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version 1.9, 2004/01/29 19:37:42

version 1.33, 2008/01/25 15:51:41

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<s id="id000380">Si quis dicat in qua proportione &longs;unt infinitæ <lb/>quantitates analogæ cum 12, quæiunctæ efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, & in ea pro<lb/>portione <expan abbr="er&utilde;t">erunt</expan> illæ quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, & 132 diuide per 12, exeunt 11 & 5, & ita eruntin proportione 11 <lb/>ad 5 experiaris, & inuenies, & demon&longs;tratur ex prioribus.</s>

<s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>uæftio.<emph.end type="italics"/></s>

<s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>uæ&longs;tio.<emph.end type="italics"/></s>

<s id="id000382">Propo&longs;itio decimanona.</s>

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<s id="id000385">Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; 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 &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; 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">aut</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">oes</expan> <lb/>fiant &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="id000387">E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti

<s id="id000387">E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, ea ra<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, ante penultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. </s>

<s id="id000388">Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propo&longs;itas a b c d e f g h quan<lb/>titates primi ordinis, &longs;ed quantitates æquales quæ <expan abbr="con&longs;tãt">con&longs;tant</expan> ex quan <lb/>titatis. </s>

<s id="id000389">primi ordinis, & &longs;upplementis, appellabo quantitates &longs;ecun<lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e<lb/>ptem, & &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>

<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, eara<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, antepenultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. </s>

<s id="id000389">primi ordinis, & fupplementis, appellabo quantitates &longs;ecun<lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e<lb/>ptem, & &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>

<s id="id000390">Erunt er <lb/>go &longs;upplementa i k l m n o p, quætanto erunt maiora quanto quan <lb/>titates primi ordinis &longs;unt minores, & contrà tanto maiora, quanto <lb/><expan abbr="quãtitates">quantitates</expan> primi ordinis &longs;unt maiores. </s>

<s id="id000391">quantitates <expan abbr="a&utilde;t">aunt</expan> &longs;ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>

<s id="id000391">quantitates <expan abbr="a&utilde;t">aut</expan> &longs;ecundi ordi<lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>

<s id="id000392">Hæcuolui pluribus <lb/>agere, ut dilucidior e&longs;&longs;et propo&longs;itio. </s>

<s id="id000393">quæ licet <expan abbr="nõ">non</expan> &longs;it difficilis, e&longs;t <expan abbr="tam&etilde;">tamen</expan> <lb/>confu&longs;a ualde propter multitudinem <expan abbr="quantitat&utilde;">quantitatum</expan> & ordinum. </s>

<s id="id000394">Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, & utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. & 9.4 & 1. quæ iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, quæ &longs;unt octies 64, & eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, & erunt 556, &longs;imul iuncta & eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti<lb/>tatum omnium primi ordinis, & e&longs;t tale <expan abbr="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or<lb/>dinis unius demon&longs;tratio h&etail;c e&longs;t. </s>

<s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con <lb/>&longs;tant ex quatuor partibus quarum du&etail; &longs;unt quadrata partium, reli<lb/>quæ duæ &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit æqua<lb/>lis 1, & p &etail;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&etail;qualia">fuerunt&etail;qualia</expan> mutuò quanti <lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="a&utilde;t">aunt</expan> &etail;qualis m. </s>

<s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con<lb/>&longs;tant ex quatuor partibus quarum du&etail; &longs;unt quadrata partium, reli<lb/>quæ duæ &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit æqua<lb/>lis 1, & p &etail;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&etail;qualia">fuerunt &etail;qualia</expan> mutuò quanti<lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="a&utilde;t">aut</expan> &etail;qualis m. </s>

<s id="id000396"><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis hab entibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutuò æqualia ip&longs;is quan <lb/>titatibus quod quadrata partium <expan abbr="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam & h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua

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<s id="id000400">Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb/>&longs;ecundi ordinis, prout &longs;unt diui&longs;a in partes addito quadrato a, &longs;int <lb/>dupla quadratis primarum quantítatum, &longs;imul pariter acceptis. </s>

<s id="id000401">Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e æquale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;<lb/>dem primi ordinis pariter acceptis. </s>

<s id="id000402">Con&longs;tatigitur, quod duplum i<lb/>in b e&longs;t æquale duplo h in ip&longs;um b, quia h & i &longs;unt æquales, & du<lb/>plum k in ip&longs;um c, e&longs;t æquale quadruplo h in idem c, quia k e&longs;t du<lb/>pla h, & &longs;imiliter duplum l in ip&longs;um d e&longs;t æquale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, & ita procedendo erunt illa dupla producta æ<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio quòd &longs;upplementum l continet h ter, duplicabo tria & fi<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d æquale e&longs;t &longs;excuplo h in ip&longs;um d. </s>

<s id="id000403">Quo con<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> e&longs;&longs;e æqualia quadratis ip&longs;a<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan<lb/>titates productis illis prioribus, & fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede<lb/>cies, & in h quindecies æquale duplo producti uniu&longs;cuiu&longs;que quan<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a<lb/>rum quantitarum, at quadratum a e&longs;t &etail;quale producto ex h in eam, <lb/>qu&etail; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"/><lb/>h per demon&longs;trata ab Euclide, & pariter de quadrato b, quod e&longs;t &etail;<lb/>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con <lb/>tinet h, & ita quadratum c æquale e&longs;t ei, quod continetur &longs;ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & &longs;imiliter de <lb/>quadrato c & omnibus reliquis, u&longs;que ad h ip&longs;um. </s>

<s id="id000403">Quo con<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> e&longs;&longs;e æqualia quadratis ip&longs;a<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan<lb/>titates productis illis prioribus, & fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede<lb/>cies, & in h quindecies æquale duplo producti uniu&longs;cuiu&longs;que quan<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a<lb/>rum quantitatum, at quadratum a e&longs;t &etail;quale producto ex h in eam, <lb/>qu&etail; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"/><lb/>h per demon&longs;trata ab Euclide, & pariter de quadrato b, quod e&longs;t &etail;<lb/>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con<lb/>tinet h, & ita quadratum c æquale e&longs;t ei, quod continetur &longs;ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & &longs;imiliter de <lb/>quadrato c & omnibus reliquis, u&longs;que ad h ip&longs;um. </s>

<s id="id000404">Gratia ergo exem

<s id="id000404">Gratia ergo exem<pb pagenum="20" xlink:href="015/01/039.jpg"/>pli quadratum a, erit æquale producto ex h in omnes quantitates &longs;e<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, & &longs;imiliter quotus e&longs;t numerus quan <lb/>títatum incipiendo à b, & quotus e&longs;t numerus quantitatum incipi<lb/>endo à c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om<lb/>nium quantitatum &longs;imul iuncta &longs;unt æqualia productis ex h in &longs;in<lb/>gulas illarum toties &longs;umptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uer&longs;us a. <lb/></s>

<pb pagenum="20" xlink:href="015/01/039.jpg"/>pli quadratum a, erit æquale producto ex h in omnes quatitates &longs;e<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, & &longs;imiliter quotus e&longs;t numerus quan <lb/>títatum incipiendo à b, & quotus e&longs;t numerus quantitatum incipi<lb/>endo à c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om<lb/>nium quantitatum &longs;imul iuncta &longs;unt æqualia productis ex h in &longs;in<lb/>gulas illarum toties &longs;umptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uer&longs;us a. <lb/></s>

<s id="id000405">Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &etail;quale e&longs;t producto <lb/>eiu&longs;dem quantitatis, & dupli omnium &longs;equentium primi ordinis in <lb/>ip&longs;am minimam quantitatem, uelut quadratum a e&longs;t æquale produ <lb/>cto ex h in a, & in duplum b c d e f g h, hoc <expan abbr="aut&etilde;">autem</expan> facile e&longs;t probare in <lb/>his quantitatibus, quia &longs;i quadratum a e&longs;t æquale producto h in o<lb/>mnes quantitates &longs;ecundi ordinis, & omnes quantitates &longs;ecundi or <lb/>dinis &longs;imul &longs;umptæ &longs;unt &etail;quales ip&longs;i a, & duplo <expan abbr="reliquar&utilde;">reliquarum</expan> primi or <lb/>dinis, quia tales quantitates &longs;unt æquales &longs;uis &longs;upplementis uici&longs;<lb/>&longs;im, ut h cum i, k cum g, f cum l, e <expan abbr="c&utilde;">cum</expan> m, ergo tam &longs;upplementa, quàm <lb/>quantitates primi ordinis &longs;unt dimidium quantitatum &longs;ecundi or<lb/>dinis, ergo duplum quantitatum primi ordinis e&longs;t dimidium quan <lb/>titatum &longs;ecundi ordinis, uerùm de b dico idem accidere, quia qua<lb/>dratum b e&longs;t &etail;quale producto ex h in b, & in duplum reliquarum à <lb/>b, &longs;cilicet duplum c d e f g h, & hoc e&longs;t o&longs;tendere, quod i&longs;t&etail; quantita <lb/>tes &longs;unt dimidium totidem quantitatum æqualium b, nam c e&longs;t mi<lb/>nor b in h, & &longs;upplementum p quod e&longs;t æquale ip&longs;i b, &longs;i tota h p fiat <lb/>æqualis ip&longs;i b, ut pote h q erit ip&longs;a q dempta h æqualis ip&longs;i c, ergo <lb/>quantitates primi ordinis &longs;emper &longs;unt æquales &longs;upplementis non <lb/>ueris, &longs;ed prioris quantitatis a&longs;&longs;umptæ, &longs;eu in comparatione ad il<lb/>lam, quadratum igitur b e&longs;t æquale producto ex h in b, & in duplum <lb/>c d e f g h, & &longs;imiliter per eadem, quadratum c e&longs;t æquale producto <lb/>ex h in c, & in duplum d e f g h, & &longs;ic de alijs. </s>

<s id="id000406">Habemus ergo, quod <lb/>quadrata a b c d e f g h &longs;imul iuncta &longs;unt æqualia producto ex h in <lb/>a, & in duplum reliquarum, & ex h in b, & in duplum reliquarum <lb/>&longs;equentium, & producto ex h in c &longs;emel, & in duplum &longs;equentium <lb/>u&longs;que ad h, & ita de reliquis. </s>

<s id="id000407">hoc enim e&longs;t, quod nuper demon&longs;traui<lb/>mus. </s>

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<s id="id000435">Sit proportio a ad b maior quàm c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, quàm c ad d, et minor quàm <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confu&longs;a minor quàm a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo quòd e ha<lb/>bet proportionem ad b, quam c ad d, quæ <expan abbr="aut&etilde;">autem</expan> c ad d minor, quám <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, quàm a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi<lb/>nor, quàm confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba<lb/>bitur, & primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma<lb/>ior proportio a f ad b d, quàm a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, quàm a ad b.</s>

<s id="id000435">Sit proportio a ad b maior quàm c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, quàm c ad d, et minor quàm <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiam decimam ha<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confu&longs;a minor quàm a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo quòd e ha<lb/>bet proportionem ad b, quam c ad d, quæ <expan abbr="aut&etilde;">autem</expan> c ad d minor, quám <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, quàm a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi<lb/>nor, quàm confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba<lb/>bitur, & primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma<lb/>ior proportio a f ad b d, quàm a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>eandem tertiam decimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, quàm a ad b.</s>

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<s id="id000487">Sed <lb/>quia fit per mu&longs;culos, qui trahuntur: hic autem motus e&longs;t uiolen<lb/>tus, ideò per con&longs;equentiam fatigat. </s>

<s id="id000488">Qui uerò naturalis, e&longs;t ut re<lb/>deat corpus ad &longs;uum locum, igitur naturalis e&longs;t ad locum. </s>

<s id="id000489">Sed <lb/>uiolenti finis e&longs;t, ut protrudatur ex loco in quo e&longs;t, non habens cer<lb/>tum finem. </s>

<s id="id000490">licet enim qui trahit, ad &longs;uum locum trabat, non tamen <lb/>ad locum mobilis.</s>

<s id="id000490">licet enim qui trahit, ad &longs;uum locum trahat, non tamen <lb/>ad locum mobilis.</s>

<s id="id000491">Propo&longs;itio uige&longs;imaoctaua.</s>

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<s id="id000518">Cùm enim ex præcedenti augeantur &longs;emper ob medium, & cau<lb/>fa, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelo cior in fine quàm in alia <lb/>parte temporis. </s>

<s id="id000518">Cùm enim ex præcedenti augeantur &longs;emper ob medium, & cau<lb/>&longs;a, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelocior in fine quàm in alia <lb/>parte temporis. </s>

<s id="id000519">In uiolento autem, cùm perueniat ad finem de&longs;init </s>

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<s id="id000532">Propo&longs;itio trige&longs;imatertia.</s>

<s id="id000532">Propo&longs;itio trige&longs;ima ertia.</s>

<s id="id000533">Omnia duo mobilia æqualis undique magnitudinis, quæ æquali <lb/>in tempore æqualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"/></s>

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<s id="id000550">Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="id.015.01.047.2.jpg" xlink:href="015/01/047/2.jpg"/><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s>

<s id="id000551">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 id="id000551">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/>etiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>

<s id="id000552"><arrow.to.target n="marg90"/><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e & d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo æqualis a b, igitur cum a c, e & d &longs;int analogæ, erunt & a b c, <lb/>a b, & d analogæ, quod fuit demon&longs;trandum.</s>

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<s id="id000695">& dico quae in <lb/>animalibus tres &longs;unt cau&longs;æ, una, quia e&longs;t mi&longs;tus, & habet repugnan<lb/>tiam: &longs;ecunda, quia e&longs;t de loco ad locum, motus autem c&oelig;li e&longs;t in lo <lb/>co: tertia e&longs;t communis etiam c&oelig;lo, et e&longs;t, <expan abbr="quoniã">quoniam</expan> non e&longs;t ratio finis. <lb/></s>

<s id="id000696">Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. </s>

<s id="id000697">Quid e&longs;t ergo <lb/>proportio, <expan abbr="c&utilde;">cum</expan> &longs;it <expan abbr="ultim&utilde;">ultimum</expan> uoluntatis uit&etail;, ut obtemperet primæ cau&longs;æ, <lb/>ideo illud e&longs;t <expan abbr="ultim&utilde;">ultimum</expan>, &qring; mouet. </s>

<s id="id000698">E&longs;t <expan abbr="a&utilde;t">aunt</expan> idem uelle, & po&longs;&longs;e. </s>

<s id="id000698">E&longs;t <expan abbr="a&utilde;t">aut</expan> idem uelle, & po&longs;&longs;e. </s>

<s id="id000699">In natura

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<s id="id000714">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&etail; 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. </s>

<s id="id000714">Hoch declarato ponatur m spatium compositum 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&etail; 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. </s>

<s id="id000715">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>

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<s id="id000740">Sit rur&longs;us ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per &longs;uos <lb/>denominatores, & erit ut a in quinque annis. </s>

<s id="id000741">b in decem, c in decem<lb/>&longs;eptem circuant, & redeant ad idem punctum, & quia quin que nu<lb/>merat decem, & decem, & decem&longs;eptem &longs;unt numeri inuicem pri<lb/>mi, ducam decem in decem&longs;eptem fiunt centum &longs;eptuaginta. </s>

<s id="id000742">Con<lb/>&longs;tat igitur c quadragíes, b quinquagies &longs;emel, a &longs;exagies octies cir<lb/>cumuerti, & redire ad idem punctum: ergo rur&longs;us coibunt po&longs;t tot <lb/>annos in eo, dico modo, quod non ante: nam &longs;i non &longs;it, ut in trigin<lb/>ta tribus annis. </s>

<s id="id000743">gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decem&longs;eptem</expan>, decem, & quin<lb/>que, & relinquentur &longs;exdecim tria & tria, & rur&longs;us ex &longs;exdecim tres <pb pagenum="39" xlink:href="015/01/058.jpg"/>circuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem &longs;i iam &longs;upponi<lb/>mus 17 & 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione./><lb/></s>

<pb pagenum="39" xlink:href="015/01/058.jpg"/>cir cuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem &longs;i iam &longs;upponi/><lb/>mus 17 & 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione./><lb/></s>

<s id="id000744">Igitur &longs;equuntur eadem corrolaria, quæ dicta &longs;unt.</s>

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<s id="id000767">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. </s>

<s id="id000768">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/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k l. </s>

<s id="id000768">Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cuique<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/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k l. </s>

<s id="id000769">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>

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<pb pagenum="41" xlink:href="015/01/060.jpg"/>&longs;a a c, quia multiplex ei. </s>

<s id="id000777">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre. <lb/></s>

<s id="id000777">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre<lb/></s>

<s id="id000778">gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom <lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>

<s id="id000779">Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, & <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommen&longs;a a k & a c, & circulo ad dita a k. </s>

<s id="id000780">Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, & &longs;i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra<lb/>ctis, ut a k & a l reci&longs;a: & a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu <lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi<lb/>cæ numeri, &longs;eu latera corporum rhete, a k uero & a l, & huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"/></s>

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<s id="id000803">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" xlink:href="015/01/061.jpg"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b &longs;unt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>

<pb pagenum="42" xlink:href="015/01/061.jpg"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b &longs;unt &longs;ex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>

<s id="id000804">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. </s>

<s id="id000805">Cum enim tempora di<lb/>uer&longs;a diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"/><lb/>numeri in eadem ratione. </s>

<s id="id000806">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. </s>

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<s id="id000815">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

<s id="id000815">Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & &longs;int a d, a e incommen&longs;æ, dico quòd a b c nunquam con<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à primo: &longs;i non con<pb pagenum="43" xlink:href="015/01/062.jpg"/><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"/><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/></s>

<pb pagenum="43" xlink:href="015/01/062.jpg"/><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"/><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/></s>

<s id="id000816">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. </s>

<s id="id000817">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>

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<pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini <lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>corundem.<lb/><arrow.to.target n="marg143"/></s>

<pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini<lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>eorundem.<lb/><arrow.to.target n="marg143"/></s>

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<pb pagenum="45" xlink:href="015/01/064.jpg"/>momum autem, & gingiber numerantur inter medicinas calídas <lb/>tertij gradus, & hoc opus comparatur ad corpus &longs;icut dicit Gale<lb/>nus, & Serapio non ad linguam, ut medici no&longs;tri temporis interpre <lb/>tantur. </s>

<s id="id000843">Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordl <lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>

<s id="id000843">Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordi<lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>

<s id="id000844">Sed non oportet h&etail;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam quòd con&longs;u&longs;a &longs;it per &longs;eip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en <lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>

<s id="id000844">Sed non oportet h&etail;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam quòd confu&longs;a &longs;it per &longs;e ip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en<lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>

<s id="id000845">Nam de &longs;icco, <lb/>& humido, cum &longs;int longè minoris actionis, quàm calidum, & fri<lb/>gidum, & præcipuè humidum, non uideo quomodo po&longs;sit Gale<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, quæ de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s>

<s id="id000846">Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"/><lb/>& enulam & zelim in tertio ordine calidorum & humidorum: & <lb/>inter frigidas, & humidas in tertio portulacam, aizoum, & uirgam <lb/>pa&longs;toris, & fungos. </s>

<s id="id000847">Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&etail; &longs;int humidæ. </s>

<s id="id000848">&longs;ecundum, <lb/>quando dicit medicinas calídas, aut frigidas, atque humídas in ter<lb/>tio ordine, intelligit &longs;olum de qualitate actiua &longs;cilicet caliditate, uel <lb/>frigiditate, & non de humida qualitate, quod o&longs;tendit de gingibe<lb/>re, & enula, dicens, quod &longs;unt calidæ in tertio ordine, & humidæ <lb/>humido crudo, non au&longs;us addere ordinem, quia non uídit ratio<lb/>nem, qua po&longs;&longs;ent dici humidæ in tertio. </s>

<s id="id000849">Et clarius in capite de zei<lb/>len, quem &longs;tatuerat inter medicinas calidas, & humidas in tertio, di <lb/>cit quod e&longs;t calida in tertio, & humida in primo, ergo non intelligit <lb/>per medicinas calidas & humidas in tertio ordine, quod &longs;int humi<lb/>dæ in tertio ordine. </s>

<s id="id000850">Clarius etiam de frigidis & humidis, nam por<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, & quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, & hu<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad<lb/>ijcit, quod e&longs;t &longs;iccum parum, & de uirga pa&longs;toris nihil dicit de hu<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun<lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, & portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, quàm caliditas, & calida magis hu<pb pagenum="46" xlink:href="015/01/065.jpg"/>mectant, quia magis penetrat uis medicamenti, & hæc regula de <lb/>humido, & &longs;icco e&longs;t generalis apud Serapionem, quod non intelli<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicinæ &longs;iccæ in tertio, & for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, quæ in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con<lb/>&longs;iderandum relinquo.</s>

<pb pagenum="46" xlink:href="015/01/065.jpg"/>mectant, quia magis penetrat uis medicamenti, & hæc regula de <lb/>humido, & &longs;icco e&longs;t generalis apud Serapionem, quod non intelli<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicinæ &longs;iccæ in tertio, & for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, quæ in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con <lb/>&longs;i derandum relinquo.</s>

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<s id="id000859">Suppo&longs;ito ergo quod primò ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="id.015.01.065.1.jpg" xlink:href="015/01/065/1.jpg"/><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" xlink:href="015/01/066.jpg"/>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. </s>

<pb pagenum="47" xlink:href="015/01/066.jpg"/>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/>locum, “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. </s>

<s id="id000860">Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s>

<s id="id000861">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. </s>

<s id="id000862">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. </s>

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<s id="id000864">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 id="id000865">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/><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"/><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

<s id="id000865">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/><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"/><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 relinquuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag<pb pagenum="48" xlink:href="015/01/067.jpg"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio &longs;ecundi ordinis. </s>

<pb pagenum="48" xlink:href="015/01/067.jpg"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio &longs;ecundi ordinis. </s>

<s id="id000866">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. </s>

<s id="id000867">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>

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<s id="id000873">Cum enim proportionis medium &longs;it latus numeri eo quod ex bi <lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, quæ demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmeticæ de omnibus binomijs cum &longs;uis </s>

<s id="id000874"><arrow.to.target n="marg151"/><lb/>reci&longs;is, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & &longs;uum reci&longs;um, igitur cum proportio producto<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci<lb/><arrow.to.target n="marg152"/><lb/>&longs;a crunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo <02> nu <lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, & omnia com<lb/>men&longs;a illi, e&longs;t duplicata ei quæ ad <02> numeri.<lb/><arrow.to.target n="marg154"/></s>

<s id="id000874"><arrow.to.target n="marg151"/><lb/>reci&longs;is, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & &longs;uum reci&longs;um, igitur cum proportio producto<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci<lb/><arrow.to.target n="marg152"/><lb/>&longs;a erunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo <02> nu<lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, & omnia com<lb/>men&longs;a illi, e&longs;t duplicata ei quæ ad <02> numeri.<lb/><arrow.to.target n="marg154"/></s>

<s id="id000875"><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro<lb/>po&longs;. </s>

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<s id="id000893">Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. </s>

<s id="id000894">At ca<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean<lb/>dem partem, &longs;eu aliam. </s>

<s id="id000895">Qui uerò naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><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. </s>

<s id="id000895">Qui uerò naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><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, quandoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latent nos. </s>

<s id="id000896">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 aëris <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. </s>

<s id="id000896">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 aëris <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, itaque <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">aeris</expan> e&longs;&longs;et at que continuus. </s>

<s id="id000897">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. </s>

<s id="id000898">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. </s>

<s id="id000899">A'&longs;en&longs;u quidem, quoniam nebul&etail;, <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&etail; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. </s>

<s id="id000900">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. </s>

<s id="id000900">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. </s>

<s id="id000901">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. </s>

<s id="id000902">Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. </s>

<s id="id000903">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>

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<s id="id000912">Dico etiam, quod tardius ad c quàm d. </s>

<s id="id000913">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. </s>

<s id="id000914">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. </s>

<s id="id000915">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. </s>

<s id="id000915">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/>tum ob cau&longs;am dictam. </s>

<s id="id000916">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 id="id000931">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>

<s id="id000931">Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub<lb/>&longs;tantia, &longs;eu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</s>

<s id="id000932">Propo&longs;itio &longs;exage&longs;imaprima.</s>

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<s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><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. </s>

<s id="id000979">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. </s>

<s id="id000979">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. </s>

<s id="id000980">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>

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<s id="id000999"><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uerfim moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uer&longs;im moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer<lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001001">Por

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</row>

</table>

<s id="id001070"><arrow.to.target n="marg196"/><lb/>co o p q r s in proportione b ad a, & tuxyz in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæt <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001070"><arrow.to.target n="marg196"/><lb/>eo o p q r s in proportione b ad a, & t u x y z in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæ t <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001071">Quia ergo propor<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, quæ e&longs;t t ad <lb/>o, & ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, & o ad <lb/><arrow.to.target n="marg201"/><lb/>n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, & u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&etail; <lb/>e&longs;t t ad o erit h ad e, duplicata ei quæ e&longs;t g ad b, & ita de reliquis, & <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, quæ e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"/><lb/>ad u duplicatam ei, quæ e&longs;t o ad t. </s>

<s id="id001072">Aliter & euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e & h duplicata ei quæ e&longs;t b & g <lb/>ad a, ut &longs;upra, & quadrati b ad quadratum a, & quadrati g ad qua<lb/><arrow.to.target n="marg203"/><lb/>dratum a duplicata his quæ b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. </s>

<s id="id001073">Et conuertendo qua<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"/>hic & erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s>

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<s id="id001094">Quilibet &longs;ector &longs;phæræ æqualis e&longs;t cono, cuius ba&longs;is e&longs;t circu<lb/>lus æqualis &longs;uperficiei eiu&longs;dem portionis, altitudo uerò &longs;phæræ &longs;e<lb/>midiameter. </s>

<s id="id001095">Proportio &longs;phæræ ad &longs;ectorem datum, e&longs;t duplica<lb/>ta ei, qu&etail; e&longs;t dimetientis ad lineam, quæ à uertice portionis ad lim<lb/>bum. </s>

<s id="id001096">Cum enim &longs;phæra &longs;it æqualis cono, cuius ba&longs;is e&longs;t maior cir<lb/>culus, altitudo uerò dupla dimetienti per tertiam harum, quæ hic <lb/><arrow.to.target n="marg209"/><lb/>proponuntur: erit &longs;phæra æqualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it æqualis diametro &longs;phæræ, altitudo uerò &longs;e<lb/>midiameter &longs;phæræ. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;emidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. </s>

<s id="id001098">Sed proportio talium circulorum e&longs;t duplicata propor<lb/><arrow.to.target n="marg211"/><lb/>tioni &longs;emidimetientium, igitur proportio &longs;phæræ ad &longs;uum &longs;ecto<lb/>rem e&longs;t ueluti dimetientis &longs;phæræ ad lineam, quæ á uertice portio<lb/><arrow.to.target n="marg212"/><lb/>nis ad limbum duplicata. </s>

<s id="id001099">Cuicunque portioni &longs;phæræ conus ille <lb/>habetur æqualis, qui ba&longs;im hab eat eandem cum portione, altitudi<lb/>nem uerò lineam rectam, quæ ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;phæræ unà cum alti<lb/>tudine reliquæ portionis habet ad eandem reliquæ portionis alti<lb/><arrow.to.target n="marg213"/><lb/>tudinem. </s>

<s id="id001100">Earum &longs;phæræ portionum, quæ æqualibus &longs;uperfi<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas &longs;phæræ maxima exi&longs;tit. </s>

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<s id="id001163">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;ectionem/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001163">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;ectionem <lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001164"><arrow.to.target n="marg238"/><lb/>metro æquidi&longs;tanti ducta. </s>

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<pb pagenum="62" xlink:href="015/01/081.jpg"/><arrow.to.target n="marg248"/><lb/>miles, &longs;ed unam per&longs;æpe in utri&longs; que e&longs;&longs;e uult. </s>

<s id="id001179">Sed & hoc Archime<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axem/> coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

<s id="id001179">Sed & hoc Archime<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axem coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

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<s id="id001203">Propo&longs;itio &longs;eptuage&longs;imaprima.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;penfionem inuenire.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;pen&longs;ionem inuenire.</s>

<s id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen <lb/>dens per a b leuius quam per b <expan abbr="crectã">crectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem <lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat folum particu<lb/>lam ex decem millibus recti ponderis.</s>

<s id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen<lb/>dens per a b leuius quam per b <expan abbr="crectã">c rectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem<lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat &longs;olum particu<lb/>lam ex decem millibus recti ponderis.</s>

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Line 4750

<s id="id001245"><arrow.to.target n="marg265"/><lb/>uitas ictus, ponderis, fractura, mollities eius, quodicitur, mollities <lb/>eius, quod excipit ictum, motus eiu&longs;dem, & figura lata, & inæqua<lb/>lis. </s>

<s id="id001246">Durities ergo, quatenus fracturæ opponitur, aliud e&longs;t, quam ut <lb/>molliciei: & utra que e&longs;t cau&longs;a, quæ augetictum, ut reliquæ <lb/> oppo&longs;itæ minuunt, dicemus autem de his inferius.</s>

<s id="id001246">Durities ergo, quatenus fracturæ opponitur, aliud e&longs;t, quam ut <lb/>mollitiei: & utra que e&longs;t cau&longs;a, quæ auget ictum, ut reliquæ <lb/> oppo&longs;itæ minuunt, dicemus autem de his inferius.</s>

<s id="id001248">Propo&longs;itio &longs;eptuage&longs;imaquints.</s>

<s id="id001248">Propo&longs;itio &longs;eptuage&longs;ima quinta.</s>

<s id="id001249">Proportionem immoti in aqua ad immotum in terra in excipien <lb/>do ictum inuenire.</s>

