;itio cente&longs;imatrige&longs;ima.</s>

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<arrow.to.target n="marg448"></arrow.to.target><lb/>notus, & ita extendemus per totum annum. Cum uerò fuerit g ele<lb/>uatus erit, ut <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t, in circulo magno uerticali, ergo an<lb/>gulus fiet in eodem circulo, quia gnomo e&longs;t etiam in illius &longs;uperfi<lb/>cie. Ergo angulus erit æqualis angulo, quem faceret &longs;ol, &longs;i oriretur <lb/>

<arrow.to.target n="marg449"></arrow.to.target><lb/>

<arrow.to.target n="fig97"></arrow.to.target><lb/>in puncto horizontis, quem &longs;ecat circulus <lb/>uerticalis &longs;ub ea altitudine: &longs;ed his e&longs;t no<lb/>tus: nam in priore figura g h f e&longs;t notus ea<lb/>

<figure id="fig97"></figure><lb/>in puncto horizontis, quem &longs;ecat circulus <lb/>uerticalis &longs;ub ea altitudine: &longs;ed his e&longs;t no<lb/>tus: nam in priore figura g h f e&longs;t notus ea<lb/>

<arrow.to.target n="marg450"></arrow.to.target><lb/><expan abbr="d&etilde;">dem</expan> ratione, qua f, & ideò ei oppo&longs;itus k h n, <lb/>& k rectus, e&longs;t enim f polus b d, & h k decli <lb/>natio nota ergo k n, & h n notæ. At e k, & <lb/>g h fuere notæ. Ergo e n, & g n, quare re&longs;i<lb/>duæ n l & n b notæ. E&longs;t autem angulus l <lb/>rectus. ergo ortus amplitudo punctil nota <lb/>&longs;cilicet arcus l b, ergo in præ&longs;enti figura angulus m h b, ergo k h l. <lb/>igitur poterimus &longs;tatuere angulos umbrarum, & iam po&longs;&longs;umus <lb/>determinare magnitudinem: ergo punctum ad <expan abbr="ungu&etilde;">unguem</expan> umbr&ecedil; qua<lb/>libet hora, & parte horæ &longs;ingulis diebus in quacunque regione datæ <lb/>altitudinis poli uer&longs;a, & rects. In cylindrica autem eodem modo &longs;i<lb/>cut in uer&longs;a, e&longs;t enim &longs;pecies umbr&ecedil; uer&longs;&ecedil;, ni&longs;i quod analema ob ob<lb/>liquitatem cylindri melius aptatur, rotundum &longs;cilicet cum <expan abbr="rot&utilde;do">rotundo</expan>.</s>

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<s>Propo&longs;itio cente&longs;imatrige&longs;imaprima.</s>

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<s>Sit a b linea, cui adiecta &longs;it b c, & rur&longs;us ad b c c d <expan abbr="æ&qacute;ualis">æqualis</expan> b c <lb/>dico, quod proportio a c ad a b e&longs;t maior, quàm a d ad a c. Propor <lb/>

<arrow.to.target n="marg451"></arrow.to.target><lb/>tio enim c d ad c a minor e&longs;t, quàm ad a b per octauam quinti E<lb/>lementorum. Ergo minor d c ad c a quàm c b ad a b, quia b c & c d <lb/>&longs;unt æquales, ideò <expan abbr="æqual&etilde;">æqualem</expan> habent <expan abbr="proportion&etilde;">proportionem</expan> <lb/>ad a b: <expan abbr="igi&ttilde;">igitur</expan> coniungendo per 28. Quinti propor <lb/>

<arrow.to.target n="fig98"></arrow.to.target><lb/>tio d a ad a c minor, quam c a ad a b, quod erat demon&longs;trandum.<lb/>

<figure id="fig98"></figure><lb/>tio d a ad a c minor, quam c a ad a b, quod erat demon&longs;trandum.<lb/>

<arrow.to.target n="marg452"></arrow.to.target></s>

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<s>Propo&longs;itio cente&longs;imatrige&longs;ima&longs;ecunda.</s>

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<s>Sint duæ line&ecedil; a b, & c d. & &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/>

<arrow.to.target n="fig99"></arrow.to.target><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adij cia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t du pla ad a b, tota igitur

<figure id="fig99"></figure><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adij cia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t du pla ad a b, tota igitur

<pb pagenum="127"/>g c duplatoti e a. quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, & per <lb/>euer&longs;am ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="cõponitur">componitur</expan> ex g e ad <lb/>f e, & f c ad c d, igitur e a ad c b componitur ex ei&longs;dem. Proportio <lb/>autem g c ad f c e&longs;t minor, quam f c ad c d, igitur minor quàm du<lb/>plicata f c ad c d. con&longs;tat uerò ex ei&longs;dem, quod proportio c a ad a b <lb/>maior e&longs;t duplicata g c ad f c.</s>

<s>Propo&longs;itio cente&longs;imatrige&longs;imatertia.</s>

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<s>Secundum &longs;ic per eadem, addito enim duplo f c ip&longs;i <lb/>

<arrow.to.target n="fig100"></arrow.to.target><lb/>a b ut in &longs;ecunda figura, & &longs;int a m, & m n erit f d ad c d, <lb/>ut n a ad a b, quare cum n a ad a b &longs;it minor duplicata per <lb/>præcedentem in b ad a b, & a b ad e b &longs;it maior, ut demon <lb/>&longs;tratum e&longs;t in uige&longs;ima tertia huius, quàm m b ad a b, erit <lb/>f d ad d c multo minor duplicata a b ad b e, quod e&longs;t &longs;e<lb/>cundum.</s>

<figure id="fig100"></figure><lb/>a b ut in &longs;ecunda figura, & &longs;int a m, & m n erit f d ad c d, <lb/>ut n a ad a b, quare cum n a ad a b &longs;it minor duplicata per <lb/>præcedentem in b ad a b, & a b ad e b &longs;it maior, ut demon <lb/>&longs;tratum e&longs;t in uige&longs;ima tertia huius, quàm m b ad a b, erit <lb/>f d ad d c multo minor duplicata a b ad b e, quod e&longs;t &longs;e<lb/>cundum.</s>

<s>Propo&longs;itio cente&longs;imatrige&longs;imaquarta.</s>

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<s>Sit rectangulum a c cuius tertia pars c e &longs;it quadrata, dico quod <lb/>

<arrow.to.target n="marg455"></arrow.to.target><lb/>corpus, quod <expan abbr="cõ&longs;tat">con&longs;tat</expan> ex e d in a b e&longs;t maius omni corpore, quod fue <lb/>rit ex latere partis &longs;uperficiei a b in reliquam <expan abbr="part&etilde;">partem</expan>. Si non diuidatur <lb/>uel &longs;upra uel infra, & primo in f erit <expan abbr="aut&etilde;">autem</expan> proportio e d ad d f, ut e c ad

<pb pagenum="128"/>e k, & f a ad a e, ut &longs;uperficierum ip&longs;a<lb/>

<arrow.to.target n="fig101"></arrow.to.target><lb/>rum per primam &longs;exti Elementorum: at <lb/>per præcedentem maior e&longs;t proportio <lb/>e d ad d f, quàm a f ad a e, duplicata igi<lb/>tur maior e&longs;t proportio e d ad eam, qu&ecedil; <lb/>pote&longs;t &longs;uper f c &longs;uperficiem, quam f a ad <lb/>a e, igitur maior, quàm a k ad a b ex pri<lb/>ma &longs;exti Elementorum: igitur per trige <lb/>&longs;imam quartam undecimi. Parallelipe<lb/>dum ex e d in a b maius e&longs;t parallelipedo ex ea, quæ pote&longs;t in f c &longs;u<lb/>perficiem in ip&longs;am &longs;uperficiem a k. Si uerò diui&longs;io facta fuerit in g, <lb/>con&longs;tat ex præcedenti, quod minor e&longs;t proportio g e ad e d, quàm <lb/>&longs;it duplicata e a ad a d a g, eam igitur minor proportio eius lineæ, <lb/>quæ pote&longs;t in g e &longs;uperficiem ad e d quam a b ad a h, igitur paralle<lb/>lipedum ex e d in a b e&longs;t maius parallelipedo ex ea, quæ pote&longs;t g c <lb/>in a h cum &longs;it a b ad a h, ut dictum e&longs;t, uelut a e ad a g.<lb/>

<figure id="fig101"></figure><lb/>rum per primam &longs;exti Elementorum: at <lb/>per præcedentem maior e&longs;t proportio <lb/>e d ad d f, quàm a f ad a e, duplicata igi<lb/>tur maior e&longs;t proportio e d ad eam, qu&ecedil; <lb/>pote&longs;t &longs;uper f c &longs;uperficiem, quam f a ad <lb/>a e, igitur maior, quàm a k ad a b ex pri<lb/>ma &longs;exti Elementorum: igitur per trige <lb/>&longs;imam quartam undecimi. Parallelipe<lb/>dum ex e d in a b maius e&longs;t parallelipedo ex ea, quæ pote&longs;t in f c &longs;u<lb/>perficiem in ip&longs;am &longs;uperficiem a k. Si uerò diui&longs;io facta fuerit in g, <lb/>con&longs;tat ex præcedenti, quod minor e&longs;t proportio g e ad e d, quàm <lb/>&longs;it duplicata e a ad a d a g, eam igitur minor proportio eius lineæ, <lb/>quæ pote&longs;t in g e &longs;uperficiem ad e d quam a b ad a h, igitur paralle<lb/>lipedum ex e d in a b e&longs;t maius parallelipedo ex ea, quæ pote&longs;t g c <lb/>in a h cum &longs;it a b ad a h, ut dictum e&longs;t, uelut a e ad a g.<lb/>

<arrow.to.target n="marg456"></arrow.to.target></s>

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<s>Manife&longs;tum e&longs;t autem, quòd tale corpus e&longs;t æquale duplo cubi <lb/>lateris partis tertiæ quadratæ.</s>

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<s>Sit a c dupla b c, & &longs;it quadratum ad ip&longs;ius a c, dico parallelipe<lb/>

<arrow.to.target n="fig102"></arrow.to.target><lb/>dum ex b c in a d maius e&longs;&longs;e quouis alio ex <lb/>diui&longs;ione lineæ a b &longs;imiliter creato. Secetur <lb/>primo in e, & fiat quadratum a f, eritque per <lb/>uige&longs;imam quintam. Huius proportio c b <lb/>ad b c maior duplicata a e ad a c, quare ma<lb/>ior, quam a f ad a d per uige&longs;imam &longs;exti Ele <lb/>mentorum, igitur per trige&longs;imam quartam <lb/>undecimi, Parallelipedum ex b c in a d maius e&longs;t parallelipedo e b <lb/>in a f, quod e&longs;t demon&longs;trandum. Si uerò diui&longs;io cadat in g, fiat qua<lb/>dratum a h, et erit per uige&longs;imamtertiam huius proportio g c ad c b <lb/>minor, quam duplicata c a ad a g: igitur minor, quàm a d ad a h, igi<lb/>tur per eandem parallelipedum ex c b in a d maius e&longs;t parallelipe<lb/>do ex g b in a h.</s>

<figure id="fig102"></figure><lb/>dum ex b c in a d maius e&longs;&longs;e quouis alio ex <lb/>diui&longs;ione lineæ a b &longs;imiliter creato. Secetur <lb/>primo in e, & fiat quadratum a f, eritque per <lb/>uige&longs;imam quintam. Huius proportio c b <lb/>ad b c maior duplicata a e ad a c, quare ma<lb/>ior, quam a f ad a d per uige&longs;imam &longs;exti Ele <lb/>mentorum, igitur per trige&longs;imam quartam <lb/>undecimi, Parallelipedum ex b c in a d maius e&longs;t parallelipedo e b <lb/>in a f, quod e&longs;t demon&longs;trandum. Si uerò diui&longs;io cadat in g, fiat qua<lb/>dratum a h, et erit per uige&longs;imamtertiam huius proportio g c ad c b <lb/>minor, quam duplicata c a ad a g: igitur minor, quàm a d ad a h, igi<lb/>tur per eandem parallelipedum ex c b in a d maius e&longs;t parallelipe<lb/>do ex g b in a h.</s>

<s>

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<s>Propo&longs;itis quibu&longs;uis numeris utpotè 916132832, uolo detrahere <lb/><02> relatam primam, primum habebo in tabula de&longs;cripta relata pri<lb/>ma numerorum &longs;implicium u&longs;que ad 10 uelut in exemplo. Dein de <lb/>

<arrow.to.target n="fig103"></arrow.to.target><lb/>&longs;ub&longs;cribam pun<lb/>ctum &longs;ub prima <lb/>nota à dextra, & <lb/>quia e&longs;t quarta in <lb/>ordine hoc, &longs;eu quinta denominatio &longs;ecun<lb/>dum no&longs;trum, omittam quatuor notas in<lb/>ter medias, & &longs;ub&longs;cribam punctum aliud, <lb/>& ita facerem &longs;i e&longs;&longs;ent plures quàm decem <lb/>notæ: relinquitur ergo ad <expan abbr="p&utilde;ctum">punctum</expan> primum <lb/>à &longs;ini&longs;tra 9161, cuius qu&ecedil;ro <02> relatam pri<lb/>mam in tabula, quam inuenio e&longs;&longs;e 6, nam <lb/>7776 eius relatum primum e&longs;t <lb/>proximius ex minoribus ad 9161, <lb/>detraho igitur 7776, ex numero <lb/>propo&longs;itio relinquitur. Dein de <lb/>póno 6 & quadratum eius, & cub. & quadratum <lb/>quadrati, quia, ut dixi, e&longs;t quarta denominatio a<lb/>pud illum, & è regione numeros præcedentes in<lb/>uentos relati primi ex præcedenti propo&longs;itione: & duco &longs;ingulos <lb/>cum &longs;uis collateralibus, ut uides etiam in figura, et cum ultimo pro<lb/>ducto, &longs;cilicet 64800000 diuido 138532832 exit 2, huius accipio o<lb/>mnes numeros ad relatum primum u&longs;que ut uides, & pono minores <lb/>è regione maiorum, utpotè 2 è regione 1296 & 50000, & 4 è regio

<figure id="fig103"></figure><lb/>&longs;ub&longs;cribam pun<lb/>ctum &longs;ub prima <lb/>nota à dextra, & <lb/>quia e&longs;t quarta in <lb/>ordine hoc, &longs;eu quinta denominatio &longs;ecun<lb/>dum no&longs;trum, omittam quatuor notas in<lb/>ter medias, & &longs;ub&longs;cribam punctum aliud, <lb/>& ita facerem &longs;i e&longs;&longs;ent plures quàm decem <lb/>notæ: relinquitur ergo ad <expan abbr="p&utilde;ctum">punctum</expan> primum <lb/>à &longs;ini&longs;tra 9161, cuius qu&ecedil;ro <02> relatam pri<lb/>mam in tabula, quam inuenio e&longs;&longs;e 6, nam <lb/>7776 eius relatum primum e&longs;t <lb/>proximius ex minoribus ad 9161, <lb/>detraho igitur 7776, ex numero <lb/>propo&longs;itio relinquitur. Dein de <lb/>póno 6 & quadratum eius, & cub. & quadratum <lb/>quadrati, quia, ut dixi, e&longs;t quarta denominatio a<lb/>pud illum, & è regione numeros præcedentes in<lb/>uentos relati primi ex præcedenti propo&longs;itione: & duco &longs;ingulos <lb/>cum &longs;uis collateralibus, ut uides etiam in figura, et cum ultimo pro<lb/>ducto, &longs;cilicet 64800000 diuido 138532832 exit 2, huius accipio o<lb/>mnes numeros ad relatum primum u&longs;que ut uides, & pono minores <lb/>è regione maiorum, utpotè 2 è regione 1296 & 50000, & 4 è regio

<pb pagenum="133"/>ne 216 & 10000, & 8 è regione 36 & 10000, & 16 è regione 6, & 50, <lb/>& duco 6 in 50 fit 300, duco in 16 fit 4800, duco 36 in 1000 fit <lb/>36000, duco 36 in 8 fit 288000, duco etiam 216 in 10000 & fit <lb/>2160000, & duco hos per 4 fit 86400000, duco rur&longs;us 1296 in <lb/>50000 fit 64800000, duco in 2 fit 129600000. Demum addo 32 re<lb/>latum primum 2, & fit &longs;umma omnium 138532832, & ita habemus <lb/>radicem relatam primam dictinumeri e&longs;&longs;e 62. Et &longs;i numerus produ <lb/>ctus fui&longs;&longs;et maior oportui&longs;&longs;et accipere proximo minorem. Inde per <lb/>regulam &longs;equentem addere minutias.</s>

<s>Propo&longs;itio cente&longs;imaquadrage&longs;ima.</s>

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<s>Tertius modus e&longs;t &longs;ubtilior, tu &longs;cis, &qring;d duo decima denominatio <lb/>e&longs;t quadrata &longs;ext&ecedil;, & quadrata quad, tertiæ, & cuba quarti, quarta <lb/>autem e&longs;t inter <expan abbr="tertiã">tertiam</expan> & &longs;extam &longs;ecunda quantitas in continua pro<lb/>portione: ergo inuenta <02> numeri propo&longs;iti & <02> radicis inuentæ <lb/><expan abbr="reducã">reducam</expan> ad unam denominationem, et inter numeratores collo cabo <lb/>duas quantitates, quod facile erit &longs;en&longs;im procedendo, & habebo <02><lb/>cu. quæ&longs;itam, &longs;cilicet minorem ex duabus intermedijs. Et &longs;imiliter <lb/>pro relata prima, capiam &longs;exaginta denominationes, & &longs;cis, quòd <lb/>quintadecima e&longs;t <02> <02> &longs;exage&longs;im&ecedil;, & decima e&longs;t <02> cu. <02> &longs;exage&longs;im&ecedil;, <lb/>& duodecima <02> relata prima &longs;exage&longs;imæ per eandem inuenta, er<lb/>go <02> numeri propo&longs;iti tanquam ille &longs;it &longs;exage&longs;ima denominatio, <lb/>inueniam illius radicis inuentæ <02> quadratam, & cubicam, & <lb/>quia duodecima quantitas quæ e&longs;t <02> relata prima numeri e&longs;t <lb/>&longs;ecunda, quatuor intermediarum inter ponam inter <02> quadra<lb/>tum, quadratum, & cubicam quadratam quatuor numeros in <lb/>continua proportione, & &longs;ecundus ex minoribus erit <02> relata <lb/>prima numeri propo&longs;iti. Exemplum cubicæ uolo <02> cu: 5 habui <02><lb/>quadratam eius 2 5/21 &longs;ed uolo proximiorem diuidendo 4/441 per 4, <lb/>quod e&longs;t fermè duplum 2 5/21 exit 1/441 detraho ex 2 5/21 relinquitur ualde <lb/>proxima <02> 5. 2 104/441 huius igitur radix quadrata, primo inuenta e&longs;t 1 1/2 <lb/>&longs;ecunda proximior e&longs;t 1 41/84 reduco ad eandem denominationem fi<lb/>ent 284/9261 2 416/1764 & 1 861/1764 inter 3944, & 2625, inueniemus duos nume<lb/>ros in continua proportione, ut uides, & erit &longs;ecunda quantitas <lb/>