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<s id="id001273">Sit ergo ut a b nauis tangat ro&longs;tro b c &longs;ic ut <lb/>diffugiat, erit hypomochlium c, & &longs;i tangat <lb/>e f hypomochlium e&longs;t in d dupla, ergo e&longs;t c b <lb/>ip&longs;i d e, igitur ictus duplo minor excipitur à <lb/>c b quàm ef. </s>

<s id="id001274">E&longs;t etiam tempus longè maius, <lb/>quo excipit ictum ef, quàm b c: &longs;tatim enim di&longs;cedit b c occurrit que<lb/>alijs partibus, in c f autem impingit, & angulus a d c e&longs;t longè ma<lb/>ior recto, quàm a b f: ob hæc igitur longè maior e&longs;t ictus c f quàm <lb/>b c: uocant autem hoc declinationem.</s>

<s id="id001274">E&longs;t etiam tempus longè maius, <lb/>quo excipit ictum ef, quàm b c: &longs;tatim enim di&longs;cedit b c occurritque<lb/>aliis partibus, in c f autem impingit, & angulus a d c e&longs;t longè ma<lb/>ior recto, quàm a b f: ob hæc igitur longè maior e&longs;t ictus c f quàm <lb/>b c: uocant autem hoc declinationem.</s>

<s id="id001275">Propo&longs;itio &longs;eptuage&longs;ima&longs;eptima.</s>

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<s id="id001279">Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di

<s id="id001279">Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di<pb pagenum="67" xlink:href="015/01/086.jpg"/>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 nunc, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><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. </s>

<pb pagenum="67" xlink:href="015/01/086.jpg"/>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/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><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. </s>

<s id="id001280">Quod autem in æquali tempore mouebi<lb/>tur nauis per c k & h c patet ex a&longs;&longs;umpto inferius declarando.</s>

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<pb pagenum="71" xlink:href="015/01/090.jpg"/>tem &longs;ubmergeretur. </s>

<s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardiu, moueretur. </s>

<s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardius, moueretur. </s>

<s id="id001340">Pro<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maloribus no&longs;tris |fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien <lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>

<s id="id001340">Pro<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maioribus no&longs;tris |fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien<lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>

<s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini<lb/>mum, ut nauis &longs;acile ab eo inuerteretur. </s>

<s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini<lb/>mum, ut nauis facile ab eo inuerteretur. </s>

<s id="id001342">Vnde etiam nunc minus <lb/>minime habent tam quantitate, quam etiam altitudine, quod uo<lb/>cant Trinehetum, &longs;olum enim &longs;u&longs;tinet nauim, quæ à uentis, uel un<lb/>dis mergi &longs;olet: ab undis ubi humilior e&longs;t, à uentis à lateribus, et an<lb/>teriore parte. </s>

<s id="id001343">Vnde humile, & exiguum uelum efficit, ut nauis ante<lb/>riore parte leuis, nec mergatur prona à uentis, nec aquas ea exci<lb/>piat, nec tamen impelli pote&longs;t nauis in &longs;copulos, nec euerti ob cau<lb/>&longs;as dictas: ob quæ in magnis tempe&longs;tatibus hoc ip&longs;o duntaxat uti <lb/>&longs;olent. </s>

<s id="id001344">Quod et&longs;i nimium &longs;æuierint, etiam illud demittunt, & &longs;i <lb/>fieri pote&longs;t, etiam malum ip&longs;am quamuis &longs;ine uelo &longs;it. </s>

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<s id="id001351">Vltimo loco, quæ qua&longs;i mediæ, ante quidem <lb/>tenues, pò&longs;t latiores ad uelocem cur&longs;um, & ferendum onera aptæ, <lb/>& humiles altis: & leui ex ligno. </s>

<s id="id001352">Sed nos de uelorum uarieta<lb/>te loquimur, non ea', quæ ad malos pertinet. </s>

<s id="id001353">Con&longs;tat enim me<lb/>dio loco plus mouere, quam in extremis, ut infrà docebi<lb/>mus. </s>

<s id="id001354">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam &longs;yderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu<lb/>lum attuli&longs;&longs;ent. </s>

<s id="id001354">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam &longs;ijderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu<lb/>lum attuli&longs;&longs;ent. </s>

<s id="id001355">At nunc inuenta pyxide, & lapidis Her

<pb pagenum="72" xlink:href="015/01/091.jpg"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, & pote&longs;tate deformatum, ad amu&longs;<lb/>&longs;im contrahant. </s>

<s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uper&longs;iciei ueli ad uelum longitudine quidem, </s>

<s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uperficiei ueli ad uelum longitudine quidem, </s>

<s id="id001357"><arrow.to.target n="marg285"/><lb/>ac latitudine conflata per multiplicationem. </s>

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<s id="id001380">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. </s>

<s id="id001381">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/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><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. </s>

<s id="id001381">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/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge<lb/>nere, ubi uim fecundam 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. </s>

<s id="id001382">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. </s>

<s id="id001382">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 uehementer, 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. </s>

<s id="id001383">Cuius indi<lb/>cio e&longs;t Oceani maris, aquarum fluxus & refluxus. </s>

<s id="id001384">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. </s>

<s id="id001385">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

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<s id="id001399">Sit aurum a, & liquor b, quæ repleant uas c, & <lb/>pondus amborum &longs;it librarum quadraginta, & <lb/><figure id="id.015.01.093.1.jpg" xlink:href="015/01/093/1.jpg"/><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 duodecuplum ponderi liquoris, & <pb pagenum="75" xlink:href="015/01/094.jpg"/>&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 libræ ij, at auri pondus e&longs;t <lb/>libræ xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu<lb/>pla. </s>

<pb pagenum="75" xlink:href="015/01/094.jpg"/>&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. </s>

<s id="id001400">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>

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<s id="id001493">Et ita mediæ ad minorem quintupla, & ideò maioris ad

<pb pagenum="81" xlink:href="015/01/100.jpg"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>pcriculo&longs;a, ut in rotis moletrinis, & &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelo cior minor media, quanto media <lb/>maiore. </s>

<pb pagenum="81" xlink:href="015/01/100.jpg"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>periculo&longs;a, ut in rotis moletrinis, & &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelocior minor media, quanto media <lb/>maiore. </s>

<s id="id001494">Rur&longs;us proportio partium maioris ad mediæ partes tripla <lb/>e&longs;t &longs;cilicet ccclx ad cxx, & mediæ ad <expan abbr="minor&etilde;">minorem</expan> tripla cxx ad xl, & pro<lb/>portio e&longs;t &longs;excupla, iterum igitur partes maioris ad mediam, & me<lb/>diæ ad minorem erunt in dupla proportione, utrobique, & e&longs;t pul<lb/>chrum. </s>

<s id="id001495">Ideò partes etiam minimæ rotæ erunt &longs;atis magnæ: nam <lb/>cum diameter &longs;it bes pedis, ambitus peripheriæ erit duorum pe<lb/>dum. </s>

<s id="id001496">1. unciarum uigintiquatuor: igitur diui&longs;a peripheria in xl par<lb/>ter, unaquæ que pars erit maior dimidia uncia.</s>

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<s id="id001509">Contingit hoc multis modis: primum acceptis duabus &longs;phæru<lb/>lis æqualibus ex cry&longs;tali&longs;ub&longs;tantia unaque demi&longs;&longs;a ab alti&longs;sima turri, <lb/>& men&longs;urato ictu per in&longs;trumentum præcedens, & &longs;ub totidem <lb/>momentis alia demi&longs;&longs;a in aquam, in de &longs;ub eodem tempore dimen<lb/>&longs;a altitudine, erit proportio &longs;patij ad &longs;patium, ut den&longs;itatis aquæ, ad <lb/>den&longs;itatem aëris. </s>

<s id="id001510">Item emi&longs;&longs;a &longs;phærula per in&longs;trumentum in aërem, <lb/>in de in aquam: & fumpta proportione. </s>

<s id="id001510">Item emi&longs;&longs;a &longs;phærula per in&longs;trumentum in aërem, <lb/>in de in aquam: & &longs;umpta proportione. </s>

<s id="id001511">Et uidimus &longs;corpionem, <lb/>qui <expan abbr="&longs;phærulã">&longs;phærulam</expan> creteam emittebat pedibus lxx, & in aqua per unum <lb/>& dimidium adeò, ut proportio fuerit, ut quinquaginta ad unum: <lb/>ideò e&longs;t fallax experimentum in uiolento motu: nam cum emitte<lb/>batur in aquam erat propè, & ob id in &longs;ummo robore: cùm in aë<lb/>rem, emittitur &longs;en&longs;im uis. </s>

<s id="id001512">De hoc ergo loquar.</s>

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<s id="id001544">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/></s>

<s id="id001545">Eademque ratio aliorum planorum. </s>

<s id="id001546">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>. </s>

<s id="id001547">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 id="id001547">Et ideo uincula in rami cibus duplicia dextra, & &longs;ini<lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamën longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s>

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<s id="id001550">Ex hoc etiam &longs;equitur, <lb/><figure id="id.015.01.104.1.jpg" xlink:href="015/01/104/1.jpg"/><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. </s>

<s id="id001551">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. </s>

<s id="id001552">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. </s>

<s id="id001553">Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s>

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<s id="id001558">Propo&longs;itio nonage&longs;ima&longs;ecunda.</s>

<s id="id001559">Propprtionem ponderis æqualis iuxta longitu dinis compara<lb/>tionem demon&longs;trare.</s>

<s id="id001559">Proportionem ponderis æqualis iuxta longitudinis compara<lb/>tionem demon&longs;trare.</s>

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<s id="id001578"><arrow.to.target n="marg317"/><lb/>profunditatem, & aquæ den&longs;itatem, &longs;ed quanto aqua den&longs;ior e&longs;t, <lb/>tanto minus nauis de&longs;cendit, & quanto minus den&longs;a, tanto magis: <lb/>ergo pari modo fermè redduntur mobiles, & in aqua dulci & &longs;al&longs;a, <lb/>ubi naues &longs;int &longs;imiles forma, pondere, magnitudine. </s>

<s id="id001579">Quia crgo ne<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe<lb/>netret, cum ergo quod facilius e&longs;t, præcedat, difficilius ergo naues

<s id="id001579">Quia ergo ne<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe<lb/>netret, cum ergo quod facilius e&longs;t, præcedat, difficilius ergo naues

<pb pagenum="87" xlink:href="015/01/106.jpg"/>utrinque mouebuntur, & quia inter duos quo&longs;cunque motus contra<lb/>rios <expan abbr="nõ">non</expan> e&longs;&longs;eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, & in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uelloco moueatur, uel ce<lb/><arrow.to.target n="marg318"/><lb/>dat, & ictus penetret, uel aër non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, & ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, & &longs;i durum &longs;it confringatur. <lb/></s>

<pb pagenum="87" xlink:href="015/01/106.jpg"/>utrinque mouebuntur, & quia inter duos quo&longs;cunque motus contra<lb/>rios <expan abbr="nõ">non</expan> e&longs;&longs;e eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, & in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uel loco moueatur, uel ce<lb/><arrow.to.target n="marg318"/><lb/>dat, & ictus penetret, uel aër non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, & ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, & &longs;i durum &longs;it confringatur. <lb/></s>

<s id="id001580">Proportio ergo rece&longs;&longs;us ad ictum e&longs;t ut temporis, & magnitudinis <lb/>partis, quæ cedit, & retro ce&longs;&longs;us po&longs;ito ictu tanquam monade.</s>

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<s id="id001598"><margin.target id="marg320"/>E<emph type="italics"/>x &longs;ecunda <lb/>animi com<lb/>muni &longs;enter <lb/>tia.<emph.end type="italics"/></s>

<s id="id001598"><margin.target id="marg320"/>E<emph type="italics"/>x &longs;ecunda <lb/>animi com<lb/>muni &longs;enten<lb/>tia.<emph.end type="italics"/></s>

<s id="id001599">Sed &longs;i &longs;it pars, erit &longs;umpta &longs;ecundum partem monadis ip&longs;ius, <lb/>ideò erit cognita iuxta nomen, uelut dimidium e&longs;t dimidium mo<lb/>nadis, dimi dium tertiæ partis monadis e&longs;t cognitum, quia tertia <lb/>pars e&longs;t cognita, & &longs;cimus, quanta pars a&longs;&longs;umatur illius. </s>

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<s id="id001622"><arrow.to.target n="marg328"/><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"/><lb/><figure id="id.015.01.107.2.jpg" xlink:href="015/01/107/2.jpg"/><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. </s>

<s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a cigi

<s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a c igi<pb pagenum="89" xlink:href="015/01/108.jpg"/>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. </s>

<pb pagenum="89" xlink:href="015/01/108.jpg"/>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. </s>

<s id="id001624">Ergo per <lb/>eandem omnia nota &longs;cilicet b c ad b a, & b c ad c a. </s>

<s id="id001625">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. </s>

<s id="id001626">Sed hoc e&longs;t æ<lb/><arrow.to.target n="marg330"/><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. </s>

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<pb pagenum="91" xlink:href="015/01/110.jpg"/>gnus &longs;plendor, referens corpus Solis. </s>

<s id="id001667">Propterea dico, quòd uel ac<lb/>eidit, quia homo non pote&longs;t ea hora intueri Solem, & etiam e&longs;t im<lb/>peditus à radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellulæ in centro Lun&etail;: & hic e&longs;t <lb/>&longs;plen dor Solis collectus in centro Lunæ. </s>

<s id="id001667">Propterea dico, quòd uel ac<lb/>cidit, quia homo non pote&longs;t ea hora intueri Solem, & etiam e&longs;t im<lb/>peditus à radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellulæ in centro Lun&etail;: & hic e&longs;t <lb/>&longs;plendor Solis collectus in centro Lunæ. </s>

<s id="id001668">po&longs;&longs;et etiam dici, quòd <lb/>Luna circa medium propter maculam non admitteret lumen, & ita <lb/>e&longs;&longs;et inæqualium partium.</s>

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<s id="id001719">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. </s>

<s id="id001720">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. </s>

<s id="id001721">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/><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"/><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/></s>

<s id="id001722">Siue &longs;it parabole, &longs;iue hiperbole, &longs;eu &longs;it alia coincidentium.</s>

<s id="id001722">Siue &longs;it parabole, &longs;iue hyperbole, &longs;eu &longs;it alia coincidentium.</s>

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<s id="id001728"><arrow.to.target n="marg351"/><lb/>inuer&longs;io, & quàm diuer&longs;a &longs;it rur&longs;us, & quòd attractio e&longs;t dimidium <lb/><arrow.to.target n="marg352"/><lb/>ponderis eleuati. </s>

<s id="id001729">Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, & pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"/><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diuifa proportione ponde <lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s>

<s id="id001729">Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, & pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"/><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diui&longs;a proportione ponde<lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s>

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<s id="id001759">Propterea dicendum e&longs;t, quòd qui ex commu<lb/>nibus propria nituntur demon&longs;trare, omnes corrumpunt di&longs;cipli<lb/>nas. </s>

<s id="id001760">Nihil deterius e&longs;t his mon&longs;tris. </s>

<s id="id001761">Nam et&longs;i hæc ratio uera e&longs;&longs;et: <lb/>non tamen reddit cau&longs;am, quia non e&longs;t ex proprijs principijs. </s>

<s id="id001762">Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, quàm b, ergo plus grauat cip&longs;um a <lb/>quàm b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, quòd ma<lb/>gis de&longs;ce&longs;&longs;erit. </s>

<s id="id001762">Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, quàm b, ergo plus grauat c ip&longs;um a <lb/>quàm b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, quòd ma<lb/>gis dece&longs;&longs;erit. </s>

<s id="id001763">Cau&longs;a ergo erroris e&longs;t, quod &longs;i ponatur angulus f b a <lb/>æqualis angulo f a b, & ponatur b f &etail;qualis b c, tun c in eodem tem<lb/>pore, in quo tran&longs;it dimidium c in e, tran&longs;ibit aliud dimidium c in f. <lb/></s>

<s id="id001764">quia &longs;eparat&etail; partes grauiores &longs;unt in c b, quàm c a, propter di&longs;tan<lb/>tiam ab hypomochlio, &longs;ed tunc uelo cius mouentur, & angulus fit <lb/>&etail;qualis. </s>

<s id="id001765">Sed quando pondus e&longs;t unum, & c de&longs;cendit ad e, cum de<lb/>&longs;cendat inæquali tempore, & peragat maiorem angulum compa<lb/>ratione a, quam b, &longs;equitur, ut uelo cius moueatur comparatione a <lb/>quàm b. </s>

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<s id="id001793">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 &etail;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"/><lb/>f b ut d e ad d b. </s>

<s id="id001794">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. </s>

<s id="id001795">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. </s>

<s id="id001796">Rur&longs;us ponamus, <lb/><arrow.to.target n="marg368"/><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. </s>

<s id="id001797">Ductis rectis a c & b d

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<s id="id001807"><margin.target id="marg366"/>I<emph type="italics"/>n<emph.end type="italics"/> 16. <emph type="italics"/>de<emph.end type="italics"/><lb/>S<emph type="italics"/>ubtil.<emph.end type="italics"/></s>

<s id="id001808"><margin.target id="marg367"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>Elem.<emph.end type="italics"/></s>

<s id="id001809"><margin.target id="marg368"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id001813">Sit modo obtu&longs;i angulus a b c, & nota latera &longs;ingula, & angu<lb/>lus a b c, & producantur latera ad perpendicu<lb/><figure id="id.015.01.118.1.jpg" xlink:href="015/01/118/1.jpg"/><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"/><lb/>quales. </s>

<s id="id001814">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 &etail;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. </s>

<s id="id001815">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"/><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>

<s id="id001815">Quod uerò propositurà Monteregio de cognitione an<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/><arrow.to.target n="marg373"/><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>

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<s id="id001830">Sint du&etail; diametri a b, c d ad perpendi <lb/>culum &longs;ecantes &longs;e in centro, & <expan abbr="duc&utilde;tur">ducuntur</expan> <lb/>&longs;upr f g k h, & infra m l ad perpendicu<lb/>lum &longs;upra a b: dico f g e&longs;&longs;e maiorem f a, <lb/>& k h k a, & contrà minorem m l, quàm <lb/>m a. </s>

<s id="id001831">Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a æquale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, & ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="a&utilde;t">aunt</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>

<s id="id001831">Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a æquale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, & ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="a&utilde;t">aut</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>

<s id="id001832">Suppo&longs;ito etiam, quòd <lb/><arrow.to.target n="marg375"/><lb/>a g arcus &longs;it dimidium a c, dico a f <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e f e, nam quadratum e <lb/><arrow.to.target n="marg376"/><lb/>g æquale e&longs;t quadratis f e, & f g, & <expan abbr="quadrat&utilde;">quadratum</expan> a g quadratis f g & f a <lb/>& e g e&longs;t &etail;qualis lateri exagoni, & a g latus octogoni, igitur e g ma<lb/><arrow.to.target n="marg377"/><lb/>ior g a, & duo quadrata e f & f g maiora duobus quadratis f g & <lb/>f a, detracto igitur communi f g quadrato, patet propo&longs;itum.<lb/><arrow.to.target n="marg378"/></s>

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<s id="id001843">Per præcedentem moto puncto a uer&longs;us c &longs;emper u&longs; que ad e, c ma <lb/><arrow.to.target n="marg380"/><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/>&etail;qualis ea, maius fit in crementum in a e, quàm re&longs;pectu lineæ tran&longs;<lb/>uer&longs;alis. </s>

<s id="id001844">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 &etail;quidi&longs;tan<pb pagenum="101" xlink:href="015/01/120.jpg"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"/><lb/>& anguli f e a, & f e c &etail;quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &etail;qua <lb/><arrow.to.target n="marg383"/><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/><arrow.to.target n="marg384"/><lb/>æquale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h e&longs;t æqualis k g ex eisdem o&longs;tendi<lb/>tur f l m k quadratum e&longs;&longs;e. </s>

<pb pagenum="101" xlink:href="015/01/120.jpg"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"/><lb/>& anguli f e a, & f e c &etail;quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &etail;qua <lb/><arrow.to.target n="marg383"/><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/><arrow.to.target n="marg384"/><lb/>æquale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h e&longs;t æqualis k g ex eisdem o&longs;tendi<lb/>tur f l m k quadratum e&longs;&longs;e. </s>

<s id="id001845">Quia ergo <lb/>k h e&longs;t æqualis k g, & k l æqualis k m, erit l g æqualis m h. </s>

<s id="id001846">Er<lb/>go de&longs;cendendo ex g in f, quantum f l &longs;uperat l g, tantum de&longs;cen<lb/>dendo ex f in h, f m &longs;uperat m h per communem animi &longs;ententi<lb/>am. </s>

<s id="id001847">At f m e&longs;t de&longs;cen&longs;us f in linea a e, & m h di&longs;tantia, quæ acqui<lb/>ritur in linea f r, n m enim e&longs;t æqualis f r, igitur n h excedit f r in <lb/>h m, & ita a n excedit a r in n r &etail;quali f m. </s>

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<pb pagenum="103" xlink:href="015/01/122.jpg"/>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/></s>

<s id="id001876">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. </s>

<s id="id001876">Ergo iuxta hoc fient libræ, quæ examinabunt decimam, & uige&longs;i<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in pretio&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. </s>

<s id="id001877">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>

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<s id="id001885">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. </s>

<s id="id001886">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. </s>

<s id="id001887">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&etail;, &longs;eu mundi, e&longs;t in&longs;en&longs;ibilis omnino. <lb/></s>

<s id="id001888">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. </s>

<s id="id001888">Circa hæc <expan abbr="notand&utilde;">notandum</expan> i&longs;tud mirabile &longs;cilicet, 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. </s>

<s id="id001889">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&etail; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em

<s id="id001889">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">aut</expan> medium <lb/>grauis &longs;it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu&etail; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em<pb pagenum="104" xlink:href="015/01/123.jpg"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>

<pb pagenum="104" xlink:href="015/01/123.jpg"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>

<s id="id001890">Ergo &longs;ola in clinatio ad hoc ut <expan abbr="medi&utilde;">medium</expan> grauis &longs;it in li<lb/>nea <expan abbr="centror&utilde;">centrorum</expan> grauitatis & terræ, &longs;ufficit. </s>

<s id="id001891">E&longs;t ergo principium in &longs;ei<lb/>p&longs;o. </s>

<s id="id001892">In appen&longs;is &longs;imiliter. </s>

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<s id="id001898">Propo&longs;itio cente&longs;imadecima.</s>

<s id="id001899">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/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s>

<s id="id001899">Si duæ &longs;phæræ ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">ae</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s>

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<s id="id001903">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. </s>

<s id="id001904">Nam a propor<lb/>tionem habet ad b, ut uiginti&longs;eptem ad unum. </s>

<s id="id001905">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. </s>

<s id="id001906">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. </s>

<s id="id001906">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">aer</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. </s>

<s id="id001907">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. </s>

<s id="id001908">Similiter & in aqua:

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<pb pagenum="106" xlink:href="015/01/125.jpg"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>

<s id="id001922">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. </s>

<s id="id001923">In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;tcteles. </s>

<s id="id001923">In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;toteles. </s>

<s id="id001924">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/><arrow.to.target n="marg396"/><lb/>rem &longs;cilicet, & altiorem in medio con&longs;tituerunt.</s>

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<s id="id001941">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. </s>

<s id="id001942">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. </s>

<s id="id001943">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. </s>

<s id="id001944">Eadem quoque ratio e&longs;t omnium machinarum: ideò oblon<lb/>g&etail;longius eiaculantur, quoniam proportionem &longs;eruant ad cana<lb/><arrow.to.target n="marg400"/><lb/>iem. </s>

<s id="id001944">Eadem quoque ratio e&longs;t omnium machinarum: ideò ob lon<lb/>g&etail; longius eiaculantur, quoniam proportionem &longs;eruant ad cana<lb/><arrow.to.target n="marg400"/><lb/>lem. </s>

<s id="id001945">Sed de hoc inferius agetur.</s>

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<pb pagenum="109" xlink:href="015/01/128.jpg"/>axungia. </s>

<s id="id001979">Et quae rota magna &longs;it: quoniam cum <expan abbr="nõ">non</expan> rota, &longs;ed axis traha<lb/>tur in æquali tempore & magna, & parua trahitur: utra que uerò una <lb/>conuer&longs;ione tantam <expan abbr="lineã">lineam</expan> rectam &longs;uperat, quanta e&longs;t rotæ periphe<lb/>ria. </s>

<s id="id001980">Quod &longs;i plures &longs;int rotæ celerius feruntur, quia axis minus tan<lb/>to <expan abbr="rotã">rotam</expan> premit. </s>

<s id="id001981">Et &longs;i rectus &longs;it axis, & bene rotundus, & foramen ro <lb/>tundum, & latius, & è duri&longs;simo ligno, ut non po&longs;sit in clinari: & <lb/>rota ip&longs;a in ambitu æqualis, omnia hæc faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s>

<s id="id001981">Et &longs;i rectus &longs;it axis, & bene rotundus, & foramen ro<lb/>tundum, & latius, & è duri&longs;simo ligno, ut non po&longs;sit in clinari: & <lb/>rota ip&longs;a in ambitu æqualis, omnia hæc faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s>

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<s id="id001985"><foreign lang="greek">I)/xnia tu/pte w_o/dessi w_a/r & ko/nin a)|mfi xuqu_nai</foreign>.</s>

<s id="id001985"><foreign lang="greek">i)/xnia tu/pte o/deosi a/r & ko/nin a)mfixuqῡnai</foreign>.</s>

<s id="id001986">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. </s>

<s id="id001987">Principalis <lb/>autem cau&longs;a uelo citatis e&longs;t agens, uelut equi. </s>

<s id="id001988">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. </s>

<s id="id001988">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, &longs;ci<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 utrunque 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. </s>

<s id="id001989">Trahitur ergo <lb/><figure id="id.015.01.128.1.jpg" xlink:href="015/01/128/1.jpg"/><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri<lb/>bus. </s>

<s id="id001990">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&etail;. </s>

<s id="id001991">Quare leuius quàm in curru, difficilius quàm <lb/>in rota uel &longs;phæra à &longs;uperficie extima circumacta. </s>

<s id="id001992">Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. </s>

<s id="id001993">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. </s>

<s id="id001994">Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rotæ. </s>

<s id="id001995">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. </s>

<s id="id001995">Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir<lb/>cumferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta<lb/>tem. </s>

<s id="id001996">Sed & trochlea caret colloppibus. </s>

<s id="id001997">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

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<s id="id002021"><arrow.to.target n="marg408"/><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. </s>

<s id="id002022">Ex <lb/>quo nece&longs;&longs;e e&longs;&longs;et, ut quod longius ferretur, etiam ualidiores ictus <lb/><figure id="id.015.01.130.1.jpg" xlink:href="015/01/130/1.jpg"/><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. </s>

<s id="id002023">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. </s>

<s id="id002023">Præterea illud o&longs;tendit 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. </s>

<s id="id002024">Non e&longs;t ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. </s>

<s id="id002025">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. </s>

<s id="id002026">Sed uide

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<s id="id002032">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. </s>

<s id="id002033">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. </s>

<s id="id002034">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: itaque 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. </s>

<s id="id002035">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. </s>

<s id="id002035">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">uis</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>, quia, ut dixi, <expan abbr="tãto">tanto</expan> grauius, e&longs;t quod ferit. </s>

<s id="id002036">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>

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<s id="id002044">Nam <lb/>qui &longs;uccutiunt, ubi pondus, aut coniunctio maior &longs;it, quàm ut di<lb/>&longs;trahere po&longs;sint, conden&longs;ant partes proximiores, & rimas faciunt, <lb/>per quas exhalat ignis aut omnino extinguitur, aut conden&longs;atur. <lb/></s>

<s id="id002045">At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb/>go canali, ob id ergo maiore impetu feruntur per illas, quàm per <lb/>breuiores, etiam quòd minor &longs;it puluis, minor &longs;it ignis. </s>

<s id="id002046">Experimen <lb/>tum facies in canali, ubi &longs;ambuci medulla pro globulo flatu impel<lb/>lente expellitur ab&longs; que periculo: nam quanto minor fuerit canalis <lb/>ambitu ac longior eo maiore impetu pellitur. </s>

<s id="id002047">For&longs;an qui&longs;piam nos <lb/>meritò poterit uideri <expan abbr="repreh&etilde;di&longs;&longs;e">reprehendi&longs;&longs;e</expan>, quòd inanis gloriæ &longs;tudio per<lb/>nitio&longs;a humano generi do ceam. </s>

<s id="id002048">Quibus re&longs;pondeo, me nihil do cu <lb/>i&longs;&longs;e, quod ín humani generis detrimentum cedat, huiu&longs;mo di que pr&etail;<lb/>cepta iam ob&longs;cura&longs;&longs;e, ut ne quid mali accidere po&longs;&longs;et hominibus ex <lb/>his: <expan abbr="nã">nam</expan> quòd ad ea, quæ declarata, &longs;unt, cau&longs;as &longs;olùm retuli, effectus <lb/>ip&longs;imodi artis <expan abbr="nimi&utilde;">nimium</expan> feruntur, ac nimio plu&longs;quam <expan abbr="uell&etilde;">uellem</expan> in telligun<lb/>tur. </s>

<s id="id002049">Vt cum ad copiam, ad magnitudinem, ad coacta imperia mi&longs;e<lb/>rorum re&longs;picio, nihil plus po&longs;sit addi. </s>

<s id="id002050">Omnia enim hucu&longs; que <expan abbr="&longs;pectãt">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>

<s id="id002050">Omnia enim huiu&longs;que <expan abbr="&longs;pectãt">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>

<s id="id002051">An ergo &longs;uccurrere afflictis, ob&longs;e&longs;sis, <lb/>cinctis, æquare <expan abbr="condition&etilde;">conditionem</expan>, liberare à &longs;eruitute etiam rebelles <expan abbr="nõ">non</expan> li<lb/>cebit? </s>

<s id="id002052">Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb/>commodum hominum, ea uer&longs;a e&longs;t per uim in <expan abbr="Tyrannid&etilde;">Tyrannidem</expan>. </s>

<s id="id002053">Subtili <lb/>ergo ratione <expan abbr="occurrend&utilde;">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni<lb/>mis, ut dixi, qu&etail; ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinar&utilde;">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem &longs;pacij, per quos globus ille de<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>

<s id="id002053">Subtili <lb/>ergo ratione <expan abbr="occurrend&utilde;">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni<lb/>mis, ut dixi, qu&etail; ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinar&utilde;">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem &longs;patij, per quos globus ille de<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>

<s id="id002054">Multa <lb/>damnamus, <expan abbr="&qtilde;">quae</expan> Deus e&longs;&longs;e uult: boni uiri e&longs;t <expan abbr="nõ">non</expan> ni&longs;i opitulari homini<lb/>bus, <expan abbr="etiã">etiam</expan> malis modo bonis futuri <expan abbr="nõ">non</expan> &longs;int <expan abbr="impedim&etilde;to">impedimento</expan>: <expan abbr="quamobr&etilde;">quamobrem</expan>

Line 6443

Line 6410

<s id="id002059">Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui &longs;i <lb/><arrow.to.target n="marg410"/><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>. </s>

<s id="id002060">Quæritur ergo ex b d e muro <lb/>qualis excipietur? </s>

<s id="id002061">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>. </s>

<s id="id002061">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">aut</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;">decem nouem</expan> ad <expan abbr="un&utilde;">unum</expan>. </s>

<s id="id002062">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&etail;c e&longs;t. </s>

<s id="id002063">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. </s>

<s id="id002064">ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu&etail; 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. </s>

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<s id="id002075">Quantum ictus machin&etail; procliuis ad <expan abbr="angul&utilde;">angulum</expan> <expan abbr="minua&ttilde;">minuatur</expan> explorare.</s>

<s id="id002076">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"/><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &etail;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/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&etail; <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. </s>

<s id="id002076">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"/><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &etail;quali angulo a b c, longè <expan abbr="tñ">tamen</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/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&etail; <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. </s>

<s id="id002077">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. </s>

<s id="id002078">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="uiri&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. </s>

<s id="id002078">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="uiri&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">tamen</expan>, & <expan abbr="ictu&utilde;">ictuum</expan> mul<lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. </s>

<s id="id002079">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. </s>

<s id="id002080"><expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen<lb/>diculo, et &etail;quidi&longs;tanti ad <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. </s>

<s id="id002081">Venetus. </s>

<s id="id002082">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="id.015.01.134.2.jpg" xlink:href="015/01/134/2.jpg"/><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/></s>

<s id="id002083">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&etail;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&etail; fieri non pote&longs;t, <lb/>utictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. </s>

<s id="id002083">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&etail;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">aut</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&etail; fieri non pote&longs;t, <lb/>ut ictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. </s>

<s id="id002084">Eam ob cau&longs;am <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&etail;difici&utilde;">&etail;dificium</expan> ul<lb/><figure id="id.015.01.134.3.jpg" xlink:href="015/01/134/3.jpg"/><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&etail;c urbs, ne tota &etail;dificijs euer<lb/>&longs;is concidat. </s>

<s id="id002085"><expan abbr="Turcar&utilde;">Turcarum</expan> enim Princeps di<lb/>dicit, ut in Nouo ca&longs;tro in Melit&etail; In&longs;ul&etail; <lb/>arce S. </s>

<s id="id002086">Elmi appellata plu&longs; <08> mille icti<lb/>bus in &longs;ingulos dies imo M D obtundere

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<s id="id002097">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. </s>

<s id="id002098">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/></s>

<s id="id002099">Ob id præcipuum e&longs;t auxilium pr&etail;ter h&etail;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>. </s>

<s id="id002100">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/></s>

<s id="id002100">Serio <expan abbr="a&utilde;t">aut</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/></s>

<s id="id002101">per&longs;&etail;pe <expan abbr="etiã">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. </s>

<s id="id002102">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. </s>

<s id="id002102">Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à capto 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. </s>

<s id="id002103">nam <expan abbr="experiment&utilde;">experimentum</expan> frequens do cuit, ubi illæ ma <lb/>gn&etail; 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/></s>

<s id="id002104">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="tolli&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="&etail;dificior&utilde;">&etail;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. </s>

<s id="id002105"><expan abbr="&longs;atisfact&utilde;">&longs;atisfactum</expan> puto, pr&etail;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;&etail;: nihil. </s>

<s id="id002107">re&longs;i&longs;tit immen&longs;&etail; illi pote&longs;tati, & crudelitati <expan abbr="&longs;&etail;ui&longs;simor&utilde;">&longs;&etail;ui&longs;simorum</expan> ty <lb/><expan abbr="rãnor&utilde;">rannorum</expan>. </s>

<s id="id002108"><expan abbr="Ver&utilde;">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificio&longs;is ignib. </s>

<s id="id002109">eru

<s id="id002109">eru<pb pagenum="117" xlink:href="015/01/136.jpg"/>untur. </s>

<pb pagenum="117" xlink:href="015/01/136.jpg"/>untur. </s>

<s id="id002110">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. </s>

<s id="id002111">Pontes euertunt machi <lb/>næ, ignes que. </s>

<s id="id002112">Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis con&longs;i&longs;tere non po&longs;&longs;unt. </s>

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<s id="id002119">Di <lb/>ces, h&etail;c quid ad proportionem? </s>

<s id="id002120">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. </s>

<s id="id002121">Sed puerulo indito luxus, ignauiæ, crudelitatis at que<lb/>&longs;tultiti&etail; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. </s>

<s id="id002122">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&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002122">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, quod 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&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002123">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. </s>

<s id="id002124">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&etail;, 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&etail;be&ttilde;">pr&etail;betur</expan>: imò percul&longs;i defen

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<pb pagenum="119" xlink:href="015/01/138.jpg"/>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. </s>

<s id="id002135">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. </s>

<s id="id002136">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. </s>

<s id="id002136">Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>cra&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. </s>

<s id="id002137">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>

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<s id="id002140">Sit flabellum a b c appen&longs;um, ut &longs;olet, in a, & moueatur motu </s>

<s id="id002141"><arrow.to.target n="marg413"/><lb/>qua&longs;i circa axem p a q in parte inferiore, & aër comprehen&longs;us &longs;ub <lb/>b h k, & &longs;patium &longs;it 1 m figuræ nauicularis, quæ con&longs;tat e&longs;&longs;e par<lb/>tem cylindri inanis ex formatione ab Euclide &longs;cripta: nam &longs;i pro<lb/>poneretur p a q ad perpendiculum &longs;uper&longs;tans plano, fieret circum<lb/>ducta a b c &longs;uperficie, quæ e&longs;&longs;et lata &longs;uperius, &longs;icut etiam inferius <lb/><arrow.to.target n="marg414"/><lb/>cylindrus: at &longs;uperius a b tenuis e&longs;t, & angu&longs;ta, ergo fiet pars cy<lb/>lindri inanis: quia non circunuoluitur, donecredeat. </s>

<s id="id002142">Ergo per di<lb/>cta &longs;uperius &longs;ectio illius p r q s per axem e&longs;t pars cuiu&longs;dam elly<lb/><arrow.to.target n="marg415"/><lb/>p&longs;is. </s>

<s id="id002143">Et &longs;ectio quæuis planæ &longs;uperficiei æquidi&longs;tans a b cuelut tu, <lb/>item que æquidi&longs;tans axi p a q e&longs;t &longs;uperficies rectangula, quarum <lb/>una e&longs;t &longs;imilis, & æqualis b h k, e&longs;t in una &longs;uperficie cum axe p a q <lb/>alia uerò e&longs;t æquidi&longs;tans eidem axi maior aut minor æquidi&longs;tanti<lb/>um, & ip&longs;a laterum, at que rectangula ac &longs;i cylindrus &longs;tans axi plano <lb/>æquidi&longs;tanti &longs;ecaretur iuxta longitudinem &longs;eu altitudinem &longs;uam: <lb/>& manife&longs;tum e&longs;t, quod i&longs;ta duo plana, & eorum &longs;uperficies &longs;ecant <lb/>&longs;e mutuò ad rectos angulos.</s>

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<s id="id002147">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. </s>

<s id="id002148">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. </s>

<s id="id002149">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

<s id="id002149">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> refrigera<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" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>

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<s id="id002186">Nam exempli gratia, cum &longs;ol e&longs;t in initio Capricorni, & in C&oelig;li <lb/>medio, minima e&longs;t umbra eius diei, & totius anni. </s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo æd circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo ad circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

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<s id="id002260">Data regionis altitudine, & loco &longs;olis proportionem gnomo<lb/>nis tam ad umbram rectam, quam uer&longs;am, uel etiam in cylindro de<lb/>terminare.</s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenortum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenorum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002262">Plerique autem tam laborio&longs;è id conati &longs;unt de<lb/>mon&longs;trare, ut &longs;tudio&longs;os deterruerint ab opere: res autem ip&longs;a facil<lb/>lima e&longs;t. </s>

<s id="id002263">Propo&longs;ita ergo Poli exacta altitudine &longs;olis in Meridie <lb/>declinatione addita uel detracta, habebis re&longs;iduum eius ad qua<lb/>drantem f g, & &longs;imiliter habebis ex declinatione nota loci &longs;olis de<lb/>tracta à quadrante f h & iuxta horam tuam, & minutum multi<lb/><arrow.to.target n="marg446"/><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare re&longs;iduum h l, igitur punctum umbr&etail; rect&etail;, uel uer&longs;&etail; ip&longs;i<lb/>us gnomonis ad unguem, & ita con&longs;titues horologium exacti&longs;si<lb/>mum &longs;ecundum ea, quæ dixi in Corrolarijs &longs;upradictis, & quia ho<lb/><arrow.to.target n="marg447"/><lb/>rizon a b c d &longs;ecat æquinoctialem in <expan abbr="c&etilde;tro">centro</expan> terræ ducta g h k, erunt <lb/>anguli b h g, & k h l &etail;quales. </s>

<s id="id002264">Igitur po&longs;ito g ortu puncti eclypti<lb/>cæ, erit g b ortus amplitudo nota, & ideò angulus b h g, & k h l

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<s id="id002291">Sint duæ line&etail; a b, & c d. </s>

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><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

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uocetur 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, adijcia<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 dupla ad a b, tota igitur <pb pagenum="127" xlink:href="015/01/146.jpg"/>g c dupla toti e a. </s>

<pb pagenum="127" xlink:href="015/01/146.jpg"/>g c duplatoti e a. </s>

<s id="id002293">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. </s>

<s id="id002294">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. </s>

<s id="id002295">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>

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<s id="id002405">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 id="id002406">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

<s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t fun<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 f ip&longs;ius a e, quàm <pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<s id="id002407">Dico ergo quod maior e&longs;t proportio a b <lb/><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"/><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"/><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. </s>

<s id="id002408">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/><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"/><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/><arrow.to.target n="marg468"/><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"/><lb/>nor quàm a b ad e, quod fuit propo&longs;itum. </s>

<s id="id002409">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/><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"/><lb/>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160 detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 /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. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimere</expan> & cum pro<lb/>pinquitate ita facio. </s>

<s id="id002411">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 id="id002425">Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s>

<s id="id002426">Cum plerunque numeri fracti hab cantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<s id="id002426">Cum plerunque numeri fracti habeantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<pb pagenum="137" xlink:href="015/01/156.jpg"/>cum hoc &longs;it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diui&longs;iones deducere. </s>

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<s id="id002492">pro<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium re&longs;i<lb/>dui ducamus in &longs;e, & addamus aggregato quadrati a b cum ip&longs;a

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b clinea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamusu: olo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineantinter &longs;e proportionem datam.</s>

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b c linea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamus : uolo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineant inter &longs;e proportionem datam.</s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecuu <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002494">Propo&longs;itio cente&longs;imaquadrage&longs;imanona.</s>

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<s id="id002525">Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;ecunda.</s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij'ad par <lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio ad ditarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad par<lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio additarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002527">Demum <lb/>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb/>nore, uelut maioris partis ad minorem.<lb/><arrow.to.target n="marg496"/></s>

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<pb pagenum="143" xlink:href="015/01/162.jpg"/>c b ad b d, uelut g a ad a d, & hoc e&longs;t primum. </s>

<s id="id002531">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/><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"/><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>

<s id="id002532">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. </s>

<s id="id002533">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="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><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 id="id002533">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="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>cti 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;it l 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 id="id002534"><arrow.to.target n="marg497"/><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. </s>

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<pb pagenum="144" xlink:href="015/01/163.jpg"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;e quitur ad aliam. </s>

<s id="id002544">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. </s>

<s id="id002545">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. </s>

<s id="id002546">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/></s>

<s id="id002546">Archimedes <expan abbr="a&utilde;t">aut</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uocat 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/></s>

<s id="id002547">Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. </s>

<s id="id002548">Et &longs;imiliter f g ad <lb/>g d, & e&longs;t 1 præ proportione data. </s>

<s id="id002549">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. </s>

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<s id="id002570">H&etail;c propo&longs;itio e&longs;t admirabilis: & etiam de&longs;crip&longs;i, ut multa &longs;ecre<lb/>ta Dialecticæ potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propo&longs;ito <expan abbr="mult&utilde;">multum</expan> <lb/>congrueret. </s>

<s id="id002571">Ideò potius &longs;cholij cau&longs;a po&longs;ita e&longs;t quam ip&longs;ius tracta<lb/>tionis: ut <expan abbr="mod&utilde;">modum</expan> demon&longs;trandi magis quam id, &qring;d <expan abbr="demon&longs;tra&ttilde;">demon&longs;tratur</expan>, re<lb/>&longs;picere oporteat. </s>

<s id="id002572"><expan abbr="Con&longs;titua&ttilde;">Con&longs;tituatur</expan> ergo (per uiam problematis) linea a b <lb/>& proportio c ad d, & fiat d e ad c, ut c ad d, & a b ad e ut b f ad d, & <lb/>ut g ad c, eritque g media inter a f & f b, quod licet &longs;olum &longs;upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facilè demon&longs;tratur & à Commandino adie<lb/>cta e&longs;t <expan abbr="demõ">demom</expan> &longs;tratio. </s>

<s id="id002573">Concurrant ergo ex a & b du&etail; line&etail; in aliquod </s>

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<s id="id002583">Cum ergo h&etail;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. </s>

<s id="id002584">Etidem dico &longs;i k e&longs;&longs;et ultra h quod po<lb/>te&longs;t contingere. </s>

<s id="id002585">modò k a ad k b &longs;it ut c ad d & k f &longs;it &etail;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 &etail;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;&etail; erunt in <lb/>proportione c ad d. </s>

<s id="id002586">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. </s>

<s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">perila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. </s>

<s id="id002587">Et pote&longs;t demon&longs;trari Geometricè duobus uer<lb/>bis. </s>

<s id="id002589"><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 id="id002590"><arrow.to.target n="marg510"/><lb/>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb/><arrow.to.target n="marg511"/><lb/>cum ex &longs;uppo&longs;ito h f ad fb &longs;it ut c ad d, erit a ad b, ut c ad d. </s>

<s id="id002591">Et uides <lb/>Apollonium, & Pappium quanta &longs;uperflua adij ciant in hac &longs;ecun<lb/><arrow.to.target n="marg512"/><lb/>da parte demon&longs;trationis, quæ e&longs;t prima apud illos, & ducunt <expan abbr="unã">unam</expan> <lb/>lineam non nece&longs;&longs;ariam ex puncto b ad latus fh. </s>

<s id="id002592">Vt <expan abbr="antiquor&utilde;">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, quæ ferunt excel <lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">w_e?ila/mpousin</foreign> inducen <lb/>tes. </s>

<s id="id002592">Vt <expan abbr="antiquor&utilde;">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, quæ ferunt excel<lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">pẹrila/mpousin</foreign> inducen <lb/>tes. </s>

<s id="id002593">e&longs;t enim &longs;ingulare hoc exemplum.<lb/><arrow.to.target n="marg513"/></s>

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<s id="id002600">Ex hoc pater qualiter ex uera demon&longs;tratione &longs;en&longs;u o&longs;ten&longs;a per<lb/>uenimus ad quotquot imaginando, inde intellectu abiectis condi<lb/>tionibus non nece&longs;&longs;arijs facimus infinitum & uniuer&longs;ale. </s>

<s id="id002601">Demum <lb/>&longs;ine artis &longs;pe cialis auxilio o&longs;tendimus Iheorema uniuer&longs;ale (quod <lb/>etiam poterat o&longs;tendi Geometricè, &longs;ed longè pulchrius e&longs;t, ac &longs;ubli<lb/>mius per <foreign lang="greek">w_erilampousin</foreign>, qa hoc ip&longs;o infinita alia do cemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere) &longs;cilicet quod à quouis <lb/>puncto peripheri&etail; circuli, cuius &longs;emidiameter e&longs;t media proportio<lb/>ne inter totam exten&longs;am à centro u&longs;que exterius, & partem quæ' e&longs;t à <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datar&utilde;">datarum</expan> <lb/>linearum lineæ ductæ ex eo ad punctum exterius, & punctum de<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s>

<s id="id002601">Demum <lb/>&longs;ine artis &longs;pecialis auxilio o&longs;tendimus theorema uniuer&longs;ale (quod <lb/>etiam poterat o&longs;tendi Geometricè, &longs;ed longè pulchrius e&longs;t, ac &longs;ubli<lb/>mius per <foreign lang="greek">perilampousin</foreign>, qua hoc ip&longs;o infinita alia docemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere) &longs;cilicet quod à quouis <lb/>puncto peripheri&etail; circuli, cuius &longs;emidiameter e&longs;t media proportio<lb/>ne inter totam exten&longs;am à centro u&longs;que exterius, & partem quæ' e&longs;t à <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datar&utilde;">datarum</expan> <lb/>linearum lineæ ductæ ex eo ad punctum exterius, & punctum de<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s>

<s id="id002602">Propo&longs;itio cente&longs;imaquinquage&longs;imaquinta.</s>

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<s id="id002625">Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit diuidi per <expan abbr="dupl&utilde;">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrahã">detraham</expan> 4 & <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="dupl&utilde;">duplum</expan> radicis quod e&longs;t 4 exit <lb/>24, cuius quadratum &qring;d e&longs;t 576 addito 100 facit 676 <expan abbr="quadrat&utilde;">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui<lb/>di per &longs;e, duplum <02> 9 & exit 4 cuius <expan abbr="quadrat&utilde;">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s>

<s id="id002625">Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit diuidi per <expan abbr="dupl&utilde;">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrahã">detraham</expan> 4 & <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="dupl&utilde;">duplum</expan> radicis quod e&longs;t 4 exit <lb/>24, cuius quadratum qua e&longs;t 576 addito 100 facit 676 <expan abbr="quadrat&utilde;">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui<lb/>di per &longs;e, duplum <02> 9 & exit 4 cuius <expan abbr="quadrat&utilde;">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s>

<s id="id002626">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&etail; 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&etail; &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. </s>

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<s id="id002662">Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s>

<s id="id002663">Sexta regula, ponamus modò &qring;d uelim diuidere 10, <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. </s>

<s id="id002663">Sexta regula, ponamus modò quod uelim diuidere 10, <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. </s>

<s id="id002665">9 & 1, & non <expan abbr="dupl&utilde;">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002666">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. </s>

<s id="id002666">ita quod &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. </s>

<s id="id002668">fit 250/25, at 250 <expan abbr="cõponi&ttilde;">componitur</expan> aliter <lb/>ex duob. </s>

<s id="id002670"><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/></s>

<s id="id002671">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>. </s>

<s id="id002672">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>. </s>

<s id="id002673">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> compo <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>

<s id="id002673">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> comp <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>

<s id="id002674">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 id="id002685">fit <expan abbr="tot&utilde;">totum</expan>, <lb/>ergo ex medietate in <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002686">idem fit medietas, &longs;ed medietas e&longs;t nume<lb/>rus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002687">ergo 3 e&longs;&longs;et numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002688">&qring;d e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int inæ quales, & ut 6 diuidatur in duas partes in&etail;quales, hoc <lb/><expan abbr="a&utilde;t">aunt</expan> fit diuidendo quemlibet <expan abbr="numer&utilde;">numerum</expan> parem, qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. <lb/></s>

<s id="id002688">quod e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int inæ quales, & ut 6 diuidatur in duas partes in&etail;quales, hoc <lb/><expan abbr="a&utilde;t">aut</expan> fit diuidendo quemlibet <expan abbr="numer&utilde;">numerum</expan> parem, qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. <lb/></s>

<s id="id002689">numeris <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002690">nam &longs;i e&longs;&longs;et impar, <expan abbr="nõ">non</expan> po&longs;&longs;et prodire numerus integer, & <lb/><expan abbr="c&utilde;">cum</expan> prouenerit numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002691">ille erit <expan abbr="qu&etilde;">quem</expan> qu&etail;rimus, <expan abbr="nã">nam</expan> diui&longs;o 6 per to<lb/>tum <expan abbr="ill&utilde;">illum</expan> numerum, inde &qring;d prouenit multiplicato per numeros <expan abbr="&qtilde;d">quad</expan>, <lb/><expan abbr="cõponentes">componentes</expan> illum <expan abbr="numer&utilde;">numerum</expan> productum, <expan abbr="producun&ttilde;">producuntur</expan> partes 6, quæ <expan abbr="er&utilde;t">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>

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<s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 & 3, & abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, & diui <lb/>&longs;o producto relin quitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 & <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u<lb/>pere&longs;t, & duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>& ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="cõpo&longs;ito">compo&longs;ito</expan> <lb/>ex 5 & 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relin quat 5, igitur neque talis numerus potérit <expan abbr="cõponi">componi</expan> ex duo<lb/>bus quadratis, in quib. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relin quitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relin quitur <lb/>6, duc in 6 fit 36, diuide per 11, relin quitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relin quunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque<lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in &qring;&longs;upere&longs;t 2, <expan abbr="cõ">com</expan> fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in quod ̊&longs;upere&longs;t 2, <expan abbr="cõ">con</expan>fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002697"><expan abbr="duor&utilde;">duo<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerrorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa <lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>

<s id="id002697"><expan abbr="duor&utilde;">duo<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>

<s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus'">quadratus'</expan>.</s>

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<pb pagenum="152" xlink:href="015/01/171.jpg"/>qui diui&longs;us per 6 &longs;upererit 3, & in paribus qui poterit diuidi per 6. <lb/>Quia <expan abbr="componun&ttilde;">componuntur</expan> ex huiu&longs;modi: uelut 3 in &longs;e facit 9, & 25 in &longs;e facit <lb/>225, qui <expan abbr="i&utilde;cti">iuncti</expan> <expan abbr="faci&utilde;t">faciunt</expan> 234, diui&longs;o 235 per 6 exit 39, qui <expan abbr="iter&utilde;">iterum</expan> diui&longs;us per 6 <lb/>&longs;upere&longs;t 3, & &longs;imiliter capio 6 & 12, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;drata">quadrata</expan> &longs;unt 36 & 144, & <lb/><expan abbr="aggregat&utilde;">aggregatum</expan> 180, qui diui&longs;us per 6 exit 30, qui <expan abbr="iter&utilde;">iterum</expan> pote&longs;t diuidi per <lb/>6. Et hoc quia <expan abbr="quilibetillor&utilde;">quilibetillorum</expan> pote&longs;t diuidi per <expan abbr="&qtilde;drat&utilde;">quadratum</expan> 6 in paribus, <lb/>ergo aggregato diui&longs;o per 6 &qring;d prodit, <expan abbr="iter&utilde;">iterum</expan> poterit diuidi per 6. <lb/>Et in imparibus quo dlibet <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> exuperat &longs;upra &longs;enarios in 3, <lb/><expan abbr="igi&ttilde;">igitur</expan> <expan abbr="aggregat&utilde;">aggregatum</expan> diui&longs;um in 2 pariet <expan abbr="numer&utilde;">numerum</expan> qui diui&longs;us per 3, exibit <lb/>numerus impar <expan abbr="cõpo&longs;itus">compo&longs;itus</expan> ex &longs;enarijs & 3. Illud ergo <expan abbr="quadrat&utilde;">quadratum</expan>, &qring;d <lb/>prodibit, uel erit <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex &longs;enarijs, uel &longs;upererit 3. Sed <expan abbr="c&utilde;">cum</expan> 3 nume <lb/>ret 6, ergo tres <expan abbr="&qtilde;drati">quadrati</expan> numeri &longs;cilicet duo, qui <expan abbr="cõponunt">componunt</expan> <expan abbr="numer&utilde;">numerum</expan>, <lb/><arrow.to.target n="marg521"/><lb/>& qui prodit per <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> 6, erunt <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> inter &longs;e, ergo & radices il <lb/>lorum. </s>

<s id="id002701"><expan abbr="Igi&ttilde;">Igitur</expan> radix numeri <expan abbr="&qtilde;drati">quadrati</expan>, qui prouenit diui&longs;o aggregato <expan abbr="quadrator&utilde;">qua<lb/>dratorum</expan> per 6 e&longs;t ex <expan abbr="eod&etilde;">eodem</expan> ordine <expan abbr="impari&utilde;">imparium</expan>, &longs;i impares numeri <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="fuer&utilde;t">fuerunt</expan>, aut <expan abbr="pari&utilde;">parium</expan> &longs;i pares. </s>

<s id="id002702">At hoc e&longs;&longs;e <expan abbr="nõ">non</expan> pote&longs;t, <expan abbr="nã">nam</expan> fracti illi numeri, <lb/>qui <expan abbr="er&utilde;t">erunt</expan> radices, <expan abbr="nõ">non</expan> <expan abbr="er&utilde;t">erunt</expan> minimi, &longs;ed diui&longs;i per 3 o&longs;tendent minores, <lb/>quod e&longs;t contra &longs;uppo&longs;itum, quare nullo modo 6 pote&longs;t diuidi in <lb/>duos numeros quadratos, neque integros, neque fractos, quod erat <lb/>demon&longs;trandum. </s>

<s id="id002703">Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="nõ">non</expan> po&longs;sit diuidi, & quado po&longs;sit, quod po&longs;sit, & quomodo &longs;imul.</s>

<s id="id002703">Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="nõ">non</expan> po&longs;sit diuidi, & quando po&longs;sit, quod po&longs;sit, & quomodo &longs;imul.</s>

<s id="id002704"><margin.target id="marg521"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>&longs;e<lb/>ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id002708">Contingit quando que &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem <lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aunt</expan> maius efficere: id <lb/>duob. </s>

<s id="id002708">Contingit quandoque &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem<lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aut</expan> maius efficere: id <lb/>duob. </s>

<s id="id002709">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="tardior&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>. </s>

<s id="id002710">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&etail; rot&etail; 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&etail; rot&etail; circumuoluentur. </s>

<s id="id002710">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">aut</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&etail; rot&etail; 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&etail; rot&etail; circumuoluentur. </s>

<s id="id002711">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&etail; <expan abbr="er&utilde;t">erunt</expan> ad <expan abbr="expl&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" xlink:href="015/01/172.jpg"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aunt</expan>, & qu&etail; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. </s>

<pb pagenum="153" xlink:href="015/01/172.jpg"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aut</expan>, & qu&etail; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. </s>

<s id="id002712">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. </s>

<s id="id002712">Ponamus ergo po&longs;t<08> eadem e&longs;t ratio celerioris & <lb/>æqué uelocis, ponderis <expan abbr="a&utilde;t">aut</expan> tardius de&longs;cendentis, aut <expan abbr="cõtrà">contrà</expan> tardio<lb/>ris, aut æqualiter circumducti in dicis, celerioris <expan abbr="a&utilde;t">aut</expan> de&longs;cen&longs;us pon<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. </s>

<s id="id002713">Sit ergo ut pon <lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aunt</expan> &etail;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 id="id002713">Sit ergo ut pon<lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aut</expan> &etail;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/>tum <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 id="id002714">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/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. </s>

<s id="id002714">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">tur</expan>, qua<lb/>ter decies, & <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno circuitu 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/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. </s>

<s id="id002715">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="quoniã">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 id="id002715">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="quoniã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tn</expan> tardè quod &longs;it pro paruitatis circuitus ratione.</s>

<s id="id002716">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;&etail;">appen&longs;&etail;</expan> du&etail; trochle&etail; g & h, & fu <lb/>nis una parte tro chle&etail; 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&etail;, 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

<s id="id002716">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;&etail;">appen&longs;&etail;</expan> du&etail; trochle&etail; g & h, & fu <lb/>nis una parte trochle&etail; appen&longs;us in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam trochleam l in&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 trochle&etail;, 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 circumuolutus exeat <lb/>ip&longs;um, & <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> ad tro<expan abbr="chleãn">chlean</expan>, &longs;ub que ea circumuolutus <expan abbr="iter&utilde;">iterum</expan> a&longs;cen<pb pagenum="154" xlink:href="015/01/173.jpg"/>dat à dextra parte, et circumuoluatur h cochle&etail; rediens ad &longs;ini&longs;tram <lb/>ibique de&longs;cendens connectatur trochleæ in inferiori in o, cuius imæ <lb/>parti annectatur pondus remorans in imo annexum parte troch<lb/>leæ p. </s>

<pb pagenum="154" xlink:href="015/01/173.jpg"/>dat à dextra parte, et circumuoluatur h co chle&etail; rediens ad &longs;ini&longs;tram <lb/>ibique de&longs;cendens connectatur tro chleæ in inferiori in o, cuius imæ <lb/>parti annectatur pondus remorans in imo annexum parte tro ch<lb/>leæp. </s>