<arrow.to.target n="fig104"></arrow.to.target><lb/>3006/7641, quod e&longs;t 167/98 proximum ad 1 5/7, <02> cubica. 5. <lb/><expan abbr="nã">nam</expan> eius cubus e&longs;t 5. 13/343 at exacti&longs;sima e&longs;t ergo 1 69/98. <lb/>ut liquet. Pro relata prima ergo ponamus, ut ue<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;imiliter <02> cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>& inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb/>& &longs;ecundus po&longs;t minimum ex illis erit <02> relata prima propinqui&longs;<lb/>&longs;ima 25. Quomodo uerò inueniantur facillimè illi termini, do<lb/>cui in &longs;exto libro operis perfecti.</s>

<figure id="fig104"></figure><lb/>3006/7641, quod e&longs;t 167/98 proximum ad 1 5/7, <02> cubica. 5. <lb/><expan abbr="nã">nam</expan> eius cubus e&longs;t 5. 13/343 at exacti&longs;sima e&longs;t ergo 1 69/98. <lb/>ut liquet. Pro relata prima ergo ponamus, ut ue<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;imiliter <02> cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>& inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb/>& &longs;ecundus po&longs;t minimum ex illis erit <02> relata prima propinqui&longs;<lb/>&longs;ima 25. Quomodo uerò inueniantur facillimè illi termini, do<lb/>cui in &longs;exto libro operis perfecti.</s>

<s>Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t &longs;un<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c fip&longs;ius a e, quàm

<pb pagenum="135"/>c d ip&longs;ius a f ex Euclide. Dico ergo quod maior e&longs;t proportio a b <lb/>

<arrow.to.target n="fig105"></arrow.to.target><lb/>ad c d, quàm a e ad e f, fiat d g ad quam &longs;it b c ut <lb/>

<figure id="fig105"></figure><lb/>ad c d, quàm a e ad e f, fiat d g ad quam &longs;it b c ut <lb/>

<arrow.to.target n="marg467"></arrow.to.target><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au<lb/>tem e&longs;t a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f quàm a b ad c d quod fuit propo&longs;itum. Simili <lb/>ter &longs;i fuerint duæ quantitates, a b & c d, quarum a b &longs;it maiore, c d <lb/>autem eadem e minor, dico, quòd dimidium aggregati a b & c d <lb/>maiorem habebit proportionem ad e, quàm c d & minor, nam iun<lb/>cta b f æquali d e ad a b, ita ut f g &longs;it dimidium totius a f, qùia ergo <lb/>

<arrow.to.target n="fig106"></arrow.to.target><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/>

<figure id="fig106"></figure><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/>

<arrow.to.target n="marg468"></arrow.to.target><lb/>f a cum &longs;it minor b a, & &longs;imiliter f g e&longs;t mi<lb/>nor a b, quia a b e&longs;t maior dimidio a f, quia <lb/>e&longs;t maior b f, ergo proportio g f ad c e&longs;t ma <lb/>ior quam b f ad e, ita quam c d ad e, & mi<lb/>

<arrow.to.target n="marg469"></arrow.to.target><lb/>nor quàm a b ad e, quod fuit propo&longs;itum. Quo ui&longs;o uolo <02> 1000 <lb/>quadratam, & quòd de quadrata dico, dico etiam de alijs radici<lb/>bus & erit ex &longs;ecunda regula harum 31 39/62 & quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regulæ 31 38/61 erit minus, & in ueritate <lb/>in eo, quod fit ducendo, ut uides, & hoc e&longs;t pro<lb/>

<arrow.to.target n="fig107"></arrow.to.target><lb/>ximum ad 1<gap/>/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160

<figure id="fig107"></figure><lb/>ximum ad 1<gap/>/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160

<arrow.to.target n="fig108"></arrow.to.target> detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 <gap/>/40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, & erit <02> exa<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita <lb/>tem, & propinquitatem in producto differentia e&longs;t 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du<lb/>plum eius erit paulo maius 1/6300. Vnde facilior e&longs;t, & breuior hæc <lb/>uia quàm per 00 ad ditus. Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. Con&longs;idero quòd 31 38/61 e&longs;t maius 1/6300 radice, di<lb/>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator e&longs;t proxi<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum <02> quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s>

<figure id="fig108"></figure> detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 <gap/>/40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, & erit <02> exa<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita <lb/>tem, & propinquitatem in producto differentia e&longs;t 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du<lb/>plum eius erit paulo maius 1/6300. Vnde facilior e&longs;t, & breuior hæc <lb/>uia quàm per 00 ad ditus. Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. Con&longs;idero quòd 31 38/61 e&longs;t maius 1/6300 radice, di<lb/>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator e&longs;t proxi<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum <02> quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s>

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<s>Quinta regula e&longs;t omnium pulcherrima, & e&longs;t communis omni <lb/>bus & fractis & integris & omnibus generibus radicum, & &longs;it ex<lb/>emplum, uolo <02> radicis &longs;upra&longs;criptæ &longs;cilicet 31 3913/6283 multiplico 31 <lb/>in 6283, & fit 194793, cui addo 3913, fit 198686 manife&longs;tum e&longs;t igi<lb/>tur, quod 198686/6283 æquiualet 31 3913/6283 hoc facto, quod e&longs;t commune om

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<s>Sit gratia exempli, in &longs;ex annis u&longs;ura rediuiua uige&longs;imæ, erit<lb/>qúe proportio 21/20, cuius numeratorem &longs;exies ducam in &longs;e primum <lb/>bis fit 441: ergo ducto 441 in &longs;e fit qúe 194481 ductum in 441 <lb/>fit 85766121 &longs;exies ductum 21, quinquies autem ducam 20 deno<lb/>

<arrow.to.target n="fig109"></arrow.to.target><lb/>minatorem in &longs;e fit bis 400, ter 8000, <lb/>quinquies ergo 3200000, diuide nume<lb/>ratorem per denominatorem abiectis <lb/>quinque notis erit 26 2566121/3200000. Quæ propor <lb/>tio e&longs;t proxima 26 4/5 ad 20, & ita ut 134 ad <lb/>100. Et &longs;i pigeret tædij autlaboris po&longs;&longs;es <lb/>pro xij annis, ducere 134 in &longs;e, & fit 17956 <lb/>diuide per 100 eadem ratione, exit 179 14/25 <lb/>& ita 100 in xij annis, fit tantundem. Et <lb/>ita pro xviij & xx annis.</s>

<figure id="fig109"></figure><lb/>minatorem in &longs;e fit bis 400, ter 8000, <lb/>quinquies ergo 3200000, diuide nume<lb/>ratorem per denominatorem abiectis <lb/>quinque notis erit 26 2566121/3200000. Quæ propor <lb/>tio e&longs;t proxima 26 4/5 ad 20, & ita ut 134 ad <lb/>100. Et &longs;i pigeret tædij autlaboris po&longs;&longs;es <lb/>pro xij annis, ducere 134 in &longs;e, & fit 17956 <lb/>diuide per 100 eadem ratione, exit 179 14/25 <lb/>& ita 100 in xij annis, fit tantundem. Et <lb/>ita pro xviij & xx annis.</s>

<s>Propo&longs;itio cente&longs;imaquadrage&longs;imatertia.</s>

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<s>Sit a c diui&longs;a in a b, b c quadratum a b &longs;it <lb/>

<arrow.to.target n="fig110"></arrow.to.target><lb/>a d, <expan abbr="quadrat&utilde;">quadratum</expan> b c, &longs;it b e <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>ex a b in b e, a f dico quòd corpora ex a b in <lb/>b e, & b c in a d æqualia &longs;unt corpori ex a c <lb/>in a f. Quia enim corpus ex a c in a f con&longs;tat <lb/>ex a b in a f, & b c in a f, per primam &longs;ecun</s>

<figure id="fig110"></figure><lb/>a d, <expan abbr="quadrat&utilde;">quadratum</expan> b c, &longs;it b e <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>ex a b in b e, a f dico quòd corpora ex a b in <lb/>b e, & b c in a d æqualia &longs;unt corpori ex a c <lb/>in a f. Quia enim corpus ex a c in a f con&longs;tat <lb/>ex a b in a f, & b c in a f, per primam &longs;ecun</s>

<s>

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<s>Sit a b diui&longs;a per æqualia in c, & per inæqua<lb/>lia in d, dico, quòd duplum cubi a c e&longs;t maius ag <lb/>

<arrow.to.target n="fig111"></arrow.to.target><lb/>gregato corporum ex a d in quadratum b d, & b d in quadratum <lb/>a cin eo quod fit ex a b in quadratum c d, nam per <expan abbr="præcedent&etilde;">præcedentem</expan> du<lb/>plum cubi a c e&longs;t æquale corpori ex a b in quadratum a c: aggrega<lb/>tum quo que corporum ex a d in quadratum b d, & b d in quadra<lb/>tum a d e&longs;t &ecedil;quale ei, quod fit ex a b in <expan abbr="rectangul&utilde;">rectangulum</expan> ex a d in d b. <expan abbr="qua-drat&utilde;">qua<lb/>dratum</expan> <expan abbr="aut&etilde;">autem</expan> a c e&longs;t maius rectangulo a d in d b quadrato c d differen <lb/>tiæ, igitur duplum cubi a c excedit aggregatum <expan abbr="corpor&utilde;">corporum</expan> <expan abbr="mutuor&utilde;">mutuorum</expan> <lb/>in corpore ex a b in quadratum c d differenti&ecedil;, quod e&longs;t <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>

<figure id="fig111"></figure><lb/>gregato corporum ex a d in quadratum b d, & b d in quadratum <lb/>a cin eo quod fit ex a b in quadratum c d, nam per <expan abbr="præcedent&etilde;">præcedentem</expan> du<lb/>plum cubi a c e&longs;t æquale corpori ex a b in quadratum a c: aggrega<lb/>tum quo que corporum ex a d in quadratum b d, & b d in quadra<lb/>tum a d e&longs;t &ecedil;quale ei, quod fit ex a b in <expan abbr="rectangul&utilde;">rectangulum</expan> ex a d in d b. <expan abbr="qua-drat&utilde;">qua<lb/>dratum</expan> <expan abbr="aut&etilde;">autem</expan> a c e&longs;t maius rectangulo a d in d b quadrato c d differen <lb/>tiæ, igitur duplum cubi a c excedit aggregatum <expan abbr="corpor&utilde;">corporum</expan> <expan abbr="mutuor&utilde;">mutuorum</expan> <lb/>in corpore ex a b in quadratum c d differenti&ecedil;, quod e&longs;t <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>

<s>

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<s>Sit linea a c diui&longs;a in b, & &longs;it differentia a b, <lb/>b c, b d, dico quod quadrata a b & b c detracto <lb/>

<arrow.to.target n="fig112"></arrow.to.target><lb/>eo quod fit ex a b in b c, æqualia &longs;unt producto a b in b c cum qua<lb/>drato b d. Quoniam. n. quadrata a b, b c æqualia quadratis a d d b <lb/>b c & productis ex a d in d b bis & quod fit ex a b in b c æquale e&longs;t <lb/>ei quod fit ex a d in &longs;e cum eo quod fit ex a d in d b, quia a d e&longs;t &ecedil;qua </s>

<figure id="fig112"></figure><lb/>eo quod fit ex a b in b c, æqualia &longs;unt producto a b in b c cum qua<lb/>drato b d. Quoniam. n. quadrata a b, b c æqualia quadratis a d d b <lb/>b c & productis ex a d in d b bis & quod fit ex a b in b c æquale e&longs;t <lb/>ei quod fit ex a d in &longs;e cum eo quod fit ex a d in d b, quia a d e&longs;t &ecedil;qua </s>

<s>

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<s>Sit linea a b diui&longs;a in c uolo eius <lb/>

<arrow.to.target n="fig113"></arrow.to.target><lb/>partibus addere lineas, ut propo&longs;i</s>

<figure id="fig113"></figure><lb/>partibus addere lineas, ut propo&longs;i</s>

<s>

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<s>Sit a, b, c, d, propo&longs;itæ line&ecedil;, <lb/>

<arrow.to.target n="fig114"></arrow.to.target><lb/>uolo diuidere a b ita in e ut <lb/>&longs;umpta &longs;ecundum proportio<lb/>nem alicuius quantitatis, puta <lb/>g ad a e &longs;ic b f ad e b & ut g ad <lb/>e b &longs;ic g a ad a e ut &longs;it propor<lb/>tio g e ad e f ut c ad d. Sint ergo <lb/>omnia <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> & &longs;it g rectan<lb/>gulum ex a e in e b, cum ergo <lb/>g a contineat a e ut g continet e b, g autem continet e b &longs;ecundum <lb/>a e, igitur g a continet a e &longs;ecundum a c, ergo ex diffinitione qua</s>

<figure id="fig114"></figure><lb/>uolo diuidere a b ita in e ut <lb/>&longs;umpta &longs;ecundum proportio<lb/>nem alicuius quantitatis, puta <lb/>g ad a e &longs;ic b f ad e b & ut g ad <lb/>e b &longs;ic g a ad a e ut &longs;it propor<lb/>tio g e ad e f ut c ad d. Sint ergo <lb/>omnia <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> & &longs;it g rectan<lb/>gulum ex a e in e b, cum ergo <lb/>g a contineat a e ut g continet e b, g autem continet e b &longs;ecundum <lb/>a e, igitur g a continet a e &longs;ecundum a c, ergo ex diffinitione qua</s>

<s>

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<s>Sit data a b quam uolo diuidere, ut proponitur &longs;ub proportio<lb/>

<arrow.to.target n="marg488"></arrow.to.target><lb/>ne c d ad e, diuido a b bifariam in f, & ab&longs;cindo <lb/>

<arrow.to.target n="fig115"></arrow.to.target><lb/>g d æqualem d e, & inter c g <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> & c e inter<lb/>pono proportione, & ut h ad c g ita a f medietatis a b ad fk. Omnia <lb/>i&longs;ta &longs;unt noti&longs;sima ex primo & &longs;exto Elemento<lb/>

<figure id="fig115"></figure><lb/>g d æqualem d e, & inter c g <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> & c e inter<lb/>pono proportione, & ut h ad c g ita a f medietatis a b ad fk. Omnia <lb/>i&longs;ta &longs;unt noti&longs;sima ex primo & &longs;exto Elemento<lb/>

<arrow.to.target n="fig116"></arrow.to.target><lb/><expan abbr="r&utilde;">rum</expan> Euclidis. Si ergo ab&longs;cindantur fk ex fa, dico <lb/>quod proportio quadratorum l k & k a ad du<lb/>plum rectanguli a k in k b e&longs;t ut c d ad d e. Quia. n. c e ad c g dupli<lb/>cata e&longs;t ei qu&ecedil; e&longs;t h ad c g, duplicata e&longs;t <expan abbr="etiã">etiam</expan> ei quæ e&longs;t f a ad fk, qua<lb/>re ut quadrati a f ad fk, ita c e ad c g, igitur di&longs;iungendo c g ad g e ut <lb/>re&longs;idui quadrati k f ad re&longs;iduum quadrati a f, quare c g ad g d ut <lb/>quadrati k f ad dimidium re&longs;idui quadrati a f, igitur coniunctim c d <lb/>ad d g ut quadrati k f & dimidij re&longs;idui quadrati a f ad ip&longs;um dimi<lb/>dium re&longs;idui. At uerò cum g d &longs;it æqualis d e, erit c d ad d e ut qua<lb/>drati k f cum dimidio re&longs;idui &longs;æpius dicti ad ip&longs;um dimidium re&longs;i<lb/>dui. Igitur etiam ut dupli quadrati k f cum re&longs;iduo ad <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;unt <lb/>enim omnia duplicata. At <expan abbr="dupl&utilde;">duplum</expan> quadrati k f <expan abbr="c&utilde;">cum</expan> re&longs;iduo e&longs;t æqua<lb/>le quadratis a f & f k, igitur quadratorum a f & f k ad differentiam <lb/>eo rum proportio e&longs;t ut c d ad d e, igitur dupli quadratorum a f & <lb/>f k ad duplum differentiæ quadratorum a f & fk ut c d ad d e. Ve<lb/>

<figure id="fig116"></figure><lb/><expan abbr="r&utilde;">rum</expan> Euclidis. Si ergo ab&longs;cindantur fk ex fa, dico <lb/>quod proportio quadratorum l k & k a ad du<lb/>plum rectanguli a k in k b e&longs;t ut c d ad d e. Quia. n. c e ad c g dupli<lb/>cata e&longs;t ei qu&ecedil; e&longs;t h ad c g, duplicata e&longs;t <expan abbr="etiã">etiam</expan> ei quæ e&longs;t f a ad fk, qua<lb/>re ut quadrati a f ad fk, ita c e ad c g, igitur di&longs;iungendo c g ad g e ut <lb/>re&longs;idui quadrati k f ad re&longs;iduum quadrati a f, quare c g ad g d ut <lb/>quadrati k f ad dimidium re&longs;idui quadrati a f, igitur coniunctim c d <lb/>ad d g ut quadrati k f & dimidij re&longs;idui quadrati a f ad ip&longs;um dimi<lb/>dium re&longs;idui. At uerò cum g d &longs;it æqualis d e, erit c d ad d e ut qua<lb/>drati k f cum dimidio re&longs;idui &longs;æpius dicti ad ip&longs;um dimidium re&longs;i<lb/>dui. Igitur etiam ut dupli quadrati k f cum re&longs;iduo ad <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;unt <lb/>enim omnia duplicata. At <expan abbr="dupl&utilde;">duplum</expan> quadrati k f <expan abbr="c&utilde;">cum</expan> re&longs;iduo e&longs;t æqua<lb/>le quadratis a f & f k, igitur quadratorum a f & f k ad differentiam <lb/>eo rum proportio e&longs;t ut c d ad d e, igitur dupli quadratorum a f & <lb/>f k ad duplum differentiæ quadratorum a f & fk ut c d ad d e. Ve<lb/>

<arrow.to.target n="marg489"></arrow.to.target><lb/>rum duplum quadratorum a f & f k æquatur quadratis b k & k a. <lb/>