<s id="id002717">Cum ergo trahitur n tro chlea, trahitur funis adeò ut pon<lb/>dus m, tandem a&longs;cendat cum tro chleal prope k: quia ergo in duo<lb/>decim horis pondus m de&longs;cenderet per k l funem reuolutionibus <lb/>circa d rotam dicamus uiginti, ergo &longs;i debet de&longs;cendere à k ad l, per <lb/>funem duplicatam k l cum ip&longs;am nece&longs;&longs;e &longs;it obequitantem d reuo<lb/>lutionibus quadraginta circumuolui d, nam tota o h n d m g l k lon <lb/>gè maior e&longs;t duplo k l, nece&longs;&longs;e e&longs;t m de&longs;cendere tardius quàm in du <lb/>plo temporis, quo de&longs;cenderet per rectum funem k l, quod erat de<lb/>mon&longs;trandum. </s>

<s id="id002718">Et hanc appendicem uidi apud Cæ&longs;arem Odonum <lb/>Apulum medicum, uirum elegantem lepidique ingenij. </s>

<s id="id002719">Memento <lb/>uerò quod ubi orbiculi non cederent funi, uel quia duriores in cir<lb/>cumuolutione, uel quia latius exciperent illum reduplicato fune <lb/>circa illos omnin o circumducuntur, &longs;ed difficilius ideò egent gra<lb/>uiori pondere.</s>

<s id="id002720">Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;eptima.</s>

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<s id="id002724">Sunt horum duo genera primum, & anti <lb/><figure id="id.015.01.173.1.jpg" xlink:href="015/01/173/1.jpg"/><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. </s>

<s id="id002725">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/></s>

<s id="id002725">Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;piræ in fine, cu<lb/>ius c annectatur ferreo circulo, qui habeatur loco 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, trahet <lb/>ergo molam, & con&longs;tringet 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/></s>

<s id="id002726">Hic ade&longs;t cap&longs;ula, & quod circumuertitur à claue non e&longs;t axis mol&etail; <lb/>&longs;ed extra molam, &longs;cilicet e h. </s>

<s id="id002727">Et quoniam hac ratione quanto mola a

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<pb pagenum="155" xlink:href="015/01/174.jpg"/>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 id="id002728">Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&etail; <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&etail;; ergo circumducto axe mol&etail; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>

<s id="id002728">Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&etail; <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">aut</expan> rot&etail;; ergo circumducto axe mol&etail; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>

<s id="id002729">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. </s>

<s id="id002730">Sed <lb/>in hoc idem e&longs;t in commodum &longs;ine remedio <lb/><figure id="id.015.01.174.1.jpg" xlink:href="015/01/174/1.jpg"/><lb/>quod fuit in priore. </s>

<s id="id002731">Vbi enim c&oelig;perit laxa<lb/>ri mola tanto tardius progrediuntur rotæ <lb/>atque index. </s>

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<s id="id002747">Hoc fieri pote&longs;t in &longs;ingulo genere horologij trium <expan abbr="de&longs;criptor&utilde;">de&longs;criptorum</expan>. <lb/></s>

<s id="id002748">Propterea &longs;ufficiat de uno o&longs;tendi&longs;&longs;e. </s>

<s id="id002749">Sed & in &longs;ingulo genere &longs;unt <lb/>multi modi, unius tamen reddidi&longs;&longs;e <expan abbr="ration&etilde;">rationem</expan> &longs;ufficiat. </s>

<s id="id002750">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. </s>

<s id="id002750">Hoc <expan abbr="a&utilde;t">aut</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. </s>

<s id="id002751">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. </s>

<s id="id002751">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">transferatur</expan> index, non ce&longs;&longs;at pul&longs;atio: imò nec <lb/>retineri pote&longs;t, donec pondus illud de&longs;cenderit. </s>

<s id="id002752">Ergo primi & ter<lb/>tij ratio hæc habeatur, cum rota qu&etail; 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,

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<pb pagenum="158" xlink:href="015/01/177.jpg"/>rantur: quinimò illud magis mirum de quo illi non mirantur quia <lb/>frequens e&longs;t, &longs;cilicet cur aut quomodo &longs;i diui&longs;æ &longs;unt ut cir <expan abbr="ç&utilde;ducto">çunducto</expan> <lb/>indice non transferatur rota mallei, <expan abbr="põdere">pondere</expan> tamen uer&longs;ata rota in<lb/>dicis in idem incidat, ut horæ quæ pul&longs;u declarantur ad unguem <lb/>& in ei&longs;dem &longs;ectionibus <expan abbr="cõueniant">conueniant</expan> cum horis quas index o&longs;ten dit.</s>

<s id="id002762">Ver&grave;m quia multis modis contingit ordinem horologiorum <lb/>peruerti: in &longs;imilibus quidem &longs;i hora indicis &longs;imul & pul&longs;us unà <lb/>circumferuntur, &longs;ed tardius ambo index traducitur ad locum debi<lb/>tum, inde ponderi aliquid additur. </s>

<s id="id002762">Verum quia multis modis contingit ordinem horologiorum <lb/>peruerti: in &longs;imilibus quidem &longs;i hora indicis &longs;imul & pul&longs;us unà <lb/>circumferuntur, &longs;ed tardius ambo index traducitur ad locum debi<lb/>tum, inde ponderi aliquid additur. </s>

<s id="id002763">Si uerò antè proce&longs;&longs;erit quam. <lb/></s>

<s id="id002764">Sol in dicet ablato pondere, &longs;ines tempus fluere u&longs;que ad indicis lo<lb/>cum &longs;ine motu horologij, pondus quoque ip&longs;um minues. </s>

<s id="id002765">At &longs;i pon<lb/>dus pul&longs;us in terram deuenerit uel propè, expecta donec &longs;uper li<lb/>nea index fuerit, inde trahe, neque. </s>

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<s id="id002792">Ex hoc etiam patet quod &longs;pacium con<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tent&utilde;">tentum</expan> à peripheria circuli nulli angulo rectilineo &etail;quale e&longs;&longs;e pote&longs;t. <lb/></s>

<s id="id002792">Ex hoc etiam patet quod &longs;patium con<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tent&utilde;">tentum</expan> à peripheria circuli nulli angulo rectilineo &etail;quale e&longs;&longs;e pote&longs;t. <lb/></s>

<s id="id002793">nam dimidium e&longs;&longs;et æquale dimidio, quod e&longs;t contra demon&longs;trata.</s>

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<s id="id002810"><margin.target id="marg538"/>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 id="id002811">Ex hoc patet quod reliqua duo &longs;pacia <lb/><arrow.to.target n="marg539"/><lb/>non po&longs;&longs;unt e&longs;&longs;e æqualia rectilineo. </s>

<s id="id002811">Ex hoc patet quod reliqua duo &longs;patia <lb/><arrow.to.target n="marg539"/><lb/>non po&longs;&longs;unt e&longs;&longs;e æqualia rectilineo. </s>

<s id="id002812">Nam <lb/>&longs;patium b a c demon&longs;tratum e&longs;t æquale e&longs;<lb/>&longs;e rectilineo, & b ad non e&longs;t æquale rectili<lb/>neo, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c a d non pote&longs;t e&longs;&longs;e æquale <lb/>angulo rectilineo, nam &longs;i &longs;ic &longs;it b a c &etail;quale <lb/>f g h & c a d h g k, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="tot&utilde;">totum</expan>, b a d erit &etail;quale <lb/><arrow.to.target n="marg540"/><lb/>toti f g k &qring;d e&longs;t contra <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan>, ideò neque<lb/>b a e quia b a c & d a e &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineis <lb/>per &longs;e, & <expan abbr="etiã">etiam</expan> pariter accepta. </s>

<s id="id002812">Nam <lb/>&longs;patium b a c demon&longs;tratum e&longs;t æquale e&longs;<lb/>&longs;e rectilineo, & b ad non e&longs;t æquale rectili<lb/>neo, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c a d non pote&longs;t e&longs;&longs;e æquale <lb/>angulo rectilineo, nam &longs;i &longs;ic &longs;it b a c &etail;quale <lb/>f g h & c a d h g k, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="tot&utilde;">totum</expan>, b a d erit &etail;quale <lb/><arrow.to.target n="marg540"/><lb/>toti f g k quod e&longs;t contra <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan>, ideò neque<lb/>b a e quia b a c & d a e &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineis <lb/>per &longs;e, & <expan abbr="etiã">etiam</expan> pariter accepta. </s>

<s id="id002813">Totum <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> a e&longs;t <expan abbr="&etail;&qtilde;le">&etail;quale</expan> quatuor, re<lb/>ctis ergo <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;cilicet &longs;patia c a d & b a c pariter accepta &longs;unt <expan abbr="&etail;&qtilde;lia">&etail;qua<lb/>lia</expan> rectilineis &longs;patijs, &longs;ed <expan abbr="&longs;pati&utilde;">&longs;patium</expan> e a d non e&longs;t <expan abbr="æ&qtilde;le">æquale</expan> rectilineo, ergo per<lb/>demon&longs;trata hic, nec b a e, <expan abbr="nã">nam</expan> &longs;i &longs;it, &longs;it ergo b a e æquale h g k & quia <lb/>ambo &longs;patia b a e & c a d &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineo ex demon&longs;tratis, &longs;it <lb/>ergo æqualia f g k, erit ergo ex communi animi &longs;ententia &longs;patium f <lb/>g h æquale &longs;pacio c a d, quod e&longs;t contra primam partem corrolarij.</s>

<s id="id002813">Totum <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> a e&longs;t <expan abbr="&etail;&qtilde;le">&etail;quale</expan> quatuor, re<lb/>ctis ergo <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;cilicet &longs;patia c a d & b a c pariter accepta &longs;unt <expan abbr="&etail;&qtilde;lia">&etail;qua<lb/>lia</expan> rectilineis &longs;patijs, &longs;ed <expan abbr="&longs;pati&utilde;">&longs;patium</expan> e a d non e&longs;t <expan abbr="æ&qtilde;le">æquale</expan> rectilineo, ergo per<lb/>demon&longs;trata hic, nec b a e, <expan abbr="nã">nam</expan> &longs;i &longs;it, &longs;it ergo b a e æquale h g k & quia <lb/>ambo &longs;patia b a e & c a d &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineo ex demon&longs;tratis, &longs;it <lb/>ergo æqualia f g k, erit ergo ex communi animi &longs;ententia &longs;patium f <lb/>g h æquale &longs;patio c a d, quod e&longs;t contra primam partem corrolarij.</s>

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<s id="id002828">SCHOLIVM.</s>

<s id="id002829">Atque hæc omnia &longs;equuntur de mente Euclidis, quæ tamen ui<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe

<s id="id002829">Atque hæc omnia &longs;equuntur de mente Euclidis, quæ tamen ui<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe<pb pagenum="161" xlink:href="015/01/180.jpg"/>ria & recta &longs;unt ex genere quantitatis continuæ, & quòd detur ma<lb/>ius & minus & nunquam detur &etail;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. </s>

<pb pagenum="161" xlink:href="015/01/180.jpg"/>ria & recta &longs;unt ex genere quantitatis continuæ, & quòd detur ma<lb/>ius & minus & nunquam detur &etail;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. </s>

<s id="id002830">Et maximè quod etiam anguli ex peripheria & recta <lb/>&longs;unt diuer&longs;orum generum inter &longs;e & infinitorum. </s>

<s id="id002831">Pr&etail;terea i&longs;tud re<lb/>pugnare uidetur ip&longs;imet Euclidi, dicenti duabus magnitu dinibus <lb/><arrow.to.target n="marg547"/><lb/><arrow.to.target n="marg548"/><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. </s>

<s id="id002832">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/><figure id="id.015.01.180.1.jpg" xlink:href="015/01/180/1.jpg"/><lb/>ra a lunari, & quadrangulo b. </s>

<s id="id002833">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/></s>

<s id="id002834">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="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e & a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><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"/><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. </s>

<s id="id002834">Proponantur ergo duo circuli a d e & a f g qui &longs;e contingant in a, & <lb/>eorum centra &longs;int b & c & ducantur rectæ a f d & a g e & con&longs;tat <lb/>qui portiones a d & a f &longs;imiles &longs;unt, <lb/><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e & a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><lb/>per centra circulorum ex contactu <lb/>tran&longs;ibit per illa: quare anguli h a g <lb/>& h a e &longs;unt ijdem & &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"/><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. </s>

<s id="id002835">Quia ergo &longs;unt du&etail; 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"/><lb/>a d e peripheria e&longs;t maior recta a g e. </s>

<s id="id002836">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. </s>

<s id="id002837">igitur angulus peripheria<lb/>rum e&longs;t maior angulo à rectis contento. </s>

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<s id="id002882"><margin.target id="marg566"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002883"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff<lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002883"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff <lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002884"><margin.target id="marg568"/>P<emph type="italics"/>er præce<lb/>dentem.<emph.end type="italics"/></s>

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<s id="id002929">Propo&longs;itionem motuum omnium <expan abbr="&longs;tellar&utilde;">&longs;tellarum</expan> ad &longs;olem con&longs;iderare.</s>

<s id="id002930">Videtur Sol qua&longs;i Rex in C&oelig;lo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, & uideturres admiratione digna his, qui non <lb/>nouerunt, quanta &longs;it concordia omnium rerum, de qua infrà dice<lb/>mus. </s>

<s id="id002930">Videtur Sol qua&longs;i Rex in C&oelig;lo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, & uidetur es admiratione digna his, qui non <lb/>nouerunt, quanta &longs;it concordia omnium rerum, de qua infrà dice<lb/>mus. </s>

<s id="id002931">Ergo Luna primum hoc habet, ut linea æqualis motu Solis <lb/>&longs;emper media &longs;it inter lineam æqualis motus Lun&etail; & loci maximè <lb/>inæqualitatis motus eius, ubi &longs;cilicet tardi&longs;simè mouetur, Veneris <lb/>autem & Mercurij ut motus æquales idem &longs;emper &longs;int cum motu <lb/>æquali, & locus cumloco ip&longs;ius Solis ad unguem præterid quod <lb/>infrà dicemus. </s>

<s id="id002932">Trium uerò <expan abbr="&longs;uperior&utilde;">&longs;uperiorum</expan> ratio &longs;ic <expan abbr="cõ&longs;tat">con&longs;tat</expan> ad Solem ut à <lb/>Prolem&etail;o <expan abbr="ob&longs;eruat&utilde;">ob&longs;eruatum</expan> e&longs;t ex Hipparcho. </s>

<s id="id002932">Trium uerò <expan abbr="&longs;uperior&utilde;">&longs;uperiorum</expan> ratio &longs;ic <expan abbr="cõ&longs;tat">con&longs;tat</expan> ad Solem ut à <lb/>Ptolem&etail;o <expan abbr="ob&longs;eruat&utilde;">ob&longs;eruatum</expan> e&longs;t ex Hipparcho. </s>

<s id="id002933">In omnire&longs;titutione cuiu&longs;<lb/>libet planet&etail; &longs;uperioris numerus <expan abbr="reuolution&utilde;">reuolutionum</expan> Solis &etail;qualis e&longs;t nu<lb/>mero <expan abbr="re&longs;titution&utilde;">re&longs;titutionum</expan> planet&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <expan abbr="mot&utilde;">motum</expan> æqualitatis & in&etail;qualita <lb/>tis pariter acceptis. </s>

<s id="id002934">Velut Saturnus in annis quinquaginta nouem <lb/>die una & horis decem octo quinquage&longs;ies &longs;epties per motum in&etail;<lb/>qualem ad <expan abbr="ungu&etilde;">unguem</expan>, per æqualem autem duabus reuolutionibus par <lb/>te in&longs;uper una & quadraginta quin que minutijs, quæ re&longs;pondent di<lb/>ei uni, & horis decem octo ex motu Solis, & ita bis Saturnus reuol <lb/>uitur &longs;ecundum motum æqualitatis & quinquage&longs;ies &longs;epties per <lb/>motum inæqualem & &longs;imiliter. </s>

<s id="id002935">Iupiter in annis 70, diebus trecen<lb/>tis &longs;exaginta, horis quatuor, &longs;exaginta quinque reuolutiones in&etail;qua <lb/>les perficiet & &longs;ex &etail;quales, deficientibus ex &etail;qualibus quatuor par<lb/>tibus & dextante quod e&longs;t <expan abbr="quãtum">quantum</expan> peragraret Solin quatuor die<lb/>bus, & dextante diei ad perfectionem &longs;cilicet annorum &longs;eptuaginta <lb/>atque unius. </s>

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<s id="id002938">Mercurij quo que &longs;tella in quadraginta &longs;ex annis & una die <lb/>& hora una fermè quadraginta &longs;ex fermè perficit reuolutiones æ<lb/>qualis motus & in&longs;uper gradum unum cum portione re&longs;pondenti <lb/>portioni temporis, id e&longs;t, horæ fermè uni: in æqualitatis autem cen<lb/>&longs;um quadraginta quin que. </s>

<s id="id002939">Atque h&etail;c &longs;unt manife&longs;ti&longs;sima et ut dixi ad<lb/>miranda &longs;unt, præterea alia minus generalia, aut minus manife&longs;ta <lb/>aut non tanti momenti quæ con&longs;ultò prætermitto, non e&longs;t. </s>

<s id="id002941">locus <lb/>hic do cendi artes &longs;ingulas &longs;ed &longs;olum ea tra ctandi quæ ad argumen

<s id="id002941">locus <lb/>hic docendi artes &longs;ingulas &longs;ed &longs;olum ea tractandi quæ ad argumen<pb pagenum="168" xlink:href="015/01/187.jpg"/>tum pertinent. </s>

<pb pagenum="168" xlink:href="015/01/187.jpg"/>tum pertinent. </s>

<s id="id002942">Igitur ut ad rem redeam. </s>

<s id="id002943">Solis cum octauo Orbe ea <lb/>ratio e&longs;t, ut linea quam ille permeat eadem &longs;it quam qu&etail; fix&etail; &longs;tellæ, <lb/>non. </s>

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<s id="id002957">omnia <expan abbr="cõprehendi">comprehendi</expan> uidentur. </s>

<s id="id002958">1. omnes uo <expan abbr="c&utilde;">cum</expan> differentiæ, <lb/><expan abbr="quanquã">quanquam</expan> ex octo <expan abbr="tant&utilde;">tantum</expan> uo cibus con&longs;tet. </s>

<s id="id002959">Pò&longs;t &longs;unt quæ in <expan abbr="&qtilde;drupla">quadrupla</expan>, <lb/>unde bis diapa&longs;on, po&longs;t quæ in tripla, nam propior e&longs;t monadi &longs;eu &etail;<lb/>qualitati: &longs;ed non adeò &longs;implex ut bis diapa&longs;on. </s>

<s id="id002960">Vocant <expan abbr="a&utilde;t">aunt</expan> hanc <lb/>diapa&longs;on diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">&longs;ub&longs;equitur</expan> octupla qu&etail; uix in uocib. </s>

<s id="id002960">Vocant <expan abbr="a&utilde;t">aut</expan> hanc <lb/>diapa&longs;on diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">&longs;ub&longs;equitur</expan> octupla qu&etail; uix in uocibus </s>

<s id="id002961">huma<lb/>nis habetur: <expan abbr="frequ&etilde;s">frequens</expan> in in&longs;trumentis, uo <expan abbr="ca&ttilde;que">caturque</expan> tris diapa&longs;on inde &longs;ex<lb/>cupla, &longs;eu bis diapa&longs;on diapente. </s>

<s id="id002962">Quintupla <expan abbr="a&utilde;t">aunt</expan> minus <expan abbr="cõcors">concors</expan> e&longs;t: <lb/>&longs;ed de hac inferius dicemus, atque de multiplicib. </s>

<s id="id002962">Quintupla <expan abbr="a&utilde;t">aut</expan> minus <expan abbr="cõcors">concors</expan> e&longs;t: <lb/>&longs;ed de hac inferius dicemus, atque de multiplicibus </s>

<s id="id002963">dicta &longs;unto. </s>

<s id="id002964">Sed de <lb/><expan abbr="cõ">com</expan> centu ex particula &longs;uperaddita &longs;exquialtera &longs;exquitertia atque alijs <lb/>nunc agendum. </s>

<s id="id002965">Clarum e&longs;t. </s>

<s id="id002967">has e&longs;&longs;e &longs;implici&longs;simas. </s>

<s id="id002968">Cum ergo du <lb/>pla proportio non magis po&longs;sit diuidi æqualibus interuallis atque<lb/>&longs;implicibus proportionibus quàm in &longs;exquialteram & &longs;exquiter<lb/>tiam, uelutinter 4 & 2 interpo&longs;ito 3. nam proportio 3 ad 2 e&longs;t &longs;ex<lb/>quialtera, & 4 ad 3 &longs;exquitertia: nec melius pote&longs;t diuidi, at &longs;exqui<lb/>alteram & &longs;exquitertiam quantumuis magnis numeris diuidere <lb/>non licebat melius aut commodius quam per &longs;exquioctauas: uelu<lb/>ti &longs;umpto numero 64 cui duplus e&longs;t 128, inter medius 96 qui cum <lb/>64 &longs;exquialteram facit proportionem, quæ &longs;uaui&longs;sima e&longs;t omni<lb/>um deductis multiplicibus, uo caturque diapente. </s>

<s id="id002969">At quæ e&longs;t 128 ad <lb/>96 &longs;exquitertia e&longs;t minu&longs;que benè &longs;onat per &longs;e, &longs;ed in acutioribus uo<lb/>cibus &longs;olum cum alijs benè &longs;onat, uelut cum diapente, perficiens <lb/>diapa&longs;on, interuallum, ergo inter 96 & 64 diui&longs;um per &longs;exquiocta<pb pagenum="169" xlink:href="015/01/188.jpg"/>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 e&longs;t &longs;exquio<expan abbr="ctau&utilde;">ctauum</expan>, &longs;icut 81 ad 72. uerùm <lb/>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb/>& multo minus 96 ad 81, quare ui&longs;um e&longs;t Ptolem&etail;o ut &longs;ubtracta mona<lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, & 96, proportio <expan abbr="a&utilde;t">aut</expan> 80 ad 64 <expan abbr="cõ&longs;tituit">con&longs;tituit</expan> &longs;exqui<lb/><expan abbr="quartã">quartam</expan> atque <expan abbr="diton&utilde;">ditonum</expan>, proportio quoque 96 ad 72 <expan abbr="&longs;exquitertiã">&longs;exquitertiam</expan> <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> que. <lb/></s>

<s id="id002970">Rur&longs;us proportio 128 ad 64 <expan abbr="cõponi&ttilde;">componitur</expan> ex proportionibus </s>

<pb pagenum="169" xlink:href="015/01/188.jpg"/>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 e&longs;t &longs;exquio <expan abbr="ctau&utilde;">ctauum</expan>, &longs;icut 81 ad 72. uerùm <lb/>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb/>& multominus 96 ad 81, quare ui&longs;um e&longs;t Ptolem&etail;o ut &longs;ubtracta mona <lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, & 96, proportio <expan abbr="a&utilde;t">aunt</expan> 80 ad 64 <expan abbr="cõ&longs;tituit">con&longs;tituit</expan> &longs;exqui <lb/><expan abbr="quartã">quartam</expan> atque <expan abbr="diton&utilde;">ditonum</expan>, proportio quo que 96 ad 72 <expan abbr="&longs;exquitertiã">&longs;exquitertiam</expan> <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> que. <lb/></s>

<s id="id002970">Rur&longs;us proportio 128 ad 64 <expan abbr="cõponi&ttilde;">componitur</expan> ex proportionib. </s>

<s id="id002971">80 ad 64, <expan abbr="&qtilde;">quae</expan> <expan abbr="habe&ttilde;">habetur</expan> <lb/>pro ditono ut <expan abbr="dict&utilde;">dictum</expan> e&longs;t, & e&longs;t &longs;exquiquarta proportio. </s>

<s id="id002972">At 128 cum 80 e&longs;t in <lb/>proportione &longs;uperpartiente tres quintas, <expan abbr="&qtilde;">quae</expan> <expan abbr="iter&utilde;">iterum</expan> e&longs;t con&longs;ona. </s>

<s id="id002973">Regula <expan abbr="e&mtilde;">emm</expan> <lb/>e&longs;t quae ubi con&longs;onantia uo <expan abbr="c&utilde;">cum</expan> <expan abbr="diuida&ttilde;">diuidatur</expan> in duas partes, <expan abbr="quar&utilde;">quarum</expan> una &longs;it con&longs;o <lb/>nans, <expan abbr="reliquã">reliquam</expan> <expan abbr="etiã">etiam</expan> e&longs;&longs;e <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan>, at <expan abbr="nõ">non</expan> <expan abbr="cõuerti&ttilde;">conuertitur</expan>. </s>

<s id="id002974">S&etail;pe. </s>

<s id="id002976">fit ut ex duabus <lb/></s>

<s id="id002977">con&longs;onantibus di&longs;&longs;onans <expan abbr="cõpo&longs;itio">compo&longs;itio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="c&utilde;">cum</expan> ditono, &longs;ed ut ad <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> reuertar, alia diapa&longs;on e&longs;t inter 80 <lb/>& 40, at inter 48 & 40 e&longs;t &longs;emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">o&longs;ten&longs;um</expan> e&longs;t, uelut inter 96 & <lb/>80, nam inter 45 & 40 e&longs;t proportio &longs;exquioctaua, inter 48 <expan abbr="a&utilde;t">aunt</expan> & 45 &longs;ex<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> e&longs;t &longs;uperbi<lb/>partiens tertias &longs;eu &longs;olida <expan abbr="c&utilde;">cum</expan> be&longs;&longs;e &longs;eu &longs;exta maior erit <expan abbr="cõ&longs;onans">con&longs;onans</expan>. </s>

<s id="id002977">con&longs;onantibus di&longs;&longs;onans <expan abbr="cõpo&longs;itio">compo&longs;itio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="c&utilde;">cum</expan> ditono, &longs;ed ut ad <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> reuertar, alia diapa&longs;on e&longs;t inter 80 <lb/>& 40, at inter 48 & 40 e&longs;t &longs;emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">o&longs;ten&longs;um</expan> e&longs;t, uelut inter 96 & <lb/>80, nam inter 45 & 40 e&longs;t proportio &longs;exquioctaua, inter 48 <expan abbr="a&utilde;t">aut</expan> & 45 &longs;ex<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> e&longs;t &longs;uperbi<lb/>partiens tertias &longs;eu &longs;olida <expan abbr="c&utilde;">cum</expan> be&longs;&longs;e &longs;eu &longs;exta maior erit <expan abbr="cõ&longs;onans">con&longs;onans</expan>. </s>

<s id="id002978">Iam er <lb/>go uidemus detractione aut additione &longs;exquio ctuage&longs;imæ, concinnas <lb/>reddi uulgatiores armonias: <expan abbr="tertiã">tertiam</expan> utran que <expan abbr="maior&etilde;">maiorem</expan> &longs;cilicet & <expan abbr="minor&etilde;">minorem</expan>, ac <lb/>rur&longs;us <expan abbr="&longs;extã">&longs;extam</expan> <expan abbr="maior&etilde;">maiorem</expan> atque minore <expan abbr="&qtilde;">quae</expan> in minoribus numeris &longs;cilicet à mo<lb/>nade ad octo po&longs;itæ &longs;unt. </s>

<s id="id002979">Vides præterea <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> in &longs;exquiquinta <lb/><arrow.to.target n="table21"/><lb/><expan abbr="cõ&longs;tare">con&longs;tare</expan>: adeò ut à &longs;enario infra nihil inutile <lb/><figure id="id.015.01.188.1.jpg" xlink:href="015/01/188/1.jpg"/>reddatur. </s>

<s id="id002980">Diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> cum primum di <lb/>uidi pote&longs;t, &longs;i &longs;ecus diuidatur <08> in <expan abbr="diton&utilde;">ditonum</expan> <lb/>& <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, aut in &longs;emiditonum & <expan abbr="ton&utilde;">tonum</expan>, <lb/>&longs;cilicet in duo <expan abbr="tant&utilde;">tantum</expan> interualla, non <expan abbr="cõmodius">commo<lb/>dius</expan> <expan abbr="quã">quam</expan> inter octo & &longs;eptem & &longs;ex diuidi <lb/>pote&longs;t. </s>

<s id="id002980">Diate&longs;&longs;aron <expan abbr="a&utilde;t">aut</expan> cum primum di <lb/>uidi pote&longs;t, &longs;i &longs;ecus diuidatur <08> in <expan abbr="diton&utilde;">ditonum</expan> <lb/>& <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, aut in &longs;emiditonum & <expan abbr="ton&utilde;">tonum</expan>, <lb/>&longs;cilicet in duo <expan abbr="tant&utilde;">tantum</expan> interualla, non <expan abbr="cõmodius">commo<lb/>dius</expan> <expan abbr="quã">quam</expan> inter octo & &longs;eptem & &longs;ex diuidi <lb/>pote&longs;t. </s>

<s id="id002981">Cum ergo octo ad <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> di&longs;&longs;ona &longs;it, <lb/>quippe nimis remota e&longs;t h&etail;c proportio à &longs;en <lb/>&longs;u humano: <expan abbr="quamobr&etilde;">quamobrem</expan> ex regula data, ne<lb/>que proportio <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> ad &longs;ex. </s>

<s id="id002982">Sed dubitabis <lb/>meritò, quia <expan abbr="c&utilde;">cum</expan> diate&longs;&longs;aron diuidatur <expan abbr="bifariã">bifa<lb/>riam</expan>, in <expan abbr="diton&utilde;">ditonum</expan> & <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, ac rur&longs;us in <expan abbr="&longs;emiditon&utilde;">&longs;e<lb/>miditonum</expan> & <expan abbr="ton&utilde;">tonum</expan>, quarum altera <expan abbr="cõ&longs;onans">con&longs;onans</expan> e&longs;t, reliqua <expan abbr="nõ">non</expan>. </s>

<s id="id002983"><expan abbr="Vide&ttilde;">Videtur</expan> ergo <lb/>infirmari regula illa, quae con&longs;onantia diui&longs;a &longs;i una pars <expan abbr="cõ&longs;onet">con&longs;onet</expan>, alia non <lb/>po&longs;sit e&longs;&longs;e di&longs;&longs;onans, <expan abbr="nã">nam</expan> con&longs;tat <expan abbr="coni&utilde;">conium</expan> & <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan> tam per &longs;e quam in <lb/><expan abbr="cõpo&longs;itione">compo&longs;itione</expan> di&longs;&longs;onare: & <expan abbr="nõ">non</expan> <expan abbr="par&utilde;">parum</expan> &longs;ed acerbè. </s>

<s id="id002984"><expan abbr="Ver&utilde;">Verum</expan> re&longs;pondeo diate&longs;&longs;a <lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">emm</expan> per <lb/>fe e&longs;t, di&longs;&longs;onans e&longs;t: at que &longs;ic in <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan> & di&longs;&longs;onantem diuidi pote&longs;t: <lb/>quatenus <expan abbr="a&utilde;t">aunt</expan> pars e&longs;t diapa&longs;on <expan abbr="cõ&longs;onans">con&longs;onans</expan> in acutis: quan <08> <expan abbr="etiã">etiam</expan> adiecta <lb/>ditono aut &longs;emiditono &longs;uprà efficiat <expan abbr="&longs;extã">&longs;extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>benè &longs;onantes. </s>

<s id="id002984"><expan abbr="Ver&utilde;">Verum</expan> re&longs;pondeo diate&longs;&longs;a<lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">emm</expan> per <lb/>&longs;e e&longs;t, di&longs;&longs;onans e&longs;t: at que &longs;ic in <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan> & di&longs;&longs;onantem diuidi pote&longs;t: <lb/>quatenus <expan abbr="a&utilde;t">aut</expan> pars e&longs;t diapa&longs;on <expan abbr="cõ&longs;onans">con&longs;onans</expan> in acutis: quan <08> <expan abbr="etiã">etiam</expan> adiecta <lb/>ditono aut &longs;emiditono &longs;uprà efficiat <expan abbr="&longs;extã">&longs;extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>benè &longs;onantes. </s>

<s id="id002985">At quintupla proportio ut ab initio propo&longs;itum e&longs;t, <expan abbr="cõ&longs;tat">con&longs;tat</expan> <lb/>bis diapa&longs;on, & &longs;exquiquarta, ut planè <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t: &longs;exquiquarta <expan abbr="a&utilde;t">aunt</expan>

<pb pagenum="170" xlink:href="015/01/189.jpg"/>ditonus: bis diapa&longs;on <expan abbr="a&utilde;t">aunt</expan> quindecim uo cibus. </s>

<pb pagenum="170" xlink:href="015/01/189.jpg"/>ditonus: bis diapa&longs;on <expan abbr="a&utilde;t">aut</expan> quindecim uo cibus. </s>

<s id="id002986">Omnes igitur decem, & <lb/><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> uoces, <expan abbr="&qtilde;">quae</expan> &longs;exdecim interuallis <expan abbr="di&longs;tinguun&ttilde;">di&longs;tinguuntur</expan>, con&longs;onantes &longs;unt: & ex <lb/>genere ditoni, & &longs;exquiquartæ, &longs;ed paulo minus benè <expan abbr="&longs;onãt">&longs;onant</expan> <08> ditonus <lb/>ip&longs;e. </s>

<s id="id002987">Igitur <expan abbr="quintuplã">quintuplam</expan> multiplicem ad &longs;ex <expan abbr="quiquartã">quiquartam</expan> reduximus. </s>

<s id="id002988">Verum <lb/>ut o&longs;ten&longs;um e&longs;t & decima&longs;eptima, <expan abbr="&qtilde;">quae</expan> bis diapa&longs;on <expan abbr="cõ&longs;tat">con&longs;tat</expan>, & &longs;emiditono <lb/>benè &longs;onat, h&etail;c <expan abbr="a&utilde;t">aunt</expan> inter non aginta &longs;ex & uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> e&longs;t & <lb/>&longs;uperquadripartiens quintas. </s>

<s id="id002988">Verum <lb/>ut o&longs;ten&longs;um e&longs;t & decima&longs;eptima, <expan abbr="&qtilde;">quae</expan> bis diapa&longs;on <expan abbr="cõ&longs;tat">con&longs;tat</expan>, & &longs;emiditono <lb/>benè &longs;onat, h&etail;c <expan abbr="a&utilde;t">aut</expan> inter nonaginta &longs;ex & uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> e&longs;t & <lb/>&longs;uperquadripartiens quintas. </s>

<s id="id002989">Diapa&longs;on quo que cum &longs;exta maiore & mi<lb/>nore eandem habentrationem quam 16 ad 5, & 10 ad 3, triplam utranque, <lb/>&longs;ed altera &longs;exquiquinta, altera &longs;exquitertia: bis diapa&longs;on uerò <expan abbr="c&utilde;">cum</expan> ei&longs;dem <lb/>ut uiginti ad tria, & 32 ad quin que &longs;excupla utraque: &longs;ed altera &longs;uperbipar<lb/>tiens tertias, altera quintas. </s>

<s id="id002990"><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t igitur hanc diui&longs;ionem <expan abbr="nõ">non</expan> &longs;o<lb/>lum concinnam magis e&longs;&longs;e & &longs;uauem &longs;ed omnem <expan abbr="tonor&utilde;">tonorum</expan> & &longs;emitonio<lb/>rum <expan abbr="nece&longs;sitat&etilde;">nece&longs;sitatem</expan> effugere. </s>

<s id="id002991">Quòd uerò in cau&longs;a fuit ut toni & &longs;emitonia <lb/>in u&longs;u e&longs;&longs;ent, id e&longs;t, quoniam in <expan abbr="di&longs;c&etilde;do">di&longs;cendo</expan> nece&longs;&longs;e e&longs;t eandem &longs;eruari ratio<lb/>nem in <expan abbr="crementor&utilde;">crementorum</expan>, ne que arithmeticam &longs;ed <expan abbr="geometricã">geometricam</expan>. </s>

<s id="id002992">Ideò <expan abbr="a&longs;c&etilde;&longs;us">a&longs;cen&longs;us</expan> per <lb/>tonos & &longs;emitonia <expan abbr="cõmodus">commodus</expan> fuit, nam duplicem <expan abbr="&longs;ol&utilde;">&longs;olum</expan> differentiam pue <lb/>ri u&longs;u a&longs;&longs;equi coguntur. </s>

<s id="id002993">At uerò poterat & per &longs;exqui&longs;extam diuidi dia <lb/>te&longs;&longs;aron, ut inter triginta &longs;ex & quadraginta nouem interpo&longs;itis 42, ue<lb/>rùm triplex <expan abbr="&longs;equeba&ttilde;">&longs;equebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diate&longs;&longs;aron ad amu&longs;sim <lb/>non &longs;eruaretur, &longs;ed incidebat in cacophoniam, addita quadrage&longs;ima o<lb/>ctaua parte: deficiente <expan abbr="a&utilde;t">aunt</expan> in duabus &longs;exqui&longs;eptimis numeris &longs;eu propor<lb/>tione &longs;exquitertia: ut inter 49 & 64 loco 48 & 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb/>36, additaigitur monade in termino medio utrin que fit di&longs;&longs;onantia. </s>

<s id="id002993">At uerò poterat & per &longs;exqui&longs;extam diuidi dia<lb/>te&longs;&longs;aron, ut inter triginta &longs;ex & quadraginta nouem interpo&longs;itis 42, ue<lb/>rùm triplex <expan abbr="&longs;equeba&ttilde;">&longs;equebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diate&longs;&longs;aron ad amu&longs;sim <lb/>non &longs;eruaretur, &longs;ed incidebat in cacophoniam, addita quadrage&longs;ima o<lb/>ctaua parte: deficiente <expan abbr="a&utilde;t">aut</expan> in duabus &longs;exqui&longs;eptimis numeris &longs;eu propor<lb/>tione &longs;exquitertia: ut inter 49 & 64 loco 48 & 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb/>36, addita igitur monade in termino medio utrin que fit di&longs;&longs;onantia. </s>

<s id="id002994">Se<lb/>cundum inconueniens, e&longs;t quae &longs;ic diuidente non &longs;eruabatur ratio &longs;exqui<lb/>quartæ & &longs;exquiquintæ &longs;eu ditoni & &longs;emiditoni, quæ uoces benè &longs;o<lb/>nant. </s>

<s id="id002995">Tertium inconueniens erat, quòd hæcratio diuidendi diapentes <lb/>minimè &longs;atisfaciebat, uelutinter 324 & 216. Interponere enim nece&longs;&longs;e <lb/>erat 252 & 294, unde incongrua rur&longs;us erat diui&longs;io. </s>

<s id="id002996">His tot cau&longs;is cum <lb/>proportiones maiores non fatisfacerent ut &longs;exqui quinta quæ diate&longs;&longs;a<lb/>ron nullo modo æqualiter diuidere pote&longs;t, & in diapente deficit &longs;exqui <lb/>uige&longs;imaquarta, ut inter 25 & 36, coacti &longs;unt cum nec &longs;exqui&longs;exta nec <lb/>&longs;exqui&longs;eptima idoneæ e&longs;&longs;ent ad &longs;exquio ctauam confugere.</s>

Line 8799

Line 8745

</table>

<s id="id002997">E&longs;t & alia diui&longs;io toni in &longs;emitonia, <expan abbr="&qtilde;">quae</expan> e&longs;t uaria <expan abbr="pon&etilde;do">ponendo</expan> <expan abbr="ton&utilde;">tonum</expan> inter 18 <lb/>& 16, media uox e&longs;t 17 &longs;emitonium maius inter 17 & 16, &longs;ed minus inter <lb/>18 & 17, <expan abbr="quor&utilde;">quorum</expan> differentia e&longs;t 1/288. Hic &longs;ubit admiratio quomodo <expan abbr="&longs;emitoni&utilde;">&longs;emi<lb/>tonium</expan> minus <expan abbr="apte&ttilde;">aptetur</expan> tam gratè in &longs;ymphonijs, maius <expan abbr="a&utilde;t">aunt</expan> <expan abbr="nequaquã">nequaquam</expan>. </s>

<s id="id002998">Ptole <lb/>m&etail;us hoc negaret, quia &longs;exquiquinta &longs;eu &longs;emiditonus <expan abbr="cõ&longs;tat">con&longs;tat</expan> tono inte<lb/>gro, qui e&longs;t inter 90 & 80, & &longs;emitonio <expan abbr="plu&longs;quã">plu&longs;quam</expan> maiore quod e&longs;t inter <lb/>96 & 90, & e&longs;t &longs;exquiquinta decima: <expan abbr="&qtilde;">quae</expan> maior e&longs;t tono maiore 1/255. Pro<lb/>pterea dicemus cau&longs;am e&longs;&longs;e quae po&longs;ito &longs;emiditono inter 81 & 96, id e&longs;t, <lb/>27 & 32 &longs;ublato tono, id e&longs;t, 234 & 216, remanebit 13 differentia 256 ad <lb/>243, &longs;eu qualis e&longs;t 96 ad 91 & 1/8 quæ e&longs;t ut 768 ad 729 et redit ad <expan abbr="id&etilde;">idem</expan>, &longs;cili

<s id="id002998">Ptole<lb/>m&etail;us hoc negaret, quia &longs;exquiquinta &longs;eu &longs;emiditonus <expan abbr="cõ&longs;tat">con&longs;tat</expan> tono inte<lb/>gro, qui e&longs;t inter 90 & 80, & &longs;emitonio <expan abbr="plu&longs;quã">plu&longs;quam</expan> maiore quod e&longs;t inter <lb/>96 & 90, & e&longs;t &longs;exquiquinta decima: <expan abbr="&qtilde;">quae</expan> maior e&longs;t tono maiore 1/255. Pro<lb/>pterea dicemus cau&longs;am e&longs;&longs;e quae po&longs;ito &longs;emiditono inter 81 & 96, id e&longs;t, <lb/>27 & 32 &longs;ublato tono, id e&longs;t, 234 & 216, remanebit 13 differentia 256 ad <lb/>243, &longs;eu qualis e&longs;t 96 ad 91 & 1/8 quæ e&longs;t ut 768 ad 729 et redit ad <expan abbr="id&etilde;">idem</expan>, &longs;cili<pb pagenum="171" xlink:href="015/01/190.jpg"/>cet, ut 256 ad 243, 13 autem e&longs;t paulo plus decimanona, ergo multo mi<lb/>nus &longs;emitonio minore. </s>

<pb pagenum="171" xlink:href="015/01/190.jpg"/>cet, ut 256 ad 243, 13 autem e&longs;t paulo plus decimanona, ergo multo mi<lb/>nus &longs;emitonio minore. </s>

<s id="id002999">&longs;ecundum <expan abbr="m&etilde;tem">mentem</expan> ergo Ptolemæi, po&longs;ito tono <lb/>inter 135, & 120, & &longs;emitonio maiore inter 128 & 120 remanebit &longs;emito<lb/>nium minus fermè inter 19 & 18, id e&longs;t, 133 & 126, qu&etail; proportio differt <lb/>à 135 & 138. Si quis autem bene animaduertat, &longs;exquioctuage&longs;ima illa <lb/>adimitur, ex tono & additur &longs;emitonio minori, & hæc e&longs;t cau&longs;a quòd <lb/>&longs;emitonium maius Ptolemæi &longs;it concinnum, quia additur tonis imper <lb/>fectis. </s>

<s id="id003000">Dimidium autem &longs;emitonij minoris e&longs;t inter 36 & 35, & uocatur <lb/><expan abbr="cõma">comma</expan>: & e&longs;t minus & maius: maius e&longs;t inter 35 & 34, rur&longs;us <expan abbr="cõma">comma</expan> mi<lb/>nus diuiditur in duas die&longs;es, minorem, quæ e&longs;t inter 72 & 71, & maio<lb/>rem, qu&etail; e&longs;t inter 71 & 70, & ideò manet difficultas quomodo intenta <lb/>uoce per die&longs;im fiat melior con&longs;onantia? </s>

<s id="id003001">nam de remi&longs;sione po&longs;&longs;emus <lb/>dicere quòd accipitur loco &longs;exquio ctuage&longs;imæ: &longs;ed in &longs;exquioctuage<lb/>&longs;ima remittitur de tono &longs;ecundum mentem Ptolemæi, in die&longs;i intendi<lb/>tur &longs;emitonium minus, &longs;icut o&longs;tendit experimentum, &longs;ed for&longs;an conue <lb/>niunt quia intentio &longs;emitonij minoris deducit &longs;emiditonum ad &longs;exqui <lb/>quintam: e&longs;t enim differentia &longs;emitonij minoris intenti hoc modo ad <lb/>&longs;emitonium minus, ut 136 ad 135: &longs;ed hoc e&longs;t longè minus &longs;exquioctua <lb/>ge&longs;ima, unum &longs;at e&longs;t, hanc e&longs;&longs;e ultimam diui&longs;ionem toni in octo par<lb/>tes, & ut in diatonico toni dominantur, ita in chromatico &longs;emitonia in <lb/>enarmonico die&longs;es, &longs;ed die&longs;es fugitando (utita dicam) ac aures uelli<lb/>cando, mirum in modum oblectant audientes: uelut toni &longs;tando, un<lb/>de etiam nomen, &longs;emitonia medium modum obtinent.</s>

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<s id="id003004">Virum enim hunc &longs;upra omnem humani ingenij <expan abbr="metã">metam</expan> fui&longs;&longs;e <expan abbr="nõ">non</expan> nega<lb/>mus: &longs;ed hanc partem quam hic agit, adeò infeliciter tractat, ut malim <lb/>credere <expan abbr="tot&utilde;">totum</expan> illum tertium <expan abbr="libr&utilde;">librum</expan> fui&longs;&longs;e ab aliquo alio <expan abbr="adiect&utilde;">adiectum</expan>. </s>

<s id="id003005">Etenim <lb/>quid turpius &longs;apienti homini <08> imitari uulgares illos? </s>

<s id="id003006"><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> planetæ, <lb/>&longs;eptem mundi miracula, <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> artes liberales: quid enim &longs;imilitudo nu<pb pagenum="172" xlink:href="015/01/191.jpg"/>meri iuuare pote&longs;t, aut quàm afferre utilitatem? </s>

<pb pagenum="172" xlink:href="015/01/191.jpg"/>meri iuuare pote&longs;t, aut quàm afferre utilitatem? </s>

<s id="id003007">nimis certè in <expan abbr="dign&utilde;">dignum</expan> e&longs;t <lb/>uti <expan abbr="argum&etilde;to">argumento</expan> à &longs;imilitudine &longs;umpto: tum maximè adeò leui. </s>

<s id="id003008">Sed quo<lb/>niam con&longs;tat omnia quæ in mundo &longs;unt ordine coniuncta e&longs;&longs;e, & ne<lb/>ce&longs;sitate uinciri, ideò cùm finis ip&longs;e uerus &longs;it, non tam debemus Ptole<lb/>mæum damnare, quae non probauerit, quàm laudare, quod <expan abbr="ueritat&etilde;">ueritatem</expan> &longs;ine <lb/>ratione &longs;it a&longs;&longs;ectus. </s>

<s id="id003009">Sæpe enim accidit huiu&longs;modi uiris adeò pr&etail;&longs;tan<lb/>tibus ut ueritas detegatur, quam cùm illi, ut mos e&longs;t <expan abbr="homin&utilde;">hominum</expan>, rationi<lb/>bus adornare nituntur, tran&longs;gredientes metam muneris, in ab&longs;urda & <lb/>ineptias <expan abbr="incid&utilde;t">incidunt</expan>. </s>

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<s id="id003011">Verùm dubium e&longs;t, an &longs;oni propter nume <lb/>ros iucundi &longs;int, an propter aliud? </s>

<s id="id003012">& &longs;i propter aliud, cur ergo numeri <lb/>ad hoc &longs;unt nece&longs;&longs;arij? </s>

<s id="id003013">& cur ob&longs;eruare eos oportet ne ab illorum ordi <lb/>ne di&longs;iungi po&longs;sint? </s>

<s id="id003014">Hoc <expan abbr="a&utilde;t">aunt</expan> perfacilè <expan abbr="intelligi&ttilde;">intelligitur</expan>, & à nobis aliâs decla<lb/>ratum e&longs;t, &longs;cilicet delectare nos, quæ percipiuntur quæque ratione facta <lb/>uidentur, <expan abbr="quoniã">quoniam</expan> in his naturæ uis relucet & imago uniuer&longs;i, ergo dele <lb/>ctant nos, quoniam natur&etail; ordine nos con&longs;tamus. </s>

<s id="id003014">Hoc <expan abbr="a&utilde;t">aut</expan> perfacilè <expan abbr="intelligi&ttilde;">intelligitur</expan>, & à nobis aliâs decla<lb/>ratum e&longs;t, &longs;cilicet delectare nos, quæ percipiuntur quæque ratione facta <lb/>uidentur, <expan abbr="quoniã">quoniam</expan> in his naturæ uis relucet & imago uniuer&longs;i, ergo dele<lb/>ctant nos, quoniam natur&etail; ordine nos con&longs;tamus. </s>

<s id="id003015">Illud difficilius lon <lb/>gè &qring;d <expan abbr="tam&etilde;">tamen</expan> diligenti ob&longs;eruatione <expan abbr="dign&utilde;">dignum</expan> uidetur, &longs;cilicet, quonam pa <lb/>cto harmonia cum rebus c&oelig;le&longs;tibus aut humanis <expan abbr="cõiuncta">coniuncta</expan> &longs;it. </s>

<s id="id003016">For&longs;an <lb/>& illud ab re non e&longs;&longs;et intelligere, cur nullum animal pr&etail;ter hominem <lb/>capax &longs;it harmoniæ? </s>

<s id="id003017">an for&longs;an <expan abbr="quoniã">quoniam</expan> &longs;olus homo ratione participet, <lb/>& ob id &longs;olus gaudet ratione? </s>

<s id="id003018">ordinata <expan abbr="a&utilde;t">aunt</expan> ratione <expan abbr="cõ&longs;tant">con&longs;tant</expan> aut &longs;ola aut <lb/>maximè, numerus autem quid aliud e&longs;t quàm ordinis <expan abbr="&longs;eparator&utilde;">&longs;eparatorum</expan> ima<lb/>go. </s>

<s id="id003018">ordinata <expan abbr="a&utilde;t">aut</expan> ratione <expan abbr="cõ&longs;tant">con&longs;tant</expan> aut &longs;ola aut <lb/>maximè, numerus autem quid aliud e&longs;t quàm ordinis <expan abbr="&longs;eparator&utilde;">&longs;eparatorum</expan> ima<lb/>go. </s>

<s id="id003019">Porrò hæc accipienda &longs;unt ex his quæ &longs;en&longs;ibus deprehenduntur, <lb/>qualia &longs;unt quae animus mouetur & uarios affectus in duit iuxta harmo<lb/>niæ diuer&longs;itatem lætiti&etail;, tri&longs;titi&etail;, impetus, remi&longs;sionis, timoris, &longs;pei, ira<lb/>cundiæ, & commi&longs;erationis. </s>

<s id="id003020">Nos enim maximè octo affectus mouent <lb/>mu&longs;icæ modulationes. </s>

<s id="id003021">Secundum quid autem mouent? </s>

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<s id="id003026">Nam <lb/>duo in uniuer&longs;um ex ui&longs;u ad animi affectus mouendos habentur, tene <lb/>bræ ad tri&longs;titiam & metum, pictura regionum <expan abbr="am&oelig;nar&utilde;">am&oelig;narum</expan> ad iucundita <lb/>tem, &longs;ed <expan abbr="irã">iram</expan> quæ moueant picturæ alacritatemúe aut <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan>, <lb/>non habemus. </s>

<s id="id003027">Videtur ergo ob hæc &longs;onus ip&longs;e magis animæ intimus <lb/><08> ullum aliud &longs;en&longs;ile. </s>

<s id="id003028">Quod &longs;i odoratus e&longs;t in <expan abbr="app&etilde;dicibus">appendicibus</expan> cerebri, ui <lb/>&longs;us in pupilla oculi, gu&longs;tus in linguæ neruis, ueri&longs;imile e&longs;t magis inti<lb/>mum e&longs;&longs;e auditum, &longs;cilicet in cerebro ip&longs;o, atque ob id magis ab illo mo<lb/>ueri animam. </s>

<s id="id003029">Neque <expan abbr="e&mtilde;">emm</expan> in <expan abbr="a&etilde;re">aerre</expan> concepto à concauitatibus auris, qui no <lb/>&longs;tri pars non e&longs;t: neque à tympano, cùm &longs;uperflua fui&longs;&longs;et cauitas interior <lb/>omnis: neque enim inter pupillam & cerebrum pars ulla cernitur ad ui<lb/>&longs;um adiuuandum idonea: &longs;ed &longs;olus &longs;ufficit con&longs;en&longs;us pupill&etail; cum cere <lb/>bro: nam ad nos per &longs;piritus deffertur imago, non <expan abbr="e&mtilde;">emm</expan> ui&longs;us e&longs;&longs;et unus, <lb/>nec in uno tempore fieret, &longs;ed ueluti è <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> &longs;peculo & decimo &longs;imul, <lb/>& eodem tempore reflectitur imago, ut à primo ita &longs;en&longs;us ui&longs;us ex pu<lb/>pilla in cerebro & in corde & anima &longs;imul relucet. </s>

<s id="id003029">Neque <expan abbr="e&mtilde;">emm</expan> in <expan abbr="a&etilde;re">aere</expan> concepto à concauitatibus auris, qui no<lb/>&longs;tri pars non e&longs;t: neque à tympano, cùm &longs;uperflua fui&longs;&longs;et cauitas interior <lb/>omnis: neque enim inter pupillam & cerebrum pars ulla cernitur ad ui<lb/>&longs;um adiuuandum idonea: &longs;ed &longs;olus &longs;ufficit con&longs;en&longs;us pupill&etail; cum cere<lb/>bro: nam ad nos per &longs;piritus differtur imago, non <expan abbr="e&mtilde;">emm</expan> ui&longs;us e&longs;&longs;et unus, <lb/>nec in uno tempore fieret, &longs;ed ueluti è <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> &longs;peculo & decimo &longs;imul, <lb/>& eodem tempore reflectitur imago, ut à primo ita &longs;en&longs;us ui&longs;us ex pu<lb/>pilla in cerebro & in corde & anima &longs;imul relucet. </s>

<s id="id003030">At ergo non potuit <lb/>in tympano uel neruo den&longs;iore fieri auditus, &longs;ed in cerebro ip&longs;o, ob &qring;d <lb/>magis moueret affectus. </s>

<s id="id003031">Sed & magis incorporeus e&longs;t &longs;onus, ut qui <lb/>in&longs;trumentum proprium non afficiat, ni&longs;i cum immoderatus fuerit, at <lb/>omnis color, omnis lux oculum afficit, ac, ut ita dicam, tingit, neque &longs;uc<lb/>ce&longs;siones illas ob id adeò minutas oculus percipere pote&longs;t ut auris, <lb/>&longs;ed coinquinatur, ut ita dicam, priorum obiectorum reliquijs atque ima <lb/>ginibus. </s>

<s id="id003032">Vt in uniuer&longs;um con&longs;tet puriorem e&longs;&longs;e auditus &longs;en&longs;um etiam <lb/>animæ no&longs;træ propiorem quàm ui&longs;um.</s>

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<pb pagenum="174" xlink:href="015/01/193.jpg"/>nantiarum percipit perfectam diapa&longs;on. </s>

<s id="id003038">Videamus ergo an aliquid &longs;it <lb/>&longs;imile in animæ facultatibus, nec <expan abbr="dubi&utilde;">dubium</expan> e&longs;t quin ex &longs;en&longs;ib. </s>

<s id="id003038">Videamus ergo an aliquid &longs;it <lb/>&longs;imile in animæ facultatibus, nec <expan abbr="dubi&utilde;">dubium</expan> e&longs;t quin ex &longs;en&longs;ibus. </s>

<s id="id003039">exterioribus <lb/>atque interioribus fiat intelligentia. </s>

<s id="id003040">Et &longs;en&longs;us <expan abbr="quid&etilde;">quidem</expan> exteriores &longs;exquiter <lb/>tia <expan abbr="cõ&longs;tant">con&longs;tant</expan>: e&longs;t enim <expan abbr="illor&utilde;">illorum</expan> imperfecta cognitio: maior longè memori&etail; <lb/>unius & rationis reliquarumque <expan abbr="facultat&utilde;">facultatum</expan>, ex quibus <expan abbr="intellig&etilde;tia">intelligentia</expan> oritur. <lb/></s>

<s id="id003041">Iam uerò habemus exactam <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> facultatum anim&etail; human&etail;, <expan abbr="&qtilde;">quae</expan> <lb/>cogno&longs;cit. </s>

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<s id="id003048">Et <expan abbr="uide&ttilde;">uidetur</expan> mu&longs;ica nec hoc &etail;qualiter monere, &longs;ed <expan abbr="prim&utilde;">primum</expan> uideamus <lb/>an hi&longs;oli affectus &longs;int maximi, quippe dee&longs;&longs;e <expan abbr="uiden&ttilde;">uidentur</expan> amor atque odium. <lb/></s>

<s id="id003049">Et mihi dubium non e&longs;t quin hi potenti&longs;simi &longs;int <expan abbr="omni&utilde;">omnium</expan> præter <expan abbr="met&utilde;">metum</expan>. <lb/></s>

<s id="id003050">Sed metus <expan abbr="c&utilde;">cum</expan> cau&longs;a, affectus propriè <expan abbr="nõ">non</expan> e&longs;t, &longs;ed potius &longs;cientia <expan abbr="quædã">quædam</expan>. <lb/></s>

<s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor <lb/>tis, ppri&etail; aut de filio, non e&longs;t à ratione alienús, nec excedit metas, modò <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negocio: <lb/>tum maximè ob id quod nulla mu&longs;ica e&longs;t quæ <expan abbr="met&utilde;">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>

<s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor<lb/>tis, propri&etail; aut de filio, non e&longs;t à ratione alienús, nec excedit metas, modò <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negocio: <lb/>tum maximè ob id quod nulla mu&longs;ica e&longs;t quæ <expan abbr="met&utilde;">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>

<s id="id003052">Indicio e&longs;t quae potius <expan abbr="ill&utilde;">illum</expan> <lb/>excudit abrupta mu&longs;ica, &longs;icut & omnia alia quæ perturbant rationem, <lb/>ueluti <expan abbr="&longs;olan&utilde;">&longs;olanum</expan> & madrangora atque cicuta. </s>

<s id="id003053">Amorem igitur & odium <expan abbr="nõ">non</expan> <lb/>excitat mu&longs;ica, quia amor & odium alicuius &longs;unt amor & odium, mu&longs;i <lb/>ca <expan abbr="a&utilde;t">aunt</expan> generales &longs;olum mouet animi affectus. </s>

<s id="id003054">Et commi&longs;eratio, licet &longs;it <lb/>Didonis aut Phillidis, tamen e&longs;t generaliter mi&longs;erentis. </s>

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<pb pagenum="175" xlink:href="015/01/194.jpg"/>nes habentibus atque a&longs;peris. </s>

<s id="id003061">Hæc autem, ut ita dicam, nulla e&longs;t mu&longs;ica. <lb/></s>