<arrow.to.target n="marg490"></arrow.to.target><lb/>Et duplum differentiæ quadratorum a f & fk e&longs;t &ecedil;quale duplo pro <lb/>ducti b k in k a, igitur proportio quadratorum k b & k a ad <expan abbr="dupl&utilde;">duplum</expan> <lb/>producti k b in k a e&longs;t ueluti c d ad d e, quod e&longs;t propo&longs;itum.</s>

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<s><margin.target id="marg490"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s>Propo&longs;itio cente&longs;imaquinquage&longs;ima.</s>

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<s>Propo&longs;itis duabus lineis <expan abbr="lineã">lineam</expan> communem <lb/>

<arrow.to.target n="fig117"></arrow.to.target><lb/>utrique adiungere, ut &longs;it maioris ad additam pro<lb/>portio, uelut quadratorum minoris & adiectæ <lb/>ad duplum unius in alteram.</s>

<figure id="fig117"></figure><lb/>utrique adiungere, ut &longs;it maioris ad additam pro<lb/>portio, uelut quadratorum minoris & adiectæ <lb/>ad duplum unius in alteram.</s>

<s>Hæc e&longs;t qua&longs;i conuer&longs;a <expan abbr="præced&etilde;tis">præcedentis</expan>. Sit a ma<lb/>

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<s>Sit a b diui&longs;a in puncto c, & fiat c d æqualis <lb/>c b, manife&longs;tum e&longs;t quod differentia partium <lb/>

<arrow.to.target n="fig118"></arrow.to.target><lb/>e&longs;t a d, dico proportionem differentiæ quadra <lb/>torum a c & c b ad quadratum a d differentiæ partium e&longs;&longs;e ut a b ad </s>

<figure id="fig118"></figure><lb/>e&longs;t a d, dico proportionem differentiæ quadra <lb/>torum a c & c b ad quadratum a d differentiæ partium e&longs;&longs;e ut a b ad </s>

<s>

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<s>Sit propo&longs;ita a b diui&longs;a per <lb/>

<arrow.to.target n="fig119"></arrow.to.target><lb/>æqualia in c per inæqualia in <lb/>d, & &longs;it ut addantur a g & b f, <lb/>ita ut proportio c a, & a d ad a d &longs;it ueluti f d ad d b, & c b & b d ad <lb/>b d, uelut g d ad d a, & hæc e&longs;t quarta <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> Archimedis de &longs;ph&ecedil;ra, <lb/>& Cylindro: quia ergo a c & a d ad a d, ut f d ad d b erit a c ad a d, <lb/>fb ad b d. Et &longs;imiliter quia e&longs;t c b & b d ad b d, uelut g d ad d a erit

<figure id="fig119"></figure><lb/>æqualia in c per inæqualia in <lb/>d, & &longs;it ut addantur a g & b f, <lb/>ita ut proportio c a, & a d ad a d &longs;it ueluti f d ad d b, & c b & b d ad <lb/>b d, uelut g d ad d a, & hæc e&longs;t quarta <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> Archimedis de &longs;ph&ecedil;ra, <lb/>& Cylindro: quia ergo a c & a d ad a d, ut f d ad d b erit a c ad a d, <lb/>fb ad b d. Et &longs;imiliter quia e&longs;t c b & b d ad b d, uelut g d ad d a erit

<pb pagenum="143"/>c b ad b d, uelut g a ad a d, & hoc e&longs;t primum. Quia ergo c a e&longs;t æ<lb/>qualis c b, erit c a ad b d, uelut g a ad a d, & iam fuit a d ad c a, ut b d <lb/>ad f b, per conuer&longs;am igitur a d ad b d, ut g a ad a d, & ut b d ad fb, <lb/>interpo&longs;itis ergo a d & d b inter a g & b f cum compo&longs;ita &longs;it pro<lb/>portio a g ad b f ex proportione a g ad a d, & ad d b, & d b <lb/>ad b f, & proportio a d ad d b, &longs;it æqualis proportioni <lb/>

<arrow.to.target n="fig120"></arrow.to.target><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. Per de<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/>

<figure id="fig120"></figure><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. Per de<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/>

<arrow.to.target n="fig121"></arrow.to.target><lb/>a d & d b ad a c componunt proportionem produ<lb/>ducti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;itl ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<figure id="fig121"></figure><lb/>a d & d b ad a c componunt proportionem produ<lb/>ducti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;itl ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<s>

<arrow.to.target n="marg497"></arrow.to.target><lb/>l &longs;unt æquales, quia producuntur ex ei&longs;dem, igitur per demon&longs;tra<lb/>ta in quinto Elementorum Euclidis, k e&longs;t æquale m, ergo a c e&longs;t me<lb/>dia pro portione inter b f & g a, quod e&longs;t tertium. Quia uerò ex pri<lb/>mo demon&longs;trato e&longs;t fb ad b d, ut a c ad a d, & c b ad idem b d, ut g a <lb/>ad idem a d erit coniungendo fb & b c ad b d, ut coniun<lb/>

<arrow.to.target n="fig122"></arrow.to.target><lb/>gendo g a & a c ad a d, &longs;ed fb & b c componunt f c & g a, <lb/>& a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er<lb/>go permutando g c ad f c, ut a d ad b d, quod e&longs;t quartum.</s>

<figure id="fig122"></figure><lb/>gendo g a & a c ad a d, &longs;ed fb & b c componunt f c & g a, <lb/>& a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er<lb/>go permutando g c ad f c, ut a d ad b d, quod e&longs;t quartum.</s>

<s><margin.target id="marg497"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> P<emph type="italics"/>rop.<emph.end type="italics"/> 23 <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 9.</s>

<s>Cum ergo punctum d fuerit datum, licet inuenire a g & b f, faci<lb/>lè, ut Archimedes præ&longs;up ponit proportionem g d ad d f datam & <lb/>quærit eam, quæ e&longs;t a d ad d b, & peruenitur ad res numero triplo <lb/>quadrati dimidij lineæ a&longs;&longs;umptæ æquales cubo & numero, qui &longs;it <lb/>ex duplo cubi dimidij in 1 m: ip&longs;a proportione, & quod produci<lb/>tur diui&longs;o per 1 p: ip&longs;a proportione. Veluti po&longs;ita a b 10, & propor<lb/>tione quam uolo g d ad d f &longs;excupla, duco 5 dimidium 10 in &longs;e fit 25, <lb/>& triplico, fit 75 numerus rerum. Inde duco 5 idem dimidium ad <lb/>cubum fit 125, duplico fit 250, duco in 5, qui e&longs;t 1 m: proportione fit <lb/>1250, diuido per 7, qui e&longs;t 1 p: proportione exit 178 4/7 numerus, qui <lb/>cum cubo æquatur 75 rebus. Cum ergo con&longs;tituta fuerit diui&longs;io in <lb/>c, non recipit proportionem g d ad f d quam uolueris, &longs;ed &longs;equitur <lb/>una &longs;ola ad <expan abbr="illã">illam</expan>, & e&longs;t mirabile, quoniam line&ecedil; uidentur &longs;umi liberè. <lb/>Sed non e&longs;t ita. Et <expan abbr="etiã">etiam</expan> quia Archimedes <expan abbr="uide&ttilde;">uidetur</expan> a&longs;&longs;umere <expan abbr="aliã">aliam</expan> lineam, <lb/>&longs;ed non inue &longs;tigat eam, imò o&longs;tendit eam ex a&longs;&longs;umptis. At Euto ci<lb/>us o&longs;ten dit ambas, <expan abbr="unã">unam</expan> ex propria inuentione, aliam ex Diocle, &longs;ed

<pb pagenum="144"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;e quitur ad aliam. Ex hoc pa<lb/>tet cur Dio cles a&longs;&longs;ump&longs;erit lineam unam, quæ e&longs;t a c, quæ &longs;e ha<lb/>bet ad a d, & d b, ut uici&longs;sim a d, & d b ad additas, quod e&longs;t pri<lb/>mum demon&longs;tratum. Sic enim omittit primum quod proponit Ar <lb/>chimedes, & a&longs;&longs;umit quod proximum e&longs;t: & ideò Archimedes non <lb/>pro bat, nec præ&longs;upponit, quod à Diocle probatur, &longs;cilicet datum <lb/>e&longs;&longs;e punctum d in linea a b, &longs;ed &longs;olum in linea g f, ideò cogitur pro<lb/>bare &longs;ecundum quod demon&longs;tratur ab Eutocio, & à nobis demon <lb/>&longs;tratum e&longs;t &longs;uprà. Archimedes <expan abbr="a&utilde;t">aunt</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uo cat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/>Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. Et &longs;imiliter f g ad <lb/>g d, & e&longs;t 1 præ proportione data. Vnde notandum quod datum <lb/>dicitur, &longs;impliciter cognitum alio modo, dicitur datum po&longs;itione, <lb/>quod e&longs;t certum & tale, uelut &longs;i quis dicat, diuide 10 in duos nume<lb/>ros quadratos: hoc non e&longs;t datum, pote&longs;t enim diuidi pluribus mo <lb/>dis. At &longs;i dicas ut una pars &longs;it alterius <expan abbr="quadrat&utilde;">quadratum</expan>, i&longs;tud antequàm &longs;ci <lb/>untur partes, dicitur datum po&longs;itione. Ergo datum po&longs;itione e&longs;t du <lb/>plex, uel ut ratio nota &longs;it, non autem quantitas, ut &longs;i dicam a b e&longs;t du <lb/>pla ad b c, utra que dicitur nota po&longs;itione, quo<lb/>niam ne&longs;cio quanta &longs;it a b. Vel &longs;i quantitas e&longs;t <lb/>

<arrow.to.target n="fig123"></arrow.to.target><lb/>nota proportio ignota &longs;it, ut &longs;i a c &longs;it 10, & &longs;it, <lb/>ut b c &longs;it <02> relata, a b erit punctus b, & proportio a b ad b c data po <lb/>&longs;itione, non tamen nota. Et &longs;i dicas igitur omnia, quæ habent deter <lb/>minationem erunt data po&longs;itione? Dico quod non, quia oportet, <lb/>ut illa determinatio comprehendatur &longs;ub una ratione, eaque &longs;altem <lb/>generaliter co gnita.</s>

<figure id="fig123"></figure><lb/>nota proportio ignota &longs;it, ut &longs;i a c &longs;it 10, & &longs;it, <lb/>ut b c &longs;it <02> relata, a b erit punctus b, & proportio a b ad b c data po <lb/>&longs;itione, non tamen nota. Et &longs;i dicas igitur omnia, quæ habent deter <lb/>minationem erunt data po&longs;itione? Dico quod non, quia oportet, <lb/>ut illa determinatio comprehendatur &longs;ub una ratione, eaque &longs;altem <lb/>generaliter co gnita.</s>

<s>Propo&longs;itio cente&longs;imaquinquage&longs;imatertia.</s>

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<s>

<arrow.to.target n="marg499"></arrow.to.target><lb/>quod ambabus manibus uis conduplicatur, & ma<lb/>

<arrow.to.target n="fig124"></arrow.to.target><lb/>ior redditur, quanta e&longs;t proportio totius ad exce&longs;<lb/>&longs;um: uelut &longs;it a quod mouetur ab una manu uiribus <lb/>ut b, quæ &longs;unt exce&longs;&longs;us b d &longs;upra a, cum ergo propor <lb/>tio c b d ad a &longs;it compo&longs;ita ex proportionibus c & <lb/>b d ad a manife&longs;tum e&longs;t, quod erit producta ex pro<lb/>portione c b d ad b d, & b d ad a, &longs;ed e b d e&longs;t dupla <lb/>ad b d, quia e e&longs;t æqualis, cigitur proportio c b d ad <lb/>

<figure id="fig124"></figure><lb/>ior redditur, quanta e&longs;t proportio totius ad exce&longs;<lb/>&longs;um: uelut &longs;it a quod mouetur ab una manu uiribus <lb/>ut b, quæ &longs;unt exce&longs;&longs;us b d &longs;upra a, cum ergo propor <lb/>tio c b d ad a &longs;it compo&longs;ita ex proportionibus c & <lb/>b d ad a manife&longs;tum e&longs;t, quod erit producta ex pro<lb/>portione c b d ad b d, & b d ad a, &longs;ed e b d e&longs;t dupla <lb/>ad b d, quia e e&longs;t æqualis, cigitur proportio c b d ad <lb/>

<arrow.to.target n="marg500"></arrow.to.target><lb/>a e&longs;t maior multo quàm duorum exce&longs;&longs;uum, qui mo <lb/>uerent in proportione dupla: uelut &longs;i adderemus f

<pb pagenum="145"/>ad d b æqualem b, multo maior e&longs;t ex communi animi &longs;ententia e f <lb/>b d <expan abbr="quã">quam</expan> f b d, quia e continet f, & quantum e&longs;t d in&longs;uper: cum ergo <lb/>b cum d moueat a in proportione b d ad a & f cum d mouebit a in <lb/>proportione eadem qua b d, ergo per uiam additionis duplo ue<lb/>locius, quàm dupla proportione, uerùm dupla comparatione ad <lb/>proportionem b d ad a, non autem duplicata &longs;ed dupla, ut dixi, qu&ecedil; <lb/>erit maior quàm dupla per <expan abbr="addition&etilde;">additionem</expan> exce&longs;&longs;us. Ergo &longs;i addatur al<lb/>ter homo, erit dupla ad illam duplam, ueluti addendo æqualem d b <lb/>f e, adeò ut &longs;i proportio d b f e e&longs;&longs;et quintupla, mouerent illi duo in <lb/>proportione decupla. Sed annexo baculo aut lima aut &longs;erra annu<lb/>lo h, ita ut circunuolui po&longs;sit h æquabit uires non &longs;olum d b f e &longs;ed <lb/>multorum hominum. igitur multo plus aget homo ambabus ma<lb/>nibus radendo aut &longs;ecando cum g, quàm quadrupla proportione <lb/>unius manus, & hocincrementum e&longs;t non &longs;olum magnæ <lb/>utilitatis, &longs;ed ualde <expan abbr="accõmodatum">accommodatum</expan> in actionibus artificum <lb/>operum grauiorum. Et huiu&longs;modi conduplicatio e&longs;t ratio <lb/>limæ quam &longs;urdam uocamus.</s>

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<arrow.to.target n="marg505"></arrow.to.target><lb/>a f ad g ita g ad b f ex &longs;uppo&longs;ito: & ut a f ad g, it a h a ad h b, ex &longs;uppo

<pb pagenum="146"/>&longs;ito igitur ut a h ad h b ita h b ad b l, &longs;ed angulus a h b e&longs;t æqualis <lb/>angulo h b l, ergo triangulus a h b e&longs;t <lb/>&longs;imilis triangulo h b l, quare angulus <lb/>b h l e&longs;t &ecedil;qualis angulo h a f, igitur du <lb/>orum triangulorum f a h, & fb h duo <lb/>

<arrow.to.target n="marg506"></arrow.to.target><lb/>anguli unius a & f &longs;unt æquales duo<lb/>bus angulis, alterius igitur propor<lb/>

<arrow.to.target n="fig125"></arrow.to.target><lb/>tio a f ad fh re&longs;picientium angulos &ecedil;<lb/>

<figure id="fig125"></figure><lb/>tio a f ad fh re&longs;picientium angulos &ecedil;<lb/>

<arrow.to.target n="marg507"></arrow.to.target><lb/>quales ut a h ad h b re&longs;picientium an<lb/>

<arrow.to.target n="marg508"></arrow.to.target><lb/>gulum f, &longs;ed a h ad h b ut c ad d, ex &longs;up <lb/>po&longs;ito igitur a f ad f h, ut c ad d, &longs;ed ut c ad d ita a f ad g, ex &longs;uppo&longs;ito <lb/>ergo h f e&longs;t æqualis g.<lb/>

<arrow.to.target n="marg509"></arrow.to.target></s>

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<s>Cum ergo h&ecedil;c demon&longs;tratio &longs;it ex &longs;en&longs;u in uno puncto h, ideò ad <lb/>quælibet puncta traduci pote&longs;t, quæ potero imaginari, & ita pri<lb/>ma uo cabitur &longs;en&longs;us, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demon&longs;tran<lb/>do non a&longs;&longs;umimus aliquid, quod &longs;it proprium alicui puncto, ni&longs;i <lb/>proportionem h a ad h b &longs;imilem e&longs;&longs;e c ad d, ideo hoc pertinet ad <lb/>intellectum, & e&longs;t tertium. Etidem dico &longs;i k e&longs;&longs;et ultra h quod po<lb/>te&longs;t contingere. modò k a ad k b &longs;it ut c ad d & k f &longs;it &ecedil;qualis g idem <lb/>&longs;equetur, & comprehenditur &longs;ub tertio & pertinet ad intellectum, <lb/>& quoniam demon&longs;tratur quod punctum k ubicun que &longs;umatur, e&longs;t <lb/>in &ecedil;quali <expan abbr="di&longs;tãtia">di&longs;tantia</expan> à puncto f&longs;cilicet per g lineam, erit &longs;emper in peri<lb/>pheria circuli, & hoc pote&longs;t e&longs;&longs;e in infinitis locis &longs;impliciter & extra <lb/>infinitum nihil e&longs;t, igitur &longs;ub hoc continetur conuer&longs;um &longs;cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a & b ip&longs;&ecedil; erunt in <lb/>proportione c ad d. Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">w_erila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. Et pote&longs;t demon&longs;trari Geometricè duobus uer<lb/>bis. Quia. n. <expan abbr="f&longs;upponi&ttilde;">f&longs;upponitur</expan> æqualis g eo quòd h e&longs;t in peripheria circu<lb/>li erit media inter a f & f b, quare cum angulus f &longs;it communis, erit <lb/>proportio a h ad h b, laterum re&longs;picientium angulum f in utroque </s>