<s id="id003062">Sed neque mu&longs;ica ulla tri&longs;titiam gignit, cum ut dixi, tri&longs;titia nil aliud &longs;it <08><lb/>mortis imago, mu&longs;ica <expan abbr="a&utilde;t">aunt</expan> uitam fouet. </s>

<s id="id003063">Vnde <expan abbr="nõ">non</expan> immeritò fertur Xeno <lb/>philus mu&longs;icus <expan abbr="cent&utilde;">centum</expan> quinque annis &longs;ine aliquo <expan abbr="incõmodo">incommodo</expan> uixi&longs;&longs;e, quod <lb/>&longs;ingulare e&longs;&longs;e exemplum in humana uita refert Plinius. </s>

<s id="id003064">Relin quitur igi <lb/>tur tandem, ut mu&longs;ica maximè moueat tres affectus lætitiam, remi&longs;sio<lb/>nem & mi&longs;ericordiam. </s>

<s id="id003065">Et quod ex his po&longs;tmodum ad labores in&longs;urga<lb/>mus intentius, hoc non e&longs;t ex mu&longs;ic&etail; ui aut facultate, &longs;ed <expan abbr="cõ&longs;equentibus">con&longs;equentibus</expan> <lb/>ad illa alia cau&longs;is. </s>

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<s id="id003069"><arrow.to.target n="marg580"/><lb/>affectuum, <expan abbr="c&utilde;">cum</expan> remittuntur uoces aut intendantur ad <expan abbr="ear&utilde;">earum</expan> intentionem. <lb/></s>

<s id="id003070">Sed non e&longs;t æqualis ratio, quoniam natura no&longs;tra ad <expan abbr="remi&longs;sion&etilde;">remi&longs;sionem</expan> natu<lb/>raliter inclinata e&longs;t, ad intentionem non ita, &longs;ed per uim <expan abbr="quandã">quandam</expan> aut me<lb/>dio uoluptatis, aut cum anima purior e&longs;t à corporis impedimentis. </s>

<s id="id003071">Et <lb/>ob id ad &longs;tudia nil aptius e&longs;t pura &longs;obrietate: nihil ineptius crapula atque<lb/>temulentia. </s>

<s id="id003072">At l&etail;titi&etail; cau&longs;&etail; &longs;unt, & <expan abbr="cõ">com</expan> cordia uo <expan abbr="c&utilde;">cum</expan>, & mutatio ex a&longs;pera <lb/>in &longs;uauem, <expan abbr="nõ">non</expan> &longs;ecus ac eius qui euadit è paupertate uel è mole&longs;tia aliqua <lb/>aut dolore aut alio <expan abbr="incõmodo">incommodo</expan>, tum inten&longs;io uo <expan abbr="c&utilde;">cum</expan> ac liber &longs;onus. </s>

<s id="id003072">At l&etail;titi&etail; cau&longs;&etail; &longs;unt, & <expan abbr="cõ">con</expan>cordia uo <expan abbr="c&utilde;">cum</expan>, & mutatio ex a&longs;pera <lb/>in &longs;uauem, <expan abbr="nõ">non</expan> &longs;ecus ac eius qui euadit è paupertate uel è mole&longs;tia aliqua <lb/>aut dolore aut alio <expan abbr="incõmodo">incommodo</expan>, tum inten&longs;io uo <expan abbr="c&utilde;">cum</expan> ac liber &longs;onus. </s>

<s id="id003073">Vnde <lb/>in l&etail;titia &longs;olent homines exclamare. </s>

<s id="id003074">At ad <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan> mouendam <lb/>omnia remitti oportet ex magna in parua, adeoque deficientem ex a&longs;pera <lb/>in leuem, ex ueloci in tardam, ex di&longs;&longs;ona in con&longs;onantem. </s>

<s id="id003075">Antiqui ergo <lb/>(ut author e&longs;t Cælius Rhodiginius) Dorico ad temperantiam & mode <lb/><arrow.to.target n="marg581"/><lb/>rationem utebantur, &longs;cilicet quòd non haberet præcipites lap&longs;us, neque<lb/>arduas intentiones: Phrygio ad impetum & bellicum ardorem, &longs;cilicet <lb/>per a&longs;peras intentiones: Lydio ad fletus & lamentationes per ca&longs;us & <lb/>remi&longs;siones longas ac &longs;uaues: ideo funeribus peculiaris: Mixolydio ad <lb/>commi&longs;erationem, ut defectiones interponantur & breues abruptæque<lb/>remi&longs;siones, iuuantque in hoc plurimum & &longs;en&longs;us uerborum, familiaris <lb/>hic tragædijs: Aeolicus qui & Ionicus tranquillitatis animi author e&longs;t <lb/>&longs;o mnumque conciliat: Dorico non ab&longs;imilis &longs;ed &longs;uauior & mollior: ideò <lb/>chromatici generis. </s>

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<s id="id003097">Ergo <lb/>ut ad rem ueniam: mu&longs;ica proportio bifariam <expan abbr="inueni&ttilde;">inuenitur</expan> in &longs;aporibus: &longs;im<lb/>pliciter, & ex comparatione, & &longs;impliciter quidem &longs;umma &longs;uauitas ad <lb/>diapa&longs;on refertur: e&longs;t enim &longs;uaui&longs;simus concen&longs;us in &longs;aporibus, ergo <lb/>dulce ei <expan abbr="re&longs;põdet">re&longs;pondet</expan>, ut &longs;implex, quid enim &longs;uauius e&longs;&longs;e pote&longs;t in utro que ge <lb/>nere. </s>

<s id="id003098">At pinguis, qualis in carnibus & ouis benè pr&etail;paratis ad <expan abbr="diap&etilde;te">diapente</expan> <lb/>refertur, e&longs;t enim & ip&longs;e &longs;uaui&longs;simus po&longs;t dulce, at que in &longs;uo genere perfe <lb/>ctus, diate&longs;&longs;aron uerò optimè &longs;al&longs;o <expan abbr="cõuenit">conuenit</expan>. </s>

<s id="id003099">Hic enim per &longs;e improbus <lb/>e&longs;t & in&longs;uauis, &longs;icut etiam &longs;apor &longs;al&longs;us e&longs;t, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> cum diapente <lb/>perficit diapa&longs;on, & cum diapa&longs;o inutile e&longs;t, et di&longs;cordat, ita &longs;apor &longs;al&longs;us <lb/>cum pingui &longs;ummam delectationem affert: cum dulci adeò parum con <lb/>gruit, ut melius &longs;ocietur <expan abbr="c&utilde;">cum</expan> amaro, uelut in oliuis benè &longs;al&longs;is. </s>

<s id="id003100">Ergo &longs;al<lb/>&longs;us &longs;apor cum diate&longs;&longs;aro ad <expan abbr="ungu&etilde;">unguem</expan> congruit rur&longs;us &longs;emiditonus <expan abbr="c&utilde;">cum</expan> in&longs;i<lb/>pido, & a&longs;tringens cum ditono conueniunt ad unguem, nam uterque <expan abbr="nõ">non</expan> <lb/>illepidus, & cum dulci conuenit, ita &longs;emiditonus & ditonus cum diapa<pb pagenum="171 [=177]" xlink:href="015/01/196.jpg"/>&longs;o conueniunt, uterque etiam horum &longs;aporum parum mouet &longs;en<lb/>&longs;um, & inter &longs;e &longs;unt qua&longs;i &longs;imiles quod ditono accidit & &longs;emidito<lb/>no, &longs;ed & neuter horum cum pingui conuenit, neque ditonus aut &longs;e<lb/>miditonus cum diapente congruit, di&longs;cordat enim h&etail;c compo&longs;itio <lb/>non parum. </s>

<pb pagenum="171 [=177]" xlink:href="015/01/196.jpg"/>&longs;o conueniunt, uterque etiam horum &longs;aporum parum mouet&longs;en<lb/>&longs;um, & inter &longs;e &longs;unt qua&longs;i &longs;imiles quod ditono accidit & &longs;emidito<lb/>no, &longs;ed & neuter horum cum pingui conuenit, neque ditonus aut &longs;e<lb/>miditonus cum diapente congruit, di&longs;cordat enim h&etail;c compo&longs;itio <lb/>non parum. </s>

<s id="id003101">Rur&longs;us & in hoc &longs;imiles &longs;unt quod diate&longs;&longs;aron cum di<lb/>tono & &longs;emiditono plurimum conuenit, ita & in&longs;ipidum, & a&longs;trin<lb/>gens cum &longs;al&longs;o bellè <expan abbr="cõueniunt">conueniunt</expan>. </s>

<s id="id003102">Diate&longs;&longs;aron enim cum ditono &longs;ex<lb/>tam efficit maiorem, & cum &longs;emiditono minorem qu&etail; utrique con&longs;o <lb/>nant, non tamen plus &longs;uaues per &longs;e &longs;unt, quòd dulci & pingui care<lb/>ant, ut nec &longs;exta maior aut minor, &qring;d neque diapa&longs;on perficiant neque<lb/>diapente: Acris <expan abbr="aut&etilde;">autem</expan> &longs;apor &longs;exta maiori &longs;imilis e&longs;t, acidus minori: <lb/>mutuo conueniunt cum in&longs;ipido acris, & cum a&longs;tringente acidus, <lb/>quemadmodum & &longs;exta maior cum &longs;emiditono, & minor cum di<lb/>tono copulatur perficientes diapa&longs;on: &longs;ed minus &longs;uauem, quia ab<lb/>e&longs;t diapente ibi, quia abe&longs;t pingue: au&longs;terum uero cum acri mode<lb/>rato conuenit, propterea bene uterque cum in&longs;ipido iungitur, unde <lb/>illud Epigrammatici:</s>

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<s id="id003104">Piper enim acre e&longs;t, & uinum au&longs;terum e&longs;t. </s>

<s id="id003105">Et iu&longs;ta querela Cicero<lb/>nis in Epi&longs;tolis familiaribus, qui à maluis fatetur &longs;e uictum, ut deci<lb/>derit in lienteriam: conueniunt ambo hi &longs;apores <expan abbr="c&utilde;">cum</expan> dulci & pingui, <lb/>uelut & utraque &longs;exta maior & minor cum diapa&longs;on & diapente, at <lb/>neuter cum &longs;al&longs;o, nam neque diate&longs;&longs;aron cum &longs;extamaiore uel mino<lb/>re iungi pote&longs;t. </s>

<s id="id003106">Amarus autem &longs;apor tono per&longs;imilis e&longs;t, di&longs;&longs;onus <lb/>enim per &longs;e e&longs;t &longs;emper, & amarus per&longs;e odio&longs;us tonus origo e&longs;t o<lb/>mnium <expan abbr="con&longs;onantiar&utilde;">con&longs;onantiarum</expan>, ita omnes fructus, &longs;eu dulces &longs;eu a&longs;tringen<lb/>tes, &longs;eu acidi, &longs;eu acres prius amari &longs;unt: tonus præterea nulla cum <lb/>con&longs;onantia peius coit quàm cum diapa&longs;o, ita neque amarus &longs;apor <lb/>infelicius iungnur quàm cum dulci, amarus quo que &longs;apor cum nul<lb/>lo magis conuenit <expan abbr="quã">quam</expan> cum &longs;al&longs;o, ita tonus additus diate&longs;&longs;aro, perfi <lb/>cit diapente dulci&longs;simam con&longs;onantiam, ut multi oliuas benè&longs;al&longs;as <lb/>prætulerint fa&longs;ianis: tantum conuenit &longs;al&longs;o cum amaro, amarus, <lb/>quo que &longs;apor leuis non abhorret à pingui, deteriorem <expan abbr="tam&etilde;">tamen</expan> aliquan <lb/>to efficit, ut intortis ex ab&longs;ynthio ouis & ca&longs;eo, atque in uitibus in <lb/>quibus coma ab&longs;ynthij in cocta fuit parum, degenerat tamen &longs;apor <lb/>ille à pingui: ita tono addito ad diapente fit &longs;exta maior, non adeò <lb/>&longs;uauis ut diapente, attamen <expan abbr="nõ">non</expan> pror&longs;us in&longs;uauis. </s>

<s id="id003106">Amarus autem &longs;apor tono per&longs;imilis e&longs;t, di&longs;&longs;onus <lb/>enim per &longs;e e&longs;t &longs;emper, & amarus per&longs;e odio&longs;us tonus origo e&longs;t o<lb/>mnium <expan abbr="con&longs;onantiar&utilde;">con&longs;onantiarum</expan>, ita omnes fructus, &longs;eu dulces &longs;eu a&longs;tringen<lb/>tes, &longs;eu acidi, &longs;eu acres prius amari &longs;unt: tonus præterea nulla cum <lb/>con&longs;onantia peius coit quàm cum diapa&longs;o, ita neque amarus &longs;apor <lb/>infelicius iungitur quàm cum dulci, amarus quo que &longs;apor cum nul<lb/>lo magis conuenit <expan abbr="quã">quam</expan> cum &longs;al&longs;o, ita tonus additus diate&longs;&longs;aro, perfi<lb/>cit diapente dulci&longs;simam con&longs;onantiam, ut multi oliuas benè&longs;al&longs;as <lb/>prætulerint fa&longs;ianis: tantum conuenit &longs;al&longs;o cum amaro, amarus, <lb/>quo que &longs;apor leuis non abhorret à pingui, deteriorem <expan abbr="tam&etilde;">tamen</expan> aliquan<lb/>to efficit, ut intortis ex ab&longs;ynthio ouis & ca&longs;eo, atque in uitibus in <lb/>quibus coma ab&longs;ynthij in cocta fuit parum, degenerat tamen &longs;apor <lb/>ille à pingui: ita tono addito ad diapente fit &longs;exta maior, non adeò <lb/>&longs;uauis ut diapente, at tamen <expan abbr="nõ">non</expan> pror&longs;us in&longs;uauis. </s>

<s id="id003107">Similiter &longs;i tonus <lb/>addatur ad &longs;emiditonum aut ad ditonum ex altero fit diate&longs;&longs;aron, <lb/>qui non concordat ex reliquo tritonus omnium a&longs;perrimus. </s>

<s id="id003108">Ergo <lb/>cum idem fiat coniuncto amaro cum in&longs;ipido, ac deterius <expan abbr="c&utilde;">cum</expan> a&longs;trin

<pb pagenum="172 [=178]" xlink:href="015/01/197.jpg"/>gente, uelut in acerbis glandibus, quibus nihil tri&longs;tius gu&longs;tari po<lb/>te&longs;t. </s>

<s id="id003109">Manife&longs;tum e&longs;tigitur optimè conuenire hano &longs;aporum diui<lb/>&longs;ionem cum mu&longs;ica proportione.</s>

<s id="id003109">Manife&longs;tum e&longs;t igitur optimè conuenire hanc &longs;aporum diui<lb/>&longs;ionem cum mu&longs;ica proportione.</s>

<s id="id003110">Cumque &longs;apores ex &longs;eptem planetis pendent manife&longs;tè, Saturnus <lb/><expan abbr="e&mtilde;">emm</expan> habet a&longs;tringens, quoniam frigidus e&longs;t & &longs;iccus. </s>

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<s id="id003116">Huiu&longs;modi ergo ratione con&longs;iderata <lb/>Luna ad <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> pertinebit Mercurius ad <expan abbr="&longs;extã">&longs;extam</expan> minorem, Sol <lb/>ad &longs;extam maiorem, Mars ad <expan abbr="tetrachord&utilde;">tetrachordum</expan>, Saturnus ad ditonum, <lb/>Iupiter ad diapente, Venus ad diapa&longs;on, unde plena illius dona uul <lb/>garis felicitatis opum honoris amoris & uoluptatis, po&longs;t quem e&longs;t <lb/>Iupiter, ut &longs;ine his duobus omnino nulla po&longs;sit e&longs;&longs;e felicitas.</s>

<s id="id003117">Sed & in circulo &longs;igniferi aliquam mu&longs;ica proportio habebit ra<lb/>tionem: diapa&longs;on <expan abbr="e&mtilde;">emm</expan> erit & totius ad dimidium, & be&longs;sis ad trien<lb/>tem, & dimidij ad quadrantem, & trientis ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diapente <expan abbr="a&utilde;t">aunt</expan> <lb/>totius circuli ad be&longs;&longs;em, & dodrantis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & dimidij ad tri<lb/>entem, & <expan abbr="quadrãtis">quadrantis</expan> ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> totius circuli ad do <lb/>drantem, & be&longs;sis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & trientis ad <expan abbr="quadrãtem">quadrantem</expan>: itaque in hoc <lb/>&longs;olo <expan abbr="c&utilde;">cum</expan> Ptolem&etail;o concordamus, in reliquis duobus ne&longs;cio qua ra<lb/>tione Ptolem&etail;us omi&longs;erit unam <expan abbr="cõiugationem">coniugationem</expan>, nam <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent qua<lb/>tuor in diapa&longs;on & diapente, tres tantum numerauit. </s>

<s id="id003117">Sed & in circulo &longs;igniferi aliquam mu&longs;ica proportio habebit ra<lb/>tionem: diapa&longs;on <expan abbr="e&mtilde;">emm</expan> erit & totius ad dimidium, & be&longs;sis ad trien<lb/>tem, & dimidij ad quadrantem, & trientis ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diapente <expan abbr="a&utilde;t">aut</expan> <lb/>totius circuli ad be&longs;&longs;em, & dodrantis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & dimidij ad tri<lb/>entem, & <expan abbr="quadrãtis">quadrantis</expan> ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> totius circuli ad do<lb/>drantem, & be&longs;sis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & trientis ad <expan abbr="quadrãtem">quadrantem</expan>: itaque in hoc <lb/>&longs;olo <expan abbr="c&utilde;">cum</expan> Ptolem&etail;o concordamus, in reliquis duobus ne&longs;cio qua ra<lb/>tione Ptolem&etail;us omi&longs;erit unam <expan abbr="cõiugationem">coniugationem</expan>, nam <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent qua<lb/>tuor in diapa&longs;on & diapente, tres tantum numerauit. </s>

<s id="id003118">Reliquas <expan abbr="a&utilde;t">aunt</expan> <lb/>quatuor per integra &longs;igna numerare licebit, ad <expan abbr="ration&etilde;">rationem</expan>, tamen a&longs;pe<lb/>ctuum deducere non po&longs;&longs;umus, propterea efficaciam quandam ha <lb/>bent etiam &longs;ignorum mutationes, &longs;ed harmoniam non perficiunt, <lb/>nam & &longs;i &longs;umamus &longs;exquiquartam & &longs;exquiquintam, ut in his &longs;ex<lb/>quialteram, &longs;eu diapente con&longs;tituamus, aut tria aut &longs;ex &longs;igna acci<lb/>pere oportebit: utrunque fuerit, reliqua pars ad diate&longs;&longs;aron pertinere <lb/>minimè pote&longs;t: quamobrem conuenientius e&longs;&longs;et meo iudicio, ut to <lb/>tus circulus non ad diapa&longs;on, uelut Ptolemæus, referretur, &longs;ed po<lb/>tius ad diapa&longs;on diapente: ita enim con&longs;titutis quatuor, quinque, <lb/>&longs;ex, duo decimque numeris, con&longs;taret tota ratio harmonica, diui&longs;o e<lb/>tiam diapente in ditonum & &longs;emiditonum. </s>

<s id="id003119">&longs;ed de hoc &longs;atis.</s>

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<s id="id003165">In capitulis enim dextra ac &longs;ini&longs;tra <lb/>&longs;unt foramina homotonorum, per qu&etail; tenduntur ergatis aut &longs;ucu<lb/>lis & uectibus è neruo torti funes, qui non præcluduntur, nec præ<lb/>ligantur ni&longs;i &longs;onitus ad artificis aures certos & &etail;quales fuerint. </s>

<s id="id003166">Bra<lb/>chia enim quæ in eas tentiones includuntur cum exten duntur æ<lb/>qualiter & parter utraque plagam emittere debent. </s>

<s id="id003167">Quod &longs;i non ho<lb/>motona fuerint, impedient directam telorum mi&longs;sionem. </s>

<s id="id003168">Item the<lb/>atris ua&longs;a ærea, qu&etail; in cellis &longs;ub gradib. </s>

<s id="id003168">Item the<lb/>atris ua&longs;a ærea, qu&etail; in cellis &longs;ub gradibus. </s>

<s id="id003169">mathematica ratione <expan abbr="collocan&ttilde;">collo<lb/>cantur</expan>, & <expan abbr="&longs;onit&utilde;">&longs;onitum</expan> di&longs;crimina, qu&etail; Gr&etail;ci <foreign lang="greek">)hx=eia</foreign> <expan abbr="uocãt">uocant</expan>, ad &longs;ymphonias mu <lb/>&longs;icas &longs;iue concentus <expan abbr="componun&ttilde;">componuntur</expan>, diui&longs;a in circinatione diate&longs;&longs;aron <lb/>& diapente & diapa&longs;on, uti uox &longs;c&oelig;nici &longs;onitus <expan abbr="cõueniens">conueniens</expan> in di&longs;po <lb/>&longs;itionibus, tactu <expan abbr="c&utilde;">cum</expan> o&longs;tenderit aucta <expan abbr="c&utilde;">cum</expan> <expan abbr="increm&etilde;to">incremento</expan> clarior et &longs;uauior <lb/>ad <expan abbr="&longs;pectator&utilde;">&longs;pectatorum</expan> perueniat aures. </s>

<s id="id003170">Hydraulicas quo que machinas & cæ<lb/>tera <expan abbr="&qtilde;">quae</expan> &longs;unt &longs;imilia his organis &longs;ine mu&longs;icis rationib. </s>

<s id="id003170">Hydraulicas quo que machinas & cæ<lb/>tera <expan abbr="&qtilde;">quae</expan> &longs;unt &longs;imilia his organis &longs;ine mu&longs;icis rationibus. </s>

<s id="id003171">efficere nemo <lb/>poterit. </s>

<s id="id003172">Capiamus ergo primum illud &qring;d e&longs;t manife&longs;tius, &longs;cilicet de <lb/>hydraulicis organis quorum meminit Suetonius in Nerone: Reli<lb/>quam diei partem per organa hydraulica noui & ignoti generis cir <lb/>cunduxit, o&longs;tenden&longs;que &longs;ingula de ratione ac difficultate cuiu&longs;que di&longs;<lb/>&longs;erens iam &longs;e prolaturum, ut con&longs;tet illa fui&longs;&longs;e magni opificij quæ <lb/>no&longs;tra &etail;tate de&longs;iere.” Re&longs;tat unicum & ualde leue <expan abbr="exempl&utilde;">exemplum</expan> auiculæ <lb/>æneæ uelligneæ re&longs;onantis. </s>

<s id="id003173">Certum e&longs;t <expan abbr="a&etilde;re">aerre</expan> effici &longs;onum, &longs;ed ita mi <lb/>&longs;ceri aquæ, ut dulcior & mollior non &longs;olum euadat, &longs;ed etiam acuti<lb/>or ac modulatior. </s>

<s id="id003173">Certum e&longs;t <expan abbr="a&etilde;re">aere</expan> effici &longs;onum, &longs;ed ita mi<lb/>&longs;ceri aquæ, ut dulcior & mollior non &longs;olum euadat, &longs;ed etiam acuti<lb/>or ac modulatior. </s>

<s id="id003174">Eadem autem ratio maris: &longs;ed cum aquæ corpus <lb/>moueatur, uidetur difficile &longs;eruare proportionem. </s>

<s id="id003175">ea prima diffi<lb/>cultas. </s>

<s id="id003176">&longs;ecunda e&longs;t, quod cùm aqua moueatur, uix ficri po&longs;&longs;e uide<lb/>tur ut totum &longs;eruet uocis integrum tenorem. </s>

<s id="id003176">&longs;ecunda e&longs;t, quod cùm aqua moueatur, uix fieri po&longs;&longs;e uide<lb/>tur ut totum &longs;eruet uocis integrum tenorem. </s>

<s id="id003177">tertia ob illius con

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<s id="id003189">“De Va&longs;is uerò æneis theatri quod melius e&longs;t quàm ut eundem <lb/>authorem con&longs;ulamus, dicentem ua&longs;a &etail;rea pro ratione magnitudi<lb/>nis theatri ita fabricentur, ut cum <expan abbr="tang&utilde;tur">tanguntur</expan>, &longs;onitum facere po&longs;sint <lb/>inter &longs;e diate&longs;&longs;aron diapent, ex ordine addit diapa&longs;on, po&longs;tea inter <lb/>&longs;edes theatri con&longs;titutis cellis ratione mu&longs;ica ibi collo centur: ita uti <lb/>nullum parietem tangant circaque habeant locum <expan abbr="uacu&utilde;">uacuum</expan> et à &longs;ummo <lb/>capite &longs;patium, ponantque inuer&longs;a & hab eant in parte qu&etail; &longs;pectat ad <lb/>&longs;cenam &longs;uppo&longs;itos cuneos ne minus alios &longs;emipede, contraque eas <lb/>cellas relinquantur apertur&etail; inferiorum graduum cubilibus lon<lb/>g&etail; pedes duos altæ &longs;emipedem. </s>

<s id="id003189">“De Va&longs;is uerò æneis theatri quod melius e&longs;t quàm ut eundem <lb/>authorem con&longs;ulamus, dicentem ua&longs;a &etail;rea pro ratione magnitudi<lb/>nis theatri ita fabricentur, ut cum <expan abbr="tang&utilde;tur">tanguntur</expan>, &longs;onitum facere po&longs;sint <lb/>inter &longs;e diate&longs;&longs;aron diapente, ex ordine addit diapa&longs;on, po&longs;tea inter <lb/>&longs;edes theatri con&longs;titutis cellis ratione mu&longs;ica ibi collocentur: ita uti <lb/>nullum parietem tangant circaque habeant locum <expan abbr="uacu&utilde;">uacuum</expan> et à &longs;ummo <lb/>capite &longs;patium, ponantque inuer&longs;a & habeant in parte qu&etail; &longs;pectat ad <lb/>&longs;cenam &longs;uppo&longs;itos cuneos ne minus alios &longs;emipede, contraque eas <lb/>cellas relinquantur apertur&etail; inferiorum graduum cubilibus lon<lb/>g&etail; pedes duos altæ &longs;emipedem. </s>

<s id="id003190">Et &longs;i non erit ampla magnitudine <lb/>theatrum, media altitudinis tran&longs;uer&longs;aregio de&longs;ignetur, & in ea tre <lb/>decim cellæ duo decim æqualib.