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<s>Secunda regula, cum uolueris propo&longs;ito uno numero quadra<lb/>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb/>quemuis numerum quadratum per primum exemplum regul&ecedil; pri <lb/>mæ, & cum eo diuide numerum propo&longs;itum, & qui proueniet erit <lb/>quadratus, <expan abbr="h&utilde;c">hunc</expan> ergo duces in partes numeri quadrati qu&ecedil; &longs;unt nu<lb/>meri <expan abbr="&qtilde;drati">quadrati</expan>, & fient duo quadrati numeri, & illi <expan abbr="compon&etilde;t">component</expan> <expan abbr="numer&utilde;">numerum</expan> <lb/><expan abbr="quadrat&utilde;">quadratum</expan> <expan abbr="prior&etilde;">priorem</expan> quem diui&longs;i&longs;ti. quia multipli catio fit per <expan abbr="eo&longs;d&etilde;">eo&longs;dem</expan> nu<lb/>meros qui &longs;unt partes diui&longs;oris. Velut uolo facere de 4 duas partes <lb/>qu&ecedil; &longs;int <expan abbr="&qtilde;drati">quadrati</expan> numeri, capio <expan abbr="numer&utilde;">numerum</expan> <expan abbr="&qtilde;drat&utilde;">quadratum</expan> qui <expan abbr="cõpona&ttilde;">componatur</expan> ex duo<lb/>bus <expan abbr="&qtilde;dratis">quadratis</expan>, uelut 25, diuido 4 per 25 exit 4/25 <expan abbr="h&utilde;c">hunc</expan> duco per 9 & 16 <expan abbr="&qtilde;dra-tos">quadra<lb/>tos</expan> numeros <expan abbr="cõponentes">componentes</expan> 25 <expan abbr="fi&utilde;t">fiunt</expan> 1 11/25 & 2 14/25 <expan abbr="&qtilde;drati">quadrati</expan> 1 2/5 & 1 3/5 Et hi <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="cõponunt">componunt</expan> 4. Et ita po&longs;&longs;es diuidere infinitis modis, puta per 17 13/36 & <lb/>per 169. Tertia regula cum unus numerus additus <lb/>

<arrow.to.target n="fig126"></arrow.to.target><lb/>primo & detractis à <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> facit ambo quadrata, <expan abbr="id&etilde;">idem</expan> <lb/>numerus coniunctus cum differentia illorum nume<lb/>rorum & detractus à primo & additus &longs;ecundo facit <lb/>eo&longs;dem numeros quadratos, ueluti capio 10 primum <lb/>3 &longs;ecundum 6 additus ad 10 & detractus à 7 efficit 6 <lb/>& 1 quadratos dico quod iunctus 16 cum 3 differen<lb/>tia 10 & 7 fit 9, qui detractus à 10 & additus ad 7 effi<lb/>cit 1 & 16 numeros quadratos priores.</s>

<figure id="fig126"></figure><lb/>primo & detractis à <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> facit ambo quadrata, <expan abbr="id&etilde;">idem</expan> <lb/>numerus coniunctus cum differentia illorum nume<lb/>rorum & detractus à primo & additus &longs;ecundo facit <lb/>eo&longs;dem numeros quadratos, ueluti capio 10 primum <lb/>3 &longs;ecundum 6 additus ad 10 & detractus à 7 efficit 6 <lb/>& 1 quadratos dico quod iunctus 16 cum 3 differen<lb/>tia 10 & 7 fit 9, qui detractus à 10 & additus ad 7 effi<lb/>cit 1 & 16 numeros quadratos priores.</s>

<s>SCHOLIVM</s>

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<s>Quarta regula, <expan abbr="c&utilde;">cum</expan> uolueris <expan abbr="numer&utilde;">numerum</expan> aliquem non quad. qui bifa <lb/><expan abbr="riã">riam</expan> <expan abbr="compona&ttilde;">componatur</expan> ex duob. <expan abbr="&qtilde;d">quad</expan>. uelut 10 ex 25, & 25 & 49 & 1, <lb/>

<arrow.to.target n="fig127"></arrow.to.target><lb/>& <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir <expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&ecedil;&qtilde;lis">&ecedil;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cunf</expan> qua<lb/>drato &ecedil;qualia duob. <expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="c&utilde;">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. Maio<lb/>ra <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="dupl&utilde;">duplum</expan> quad. c d <expan abbr="c&utilde;">cum</expan> f qua<lb/>drato &ecedil;qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. Ergo propo&longs;ito numero, putà 3 ducam in &longs;e <lb/>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in &longs;e fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinqui&ttilde;">relinquitur</expan> <lb/>1 numerus <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dicã">dicam</expan> &qring;d 3 <expan abbr="c&utilde;">cum</expan> duplo 2, & erit <expan abbr="tot&utilde;">totum</expan> 7, e&longs;t unus <lb/>numerus, alter <02> 1. 1. 1, & <expan abbr="hor&utilde;">horum</expan> <expan abbr="&qtilde;d">quad</expan>. <expan abbr="cõponunt">componunt</expan> 50, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 5. Et &longs;imi <lb/>liter capio 6 <expan abbr="&qtilde;d">quad</expan>. 36 <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 4. 32 differentia 4, numerus <expan abbr="&qtilde;d">quad</expan>. 2, ideo <lb/>6 <expan abbr="c&utilde;">cum</expan> duplo 4, & e&longs;t 14, e&longs;t unus numerus, alter 2, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;d">quad</expan>. &longs;unt 200, <lb/><expan abbr="dimidi&utilde;">dimidium</expan> e&longs;t 100 <expan abbr="&qtilde;d">quad</expan>. 10 <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> ex 6 & 4. Et ita capio 9, <expan abbr="&qtilde;d">quad</expan>. eius 81 du <lb/><expan abbr="pl&utilde;">plum</expan> <expan abbr="&qtilde;d">quad</expan>. 6. 72 differentia 9 numerus <expan abbr="&qtilde;d">quad</expan>. <expan abbr="igi&ttilde;">igitur</expan> cum duplo 6, & e&longs;t 21, e&longs;t <lb/>unus <expan abbr="illor&utilde;">illorum</expan>, alter 3 <expan abbr="&qtilde;d">quad</expan>. 450, <expan abbr="dupl&utilde;">duplum</expan> 225 <expan abbr="&qtilde;d">quad</expan>. 15, qui con&longs;tat ex 9 & 6. Et <lb/>ita capio 11 <expan abbr="&qtilde;d">quad</expan>. cuius e&longs;t 121, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 6 e&longs;t 72 differentia, 72 & 21 e&longs;t <lb/>49 numerus <expan abbr="&qtilde;d">quad</expan>. 7, <expan abbr="igi&ttilde;">igitur</expan> 23 qui con&longs;tat ex 11, & duplo 6 numeri mino <lb/>ris e&longs;t unus numerus, alter e&longs;t 7 <expan abbr="&qtilde;d">quad</expan>. <expan abbr="quor&utilde;">quorum</expan> &longs;unt 578. <expan abbr="dupl&utilde;">duplum</expan> 289, <expan abbr="&qtilde;d">quad</expan>. <lb/>17, qui con&longs;tat ex 11 & 6. Quinta regula, per hoc inueniemus infini <lb/>tos numeros <expan abbr="&qtilde;d">quad</expan>. <expan abbr="cõponentes">componentes</expan> 32, nam <expan abbr="c&utilde;">cum</expan> 32 &longs;it duplus <expan abbr="&qtilde;d">quad</expan>. <expan abbr="diuidã">diuidam</expan> per<lb/>unum <expan abbr="aggregat&utilde;">aggregatum</expan> ex inuentis puta 578, & quia ambo ex &longs;uppo&longs;ito <lb/>&longs;unt dupli ad <expan abbr="&qtilde;d">quad</expan>. qui proueniet erit <expan abbr="&qtilde;d">quad</expan>. &longs;cilicet 16/289, duc in numeros <expan abbr="&qtilde;-dratos">qua<lb/>dratos</expan> qui componunt 578, & &longs;unt 529 & 49, & fient 2 206/289 & 29 83/289, <lb/>& hi iuncti <expan abbr="fi&utilde;t">fiunt</expan> 32, quia &longs;unt multiplicatæ partes numeri, per quem <lb/>e&longs;t <gap/>iui&longs;us numerus. Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s>

<figure id="fig127"></figure><lb/>& <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir <expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&ecedil;&qtilde;lis">&ecedil;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cunf</expan> qua<lb/>drato &ecedil;qualia duob. <expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="c&utilde;">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. Maio<lb/>ra <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="dupl&utilde;">duplum</expan> quad. c d <expan abbr="c&utilde;">cum</expan> f qua<lb/>drato &ecedil;qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. Ergo propo&longs;ito numero, putà 3 ducam in &longs;e <lb/>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in &longs;e fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinqui&ttilde;">relinquitur</expan> <lb/>1 numerus <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dicã">dicam</expan> &qring;d 3 <expan abbr="c&utilde;">cum</expan> duplo 2, & erit <expan abbr="tot&utilde;">totum</expan> 7, e&longs;t unus <lb/>numerus, alter <02> 1. 1. 1, & <expan abbr="hor&utilde;">horum</expan> <expan abbr="&qtilde;d">quad</expan>. <expan abbr="cõponunt">componunt</expan> 50, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 5. Et &longs;imi <lb/>liter capio 6 <expan abbr="&qtilde;d">quad</expan>. 36 <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 4. 32 differentia 4, numerus <expan abbr="&qtilde;d">quad</expan>. 2, ideo <lb/>6 <expan abbr="c&utilde;">cum</expan> duplo 4, & e&longs;t 14, e&longs;t unus numerus, alter 2, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;d">quad</expan>. &longs;unt 200, <lb/><expan abbr="dimidi&utilde;">dimidium</expan> e&longs;t 100 <expan abbr="&qtilde;d">quad</expan>. 10 <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> ex 6 & 4. Et ita capio 9, <expan abbr="&qtilde;d">quad</expan>. eius 81 du <lb/><expan abbr="pl&utilde;">plum</expan> <expan abbr="&qtilde;d">quad</expan>. 6. 72 differentia 9 numerus <expan abbr="&qtilde;d">quad</expan>. <expan abbr="igi&ttilde;">igitur</expan> cum duplo 6, & e&longs;t 21, e&longs;t <lb/>unus <expan abbr="illor&utilde;">illorum</expan>, alter 3 <expan abbr="&qtilde;d">quad</expan>. 450, <expan abbr="dupl&utilde;">duplum</expan> 225 <expan abbr="&qtilde;d">quad</expan>. 15, qui con&longs;tat ex 9 & 6. Et <lb/>ita capio 11 <expan abbr="&qtilde;d">quad</expan>. cuius e&longs;t 121, <expan abbr="dupl&utilde;">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. 6 e&longs;t 72 differentia, 72 & 21 e&longs;t <lb/>49 numerus <expan abbr="&qtilde;d">quad</expan>. 7, <expan abbr="igi&ttilde;">igitur</expan> 23 qui con&longs;tat ex 11, & duplo 6 numeri mino <lb/>ris e&longs;t unus numerus, alter e&longs;t 7 <expan abbr="&qtilde;d">quad</expan>. <expan abbr="quor&utilde;">quorum</expan> &longs;unt 578. <expan abbr="dupl&utilde;">duplum</expan> 289, <expan abbr="&qtilde;d">quad</expan>. <lb/>17, qui con&longs;tat ex 11 & 6. Quinta regula, per hoc inueniemus infini <lb/>tos numeros <expan abbr="&qtilde;d">quad</expan>. <expan abbr="cõponentes">componentes</expan> 32, nam <expan abbr="c&utilde;">cum</expan> 32 &longs;it duplus <expan abbr="&qtilde;d">quad</expan>. <expan abbr="diuidã">diuidam</expan> per<lb/>unum <expan abbr="aggregat&utilde;">aggregatum</expan> ex inuentis puta 578, & quia ambo ex &longs;uppo&longs;ito <lb/>&longs;unt dupli ad <expan abbr="&qtilde;d">quad</expan>. qui proueniet erit <expan abbr="&qtilde;d">quad</expan>. &longs;cilicet 16/289, duc in numeros <expan abbr="&qtilde;-dratos">qua<lb/>dratos</expan> qui componunt 578, & &longs;unt 529 & 49, & fient 2 206/289 & 29 83/289, <lb/>& hi iuncti <expan abbr="fi&utilde;t">fiunt</expan> 32, quia &longs;unt multiplicatæ partes numeri, per quem <lb/>e&longs;t <gap/>iui&longs;us numerus. Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s>

<s>Sexta regula, ponamus modò &qring;d uelim diuidere 10, <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. <expan abbr="&qtilde;d">quad</expan>. 9 & 1, & non <expan abbr="dupl&utilde;">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. ita &qring;d &longs;it diui&longs;us in alios <lb/>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex duob. <expan abbr="&qtilde;d">quad</expan>. fit 250/25, at 250 <expan abbr="cõponi&ttilde;">componitur</expan> aliter <lb/>ex duob. quad. <08> 225/25 & 25/25, &longs;cilicet 169/25 & 81/25, id e&longs;t 6 19/25 & 3 6/25, qui &longs;unt <expan abbr="&qtilde;d">quad</expan>. <lb/>2 3/5 & 1 4/5, & ita uolo diuidere 13 in duo alia <expan abbr="&qtilde;drata">quadrata</expan> <08> 9 & 4, duco 13 in <lb/>25 & fit 325/25, qui nece&longs;&longs;ario <expan abbr="cõponi&ttilde;">componitur</expan> ex 225/25 & 100/25, &longs;ed ego uolo &qring;d <expan abbr="cõpo">compo</expan> <lb/><expan abbr="na&ttilde;">natur</expan> aliter, uelut ex 289/25 & 63/25, & ita ex 11 14/25 & 1 11/25, qui &longs;unt numeri <expan abbr="&qtilde;d">quad</expan>. com <lb/>ponentes 13, & <02> &longs;unt 3 2/5 & 1 1/5, & in his opus e&longs;t in du&longs;tria, &longs;cilicet ut <lb/><expan abbr="multiplice&ttilde;">multiplicetur</expan> per numeros <expan abbr="&qtilde;d">quad</expan>. ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> compo <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. Vt uerò uideamus <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, proponamus quae uelim diui <lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="prim&utilde;">primum</expan> &longs;cire debes &qring;d non po&longs;&longs;unt e&longs;&longs;e

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<s>Contingit quando que &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem <lb/>

<arrow.to.target n="fig128"></arrow.to.target><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aunt</expan> maius efficere: id <lb/>duob. modis po&longs;&longs;umus, <expan abbr="quor&utilde;">quorum</expan> unus diffici<lb/>lior e&longs;t &longs;ed perpetuus, & longè nobilior, nam <lb/>grauitas ponderis uer&longs;atilis efficit <expan abbr="quid&etilde;">quidem</expan> <expan abbr="tar-dior&etilde;">tar<lb/>diorem</expan>, &longs;ed di fficilius <expan abbr="mobil&etilde;">mobilem</expan>, & ob id grauio<lb/>re <expan abbr="põdere">pondere</expan> in <expan abbr="digent&etilde;">digentem</expan>. Sit ergo rota a b uer&longs;ati<lb/>lis, quæ certam men&longs;uram exigit pro quacunque funis parte corre&longs;peron <lb/>dentis uni denti ex centum, in quos di&longs;tincta &longs;it, curriculum <expan abbr="a&utilde;t">aunt</expan> c d <lb/>quinque <expan abbr="denti&utilde;">dentium</expan>, per &qring;drota &longs;exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="cõuer&longs;ione">conuer&longs;ione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim&ecedil; rot&ecedil; uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rur&longs;us ad <expan abbr="hãc">hanc</expan> <expan abbr="&longs;ecundã">&longs;ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo &longs;ex <expan abbr="denti&utilde;">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> &longs;eptuaginta duo, ut in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> &longs;int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> prim&ecedil; rot&ecedil; circumuoluentur. Iam <lb/>uerò tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditat&etilde;">tarditatem</expan> tem <lb/>poris uer&longs;atilis: <expan abbr="quãto">quanto</expan> <expan abbr="igi&ttilde;">igitur</expan> pondero&longs;ius fuerit illud <expan abbr="t&etilde;pus">tempus</expan>, tanto tar<lb/>dius <expan abbr="mouebi&ttilde;">mouebitur</expan>, pauciores que circumuolutiones nece&longs;&longs;ari&ecedil; <expan abbr="er&utilde;t">erunt</expan> ad <expan abbr="ex-pl&etilde;dam">ex<lb/>plendam</expan> unam <expan abbr="di&etilde;">diem</expan>: id e&longs;t horas 24, &longs;ed hoc in <expan abbr="cõmodi">commodi</expan> accedet, quòd <lb/>reuolutio indicis tanto tardior erit, ut <expan abbr="nõ">non</expan> iu&longs;tè o&longs;ten dat horas: pro

<figure id="fig128"></figure><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aunt</expan> maius efficere: id <lb/>duob. modis po&longs;&longs;umus, <expan abbr="quor&utilde;">quorum</expan> unus diffici<lb/>lior e&longs;t &longs;ed perpetuus, & longè nobilior, nam <lb/>grauitas ponderis uer&longs;atilis efficit <expan abbr="quid&etilde;">quidem</expan> <expan abbr="tar-dior&etilde;">tar<lb/>diorem</expan>, &longs;ed di fficilius <expan abbr="mobil&etilde;">mobilem</expan>, & ob id grauio<lb/>re <expan abbr="põdere">pondere</expan> in <expan abbr="digent&etilde;">digentem</expan>. Sit ergo rota a b uer&longs;ati<lb/>lis, quæ certam men&longs;uram exigit pro quacunque funis parte corre&longs;peron <lb/>dentis uni denti ex centum, in quos di&longs;tincta &longs;it, curriculum <expan abbr="a&utilde;t">aunt</expan> c d <lb/>quinque <expan abbr="denti&utilde;">dentium</expan>, per &qring;drota &longs;exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="cõuer&longs;ione">conuer&longs;ione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim&ecedil; rot&ecedil; uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rur&longs;us ad <expan abbr="hãc">hanc</expan> <expan abbr="&longs;ecundã">&longs;ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo &longs;ex <expan abbr="denti&utilde;">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> &longs;eptuaginta duo, ut in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> &longs;int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> prim&ecedil; rot&ecedil; circumuoluentur. Iam <lb/>uerò tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditat&etilde;">tarditatem</expan> tem <lb/>poris uer&longs;atilis: <expan abbr="quãto">quanto</expan> <expan abbr="igi&ttilde;">igitur</expan> pondero&longs;ius fuerit illud <expan abbr="t&etilde;pus">tempus</expan>, tanto tar<lb/>dius <expan abbr="mouebi&ttilde;">mouebitur</expan>, pauciores que circumuolutiones nece&longs;&longs;ari&ecedil; <expan abbr="er&utilde;t">erunt</expan> ad <expan abbr="ex-pl&etilde;dam">ex<lb/>plendam</expan> unam <expan abbr="di&etilde;">diem</expan>: id e&longs;t horas 24, &longs;ed hoc in <expan abbr="cõmodi">commodi</expan> accedet, quòd <lb/>reuolutio indicis tanto tardior erit, ut <expan abbr="nõ">non</expan> iu&longs;tè o&longs;ten dat horas: pro

<pb pagenum="153"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aunt</expan>, & qu&ecedil; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. Ponamus ergo po&longs;t<08> eadem e&longs;t ratio celerioris & <lb/>æqué uelocis, ponderis <expan abbr="a&utilde;t">aunt</expan> tardius de&longs;cendentis, aut <expan abbr="cõtrà">contrà</expan> tardio<lb/>ris, aut æqualiter cir cumducti in dicis, celerioris <expan abbr="a&utilde;t">aunt</expan> de&longs;cen&longs;us pon<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. Sit ergo ut pon <lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aunt</expan> &ecedil;qualiter circumferri, dico <lb/>quod ex tempore mobili &longs;eu uer&longs;atili (& e&longs;t ferrum, quod in &longs;um<lb/>mo horologij citra ultraque <expan abbr="fer&ttilde;">fertur</expan> tam in horologijs ponderum <08> mo <lb/>læ) id fieri non pote&longs;t: nam quantum tardabitur rota tertia &longs;ecunda <lb/>& prima, atque ob id de&longs;cen&longs;us ponderum, tantum remorabitur rota <lb/>prima quæ indicem o&longs;tendit, ergo tantum index tardabitur quan<lb/>rum <expan abbr="põdera">pondera</expan>, & ut uno uerbo dicam, cùm <expan abbr="ead&etilde;">eadem</expan> rota index circumfe<lb/>ratur, & <expan abbr="põdus">pondus</expan> de&longs;cendat, <expan abbr="quant&utilde;">quantum</expan> unum tardatur tantum & aliud.</s>

<s>Secundus modus e&longs;t, ut rota una totum tempus cum indice in ui <lb/>gintiquatuor horis circumuoluatur, & currulis in quo funis minor <lb/>fiat: nece&longs;&longs;e e&longs;t <expan abbr="igi&ttilde;">igitur</expan>, ut circumuoluta rota aut &longs;emel aut bis, <expan abbr="&ttilde;er">turer</expan>, qua<lb/>ter decies, & <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno cir cuitu index, et &longs;ine errore: quo<lb/>niam tempus & dentes men&longs;uræ re&longs;pondent: igitur &longs;ub ei&longs;dem cir<lb/>cuitibus numero eodemque tempore minus ex fune <expan abbr="de&longs;cend&etilde;t">de&longs;cendent</expan> in cur <lb/>ruli paruo <08> magno: quare mutatione indiget currulis, aut ut funis <lb/>circumuoluens rotam curriculum habeat <expan abbr="annex&utilde;">annexum</expan> rotæ o&longs;ten denti <lb/>horas, in qua pauciores &longs;int dentes: nam in eodem tempore, & cir<lb/>cuitu paucioribus uicibus circumuoluitur rota funis quæ grauita<lb/>te temporis, & multitudine <expan abbr="denti&utilde;">dentium</expan> certam <lb/>

<arrow.to.target n="fig129"></arrow.to.target><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. Sed in hoc nece&longs;&longs;e e&longs;t gra <lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quo-niã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tnm</expan> tardè <08> &longs;it pro paruitatis circuitus ratione.</s>

<figure id="fig129"></figure><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. Sed in hoc nece&longs;&longs;e e&longs;t gra <lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quo-niã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tnm</expan> tardè <08> &longs;it pro paruitatis circuitus ratione.</s>

<s>Tertius modus facilior e&longs;t, & magis com <lb/><expan abbr="p&etilde;dio&longs;us">pendio&longs;us</expan>: Sit horologium a b c, in quo rota <lb/>d quæ funem <expan abbr="cõtinet">continet</expan> ba&longs;is horologij e f, cui <lb/>firmiter &longs;int <expan abbr="app&etilde;&longs;&ecedil;">appen&longs;&ecedil;</expan> du&ecedil; trochle&ecedil; g & h, & fu <lb/>nis una parte tro chle&ecedil; appen&longs;us in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam tro chleam lin&longs;eraturque<lb/>ibi orbiculo &longs;uo, & redeat à dextra &longs;uperius <lb/><expan abbr="in&longs;era&ttilde;que">in&longs;eraturque</expan> orbiculo &longs;uperioris tro chle&ecedil;, dedu <lb/><expan abbr="ca&ttilde;que">caturque</expan> uer&longs;us <expan abbr="&longs;ini&longs;trã">&longs;ini&longs;tram</expan>: atque ibi <expan abbr="de&longs;cend&etilde;s">de&longs;cendens</expan> habe <lb/>at <expan abbr="põdus">pondus</expan> tractorium in m, <expan abbr="deduca&ttilde;que">deducaturque</expan> &longs;upra <lb/>ad <expan abbr="rotã">rotam</expan> horologij d, et cir cumuolutus exeat <lb/>ip&longs;um, & <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> ad tro <expan abbr="chleãn">chleann</expan>, &longs;ub que ea circumuolutus <expan abbr="iter&utilde;">iterum</expan> a&longs;cen

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Line 7690

<s>Sunt horum duo genera primum, & anti <lb/>

<arrow.to.target n="fig130"></arrow.to.target><lb/>quius licet multo po&longs;terius eo quod pon<lb/>deribus ducitur, quod funiculo ex inte&longs;ti<lb/>nis ouium &longs;eu fidibus liræ agitur. Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;piræ in fine, cu <lb/>ius cannectatur ferreo circulo, qui habeatur lo co cap&longs;ulæ b c, quæ <lb/>circumuolui po&longs;sit: huic <expan abbr="circ&utilde;ductus">circunductus</expan> funis d e multipliciter in pun <lb/>cto g, &longs;it autem e h in modum pyramidis &longs;en&longs;im in acutum, &longs;ed non <lb/>ualde per <expan abbr="&longs;pirã">&longs;piram</expan> exculptam de&longs;inentis, cui rota in uertice in&longs;erta den <lb/>&longs;iculo, & uertatur h e, colligens funiculum tractum in &longs;pira uer&longs;us <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ulã">cap&longs;ulam</expan> uer&longs;us c, traher <lb/>ergo molam, & con&longs;trin get uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb/>quæ circumuolui pote&longs;t a b e ad h: & cum trahitur in d eremittitur, <lb/>non pote&longs;t mola &longs;tatim retrahere reluctantibus denticulis h l rotæ, <lb/>& alijs quæ implicantur curriculo m, a igitur mola con&longs;tructa uio<lb/>lenter mouet b g d, cap&longs;ulam motu contrario à c in d & in g & in b, <lb/>quare funis d e trahitur, & trahit e h illum circumuoluendo contra<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rotã">rotam</expan>, & &longs;ic deinceps donec tempus moueatur, & rota indicis. <lb/>Hic ade&longs;t cap&longs;ula, & quod circumuertitur à claue non e&longs;t axis mol&ecedil; <lb/>&longs;ed extra molam, &longs;cilicet e h. Et quoniam hac ratione quanto mola a

<figure id="fig130"></figure><lb/>quius licet multo po&longs;terius eo quod pon<lb/>deribus ducitur, quod funiculo ex inte&longs;ti<lb/>nis ouium &longs;eu fidibus liræ agitur. Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;piræ in fine, cu <lb/>ius cannectatur ferreo circulo, qui habeatur lo co cap&longs;ulæ b c, quæ <lb/>circumuolui po&longs;sit: huic <expan abbr="circ&utilde;ductus">circunductus</expan> funis d e multipliciter in pun <lb/>cto g, &longs;it autem e h in modum pyramidis &longs;en&longs;im in acutum, &longs;ed non <lb/>ualde per <expan abbr="&longs;pirã">&longs;piram</expan> exculptam de&longs;inentis, cui rota in uertice in&longs;erta den <lb/>&longs;iculo, & uertatur h e, colligens funiculum tractum in &longs;pira uer&longs;us <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ulã">cap&longs;ulam</expan> uer&longs;us c, traher <lb/>ergo molam, & con&longs;trin get uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb/>quæ circumuolui pote&longs;t a b e ad h: & cum trahitur in d eremittitur, <lb/>non pote&longs;t mola &longs;tatim retrahere reluctantibus denticulis h l rotæ, <lb/>& alijs quæ implicantur curriculo m, a igitur mola con&longs;tructa uio<lb/>lenter mouet b g d, cap&longs;ulam motu contrario à c in d & in g & in b, <lb/>quare funis d e trahitur, & trahit e h illum circumuoluendo contra<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rotã">rotam</expan>, & &longs;ic deinceps donec tempus moueatur, & rota indicis. <lb/>Hic ade&longs;t cap&longs;ula, & quod circumuertitur à claue non e&longs;t axis mol&ecedil; <lb/>&longs;ed extra molam, &longs;cilicet e h. Et quoniam hac ratione quanto mola a

<pb pagenum="155"/>magis <expan abbr="explicabi&ttilde;">explicabitur</expan>, tanto lentius trahet, & uertet e h, ideò hoc ex &longs;tru <lb/>ctura auxilium præ&longs;tatur, ut funis in inferiore parte <expan abbr="cõplexus">complexus</expan> latio<lb/>res orbes, & è regione tanto uehementius uertat e h: & ita uis quæ <lb/>remittitur ob molæ laxitatem, augetur tantundem ob &longs;itum & ma<lb/>gnitudinem &longs;pirarum ut di&longs;tantiorum &longs;ua extremitate ab hypomo <lb/>chlio, quod e&longs;t axis coni e h, &longs;eu in&longs;tar axis.</s>

<s>Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&ecedil; <lb/>habet <expan abbr="rotã">rotam</expan> plano &longs;ub &longs;tratam, plenam denticulis axis, quo circum<lb/>agitur uiolenter, non e&longs;t extra molam, &longs;ed ei annexa e&longs;t mola intus, <lb/>exterius <expan abbr="a&utilde;t">aunt</expan> rot&ecedil;; ergo circumducto axe mol&ecedil; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. Vbi mola quan<lb/>tum decet con&longs;tricta e&longs;t &longs;ublato clauo &longs;tatim &longs;ecum trahit rotam, & <lb/>illa <expan abbr="curricul&utilde;">curriculum</expan> rotas que alias, & tempus agitur, & index uertitur. Sed <lb/>in hoc idem e&longs;t in commodum &longs;ine remedio <lb/>

<arrow.to.target n="fig131"></arrow.to.target><lb/>quod fuit in priore. Vbi enim c&oelig;perit laxa<lb/>ri mola tanto tardius progrediuntur rotæ <lb/>atque index. Veluti axis a b cui &longs;ecun dum lon <lb/>gitudinem molæ caput interius annexum <lb/>e&longs;t altero circulo rotæ in c d curriculum rotæ e, implexum rotæ f <lb/>clauus rotam retinens, donec circumducto a b mola con&longs;tringa<lb/>tur, & latus eius trahat rotam ex c. Inde &longs;ublato clauo circulus, &longs;eu <lb/>rota trahitur ex c in g, & in famola, quæ etiam &longs;ecundum eandem <lb/>partem circumuoluta e&longs;t: igitur d circumagetur à rota & reliqua. <lb/>Sed ut dixi con&longs;tructio hæc non &longs;atisfacit.</s>

<figure id="fig131"></figure><lb/>quod fuit in priore. Vbi enim c&oelig;perit laxa<lb/>ri mola tanto tardius progrediuntur rotæ <lb/>atque index. Veluti axis a b cui &longs;ecun dum lon <lb/>gitudinem molæ caput interius annexum <lb/>e&longs;t altero circulo rotæ in c d curriculum rotæ e, implexum rotæ f <lb/>clauus rotam retinens, donec circumducto a b mola con&longs;tringa<lb/>tur, & latus eius trahat rotam ex c. Inde &longs;ublato clauo circulus, &longs;eu <lb/>rota trahitur ex c in g, & in famola, quæ etiam &longs;ecundum eandem <lb/>partem circumuoluta e&longs;t: igitur d circumagetur à rota & reliqua. <lb/>Sed ut dixi con&longs;tructio hæc non &longs;atisfacit.</s>

<s>Aliam ergo oportuit excogitare qu&ecedil; huiu&longs;modi e&longs;t. Sub axe a b, <lb/>qui cir cumuertitur ad molam contrahendam rotam, collocant par <lb/>uam quæ e&longs;t, ut ita dicam, pars axis ima cui in&longs;eruntur dentes in am <lb/>bitu ea ratione, ut dum mola ten ditur, premant denticulos interio<lb/>res, atque ita elabitur, totiesque circumducitur manente g f, donec <lb/>colligatur mola, quæ non ut in priore reliquo extremo ulli rotæ <lb/>affixa e&longs;t, &longs;ed columnæ in continenti <lb/>opercula horologij. Cum ergo mola <lb/>tenta retrahat axem a b contrario mo<lb/>

<arrow.to.target n="fig132"></arrow.to.target><lb/>tu, & ille rotam mobilem, quæ cum <lb/>non po&longs;sit regredi propter auer&longs;os <lb/>dentes, mouet rotam f g contrario mo <lb/>tu, quæ circumacta per denticulos &longs;u<lb/>os curriculum agit, & reliqua omnia <lb/>nece&longs;&longs;aria. Cur autem cum laxatur mo <lb/>la, & uertit lentius c e rotam coniun<lb/>ctam, ideoque g f, & reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, & circumuo

<figure id="fig132"></figure><lb/>tu, & ille rotam mobilem, quæ cum <lb/>non po&longs;sit regredi propter auer&longs;os <lb/>dentes, mouet rotam f g contrario mo <lb/>tu, quæ circumacta per denticulos &longs;u<lb/>os curriculum agit, & reliqua omnia <lb/>nece&longs;&longs;aria. Cur autem cum laxatur mo <lb/>la, & uertit lentius c e rotam coniun<lb/>ctam, ideoque g f, & reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, & circumuo

<pb pagenum="156"/>lutio indicis cau&longs;a e&longs;t alia longè quàm in priore, nam mola longior <lb/>fit cra&longs;sior, & durior adeoque robu&longs;ta, & rotæ leues, ac tempus dum <lb/>laxata fuerit munus &longs;uum iu&longs;to in tempore obeant: quare nece&longs;&longs;e <lb/>e&longs;t, ut ab initio uehementius agat, & celerius rotam cum axe qui <gap/><lb/>hitur à mola. Ergo excogitarunt aliud genus retinaculi forma <gap/>o<lb/>chleæ quod ab initio moratur <expan abbr="uehem&etilde;ter">uehementer</expan> axem ne circumagatur, et <lb/>quanto magis mola explicatur eo minus retinet <expan abbr="impet&utilde;">impetum</expan> illius <gap/>deo <lb/>ut uehementer retineat uehementem concitationem medio criter <lb/>moderatam, &longs;egniter lentam, nullo modo iu&longs;tam: ita fit, ut &longs;emper <lb/>fermè æqualiter moueatur. Difficile e&longs;t tamen ad unguem &longs;eruare <lb/>moderationem, & æqualitatem, & magis etiam in his horologijs, <lb/>quæ uno circuitu molæ tempus <expan abbr="lõgius">longius</expan> exigunt: at difficilius etiam <lb/>efficere molam, quæ longo tempore duret, cum intenta ualde cele<lb/>rius moueat rotas, & ob id breui ab&longs;oluat circuitum, mollior au<lb/>tem citò remittatur. Et ob id longior & non adeò <lb/>dura melior e&longs;t. Ratio autem cochleæ ita &longs;e habet. <lb/>

<arrow.to.target n="fig133"></arrow.to.target><lb/>Circa axem molæ d deducitur cochlea a b c, quæ <lb/>dum laxatur mola cochlea mouetur ex b in c, at que<lb/>ita pariter laxatur uis cochleæ retinentis axem.</s>

<figure id="fig133"></figure><lb/>Circa axem molæ d deducitur cochlea a b c, quæ <lb/>dum laxatur mola cochlea mouetur ex b in c, at que<lb/>ita pariter laxatur uis cochleæ retinentis axem.</s>

<s>Propo&longs;itio cente&longs;imaquinquage&longs;imaoctaua.</s>

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<s>Hoc fieri pote&longs;t in &longs;ingulo genere horologij trium <expan abbr="de&longs;criptor&utilde;">de&longs;criptorum</expan>. <lb/>Propterea &longs;ufficiat de uno o&longs;tendi&longs;&longs;e. Sed & in &longs;ingulo genere &longs;unt <lb/>multi modi, unius tamen reddidi&longs;&longs;e <expan abbr="ration&etilde;">rationem</expan> &longs;ufficiat. Hoc <expan abbr="a&utilde;t">aunt</expan> qua<lb/>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb/>indice: &longs;ecunda ut conuer&longs;o indice conuertatur, & rota ictuum: ter <lb/>tia ut ictuum numerus cum numero indicis conueniat. Vnde mul<lb/>ta &longs;unt horologia, in quibus ictus unus &longs;olum auditur &longs;ingulis ho<lb/>ris, atque hic modus facilis e&longs;t: quarta cur in horum pleri&longs; que &longs;i non <lb/>pul&longs;ata &longs;tatim hora <expan abbr="transfera&ttilde;ur">transferaturur</expan> index, non ce&longs;&longs;at pul&longs;atio: imò nec <lb/>retineri pote&longs;t, donec pondus illud de&longs;cenderit. Ergo primi & ter<lb/>tij ratio hæc habeatur, cum rota qu&ecedil; indicis rotam circumagit, per<lb/>uenerit ad horæ finem, denticulo &longs;oluit aliam, eleuans obicem, illa <lb/>mouetur à pondere proprio alio, &longs;cilicet ab illo quod tempus agit: <lb/>aut &longs;i &longs;it horologium molæ à mola alia propria, quæ malleos cir<lb/>cumacta perpetuò mouet, atque motura e&longs;&longs;et &longs;emper, donec pondus <lb/>ad terram de&longs;cenderet: uerum dum mouetur de&longs;cendit ferrum pro <lb/>quouis ictu quod in rotæ limbum incidit, & donec inciderit in eam <lb/>partem quæ lenis e&longs;t dilabitur, nec retinetur, & ita eleuatur rur&longs;us,

<pb pagenum="157"/>at uero cum in concauam partem incidit retineri nece&longs;&longs;e e&longs;t: atque ita <lb/>pondus non amplius de&longs;cendit, rota &longs;i&longs;titur, malleus manet immo<lb/>bilis: &longs;patia ergo quæ &longs;unt inter cauitates &longs;unt &longs;ecundum magnitu<lb/>dinem proportionis numerórum <expan abbr="horar&utilde;">horarum</expan>, uel ad &longs;ex, uel ad duode<lb/>cim, uel ad uiginti<lb/>