<s id="id003190">Et &longs;i non erit ampla magnitudine <lb/>theatrum, media altitudinis tran&longs;uer&longs;aregio de&longs;ignetur, & in ea tre<lb/>decim cellæ duodecim æqualibus.

interuallis di&longs;tantes <expan abbr="confornicen&ttilde;">confornicentur</expan> <lb/>uti ea echea quæ &longs;upra &longs;cripta &longs;unt, ad neten hyperboleon &longs;onan<lb/>tia in cellis quæ &longs;untin cornibus extremis utraque parte prima col<lb/>locentur, &longs;ecunda ab extremis diate&longs;&longs;aron ad <expan abbr="net&etilde;">netem</expan> diezeugmenon, <lb/>tertia diate&longs;&longs;aron ad neten parame&longs;on, quarta ad neten &longs;ynemme<lb/>non, quinta diate&longs;&longs;aron ad me&longs;en, &longs;exta diate&longs;&longs;aron ad hypaten me<lb/>&longs;en in medio unum diate&longs;&longs;aron ad hypaten hypaton. </s>

<s id="id003191">Quæ fequun<lb/>tur & ad intelligentiam prædictorum melius ex Gulielmo Philan<lb/>dro emendata &longs;ic tran&longs;cribemus: Eas regiones in tredecim cellas <lb/>diuidit æqualibus interuallis: id e&longs;t, cellas paribus uici&longs;sim inter

<s id="id003191">Quæ sequun<lb/>tur & ad intelligentiam prædictorum melius ex Gulielmo Philan<lb/>dro emendata &longs;ic tran&longs;cribemus: Eas regiones in tredecim cellas <lb/>diuidit æqualibus interuallis: id e&longs;t, cellas paribus uici&longs;sim inter

<pb pagenum="178 [=184]" xlink:href="015/01/203.jpg"/>&longs;ticijs di&longs;po&longs;itas di&longs;tribuit &longs;ex hinc atque hinc & unam mediam, quæ <lb/>tamen non u&longs;us, &longs;ed partitionis & re&longs;pon&longs;us cau&longs;a fit in media pr&etail;<lb/>cinctione. </s>

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<s id="id003213">In media quæ &longs;unt ad me&longs;en, quod ea ad proslambanomenon <lb/>habet con&longs;onantiam diapa&longs;on, & ad lychanon hypaton diapente.” </s>

<s id="id003214"><arrow.to.target n="marg591"/><lb/>Hæc autem ex &longs;igura patent in opere de Subtilitate de&longs;cripta.</s>

<s id="id003214"><arrow.to.target n="marg591"/><lb/>Hæc autem ex figura patent in opere de Subtilitate de&longs;cripta.</s>

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<s id="id003220">Coniugationes cuiu&longs;uis numeri breuiter inuenire.</s>

<s id="id003221">Sint gratia exempli <expan abbr="dec&etilde;">decem</expan> homines, & patet quod po&longs;&longs;ent e&longs;&longs;e &longs;in <lb/><arrow.to.target n="marg592"/><lb/>guli, & hoc <expan abbr="dec&etilde;">decem</expan> modis, quia &longs;unt <expan abbr="dec&etilde;">decem</expan>, ut Petrus & Ioannes: item, <lb/>po&longs;&longs;unt e&longs;&longs;e omnes &longs;imul, & hoc uno modo tantum, & po&longs;&longs;unt e&longs;&longs;e <lb/>duo, & hoc pote&longs;t uariari <expan abbr="&qtilde;">quae</expan> draginta quinque modis: & po&longs;&longs;unt e&longs;&longs;e <lb/>octo, & manife&longs;tum e&longs;t, quod <expan abbr="totid&etilde;">totidem</expan> modis uariantur, &longs;cilicet qua<lb/>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinqu&utilde;tur">quirelinquuntur</expan>, uariari <lb/>po&longs;&longs;unt 45 modis, ergo & illi octo ad <expan abbr="ungu&etilde;">unguem</expan> totidem modis. </s>

<s id="id003221">Sint gratia exempli <expan abbr="dec&etilde;">decem</expan> homines, & patet quod po&longs;&longs;ent e&longs;&longs;e &longs;in<lb/><arrow.to.target n="marg592"/><lb/>guli, & hoc <expan abbr="dec&etilde;">decem</expan> modis, quia &longs;unt <expan abbr="dec&etilde;">decem</expan>, ut Petrus & Ioannes: item, <lb/>po&longs;&longs;unt e&longs;&longs;e omnes &longs;imul, & hoc uno modo tantum, & po&longs;&longs;unt e&longs;&longs;e <lb/>duo, & hoc pote&longs;t uariari <expan abbr="&qtilde;">qua</expan>draginta quinque modis: & po&longs;&longs;unt e&longs;&longs;e <lb/>octo, & manife&longs;tum e&longs;t, quod <expan abbr="totid&etilde;">totidem</expan> modis uariantur, &longs;cilicet qua<lb/>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinqu&utilde;tur">qui relinquuntur</expan>, uariari <lb/>po&longs;&longs;unt 45 modis, ergo & illi octo ad <expan abbr="ungu&etilde;">unguem</expan> totidem modis. </s>

<s id="id003222">Et &longs;i<lb/>militer tres quot modis uariantur tot modis <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>, & quot modis <lb/>quatuor tot&longs;ex: quinque autem quia &longs;unt dimidium decem, pluribus <lb/>modis uariantur. </s>

<s id="id003223">Etideò pro ordine huius detrahes <expan abbr="un&utilde;">unum</expan>, ut &longs;i &longs;int <lb/>undecim uiri pones decem, &longs;i decem pones <expan abbr="nou&etilde;">nouem</expan>, & colliges natu<lb/>ralem &longs;eriem numerorum, utinfrà uides uno &longs;emper termino defi<lb/>ciente: & expriore ordine, ubi uidebis &longs;emper <expan abbr="etiã">etiam</expan> duplicari nume<lb/>ros: ut 3. 6. in de &longs;ub 6. 10. & 20 àlatere, & &longs;ub 20 35. & à latere 70 du<lb/>plum 35, & &longs;ub <lb/><arrow.to.target n="table23"/><lb/><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"/>70 126, & à late<lb/>re 252, & hoc pro<lb/>cognitione &qring;d <lb/>rectè &longs;is opera<lb/>tus. </s>

<s id="id003223">Et ideò pro ordine huius detrahes <expan abbr="un&utilde;">unum</expan>, ut &longs;i &longs;int <lb/>undecim uiri pones decem, &longs;i decem pones <expan abbr="nou&etilde;">nouem</expan>, & colliges natu<lb/>ralem seriem numerorum, ut infrà uides uno &longs;emper termino defi<lb/>ciente: & ex priore ordine, ubi uidebis &longs;emper <expan abbr="etiã">etiam</expan> duplicari nume<lb/>ros: ut 3. 6. in de &longs;ub 6. 10. & 20 àlatere, & &longs;ub 20 35. & à latere 70 du<lb/>plum 35, & &longs;ub <lb/><arrow.to.target n="table23"/><lb/><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"/>70 126, & à late<lb/>re 252, & hoc pro <lb/>cognitione &qring;d <lb/>rectè &longs;is opera<lb/>tus. </s>

<s id="id003224">Secundò a<lb/>nimaduertes <expan abbr="&longs;equ&etilde;tes">&longs;e<lb/>quentes</expan> ordines <lb/>fieri ex recta li<lb/>nea priorum, ue <lb/>lut &longs;extus ordo e&longs;t 7. 28. 84. 210. 462. ita incipiendo in primo ordi<lb/>ne à 7, & tendendo ad dextram, inuenies illos eo&longs;dem numeros ad <lb/>unguem, & ita in &longs;eptimo ordine 8. 36. 120. 330. à &longs;ini&longs;tra inuento 8 <lb/>in primo ordine, & procedendo ad dextram, inuenies 36. 120. & <lb/>330. Tertium e&longs;t quod numeri ultimi à medio &longs;unt ijdem, ut 462 & <lb/>462. 330 & 330. 165 & 165. 55 & 55. 11 & 11. Et &longs;eor&longs;um, ut dixi, rema<lb/>net 1. Oportetigitur colligere numeros angulares, ut à latere ui<lb/>des, & fit 2047 numerus coniugationum, tot enim modis po&longs;&longs;unt <lb/>uariari. </s>

<s id="id003225">Et &longs;i e&longs;&longs;ent decem tantum, ut ab initio propo&longs;ui, primus or<lb/>do finitur ad 10, &longs;ecundus ad 45, tertius ad 120, quartus ad 210, quin <lb/>tus ad 252, &longs;extus redit ad 210, &longs;eptimus ad 120, octauus ad 45, no<lb/>nus ad 10, decimus ad 1. Etita colligeretur &longs;umma ex extremis nu<lb/>meris angularibus 1023. Et tot erunt coniugationes. </s>

<s id="id003226">Hic uides quia <lb/>numerus 10 e&longs;t par, et quod adempta monade, relinquitur 9, qui e&longs;t <lb/>impar quòd medius qui pertinet ad quintum ordinem e&longs;t maxi

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<s id="id003232">Hæc autem ratio &longs;atisfacit multum, & e&longs;t ne<lb/>ce&longs;&longs;aria in temperiebus corporis humani. </s>

<s id="id003233">Vt in <lb/>&longs;ecundo, De dentibus. </s>

<s id="id003234">Et etiam ut quælibet di<lb/>&longs;ciplina quàm breui&longs;simè tradi po&longs;sit, ut gratia <lb/>exempli, medicina tota in una pagina, dico me<lb/>dicina <expan abbr="nõ">non</expan> &longs;olum Græcorum, &longs;ed etiam Arabum <lb/>& Latinorum, & etiam longè plus: nam &longs;i tradatur uigintiquatuor <lb/>regulis fimplicibus, & ex illis fiant coniugationes 16777215, mani <lb/>fe&longs;tum e&longs;t quod erunt regulæ omnes hæ multo plures, quàm con<lb/>tineantur in omnibus libris Græcorum, & Arabum, & Latino<lb/>rum, qui extant. </s>

<s id="id003234">Et etiam ut quælibet di<lb/>&longs;ciplina quàm breui&longs;simè tradi po&longs;sit, ut gratia <lb/>exempli, medicina tota in una pagina, dico me<lb/>dicina <expan abbr="nõ">non</expan> &longs;olum Græcorum, &longs;ed etiam Arabum <lb/>& Latinorum, & etiam longè plus: nam &longs;i tradatur uiginti quatuor <lb/>regulis simplicibus, & ex illis fiant coniugationes 16777215, mani <lb/>fe&longs;tum e&longs;t quod erunt regulæ omnes hæ multo plures, quàm con<lb/>tineantur in omnibus libris Græcorum, & Arabum, & Latino<lb/>rum, qui extant. </s>

<s id="id003235">Et tamen per&longs;picuum e&longs;t, uigintiquatuor regulas <lb/>una pagina commodi&longs;simè contineri. </s>

<s id="id003236">Et hoc aliâs docui, quan<lb/>quàm credam me erra&longs;&longs;e in &longs;upputatione, nam locum inuenire non <lb/>potui. </s>

<s id="id003237">Vnum e&longs;t id certum, quòd hæc ratio quàm nunc explicabo, <lb/>e&longs;t uera & demon&longs;tratiua, & facillima.</s>

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<s id="id003239">Et ideò inueni hanc, quæ (ut <lb/>dixi) facillima e&longs;t: adde numero propo&longs;ito monadem, in de confla<lb/>ri inuenias numerum à monade in eodem ordine, & ab eo detra<lb/>cta monade habes numerum coniugationum. </s>

<s id="id003240">Exemplum, &longs;i &longs;int <lb/>10 adde 1 fit 11. Vndecimus ergo numerus in proportione dupla <lb/>e&longs;t 1024, detrahe 1 & relinquantur 1023 numerus coniugationum, <lb/>ut in priore &longs;upputatione. </s>

<s id="id003241">Item &longs;i &longs;int 11 numeri adde 1 fit 12, duo de<lb/>cimus ergo numerus in proportione dupla e&longs;t 2048, detrahe 1 re<lb/>lin quuntur 2047, coniugationes 11, ut prius in &longs;uprà &longs;cripto exem<lb/>plo. </s>

<s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uige&longs;imus quin<lb/>cus igitur numerus in ordine duplæ proportionis à monade e&longs;t <lb/>16777216, ergo detracta monade relin quitur numerus (ut dixi) re<lb/>gularum & coniugationum uigintiquatuor regularum, quæ ta<lb/>men non &longs;int contrariæ inuicem: nam tunc e&longs;&longs;ent pauciores. </s>

<s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uige&longs;imus quin<lb/>tus igitur numerus in ordine duplæ proportionis à monade e&longs;t <lb/>16777216, ergo detracta monade relinquitur numerus (ut dixi) re<lb/>gularum & coniugationum uiginti quatuor regularum, quæ ta<lb/>men non &longs;int contrariæ inuicem: nam tunc e&longs;&longs;ent pauciores. </s>

<s id="id003243">Et <lb/>quia in i&longs;tis numeris duplicandis po&longs;&longs;es facile incidere in errorem, <lb/>diuide ultimum per 16, & &longs;i nihil &longs;upere&longs;t, rectè proce&longs;sit opus: &longs;in

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<s id="id003414">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 id="id003415">Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire do drantem

<s id="id003415">Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire dodrantem <pb pagenum="200" xlink:href="015/01/219.jpg"/>

qualis relinquatur perfectionis, duc quadrantem in 10 fit 30, duc 12 <lb/>in 7 fit 84, detrahe 30 ex 84, relinquitur 54, divide 54 per 9, re&longs;idu<lb/>um 12 & 3, exit 6 perfectio re&longs;idui.</s>

<lb/>

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<s id="id003425">Sint duæ quanitates a b & k l, & ab&longs;ci&longs;&longs;æ duæ partes &longs;imiles ex <lb/>utragque b c & l m, & &longs;it propo&longs;ita aliqua proportio, quæ &longs;it h, & <lb/>&longs;umatur portio b d ip&longs;ius b c, &longs;ecundum proportionem h, & &longs;i<lb/>

<s id="id003425">Sint duæ quanitates a b & k l, & ab&longs;ci&longs;&longs;æ duæ partes &longs;imiles ex <lb/>utraque b c & l m, & &longs;it propo&longs;ita aliqua proportio, quæ &longs;it h, & <lb/>&longs;umatur portio b d ip&longs;ius b c, &longs;ecundum proportionem h, & &longs;i<lb/>

<pb pagenum="201" xlink:href="015/01/220.jpg"/>militer l n ip&longs;ius l m, iuxta pro<lb/><figure id="id.015.01.220.1.jpg" xlink:href="015/01/220/1.jpg"/><lb/>portionem h, &longs;umatur rur&longs;us <lb/><arrow.to.target n="marg626"/><lb/>de ip&longs;ius a b pars &longs;ecundum h, <lb/><arrow.to.target n="marg627"/><lb/>& n o ip&longs;ius k l, &longs;ecundum ean <lb/>dem proportionem. </s>

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<pb pagenum="204" xlink:href="015/01/223.jpg"/>portionem b e ad c k data e&longs;t, & c f ad e d, ut b c ad b a, erit ut a c ad <lb/>l e contineat a b ad b k, ut f e ad e l, c b ad b k, &longs;ed a b ad a d, ut d c ad <lb/>d h, igitur a b ad b d, ut d e ad c h. </s>

<s id="id003485">Sunt ergo duæ quantitates a b & <lb/>d c, quæ eandem habent compo&longs;itam proportionem ad g k & k b, <lb/>& h l & l e, quare per præcedentem proportionis h l ad l e, ad pro<lb/>portinem g k ad k b, ut h l detracto prouentu d e, diui&longs;i per propor<lb/>tionem ad d e ad proportionem g k, detracto prouentu a b, diui&longs;i <lb/>per eandem proportionem ad ip&longs;um a b. </s>

<s id="id003485">Sunt ergo duæ quantitates a b & <lb/>d c, quæ eandem habent compo&longs;itam proportionem ad g k & k b, <lb/>& h l & l e, quare per præcedentem proportionis h l ad l e, ad pro<lb/>portionem g k ad k b, ut h l detracto prouentu d e, diui&longs;i per propor<lb/>tionem ad d e ad proportionem g k, detracto prouentu a b, diui&longs;i <lb/>per eandem proportionem ad ip&longs;um a b. </s>

<s id="id003486">Si igitur nota e&longs;t l e & h l, <lb/>erit nota proportio re&longs;idui h l detracto prouentu d e diui&longs;i per pro<lb/>portionem, quare nota detractio g k detracto prouentu a b diui&longs;i <lb/>per eandem proportionem ad a b. </s>

<s id="id003487">E&longs;t autem a b nota, & propor<lb/>tio nota, & ideo prouentus, & cum &longs;it proportio nota, erit ergo <lb/>re&longs;iduum notum, cui addito prouentu fit tota g k nota, quod fuit <lb/>demon&longs;trandum.</s>

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<s id="id003546">Vt con&longs;tet iam in natura animalium <lb/>& in genere caloris habentis magnum motum, & &longs;ub&longs;tantiam te<lb/>nuem hoc inueniri po&longs;&longs;e, ut excludamus plantas de <expan abbr="quar&utilde;">quarum</expan> uita lon<lb/>gi&longs;sima &longs;atis con&longs;tat, &longs;ed quia caret motu euidenti calor in illis, & <lb/>&longs;ub&longs;tantia e&longs;t cra&longs;&longs;a animalium comparatione, non laboro. </s>

<s id="id003547">At de <lb/>elephanto omnes confitentur quòd &longs;it omnium ingenio &longs;i&longs;simum, <lb/>adeò ut multi homines illo indu&longs;tria & cognitione inferiores e&longs;&longs;e <lb/>uideantur. </s>

<s id="id003548">Neque etiam ueri&longs;imile e&longs;t quod natura hominem fecerit <lb/>hac in parte illo inferiorem, præ&longs;ertim cum de nullo alio animali <lb/>apud Ari&longs;totelem dubium &longs;it, & ubi modo aliquod dubium e&longs;&longs;et <lb/>propter querelam Theophra&longs;ti, & illud quod &longs;olet prædicari de <lb/>ceruis, tanto magis ueri&longs;imile e&longs;t indignum fui&longs;&longs;e hominem conce<lb/>dere tot animalibus in diuturnitate uitæ. </s>

<s id="id003549">Quam obrem cum hæc <lb/>tractatio ad libros de tuenda Sanitate &longs;pectaret, homines ad eos re<lb/>leego, nam ob id illos con&longs;crip&longs;i quòd uiderem Galenum nec hoc

<s id="id003549">Quam ob rem cum hæc <lb/>tractatio ad libros de tuenda Sanitate &longs;pectaret, homines ad eos re<lb/>lego, nam ob id illos con&longs;crip&longs;i quòd uiderem Galenum nec hoc <pb pagenum="210" xlink:href="015/01/229.jpg"/>uidi&longs;&longs;e nec multa alia, &longs;ed eorum loco longas & inutiles di&longs;putatio<lb/>nes inter&longs;erui&longs;&longs;e. </s>

<pb pagenum="210" xlink:href="015/01/229.jpg"/>uidi&longs;&longs;e nec multa alia, &longs;ed eorum loco longas & inutiles di&longs;putatio<lb/>nes inter&longs;erui&longs;&longs;e. </s>

<s id="id003550">Verùm etiam, quoniam eam tractationem diuul<lb/>&longs;it, ut alia cogamus quærere in libris de Alimentis, alia, de cibis bo<lb/>ni & mali &longs;ucci: tum uerò & tractatio ip&longs;a eduliorum e&longs;t imperfe<lb/>cta, & multa etiam deficiunt circa genera: in quo e&longs;t ex cu&longs;andus ob <lb/>uarietatem regionis & ætatis. </s>

<s id="id003551">Dee&longs;t præterea maxima pars, quæ <lb/>nec ibi nec alibi habetur, &longs;cilieet, de ciborum præparatione. </s>

<s id="id003551">Dee&longs;t præterea maxima pars, quæ <lb/>nec ibi nec alibi habetur, &longs;cilicet, de ciborum præparatione. </s>

<s id="id003552">Quod <lb/>etiam hæc latuerint tot præclaros uiros, quid mirum? </s>

<s id="id003553">cum Hippo<lb/>crates uixerit &longs;eculo illo agre&longs;ti, in quo non e&longs;t mirandum, quod ali <lb/>quid, pauca quædam & ab&longs;tru&longs;a omi&longs;erit, &longs;ed quod tam multa tam <lb/>bene inuenerit, ut fuerit, &longs;icut de Pindaro dicitur, imò longè uerius <lb/>quam de Pindaro inimitabilis. </s>

<s id="id003554">De Galeno quid mirum, qui non <lb/>ni&longs;i ueterum &longs;cripta collegit, atque utinam <expan abbr="&longs;alt&etilde;">&longs;altem</expan> bene. </s>

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<s id="id003582"><expan abbr="Declarat&utilde;">Declaratum</expan> e&longs;t. </s>

<s id="id003584">&longs;uperius quod unus motus in <lb/><expan abbr="eod&etilde;">eodem</expan> mobili <expan abbr="ali&utilde;">alium</expan> impedit & retardat. </s>

<s id="id003585">Cum ergo h k &longs;it in &longs;pacio a b <lb/>l m & aqua <expan abbr="rapia&ttilde;">rapiatur</expan> motu, dico ad d e mouebit ad d e, & motu dico <lb/>qui uidetur circularis, nam mouetur motu eius à quo <expan abbr="&longs;u&longs;tine&ttilde;">&longs;u&longs;tinetur</expan>. </s>

<s id="id003585">Cum ergo h k &longs;it in &longs;patio a b <lb/>l m & aqua <expan abbr="rapia&ttilde;">rapiatur</expan> motu, dico ad d e mouebit ad d e, & motu dico <lb/>qui uidetur circularis, nam mouetur motu eius à quo <expan abbr="&longs;u&longs;tine&ttilde;">&longs;u&longs;tinetur</expan>. </s>

<s id="id003586">Mo<lb/>uetur etiam ad d e, quoniam pars illa e&longs;t humilior, nam &longs;emper de<lb/>&longs;cendit, omne <expan abbr="a&utilde;t">aunt</expan> quod mouetur partim e&longs;t in termino, à quo, par<lb/>tim ad quem, ideo partim iam aqua illa cum de&longs;cendat humilior e&longs;t <lb/>locus, igitur nauis ad <expan abbr="ill&utilde;">illum</expan> locum feretur. </s>

<s id="id003586">Mo<lb/>uetur etiam ad d e, quoniam pars illa e&longs;t humilior, nam &longs;emper de<lb/>&longs;cendit, omne <expan abbr="a&utilde;t">aut</expan> quod mouetur partim e&longs;t in termino, à quo, par<lb/>tim ad quem, ideo partim iam aqua illa cum de&longs;cendat humilior e&longs;t <lb/>locus, igitur nauis ad <expan abbr="ill&utilde;">illum</expan> locum feretur. </s>

<s id="id003587">Tertio, quia latus k impelli <lb/>tur, in maiore circulo, ideo maiore impetu <lb/><figure id="id.015.01.231.1.jpg" xlink:href="015/01/231/1.jpg"/><lb/>quàm h, quare <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan> & circulo mouebi<lb/>tur, <expan abbr="nã">nam</expan> &longs;i h quie&longs;ceret <expan abbr="palã">palam</expan> e&longs;t, &qring;d nauis circu <lb/>lariter <expan abbr="mouere&ttilde;">moueretur</expan>, &longs;ed h fungitur uice <expan abbr="quie&longs;c&etilde;tis">quie&longs;cen<lb/>tis</expan>, quia tardius <expan abbr="moue&ttilde;">mouetur</expan> <expan abbr="quã">quam</expan> k, <expan abbr="igi&ttilde;">igitur</expan> k moue<lb/>bitur ad d e & motu circulari aut participe <lb/>eius. </s>

<s id="id003588">Quarta cau&longs;a e&longs;t, quoniam h cupit <expan abbr="de&longs;c&etilde;dere">de<lb/>&longs;cendere</expan>, ut graue. </s>

<s id="id003589">ergo ferri, ubi minus impe <lb/>diatur à motu <expan abbr="uiol&etilde;to">uiolento</expan>, at minus <expan abbr="impedi&ttilde;">impeditur</expan> in <lb/>circulo, de qua a b, qa a b <expan abbr="c&utilde;">cum</expan> maioris &longs;it ambitus a qua in co ulterius <lb/><expan abbr="fer&ttilde;">fertur</expan> <expan abbr="quã">quam</expan> in d e, ob hæc <expan abbr="o&itilde;a">oina</expan> & in mari & fluminibus ac lacubus <expan abbr="c&utilde;">cum</expan> na <lb/>ues fuerint in ambitu uorticis <expan abbr="iã">iam</expan> <expan abbr="rapiun&ttilde;">rapiuntur</expan> ad <expan abbr="ill&utilde;">illum</expan>, & circulari motu: <lb/>isque motus e&longs;t <expan abbr="indici&utilde;">indicium</expan> &longs;ubmer&longs;ionis, <expan abbr="quoniã">quoniam</expan> indicat <expan abbr="aquã">aquam</expan>, ibi propè <lb/><expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> rectà uer&longs;us <expan abbr="c&etilde;tr&utilde;">centrum</expan>, & ob id <expan abbr="prud&etilde;tes">prudentes</expan> naut&etail; magna ui uen <lb/>toru & <expan abbr="remor&utilde;">remorum</expan> &longs;&etail;pe <expan abbr="&longs;eruãt">&longs;eruant</expan> &longs;e, pr&etail;o <expan abbr="ccupãtes">ccupantes</expan> <expan abbr="mot&utilde;">motum</expan> <expan abbr="elic&utilde;">elicum</expan> recto motu. <lb/></s>

<s id="id003589">ergo ferri, ubi minus impe<lb/>diatur à motu <expan abbr="uiol&etilde;to">uiolento</expan>, at minus <expan abbr="impedi&ttilde;">impeditur</expan> in <lb/>circulo, de qua a b, qua a b <expan abbr="c&utilde;">cum</expan> maioris &longs;it ambitus a qua in co ulterius <lb/><expan abbr="fer&ttilde;">fertur</expan> <expan abbr="quã">quam</expan> in d e, ob hæc <expan abbr="o&itilde;a">oina</expan> & in mari & fluminibus ac lacubus <expan abbr="c&utilde;">cum</expan> na<lb/>ues fuerint in ambitu uorticis <expan abbr="iã">iam</expan> <expan abbr="rapiun&ttilde;">rapiuntur</expan> ad <expan abbr="ill&utilde;">illum</expan>, & circulari motu: <lb/>isque motus e&longs;t <expan abbr="indici&utilde;">indicium</expan> &longs;ubmer&longs;ionis, <expan abbr="quoniã">quoniam</expan> indicat <expan abbr="aquã">aquam</expan>, ibi propè <lb/><expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> rectà uer&longs;us <expan abbr="c&etilde;tr&utilde;">centrum</expan>, & ob id <expan abbr="prud&etilde;tes">prudentes</expan> naut&etail; magna ui uen<lb/>toru & <expan abbr="remor&utilde;">remorum</expan> &longs;&etail;pe <expan abbr="&longs;eruãt">&longs;eruant</expan> &longs;e, pr&etail;o<expan abbr="ccupãtes">ccupantes</expan> <expan abbr="mot&utilde;">motum</expan> <expan abbr="elic&utilde;">elicum</expan> recto motu. <lb/></s>

<s id="id003590">Cur <expan abbr="a&utilde;t">aunt</expan> aqua <expan abbr="&qtilde;">quae</expan> e&longs;t in a, non potius <expan abbr="fera&ttilde;">feratur</expan> per obliquam lineam ad d <lb/>uel g, <08> ad e uel c inde ex illis ad d uel g, præ&longs;ertim <expan abbr="c&utilde;">cum</expan> ad&longs;it breuior

<s id="id003590">Cur <expan abbr="a&utilde;t">aut</expan> aqua <expan abbr="&qtilde;">quae</expan> e&longs;t in a, non potius <expan abbr="fera&ttilde;">feratur</expan> per obliquam lineam ad d <lb/>uel g, <08> ad e uel c inde ex illis ad d uel g, præ&longs;ertim <expan abbr="c&utilde;">cum</expan> ad&longs;it breuior

<pb pagenum="213" xlink:href="015/01/232.jpg"/>a e & e d et a g breuior a e et c (ut docet Euclides) cau&longs;a e&longs;t quia aqua <lb/>quæ de&longs;cendit per e d & c g maiore impetu de&longs;cendit quàm per ad <lb/>uel a g ut demon&longs;tratum e&longs;t, ergo non poterit quæ e&longs;t in e d uel e g <lb/>loco dimoueri, nec cedere aquæ per obliquam lineam de&longs;cendenti.</s>

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Line 10663

<s id="id003598">Sed cur rur&longs;us contractiora dum &longs;unt crura, homo facilius exur<lb/>git? </s>

<s id="id003599">Proponantur c f contracta ad perpendiculum, & in clinetur b a <lb/>in o ut fiant b o & f e equidi&longs;tantes, ita enim commodius &longs;urgimus: <lb/>nec aliter qui &longs;unt imbecilliores: quia ergo b e&longs;t in directo f, ideo <lb/>mu&longs;culi femoris inferiores ob crus, & &longs;uperiores ob pectus &longs;unt <lb/>magis ten&longs;i & anteriores cruris itidem, ideo maiore ui trahunt par <lb/>ticulam. </s>

<s id="id003599">Proponantur c f contracta ad perpendiculum, & inclinetur b a <lb/>in o ut fiant b o & f e aequidi&longs;tantes, ita enim commodius &longs;urgimus: <lb/>nec aliter qui &longs;unt imbecilliores: quia ergo b e&longs;t in directo f, ideo <lb/>mu&longs;culi femoris inferiores ob crus, & &longs;uperiores ob pectus &longs;unt <lb/>magis ten&longs;i & anteriores cruris itidem, ideo maiore ui trahunt par<lb/>ticulam. </s>

<s id="id003600">Vnde manente fixo f & capite etiam & pectore grauitate <lb/>&longs;ua adiuuantibus, facilius homo exurgit quam ad latos angulos <lb/>cum contractio, ut dixi, mu&longs;culorum et inclinatio partium &longs;uperio<lb/>rum fiat maior.</s>

<s id="id003601">Rur&longs;us pro prima parte problematis, dico quòd quanto altior <lb/>e&longs;t b f tanto facilius exurgit, nam &longs;upponatur angu<lb/><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"/><lb/>lus reflixionis a h e æqualis a h c, & b c k æqualis h k f, <lb/>igitur cum b f &longs;it breuior b f, erit h k breuior b c & f k, <lb/>f c. quare b c femur, & f c crus erunt uiolentius exten<lb/>&longs;a quàm in &longs;itu h k, k f ergo, mu&longs;culi fa cilius erigent <lb/>&longs;edentem altiore loco quàm humiliore, quod erat de<lb/>mon&longs;trandum.</s>

<s id="id003601">Rur&longs;us pro prima parte problematis, dico quòd quanto altior <lb/>e&longs;t b f tanto facilius exurgit, nam &longs;upponatur angu<lb/><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"/><lb/>lus reflexionis a h e æqualis a h c, & b c k æqualis h k f, <lb/>igitur cum b f &longs;it breuior b f, erit h k breuior b c & f k, <lb/>f c. quare b c femur, & f c crus erunt uiolentius exten<lb/>&longs;a quàm in &longs;itu h k, k f ergo, mu&longs;culi facilius erigent <lb/>&longs;edentem altiore loco quàm humiliore, quod erat de<lb/>mon&longs;trandum.</s>

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Line 10803

<s id="id003652">Sint pondera b minus, c maius, & ducatur aggre<lb/><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"/><lb/>gatum ex a uiribus & b minore pondere in e, & ei <lb/>addatur quadratum dimidij a, dico quod radix &longs;eu<lb/>latus huius detracto dimidio a e&longs;t pondus auxiliare <lb/>æquale, &longs;it productum a b in e &longs;uperficies & quadra<lb/>tum dimidij a &longs;it e, ita quod tota d e &longs;it &longs;uperficies <lb/>quadrata, cuius latus &longs;it f g: f h autem dimidium a di<lb/>co h g e&longs;&longs;e pondus auxiliare æquale. </s>

<s id="id003652">Sint pondera b minus, c maius, & ducatur aggre<lb/><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"/><lb/>gatum ex a uiribus & b minore pondere in e, & ei <lb/>addatur quadratum dimidij a, dico quod radix &longs;eu <lb/>latus huius detracto dimidio a e&longs;t pondus auxiliare <lb/>æquale, &longs;it productum a b in e &longs;uperficies & quadra<lb/>tum dimidij a &longs;it e, ita quod tota d e &longs;it &longs;uperficies <lb/>quadrata, cuius latus &longs;it f g: f h autem dimidium a di<lb/>co h g e&longs;&longs;e pondus auxiliare æquale. </s>