<arrow.to.target n="fig134"></arrow.to.target><lb/>quatuor terminan<lb/>tium. Ita quod, gra<lb/>tia exempli, &longs;it iam <lb/>in cauitate a duode<lb/>cim&ecedil; horæ uncus, di <lb/>uidam circulum to<lb/>tum in duas partes <lb/>æquales, quia in &longs;in <lb/>gulis medietatibus <lb/>propo&longs;itum e&longs;t, duo <lb/>decim facere cauita<lb/>tes pro unco retinen<lb/>do. Et quia in una<lb/>quaque medietate o<lb/>portet, ut pul&longs;ent ho <lb/>ræ lxxviij, & præterea &longs;int ibi &longs;ex &longs;patia cauitatum, quarum &longs;ingulæ <lb/>contineant, gratia exempli, duo &longs;patia unius ictus, ut certius retinea <lb/>tur uncus, <expan abbr="er&utilde;t">erunt</expan> igitur &longs;patia omnia nonaginta: diuidemus ergo me<lb/>dietatem circuli utranque in nonaginta partes æquales in cipiendo <lb/>ab a, & dabimus b primæ hor&ecedil; quod &longs;patium e&longs;t unius tantum par <lb/>tis ex nonaginta, po&longs;t de&longs;cribemus c cauitatem duarum partium, <lb/>ita ubi ictum unum dederit uncus, retinebitur in c, pò&longs;t accipiemus <lb/>duo &longs;patia, & &longs;int &longs;ignificata d litera, po&longs;t qu&ecedil; faciemus cauitatem e: <lb/>& ita uncus bis cadet in d, & pul&longs;abunt duo ictus, & pò&longs;t retinebi<lb/>tur uncus in e. Et po&longs;t accipiam &longs;patium trium partium, quod &longs;it f, <lb/>& po&longs;t de&longs;cribam cauitatem g duarum partium, atque ita procedam <lb/>u&longs;que ad duodecim.</s>

<figure id="fig134"></figure><lb/>quatuor terminan<lb/>tium. Ita quod, gra<lb/>tia exempli, &longs;it iam <lb/>in cauitate a duode<lb/>cim&ecedil; horæ uncus, di <lb/>uidam circulum to<lb/>tum in duas partes <lb/>æquales, quia in &longs;in <lb/>gulis medietatibus <lb/>propo&longs;itum e&longs;t, duo <lb/>decim facere cauita<lb/>tes pro unco retinen<lb/>do. Et quia in una<lb/>quaque medietate o<lb/>portet, ut pul&longs;ent ho <lb/>ræ lxxviij, & præterea &longs;int ibi &longs;ex &longs;patia cauitatum, quarum &longs;ingulæ <lb/>contineant, gratia exempli, duo &longs;patia unius ictus, ut certius retinea <lb/>tur uncus, <expan abbr="er&utilde;t">erunt</expan> igitur &longs;patia omnia nonaginta: diuidemus ergo me<lb/>dietatem circuli utranque in nonaginta partes æquales in cipiendo <lb/>ab a, & dabimus b primæ hor&ecedil; quod &longs;patium e&longs;t unius tantum par <lb/>tis ex nonaginta, po&longs;t de&longs;cribemus c cauitatem duarum partium, <lb/>ita ubi ictum unum dederit uncus, retinebitur in c, pò&longs;t accipiemus <lb/>duo &longs;patia, & &longs;int &longs;ignificata d litera, po&longs;t qu&ecedil; faciemus cauitatem e: <lb/>& ita uncus bis cadet in d, & pul&longs;abunt duo ictus, & pò&longs;t retinebi<lb/>tur uncus in e. Et po&longs;t accipiam &longs;patium trium partium, quod &longs;it f, <lb/>& po&longs;t de&longs;cribam cauitatem g duarum partium, atque ita procedam <lb/>u&longs;que ad duodecim.</s>

<s>Ex quo manife&longs;tum e&longs;t pondus quod agit rotam uolæ non de</s>

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<s>Sit angulus a & circulus b c, dico non po&longs;&longs;e aliquem angulum <lb/>

<arrow.to.target n="marg527"></arrow.to.target><lb/>contentum recta & circuli portione e&longs;&longs;e illi <lb/>

<arrow.to.target n="fig135"></arrow.to.target><lb/>æqualem. &longs;i enim e&longs;&longs;e po&longs;sit, &longs;it c b e. duca<lb/>tur recta b d faciens rectilineum d b c &ecedil;qua <lb/>

<figure id="fig135"></figure><lb/>æqualem. &longs;i enim e&longs;&longs;e po&longs;sit, &longs;it c b e. duca<lb/>tur recta b d faciens rectilineum d b c &ecedil;qua <lb/>

<arrow.to.target n="marg528"></arrow.to.target><lb/>lem a, erit igitur d b c &ecedil;qualis e b c per com<lb/>munem animi &longs;ententiam, &longs;eu ergo b d ca<lb/>dat intra circulum &longs;eu extra, erit pars &ecedil;qua<lb/>lis toti quod e&longs;&longs;e non pote&longs;t. Sed neque po<lb/>te&longs;t cadere recta &longs;uper b e. namid e&longs;t contra demon&longs;trata ab Eucli<lb/>

<arrow.to.target n="marg529"></arrow.to.target><lb/>de. At &longs;i &longs;it angulus c b e exterior &longs;imiliter producta b d, &longs;eu intus, <lb/>&longs;eu extrà cadat, pars erit æqualis toti quod e&longs;&longs;e non pote&longs;t.</s>

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<s>Ex hoc patet quod nullus angulus peripheria circuli & recta <expan abbr="cõ-">con<lb/></expan>

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<s>Et rur&longs;us nullus angulus peripheria & <lb/>

<arrow.to.target n="marg531"></arrow.to.target><lb/>

<arrow.to.target n="fig136"></arrow.to.target><lb/>recta contentus à recta linea per æqualia <lb/>diuidi pote&longs;t, patet quia una pars e&longs;&longs;et an<lb/>gulus rectilineus, alia contentus recta & pe <lb/>ripheria: i&longs;ti <expan abbr="aut&etilde;">autem</expan> non po&longs;&longs;unt e&longs;&longs;e æquales, <lb/>quare nec prior potuit per æqualia diuidi.</s>

<figure id="fig136"></figure><lb/>recta contentus à recta linea per æqualia <lb/>diuidi pote&longs;t, patet quia una pars e&longs;&longs;et an<lb/>gulus rectilineus, alia contentus recta & pe <lb/>ripheria: i&longs;ti <expan abbr="aut&etilde;">autem</expan> non po&longs;&longs;unt e&longs;&longs;e æquales, <lb/>quare nec prior potuit per æqualia diuidi.</s>

<s>Ex hoc etiam patet quod &longs;pacium con<lb/>

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<s>

<arrow.to.target n="marg534"></arrow.to.target><lb/>in a, & ducatur ex centro inferioris d a & <lb/>

<arrow.to.target n="fig137"></arrow.to.target><lb/>a d, & ad d a cathetus a e, dico quòd a e di<lb/>uidet angulum b a c ducatur ex centro &longs;u<lb/>

<figure id="fig137"></figure><lb/>a d, & ad d a cathetus a e, dico quòd a e di<lb/>uidet angulum b a c ducatur ex centro &longs;u<lb/>

<arrow.to.target n="marg535"></arrow.to.target><lb/>perioris a c b quod &longs;it f, fa cui cathetus a g, <lb/>quia ergo e a cadit infra a g, & inter a g & <lb/>

<arrow.to.target n="marg536"></arrow.to.target><lb/>a b non pote&longs;t duci recta, igitur e a cadit in<lb/>

<arrow.to.target n="fig138"></arrow.to.target><lb/>tra a c b circulum. Rur&longs;us tangant &longs;e circuli <lb/>c d & c e, & ducatur a b per centra <expan abbr="eor&utilde;">eorum</expan> qu&ecedil; <lb/>applicabit ad c, ex c ducatur cathetus c f & <lb/><expan abbr="quoniã">quoniam</expan> c f contangit <expan abbr="circul&utilde;">circulum</expan> c e, ligitur, du<lb/>cta quauis linea infra c f, cadet intra <expan abbr="circul&utilde;">circulum</expan> <lb/>c e. Non ergo poterit cadere inter c d & c e.</s>

<figure id="fig138"></figure><lb/>tra a c b circulum. Rur&longs;us tangant &longs;e circuli <lb/>c d & c e, & ducatur a b per centra <expan abbr="eor&utilde;">eorum</expan> qu&ecedil; <lb/>applicabit ad c, ex c ducatur cathetus c f & <lb/><expan abbr="quoniã">quoniam</expan> c f contangit <expan abbr="circul&utilde;">circulum</expan> c e, ligitur, du<lb/>cta quauis linea infra c f, cadet intra <expan abbr="circul&utilde;">circulum</expan> <lb/>c e. Non ergo poterit cadere inter c d & c e.</s>

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<s><margin.target id="marg536"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s>LEMMA SECVNDVM.</s>

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<s>Sit angulus a b c duabus peripherijs æqualium circulorum con <lb/>

<arrow.to.target n="marg537"></arrow.to.target><lb/>tentus, uolo ei æqualem rectilineum fabricare, ducantur b d & b e <lb/>

<arrow.to.target n="marg538"></arrow.to.target><lb/>æquales, ut pote facto b centro eritque angulus d b a æqualis angu<lb/>lo e b c, addito utrique communi d b e ex peri <lb/>

<arrow.to.target n="fig139"></arrow.to.target><lb/>pheria & recta, fiet angulus d b e ex rectis <lb/>æqualis a b c ex peripherijs, quod crat de<lb/>mon&longs;trandum.</s>

<figure id="fig139"></figure><lb/>pheria & recta, fiet angulus d b e ex rectis <lb/>æqualis a b c ex peripherijs, quod crat de<lb/>mon&longs;trandum.</s>

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<s><margin.target id="marg538"></margin.target>P<emph type="italics"/>er modum<emph.end type="italics"/><lb/>8. <emph type="italics"/>primi<emph.end type="italics"/> E<emph type="italics"/>l.<emph.end type="italics"/></s>

<s>Ex hoc patet quod reliqua duo &longs;pacia <lb/>

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<s>

<arrow.to.target n="marg542"></arrow.to.target><lb/>ducere. Sit circulus datus a b rectilineus <lb/>

<arrow.to.target n="fig140"></arrow.to.target><lb/>angulus c d e, uolo illum diuidere circuli <lb/>periferia data b f, duco perpendicularem <lb/>d g ex, d &longs;uper d c, & facio g d æqualem a b <lb/>

<figure id="fig140"></figure><lb/>angulus c d e, uolo illum diuidere circuli <lb/>periferia data b f, duco perpendicularem <lb/>d g ex, d &longs;uper d c, & facio g d æqualem a b <lb/>

<arrow.to.target n="marg543"></arrow.to.target><lb/>& duco circulum per d qui &longs;it d h qui cadet <lb/>infra d c & ob id etiam &longs;upra d e, igitur di<lb/>uidet angulum c d e, quare cum circulus d h &longs;it æqualis circulo b f <lb/>

<arrow.to.target n="marg544"></arrow.to.target><lb/>patet propo&longs;itum.</s>

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<s>Ex hoc patet quod infinitis modis pote&longs;t diuidi angulus c d e <lb/>

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<pb pagenum="161"/>ria & recta &longs;unt ex genere quantitatis continuæ, & quòd detur ma<lb/>ius & minus & nunquam detur &ecedil;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. Et maximè quod etiam anguli ex peripheria & recta <lb/>&longs;unt diuer&longs;orum generum inter &longs;e & infinitorum. Pr&ecedil;terea i&longs;tud re<lb/>pugnare uidetur ip&longs;imet Euclidi, dicenti duabus magnitu dinibus <lb/>

<arrow.to.target n="marg547"></arrow.to.target><lb/>

<arrow.to.target n="marg548"></arrow.to.target><lb/>propo&longs;itis inæqualibus, &longs;i de maiore earum plus dimidio detraha<lb/>tur, atque iterum de re&longs;iduo maius dimidio, & rur&longs;us de eo quod re<lb/>linquitur plus dimidio, nece&longs;&longs;e erit ut tandem minor minore quan<lb/>titas relinquatur. Neque illud argumentum uidetur concludere an<lb/>gulus contactus, ex recta, & circuli circumferentia non pote&longs;t recta <lb/>diuidi, & rectilineus pote&longs;t diuidi, ergo rectilin eus &longs;emper e&longs;t ma<lb/>ior angulo contactus, quia hoc contingit in angulo contactus pro <lb/>pter modum anguli, non paruitatem: &longs;i cut etiam non ualet de figu<lb/>

<arrow.to.target n="fig141"></arrow.to.target><lb/>ra a lunari, & quadrangulo b. nam pote&longs;t b diuidi <lb/>ab angulo ad angulum recta & a non pote&longs;t, & <lb/>tamen a maius e&longs;t quam b, cum contineat ip&longs;am. <lb/>Proponantur ergo duo circuli a d e & a f g qui &longs;e contingant in a, & <lb/>corum centra &longs;int b & c & ducantur rectæ a f d & a g e & con&longs;tat <lb/>&qring;d portiones a d & a f &longs;imiles &longs;unt, <lb/>

<figure id="fig141"></figure><lb/>ra a lunari, & quadrangulo b. nam pote&longs;t b diuidi <lb/>ab angulo ad angulum recta & a non pote&longs;t, & <lb/>tamen a maius e&longs;t quam b, cum contineat ip&longs;am. <lb/>Proponantur ergo duo circuli a d e & a f g qui &longs;e contingant in a, & <lb/>corum centra &longs;int b & c & ducantur rectæ a f d & a g e & con&longs;tat <lb/>&qring;d portiones a d & a f &longs;imiles &longs;unt, <lb/>

<arrow.to.target n="fig142"></arrow.to.target><lb/>itemque a e & a g, ducta enim a b c <lb/>

<figure id="fig142"></figure><lb/>itemque a e & a g, ducta enim a b c <lb/>

<arrow.to.target n="marg549"></arrow.to.target><lb/>per centra circulorum ex contactu <lb/>tran&longs;ibit per illa: quare anguli h a g <lb/>& h a e &longs;untijdem & &longs;imiliter h a f <lb/>& h a d ijdem, portiones ergo af & <lb/>a d itemque a g & a e &longs;imiles &longs;unt: an<lb/>gulus igitur g a e ex peripherijs & <lb/>

<arrow.to.target n="marg550"></arrow.to.target><lb/>e a d ex rectis &longs;unt ijdem in puncto <lb/>a: &longs;ed quod ad ba&longs;sim maior e&longs;t ba<lb/>&longs;is g e quam e d: hoc enim &longs;uppono <lb/>quod per &longs;e e&longs;t manife&longs;tum toties <lb/><expan abbr="diuid&etilde;do">diuidendo</expan> arcum d e ut fiat minor recta g e. Quia ergo &longs;unt du&ecedil; ma<lb/>gnitudines, quarum ter mini &longs;unt ijdem ex una parte, &longs;cilicet pun<lb/>ctum a, ex alia autem unus e&longs;t maior altero, &longs;cilicet g e quam e f & <lb/>

<arrow.to.target n="marg551"></arrow.to.target><lb/>a d e peripheria e&longs;t maior recta a g e. Ergo per regulam dialecti<lb/>cam &longs;i &longs;ub eadem proportione procederent, maius e&longs;&longs;et &longs;patium <lb/>&longs;emper inter peripherias quàm rectas. igitur angulus peripheria<lb/>rum e&longs;t maior angulo à rectis contento. Cum angulus non &longs;it <lb/>ni&longs;i quidam habitus propinquitatis linearum, &longs;ed angulus con<lb/>tactus ex recta & peripheria maior e&longs;t contento ex peripherijs cum <lb/>habeat rationem totius ad partem, igitur angulus contactus e&longs;t <lb/>maior dato angulo rectilineo.</s>

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<s>Propo&longs;itio cente&longs;ima&longs;exage&longs;ima.</s>

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<s>

<arrow.to.target n="marg553"></arrow.to.target><lb/>g b, ut d ad e & g b ad g c, ut e ad f. Per pr&ecedil;ceden <lb/>

<arrow.to.target n="fig143"></arrow.to.target><lb/>tia inuenio circulum ex cuius peripheria omni<lb/>bus ex punctis ductæ lineæ ad a b &longs;int in pro<lb/>portione d ad e, & per idem circulum ex cuius <lb/>peripheria quælibet lineæ ductæ ad b c puncta <lb/>&longs;int in proportione c ad f, &longs;i igitur i&longs;ti duo circu<lb/>li &longs;e &longs;ecabunt in aliquo puncto puta g: liquet <lb/>quod lineæ ductæ ex g ad a b c, erunt in propor <lb/>tione d e f.<lb/>

<figure id="fig143"></figure><lb/>tia inuenio circulum ex cuius peripheria omni<lb/>bus ex punctis ductæ lineæ ad a b &longs;int in pro<lb/>portione d ad e, & per idem circulum ex cuius <lb/>peripheria quælibet lineæ ductæ ad b c puncta <lb/>&longs;int in proportione c ad f, &longs;i igitur i&longs;ti duo circu<lb/>li &longs;e &longs;ecabunt in aliquo puncto puta g: liquet <lb/>quod lineæ ductæ ex g ad a b c, erunt in propor <lb/>tione d e f.<lb/>

<arrow.to.target n="marg554"></arrow.to.target></s>

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<s>Ex quo liquet quod &longs;i uoluero ducere ad tria puncta data, tres <lb/>lineas in continua proportione data d ad e, &longs;ubijciam tertiam uel in <lb/>terponam, &longs;i uoluero mediam. Et &longs;i uellem, ut e&longs;&longs;et a g ad g b dupli<lb/>cata ei quæ e&longs;t g b ad b c, & uellem quòd proportio d ad a d f data <lb/>e&longs;&longs;et, oporteret inuenire duas medias proportione inter d & f, in de <lb/>operari cum una earum per modum propo&longs;itum. Differt corrola<lb/>rium hoc à propo&longs;itione in hoc, quod in propo&longs;itione non quæri<lb/>mus ni&longs;i proportionem g a ad g b & g b ad b c, non g a ad g c, neque<lb/>comparationem proportionum: at in corrolario quærimus tres <lb/>proportiones g a g b & g c, & comparationem proportionum in<lb/>ter &longs;e, &longs;cilicet æqualitatem.</s>

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<s>Si fuerint duo trianguli quorum ba&longs;es in eadem linea &longs;int con<lb/>&longs;tituti & æquales & ad unum punctum terminati, & latus unum <lb/>commune inter reliqua quantita<lb/>

<arrow.to.target n="fig144"></arrow.to.target><lb/>te medium, nece&longs;&longs;e e&longs;t angulum à <lb/>maioribus lineis contentum mi<lb/>norem e&longs;&longs;e.</s>

<figure id="fig144"></figure><lb/>te medium, nece&longs;&longs;e e&longs;t angulum à <lb/>maioribus lineis contentum mi<lb/>norem e&longs;&longs;e.</s>

<s>Sint duo trianguli a b c, a c d, </s>

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<s>His demon&longs;tratis quis dicere po&longs;&longs;et ex &longs;uperius expo&longs;itis quod <lb/>

<arrow.to.target n="marg562"></arrow.to.target><lb/>angulus rectilineus &longs;emper e&longs;&longs;etmaior angulo contactus? quia an<lb/>gulus contactus non pote&longs;t diuidi ni&longs;i obliqua linea, recti lineus <lb/>autem tam obliqua quam recta. Propter hoc exponantur circuli <lb/>

<arrow.to.target n="fig145"></arrow.to.target><lb/>tres &longs;e tangentes a b, a c, a d hac rati<lb/>one ut a b, b c, c d &longs;int æquales, erunt <lb/>

<figure id="fig145"></figure><lb/>tres &longs;e tangentes a b, a c, a d hac rati<lb/>one ut a b, b c, c d &longs;int æquales, erunt <lb/>

<arrow.to.target n="marg563"></arrow.to.target><lb/>enim centra omnia in linea conta<lb/>ctus, & ducatur a e f g recta quomo <lb/>

<arrow.to.target n="marg564"></arrow.to.target><lb/>dolibet: & erunt ductis lineis b c, <lb/>

<arrow.to.target n="marg565"></arrow.to.target><lb/>c f, d g anguli e f g recti, quare om<lb/>nes trigoni a b e, a c f, a d g, &longs;imiles <lb/>

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<s><margin.target id="marg568"></margin.target>P<emph type="italics"/>er præce<lb/>dentem.<emph.end type="italics"/></s>

<s>SCHOLIVM.</s>

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<s>Ratio autem quòd omnis angulus contactus indiuiduus &longs;it, &longs;eu <lb/>duorum circulorum, &longs;eu circuli cum recta e&longs;t, quoniam cum fuerint <lb/>duæ rationes contrariæ, & una perpetuò minuitur, alia manet ne<lb/>ce&longs;&longs;e e&longs;t, ut tandem, quæ minuitur, &longs;uperetur ab ea quæ manet: cum <lb/>ergo circuli curuitas maneat, & angulus tendat in punctum perpe<lb/>tua diminutione nece&longs;&longs;e e&longs;t, ut curuitas circuli impediat diui&longs;io<lb/>nem rectè: &longs;ed hoc habet duplicem obicem. Primum, quia nullus <lb/>angulus ex circumferentia & recta po&longs;&longs;et diuidi: hoc autem fal&longs;um <lb/>e&longs;t manife&longs;tè, cum &longs;olus ille qui fit ex contactu lineæ, quæ non di<lb/>uidit circulum, diuidi non po&longs;sit. Secundò, quod angulus conta<lb/>ctus duorum circulorum &longs;e exterius tangentium multo minus <lb/>po&longs;&longs;et diuidi angulo contactus interioris duorum circulorum, <lb/>quod tamen fal&longs;um e&longs;t: & hoc animaduertit Campanus no&longs;ter, uir <lb/>acutus. Dico ergo quòd in his qui &longs;e tangunt exterius, non fit diui<lb/>&longs;io ni&longs;i &longs;emel: & quamuis inclinentur mutuò, tamen in concur&longs;u <lb/>non aptantur, ut cum obuiat rectæ aut cauæ parti circuli quia ne<lb/>ce&longs;&longs;e e&longs;t, ut accedat, in alio autem di&longs;cedat: indicio e&longs;t quod circu<lb/>los &longs;e exterius tangentes, in puncto facilè de&longs;cribes, interius uix fie<lb/>ri pote&longs;t, &longs;ed uidentur coniuncti <lb/>

<arrow.to.target n="fig146"></arrow.to.target><lb/>per longum interuallum. Ad aliud <lb/>dico, quòd ille angulus ex recta & <lb/>peripheria conuexa circuli propter <lb/>di&longs;ce&longs;&longs;um &longs;eruat maiorem inclina<lb/>tionem in quocunque puncto, quàm <lb/>&longs;it acce&longs;&longs;us conuexæ partis exterio<lb/>ris circuli.</s>

<figure id="fig146"></figure><lb/>per longum interuallum. Ad aliud <lb/>dico, quòd ille angulus ex recta & <lb/>peripheria conuexa circuli propter <lb/>di&longs;ce&longs;&longs;um &longs;eruat maiorem inclina<lb/>tionem in quocunque puncto, quàm <lb/>&longs;it acce&longs;&longs;us conuexæ partis exterio<lb/>ris circuli.</s>

<s>Propo&longs;itio cente&longs;ima&longs;exage&longs;ima <lb/>&longs;ecunda.</s>

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<s>Proportionem duorum orbium <lb/>

<arrow.to.target n="fig147"></arrow.to.target><lb/>quorum diametrorum <expan abbr="cõuexæ">conuexæ</expan> par <lb/>tis, & concauæ proportiones datæ <lb/>&longs;int, inue&longs;tigare.</s>

<figure id="fig147"></figure><lb/>quorum diametrorum <expan abbr="cõuexæ">conuexæ</expan> par <lb/>tis, & concauæ proportiones datæ <lb/>&longs;int, inue&longs;tigare.</s>

<s>Sint duo orbes a b c d & e f g h, <lb/>

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<s>

<arrow.to.target n="marg583"></arrow.to.target><lb/>ones, & cogitationes, &longs;ed non ni&longs;i ut per corpora &longs;ignificantur: ut <lb/>ergo corpora ip&longs;a referamus. coloribus opus e&longs;t, nam corpora, co<lb/>lorata &longs;unt, &longs;ecundò ip&longs;a rerum natura &longs;cientiaque illarum, unde pi<lb/>ctorem multi&longs;cium e&longs;&longs;e nece&longs;&longs;e e&longs;t. tertium e&longs;t, ut minimas earum <lb/>differentias explicare norit. quartum, ut affectiones, uelut in ira

<pb pagenum="180"/>to ruborem, ciliorum <expan abbr="cõtractionem">contractionem</expan>, tumorem faciei in ambulante <lb/>in clinationem quandam, flexionem cruris atque &longs;imilia. quintum e&longs;t <lb/>lux coloribus <expan abbr="exhib&etilde;da">exhibenda</expan>, &longs;ed de horum nullo propo&longs;itum e&longs;t hic lo<lb/>qui, quando quidem hæc u&longs;u magis & con&longs;ideratione, quàm ratio<lb/>ne con&longs;tent proportioneúe, nec &longs;int adeò admiranda ut neque &longs;im<lb/>plex magnitudo <expan abbr="quã&longs;exto">quan&longs;exto</expan> loco reponere po&longs;&longs;umus. Tria ergo ui<lb/>dentur e&longs;&longs;e præcipua quorum nunc ratio habenda e&longs;&longs;et, ut &longs;int in <lb/>totum nouem, &longs;ed unum ex his relinquemus, tum quia alienum ab <lb/>hac con&longs;ideratione, tum quia alibi pertractatum atque etiam ab alijs, <lb/>neque adeò admiratione dignum &longs;cilicet magnitudo picturarum re<lb/>&longs;pondens magnitudini corporum iuxta &longs;itus differentiam, nam <lb/>qu&ecedil; altiores &longs;unt paulo latiores atque in &longs;uperiori magis parte quam <lb/>in inferiore, multò autem longiores e&longs;&longs;e oportet, &longs;ic & quæ à latere <lb/>erunt eadem ratione iuxta a&longs;pectus ingredientium rationem. Ve<lb/>rum hoc ut dixi omittamus, & de duplici miraculo in pictura lo<lb/>quamur, &longs;cilicet di&longs;tantia magna quam in parua tabella referimus, <lb/>et corporeitate quam in plano repr&ecedil;&longs;entamus. Horum autem duo<lb/>rum aliqua communia &longs;unt aliqua propria. Dicemus ergo <expan abbr="prim&utilde;">primum</expan> <lb/>de corpore ita pingendo, ut palàm extra tabulam prominere uide <lb/>atur. Hoc autem primum ex forma &longs;umitur, nam &longs;i corpus in plano <lb/>&longs;it nece&longs;&longs;e e&longs;t, ut partes illius quædam pror&longs;us ab&longs;condantur, par<lb/>tes aliæ non pror&longs;us, aliæ pror&longs;us &longs;int in con&longs;picuo. Ergo pictu<lb/>ram talem fingere oportebit, quæ partes &longs;ingulas pro ratione o&longs;ten <lb/>dat aut occultet. <expan abbr="Sec&utilde;da">Secunda</expan> ratio e&longs;t quodima corporis ob&longs;cura &longs;unt, <lb/>&longs;umm&ecedil; partes lucid&ecedil; & claræ aclumine qua&longs;i dealbatæ: media, me<lb/>dia quadam ratione ut in columnis, tantumque pote&longs;t hæc ratio, ut <lb/>uel &longs;ola picturas fallere nos faciat corpora eas e&longs;&longs;e putantes. Opor<lb/>tet autem imum e&longs;&longs;e ad unguem &longs;imile in colore colori anguli loci <lb/>& &longs;ummum parti quæ &longs;e oculis maximè &longs;ubiectam præbet & cla<lb/>ram: media uerò qualia ex umbris ob&longs;curari &longs;olent. Tertia ratio e&longs;t <lb/>pro modo partium iuxta <expan abbr="obliquitat&etilde;">obliquitatem</expan> a&longs;pectus: nam in&longs;picienti a b <lb/>in c d ex e oculo: depingemus in c d iuxta obli<lb/>

<arrow.to.target n="fig148"></arrow.to.target><lb/>quitatem &longs;uam, quia cum c d uideatur per line<lb/>as e a c & e b d, & eleuatum in &longs;itu a b, nece&longs;&longs;e e&longs;t <lb/>ut uideatur in &longs;itu a b, ergo eleuatum à c d. E&longs;t <lb/>& alia con&longs;ideratio proportionis ad proxima <lb/>remotaque, grati a exempli, &longs;i homo e&longs;&longs;et po&longs;t co<lb/>lumnam a b, lateret eius pars, quæ e&longs;t propinquior parieti c d, ergo <lb/>&longs;i depinxerimus hominis partes tantum dextram, reliquum &longs;ub um <lb/>bra, cogitur oculus iudicare columnam eleuatam a pariete. De<lb/>mum omnia hæc ita &longs;unt &longs;ubijcienda oculis, & per minimas diffe

<figure id="fig148"></figure><lb/>quitatem &longs;uam, quia cum c d uideatur per line<lb/>as e a c & e b d, & eleuatum in &longs;itu a b, nece&longs;&longs;e e&longs;t <lb/>ut uideatur in &longs;itu a b, ergo eleuatum à c d. E&longs;t <lb/>& alia con&longs;ideratio proportionis ad proxima <lb/>remotaque, grati a exempli, &longs;i homo e&longs;&longs;et po&longs;t co<lb/>lumnam a b, lateret eius pars, quæ e&longs;t propinquior parieti c d, ergo <lb/>&longs;i depinxerimus hominis partes tantum dextram, reliquum &longs;ub um <lb/>bra, cogitur oculus iudicare columnam eleuatam a pariete. De<lb/>mum omnia hæc ita &longs;unt &longs;ubijcienda oculis, & per minimas diffe

<pb pagenum="181"/>rentias & animaduer&longs;iones ita dijudicanda, atque experimento &longs;ub<lb/>ijcienda, tum proprio, tum aliorum non artis in expertium, ut re<gap/><lb/>pror&longs;us ab&longs;oluta uideatur, atque in hoc multum refert multiplices <lb/>partes &longs;ecundum longitudinem coloribus di&longs;tinguere ad hoc a<lb/>ptis, qui &longs;unt ob&longs;curus, &longs;ub ob&longs;curus, cinereus, qualis &longs;ilicis candi<lb/>dus &longs;ine luce, demum etiam aliquid nigri adijciendum, nam diui&longs;io <lb/>&longs;ecundum longitudinem multum impedit, hanc repræ&longs;entationem <lb/>iuuant, & extrema benè coaptata, uelut &longs;capi imi, & capitula & &longs;u<lb/>premi, <expan abbr="t&utilde;">tum</expan> trabeationes ex materia coronæ, zofoni, t&oelig;nia, epi&longs;tylia, <lb/>plinthi, echini, hypotrachelia, a&longs;tagali, apophyges. Quæ etiam in <lb/>parte inferiore <expan abbr="c&utilde;">cum</expan> &longs;pira &longs;eu ba&longs;i & limbo & toro & plintho inferio<lb/>re, & &longs;tylobata, et alia t&oelig;nia &longs;umma diligentia, & cum eleuatione ac <lb/>magnitudine ultra columnæ limites extendantur. Sicin &longs;tylobata <lb/>ratio diapente con&longs;tat, cui &longs;olet addi utrinque &longs;exta pars pro coro<lb/>nice, manife&longs;tum e&longs;t autem, quod in ea con&longs;tat mu&longs;ica ratio diapa<lb/>&longs;on ex diapente & diate&longs;&longs;aro, compo&longs;iti nam duæ &longs;extæ partes, alte <lb/>ra utrinque adiecta tertiam conficiunt ut &longs;it diate&longs;&longs;aron &longs;uprà diapen <lb/>te. In regionibus autem & &longs;patijs depingendis eadem fermè &longs;eruan <lb/>da &longs;unt duobus tamen adiectis, <expan abbr="quor&utilde;">quorum</expan> unum e&longs;t ut longinqui&longs;sima <lb/>pars, <expan abbr="nõ">non</expan> per nigrum aut ob&longs;curum, &longs;ed c&oelig;ruleum <expan abbr="color&etilde;">colorem</expan>, qualis in <lb/>c&oelig;lo determinanda e&longs;t (ni&longs;i nox fingatur) nam c&oelig;lum longi&longs;simè <lb/>à nobis di&longs;tat, ita nubes coloribus proprijs, & montes cum niui<lb/>bus, & &longs;patia uelut fluminis alueus, mare, lacus, atque hæc omnia <lb/>per colores di&longs;tantiæ finguntur, uelut fluminis pars propior clara <lb/>& lympida, & colore aqueo cernitur remota ob&longs;cura, quæ maxi<lb/>mè procul abe&longs;t nigra. Sed maxima e&longs;t confirmatio in compara<lb/>tionibus: ut &longs;i arbores propè magnæ &longs;int, & homines & animalia, <lb/>in remotiore autem parte minimi, ac qua&longs;i puncti magnitudinem <lb/>referentes, atque ut in his mu&longs;ica non geometrica aut arithmeti<lb/>ca proportio &longs;eruetur. Equidem &longs;i quis iudicio hæc con&longs;equa<lb/>tur, ac diligentia quæ &longs;cribi non po&longs;&longs;unt, &longs;ed contemplatione ha<lb/>bentur, &longs;en&longs;u quoque, quem experimentum docet, necip&longs;um man<lb/>dare literis, licet ex rationibus tamen, quas hic docemus intelli<lb/>get parum differre repræ&longs;entationem à re ip&longs;a corporea. Sed de <lb/>his hactenus, quæ &longs;i diligentius quis per&longs;equi uelit &longs;ine <lb/>artis experientia, plus adimet perfectioni rei, <lb/>quam adijciet. Hoc enim aliâs <lb/>

<arrow.to.target n="marg584"></arrow.to.target><lb/>declarauimus.</s>

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<s><margin.target id="marg584"></margin.target>I<emph type="italics"/>n prima<emph.end type="italics"/><lb/>D<emph type="italics"/>islcfficæ.<emph.end type="italics"/></s>

<s>Propo&longs;itio cente&longs;ima&longs;exage&longs;imanona.</s>

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<s>Porrò quod ad machinas attinet. Sit catapulta, cuius rudens a b <lb/>quam oportet trahere, &longs;i emittere debeat lapi<lb/>

<arrow.to.target n="fig149"></arrow.to.target><lb/>dem, aut &longs;corpio &longs;agittam ad aliquod &longs;ignum <lb/>puta c, cum ergo &longs;onus c a & c b homotenus fue <lb/>rit, non &longs;olum æqualiter pertractæ erunt c a & <lb/>c b, &longs;ed etiam æquales: nam &longs;i æquales e&longs;&longs;ent, & <lb/>in&ecedil;qualiter tractæ, aut in&ecedil;quales & inæqualiter <lb/>tract&ecedil; <expan abbr="&longs;on&utilde;">&longs;onum</expan> diuer&longs;um <expan abbr="redd&etilde;t">reddent</expan> euidenter. At &longs;i in<lb/>&ecedil;quales & <expan abbr="&ecedil;qual&etilde;">&ecedil;qualem</expan> &longs;onum reddant, erit <expan abbr="tñ">tnm</expan> ut fidis <lb/>notæ quæ &longs;trepitum edit duplicem, & effigiem <lb/>oculis <expan abbr="multiplic&etilde;">multiplicem</expan>, unde &longs;agitta in partem aduer<lb/>&longs;am dirigitur <expan abbr="rud&etilde;tis">rudentis</expan> intentioris, atque hæc ex Vitruuio eodem dum <lb/>de his agit.</s>

<figure id="fig149"></figure><lb/>dem, aut &longs;corpio &longs;agittam ad aliquod &longs;ignum <lb/>puta c, cum ergo &longs;onus c a & c b homotenus fue <lb/>rit, non &longs;olum æqualiter pertractæ erunt c a & <lb/>c b, &longs;ed etiam æquales: nam &longs;i æquales e&longs;&longs;ent, & <lb/>in&ecedil;qualiter tractæ, aut in&ecedil;quales & inæqualiter <lb/>tract&ecedil; <expan abbr="&longs;on&utilde;">&longs;onum</expan> diuer&longs;um <expan abbr="redd&etilde;t">reddent</expan> euidenter. At &longs;i in<lb/>&ecedil;quales & <expan abbr="&ecedil;qual&etilde;">&ecedil;qualem</expan> &longs;onum reddant, erit <expan abbr="tñ">tnm</expan> ut fidis <lb/>notæ quæ &longs;trepitum edit duplicem, & effigiem <lb/>oculis <expan abbr="multiplic&etilde;">multiplicem</expan>, unde &longs;agitta in partem aduer<lb/>&longs;am dirigitur <expan abbr="rud&etilde;tis">rudentis</expan> intentioris, atque hæc ex Vitruuio eodem dum <lb/>de his agit.</s>

<s>Propo&longs;itio cente&longs;ima&longs;eptuage&longs;ima.</s>

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<s>Secunda notha harmonica e&longs;t, ut &longs;it propor<lb/>

<arrow.to.target n="fig150"></arrow.to.target><lb/>tio primæ ad tertiam, uelut differentiæ primæ à <lb/>tertia ad differentiam &longs;ecundæ à tertia, ponatur <lb/>25, prima 21, &longs;ecunda 15, tertia proportio 25 ad 15 <lb/>e&longs;t uelut 10 differentiæ prim&ecedil; à tertia ad b differen <lb/>tiam &longs;ecundæ à tertia.</s>