<s id="id003653">Quia enim f g

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Line 10915

<s id="id003693">Omitto modo de motu aucto per <lb/>uim humanam, de quo uidetur quærere Ari&longs;toteles, quili<lb/>bet enim aër addit &longs;uper motum iam acqui&longs;itum & fit hoc <lb/>argumentum centies ac millies maius, quoniam m e&longs;t qui <lb/>diuidit, pondus autem non ponetrat. </s>

<s id="id003694">Sicut ergo cuneus <lb/>magis diuidit lignum quam claua, ita quod mouetur &longs;ine <lb/>proportione (ut ita dicam) non &longs;olum ob <expan abbr="impet&utilde;">impetum</expan> nece&longs;&longs;e <lb/>e&longs;t ut uehementer diuidat lignum aut lapidem &longs;ubiectum, <lb/>& non in proportione di&longs;tanti&etail;. </s>

<s id="id003695">Sicut &longs;i pondus in forma <lb/>&longs;ecuris, & ip&longs;a &longs;ecuris diuidit longe magis ligna quam cla<lb/>uis maioris ponderis & maiore ui de&longs;cendens: ita pondus motum <lb/>quam immotum. </s>

<s id="id003696">Hoc adeò per&longs;picuam habet cau&longs;&longs;am, ut quanto <lb/>plura uerba addererentur, eo redderetur res difficilior. </s>

<s id="id003696">Hoc adeò per&longs;picuam habet cau&longs;&longs;am, ut quanto <lb/>plura uerba adderentur, eo redderetur res difficilior. </s>

<s id="id003697">Habet ergo <lb/>propriam &longs;olum grauitatem & motum occultum. </s>

<s id="id003698">C&etail;terum e&longs;t ter<lb/>tium, genus <expan abbr="medi&utilde;">medium</expan>, cum idem pondus appen&longs;um e&longs;t, ue<lb/><figure id="id.015.01.237.2.jpg" xlink:href="015/01/237/2.jpg"/><lb/>lut f quod dico e&longs;&longs;e maius & minus occultum quam &longs;i ia<lb/>ceret in plano, quoniam &longs;icut tuber & cauitas in qua iacet <lb/>&longs;imul tempore &longs;unt, natura tamen tuber e&longs;t prius cauitate, <lb/>ita pondus appen&longs;um prius e&longs;t, contrà nixum uinculi na<lb/>tura & quodammodo tempore, &longs;emper enim grauat, & illud &longs;em<lb/>per re&longs;i&longs;tit &longs;upra illius grauitatem: Sed pondus quod e&longs;t in plano <lb/>occultam omnino habet actionem bifariamque di&longs;ting uitur a pon<lb/>dere &longs;u&longs;pen&longs;o: Primum quòd pondus quod quie&longs;cit & contra in<lb/>tendi principium &longs;imul non &longs;olum &longs;unt tempore &longs;ed etiam natu<lb/>ra. </s>

<s id="id003699">Sed in appen&longs;o, ut dixi, pondus prius grauat quam uincu

Line 11046

Line 10978

<s id="id003723"><arrow.to.target n="marg698"/><lb/>&longs;it dupla a c, & <foreign lang="greek">a b a g</foreign> dupla, & diuidantur a b a c & <foreign lang="greek">a b a g</foreign> in quotuis <lb/>partes &etail;quales inuicem, nam &longs;upponitur a b &etail;qualis <foreign lang="greek">a b</foreign> & a c æqua<lb/>lis <foreign lang="greek">a g</foreign>, & ducantur rectæ lineæ decu&longs;&longs;atim & ad rectos angulos, & <lb/><expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> id &longs;tatuantur re&longs;tes, quod decu&longs;&longs;a<lb/><figure id="id.015.01.239.1.jpg" xlink:href="015/01/239/1.jpg"/><lb/>tim po&longs;itæ utiliores <expan abbr="er&utilde;t">erunt</expan>, omitto quod de<lb/>centius ob &longs;patiorum minorem differenti<lb/>am. </s>

<s id="id003724">Adducam &longs;olùm tres Philo&longs;ophi ratio<lb/>nes: prima, quoniam ligna non adeò facilè <lb/>finduntur nec incuruantur tran&longs;uer&longs;im tra<lb/>cta, ut recta & &longs;ecundum longitudinem, Et <lb/>ideò longè plus durabit <foreign lang="greek">a b g d</foreign> <expan abbr="quã">quam</expan> a b c d, <lb/>& cum &longs;pondis rectoribus, & ideò etiam <lb/>cum re&longs;tibus magis intentis: & erit firmior <lb/>& pulchrior. </s>

<s id="id003725">Secunda ratio e&longs;t, quod cum <lb/>re&longs;tes in &longs;ecunda con&longs;titutione æquales inuicem &longs;int, in prima quæ <lb/>&longs;ecundum latitudinem dupl&etail;, qu&etail; longiores erunt magis laxabun<lb/>tur tran&longs;uer&longs;alibus, & ita turpiores & incommodæ breui redden<lb/>tur, & in &longs;ecunda con&longs;titutione &etail;qualiter &longs;u&longs;tinebunt pondus & re<lb/>uolutionem cubantis, tum ob æqualitatem longitudinis inter &longs;e, <lb/>tum ob &longs;itum &longs;imilem inter &longs;e, tum ad humanum decubitum <expan abbr="di&longs;simil&etilde;">di&longs;si<lb/>milem</expan>, nam (ut o&longs;ten&longs;um e&longs;t) in præcedenti magis grauat pondus in <lb/>extremis quam in medio, & magis laxantur ob id quæ &longs;unt &longs;ecun<lb/>dum eundem fitum. </s>

<s id="id003726">Et hanc cau&longs;&longs;am expo&longs;itores non intellexe<lb/>runt multi, multo minus tertiam, in qua faciunt demon&longs;trationem <lb/>Geometricam & computantrem numeris. </s>

<s id="id003726">Et hanc cau&longs;&longs;am expo&longs;itores non intellexe<lb/>runt multi, multo minus tertiam, in qua faciunt demon&longs;trationem <lb/>Geometricam & computantem numeris. </s>

<s id="id003727">Deinde non animaduer <lb/>tunt quod in &longs;ecunda figura a&longs;&longs;umunt quinque lineas, cum in prima <lb/>tantum a&longs;&longs;ump&longs;i&longs;&longs;ent quatuor. </s>

<s id="id003728">Peius omnibus e&longs;t quod demon<lb/>&longs;tratio hæc cum de tran&longs;uer&longs;is ad magis tran&longs;uer&longs;as lineas &longs;it non <lb/>e&longs;t ad propo&longs;itum Ari&longs;totelis, qui in duabus primis rationibus <lb/>tran&longs;uer&longs;as comparauit his, quæ à latere ad latus & à capite ad ca<lb/>put deducuntur, ita ubi trifariam decepti &longs;unt, ibi maximè glori<lb/>antur. </s>

<s id="id003729">Mi&longs;erum nunc philo&longs;ophandi genus: uoluntque &longs;upercilium <lb/>e&longs;&longs;e loco doctrinæ. </s>

<s id="id003730">Sint igitur lineæ ductæ ut uides, dico omnes <lb/>pariter acceptas in prima figura, e&longs;&longs;e longiores omnibus pariter ac<lb/><arrow.to.target n="marg699"/><lb/>ceptis in &longs;ecunda figura, quod intendit <expan abbr="demõ">demom</expan> &longs;trare Ari&longs;toteles. </s>

<s id="id003731">O<lb/>&longs;ten&longs;o ergo de duabus, idem &longs;uppo&longs;ito numero equali de omnibus

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Line 11018

<s id="id003741">Si duo circuli &longs;uper eodem centro eodem motu transferuntur, <lb/>æquale &longs;patiu m &longs;uperant.</s>

<s id="id003742">Sint duo circuli a b, c d &longs;uper eodem centro e qui transferantur <lb/><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"/><lb/><arrow.to.target n="marg705"/><lb/>&longs;uper axe per <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c g dum re&longs;oluitur c d, <lb/>tum ergo a erit in f, quia c d contingit pla<lb/>num c g, igitur e c e&longs;t ad <expan abbr="perp&etilde;diculum">perpendiculum</expan> c g, <lb/><arrow.to.target n="marg706"/><lb/>ergo punctum a e&longs;t in f & a f æqualis c g, <lb/><arrow.to.target n="marg707"/><lb/>igitur a b circulus &longs;olum reuolutus e&longs;t &longs;e<lb/>mel, & tantum perambulauit &longs;pacij quan<lb/>tum e d & æquali uelo citate, cùm tamen &longs;eor&longs;um &longs;it proportio &longs;pa<lb/>tij ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> ut circuli ad circulum. </s>

<s id="id003742">Sint duo circuli a b, c d &longs;uper eodem centro e qui transferantur <lb/><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"/><lb/><arrow.to.target n="marg705"/><lb/>&longs;uper axe per <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c g dum re&longs;oluitur c d, <lb/>tum ergo a erit in f, quia c d contingit pla<lb/>num c g, igitur e c e&longs;t ad <expan abbr="perp&etilde;diculum">perpendiculum</expan> c g, <lb/><arrow.to.target n="marg706"/><lb/>ergo punctum a e&longs;t in f & a f æqualis c g, <lb/><arrow.to.target n="marg707"/><lb/>igitur a b circulus &longs;olum reuolutus e&longs;t &longs;e<lb/>mel, & tantum perambulauit &longs;patij quan<lb/>tum e d & æquali uelo citate, cùm tamen &longs;eor&longs;um &longs;it proportio &longs;pa<lb/>tij ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> ut circuli ad circulum. </s>

<s id="id003743">Hæc e&longs;t &longs;ubtili&longs;sima <expan abbr="quæ&longs;tion&utilde;">quæ&longs;tionum</expan> <lb/><arrow.to.target n="marg708"/><lb/><expan abbr="propo&longs;itar&utilde;">propo&longs;itarum</expan> ab Ari&longs;to tele in mechanicis, quam &longs;ic quidam &longs;oluunt. <lb/></s>

<s id="id003744">Supponunt duo: <expan abbr="prim&utilde;">primum</expan> &longs;i quid ab aliquo mouetur nihil conferens <lb/><figure id="id.015.01.240.2.jpg" xlink:href="015/01/240/2.jpg"/><lb/>ad illum motum, <lb/>ex&longs;e ip&longs;o per tan <lb/>tum mouebitur <lb/><expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quan<lb/>tum ab illo mo<lb/>tore mouebitur: <lb/>Secundum, <expan abbr="ead&etilde;">eadem</expan> <lb/>potentia in <expan abbr="eod&etilde;">eodem</expan> <lb/>tempore diuer&longs;o <lb/>modo duo mobi <lb/>lia mouebit &etail;qua <lb/>lia, cum <expan abbr="un&utilde;">unum</expan> mo<lb/>tui a&longs;&longs;entietur aliud <expan abbr="nõ">non</expan>. </s>

<s id="id003745">quod &longs;i hæc mobilia &longs;eiuncta fui&longs;&longs;ent, quod <lb/>aptitudinem haberet <expan abbr="&longs;eiunct&utilde;">&longs;eiunctum</expan> uelo cius moueretur, quàm dum con <lb/>iunctum e&longs;t. </s>

<s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir<lb/>culo, nec conferat quic<08> ad motum, ideo tantum tran&longs;ibit &longs;pacium

<s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir<lb/>culo, nec conferat quic<08> ad motum, ideo tantum tran&longs;ibit &longs;patium <pb pagenum="222" xlink:href="015/01/241.jpg"/>c d quantum a b per primum &longs;uppo&longs;itum. </s>

<s id="id003747">Sed quoniam proposi<lb/>to circulo alio non circa idem centrum, utpote k l reuoluetur & <lb/>perueniet ad h ex demon&longs;tratis. </s>

<pb pagenum="222" xlink:href="015/01/241.jpg"/>c d quantum a b per primum &longs;uppo&longs;itum. </s>

<s id="id003747">Sed quoniam propofi<lb/>to circulo alio non circa idem centrum, utpote k l reuoluetur & <lb/>perueniet ad h ex demon&longs;tratis. </s>

<s id="id003748"><expan abbr="Re&longs;ponde&ttilde;">Re&longs;pondetur</expan> ad hoc, quod idem e&longs;t, <lb/>quia unus circulus tantum per &longs;e mouetur circa centrum, reliqui <lb/>omnes non per&longs;e circa centrum, &longs;ed ab alio circulo primo mouen<lb/>tur, ideò nihil refert &longs;eu &longs;int circa idem centrum &longs;eu circa aliud, hoc <lb/>enim fortuitum e&longs;t. </s>

<s id="id003749">Ideo ad argumentum re&longs;pondent cauillo&longs;am <lb/>e&longs;&longs;e <expan abbr="hãc">hanc</expan> di&longs;putationem, cum &longs;upponatidem ambobus circulis per <lb/>&longs;e centrum e&longs;&longs;e. </s>

<s id="id003750">Sed non e&longs;t per&longs;e, uerùm per <expan abbr="accid&etilde;s">accidens</expan>. </s>

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<s id="id003808">Sit a b c æquilaterus circulo in&longs;criptus, </s>

<s id="id003809"><arrow.to.target n="marg716"/><lb/>cuius centrum d, ducaturque ad e f rectà per <lb/>centrum, & ducantur d b & d c, eritque ex hoc <lb/><arrow.to.target n="marg717"/><lb/>triangulus a b d &etail;quilaterus triangulo a c d, <lb/><arrow.to.target n="marg718"/><lb/>quare angulus b a d æqualis c a d, igitur ar<lb/>cus b e æqualis c e, igitur arcus b e e&longs;t &longs;exta <lb/><arrow.to.target n="marg719"/><lb/>pars circuli, quare b e recta latus exagoni, <lb/>quare b e erit æqualis d e, igitur cum anguli <lb/><arrow.to.target n="marg720"/><lb/>a d f &longs;int utrin que recti, crit d f æqualis f e, itaque<lb/><arrow.to.target n="marg721"/><lb/>f d, tertia pars fa & fb dimidium a b quia b c. </s>

<s id="id003809"><arrow.to.target n="marg716"/><lb/>cuius centrum d, ducaturque ad e f rectà per <lb/>centrum, & ducantur d b & d c, eritque ex hoc <lb/><arrow.to.target n="marg717"/><lb/>triangulus a b d &etail;quilaterus triangulo a c d, <lb/><arrow.to.target n="marg718"/><lb/>quare angulus b a d æqualis c a d, igitur ar<lb/>cus b e æqualis c e, igitur arcus b e e&longs;t &longs;exta <lb/><arrow.to.target n="marg719"/><lb/>pars circuli, quare b e recta latus exagoni, <lb/>quare b e erit æqualis d e, igitur cum anguli <lb/><arrow.to.target n="marg720"/><lb/>a d f &longs;int utrin que recti, erit d f æqualis f e, itaque <lb/><arrow.to.target n="marg721"/><lb/>f d, tertia pars fa & fb dimidium a b quia b c. </s>

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<s id="id003937"><arrow.to.target n="marg766"/><lb/>h f k, &longs;i non faciant trian gulum æquicruri<lb/>um b c d &longs;uper b c, & cuius ambo latera pa <lb/>riter accepta &longs;int æqualia a b & a c. </s>

<s id="id003938">Et du<lb/>cam contingentem & habebo trapezium <lb/><arrow.to.target n="marg767"/><lb/>h b, c k. </s>

<s id="id003939">Quare &longs;i peripheria circuli b c e&longs;t <lb/><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"/><lb/>minor d b & d c pariter acceptis, habeo <expan abbr="intent&utilde;">intentum</expan>, &longs;i non toties <expan abbr="diuidã">diuidam</expan> <lb/>peripheriam per æqualia ut fiat figura polygonia &longs;uper b c æquila<lb/>tera & æquiangula, cuius differentia a peripheria &longs;it minor differen <lb/>tia d b & d c à trapezio b h, k c, id e&longs;t, tribus eius lateribus, nam cum <lb/>d h & d k &longs;int maiores h k, con&longs;tat quod d b & d e &longs;unt maiores h b, <lb/>& k c & h k igitur &longs;it differentia illa l, & <expan abbr="differ&etilde;tia">differentia</expan> peripheri&etail; à lineis <lb/>polyg oniæ minorl: igitur cum peripheria &longs;it æqualis aut maior <lb/>d b & d c, & differentia a lateribus polygoniæ minor quàm d b & <lb/>d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly<lb/><arrow.to.target n="marg768"/><lb/>goniæ quàm d b & d c ad tria latera trapezij, quare minor propor<lb/><arrow.to.target n="marg769"/><lb/>tio peripheriæ ad d b & d c quàm laterum polygoniæ ad tria latera <lb/><arrow.to.target n="marg770"/><lb/>trapezij, &longs;ed latera polygoniæ &longs;unt minora tribus laterib. </s>

<s id="id003939">Quare &longs;i peripheria circuli b c e&longs;t <lb/><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"/><lb/>minor d b & d c pariter acceptis, habeo <expan abbr="intent&utilde;">intentum</expan>, &longs;i non toties <expan abbr="diuidã">diuidam</expan> <lb/>peripheriam per æqualia ut fiat figura polygonia &longs;uper b c æquila<lb/>tera & æquiangula, cuius differentia a peripheria &longs;it minor differen<lb/>tia d b & d c à trapezio b h, k c, id e&longs;t, tribus eius lateribus, nam cum <lb/>d h & d k &longs;int maiores h k, con&longs;tat quod d b & d e &longs;unt maiores h b, <lb/>& k c & h k igitur &longs;it differentia illa l, & <expan abbr="differ&etilde;tia">differentia</expan> peripheri&etail; à lineis <lb/>polygoniæ minori: igitur cum peripheria &longs;it æqualis aut maior <lb/>d b & d c, & differentia a lateribus polygoniæ minor quàm d b & <lb/>d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly<lb/><arrow.to.target n="marg768"/><lb/>goniæ quàm d b & d c ad tria latera trapezij, quare minor propor<lb/><arrow.to.target n="marg769"/><lb/>tio peripheriæ ad d b & d c quàm laterum polygoniæ ad tria latera <lb/><arrow.to.target n="marg770"/><lb/>trapezij, &longs;ed latera polygoniæ &longs;unt minora tribus lateribus. </s>

<s id="id003940">trapezij, <lb/><arrow.to.target n="marg771"/><lb/>igitur peripheria b c e&longs;t minor d b & d e, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

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de Subtilitate, at nunc quod ad de<lb/><arrow.to.target n="marg781"/><lb/>mon&longs;trationem attinet <expan abbr="eor&utilde;">eorum</expan> &longs;ubij ciam. </s>

<s id="id003994">&longs;ingul&etail; rotul&etail; diffi <lb/>culter <expan abbr="mouen&ttilde;">mouentur</expan>, igitur nece&longs;&longs;e e&longs;t &longs;ingulas participes e&longs;&longs;e grauitatis, <lb/>igitur & totam <expan abbr="grauitat&etilde;">grauitatem</expan> e&longs;&longs;e diui&longs;am: quare ut in <expan abbr="pr&etail;ced&etilde;ti">pr&etail;cedenti</expan> facilius <lb/>moueri. </s>

<s id="id003995">Habent & rotul&etail; ip&longs;&etail; centrum &longs;eu axem hypomochlij, &longs;eu <lb/><arrow.to.target n="marg782"/><lb/>fulcimenti loco, ambitum <expan abbr="a&utilde;t">aunt</expan> iuxta &longs;emidiametrum, uelut collopes <pb pagenum="234" xlink:href="015/01/253.jpg"/>&longs;eu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="er&utilde;t">erunt</expan>, <lb/><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"/><lb/>& ut plures. </s>

<pb pagenum="234" xlink:href="015/01/253.jpg"/>&longs;eu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="er&utilde;t">erunt</expan>, <lb/><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"/><lb/>& ut plures. </s>

<s id="id003996">Vna enim alterius loco fungitur uectis. </s>

<s id="id003997">Trochlea qui<lb/>dem e&longs;t, ut uides, in&longs;trumentum longum &longs;uprà angu&longs;tius, &longs;ed non, <lb/>cra&longs;&longs;um, in quo plures orbiculi &longs;olent collo cari, unde &longs;æpe numero <lb/>trochleæ nomine intelligimus orbiculos ei in clu&longs;os, circa quos fu<lb/>nis uo catur, ut in tro chleis & orbiculi & funes in cluduntur. </s>

<s id="id003998">Succu<lb/>lis etiam &longs;olent capita funium trahi: ut uectis auxilio imò nonnun<lb/>quàm rotarum facilius pondera eleuantur.<lb/><arrow.to.target n="marg783"/></s>

<s id="id003998">Succu<lb/>lis etiam &longs;olent capita funium trahi: ut uectis auxilio imò nonnum<lb/>quàm rotarum facilius pondera eleuantur.<lb/><arrow.to.target n="marg783"/></s>

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<s id="id004039">Moueatur l m &longs;uper a c, æquidi&longs;tans &longs;emper a b, ut &longs;eruet &longs;itum <lb/>quem habebat ita, quod <expan abbr="extrem&utilde;">extremum</expan> lineæ l m &longs;it &longs;emper in linea a c, & <lb/>l punctum quod gerit uicem a, de&longs;cendat tantum in linea l m, quan<lb/>tum l extremum in linea a c: dicit Philo&longs;ophus, quod a &longs;eu l &longs;emper <lb/>de&longs;cendet in linea a d, & erit in e a. </s>

<s id="id004040">Supponatur quae latus l m fit f g, & <lb/>erit l n, f t, ducatur <expan abbr="a&utilde;t">aunt</expan> ex r puncto &longs;ectionis diametri, & lateris l m li <lb/><arrow.to.target n="marg790"/><lb/>near q, æquidi&longs;tans a f, <expan abbr="igi&ttilde;">igitur</expan> rhombus a q r f e&longs;t &longs;imilis rhombo toti <lb/>a b d c, & proportio a f ad fr, ut a c ad c d, &longs;ed a c e&longs;t &etail;qualis c d, <expan abbr="igi&ttilde;">igitur</expan> a f <lb/>e&longs;t æqualis f r, &longs;ed l de&longs;cendit in l m, <expan abbr="quant&utilde;">quantum</expan> e&longs;t a f ex &longs;uppo&longs;ito, <expan abbr="igi&ttilde;">igitur</expan> <lb/><expan abbr="punct&utilde;">punctum</expan> l &longs;emper erit in linea a d. </s>

<s id="id004041">Po&longs;t deficiunt quædam uerba: ob <lb/>quæ nemo intellexit &longs;ententiam Philo&longs;ophi, & <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo <lb/>nere lectoribus, tan<08> intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an<lb/>do, qui fal&longs;a dicat, & demon&longs;trare nitatur: produnt &longs;eip&longs;os cum <lb/>&longs;ua impudentia. </s>

<s id="id004042">Et lectoribus imponere conantur, debet ergo &longs;ic <lb/>legi (“b in ip&longs;a b c diametro latum, ubi latus b d moueatur in late<lb/>re b a, & b æqualiter uer&longs;us d in b d, æqualis enim e&longs;t ip&longs;a b e”) <lb/>Tunc enim con&longs;tat ut hic dixi, m moueri per b c rectam ut l per a d: <lb/>Dicit ergo <expan abbr="c&utilde;">cum</expan> b d <expan abbr="mouea&ttilde;">moueatur</expan> in b a, tran&longs;it unico motu <expan abbr="totã">totam</expan> b a, & pun<pb pagenum="237" xlink:href="015/01/256.jpg"/><expan abbr="ct&utilde;">ctum</expan> tamen b, quod <expan abbr="moue&ttilde;">mouetur</expan> duobus motibus, non pertran&longs;it ni&longs;i b c, <lb/>quæ pote&longs;t e&longs;&longs;e minor b a: nam <expan abbr="cõ&longs;tat">con&longs;tat</expan> quod <expan abbr="quãdo">quando</expan> m erit in a, o erit <lb/>in e, & quia m de&longs;cendit in o, in eodem tempore, ergo o erit in c, & <lb/><expan abbr="trã&longs;iuit">tran&longs;iuit</expan> &longs;emper per rectam b c: igitur m e&longs;t minus <expan abbr="mot&utilde;">motum</expan> duobus mo<lb/>tibus quàm m l unico <expan abbr="tant&utilde;">tantum</expan>. </s>

<pb pagenum="237" xlink:href="015/01/256.jpg"/><expan abbr="ct&utilde;">ctum</expan> tamen b, quod <expan abbr="moue&ttilde;">mouetur</expan> duobus motibus, non pertran&longs;it ni&longs;i b c, <lb/>quæ pote&longs;t e&longs;&longs;e minor b a: nam <expan abbr="cõ&longs;tat">con&longs;tat</expan> quod <expan abbr="quãdo">quando</expan> m erit in a, o erit <lb/>in e, & quia m de&longs;cendit in o, in eodem tempore, ergo o erit in c, & <lb/><expan abbr="trã&longs;iuit">tran&longs;iuit</expan> &longs;emper per rectam b c: igitur m e&longs;t minus <expan abbr="mot&utilde;">motum</expan> duobus mo<lb/>tibus quàm m l unico <expan abbr="tant&utilde;">tantum</expan>. </s>

<s id="id004043">Et quia aliquis dicere potui&longs;&longs;et non e&longs;t <lb/>mirum, quod m &longs;it minus motum duobus motibus quàm l m latus <lb/>unico tantum: quia m mouetur motu contrario motui lateris: nam <lb/>latus m o mouetur in latere b a a&longs;cendendo, et punctum m uer&longs;us o <lb/>in ip&longs;o m o de&longs;cendendo. </s>

<s id="id004044">Dicit Philo&longs;ophus, hoc e&longs;t mirum, quia <lb/>cum idem contingat in motu l, cuius latus mouetur per a c, & l per l <lb/>m recedendo in partem contrariam, nihilominus uelocius motum <lb/>e&longs;t l, quàm latus l m, quia a d e&longs;t longior a c. </s>

<s id="id004045">Ex quo patet, quae qu&etail;&longs;tio <lb/>Philo&longs;ophi e&longs;t una tantum, & non duæ. </s>

<s id="id004046">Et e&longs;t cur motum duobus <lb/>motibus in rhombo, in uno mouetur uelocius latere tantum moto <lb/>uno motu, in alio tardius? </s>

<s id="id004047">Et quia aliquis dicere po&longs;&longs;et, &qring;d b c po&longs;<lb/>&longs;et e&longs;&longs;e <expan abbr="lõgior">longior</expan> a c: Dicit Philo&longs;ophus, uerum e&longs;t, &longs;ed ego po&longs;&longs;um in<lb/>uenire talem rhombum, qui etiam habeat a c longiorem, & tunc ni<lb/>hilominus <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> quod dico. </s>

<s id="id004048">Aliud <expan abbr="a&utilde;t">aunt</expan>, quod docet ex hac demon<lb/>&longs;tratione, e&longs;t quae ex duobus motibus rectis diuer&longs;is pote&longs;t fieri unus <lb/>motus rectus diuer&longs;us: igitur idem punctum, puta formica poteric <lb/>&longs;imul, & &longs;emel moueri duobus motibus rectis diuer&longs;is. </s>

<s id="id004048">Aliud <expan abbr="a&utilde;t">aunt</expan>, quod docet ex hac demon<lb/>&longs;tratione, e&longs;t quae ex duobus motibus rectis diuer&longs;is pote&longs;t fieri unus <lb/>motus rectus diuer&longs;us: igitur idem punctum, puta formica poterit <lb/>&longs;imul, & &longs;emel moueri duobus motibus rectis diuer&longs;is. </s>

<s id="id004049">Et hoc e&longs;t, <lb/>quia primus motus e&longs;t rectus &longs;olum &longs;ecundum formam, & non &longs;e<lb/>cundum materiam: & alter &longs;ecundus, &longs;cilicet mi&longs;tus e&longs;t &longs;ecundum <lb/>materiam & non &longs;ecundum formam per rectam.</s>

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<s id="id004056"><margin.target id="marg793"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>primi<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id004057">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duob. </s>

<s id="id004057">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duobus </s>

<s id="id004058">motib. </s>

<s id="id004058">motibus. </s>

<s id="id004059">uno recto in linea, &longs;cilicet <lb/><arrow.to.target n="marg794"/><lb/>l m, & altero circulari. </s>

<s id="id004060">&longs;. </s>

<s id="id004061">circa <expan abbr="centr&utilde;">centrum</expan> k, & <expan abbr="tñ">tnm</expan> <expan abbr="mouebi&ttilde;">mouebitur</expan> uerè motu re<lb/>cto <expan abbr="t&mtilde;">tmm</expan> in alia linea, &longs;cilicet a c, & hoc e&longs;t <expan abbr="prim&utilde;">primum</expan> admirabile. </s>

Line 12109

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<s id="id004153">Ex hac etiam patet, quòd con&longs;tituto angulo in cen<lb/><arrow.to.target n="marg822"/><lb/>tro a b c, & ducta linea a d à puncto a, &longs;ciemus quo re&longs;i<lb/>&longs;ilit in linea b c: ducta enim c d, faciemus angulum c d e <lb/><arrow.to.target n="marg823"/><lb/>æqualem a b c, & erit angulus a d g æqualis angulo e d <lb/>h, igitur d e re&longs;ilit ex a b a d linea.</s>

<s id="id004153">Ex hac etiam patet, quòd con&longs;tituto angulo in cen<lb/><arrow.to.target n="marg822"/><lb/>tro a b c, & ducta linea a d à puncto a, &longs;ciemus quo re&longs;i<lb/>lit in linea b c: ducta enim c d, faciemus angulum c d e <lb/><arrow.to.target n="marg823"/><lb/>æqualem a b c, & erit angulus a d g æqualis angulo e d <lb/>h, igitur d e re&longs;ilit ex a b a d linea.</s>

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<s id="id004297">Et ex illo loco apparet, quod Auerroes ne&longs;ciuit Geo