<figure id="fig150"></figure><lb/>tio primæ ad tertiam, uelut differentiæ primæ à <lb/>tertia ad differentiam &longs;ecundæ à tertia, ponatur <lb/>25, prima 21, &longs;ecunda 15, tertia proportio 25 ad 15 <lb/>e&longs;t uelut 10 differentiæ prim&ecedil; à tertia ad b differen <lb/>tiam &longs;ecundæ à tertia.</s>

<s>Tertia e&longs;t &longs;imilis priori, ni&longs;i quod &longs;umitur dif<lb/>

<arrow.to.target n="fig151"></arrow.to.target><lb/>ferentia primæ à &longs;ecunda pro ultimo termino. Ex<lb/>emplum, 25 primus terminus, 19 &longs;ecundus, 15 ter<lb/>tius, proportio 25 ad 15 e&longs;t uelut 10 differentiæ pri<lb/>mæ a tertia ad b, differentiam primæ à &longs;ecunda. <lb/>Has proportiones quanquàm exiguæ utilitatis, proponere uo<lb/>lui, ut excogitatis aliquibus demon&longs;trationibus, uelut &longs;uperius <lb/>diximus, pulchra theoremata & problemata tradi po&longs;&longs;ent.</s>

<figure id="fig151"></figure><lb/>ferentia primæ à &longs;ecunda pro ultimo termino. Ex<lb/>emplum, 25 primus terminus, 19 &longs;ecundus, 15 ter<lb/>tius, proportio 25 ad 15 e&longs;t uelut 10 differentiæ pri<lb/>mæ a tertia ad b, differentiam primæ à &longs;ecunda. <lb/>Has proportiones quanquàm exiguæ utilitatis, proponere uo<lb/>lui, ut excogitatis aliquibus demon&longs;trationibus, uelut &longs;uperius <lb/>diximus, pulchra theoremata & problemata tradi po&longs;&longs;ent.</s>

<s>Propo&longs;itio cente&longs;ima&longs;eptuage&longs;imatertia.</s>

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<s>Sit a centrum circuli b c, & æqualis ei <lb/>

<arrow.to.target n="fig152"></arrow.to.target><lb/>circulus d e, centrum eius b in circumfe<lb/>rentia circuli b c, fixum ita ut ibi mouea<lb/>tur ad motum circuli b c: & moueatur b <lb/>uer&longs;us c æqualiter, & e contrario motu <lb/>etiam regulariter, & duplo uelocius ex e <lb/>uer&longs;us d, dico omnia puncta d e moue<lb/>ri in linea recta, & primum capio pun<lb/>ctum d, quod &longs;it in linea recta centro<lb/>rum: & moueatur b ad c, & &longs;i circulus d e <lb/>e&longs;&longs;et immobilis, palam e&longs;t quòd pun<lb/>ctum d cum &longs;it in una linea a b, cum b <lb/>perueniret in c, d e&longs;&longs;et in linea a c, putà in <lb/>h &longs;ecundum quantitatem, ergo b d ex </s>

<figure id="fig152"></figure><lb/>circulus d e, centrum eius b in circumfe<lb/>rentia circuli b c, fixum ita ut ibi mouea<lb/>tur ad motum circuli b c: & moueatur b <lb/>uer&longs;us c æqualiter, & e contrario motu <lb/>etiam regulariter, & duplo uelocius ex e <lb/>uer&longs;us d, dico omnia puncta d e moue<lb/>ri in linea recta, & primum capio pun<lb/>ctum d, quod &longs;it in linea recta centro<lb/>rum: & moueatur b ad c, & &longs;i circulus d e <lb/>e&longs;&longs;et immobilis, palam e&longs;t quòd pun<lb/>ctum d cum &longs;it in una linea a b, cum b <lb/>perueniret in c, d e&longs;&longs;et in linea a c, putà in <lb/>h &longs;ecundum quantitatem, ergo b d ex </s>

<s>

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<s>Ex hoc patet quòd quando b <lb/>

<arrow.to.target n="fig153"></arrow.to.target><lb/>

<arrow.to.target n="marg613"></arrow.to.target><lb/>erit in c peracta quarta circuli, ut in <lb/>&longs;ecunda figura erit per motum l e <lb/>in a: nam cum d a &longs;it dupla c b, igi<lb/>tur in eodem tempore l perueniet <lb/>ad a, in quo b perueniet ad c.</s>

<s>Dico etiam, quod <expan abbr="quãdo">quando</expan> b per<lb/>

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<s>O&longs;ten damus modo, quod pun <lb/>

<arrow.to.target n="fig154"></arrow.to.target><lb/>ctum d extra lineam centrorum, &longs;ci <lb/>licet in linea d c a f tran&longs;ibit per <expan abbr="re-ctã">re<lb/>ctam</expan> eandem, ut in tertia figura pro<lb/>ducatur c d u&longs;que ad k, ita ut c k &longs;it <lb/>æqualis c a, erit ergo punctus d pri <lb/>mæ figuræ m è regione k tertiæ, & <lb/>dum c mouetur ad e, d perueniat <lb/>ad g, erit ergo e g æqualis ea, & &longs;e<lb/>cet circulus g h rectam a d in h, & <lb/>ducatur c h. Et erit ut prius angu<lb/>lus h e g duplus h a g, ergo arcus <lb/>

<figure id="fig154"></figure><lb/>ctum d extra lineam centrorum, &longs;ci <lb/>licet in linea d c a f tran&longs;ibit per <expan abbr="re-ctã">re<lb/>ctam</expan> eandem, ut in tertia figura pro<lb/>ducatur c d u&longs;que ad k, ita ut c k &longs;it <lb/>æqualis c a, erit ergo punctus d pri <lb/>mæ figuræ m è regione k tertiæ, & <lb/>dum c mouetur ad e, d perueniat <lb/>ad g, erit ergo e g æqualis ea, & &longs;e<lb/>cet circulus g h rectam a d in h, & <lb/>ducatur c h. Et erit ut prius angu<lb/>lus h e g duplus h a g, ergo arcus <lb/>

<arrow.to.target n="fig155"></arrow.to.target><lb/>g h duplus e c, ergo g remeauit in <lb/>h in tempore quo c feretur in e, <lb/>quare d de&longs;cendit per rectam in h.</s>

<figure id="fig155"></figure><lb/>g h duplus e c, ergo g remeauit in <lb/>h in tempore quo c feretur in e, <lb/>quare d de&longs;cendit per rectam in h.</s>

<s>Dico rur&longs;us, quòd quanto ma<lb/>gis d erit propinquum lineæ d g, <lb/>tanto minus de&longs;cendet in recta, <lb/>quanto magis propinquum longi <lb/>tudinibus medijs, <expan abbr="tãto">tanto</expan> celerius mo <lb/>uebitur, adeò ut in &longs;ecunda figura <lb/>apparet motum ex d in g, non de&longs;cendit ni&longs;i per d n, & motum ex g <lb/>in l de&longs;cendit ex n in a centrum fixum. De&longs;cendat ergo ex e in h & h

Line 8914

Line 8912

<s>Sit eclyptica a b c d, & arcus regre&longs;&longs;us b c in partes <lb/>

<arrow.to.target n="fig156"></arrow.to.target><lb/>quatuor æquales diui&longs;us, & de&longs;cribantur circuli duo b <lb/>h & e k &longs;uper e & f, & &longs;upponatur orbis &longs;uperior &longs;ub <lb/>eclyptica tamen, cuius polus in f, qui circumagatur in du <lb/>plo temporis retroce&longs;&longs;us planetæ, & in di&longs;tantia circuli <lb/>e k &longs;ub puncto e eclypticæ, polus alterius orbis concen<lb/>trici inferioris, qui circumagatur in tempore retro ce&longs;&longs;us <lb/>planetæ, & planeta &longs;it in puncto 6, liquet ergo quòd pla <lb/>neta ille in uno circuitu e k circuli permeabit b c & re<lb/>meabit, & &longs;emper erit &longs;ub ip&longs;a eclyptica. Sed enim eclyptica habet <lb/>rationem rectæ lineæ, ut quiuis circulus maximus. Et &longs;i quis relu<lb/>ctetur fingamus rectam &longs;ubten&longs;am arcui b c, & aliam po&longs;tmodum <lb/>æquidi&longs;tantem in eadem &longs;uperficie, & in orbe inferiore, & tunc pa<lb/>tebit liquidò propo&longs;itum. Sed &longs;i uelim latitudinem de&longs;cribam, ma<lb/>ximam latitudinem à puncto b, & ducam circulum magnum per <lb/>punctum illud: reliqua ut prius, ad unguem: nihil enim refert quod <lb/>ad demon&longs;trationem præcedentis attinet, &longs;eu a d ponatur eclypti<lb/>ca, &longs;eu alius circulus magnus.</s>

<figure id="fig156"></figure><lb/>quatuor æquales diui&longs;us, & de&longs;cribantur circuli duo b <lb/>h & e k &longs;uper e & f, & &longs;upponatur orbis &longs;uperior &longs;ub <lb/>eclyptica tamen, cuius polus in f, qui circumagatur in du <lb/>plo temporis retroce&longs;&longs;us planetæ, & in di&longs;tantia circuli <lb/>e k &longs;ub puncto e eclypticæ, polus alterius orbis concen<lb/>trici inferioris, qui circumagatur in tempore retro ce&longs;&longs;us <lb/>planetæ, & planeta &longs;it in puncto 6, liquet ergo quòd pla <lb/>neta ille in uno circuitu e k circuli permeabit b c & re<lb/>meabit, & &longs;emper erit &longs;ub ip&longs;a eclyptica. Sed enim eclyptica habet <lb/>rationem rectæ lineæ, ut quiuis circulus maximus. Et &longs;i quis relu<lb/>ctetur fingamus rectam &longs;ubten&longs;am arcui b c, & aliam po&longs;tmodum <lb/>æquidi&longs;tantem in eadem &longs;uperficie, & in orbe inferiore, & tunc pa<lb/>tebit liquidò propo&longs;itum. Sed &longs;i uelim latitudinem de&longs;cribam, ma<lb/>ximam latitudinem à puncto b, & ducam circulum magnum per <lb/>punctum illud: reliqua ut prius, ad unguem: nihil enim refert quod <lb/>ad demon&longs;trationem præcedentis attinet, &longs;eu a d ponatur eclypti<lb/>ca, &longs;eu alius circulus magnus.</s>

<s>

Line 9042

Line 9040

<s>Sit proportio a ad compo&longs;ita ex proportionibus c <lb/>

<arrow.to.target n="marg623"></arrow.to.target><lb/>

<arrow.to.target n="fig157"></arrow.to.target><lb/>ad d & c ad e, dico quòd proportio d ad e e&longs;t, ut produ<lb/>cti ex proportione in d detracto c ad ip&longs;um c. Et nos <lb/>&longs;uperius expo&longs;uimus conuer&longs;am huius. Erit enim per <lb/><expan abbr="&longs;ecundã">&longs;ecundam</expan> demon&longs;trationem illius proportio a ad b, uelut producti <lb/>ex c in d, & e ad productum d in e: at productum d in e & in propor <lb/>tionem, e&longs;t idem quod productum proportionis in d in ip&longs;um e: igi <lb/>tur cum in uno &longs;it productum e in c, & d in c, in alio productum a b <lb/>in d in de in e, quæ &longs;unt æqualia, detracto producto e in c ex produ<lb/>cto proportionis in d & inde in e, relinquetur, productum c in d æ<lb/>quale producto a b .i. proportionis in productum d in e, detracto <lb/>numero c in e: igitur ducto c in d, & diui&longs;o per productum a b in d <lb/>numero c, exibit e, igitur cum illud productum fiat ex d, &longs;cilicetin c, <lb/>& ex e in productum proportionis in d dempto numero c, erit pro <lb/>portio d ad e, uelut producti ex d in proportionem, detracto e ad <lb/>ip&longs;um c, uelut c &longs;it 12, d 4, e 6, a b erit 5 proportio d ad e, uelut d in a b, <lb/>id e&longs;t 20, detracto c, & e&longs;t 8 ad c 12.</s>

<figure id="fig157"></figure><lb/>ad d & c ad e, dico quòd proportio d ad e e&longs;t, ut produ<lb/>cti ex proportione in d detracto c ad ip&longs;um c. Et nos <lb/>&longs;uperius expo&longs;uimus conuer&longs;am huius. Erit enim per <lb/><expan abbr="&longs;ecundã">&longs;ecundam</expan> demon&longs;trationem illius proportio a ad b, uelut producti <lb/>ex c in d, & e ad productum d in e: at productum d in e & in propor <lb/>tionem, e&longs;t idem quod productum proportionis in d in ip&longs;um e: igi <lb/>tur cum in uno &longs;it productum e in c, & d in c, in alio productum a b <lb/>in d in de in e, quæ &longs;unt æqualia, detracto producto e in c ex produ<lb/>cto proportionis in d & inde in e, relinquetur, productum c in d æ<lb/>quale producto a b .i. proportionis in productum d in e, detracto <lb/>numero c in e: igitur ducto c in d, & diui&longs;o per productum a b in d <lb/>numero c, exibit e, igitur cum illud productum fiat ex d, &longs;cilicetin c, <lb/>& ex e in productum proportionis in d dempto numero c, erit pro <lb/>portio d ad e, uelut producti ex d in proportionem, detracto e ad <lb/>ip&longs;um c, uelut c &longs;it 12, d 4, e 6, a b erit 5 proportio d ad e, uelut d in a b, <lb/>id e&longs;t 20, detracto c, & e&longs;t 8 ad c 12.</s>

<s>Ex demon&longs;tratione &longs;equitur, quod qualis e&longs;t proportio e ad a b, <lb/>

Line 9079

Line 9077

<s>Sed e&longs;t genus mi&longs;tionis, quod uocant con&longs;olationem. Veluti, <lb/>uolo ex argento perfectionis decem & &longs;eptem, & quinque, confla<lb/>re argenti ma&longs;&longs;am centum librarum perfectionis nouem, ita agen<lb/>dum e&longs;t. Detrahe 9 à 10, & omni maiori 10, relinqui<lb/>tur 1, hoc &longs;uppone 7 & 5, item detrahe 7 & 5, & omne <lb/>

<arrow.to.target n="fig158"></arrow.to.target><lb/>minus 9 à 9, relinquitur 2 & 4, iunge omnia re&longs;idua <lb/>fient 8, nam 4. 2. 11. Dicemus ergo quod 8 unci&ecedil; per<lb/>fectionis nouem componentur ex 6 uncijs perfe<lb/>ctionis decem & una &longs;eptem alia quinque. Po&longs;t di<lb/>ces, &longs;i unciæ octo fiant 100, &longs;ex & una, & una, quot fient, eruntque un<lb/>ciæ aut libræ, aut ut uo cant marchæ perfectionis decem, & duo de<lb/>cim cum dimidia, ac duodecim cum dimidia perfectionis, ut &longs;e<lb/>ptem & ut quinque: licebit etiam propo&longs;itis terminis pluribus ex <lb/>repetita operatione idem facere, ueluti &longs;int ma&longs;&longs;æ perfectionis 10. <lb/>7. 5. & 2. uolo ma&longs;&longs;am perfectionis ut 8. Tu &longs;cis quod ex 10. 7 & 5. <lb/>fit ma&longs;&longs;a perfectionis nouem data lege &longs;ub 6. 1 & 1. nunc habeo iam <lb/>perfectam ut 9, aliam ut 2, detraho 2 ex 8, relinquitur 6 & 8, x 9 re<lb/>linquitur 1, iunge fient 7, erunt ergo &longs;eptem unciæ, in <lb/>

<figure id="fig158"></figure><lb/>minus 9 à 9, relinquitur 2 & 4, iunge omnia re&longs;idua <lb/>fient 8, nam 4. 2. 11. Dicemus ergo quod 8 unci&ecedil; per<lb/>fectionis nouem componentur ex 6 uncijs perfe<lb/>ctionis decem & una &longs;eptem alia quinque. Po&longs;t di<lb/>ces, &longs;i unciæ octo fiant 100, &longs;ex & una, & una, quot fient, eruntque un<lb/>ciæ aut libræ, aut ut uo cant marchæ perfectionis decem, & duo de<lb/>cim cum dimidia, ac duodecim cum dimidia perfectionis, ut &longs;e<lb/>ptem & ut quinque: licebit etiam propo&longs;itis terminis pluribus ex <lb/>repetita operatione idem facere, ueluti &longs;int ma&longs;&longs;æ perfectionis 10. <lb/>7. 5. & 2. uolo ma&longs;&longs;am perfectionis ut 8. Tu &longs;cis quod ex 10. 7 & 5. <lb/>fit ma&longs;&longs;a perfectionis nouem data lege &longs;ub 6. 1 & 1. nunc habeo iam <lb/>perfectam ut 9, aliam ut 2, detraho 2 ex 8, relinquitur 6 & 8, x 9 re<lb/>linquitur 1, iunge fient 7, erunt ergo &longs;eptem unciæ, in <lb/>

<arrow.to.target n="fig159"></arrow.to.target><lb/>quibus &longs;ex erunt perfectionis, ut 9 & 1 perfectionis ut <lb/>2, & totum erit perfectionis ut octo. Duc ergo, ut ex<lb/>plores ueritatem, 6 in 9 fit 54, duc 2 in 1 fit 2, iunge fit 56 <lb/>diuide per 7 exit 8 perfectio quæ&longs;ita.</s>

<figure id="fig159"></figure><lb/>quibus &longs;ex erunt perfectionis, ut 9 & 1 perfectionis ut <lb/>2, & totum erit perfectionis ut octo. Duc ergo, ut ex<lb/>plores ueritatem, 6 in 9 fit 54, duc 2 in 1 fit 2, iunge fit 56 <lb/>diuide per 7 exit 8 perfectio quæ&longs;ita.</s>

<s>Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire do drantem

<pb pagenum="201"/>militer l n ip&longs;ius l m, iuxta pro<lb/>

<arrow.to.target n="fig160"></arrow.to.target><lb/>portionem h, &longs;umatur rur&longs;us <lb/>

<figure id="fig160"></figure><lb/>portionem h, &longs;umatur rur&longs;us <lb/>

<arrow.to.target n="marg626"></arrow.to.target><lb/>de ip&longs;ius a b pars &longs;ecundum h, <lb/>

<arrow.to.target n="marg627"></arrow.to.target><lb/>& n o ip&longs;ius k l, &longs;ecundum ean <lb/>dem proportionem. Et rur&longs;us <lb/>

<arrow.to.target n="marg628"></arrow.to.target><lb/>&longs;umatur e f æqualis d b, & o p <lb/>

<arrow.to.target n="marg628"></arrow.to.target><lb/>&longs;umatur e f æ